text stringlengths 4 2.78M |
|---|
---
abstract: 'I survey motivations for education and outreach initiatives in the American context and explore the value of communicating physics for physicists and for the wider society. I describe the roles of large institutions, professional organizations, and funding agencies and cite some individual actions, local activities, and coordinated national programs. I note the emergence of transnational enterprises—not only to carry out research, but to communicate physics. A brief appendix collects some useful internet resources.'
author:
- |
Chris Quigg[^1]\
[*Fermi National Accelerator Laboratory*]{}\
[*P.O. Box 500, Batavia, Illinois USA 60510*]{}
title: ' COMUNICARE FISICA ALL’AMERICANA\'
---
=11.6pt
Introductions
=============
It is my great pleasure to join Italian colleagues for this ambitious workshop on communicating physics. In my main career as a theoretical physicist, my research has addressed many topics in particle physics. My close engagement with experiment has made the Fermi National Accelerator Laboratory a stimulating scientific home for many years. Beyond my personal activities to bring the ideas and aspirations of particle physics to a broad audience, I have held a number of positions in which I have been able to encourage—and take pleasure in—the outreach and education efforts of many of my colleagues. I have served as Head of the Fermilab Theory Group, as Deputy Director of the Superconducting Super Collider Central Design Group, and as Chair of the American Physical Society’s Division of Particles and Fields. In the last capacity, I oversaw the planning and execution of *Snowmass 2001:* a summer study on the future of particle physics (see [[snowmass2001.org](http://snowmass2001.org)]{}). I am also a member of the board of trustees of the Illinois Mathematics and Science Academy, a state-supported residential high school.
Fermilab is a U.S. Department of Energy national laboratory for particle physics, operated by a consortium of 90 research universities in the United States, Canada, Italy, and Japan. The laboratory and its research community are highly international; our lab has much in common with CERN. Fermilab has some two thousand employees, an annual budget of \$300M, and welcomes 2250 experimental “users” from around the world. Our principal research instrument is the Tevatron, a superconducting proton synchrotron $2\pi$ kilometers in circumference that accelerates protons and antiprotons to nearly 1 TeV. When operated as a proton-antiproton collider, the Tevatron is the world’s most powerful microscope, giving us access to *nanonanophysics* on a scale a billion times smaller than an atom.
Many of you have visited my country, or gained an impression of it from Puccini’s opera, *La Fanciulla del West,* or from the cult classic film, *The Blues Brothers.* Nevertheless, a few remarks about the country’s character may provide useful context for today’s exploration.
The United States is large in both land area and population, and much of its area is lightly settled. It is a young nation, at least in its psyche, and it is the wealthiest country in the history of the planet. The U.S. was founded as a nation of immigrants, and the flux of people from diverse origins to our shores continues today. A great degree of social mobility is perhaps correlated with a restrained esteem for authorities (including professors). At the same time, unease with the pace of change leads many to a powerful reliance on Authority.
The U.S. is a deeply non-Napoleonic society, characterized by much local autonomy, notably in the responsibility for schools. Our university system is highly heterogeneous. The national government assumes responsibility only for a few military academies, and our most celebrated universities are largely private institutions. In America’s heartland, the state universities serve as the leading cultural institutions. Though academic science is quite young in America, research has traditionally been centered in universities—research and teaching go hand in hand—and that has led to a superb standard of postgraduate education.
In contrast to the European standard, we do not have dominant national newspapers. Our vast radio and television empires, which seem increasingly preoccupied with gossip and celebrity trials, evince little interest in science. National Public Radio, especially in its local programming, is receptive to science and culture. All of these factors—beginning with the vastness of the country—enhance the importance of local outreach and education efforts.
Incentives for Education & Outreach Efforts
===========================================
Some of the motivations I would cite for physics communication are similar to those evoked by earlier speakers. First comes the teaching imperative: the desire not only to pass on information about the minute particulars of current research, but also to cultivate an understanding of the scientific worldview, with its rejection of Authority, reliance on experiment, and celebration of doubt. Next comes the propaganda function, which I understand here in a positive light: *propaganda fide* for our attachment to reason, observation, and controlled experiment; the development of what social scientists call “diffuse good will” toward the scientific enterprise; and an attempt to influence behavior and gain acceptance and support for our undertakings.
Since we scientists devote our lives to exploration, I see outreach and education as a mode of exploring the world—an opportunity to learn from and about others. Some of my colleagues emphasize the value of “inreach”—to see ourselves as others see us! Previous speakers have spoken of engaged learning; we might well aspire to *engaged teaching.* Participation in outreach programs can open our own students to the wider world. Our communications efforts can also help to change the face of physics—to attract women in greater numbers and to show minority and new immigrant groups the appeal of careers in and around science. Finally, outreach and education programs are good for physicists: the joy that comes from sharing our numinous adventure with others is marvelous psychotherapy!
U.S. Organizations and Institutions
===================================
The American Physical Society ([[www.aps.org](http://www.aps.org)]{}), the principal professional organization for academic and industrial physicists in the United States, offers many education and outreach programs ([[www.aps.org/educ/](http://www.aps.org/educ/)]{}). Its *Physics Central* web site ([[physicscentral.org](http://physicscentral.org)]{}) aims to communicate the excitement and importance of physics to everyone. *Physical Review Focus* ([[focus.aps.org](http://focus.aps.org)]{}) stories explain selected physics research published in the APS journals *Physical Review* and *Physical Review Letters.* Several of the Divisions of the APS maintain vigorous outreach and education programs, and the Forum on Education ([[www.aps.org/units/fed/](http://www.aps.org/units/fed/)]{}) offers a meeting place for members interested in education.
The American Association of Physics Teachers ([[www.aapt.org](http://www.aapt.org)]{}), with $11\,000$ members around the world, aspires to be the leading organization for physics education. It publishes *The Physics Teacher* and the *American Journal of Physics,* and strives to provide the most current resources and up-to-date research needed to enhance a physics educator’s professional development.
The American Institute of Physics, a federation of ten Member Societies representing the spectrum of physical sciences, supports a number of educational efforts, described at [[www.aip.org/education/](http://www.aip.org/education/)]{}. The electronic newsletter *Physics News Update* ([[www.aip.org/physnews/update/](http://www.aip.org/physnews/update/)]{}) is a digest of physics news items arising from physics meetings and journals, newspapers and magazines, and other news sources. The AIP also maintains the *Physics News Graphics* image archive ([[www.aip.org/png/](http://www.aip.org/png/)]{}).
The American Association for the Advancement of Science ([[www.aaas.org](http://www.aaas.org)]{}) is the largest scientific organization in the United States. AAAS sponsors an annual meeting—in which physics plays a minor, but growing, part—that is a major event for science writers from much of the world. It also animates a great range of education programs and activities to promote public awareness of science and its role in public policy ([[www.aaas.org/programs/education/](http://www.aaas.org/programs/education/)]{}).
The Department of Energy’s Office of Science ([[www.science.doe.gov](http://www.science.doe.gov)]{}) is the largest supporter of basic research in the physical sciences in the United States. It is also the main patron of large research instruments for high-energy physics, nuclear physics, and fusion energy sciences. The Office of Science sponsors a range of education initiatives through its Workforce Development for Teachers and Scientists program ([www.scied.science.doe.gov/scied/sci\_ed.htm](http://www.scied.science.doe.gov/scied/sci_ed.htm)). The Office of High Energy Physics points with pride to educational and outreach efforts of its national laboratories and university-based research groups but does not set specific expectations or requirements for research contractors. The DOE and NSF jointly sponsor the *QuarkNet* program described in §\[subsec:qn\].
The National Science Foundation ([[www.nsf.gov](http://www.nsf.gov)]{}), which mainly funds individual investigators, is the other principal source of support for basic research in physics. Support for science and engineering education, from pre-Kindergarten through graduate school and beyond, is essential to NSF’s mission. In addition to judging the intellectual merit of scientific proposals to the NSF, reviewers are asked to take into account the “broader impacts” of the proposed research program. Specifically, *How well does the activity advance discovery and understanding while promoting teaching, training, and learning? How well does the proposed activity broaden the participation of underrepresented groups?* Last year alone, NSF grantees in particle physics reached more than $100\,000$ school-age children. NSF directly funds research into effective educational practice, and supports research participation for students and teachers.
The National Aeronautics and Space Administration ([[www.nasa.gov](http://www.nasa.gov)]{}) has an enviable record in engaging the public imagination in the value of exploration and in the astronomical sciences. NASA requires an extensive outreach and education effort for each of the missions it supports.
All of the major particle physics laboratories have extensive public affairs and education offices. Many in the *Comunicare Fisica* audience will know Judy Jackson of Fermilab ([[www.fnal.gov](http://www.fnal.gov)]{}) and Neil Calder of the Stanford Linear Accelerator Center ([[www.slac.stanford.edu](http://www.slac.stanford.edu)]{}), and the magazine ([[www.symmetrymag.org](http://www.symmetrymag.org)]{}) recently launched as a joint venture of the two institutions. Both labs have lively programs for science writing interns. Together with other particle physics laboratories around the world, U.S. institutions have launched [[interactions.org](http://interactions.org)]{}, a central resource for communicators of particle physics. To celebrate the World Year of Physics, has organized *Quantum Diaries* ([[interactions.org/quantumdiaries](http://interactions.org/quantumdiaries)]{}), a web site that follows physicists from around the world through their blogs, photos, and videos.
The International Linear Collider ([[linearcollider.org](http://linearcollider.org)]{}), which many of us see as the next great accelerator project after the Large Hadron Collider at CERN, is organized as an international design effort, with an international communications team.
The Kavli Institute for Theoretical Physics ([[www.itp.ucsb.edu](http://www.itp.ucsb.edu)]{}), on the Santa Barbara campus of the University of California, is funded by the National Science Foundation and the University of California. The KITP program of workshops encompasses research in theoretical physics very broadly understood. The Kavli Institute has created a journalist in residence program,[^2] and also has an artist in residence. The Particle Data Group ([[pdg.lbl.gov](http://pdg.lbl.gov)]{}), an international collaboration that reviews particle physics and related areas of astrophysics, compiles and analyzes data on particle properties. From its U.S. center at Lawrence Berkeley National Laboratory, the PDG produces and distributes a wealth of educational materials, including the famous standard model wall chart described in §\[subsec:smwc\].
The Division of Particles and Fields ([[www.aps.org/units/dpf/](http://www.aps.org/units/dpf/)]{}) and Division of Physics of Beams ([[www.aps.org/units/dpb/](http://www.aps.org/units/dpb/)]{}) of the American Physical Society, like many of their counterparts in other subfields, support a range of ongoing and special programs in education and outreach. Increasingly, individual experiments are making significant efforts to bring intelligible accounts of their research to public notice.
Examples from Particle Physics in the United States
===================================================
Now I would like to briefly describe a number of education and outreach initiatives carried out by American particle physicists. I have chosen these examples out of my own experiences to illustrate the lessons I will draw in §\[sec:lessons\], including significant unplanned successes. See the Appendix for additional resources.
Saturday Morning Physics at Fermilab
------------------------------------
When Leon Lederman became Director of Fermilab in 1979, he created the Saturday Morning Physics program ([[www-ppd.fnal.gov/smp-w/](http://www-ppd.fnal.gov/smp-w/)]{}) for high school students curious about science. For more than twenty-five years, we have welcomed students—and some of their teachers—to the lab on Saturday mornings for ten-week series of lectures (two hours) and in-depth tours of many activities at the laboratory. The lab runs three sessions of SMP each year, with as many as 120 students in each “class.” By now, more than six thousand students have been exposed to the ideas of particle physics, the wonders of exotic technologies, and the spirit of scientific inquiry.
We made an important discovery during the first year of Saturday Morning Physics: a number of high school science teachers accompanied their students, and devoted ten consecutive Saturday mornings to learning about particle physics. Though they didn’t all have deep preparation to teach physics, they all had sharp intelligence, inquisitive minds, and high enthusiasm. They all wanted to become better teachers; our lab could provide information and encouragement and—perhaps most important—the opportunity for them to meet and support each other. I believe that our accidental discovery of these teachers launched the initiative that blossomed, under the inspired guidance of Marge Bardeen and Stanka Jovanovic, into the lab’s Education Office ([[www-ed.fnal.gov](http://www-ed.fnal.gov)]{}) and Leon M. Lederman Science Education Center.
Physics Vans
------------
A number of university physics departments organize physics road shows for schools, shopping centers, and community events. These feature demonstrations, often performed theatrically with an amusing twist, by physics undergraduates and graduate students, and hands-on experiments for audience members. Physics vans spark curiosity and put a human face on physics. They are also great fun for the student performers!
Standard model wall chart \[subsec:smwc\]
-----------------------------------------
One of the icons of physics classrooms is the table of particles and interactions that grew out of a Conference on Teaching Modern Physics held at Fermilab in 1986.[@quandaries] The celebrated chart (see Figure \[fig:wallchart\]) of fundamental particles and interactions created by the Contemporary Physics Education Project represents the enthusiastic work of many teachers.
![Contemporary Physics Education Project standard-model wall chart. \[fig:wallchart\]](1999chart.eps){width="10cm"}
Available as wall chart, poster, and place mat, this representation of our “standard model” has had a global reach, with more than $200\,000$ distributed. It has helped move particle physics into the classroom, and it presents many essential notions of our current understanding. The wall chart project also stimulated *The Particle Adventure,* an interactive web site.
*QuarkNet* \[subsec:qn\]
------------------------
*QuarkNet* is a remarkable immersive research experience for high school teachers and their students that is based on lively ongoing partnerships with experimental research groups.[^3] Currently fifty-three *QuarkNet* centers operate in twenty-five states and Puerto Rico. They touch $100\,000$ students per year, and five hundred teachers and one hundred students are research partners with 150 physicist mentors. One of the goals is to engage teachers and students in real time with data from the Large Hadron Collider at CERN. For now, QuarkNet centers involve twelve major experiments and the computing GRID.
A key tenet of the *QuarkNet* paradigm is that the teachers and students should gain experience in assembling and commissioning real detectors. A favorite example is the construction of cosmic-ray detectors that consist of several paddles of plastic scintillator, photomultiplier tubes, and the associated trigger system. School groups are deploying these simple detectors at schools around the country in the *QuarkNet* Cosmic-Ray Detector Array. Students are learning to use the GRID to handle calculations involving large amounts of data. (I have the impression that the professionals are learning how to make the GRID a robust tool, in the process.)
Web lecture archives—unplanned outreach!
----------------------------------------
In the summer of 1999, computer scientist Chuck Severance and physicist Steve Goldfarb, then a member of the L3 collaboration at CERN’s Large Electron-Positron Collider, tested a web lecture archive of lectures for summer students. I had the good fortune to be their first experimental animal. Severance’s Sync–o–matic software framework[@weblec] is simple and functional. It synchronizes a video stream with good-resolution images of the speaker’s slides, all displayed in a web browser. A recent example from Fermilab’s web lecture archive is shown in Figure \[fig:fish3\].
[![Appearance (in a browser) of a lecture from the Fermilab web archive.[]{data-label="fig:fish3"}](ComingRevGrab.eps "fig:"){width="10cm"}]{}
When I showed the system to my friends in Fermilab’s Visual Media Services, they were impressed with the low resource cost (including modest bandwidth and storage requirements) and ease of use, and were quick to see the potential in a streaming video archive. Today at Fermilab, the streaming video archive boasts 1334 entries, including colloquia, conference talks, academic training lectures, and memorable events.[^4] We hoped that the archive might prove valuable for Fermilab’s staff and users, as indeed it has, but didn’t imagine that it would become part of the lab’s public face—and the field’s. In fact, many viewers from outside our community land at the video archive thanks to search engines, not by drilling down from the Fermilab home page.
Engaging Hispanic students
--------------------------
While living at Fermilab as a CDF postdoc, Aaron Dominguez (now at the University of Nebraska) developed an educational project in Aurora, Illinois, to improve the future and stability of the Hispanic community by supporting the educational and social accomplishments of its young people. Bilingual English/Spanish Tutors (BEST) pairs high-achieving high school student mentors with low-income elementary school children. The BEST tutors help their younger peers with homework, reading, and math after school twice a week. The tutors also gain a critical sense of responsibility for the successful education of their own Latino community. Aaron’s program has enrolled over 60 students and 35 bilingual high school mentors; the BEST model has been replicated in the neighboring community of Batavia.
Tevatron postcards
------------------
In 1997, I received an invitation to give the first Carl Sagan Memorial Lecture in the series, *Cosmos Revisited,* at the Smithsonian Institution in Washington. I wanted to give members of the audience specimens that would stimulate them to continue the conversation begun by my lecture, “The Particle Cosmos.” With Judy Jackson, we conceived an edition of eight postcards depicting significant events—the outcome of proton-antiproton collisions—from the CDF and DØ experiments.
[![ A pair of top quarks reconstructed in the DØ experiment at Fermilab. This end view shows the final decay products: two muons (turquoise), a neutrino (pink), and four jets of particles.[]{data-label="fig:fish2"}](pix_92704_14022rb.eps "fig:"){width="8cm"}]{}\
[![ A pair of top quarks reconstructed in the DØ experiment at Fermilab. This end view shows the final decay products: two muons (turquoise), a neutrino (pink), and four jets of particles.[]{data-label="fig:fish2"}](92704-14022a.eps "fig:"){width="8cm"}]{}
We began by asking the experimental groups to submit authentic event displays. An example is shown in the top panel of Figure \[fig:fish2\]. This display of a top-quark event from the DØ experiment illustrates an important reality: Experimenters are deeply attached to their detectors, so the fixed detector elements are represented by strong, assertive lines, whereas the ephemeral tracks that bear witness to noteworthy one-time occurrences are indistinct. The light green traces (one solid, one dotted) representing muons—crucial markers in the top-antitop event—are nearly invisible.
Graphic designer Bruce Kerr ([[www.kerrcom.com](http://www.kerrcom.com)]{}) preserved the authenticity of the event displays by discreetly editing the event display to emphasize the elements that signal top-pair production. Except for the background sunburst that contributes visual interest and serves as a metaphor for the conversion of energy into new forms of matter, every element of the postcard shown in the bottom panel of Figure \[fig:fish2\] is present in the original PostScript file. And every element in the original event display is preserved in the final image. Such fidelity is important;[^5] the image is instantly readable to physicists, including its creators in DØ, and intelligible—with just a bit of explanation—to laypersons. We wrote captions that would explain and initiate conversation. The postcard images ([[lutece.fnal.gov/Postcards](http://lutece.fnal.gov/Postcards)]{}) have become true icons of particle physics, with an impact far beyond their original purpose.
Snowmass 2001
-------------
The Division of Particles and Fields and the Division of Physics of Beams of the American Physical Society organized a three-week summer study on the future of particle physics in Snowmass, Colorado, in July 2001. More than 1200 physicists participated—many young, many from outside the United States—in a very broad examination of where our field should be heading. Early on, we decided to make outreach and education an essential part of the Snowmass 2001 experience. We wanted to share our love for science with the interesting mix of people in Aspen and Snowmass, and to encourage our colleagues to see each other in action. We also believed that the public interaction would reinforce the optimism and enthusiasm that participants brought to Snowmass.
Theoretical physicist Elizabeth Simmons, who chaired the outreach and education effort, has described the extraordinary results in *Physics Today.*[@lizs] The Snowmass 2001 program ([[snowmass2001.org/outreach/education2.html](http://snowmass2001.org/outreach/education2.html)]{}) included workshops for teachers and students, public lectures, a science book fair, science theater, open-air talks, astronomy activities, conversations with children, outreach workshops, and a balloon ascension to recreate Victor Hess’s discovery of cosmic rays. The centerpiece was a huge weekend science fair on the Snowmass Village Mall that attracted some 1500 members of the general public. Among the weekend’s hits was a superconducting apparatus capable of levitating an entire human (see Figure \[fig:fish\]) from the Texas Center for
[![Leon Lederman levitated by the Meissner effect at Snowmass 2001. (Photo credit: Elizabeth H. Simmons, Michigan State University)[]{data-label="fig:fish"}](levitatingleona.eps "fig:"){width="10cm"}]{}
Superconductivity and Advanced Materials ([www.tcsam.uh.edu/education\_outreach](http://www.tcsam.uh.edu/education_outreach)).
*Comunicare Fisica* participants will find special interest in the communications workshops ([[www.fnal.gov/pub/snowmass/workshops/workshop.html](http://www.fnal.gov/pub/snowmass/workshops/workshop.html)]{}) organized at Snowmass 2001.
In connection with the summer study, the DPF commissioned an illustrated thematic survey of our vision of particle physics and its future in the most ambitious intellectual terms. Within this broad vision, the document was to identify the questions we want to address over the next two decades. Like the summer study itself, the thematic survey aimed to help our community recognize and articulate what particle physics is and aspires to be, guided by the scientific imperatives. *Quarks Unbound* was a smashing success, for three principal reasons. First, we were careful to think through what we wanted to accomplish and to identify the audiences we wanted to reach. Second, we entrusted the project to a small team—not a large and representative committee—that included a science writer, a graphic designer, and a few physicists. \[We did not want the sort of bland “offends no one, delights no one” product that large committees are adept at producing!\] Third, we distributed *Quarks Unbound* widely and enthusiastically to groups and individuals. A private donor financed a second printing, so that the number of copies distributed worldwide now exceeds $60\,000$.
Some lessons \[sec:lessons\]
============================
I am continually impressed by the passion, curiosity, and faith in the value of exploration evidenced by those who attend our outreach and education programs. It may be true, as several speakers have opined, that the general population is ignorant and indifferent about science, but my experience is that we do find a receptive and engaged audience. Accordingly, the first lesson is: Respect (do not underestimate) the audience.
Leadership from established scientists and Heroes of the Field provides important validation for the efforts of others. The effectiveness of leading by example is well-established. Respected senior physicists can also discourage the misperception that engaging with the public is unworthy of serious scientists. They also have value as fundraisers, from both public and private sources.
While leaders are important, it may be even more important for the leaders to let go: to grant autonomy and resources to small groups and trust them to do wonderful things. Nothing is more stifling to creativity and innovation than repeated reviews by committees constructed to represent an average. The field is better served by original—even quirky—efforts that explore a whole range of approaches than by efforts programmed to hit the mean each time. Both those in authority and those who execute should find pleasure in experimenting and taking risks!
A familiar conceit is that “Physicists can do anything.” I like to tell my colleagues that the complete slogan concludes “…badly,” to remind them of the rewards of collaborating with professionals who know their fields as well as we physicists know physics. Working with gifted writers, editors, designers, artists, and educators can be immensely satisfying and can lead you to do things you didn’t know were possible.
Let me underline again the importance of local activities. These include organizing public lectures and symposia at the universities and labs, engaging with cultural institutions and creative people within the community, and developing relationships with local radio stations and newspapers. A record of success on the local scene may even enable effective national efforts.
One committed person can do a lot—and an individual can achieve even more with encouragement and support from colleagues. But it is also delightful to learn from others and to draw inspiration from their efforts. The *QuarkNet* model has succeeded so well because it has elements of individuality, collaboration, coordination, and the bond that grows from being part of a large and ambitious enterprise.
Always ask, Why are we doing this? Who is the audience? What are our goals? What will success mean? Be prepared to finish the task: if you prepare a new brochure about your experiment or your subject and are passive or hesitant about distributing it, you have limited your effectiveness. If following through seems like too much effort, you should notice that before you start. Expect to discover new people (not all PhD physicists). Your role may be catalytic. Leaders succeed when others become the stars. Expect also to discover new ideas, and remember that a new outreach triumph may occur when you least expect it.
Concluding remarks
==================
Experiments at the Large Hadron Collider break new ground in scale and internationalism, counting collaborators in the thousands. The outreach efforts of the ATLAS (34 countries) and CMS (31 countries) collaborations are truly transnational, and marvelously dynamic. The ATLAS experiment movie ([[atlasexperiment.org/movie/](http://atlasexperiment.org/movie/)]{}), for example, is available in ten languages and has won awards in five countries.
The impressive reach of ATLAS and CMS, depicted by the colored spaces in Figure \[fig:cms\], is a source of immense satisfaction and pleasure to
[![Global reach of the ATLAS and CMS collaborations at the LHC.[]{data-label="fig:cms"}](ATLASmap.eps "fig:"){width="11cm"}]{}\
[![Global reach of the ATLAS and CMS collaborations at the LHC.[]{data-label="fig:cms"}](CMSmap.eps "fig:"){width="11cm"}]{}
particle physicists. For me, and for many of my colleagues, the opportunity to meet and join in common cause with people from many nations is one of the great joys of our research. But there are many blank spots on the LHC maps of the world, and we should take those open spaces as a challenge. I believe that we should aspire to engage the whole world in the values and the rewards of science. Our goal should truly be physics without boundaries.
Science is more than solving the next great puzzle of particle physics; it is a set of values for contemplating the world. I draw great hope from the concluding lines of Anthony Lewis’s millennial essay in the New York *Times* of December 31, 1999. You will recognize the tradition we owe to Galileo:
> \[T\]here has been one transforming change over this thousand years. It is the adoption of the scientific method: the commitment to experiment, to test every hypothesis. But it is broader than science. It is the open mind, the willingness in all aspects of life to consider possibilities other than the received truth. It is openness to reason.
When we are at our best—when we are truest to these ideals—we do our best science, and we give our greatest gift to society.
Acknowledgements
================
I am grateful to Franco Fabbri and Rinaldo Baldini Ferroli for their kind invitation to participate in *Comunicare Fisica 2005,* and for generous hospitality in Frascati. I thank Marge Bardeen, Michael Barnett, Sharon Butler, Julia Child, Aaron Dominguez, Judy Jackson, Leon Lederman, Joe Lykken, Kate Metropolis, Helen Quinn, Liz Quigg, Randy Ruchti, Liz Simmons, Maria Spiropulu, and other colleagues for teaching me many ways to communicate physics. Fermilab is operated by Universities Research Association Inc. under Contract No. DE-AC02-76CH03000 with the U.S. Department of Energy.
[99]{} *Quarks, Quasars, and Quandaries,* edited by G. Aubrecht (American Association of Physics Teachers, College Park, MD, 1987). Charles Severance a tool for making RealMedia-based web lectures ([[www.syncomat.com](http://www.syncomat.com)]{}). See also the web lecture archive project ([[www.wlap.org](http://www.wlap.org)]{}). Elizabeth H. Simmons, “How to Popularize Physics,” Phys. Today **58,** (1) 42 (January 2005).
Appendix: Some education and outreach resources \[app\] {#appendix-some-education-and-outreach-resources-app .unnumbered}
=======================================================
[ ]{}
The page [[particleadventure.org/particleadventure/other/othersites.html](http://particleadventure.org/particleadventure/other/othersites.html)]{} contains a very extensive list of internet resources.
[^1]: Electronic mail: quigg@fnal.gov
[^2]: To my knowledge, no physics institution in the U.S.offers an experience comparable to the Woods Hole Oceanographic Institution’s Ocean Science Journalism Fellowship ([www.whoi.edu/home/news/media\_jfellowship.html](http://www.whoi.edu/home/news/media_jfellowship.html)).
[^3]: The program was established in 1998 by Michael Barnett (Lawrence Berkeley National Lab), Marjorie Bardeen (Fermilab), Keith Baker (Hampton University), and Randy Ruchti (Notre Dame), with initial funding from the National Science Foundation and continuing support from NSF and DOE.
[^4]: Archives of similar richness are maintained at CERN, SLAC, and the Kavli Institute for Theoretical Physics (see the Appendix for links). The KITP is experimenting with the new medium of enhanced podcasts.
[^5]: …but not universal in scientific illustration.
|
---
abstract: 'Inspirals and mergers of black hole (BHs) and/or neutron star (NSs) binaries are expected to be abundant sources for ground-based gravitational-wave (GW) detectors. We assess the capabilities of Advanced LIGO and Virgo to measure component masses using inspiral waveform models including spin-precession effects using a large ensemble of GW sources [**randomly oriented and distributed uniformly in volume. For 1000 sources this yields signal-to-noise ratios between 7 and 200**]{}. We make quantitative predictions for how well LIGO and Virgo will distinguish between BHs and NSs and appraise the prospect of using LIGO/Virgo observations to definitively confirm, or reject, the existence of a putative “mass gap” between NSs ($m\leq3\ M_\odot$) and BHs ($m\geq 5\ M_\odot$). We find sources with the smaller mass component satisfying $m_2 \lesssim1.5\ M_\odot$ to be unambiguously identified as containing at least one NS, while systems with $m_2\gtrsim6\ M_\odot$ will be confirmed binary BHs. Binary BHs with $m_2<5\ M_\odot$ (i.e., in the gap) cannot generically be distinguished from NSBH binaries. High-mass NSs ($2<m<3$ $M_\odot$) are often consistent with low-mass BH ($m<5\ M_\odot$), posing a challenge for determining the maximum NS mass from LIGO/Virgo observations alone. Individual sources will seldom be measured well enough to confirm objects in the mass gap and statistical inferences drawn from the detected population will be strongly dependent on the underlying distribution. If nature happens to provide a mass distribution with the populations relatively cleanly separated in chirp mass space, as some population synthesis models suggest, then NSs and BHs are more easily distinguishable.'
author:
- 'Tyson B. Littenberg'
- Ben Farr
- Scott Coughlin
- Vicky Kalogera
- 'Daniel E. Holz'
title: 'Neutron stars versus black holes: probing the mass gap with LIGO/Virgo'
---
\[sec:intro\]Introduction
=========================
Advanced LIGO [@aLIGO] and Advanced Virgo [@aVirgo] will be the most sensitive observatories in the gravitational-wave (GW) spectrum between 10 Hz to a few kHz. Binary systems comprised of compact stellar remnants, such as stellar mass black holes (BH) and neutron stars (NS) merge at frequencies between $\sim\,100$ and $\sim\,1000$ Hz and are the primary science target for the LIGO/Virgo (LV) network.
GW observations will investigate these systems in ways not accessible to electromagnetic observations. Detailed observations of individual sources will provide unparalleled insight into strong field gravity and the NS equation of state [@Flanagan:2007ix; @Read:2009yp; @Lackey:2013axa; @Wade:2014vqa; @Li:2011cg; @Sampson:2013lpa]. Inferred characteristics of the compact binary population will feed back into complicated, ill-constrained physics of compact object formation and binary evolution (@Abadie:2010cf and references therein).
We investigate the capabilities of an Advanced LV network of detectors to constrain the individual component masses of a compact binary and thereby distinguish between NSs and BHs. We identify the mass intervals for which neutron star black-hole binaries (NSBH), and binary black hole (BBH) systems can be securely distinguished. This is the first large-scale study to characterize compact binary posterior distribution functions (PDFs) including spin-precession effects over a broad range of masses and spins.
We use our results to assess LV’s role in resolving the debate over the compact object “mass gap.” Observations of X-ray binaries suggest a depletion in the mass distribution of compact remnants between the highest mass NSs ($\sim 2 {{\rm M}_\odot}$) and the lowest mass BHs ($\gtrsim 5 {{\rm M}_\odot}$) [@Ozel:2010su; @Farr:2010tu]. Inferring masses electromagnetically is challenging and systematic errors may dominate. Including variable emission from the accretion flow in the analysis of the same X-ray binaries systematically finds lower masses and disfavors the mass gap [@Kreidberg:2012ud].
The mass distribution of NSs and BHs has implications on plausible explanations for the core-collapse supernova mechanism. @Belczynski:2011bn suggests the mass gap as observational evidence that supernovae develop rapidly (within 100 to 200 ms), while a “filled gap” favors longer timescale explosion mechanisms. @Kochanek:2013yca and @Clausen:2014wia find that a bimodal mass distribution for compact remnants is a natural consequence of high mass red supergiants ending their lives as failed supernovae.
GW observations have been suggested as a means of resolving this controversy because the component masses are directly encoded in the signal. We use our study to forecast whether LV observations can confirm or falsify a mass gap between NSs and BHs. Our results are to be contrasted with the contemporaneous paper from @Mandel:2015spa, in which they assume a specific mass distribution (generated from population synthesis) and found it is possible to distinguish BHNS and BBHs and infer a gap after tens of detections.
Inferring physical parameters from gravitational-wave data {#sec:pe}
==========================================================
Inspiral waveforms are well understood from post-Newtonian (PN) expansions of the binary dynamics [e.g. @Blanchet:2006zz]. PN waveforms enable template-base data analysis methods where many trial waveforms are compared to the data. The model waveforms, or templates, are parameterized by the masses, spins, location, and orientation of the binary.
GW signals from compact binaries will be in the LV sensitivity band for tens of seconds to minutes, evolving through $N_{\rm cycle}\sim10^2$–$10^4$ cycles before exiting the band or merging. Long duration signals place strict demands on the acceptable phase difference between the template waveforms and the signal. To leading order in the PN expansion, the phase evolution depends only on the “chirp mass” [see @Peters:1963ux] ${\mathcal{M}_c}\equiv \left(m_1 m_2\right)^{3/5}\left(m_1+m_2\right)^{-1/5}$ where $m_1>m_2$ are the component masses of the binary. The uncertainty in ${\mathcal{M}_c}$ scales as $( N_{\rm cycle} )^{-1/2}$ [@Sathyaprakash:2009xs] and is thereby constrained to high precision while the component masses are completely degenerate. Higher order corrections introduce the mass ratio $q=m_2/m_1$ and couplings between the intrinsic angular momentum of the component bodies $\vec{S}_{1,2}$ and the orbital angular momenta of the system $\vec{L}$ breaking the degeneracy between $m_1$ and $m_2$, though large correlations remain [@Cutler:1994ys; @Poisson:1995ef].
@Hannam:2013uu investigated how well LV can distinguish between NSs and BHs by approximating parameter confidence intervals using the match between a proposed signal and a grid of templates. Their study used a simplified version of the full PN waveforms parameterized by a single “effective spin” and concluded that most LV detections will not provide unambiguous separation between NSs and BHs.
The effective spin approximation overstates the degeneracy between mass and spin because it ignores precession of the orbital plane and spin alignments. @Chatziioannou:2014coa revisited the topic of component-mass determination with a Markov chain Monte Carlo (MCMC) analysis using waveforms that included spin and orbital precession . Their study found that spin precession reduces mass-spin correlations, concluding that BNS systems with components consistent with known NSs in binaries would not be misidentified as either low-mass black holes or “exotic” neutron stars which they define as having masses either below $1\ {{\rm M}_\odot}$ or above $2.5\ {{\rm M}_\odot}$, and/or with dimensionless spins $\chi>0.05.$
In this work we use the same methodology of @Chatziioannou:2014coa – employing an MCMC with templates that include full two-spin precession effects to infer the PDF – but use a large ensemble of NSBH and BBH signals, paying particular attention to LV’s ability to identify sources which have one, or both, components in the mass gap.
\[sec:method\] Method
=====================
Our study uses the [*LALInference*]{} software library for recovering the parameters of compact binary systems. In our work we elect to use the MCMC implementation `lalinference_mcmc` though results do not depend on the chosen sampler. A complete description of the software is found in @LALInference.
[*LALInference*]{} is part the LSC Analysis Library (LAL) which has a wide variety of template waveforms available. [**For this work we use the SpinTaylorT2 waveform implemented in LAL. A detailed description of the waveform can be found in Appendix B of @Nitz:2013mxa**]{}. To simulate a population of plausible LV detections we draw 1000 binary parameter combinations from a uniform distribution in component masses with $m_1\geq m_2$, $m_i\geq 1\ {{\rm M}_\odot}$ and $m_1+m_2 \leq 30\ {{\rm M}_\odot}$. The maximum total mass of $30\ {{\rm M}_\odot}$ is chosen so that the merger and ring-down portion of the waveform (where the PN approximation is invalid) does not dominate the signal as observed by LV. Dimensionless spin magnitudes $\chi$ are drawn uniformly from $[0,1]$ and the spin vector orientation is randomly distributed over a sphere with respect to $\vec{L}$. Notice that our choice of simulated signals includes the possibility of neutron stars with anomalously high spins $\chi>0.7$ [@Lo:2011].
The sky location is distributed uniformly over the celestial sphere, orientations are randomly distributed, and the distance to the binary is uniform in volume. We reject sources which do not have signal-to-noise ratio $\rm{SNR}>5$ in two or more detectors. Figure \[fig:masses\] shows the mass distributions of our population. The bottom panel is a scatter plot of $m_1$ and $m_2$ colored by the source SNR over the Advanced LV network. [**It is important to note that our simulated population includes binaries at sufficiently high mass ($M\gtrsim12 {{\rm M}_\odot}$) that the merger and ringdown portion of the signal is detectable [@Buonanno:2009zt]. Our study is focused on quantifying how well masses can be determined purely from the inspiral part of the waveform, for which we have reliable waveforms appropriate for parameter estimation (i.e., valid across all parameter space). Using inspiral waveforms for both our signal simulations and parameter recovery allows us to assess the inspiral effects separate from merger-ringdown effects. However we are in urgent need of precessing waveforms that include the merger/ringdown and are valid over the full prior range to avoid systematic errors in analyses of real data. Currently no such waveforms are available. Available inspiral models that include precession and merger ringdown must be calibrated to numerical relativity simulations and to date are only valid at mass ratios $q>1/4$ [@Hannam:2013oca]. Such limited-validity waveforms lead to non-quantified, systematic biases and cannot be used for parameter estimation at present. In the absence of generic waveforms it may be necessary to apply a low-pass filter on real data to mask any merger and ringdown signals.** ]{}
[**Because GW emission is strongest along the orbital angular momentum direction, the detected binaries from a population uniform in orientation and volume is biased against systems whose orbital plane is edge-on to the observer (defined as having an inclination angle near $90^\circ$). Fig. \[fig:inclination\] shows the cumulative distribution function of our observed population (red) compared to an underlying population distributed uniformly in orientation. This selection effect has an important role in mass measurement for LV observations. Precession-induced effects on the waveforms, which are relied upon to break the degeneracy between mass ratio and spin, are less detectable for face-on systems [@Vitale:2014mka]. Previous studies which have used hand-selected populations of signals to study the effects of precession have not accounted for this selection bias, overemphasizing the role precession can play for generic signals. For example, the inclination used by @Chatziioannou:2014coa (63$^\circ$) is larger than $\sim80\%$ of our sources so the improvement they found will be fully realized for a small fraction of our simulated population.**]{}
For each binary we compute the response of Advanced LV at design sensitivity @ObservingScenarios with a low frequency cutoff at 20 Hz. Each simulated “detection” is then analyzed with [`l`alinference\_mcmc]{} which returns independent samples from the PDF for the model parameters. We do not simulate instrument noise in this study because our focus is on the degree to which LV’s frequency-dependent sensitivity, and the flexibility of the PN waveforms, limit mass measurements. Adding simulated noise to our signals introduces uncontrollable contributions to the PDF without adding any value to our assessment of each simulated signal. Posterior distributions in true detections will be altered by the particular noise realization in which the signal is embedded.
![[]{data-label="fig:masses"}](fig1.eps){width="\linewidth"}
![[]{data-label="fig:inclination"}](fig2.eps){width="\linewidth"}
\[sec:results\] Results
=======================
\[sec:m1m2\]Distinguishing black holes from neutron stars
---------------------------------------------------------
We take $m=3\ {{\rm M}_\odot}$ to be the dividing line between the masses of NSs and BHs, with all NSs located below this threshold and all BHs located above. We will use $5\ {{\rm M}_\odot}$ as the minimum mass of a BH when assuming the existence of a mass gap. The distinction between NSBHs and BBHs in our sample is determined by the measurement of the smaller mass, $m_2$. Figure \[fig:m2pe\] shows the $90\%$ credible intervals of the PDF for $m_2$ as a function of the true value from the simulated population. Each entry is colored by the SNR of the signal in the 3-detector network. Horizontal gray dashed lines denote the mass gap. We find all of our simulations with $m_2\lesssim1.5\ {{\rm M}_\odot}$ to be clearly identified as containing at least one neutron star, however our population yields only ten such systems so these are small-number statistics. Most binaries with a NS below 2 ${{\rm M}_\odot}$ constrain the NS to have a mass below 3 ${{\rm M}_\odot}$ although occasional the posterior supports the smaller object being a low-mass BH . Detections of larger mass neutron stars ($2\leq m_2 \leq 3\ {{\rm M}_\odot})$ have $90\%$ credible intervals which consistently extend into the BH regime. The tendency for recovery of high-mass neutron stars in NSBH systems to be consistent with low-mass BHs poses a challenge for determining the maximum NS mass from LV observations alone.
At what point can we *rule out* the possibility that the system contains a neutron star, and definitively declare that we have detected a binary black hole? In Figure \[fig:m2pe\] we see that the true mass of the smaller object must exceed $\sim 6\ {{\rm M}_\odot}$ before the $90\%$ credible intervals rule out a NS. [**Depending on details of spin alignment and orientation, systems with $m_2$ as low as $4\ {{\rm M}_\odot}$ can be unambiguously identified as BBHs.**]{} [**Within this range**]{} the $m_2$ posteriors for BBH sources seldom reach below $2\ {{\rm M}_\odot}$, so if a maximum NS mass were independently confirmed to be consistent with current observations LV’s classification of NSs and BHs would improve.
[**It may seem surprising in Fig \[fig:m2pe\] that the width of the credible intervals do not exhibit the $1/{\rm SNR}$ scaling predicted by Fisher matrix approximations [@Cutler:1994ys]. The Fisher approximation is only suitable at sufficiently high SNR that the posterior distribution function is well approximated by a multivariate Gaussian, in which case the inverse Fisher matrix is the covariance of the posterior. Implicit in this condition is the assumption that the model waveform is a linear function of the source parameters. See @Vallisneri:2007ev for a thorough deconstruction of Fisher matrix-based intuition being applied to GW signals. For typical LIGO/Virgo binaries these conditions are not satisfied for the mass parameters. The width of the $m_2$ credible intervals is driven by uncertainty in $q$ which, due to the degeneracy with spin, is extremely non-Gaussian and can span the entire prior range. The chirp mass, on the other hand, *is* a sufficiently well constrained parameter at the SNRs in our simulated population. In Fig. \[fig:mcpe\] we show the fractional $1-\sigma$ uncertainty in ${\mathcal{M}_c}$ inferred from the Markov chains as a function of the source value, with each event colored by the network SNR. The expected $1/\rm{SNR}$ dependence is apparent for this parameter. Notice also that the chirp mass errors grow with increasing total mass. Higher mass binaries are in band for fewer GW cycles which directly impacts measurability as discussed in Sec. \[sec:pe\].** ]{}
![[]{data-label="fig:m2pe"}](fig3.eps){width="\linewidth"}
![[]{data-label="fig:mcpe"}](fig4.eps){width="\linewidth"}
\[sec:gap\]Identifying systems in the mass gap
----------------------------------------------
Following [@Mandel:2010] we use our simulated LV detections to infer the relative fraction of NSs, BHs in the mass gap, and BHs above the mass gap. Because of the large mass-measurement uncertainties the number of detections needed to conclude BHs inhabit the mass gap is highly dependent on the underlying mass distribution. Depending on whether the observed population is heavily dominated by low- or high-mass systems, we find between ten and many hundreds of detections are necessary to conclude (at three-sigma confidence) the gap is populated.
Our initial set of simulated signals features many high-mass ratio binaries. We simulated an additional 100 sources with both objects having $3<m<5\ {{\rm M}_\odot}$ to check whether comparable-mass systems are easier to identify in the gap. The mass-gap population still suffers from large mass errors with $>95\%$ of the sources having posterior support for a NSBH system. Figure \[fig:m2\_gap\] shows the distribution of the 90% credible interval widths for $m_1$ (red, solid) and $m_2$ (blue, dotted) of the gap sources. The majority of plausible mass-gap sources yield credible intervals that are similar to or exceed the width of the mass gap.
However, $\sim 25\%$ of the mass gap sources’ $m_2$ posteriors do not reach below $2\ {{\rm M}_\odot}$. While we will not be able to say with any certainty that an individual source occupies the mass gap, we can often conclude that the binary contains either an unusually high-mass neutron star, or a pair of unusually low-mass black holes.
Three of the mass-gap sources were constrained to be $3<m<5\ {{\rm M}_\odot}$. A careful investigation of these systems revealed they were in low-probability alignments – all with the spin of the larger mass close to the orbital plane, and the best constrained having $\vec L$ nearly perpendicular to the line of sight.
![[]{data-label="fig:m2_gap"}](fig5.eps){width="\linewidth"}
\[sec:spin\]Restricting allowed spins for neutron stars
-------------------------------------------------------
All of the results thus far presented used a uniform prior on spin magnitude between $[0,1]$. The maximum upper limit on the spin of a NS is $\chi\lesssim0.7$ while observed NS spins are lower [@Lo:2011]. We investigate if using a physical prior on NS spins can improve $m_2$ measurement.
We resample the posterior by imposing a maximum spin for component masses below $3\ {{\rm M}_\odot}$, rejecting samples from the Markov chain with $\chi_2>0.7$. We found no indication that restricting the range of neutron star spins significantly improves uncertainty in the determination of $m_2$.
Figure \[fig:m1m2a2\] demonstrates the limited role of $m_2$’s spin on mass determination. Open circles mark the true values of component masses for nine representative examples. Going through each circle is the scatter plot of the posterior samples colored by $\chi_2$. The arcs traced out by the samples are lines of constant ${\mathcal{M}_c}$. The vertical and horizontal dashed lines denote the mass gap. Notice that there is no obvious correlation between position along the arc and $\chi_2$ – the spin of the smaller body is generally not constrained and therefore does not help with mass determination. Restricting the spin of the smaller mass does not help because at high mass ratios (and therefore NS-like $m_2$), the contribution to the PN phase from $\chi_2$ is suppressed. The leading order spin corrections enter the PN phase with magnitude $\chi_i m_i^2$, so $\chi_2$’s influence to the phase evolution is down-weighted relative to $\chi_1$ by $\mathcal{O}\left(({\rm mass\ ratio})^2\right)$.
![[]{data-label="fig:m1m2a2"}](fig6.eps){width="\linewidth"}
\[sec:discuss\] Discussion
==========================
In this paper we investigated the capability of the Advanced LV network to distinguish between NSs and BHs [**from the inspiral-only waveforms**]{} using a large population of plausible detections. This is the first large-scale study to characterize compact binary PDFs including spin-precession effects over a broad range of masses, mass ratios, and spins. Our study does not factor in systematic effects from real detector noise or differences between template waveforms and the true gravitational wave signal. [**Our study is limited to inspiral-only waveforms for simulation of signals and template waveforms for recovery because available precessing merger/ringdown models are not valid over the full prior volume. For many of our simulated signals the merger will be detectable and may help improve component mass estimates. Further improvements may come from including PN amplitude corrections.**]{}
We arrive at four main conclusions from our analysis:
1. *When are we certain of at least one NS?* For most systems with $m_2 \leq 2\ {{\rm M}_\odot}$, and all systems with $m_2 \leq1 .5\ {{\rm M}_\odot}$.
For larger-mass neutron stars ($2 \leq m_2 \leq 3$) the 90% credible intervals frequently extend into the low-mass BH regime. This tendency for high-mass neutron stars in NSBH systems to be consistent with low-mass black holes poses a challenge for determining the maximum NS mass from LV observations alone.
1. *When are we certain of a BBH with both masses above 3 ${{\rm M}_\odot}$?* When the mass of the smaller object exceeds $\sim 6\ {{\rm M}_\odot}$.
The $m_2$ posteriors for BBH signals seldom reach below 2 ${{\rm M}_\odot}$, so if a maximum neutron star mass were independently confirmed to be $m_{\rm max}\sim2{{\rm M}_\odot}$, then LV’s ability to discriminate between BBH and BNS/NSBH populations would be significantly improved.
1. *When is a black hole definitely not in the mass gap?* When component mass $m_2 \gtrsim 10\ {{\rm M}_\odot}$ the 90% credible intervals do not reach into the mass gap.
It may prove challenging to confirm its existence because NSs with masses above $\sim 2\ {{\rm M}_\odot}$ have error bars reaching into the $m_2\in[3,5]\ {{\rm M}_\odot}$ interval from below while BBH systems with $m_2 \lesssim 10\ {{\rm M}_\odot}$ can have credible intervals that extend below $5\ {{\rm M}_\odot}$. Our findings suggest that inferring a bimodal mass distribution based solely on the observed sample of compact binaries especially without reliable predictions for the underlying compact object mass distribution will be challenging.
1. *When are we certain of a mass-gap object?* Only in rare circumstances when the binary is nearly edge-on and spins are oriented in the orbital plane. For typical binaries component mass errors are larger than the mass gap.
We simulated an additional 100 injections with both masses in the mass gap between $[3,5]$ ${{\rm M}_\odot}$. We found $> 95\%$ of the systems had some posterior support for a NSBH system. The majority of plausible mass gap sources yield credible intervals that are similar to, or exceed the width of the mass gap itself (2 ${{\rm M}_\odot}$). [**Only sources in low-probability alignments (spins in the orbital plane and/or edge-on orientations) were constrained to be $3<m<5\ {{\rm M}_\odot}$.**]{} However, $m_2$ posteriors from mass gap BBHs do not reach below 2 ${{\rm M}_\odot}$. While we will not be able to say with any certainty that a given source occupies the mass gap, we will at least be alerted to the fact that the binary contains either an unusually high-mass neutron star, or a pair of unusually low mass black holes, with either possibility providing plenty of intrigue for the astrophysics community.
Assuming flat mass distributions, we used simulated LV detections to infer the relative fraction of NSs and BHs in and above the mass gap, and estimate how many detections would be needed to confidently conclude that gap BHs exist. Depending on whether the underlying mass distribution is heavily dominated by low- or high-mass systems, ten to hundreds of detections are needed to confirm objects in the gap. Assuming a plausible population-synthesis model where BNSs, NSBHs, and BBHs are well separated in chirp-mass space, @Mandel:2015spa found that a mass gap was statistically distinguishable with a only few tens of detectionsIt is clear that forecasts for what we may learn from LV observations of compact binaries are subject to large variance while uncertainty about the true mass distribution persists. Continued effort in theoretically understanding that distribution, and how LIGO/Virgo observations can be used to test such theories, will be of great value as we begin assembling a catalog of compact binaries coalescences.
Acknowledgments
===============
We thank Atul Adhikari, Claudeson Azurin, Brian Klein, Brandon Miller, Leah Perri, Ben Sandeen, Jeremy Vollen, Michael Zevin for help running the MCMC analysis. Will Farr, Carl-Johan Haster, and Ilya Mandel provided important comments about our calculations and results. TBL and VK acknowledge NSF award PHY-1307020. BF was supported by the Enrico Fermi Institute at the University of Chicago as a McCormick Fellow. SC thanks the US-UK Fulbright Commission for personal financial support during this research period. DEH acknowledges NSF CAREER grant PHY-1151836, the Kavli Institute for Cosmological Physics at the University of Chicago through NSF grant PHY-1125897, and an endowment from the Kavli Foundation. Computational resources were provided by the Northwestern University Grail cluster through NSF MRI award PHY-1126812.
[32]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\
12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty @noop [ ()]{}, [****, ()](\doibase 10.1088/0264-9381/32/2/024001), [****, ()](\doibase 10.1103/PhysRevD.77.021502), [****, ()](\doibase 10.1103/PhysRevD.79.124033), [****, ()](\doibase
10.1103/PhysRevD.89.043009), [****, ()](\doibase 10.1103/PhysRevD.89.103012), [****, ()](\doibase 10.1103/PhysRevD.85.082003), [****, ()](\doibase 10.1103/PhysRevD.87.102001), [****, ()](\doibase 10.1088/0264-9381/27/17/173001), [****, ()](\doibase 10.1088/0004-637X/725/2/1918), [****, ()](\doibase 10.1088/0004-637X/741/2/103), [****, ()](\doibase 10.1088/0004-637X/757/1/36), [****, ()](\doibase
10.1088/0004-637X/757/1/91), [****, ()](\doibase 10.1088/0004-637X/785/1/28), [****, ()](\doibase 10.1088/0004-637X/799/2/190), [ (), 10.1093/mnrasl/slv054](\doibase 10.1093/mnrasl/slv054), @noop [****, ()]{} [****, ()](\doibase 10.1103/PhysRev.131.435) @noop [****, ()]{}, [****, ()](\doibase 10.1103/PhysRevD.49.2658), [****, ()](\doibase 10.1103/PhysRevD.52.848), [****, ()](\doibase
10.1088/2041-8205/766/1/L14), [****, ()](\doibase 10.1088/2041-8205/798/1/L17), [****, ()](\doibase
10.1103/PhysRevD.91.042003), [****, ()](\doibase 10.1103/PhysRevD.88.124039), [****, ()](\doibase 10.1088/0004-637X/728/1/12), @noop [****, ()]{}, [****, ()](\doibase
10.1103/PhysRevLett.113.151101), [****, ()](\doibase
10.1103/PhysRevLett.112.251101), @noop [ ()]{}, [****, ()](\doibase 10.1103/PhysRevD.77.042001), [****, ()](\doibase 10.1103/PhysRevD.81.084029)
|
---
abstract: |
A number of researchers have independently introduced topologies on the set of laws of stochastic processes that extend the usual weak topology. Depending on the respective scientific background this was motivated by applications and connections to various areas (e.g. Plug–Pichler - stochastic programming, Hellwig - game theory, Aldous - stability of optimal stopping, Hoover–Keisler - model theory). Remarkably, all these seemingly independent approaches define the same *adapted weak topology* in finite discrete time. Our first main result is to construct an *adapted* variant of the empirical measure that consistently estimates the laws of stochastic processes in full generality.
A natural compatible metric for the weak adapted topology is the given by an adapted refinement of the Wasserstein distance, as established in the seminal works of Pflug-Pichler. Specifically, the adapted Wasserstein distance allows to control the error in stochastic optimization problems, pricing and hedging problems, optimal stopping problems, etc. in a Lipschitz fashion. The second main result of this article yields quantitative bounds for the convergence of the adapted empirical measure with respect to adapted Wasserstein distance. Surprisingly, we obtain virtually the same optimal rates and concentration results that are known for the classical empirical measure wrt. Wasserstein distance.
*Keywords:* empirical measure, Wasserstein distance, nested distance, weak adapted topology.\
[*Mathematics Subject Classification (2010):* 60G42, 90C46, 58E30.]{}
address:
- 'University of Twente, Department of Applied Mathematics'
- 'Vienna university, Department of Mathematics'
- 'Vienna university, Department of Mathematics'
- 'Oxford University, Department of Mathematics and St. John’s College'
author:
- 'Julio Backhoff, Daniel Bartl, Mathias Beiglböck, Johannes Wiesel'
bibliography:
- 'joint\_biblio.bib'
title: Estimating processes in adapted Wasserstein distance
---
Introduction {#sec:introduction}
============
For a Polish space $(\mathcal{X}, d)$, the (first order) Wasserstein distance on the set of Borel probabilities $\mathrm{Prob}(\mathcal{X})$, is defined by $$\begin{aligned}
\mathcal{W}(\mu,\nu):=\inf_{\pi\in\mathrm{Cpl}(\mu,\nu)}\int d(x,y)\,\pi(dx,dy).\end{aligned}$$ Here $\mathrm{Cpl}(\mu,\nu)$ is the set of couplings between $\mu$ and $\nu$, that is, probabilities $\pi\in\mathrm{Prob}(\mathcal{X}\times\mathcal{X})$ with first marginal $\mu$ and second marginal $\nu$. The Wasserstein distance is particularly well suited for many stochastic problems involing laws of *random variables*. Accordingly, studying convergence of empirical measures in Wasserstein distance has a long history; we refer to [@fournier2015rate] for results and review of the literature.
The situation drastically changes if, instead of random variables, one is interested in laws of *stochastic processes*. Consider the case of two timepoints, let $\mathcal{X}=[0,1]\times[0,1]$ and consider the probabilities $\mu=\delta_{(1/2,1)}+\delta_{(1/2,0)}$ and $\nu:=\delta_{(1/2+\varepsilon,1)}+\delta_{(1/2-\varepsilon,0)}$. Then the discrepancy of $\mu$ and $\nu$ in Wasserstein distance is of order $\varepsilon$, while, considered as laws of stochastic processes, $\mu$ and $\nu$ have very different properties.
\[TraditionalPicture\] ![Close in Wasserstein, very different as stochastic processes.[]{data-label="fig:usual.wasserstein"}](Adap4.pdf "fig:"){width="45.00000%"}
For instance, while no information is available at time $t=1$ in case of $\mu$, the whole future evolution of $\nu$ is known at time $t=1$ already. In fact, the law of an arbitrary stochastic process can be approximated in classical weak topology by laws of stochastic processes which are deterministic after the first period.
As already mentioned in the abstract, to overcome this flaw of the Wasserstein distance (or rather, the weak topology), several researchers have introduced adapted versions of the weak topology. Reassuringly, all these seemingly different definitions yield the same topology in finite discrete time, see [@BaBaBeEd19b].
Below we present an *adapted* extension of the classical Wasserstein distance which induces this topology. In analogy to its classical counterpart, it turns out to be particularly well suited to obtain a quantitative control of stochastic optimization problems, see e.g. [@glanzer2019incorporating; @BaBaBeEd19a].
Causality and adapted / nested Wasserstein distance
---------------------------------------------------
Fix $d \in \mathbb{N}$, which we interpret as the dimension of the state space, denote $T\geq 2$ the number of time points under consideration, and let $\mu, \nu$ be Borel probability measures on $\mathcal{X}=([0,1]^d)^T$. In order to account for the temporal structure of stochastic processes, it is necessary to restrict to couplings of probability measures that are non-anticipative in a specific sense:
Write $X=(X_1, \ldots, X_T)$, $Y=(Y_1, \ldots, Y_T)$ for the projections $X, Y\colon \mathcal{X} \times \mathcal X \to \mathcal X$ onto the first respectively the second coordinate. A coupling $\pi\in \mathrm{Cpl}(\mu, \nu)$ is called *causal* (in the language of Lassalle [@La18a]) if for all $t< T$ the following holds: $$\begin{aligned}
\label{CausalityDef}
\text{given } (X_1, \ldots, X_t), \quad Y_t \text{ and } (X_{t+1}, \ldots, X_T) \text{ are $\pi$-independent.}\end{aligned}$$ That is to say, in order to predict $Y_t$, the only information relevant in $X_1,\dots,X_T$ is already contained in $X_1,\dots,X_t$.
The concept of causal couplings is a suitable extension of adapted processes: a process $Z=(Z_1, \ldots, Z_T)$ on $\mathcal X$ is adapted with respect to the natural filtration if each $Z_t$ depends only on the values of $X_1, \ldots, X_t$ (so in particular is conditionally independent of $X_{t+1}, \ldots, X_T$ given $X_1,\dots,X_t$). Property represents a counterpart of adaptedness on the level of couplings rather than processes.
Similarly we call a coupling *anti-causal* if it satisfies with the roles of $X$ and $Y$ interchanged and finally we call $\pi$ *bi-causal* if it is causal as well as anti-causal. We denote the set of bi-causal couplings with marginals $\mu, \nu$ by $\mathrm{Cpl}_{\mathrm{bc}}(\mu, \nu)$.
The adapted Wasserstein distance (or nested distance) $\mathcal{AW}$ on $\mathrm{Prob}(([0,1]^d)^T)$ is defined as $$\begin{aligned}
\label{AWdef}
\mathcal{AW}(\mu,\nu) := \inf_{\pi\in\mathrm{Cpl}_{\mathrm{bc}}(\mu,\nu)}\int \sum_{t=1}^T|x_t-y_t|\,\pi(dx,dy).
\end{aligned}$$
Bi-causal couplings and the corresponding transport problem were considered by Rüschendorf [@Ru85] under the name ‘Markov-constructions’. Independently, the concept was introduced by Pflug-Pichler [@PfPi12] who realized the full potential of the modified Wasserstein distance in the context of stochastic multistage optimization problems, see also [@PfPi14; @PfPi15; @PfPi16; @glanzer2019incorporating]. Pflug-Picher refer to as *process distance* or *nested distance*. The latter name is motivated by an alternative representation of through a dynamic programming principle. For notational simplicity we state it here only for the case $T=2$ where one obtains the representation $$\begin{aligned}
\label{AWdef2}
\mathcal{AW}(\mu,\nu)
=\inf_{\gamma\in\mathrm{Cpl}(\mu_1,\nu_1) } \int |x_1-y_1| + \mathcal{W}(\mu_{x_1},\nu_{y_1}) \,\gamma(dx_1,dy_1).\end{aligned}$$ Here (and in the rest of this article), for $\mu\in\mathrm{Prob}(([0,1]^d)^T)$ and $1\leq t\leq T-1$, we denote by $\mu_1$ the first marginal of $\mu$ and by $\mu_{x_1,\dots,x_t}$ the disintegration of $\mu$, that is, $$\mu_1(\cdot)=P[X_1\in \, \cdot \, ]
\quad\text{and}\quad
\mu_{x_1,\dots,x_t}(\cdot):=P[ X_{t+1} \in \, \cdot \,| X_1=x_1,\dots,X_t=x_t]$$ for all $(x_1,\dots,x_t)\in([0,1]^d)^t$, where $X$ is a process with law $\mu$. Informally, the representation in asserts that two probabilities are close in adapted Wasserstein distance if (and only if) besides their marginals, also their kernels are similar. This is exactly what fails in the example presented in Figure \[fig:usual.wasserstein\].
Main results
------------
Let $\mu$ be a Borel probability measure on $([0,1]^d)^T$ capturing the true dynamics of the process under consideration. Furthermore let $(X^n)_{n\in\mathbb{N}}$ be an i.i.d. sample of $\mu$, defined on some fixed (sufficiently rich) abstract probability space $(\Omega,\mathcal{F},P)$, i.e., each $X^n=(X_1^n,\dots,X_T^n)$ is distributed according to $\mu$.
Set $r=(T+1)^{-1}$ for $d=1$ and $r=(dT)^{-1}$ for $d\geq 2$. For all $N\geq 1$, partition the cube $[0,1]^d$ into the disjoint union of $N^{rd}$ cubes with edges of length $N^{-r}$ and let $\varphi^N\colon[0,1]^d\to[0,1]^d$ map each such small cube to its center. Then define $$\mucausal:=\frac{1}{N}\sum_{n=1}^N \delta_{\varphi^N(X_1^n),\dots,\varphi^N(X_T^n)}.$$ for each $N\geq 1$. We call $\mucausal$ the [*adapted empirical measure*]{}.
\[TraditionalPicture\] ![The map $\varphi$ and comparison of empirical vs. adapted empirical measure for $d=1, T=2, N=8, r=1/3$.[]{data-label="fig:empirical"}](bbiilldd.pdf "fig:"){width="75.00000%"}
That is, the function $\varphi^N$ satisfies $\sup_{u\in[0,1]^d} |u-\varphi^N(u)|\leq CN^{-r}$ and its range $\varphi^N([0,1]^d)$ consist of $N^{rd}$ points. If $\varphi^N$ were the identity, then $\mucausal$ would be the *(classical) empirical measure*, which we denote by $\muempirical$.
It was first noted by Pflug-Pichler in [@PfPi16] that, in contrast the classical Wasserstein distance, $\mathcal{AW}(\mu,\muempirical)$ does *not* tend to 0 for generic choices of $\mu$ (cf. Remark \[rem:classical.empirical.doesnt.converge\] below).
Our first main Theorem is the following consistency result for the adapted empirical measure:
\[thm:almost.sure.convergence\] The adapted empirical measures is a strongly consistent estimator, that is, $$\lim_{N\to\infty} \mathcal{AW}(\mu,\mucausal) = 0$$ $P$-almost surely.
In particular, as $\mathcal{W}\leq \mathcal{AW}$ by definition, it follows that the adapted empirical measure converges in the usual weak topology as well.
In order to quantify the speed of convergence, we assume the following regularity property for the remainder of this section.
\[ass:lipschitz.kernel\] There is a version of the ($\mu$-a.s. uniquely defined) disintegration such that for every $1\leq t\leq T-1$ the mapping $$\begin{aligned}
([0,1]^d)^t
\ni (x_1, \dots, x_t)
\mapsto \mu_{x_1,\dots,x_t}
\in \mathrm{Prob}([0,1]^d)
\end{aligned}$$ is Lipschitz continuous, where $\mathrm{Prob}([0,1]^d)$ is endowed with its usual Wasserstein distance $\mathcal{W}$.
\[thm:rates.unit.cube\] Under Assumption \[ass:lipschitz.kernel\], there is a constant $C>0$ such that $$\begin{aligned}
\label{eq:mean.speed.rate}
E\Big[ \mathcal{AW}(\mu,\mucausal)\Big]
\leq C
\begin{cases}
N^{-1/(T+1)} &\text{for } d=1,\\
N^{-1/(2T)}\log(N+1) &\text{for } d=2,\\
N^{-1/(dT)} &\text{for } d\geq 3,
\end{cases}
\end{aligned}$$ for all $N\geq 1$.
In the theorem above, the constant $C$ depends on $d$, $T$, and the Lipschitz-constants in Assumption \[ass:lipschitz.kernel\].
Let us quickly compare this result with its counterpart for the classical Wasserstein distance; we refer to [@fournier2015rate] for general results and background on the problem: Ignoring the temporal structure and viewing $\mu$ as the law of a random variable on $[0,1]^{dT}$, one has $$E\Big[ \mathcal{W}(\mu,\muempirical)\Big]
\leq C
\begin{cases}
N^{-1/2} &\text{for } dT=1,\\
N^{-1/2}\log(N+1) &\text{for } dT=2,\\
N^{-1/(dT)} &\text{for } dT \geq 3,
\end{cases}$$ for all $N\geq 1$, and these rates are known to be sharp. As a consequence, for $d\geq 3$ the adapted empirical measure converges in adapted Wasserstein distance at optimal rates. For $d=2$ the rates are optimal up to a logarithmic factor and for $d=1$ the rate is (possibly) not optimal, but approaches the optimal one for large $T$.
Our final main result is the following concentration inequality: denote by $\mathop{\mathrm{rate}}(N)$ the rate for the mean speed of convergence, namely the right hand side of .
\[thm:deviation\] Under Assumption \[ass:lipschitz.kernel\], there are constants $c,C>0$ such that $$P\Big[ \mathcal{AW}(\mu,\mucausal) \geq C\mathop{\mathrm{rate}}(N)+\varepsilon \Big]
\leq 2T\exp\Big( -cN\varepsilon^2 \Big)$$ for all $N\geq 1$ and all $\varepsilon>0$.
As above, the constants $c,C$ depend on $d$, $T$, and the Lipschitz constants in Assumption \[ass:lipschitz.kernel\].
Finally, the following asymptotic regime consequence of Theorem \[thm:deviation\] holds true.
\[cor:deviation.asymptotic\] There exists a constant $c>0$ such that: For every $\varepsilon>0$ there exists $N_0(\varepsilon)$ such that $$P\Big[ \mathcal{AW}(\mu,\mucausal) \geq\varepsilon \Big]
\leq 2T \exp\Big( -cN\varepsilon^2 \Big)$$ for all $N\geq N_0(\varepsilon)$. In particular, $$\limsup_{N\to\infty} \frac{1}{N}\log P\Big[ \mathcal{AW}(\mu,\mucausal) \geq\varepsilon \Big]
\leq -c\varepsilon^2$$ for all $\varepsilon>0$.
\[ex:lipschitz.kernel\] We conclude this section by providing three simple examples in which Assumption \[ass:lipschitz.kernel\] on regularity of disintegrations is satisfied.
(a) Assume that $\mu$ is the law of a stochastic process $(X_t)_{t=1,\dots,T}$ which follows the dynamics $$X_{t+1}=F_{t+1}(X_1,\dots,X_t,\varepsilon_{t+1})$$ for $t=1,\dots,T-1$, with arbitrary $X_1$. Here $F_{t+1}\colon ([0,1]^d)^t\times\mathbb{R}^d\to[0,1]^d$ are given functions and $\varepsilon_{t+1}$ is an $\mathbb{R}^d$-valued random variable independent of $X_1,\dots,X_t$. If $(x_1,\dots,x_t)\mapsto F_{t+1}(x_1,\dots,x_t,z)$ is $L$-Lipschitz for every $z\in\mathbb{R}^d$, then Assumption \[ass:lipschitz.kernel\] holds with Lipschitz constant $L$.
(b) Assume that the probability $\mu\in\mathrm{Prob}(([0,1]^d)^T)$ has a density $f$ w.r.t. Lebesgue measure on $([0,1]^d)^T$. If $f$ is $L$-Lipschitz continuous and there is a constant $\delta>0$ for which $f\geq \delta$, then Assumption \[ass:lipschitz.kernel\] holds with Lipschitz constant $\sqrt{d}2L/\delta$.
(c) Complementing the previous point, Assumption \[ass:lipschitz.kernel\] holds if $\mu$ is supported on finitely many points.
Connection with existing literature
-----------------------------------
### Adapted topologies
A number of authors have independently introduced strengthened variants of the weak topology which take the temporal structure of processes into account.
Aldous [@Al81] introduced *extended weak convergence* as a type of convergence of stochastic processes that in particular guarantees continuity of optimal stopping problems. This line of work has been continued in [@LaPa90; @CoTo07; @HoKe84; @Ho91; @CoMeSl01; @Me03], among others. Applications to stability of SDEs/BSDEs have particularly seen a burst of activity in the last two decades. We refer to the recent article [@PaPoSa18] for an overview of the many available works in this direction.
In the economics literature, Hellwig [@He96] introduced the *information topology*. The work of Hellwig [@He96] was motivated by questions of stability in dynamic economic models/games; see [@Jo77; @VZ02; @HeSch02; @BarbieGupta] for further research in this direction.
Pflug and Pflug-Pichler [@PfPi12; @Pi13; @PfPi14; @PfPi15; @PfPi16] introduced the *nested distance* and systematically applied it to stochastic multistage optimization problems. Independently, adapted versions of the Wasserstein distance were also considered by Rüschendorf [@Ru85], Bion-Nadal and Talay [@BiTa19] and Gigli [@Gi04 Chapter 4]. Adapted distances / topologies on laws of processes are of fundamental importance in questions of stability in mathematical finance and stochastic control, see [@Do13; @BaBeHuKa17; @glanzer2019incorporating; @AcBaCa18; @BaDoDo19; @BaBaBeEd19b; @BaBePa18].
Notably, all these notions (and in fact several more that we do not discuss here) define the same topology in the present discrete time setup, see [@BaBaBeEd19a] and the work of Eder [@Ed19].
### Empirical measures and adapted Wasserstein distance
As mentioned above, it was first noted by Pflug-Pichler in [@PfPi16] that for the classical empirical measures $\muempirical$ we may not have $\mathcal{AW}(\mu,\muempirical)\to 0$ a.s. To obtain a viable estimator, the authors propose to convolute $\muempirical$ with a suitably scaled smoothing kernel. Provided the density of $\mu$ is sufficiently regular, they obtain weak consistency in adapted Wasserstein distance [@PfPi16 Theorem 25]. This is improved upon in [@glanzer2019incorporating Theorem 4] where also a deviation inequality is obtained. The main assumption in the latter result is the existence of a Lipschitz continuous density for $\mu$, which is bounded away from zero. This assumption is in line with Assumption \[ass:lipschitz.kernel\] above, needed for the deviation result of the present article in Theorem \[thm:deviation\]. Specifically, [@glanzer2019incorporating Theorem 4] is a deviation inequality as in Corollary \[cor:deviation.asymptotic\], however with $\varepsilon^2$ replaced by $\varepsilon^{2Td+4}$ (which implies slower decay as $\varepsilon^{2Td+4} <\varepsilon^2$ for small $\varepsilon$).
We stress that Theorem \[thm:almost.sure.convergence\] does not require further assumptions on the measure $\mu$ and has no predecessor in the literature.
Conceptually, the *convoluted empirical measure* considered in [@PfPi16; @glanzer2019incorporating] is related to the adapted empirical measure considered in the present article. A notable difference is that, by construction, the convoluted empirical measure is not discrete and, for practical purposes, a further discretization step may have to be considered in addition to the convolution step.
Organization of the paper
-------------------------
We start by introducing the required notation in Section \[sec:notation\]. The proof of Theorem \[thm:rates.unit.cube\] is presented in Section \[sec:proof.mean\] together with some results which will be applied in the later sections. We then proceed with the proof of Theorem \[thm:deviation\] in Section \[seq:proof.dev\], building on results of the previous section. The proof of Theorem \[thm:almost.sure.convergence\] is presented in Section \[sec:proof.as.convergence\], and again builds on (all) previous results. Finally, Section \[sec:aux\] is devoted to the proof of the examples stated in the introduction.
Notation and preparations {#sec:notation}
=========================
Throughout the paper, we fix $d \in \mathbb{N}$, $T\ge 2$, and let $\mu$ be a probability measure on $([0,1]^d)^T$. We consider $([0,1]^d)^T$ as a filtered space endowed with the canonical filtration $(\mathcal{F}_t)_t$ which is generated by the coordinate mappings. For $1\leq t\leq T$ and a Borel set $G\subset([0,1]^d)^t$ we write $\mu(G):=\mu(G\times([0,1]^d)^{T-t})$ (think of $G$ as $\mathcal{F}_t$-measurable). Note that $$\begin{aligned}
\int f(x)\,\mu(dx)
&=\iint \cdots \int f(x_1,\dots,x_T)\, \mu_{x_1,\dots,x_{T-1}}(dx_T)\cdots \mu_{x_1}(dx_2)\,\mu_1(dx_1)\end{aligned}$$ for every (bounded measurable) function $f\colon ([0,1]^d)^T\to\mathbb{R}$ which amounts to the tower property for conditional expectations and the definition of $\mu_{x_1,\dots,x_t}$ as the kernels / conditional probabilities. Here $\mu_1$ is the first marginal of $\mu$, and to ease notation, we make the convention $\mu_{x_1,\dots,x_t}:=\mu_1$ for $t=0$.
We now turn to notation more specific to this paper: For $\nu\in\mathrm{Prob}(([0,1]^d)^T)$, $1\leq t\leq T-1$, and a Borel set $G\subset ([0,1]^d)^t$, define the averaged (over $G$) kernel $$\begin{aligned}
\label{def:mu.G}
\nu_G(\cdot)
:=\frac{1}{\nu(G)} \int_{G} \nu_{x_1,\dots,x_t}(\cdot)\,\nu(dx)
\in \mathrm{Prob}([0,1]^d)\end{aligned}$$ with an arbitrary convention if we have to divide by 0; say $\nu_G=\delta_0$ in this case. In other words, if $X\sim\nu$, then $$\nu_G(\cdot) =P[X_{t+1}\in \,\cdot\, | (X_1,\dots, X_t)\in G ]$$ is the conditional distribution of $X_{t+1}$ given that $(X_1,\dots, X_t)\in G$.
Next recall the definitions of $\varphi^N$ and $r$ given in the introduction and define $$\Phi^N:=\big\{(\varphi^N)^{-1}(\{x\}) : x\in \varphi^N([0,1]^d) \big\},$$ which forms a partition of $[0,1]^d$ associated to $\varphi^N$ such that $$[0,1]^d=\bigcup_{F\in \Phi^N} F \quad\text{disjoint,}
\qquad
\sup_{F\in\Phi^N}\mathop{\mathrm{diam}} (F) \leq \frac{C}{N^r},
\qquad
| \Phi^N| \leq N^{rd}.$$ Here $\mathop{\mathrm{diam}} (F):=\sup_{x,y\in F} |x-y|$ and $| \Phi^N|$ denotes the number of elements in $\Phi^N$.
Then, for every $1\leq t\leq T-1$ and every $$G\in \Phi^N_t:=\Big\{ \prod_{1\leq s\leq t} F_s : F_s\in \Phi^N \text{ for all } 1\leq s\leq t\Big\}$$ one has $$\begin{aligned}
\label{eq:expression.kernel.estimator.partition}
\mucausal[G]
=\frac{1}{ \big| \big\{ \substack{n\in\{1,\dots,N\} \text{ s.t.}\\ (X_1^n,\dots,X_t^n)\in G} \big\} \big| } \sum_{\substack{n\in\{1,\dots,N\} \text{ s.t.}\\ (X_1^n,\dots,X_t^n)\in G}} \delta_{\varphi^N(X^n_{t+1})},
$$ where, as before, we set $\mucausal[G]=\delta_0$ if we have to divide by zero. Moreover, as $\mucausal$ charges every $G\in\Phi_t^N$ exactly once (at $\varphi^N(G):=\{ \varphi^N(g) : g\in G\}$ which consist of a single point), setting $\mucausal_g:=\mucausal_G$ for $g\in G\in \Phi_t^N$ defines a disintegration of $\mucausal$. Finally, let us already point out at this stage that the denominator in front of the sum in equals $N\muempirical(G)$.
\[rem:classical.empirical.doesnt.converge\] At least when $\mu$ has a density w.r.t. the Lebesgue measure, the probability that two observations coincide at some time is equal to zero, that is, $P[X_t^n=X_t^m \text{ for some } n\neq m \text{ and } 1\leq t\leq T]=0$. Therefore the kernels of $\muempirical$ are almost surely Dirac measures, meaning that if $Y$ is distributed according to $\muempirical$, then the entire (future) evolution of $Y$ is known already at time $1$. This implies that the classical empirical measure cannot capture any temporal structure and convergence in the weak adapted topology will not hold true. In accordance, the values of multistage stochastic optimization problems (for instance optimal stopping, utility maximization, ...) computed under $\muempirical$ will not convergence to the respective value under $\mu$ in general. In Example \[ex:opt.stop.not.cont.usual.empirical\] we illustrate this for the optimal stopping problem.
In contrast, we have just seen in that the kernels of our modified empirical measure $\mucausal$ are in general not Dirac measures and in fact behave like [*averaged kernels*]{} of the empirical measure: for a Borel set $G\subset ([0,1]^d)^t$ one has $$\begin{aligned}
\label{eq:expression.kernel.empirical.partition}
\muempirical[G]= \frac{1}{ \big|\big\{ \substack{n\in\{1,\dots,N\} \text{ s.t.}\\ (X_1^n,\dots,X_t^n)\in G} \big\}\big| } \sum_{\substack{n\in\{1,\dots,N\} \text{ s.t.}\\ (X_1^n,\dots,X_t^n)\in G}} \delta_{ X^n_{t+1}},
\end{aligned}$$ showing that $\mucausal[G]$ is indeed the push forward of $\muempirical[G]$ under $\varphi^N$.
In fact, we will show in Lemma \[lem:ingredients.indep\] that (conditionally) $\muempirical[G]$ has the same distribution as $\widehat{\mu_G}^{L_G}$, the empirical measure of $\mu_G$ with a random number $L_G:=N\widehat{\mu}^N(G)$ of observations.
In order to exclude the necessity to distinguish whether the random number $L_G$ above is positive or not, it will turn out useful to make the convention that for any probability, its empirical measure with sample size zero is just the Dirac at zero. In Section \[sec:deviation\] it is furthermore convenient to denote $\mathcal{G}_t^N:=\{ \muempirical(G) : G\in\Phi^N_t\}$.
In order to lighten notation in the subsequent proofs, we finally define $$\begin{aligned}
\label{eq:def.R}
R\colon[0,+\infty)\to[0,+\infty],
\quad R(u):=\begin{cases}
u^{-1/2}&\text{if } d=1,\\
u^{-1/2}\log(u+3) &\text{if } d=2,\\
u^{-1/d}&\text{if } d\geq 3.
\end{cases}\end{aligned}$$ The reason to go with $\log(u+3)$ in the definition of $R$ instead of $\log(u+1)$ as would have been natural in view of the statement of Theorem \[thm:rates.unit.cube\] is to guarantee that $u\mapsto uR(u)$ is concave, which simplifies notation. Also set $0R(0):=\lim_{u\to 0} u R(u)=0$.
Throughout the proofs, $C>0$ will be a generic constant depending on all sorts of external parameters, possibly increasing from line to line; e.g. $2C x^2\leq Cx^2$ for all $x\in\mathbb{R}$ but not $2x^2\leq x^2/C$ or $N\leq C$ for all $N$.
Proof of Theorem \[thm:rates.unit.cube\] {#sec:proof.mean}
========================================
We split the proof into a number of lemmas, which we will reference throughout the paper. In particular, we will sometimes (but not always) work under Assumption \[ass:lipschitz.kernel\] that the kernels of $\mu$ are Lipschitz, that is, there is a constant $L$ such that $$\mathcal{W}(\mu_{x_1,\dots,x_t},\mu_{y_1,\dots,y_t})
\leq L |(x_1,\dots,x_t)-(y_1,\dots,y_t)|$$ for all $(x_1,\dots,x_t)$ and $(y_1,\dots,y_t)$ in $([0,1]^d)^t$, and all $1\leq t\leq T-1$.
\[lem:aw.estimate.lipschitz.kernel\] Assume that the kernels of $\mu$ are Lipschitz. Then there is a constant $C>0$ such that $$\mathcal{AW}(\mu,\nu)
\leq C \mathcal{W}(\mu_1, \nu_1 ) + C\sum_{t=1}^{T-1} \int \mathcal{W}(\mu_{y_1,\dots,y_t}, \nu_{y_1,\dots,y_t} )\,\nu(d y)$$ for every $\nu\in\mathrm{Prob}(([0,1]^d)^T)$.
We first present the proof for $T=2$ which is notationally simpler: Making use of the dynamic programming principle for the adapted Wasserstein distance [@BaBeLiZa16 Proposition 5.1 and equation (5.1)], we can write $$\begin{aligned}
\label{eq:dyn.prog.AW.2period.lipschitz}
&\mathcal{AW}(\mu,\nu)
=\inf_{\gamma\in \mathrm{Cpl}(\mu_1, \nu_1)} \int |x_1-y_1| + \mathcal{W}(\mu_{x_1},\nu_{y_1}) \,\gamma(dx_1, dy_1).
\end{aligned}$$ Calling $L$ the Lipschitz constant of the kernel $x_1\mapsto \mu_{x_1}$, the triangle inequality implies $\mathcal{W}(\mu_{x_1},\nu_{y_1})\leq L |x_1-y_1| + \mathcal{W}(\mu_{y_1},\nu_{y_1})$ for all $x_1,y_1\in [0,1]^d$. Plugging this into yields the claim for $T=2$.
In case of $T\geq 2$, recall that $\mu_{x_1,\dots,x_t}:=\mu_1$ and similarly $\nu_{y_1,\dots,y_t}:=\nu_1$ for $t=0$. Further write $x_{1:t}:=(x_1,\dots,x_t)$ and $x_{t:T}:=(x_t,\cdots,x_T)$ for $x_1,\dots,x_T$ in $[0,1]^d$ and $1\leq t\leq T$. The dynamic programming principle for the adapted Wasserstein distance (see again [@BaBeLiZa16 Proposition 5.1]) asserts that $\mathcal{AW}(\mu,\nu)=V_0$, where $V_T:=0$ and, recursively $$\begin{aligned}
V_t(x_{1:t},y_{1:t})
:=\inf_{\gamma \in \mathrm{Cpl}(\mu_{x_{1:t}}, \nu_{y_{1:t}}) } \int\Big( |x_{t+1}-y_{t+1}| &+V_{t+1}(x_{1:t+1},y_{1:t+1})\Big) \\
& \gamma(dx_{t+1}, dy_{t+1})
\end{aligned}$$ for $x_{1:t}$ and $y_{1:t}$ in $([0,1]^d)^t$. We will prove the claim via backward induction, showing that for all $x_{1:t}$ and $y_{1:t}$ in $([0,1]^d)^t$ it holds that $$\begin{aligned}
\label{eq:dyn.prog.induction}
\begin{split}
&V_t(x_{1:t},y_{1:t})
\leq C\Big( |x_{1:t}-y_{1:t}|+\sum_{s=t}^{T-1}\int \mathcal{W}(\mu_{y_{1:s}}, \nu_{y_{1:s}} )\,\bar{\nu}_{y_{1:t}}(dy_{t+1:T}) \Big).
\end{split}
\end{aligned}$$ Here $\bar{\nu}_{y_{1:t}}$ is the conditional probability $P[(Y_{t+1},\dots Y_T)\in \cdot | Y_1=y_1,\dots,Y_t=y_t]$ where $Y\sim\nu$ with the convention that $\bar{\nu}_{y_{1:t}}=\nu$ for $t=0$, that is, $\bar{\nu}_{y_{1:T-1}}:=\nu_{y_{1:T-1}}$ and recursively $$\bar{\nu}_{y_{1:t}}(dy_{t+1:T})
:=\bar{\nu}_{y_{1:t+1}}(dy_{t+2:T}) \nu_{y_{1:t}}(dy_{t+1})$$ for $t=T-1,\dots,0$ and $y_{1:t}$ in $([0,1]^d)^t$.
For $t=T$, trivially holds true. Assuming that holds true for $t+1$, we compute $$\begin{aligned}
V_t(x_{1:t},y_{1:t})
&\leq C \inf_{\gamma \in \mathrm{Cpl}(\mu_{x_{1:t}}, \nu_{y_{1:t}} )}
\int\Big( \sum_{s=t+1}^{T-1}\int \mathcal{W}(\mu_{ y_{1:s} }, \nu_{ y_{1:s} } )\,\bar{\nu}_{ y_{1:t+1} }(dy_{t+2:T})\\
& + |x_{1:t+1}-y_{1:t+1}| + |x_{t+1}-y_{t+1}| \Big)\,\gamma(dx_{t+1}, dy_{t+1}).
\end{aligned}$$ By definition we have $$|x_{1:t+1}- y_{1:t+1}|
=|x_{1:t} - y_{1:t}| + |x_{t+1}-y_{t+1}|.$$ Now note that the sum over the Wasserstein distance inside the $\gamma$-integral only depends on $y$. Therefore it is independent of the choice of coupling $\gamma$ and we arrive at $$\begin{aligned}
&V_t(x_{1:t},y_{1:t})
\leq C \Big( |x_{1:t} - y_{1:t}| + \mathcal{W}(\mu_{x_{1:t}}, \nu_{y_{1:t}}) \\
&\quad+ \int \sum_{s=t+1}^{T-1}\int \mathcal{W}(\mu_{y_{1:s}}, \nu_{y_{1:s}} )\,\bar{\nu}_{y_{1:t+1}}(dy_{t+2:T})\,\nu_{y_{1:t}}(dy_{t+1}) \Big).
\end{aligned}$$ Moreover, by assumption, $$\mathcal{W}(\mu_{x_{1:t}}, \nu_{y_{1:t}})
\leq L|x_{1:t} - y_{1:t}| + \mathcal{W}(\mu_{y_{1:t}}, \nu_{y_{1:t}})$$ for all $x_{1:t}$ and $y_{1:t}$ in $([0,1]^d)^t$. Finally, recalling the definition of $\bar{\nu}$, one has that $$\begin{aligned}
&\iint \mathcal{W}(\mu_{y_{1:s}}, \nu_{y_{1:s}} )\,\bar{\nu}_{y_{1:t+1}}(dy_{t+2:T})\,\nu_{y_{1:t}}(dy_{t+1})
=\int \mathcal{W}(\mu_{y_{1:s}}, \nu_{y_{1:s}} )\,\bar{\nu}_{y_{1:t}}(dy_{t+1:T})
\end{aligned}$$ for every $t+1\leq s\leq T-1$ and $y_{1:t}$ in $([0,1]^d)^t$. This concludes the proof of .
The result now follows by setting $t=0$ in .
\[lem:integral.kernels.leq.averaged.kernel\] The following hold.
(i) We have $$\mathcal{W}(\mu_1, \mucausal[1] )
\leq \frac{C}{N^r} + \mathcal{W}(\mu_{1}, \muempirical[1] )$$ for all $N\geq 1$.
(ii) If the kernels of $\mu$ are Lipschitz, then we have $$\begin{aligned}
\int \mathcal{W}(\mu_{y_1,\dots,y_t}, \mucausal[y_1,\dots,y_t] )\,\mucausal(dy)
\leq \frac{C}{N^r} + \sum_{G\in \Phi^N_t} \muempirical(G) \mathcal{W}(\mu_G,\muempirical[G])
\end{aligned}$$ for every $N\geq 1$ and every $1\leq t\leq T-1$.
<!-- -->
(i) The triangle inequality implies $$\mathcal{W}(\mu_1, \mucausal[1] )
\leq \mathcal{W}(\mu_1, \muempirical[1] ) + \mathcal{W}( \muempirical[1], \mucausal[1] ).$$ As $\mucausal[1]$ is the push forward of $\muempirical[1]$ under the mapping $\varphi^N$, we obtain $$\begin{aligned}
\label{eq:push.forward.wasserstein}
\begin{split}
\mathcal{W}(\widehat{\mu}_1^N,\mucausal[1])
&\leq \int |x_{1}-\varphi^N(x_1)|\,\muempirical[1](dx_1)\\
&\leq \sup_{u\in[0,1]^d}|u-\varphi^N(u)|
\leq \frac{C}{N^r},
\end{split}
\end{aligned}$$ where the last inequality holds by assumption on $\varphi^N$. This proves the first claim.
(ii) For the second claim, fix $1\leq t\leq T-1$. In a first step, write $$\begin{aligned}
\label{eq:W.kernel.leq.sum.G}
\int \mathcal{W}(\mu_{y_1,\dots,y_t}, \mucausal[y_1,\dots,y_t] )\,\mucausal(dy)
\leq \sum_{G\in \Phi^N_t} \muempirical(G) \sup_{g\in G } \mathcal{W}(\mu_g, \mucausal[g] ),
\end{aligned}$$ where we used that $\mucausal(G)=\muempirical(G)$ for every $G\in\Phi_t^N$ (note that this relation only holds for $G\in\Phi_t^N$ and, of course, not for general $G\subset([0,1]^d)^t$). Recalling that $\mucausal[g]=\mucausal[G]$ for every $g\in G\in\Phi_t^N$, we proceed to estimate $$\begin{aligned}
\mathcal{W}(\mu_g, \mucausal[g] )
&\leq \mathcal{W}(\mu_g, \mu_G) + \mathcal{W}(\mu_G,\muempirical[G] )
+ \mathcal{W}(\muempirical[G], \mucausal[G] ).
\end{aligned}$$ As $\mucausal[G]$ is the push forward of $\muempirical[G]$ under the mapping $\varphi^N$ (see and the sentence afterwards), the same argument as in implies that $\mathcal{W}(\muempirical[G], \mucausal[G])\leq C N^{-r}$.
Further, convexity of $\mathcal{W}(\mu_g, \cdot)$ together with the definition of $\mu_G$ in imply that for every $G\in\Phi^N_t$ with $\mu(G)>0$, one has $$\begin{aligned}
\label{eq:estimate.mug.muG}
\begin{split}
\mathcal{W}(\mu_g, \mu_G)
&\leq \frac{1}{\mu(G)} \int_{G} \mathcal{W}(\mu_g, \mu_{z_1,\dots,z_t}) \,\mu(dz)\\
&\leq \sup_{h\in G} \mathcal{W}(\mu_g, \mu_h)
\leq L \mathop{\mathrm{diam}}(G)
\leq \frac{LC}{N^r}
\end{split}
\end{aligned}$$ for all $g\in G$, where $L$ denotes the Lipschitz constant of $(x_1,\dots,x_t)\mapsto \mu_{x_1, \dots, x_t}$.
Finally note that for every $G\in\Phi^N_t$ with $\mu(G)=0$ one has $\muempirical(G)=0$ almost surely. Hence one may restrict to those $G$ for which $\mu(G)>0$ in the sum on the right hand side of . This completes the proof.
By definition, $\muempirical[1]$ is the empirical measure of $\mu_1$ with $N$ observations. The following lemma shows that a version of this statement remains true for the averaged kernels of $\muempirical$.
\[lem:ingredients.indep\] Let $1 \le t \leq T-1$. Conditionally on $\mathcal{G}_t^N:=(\muempirical(G) )_{G\in\Phi_t^N} $, the following hold.
(i) The family $\{ \muempirical[G] : G\in\Phi^N_t\}$ is independent.
(ii) For every $G\in\Phi_t^N$, the law of $\muempirical[G] $ is the same as that of $\widehat{\mu_G}^{N\muempirical(G)}$ (the empirical measure of $\mu_G$ with sample size $N\muempirical(G)$).
To simplify notation, we agree that ‘f.a. $G$’ will always mean ‘for all $G\in\Phi^N_t$’ throughout this proof and similar for $H$; i.e. $G$ and $H$ always run through $\Phi_t^N$. For each $G$, let $L_G\in\{0,\dots,N\}$ such that $\sum_G L_G=N$. Both statements of the lemma then follow if we can show that for every family $(A_G)_{G}$ of measurable subsets of $\mathrm{Prob}([0,1]^d)$ we have $$\begin{aligned}
\label{eq:mu.G.indep}
P\Big[ \muempirical[G]\in A_G \text{ f.a.\ } G \Big| N\muempirical(H)=L_H \text{ f.a.\ } H \Big]
= \prod_G P\Big[ \widehat{\mu_G}^{L_G} \in A_G\Big]
\end{aligned}$$ whenever $P[N\muempirical(H)=L_H \text{ f.a.\ } H]\neq 0$.
To that end, fix such a family $(A_G)_{G}$ and denote by $\mathcal{I}$ the set of all partitions $(I_G)_G$ of $\{1,\dots,N\}$ such that $|I_G|=L_G$ for every $G$. We use the shorthand notation $X^I_{1:t}=\{(X^n_1,\dots,X^n_t) : n\in I\}$ for subsets $I\subset \{1,\dots,N\}$. Similarly $X_{t+1}^I:=\{ X^n_{t+1} : n\in I\}$.
(a) Fix a partition $(I_G)_G$. We first claim that $$\begin{aligned}
\label{eq:mu.G.indep.partition}
P\Big[ \muempirical[G]\in A_G \text{ f.a.\ } G \Big| X_{1:t}^{I_H}\subset H \text{ f.a.\ } H \Big]
=\prod_G P\Big[ \widehat{\mu_G}^{L_G}\in A_G \Big].
\end{aligned}$$ To see this, note that for all $G$ one has $$\begin{aligned}
\label{eq:rep.mu.on.calG}
\muempirical[G]= \frac{1}{|I_G|}\sum_{n\in I_G} \delta_{X^n_{t+1}}
\quad\text{on }\{ X^{I_H}_{1:t}\subset H \text{ f.a.\ } H\} .
\end{aligned}$$ The advantage of representation is that the dependence of $\muempirical[G]$ on $X^n_1,\dots,X^n_t$ is gone and $\muempirical[G]$ depends solely on $X^{I_G}_{t+1}$. As the $(X^n)_n$ are i.i.d. and the $I_G$ are disjoint, the pairs of random variables $$\Big( X^{I_G}_{1:t},\frac{1}{|I_G|}\sum_{n\in I_G} \delta_{X^n_{t+1}} \Big)_G
\quad\text{are independent under } P.$$ The definition of conditional expectations therefore implies that $$\begin{aligned}
\label{eq:indep.products}
\begin{split}
&P\Big[ \muempirical[G]\in A_G \text{ f.a.\ } G \Big| X^{I_H}_{1:t}\subset H \text{ f.a.\ } H \Big]\\
&=\frac{\prod_G P\big[ \frac{1}{|I_G|}\sum_{n\in I_G} \delta_{X^n_{t+1}} \in A_G \text{ and } X^{I_G}_{1:t}\subset G \big]}{\prod_G P[X^{I_G}_{1:t}\subset G ]}\\
&=\prod_G P\Big[ \frac{1}{|I_G|}\sum_{n\in I_G} \delta_{X^n_{t+1}} \in A_G \Big| X^{I_G}_{1:t}\subset G \Big].
\end{split}
\end{aligned}$$ Further, for every fixed $G$, given $\{X^{I_G}_{1:t}\subset G\}$, the family $X^{I_G}_{t+1}$ is independent with each $X^n_{t+1}$ being distributed according to $\mu_G$. Therefore, given $\{X^{I_G}_{1:t}\subset G\}$, the distribution of $\frac{1}{|I_G|}\sum_{n\in I_G} \delta_{X^n_{t+1}} $ equals the distribution of the empirical measure of $\mu_G$ with sample size $|I_G|=L_G$. We conclude that $$P\Big[ \frac{1}{|I_G|}\sum_{n\in I_G} \delta_{X^n_{t+1}} \in A_G \Big| X^{I_G}_{1:t}\subset G \Big]
= P\Big[ \widehat{\mu_G}^{L_G}\in A_G \Big].$$ Plugging this equality into yields exactly .
(b) We proceed to prove . As $\{N\muempirical(H)=L_H \text{ f.a.\ } H \}$ is the disjoint union of $\{X^{I_H}\subset H \text{ f.a.\ } H\}$ over $(I_H)_H\in\mathcal{I}$, we deduce that $$\begin{aligned}
&P\Big[ \muempirical[G]\in A_G \text{ f.a.\ } G \Big| N\muempirical(H)=L_H \text{ f.a.\ } H \Big] \\
&=\sum_{(I_H)_H\in\mathcal{I}} P\Big[ \muempirical[G]\in A_G \text{ f.a.\ } G \Big| X^{I_H}_{1:t}\subset H \text{ f.a.\ } H \Big]
\frac{P[X^{I_H}_{1:t}\subset H \text{ f.a.\ } H ] }{P[ N\muempirical(H)=L_H \text{ f.a.\ } H] }
\end{aligned}$$ By , the first term inside the sum is equal to the product of $\Pi_G P[\widehat{\mu_G}^{L_G}\in A_G]$ and in particular does not depend on the choice of $(I_H)_H$. The sum over the fractions equals 1, which shows and thus completes the proof.
\[lem:conditional.rate.expectation\] The following hold.
(a) We have $$E[\mathcal{W}(\mu_1,\muempirical[1])]
\leq C R\Big( \frac{N}{ N^{rd(T-1)}} \Big)$$ for all $N\geq 1$.
(b) For every $1\leq t\leq T-1$ we have $$E\Big[\sum_{G\in \Phi^N_t} \muempirical(G) \mathcal{W}(\mu_G,\muempirical[G])\Big| \mathcal{G}_t^N \Big]
\leq C R\Big( \frac{N}{ N^{rd(T-1)}} \Big)$$ almost surely for all $N\geq 1$.
<!-- -->
(a) As $\muempirical[1]$ is the empirical measure of $\mu_1$ with $N$ observations, [@fournier2015rate Theorem 1] implies that $E[\mathcal{W}( \mu_1,\muempirical[1] )]\leq C R( N)$ for all $N\geq 1$. The claim follows as $R$ is decreasing.
(b) For the second claim, fix $1\leq t\leq T-1$. Lemma \[lem:ingredients.indep\] implies that, conditionally on $\mathcal{G}_t^N$, the distribution of each $\muempirical[G]$ equals the distribution of the empirical measure of $\mu_G$ with sample size $N\muempirical(G)$. Therefore, estimating the mean speed of convergence of the classical empirical measure by e.g. [@fournier2015rate Theorem 1], one has that $$\begin{aligned}
\label{eq:rates.conditionally.on.muG}
E\big[\mathcal{W}(\mu_G,\muempirical[G]) \big| \mathcal{G}_t^N \big]
\leq C R(N\muempirical(G))
\end{aligned}$$ almost surely for all $N\geq 1$.
Summing over $G\in\Phi^N_t$ yields $$\begin{aligned}
&E\Big[\sum_{G\in \Phi^N_t} \muempirical(G) \mathcal{W}(\mu_G,\muempirical[G])\Big|\mathcal{G}_t^N \Big]
\leq C \sum_{G\in \Phi^N_t} \muempirical(G)R\Big(N \muempirical(G)\Big) \\
&=C \frac{|\Phi^N_t|}{N} \cdot \sum_{G\in \Phi^N_t} \frac{1}{|\Phi^N_t|} \Big( N\muempirical(G)\Big) R\Big(N \muempirical(G)\Big).
\end{aligned}$$ Now, concavity of $u\mapsto uR(u)$ implies that the latter term is smaller than $$\begin{aligned}
C \frac{|\Phi^N_t|}{N} \cdot \Big(\sum_{G\in \Phi^N_t}\frac{N \muempirical(G)}{|\Phi^N_t|} \Big) R\Big(\sum_{G\in \Phi^N_t}\frac{N \muempirical(G)}{ |\Phi^N_t| }\Big) \Big)
=C R\Big(\frac{N}{ |\Phi^N_t| }\Big).
\end{aligned}$$ Finally, using that $R$ is decreasing, one obtains $$R\Big( \frac{N}{|\Phi^N_t|} \Big)
\leq R\Big( \frac{N}{|\Phi^N_{T-1}|} \Big)
\leq R\Big( \frac{N}{ N^{rd(T-1)}} \Big).$$ This completes the proof.
By Lemma \[lem:aw.estimate.lipschitz.kernel\] one has that $$E\Big[\mathcal{AW}(\mu,\mucausal)\Big]
\leq C E[\mathcal{W}(\mu_1, \mucausal[1] )] + C\sum_{t=1}^{T-1} E\Big[\int \mathcal{W}(\mu_{y_1,\dots,y_t}, \mucausal[y_1, \dots, y_t] )\,\mucausal(d y)\Big].$$ Combining Lemma \[lem:integral.kernels.leq.averaged.kernel\] and Lemma \[lem:conditional.rate.expectation\] together with the tower property shows that $$E\Big[\mathcal{AW}(\mu,\mucausal)\Big]
\leq C\Big( \frac{1}{N^r} + R\Big(\frac{N}{N^{rd(T-1)}}\Big) \Big)$$ for every $N\geq 1$.
Lastly, by definition of $r$, one has that $$\frac{N}{N^{rd(T-1)}}
=\begin{cases}
N^{2/(T+1)} &\text{if } d=1,\\
N^{1/T} &\text{if } d\geq 2.
\end{cases}$$ Recalling the definition of $R$ in (and noting that $\log(N^{1/T} +
3) \leq C \log(N + 1)$ for all $N$ in case $d = 2$) yields the claim.
Proof of Theorem \[thm:deviation\] {#sec:deviation}
==================================
\[seq:proof.dev\]
The proof uses the following basic result for subgausian random variables, where we use the convention $x/0=\infty$ for $x>0$ and $0\cdot \infty =0$.
\[lem:subgaussian\] For an integrable zero mean random variable $Y$ and $\sigma\geq 0$ consider the following:
(i) $E[\exp(tY)]\leq \exp(t^2\sigma^2/2)$ for all $t\in\mathbb{R}$.
(ii) $P[|Y|\geq t]\leq 2 \exp(-t^2/(2\sigma^2))$ for all $t\in\mathbb{R}_+$.
Then (i) implies (ii). Moreover, (ii) implies (i) with $\sigma^2$ replaced by $8\sigma^2$ in (i).
See, for instance, [@Vershynin:2018td Proposition 2.5.2].
A zero mean random variable $Y$ which satisfies part (i) of Lemma \[lem:subgaussian\] is called subgaussian with parameter $\sigma^2$. From the definition it immediately follows that if $Y_1,\dots,Y_n$ are independent $\sigma_k^2$-subgaussian random variables, then $\sum_{k=1}^n Y_k$ is again subgaussian with parameter $\sum_{k=1}^n\sigma_k^2$. In particular, one obtains Hoeffding’s inequality $$P\Big[ \Big|\sum_{k=1}^n Y_k\Big|\geq t\Big]
\leq 2\exp\Big( \frac{-ct^2}{\sum_{k=1}^n\sigma_k^2}\Big) \quad\text{for all }t\geq 0.$$ The reason why subgaussian random variables are of interest in the proof of Theorem \[thm:deviation\] is the following:
\[lem:ingredients.subgauss\] The following hold.
(a) The random variable $$\mathcal{W}(\mu_1,\muempirical[1]) - E[\mathcal{W}(\mu_1,\muempirical[1])]$$ is subgaussian with parameter $C/N$ for all $N\geq 1$.
(b) Let $1\leq t\leq T-1$ and $G\in\Phi^N_t$. Then, conditionally on $\mathcal{G}_t^N$, the random variable $$\begin{aligned}
\muempirical(G)\Big( \mathcal{W}(\mu_G,\muempirical[G]) - E\big[\mathcal{W}(\mu_G,\muempirical[G])\big|\mathcal{G}_t^N \big] \Big)
\end{aligned}$$ is subgaussian with parameter $C\muempirical(G)/N$ for all $N\geq 1$.
<!-- -->
(a) More generally than in the statement of the lemma, let $\nu\in\mathrm{Prob}([0,1]^d)$ and $L\in\mathbb{N}$ be arbitrary. Applying McDiarmid’s inequality to the function $([0,1]^d)^L\ni x\mapsto \mathcal{W}(\nu,1/L\sum_{n=1}^L \delta_{x^n})$ shows that the random variable $$\mathcal{W}(\nu,\widehat{\nu}^L) - E[\mathcal{W}(\nu,\widehat{\nu}^L)]$$ is subgaussian with parameter $C/L$ (where $\widehat{\nu}^L$ denotes the empirical measure of $\nu$ with sample size $L$). This in particular implies point (a) of this lemma.
(b) Conditionally on $\mathcal{G}_t^N$, the distribution of $\muempirical[G]$ is the same as the distribution of the empirical measure of $\mu_G$ with $N\muempirical(G)$ observations, see Lemma \[lem:ingredients.indep\]. By (the proof of) part (a) of this lemma this implies that, conditionally on $\mathcal{G}_t^N$, the random variable $$\mathcal{W}(\mu_G,\muempirical[G]) - E\big[\mathcal{W}(\mu_G,\muempirical[G])\big|\mathcal{G}_t^N\big]$$ is subgaussian with parameter $C/(N\muempirical(G))$. Multiplying a $\sigma^2$-subgaussian random variable by a constant $a\geq 0$ yields a $\sigma^2a^2$-subgaussian random variable. This completes the proof.
\[lem:concentration.sum.mu.G\] There is a constant $c>0$ such that the following hold.
(a) We have $$P\Big[ \big| \mathcal{W}(\mu_1,\muempirical[1]) - E[\mathcal{W}(\mu_1,\muempirical[1]) ]\big|\geq \varepsilon \Big]
\leq 2\exp(-c\varepsilon^2 N)$$ for all $\varepsilon\geq 0$ and all $N\geq 1$.
(b) We have $$P\Big[ \Big|\sum_{G\in\Phi^N_t} \muempirical(G)\Big( \mathcal{W}(\mu_G,\muempirical[G]) - E[\mathcal{W}(\mu_G,\muempirical[G])| \mathcal{G}_t^N ]\Big) \Big|\geq \varepsilon \Big| \mathcal{G}_t^N \Big]
\leq 2\exp(-c\varepsilon^2 N)$$ almost surely for all $\varepsilon\geq 0$, all $N\geq 1$, and all $1\leq t\leq T-1$.
<!-- -->
(a) The proof follows immediately from Lemma \[lem:subgaussian\] and Lemma \[lem:ingredients.subgauss\].
(b) For later reference, define $$\begin{aligned}
\label{eq:def.Delta.t}
\Delta_{t}^N&:=\sum_{G\in\Phi^N_t} \Delta_{t,G}^N,\quad\text{where}\\
\nonumber
\Delta_{t,G}^N&:=\muempirical(G)\Big( \mathcal{W}(\mu_G,\muempirical[G]) - E[\mathcal{W}(\mu_G,\muempirical[G])|\mathcal{G}_t^N]\Big)
\end{aligned}$$ for every $G\in\Phi^N_t$. Conditionally on $\mathcal{G}_t^N$, Lemma \[lem:ingredients.indep\] and Lemma \[lem:ingredients.subgauss\] imply that $\{\Delta_{t,G}^N: G\in\Phi^N_t\}$ is an independent family of $C\muempirical(G)/N$-subgaussian random variables. Hence, conditionally on $\mathcal{G}_t^N$, the random variable $\Delta_{t}^N$ is subgaussian with parameter $\sum_{G\in\Phi^N_t} C\muempirical(G)/N = C/N$. We again use Lemma \[lem:subgaussian\] to conclude the proof.
By Lemma \[lem:aw.estimate.lipschitz.kernel\] and Lemma \[lem:integral.kernels.leq.averaged.kernel\] one has that $$\begin{aligned}
\mathcal{AW}(\mu,\mucausal)
&\leq C\Big( \frac{1}{N^r}+ \mathcal{W}(\mu_1,\muempirical[1]) +\sum_{t=1}^{T-1} \sum_{G\in\Phi^N_t} \muempirical(G) \mathcal{W}(\mu_G,\muempirical[G]) \Big).
\end{aligned}$$ Recalling the definition of $\Delta_{t}^N$ given in and setting $$\begin{aligned}
\label{eq:def.Delta.0}
\Delta_0^N&:=\mathcal{W}(\mu_1,\muempirical[1])- E[\mathcal{W}(\mu_1,\muempirical[1])], \\
\nonumber
S^N&:=E[\mathcal{W}(\mu_1,\muempirical[1])]+ \sum_{t=1}^{T-1}\sum_{G\in\Phi^N_t} \muempirical(G) E[\mathcal{W}(\mu_G,\muempirical[G])|\mathcal{G}_t^N]
\end{aligned}$$ for every $N\geq 1$, we can write $$\begin{aligned}
\mathcal{AW}(\mu,\mucausal)
&\leq C\Big( \frac{1}{N^r} + \sum_{t=0}^{T-1} \Delta_t^N + S^N \Big).
\end{aligned}$$ By Lemma \[lem:conditional.rate.expectation\] one has that $S^N \leq C R( N/N^{rd(T-1)} )$ almost surely for every $N\geq 1$. Recalling that $\mathop{\mathrm{rate}}(N)= R(N/N^{rd(T-1})$ and that $N^{-r}\leq\mathop{\mathrm{rate}}(N)$ (with equality for dimension $d\neq 2$), we arrive at $$\begin{aligned}
\label{eq:AW.smaller.three.terms}
\begin{split}
\mathcal{AW}(\mu,\mucausal)
&\leq C\Big( \sum_{t=0}^{T-1} \Delta_t^N + \mathop{\mathrm{rate}}(N)\Big).
\end{split}
\end{aligned}$$
Finally, Lemma \[lem:concentration.sum.mu.G\] (and the tower property) imply that $P[|\Delta_t^N|\geq \varepsilon]\leq 2\exp(-cN\varepsilon^2) $ for all $0\leq t\leq T-1$, all $\varepsilon>0$, and all $N\geq 1$. Therefore a union bound shows that $$\begin{aligned}
P\Big[ \mathcal{AW}(\mu,\mucausal) \geq C\mathop{\mathrm{rate}}(N) + \varepsilon \Big]
&\leq P\Big[\sum_{t=0}^{T-1} \Delta_t^N \geq\frac{\varepsilon}{C} \Big] \\
&\leq 2T\exp( -c N \varepsilon^2)
\end{aligned}$$ for all $N\geq 1$ and all $\varepsilon>0$, where $c>0$ is some new (small) constant. This completes the proof.
For fixed $\varepsilon>0$, choose $N_0(\varepsilon)$ so that $C\mathop{\mathrm{rate}}(N)\leq \varepsilon$ for all $N\geq N_0(\varepsilon)$ and apply Theorem \[thm:deviation\].
Proof of Theorem \[thm:almost.sure.convergence\] {#sec:proof.as.convergence}
================================================
We start by proving almost sure convergence of $\mathcal{AW}(\mu,\mucausal)$ to zero under the additional assumption that the kernels of $\mu$ admit a continuous version. Under this assumption, we can make use of the previous results and conclude almost sure convergence from the deviation inequality and a Borel-Cantelli argument. At the end of this section we show how this restriction can be removed.
\[lem:aw.estimate.continuous\] Assume that the kernels of $\mu$ are continuous. Then, for every $\delta>0$ there is a constant $C(\delta)>0$ such that $$\mathcal{AW}(\mu,\nu)
\leq \delta + C(\delta) \mathcal{W}(\mu_1,\nu_1) + C(\delta) \sum_{t=1}^{T-1} \int \mathcal{W}(\mu_{y_1,\dots,y_t}, \nu_{y_1,\dots,y_t} )\,\nu(d y)$$ for every $\nu\in\mathrm{Prob}(([0,1]^d)^T)$.
The proof is similar to the proof of Lemma \[lem:aw.estimate.lipschitz.kernel\]. For (notational) simplicity we spare the induction and restrict to $T=2$; the general case follows just as in Lemma \[lem:aw.estimate.lipschitz.kernel\]. For $T=2$ the recursive formula of $\mathcal{AW}(\mu, \nu)$ reads $$\begin{aligned}
\label{eq:dyn.prog.AW.2period}
&\mathcal{AW}(\mu,\nu)
=\inf_{\gamma\in \mathrm{Cpl}(\mu_{1}, \nu_1)} \int |x_1-y_1| + \mathcal{W}(\mu_{x_1},\nu_{y_1}) \,\gamma(dx_1, dy_1).
\end{aligned}$$ Now fix $\delta>0$. By uniform continuity of $x_1\mapsto \mu_{x_1}$ (as a continuous function with compact domain), there is $C(\delta)>0$ such that $$\begin{aligned}
\mathcal{W}(\mu_{x_1},\nu_{y_1})
&\leq \mathcal{W}(\mu_{x_1},\mu_{y_1}) + \mathcal{W}(\mu_{y_1},\nu_{y_1})\\
&\leq \delta + C(\delta)|x_1-y_1| + \mathcal{W}(\mu_{y_1},\nu_{y_1})
\end{aligned}$$ for all $x_1,y_1\in[0,1]^d$. Plugging this into yields the claim.
\[lem:integral.kernels.leq.averaged.kernel.continuous\] Assume that the kernels of $\mu$ are continuous and let $1\leq t\leq T-1$. Then, for every $\delta>0$ there is a number $N_0(\delta)$ such that $$\begin{aligned}
\int \mathcal{W}(\mu_{y_1,\dots,y_t}, \mucausal[y_1,\dots,y_t])\,\mucausal(dy)
\leq \delta + C \sum_{G\in \Phi^N_t} \muempirical(G) \mathcal{W}(\mu_G,\muempirical[G])
\end{aligned}$$ almost surely for every $N\geq N_0(\delta)$.
The proof follows exactly as in the proof of Lemma \[lem:integral.kernels.leq.averaged.kernel\]; one only needs to replace the estimate ‘$\mathcal{W}(\mu_g, \mu_G)\leq CN^{-r}$ for all $g\in G$’ (this is within that lemma) by the following:
Let $\delta>0$. By uniform continuity of $(x_1,\dots,x_t) \mapsto \mu_{x_1,\dots,x_t}$, there exists $\varepsilon>0$ such that for every $x,y\in([0,1]^d)^t$ with $|x-y|\leq\varepsilon$, one has that $\mathcal{W}(\mu_x, \mu_y)\leq\delta$. Now note that for arbitrary $G\in\Phi^N_t$ and $g\in G$ it holds that $$\begin{aligned}
\mathcal{W}(\mu_g, \mu_G)
&\leq \frac{1}{\mu(G)} \int_{G} \mathcal{W}(\mu_g, \mu_{z_1,\dots,z_t}) \,\mu(dz)
\leq \sup_{z\in G} \mathcal{W}(\mu_g, \mu_z)
\leq \delta,
\end{aligned}$$ where the last inequality holds once $\mathop{\mathrm{diam}}(G) \leq \varepsilon$. As $\mathop{\mathrm{diam}}(G) \leq C N^{-r}$ uniformly over $G\in\Phi_t^N$, this concludes the proof.
\[lem:a.s.convergence.continuous\] Assume that the kernels of $\mu$ are continuous. Then $\mathcal{AW}(\mu,\mucausal) \to 0$ almost surely.
The first part of the proof follows the proof of Theorem \[thm:deviation\]: let $\delta>0$ be arbitrary. Then, substituting Lemma \[lem:aw.estimate.continuous\] for Lemma \[lem:aw.estimate.lipschitz.kernel\] and Lemma \[lem:integral.kernels.leq.averaged.kernel.continuous\] for Lemma \[lem:integral.kernels.leq.averaged.kernel\] in the proof of Theorem \[thm:deviation\], we conclude that there exist $C(\delta)$ and $N_0(\delta)$ such that $$\begin{aligned}
\mathcal{AW}(\mu,\mucausal)
&\leq \delta +C(\delta)\Big( \sum_{t=0}^{T-1} \Delta_t^N + \mathop{\mathrm{rate}}(N)\Big),
\end{aligned}$$ almost surely for all $N\geq N_0(\delta)$; compare with . Recall that $\Delta_t^N$ was defined in for $1\leq t\leq T-1$ and in for $t=0$.
An application of Lemma \[lem:concentration.sum.mu.G\] then shows that, similar to before, $$\begin{aligned}
P\Big[ \mathcal{AW}(\mu,\mucausal) \geq \delta + \varepsilon \Big]
&\leq P\Big[ C(\delta) \sum_{t=0}^{T-1} \Delta_t^N \geq \varepsilon- C(\delta) \mathop{\mathrm{rate}}(N)\Big] \\
&\leq 2T \exp\Big( -c N \big( \frac{\varepsilon}{C(\delta)} - \mathop{\mathrm{rate}}(N)\big)^2_+ \Big)
\end{aligned}$$ for all $N\geq N_0(\delta)$, where $c>0$ is some small constant.
Let $N_1(\delta,\varepsilon)$ such that $\mathop{\mathrm{rate}}(N) \leq \varepsilon/(2C(\delta))$ for all $N\geq N_1(\delta,\varepsilon)$. Then $$\begin{aligned}
P\Big[ \mathcal{AW}(\mu,\mucausal) \geq \delta +\varepsilon\Big]
&\leq 2T \exp\Big( \frac{-c N\varepsilon^2}{4} \Big)
\end{aligned}$$ for all $N\geq \max\{N_0,N_1\}$. By a Borel-Cantelli argument, this implies that $$P\Big[ \limsup_{N\to\infty}\mathcal{AW}(\mu,\mucausal) \geq \delta +\varepsilon\Big]=0.$$ As $\varepsilon,\delta>0$ were arbitrary, we conclude that $\mathcal{AW}(\mu,\mucausal)$ converges to zero almost surely when $N\to\infty$. This completes the proof.
With this preparatory work carried out, we are now ready to prove the strong consistency of $\mucausal$.
We provide the proof for a two-period setting, that is, $T=2$. The general case follows by the same arguments, however it involves a (lengthy) backward induction just as in the proof of Lemma \[lem:aw.estimate.lipschitz.kernel\] and offers no new insights.
Let $\varepsilon>0$. We shall construct $\nu\in\mathrm{Prob}(([0,1]^d)^2)$ with continuous conditional probabilities such that $\mathcal{AW}(\mu,\nu)\leq\varepsilon$ and $\limsup_{N}\mathcal{AW}(\mucausal,\nucausal)\leq\varepsilon$ almost surely. As $\lim_N\mathcal{AW}(\nu,\nucausal)=0$ almost surely by Lemma \[lem:a.s.convergence.continuous\], the triangle inequality then implies that $\limsup_N \mathcal{AW}(\mu,\mucausal)\leq 2\varepsilon$ almost surely. Recalling that $\varepsilon>0$ was arbitrary completes the proof.
(a) By Lusin’s theorem there is a compact set $K\subset[0,1]^d$ such that $\mu(K)\geq 1-\varepsilon$ and $K\ni x_1\mapsto \mu_{x_1}$ is continuous. Extend the latter mapping to a continuous mapping $[0,1]^d\ni x_1\mapsto \nu_{x_1}$ by Tietze’s extension theorem (actually, a generalization thereof to vector valued functions: Dugundji’s theorem [@dugundji1951extension Theorem 4.1]) and define $$\nu(dx_1,dx_2):=\mu_1(dx_1)\nu_{x_1}(dx_2)\in\mathrm{Prob}(([0,1]^d)^2).$$ Then, taking the identity coupling $\gamma\in\mathrm{Cpl}(\mu_1,\nu_1)$ (that is, $\gamma=[x_1\mapsto (x_1,x_1)]_\ast \mu_1$) implies that $\mathcal{AW}(\mu,\nu)\leq \int \mathcal{W}(\mu_{x_1},\nu_{x_1})\,\mu_1(dx_1)\leq \varepsilon$.
(b) It remains to construct an i.i.d. sample of $\nu$ such that $\limsup_{N}\mathcal{AW}(\mucausal,\nucausal)\leq\varepsilon$. To that end recall that $(X^n)_n$ is an i.i.d. sample of $\mu$, and define $$Y_1^n:=X_1^n
\quad\text{and}\quad
Y_2^n:=X_2^n 1_{X_1^n \in K} + Z_2^n 1_{X_1^n \notin K}$$ for every $n$, where $Z_2^n$ satisfies that $P[Z_2^n\in \cdot |X_1^n]=\nu_{X_1^n}(\cdot)$ (and $Z^n$ is independent of $\{X^m,Y^m,Z^m : n\neq m\}$). Note that $(Y_1^n,Y^n_2)_n$ is an i.i.d. sample of $\nu$.
We again take the identity coupling between $\mucausal_1=\nucausal_1$ to obtain $$\begin{aligned}
\label{eq:AW.estimate.noncontinuous.kernel}
\mathcal{AW}(\mucausal,\nucausal)
\leq \int \mathcal{W}(\mucausal_{x_1},\nucausal_{x_1})\,\mucausal_1(dx_1)
=\sum_{G\in\Phi^N_1} \muempirical(G) \mathcal{W}(\mucausal_G,\nucausal_G).
\end{aligned}$$ In the (second) equality we also used that $\mucausal(G)=\muempirical(G)$ for every $G\in\Phi^N_1$ and that the kernels of $\mucausal$ and $\nucausal$ are constant on every $G\in\Phi_1^N$; in fact $$\begin{aligned}
\mucausal_G&= \frac{1}{N\muempirical(G)} \sum_{n\leq N \text{ s.t.\ } X_1^n\in G} \delta_{\varphi^N(X_2^n)},\\
\nucausal_G&=\frac{1}{N\muempirical(G)}\sum_{n\leq N \text{ s.t.\ } X_1^n\in G} \delta_{\varphi^N(Y_2^n)}.
\end{aligned}$$ Therefore, making use of convexity of $\alpha,\beta\mapsto \mathcal{W}(\alpha,\beta)$, we further estimate $$\begin{aligned}
\mathcal{W}(\mucausal_G,\nucausal_{G})
&\leq \frac{1}{ N\muempirical(G)} \sum_{n \leq N \text{ s.t.\ } X_n^1\in G} \mathcal{W}(\delta_{\varphi^N(X_2^n)}, \delta_{\varphi^N(Y_2^n)} ) \\
&\leq \frac{N\muempirical(G\cap K^c)}{ N\muempirical(G)},
\end{aligned}$$ where we used that $Y_2^n=X_2^n$ whenever $X_1^n\in K$. Plugging this estimate into yields $\mathcal{AW}(\mucausal, \nucausal) \leq \muempirical(K^c)$. To conclude use the strong law of large numbers which guarantees that $\lim_N \muempirical(K^c)=\mu(K^c)\leq \varepsilon$ almost surely, where the last inequality holds by choice of $K$.
Auxiliary results {#sec:aux}
=================
We start by providing a simple example showing that optimal stopping evaluated at the empirical measure does not converge to the value of the problem under the true model.
\[ex:opt.stop.not.cont.usual.empirical\] Consider a Gaussian random walk in two periods, that is, $X_0=0$, $X_1$ and $X_2-X_1$ have standard normal distribution and $X_2-X_1$ is independent of $X_1$. A classical optimal stopping problem consists of minimizing the expected cost $\int c(\tau,\cdot)\,d\mu$ over all stopping times $\tau\colon\mathbb{R}^3\to\{0,1,2\}$ (here stopping times simply means that $1_{\tau=0}$ is a function of $x_0$ and $1_{\tau=1}$ is a function of $x_0,x_1$ only), where $c\colon\{0,1,2\}\times\mathbb{R}^3\to\mathbb{R}$ is a given cost function.
Now consider the same problem under the empirical measure $\widehat{\mu}^N$ in place of $\mu$ and take for instance the cost function $c(t,x):=x_t$. As $X_1$ has Lebesgue density, it follows that $P[X_1^n=X_1^m \text{ for some } n\neq m]=0$ which means that, almost surely, the knowledge of $X_1^n$ gives perfect knowledge of $X_2^n$. In particular, for every $N\geq 1$ and almost all $\omega$, the mapping $$\tau^{N,\omega}(x_1):=\begin{cases}
1 &\text{if } x_1=X_1^n(\omega)\text{ for some } n\leq N \text{ with } X_1^n(\omega)<X_2^n(\omega) \\
2 &\text{else}
\end{cases}$$ defines a stopping time. Making use of the strong law of large numbers, we then obtain $$\inf_\tau \int c(\tau,\cdot) \, d\mu^N
\leq \int c(\tau^N,\cdot) \, d\mu^N
= \int x_1\wedge x_2\,\mu^N(dx)
\to\int x_1\wedge x_2\,\mu(dx)
<0$$ almost surely. This shows that any reasonable type of convergence (almost sure, in probability,...) towards $\inf_\tau \int c(\tau,\cdot)\,d\mu=0$ fails.
The Gaussian framework was chosen for notational convenience, the same result of course applies to absolutely continuous probabilities on the unit cube as well.
We now provide the following proof.
(a) Fix $1\leq t\leq T-1$ and let $(x_1,\dots,x_t)$, $(\tilde{x}_1,\dots,\tilde{x}_t)$ be two elements of $([0,1]^d)^t$. Define $\gamma\in \mathrm{Cpl}(\mu_{x_1,\dots,x_t},\mu_{\tilde{x}_1,\dots,\tilde{x}_t})$ by $$\gamma(A) := P\big[ \big( F_{t+1}(x_1,\dots,x_t,\varepsilon_{t+1}), F_{t+1}(\tilde{x}_1,\dots,\tilde{x}_t,\varepsilon_{t+1}) \big) \in A \big]$$ for Borel $A\subset[0,1]^d\times[0,1]^d$. Then the assumption made on $F_{t+1}$ yields $$\begin{aligned}
\mathcal{W}(\mu_{x_1,\dots,x_t},\mu_{\tilde{x_1},\dots,\tilde{x_t}})
&\leq \int |a-b|\,\gamma(da,db)\\
&\leq L|(x_1,\dots,x_t) - (\tilde{x}_1,\dots,\tilde{x}_t)|,
\end{aligned}$$ showing that Assumption \[ass:lipschitz.kernel\] is indeed satisfied.
(b) Again fix $1\leq t\leq T-1$ and let $(x_1,\dots,x_t)$, $(\tilde{x}_1,\dots,\tilde{x}_t)$ be two elements of $([0,1]^d)^t$. Then $\mu_{x_1,\dots,x_t}$ has the density $$f_{X_{t+1}|X_1,\dots,X_t}(\cdot):=\frac{ f_{X_1,\dots,X_{t+1}}(x_1,\dots,x_t,\cdot)}{f_{X_1,\dots,X_t}(x_1,\dots,x_t)}$$ w.r.t. the Lebesgue measure on $[0,1]^d$, where $f_{X_1,\dots,X_t}$ denotes the density of the distribution of $(X_1,\dots,X_t)$; similarly for $f_{X_1,\dots,X_{t+1}}$. The same goes for $\mu_{\tilde{x}_1,\dots,\tilde{x}_t}$ if $x_s$ is replaced by $\tilde{x}_s$ everywhere. Moreover, it is not hard to show that $\mathcal{W}(gdx,hdx)\leq \sqrt{d} \int |g(x)-h(x)|\,dx$ whenever $g$ and $h$ are two Lebesgue-densities on $[0,1]^d$; use e.g. the Kantorovich-Rubinstein duality and Hölder’s inequality or apply [@villani2008optimal Theorem 6.13]. Therefore one has that $$\begin{aligned}
&\mathcal{W}(\mu_{x_1,\dots,x_t},\mu_{\tilde{x_1},\dots,\tilde{x}_t}) \\
&\leq \sqrt{d} \int \Big|\frac{ f_{X_1,\dots,X_{t+1}}(x_1,\dots,x_t,u)}{f_{X_1,\dots,X_t}(x_1,\dots,x_t)} -\frac{ f_{X_1,\dots,X_{t+1}}(\tilde{x}_1,\dots,\tilde{x}_t,u)}{f_{X_1,\dots,X_t}(\tilde{x}_1,\dots,\tilde{x}_t)} \Big| \,du.
\end{aligned}$$ A quick computation using the assumptions imposed on $f$ shows that the latter can be bounded by $\sqrt{d}2L/\delta$, which completes the proof.
(c) In the case that $\mu$ is supported on finitely many points, the disintegration is uniquely defined by its value on these points. In particular, any Lipschitz continuous extension of this mapping will do, see e.g. [@johnson1986extensions].
<span style="font-variant:small-caps;">Acknowledgments:</span> Daniel Bartl is grateful for financial support through the Austrian Science Fund (FWF) under project P28661.\
Mathias Beiglböck is grateful for financial support through the Austrian Science Fund (FWF) under project Y782.\
Johannes Wiesel acknowledges support by the German National Academic Foundation.
|
---
author:
- 'C. Gielen'
- 'H. Van Winckel'
- 'M. Matsuura'
- 'M. Min'
- 'P. Deroo$^{**}$'
- 'L.B.F.M. Waters'
- 'C. Dominik'
bibliography:
- '/STER/100/cliog/disk28/Artikels/referenties.bib'
date: 'Received ; accepted '
title: 'Analysis of the infrared spectra of the peculiar post-AGB stars EPLyr and HD52961. [^1]'
---
[We aim to study in detail the peculiar mineralogy and structure of the circumstellar environment of two binary post-AGB stars, EPLyr and HD52961. Both stars were selected from a larger sample of evolved disc sources observed with Spitzer and show unique solid-state and gas features in their infrared spectra. Moreover, they show a very small infrared excess in comparison with the other sample stars.]{} [The different dust and gas species are identified on the basis of high-resolution Spitzer-IRS spectra. We fit the full spectrum to constrain grain sizes and temperature distributions in the discs. This, combined with our broad-band spectral energy distribution and interferometric measurements, allows us to study the physical structure of the disc, using a self-consistent 2D radiative-transfer disc model.]{} [We find that both stars have strong emission features due to CO$_2$ gas, dominated by $^{12}$C$^{16}$O$_2$, but with clear $^{13}$C$^{16}$O$_2$ and even $^{16}$O$^{12}$C$^{18}$O isotopic signatures. Crystalline silicates are apparent in both sources but proved very hard to model. EPLyr also shows evidence of mixed chemistry, with emission features of the rare class-C PAHs. Whether these PAHs reside in the oxygen-rich disc or in a carbon-rich outflow is still unclear. With the strongly processed silicates, the mixed chemistry and the low $^{12}$C/$^{13}$C ratio, EPLyr resembles some silicate J-type stars, although the depleted photosphere makes nucleosynthetic signatures difficult to probe. We find that the disc environment of both sources is, to a first approximation, well modelled with a passive disc, but additional physics such as grain settling, radial dust distributions, and an outflow component must be included to explain the details of the observed spectral energy distributions in both stars. ]{}
Introduction
============
The infrared spectra of post-AGB stars are often characterised by strong spectral signatures. These are formed in the gas and dust-rich circumstellar environment (CE), which is a remnant of the strong mass loss that occurred during the previous asymptotic giant branch (AGB) evolutionary phase. The chemistry in this circumstellar environment is found to be oxygen-rich or carbon-rich, depending on whether oxygen or carbon is more abundant. The less abundant of the two will be locked in the very stable CO molecule that forms in the stellar photosphere.
Typical post-AGB outflow sources that have O-rich CE not only show the well-known 9.7 and 18$\mu$m features of amorphous silicates but also narrower features, arising from crystalline silicates. [e.g. @waters96; @molster02a]. The condensates in C-rich outflows show features of carbon-species such as SiC, MgS or polycyclic aromatic hydrocarbons (PAHs) [e.g. @hony01; @hony02; @peeters02]. They are also characterised by an often very strong feature at 21$\mu$m [@kwok89; @volk99; @hony03]. The photospheres of these 21$\mu$m sources show strong enhancements of carbon and s-process elements [e.g. @reyniers04; @reyniers07b] and the 21 $\mu$m stars are recognised as post-AGB carbon stars [e.g. @vanwinckel00].
Some evolved objects show, however, features of both O-rich and C-rich dust species in their spectra. They are called mixed chemistry sources. This chemistry is detected in several sources in a wide range of different evolutionary stages. Some examples include Herbig Ae stars, or AGB stars, such as J-type carbon stars with silicate dust emission [@littlemarenin86; @lloydevans90]. Others are red giants, for example HD233517, an evolved O-rich red giant with PAHs in a circumstellar disc [@jura06]. Other examples are planetary nebulae (PNe) with evidence of silicates and PAHs [@kemper02; @gutenkunst08], or the hydrogen-poor \[WC\] central stars of PNe [@waters98; @cohen99]. Also some M supergiants are associated with emission due to PAHs [@sylvester98; @sloan08].
Post-AGB stars with evidence of mixed chemistry include HD44179, the central star of the carbon-rich Red Rectangle nebula [@cohen75]. The central star is a binary surrounded by a Keplerian O-rich circumbinary disc [e.g. @vanwinckel95; @waters98; @menshchikov02; @bujarrabal05]. Here the formation of the disc is believed to have antedated the C-rich transition of the central star [e.g. @cohen04; @witt08].
Studies have shown that these evolved binaries with circumbinary discs are much more abundant than anticipated [@deruyter06; @vanwinckel07]. Interferometric studies [@deroo06; @deroo07b] prove that the discs are indeed very compact, with radii around 50AU in the N-band. The discs are also the natural environment of the observed photospheric chemical depletion pattern in these stars [@vanwinckel98; @giridhar00], due to chemical fractionation by dust formation in the circumstellar environment [@waters92] and subsequent accretion of the gas component. The presence of a long-lived stable reservoir of dust grains also could allow for the observed strong processing of the silicate dust grains, both in size as well as in crystallinity [@molster02a; @gielen08].
Dusty RVTauri stars are a distinct class in the post-AGB stars. They cross the instability strip, and are therefore pulsating stars [@jura86; @jura99; @deruyter05]. RVTauri stars show large-amplitude photometric variations with alternating deep and shallow minima. The members are located in the high-luminosity end of the populationII instability strip, and the photometric variations are interpreted as being due to radial pulsations. Circumstellar dust emission was observed in many of them [@jura86], and this was generally acknowledged to be a decisive character to place these stars in the post-AGB phase of evolution. The grains in almost all dusty RVTauri stars are, however, not freely expanding but likely also trapped in a disc [@vanwinckel99; @deruyter05; @deruyter06].
In this paper we focus on two peculiar post-AGB stars with RVTauri pulsational characteristics: EPLyr and HD52961. These stars show unique spectral features and have very small infrared excesses in comparison to the larger sample.
The outline of the paper is as follows: We start with a short description of the programme stars in Sect. \[progstars\]. In Sect. \[observations\] we give an overview of the different observations and reduction strategies. The analysis based on the Spitzer spectra is given in Sect. \[spectralanalysis\] and subdivided in different subsections. Sect. \[silicates\] contains a description of the silicate dust features and the modelling of the Spitzer-IRS spectra. The CO$_2$ gas features are discussed in Sect. \[co2\] and the observed PAH features in EPLyr in Sect. \[pah\]. In Sect. \[sed\] we model the observed SEDs using a passive disc model, also constrained with MIDI interferometric measurements. The discussion of our different results and our conclusions are presented in Sect. \[conclusions\].
Programme stars {#progstars}
===============
In our previous study we described and modelled the Spitzer-IRS spectra of 21 sources and found that the dust around these stars is all O-rich and on average highly crystalline [@gielen08]. The two stars discussed here have the lowest $L_{\rm IR}/L_*$, respectively 12% and 3%, in the larger Spitzer sample, where an average of about 50% was found. The large infrared luminosity can be explained with a passive disc model, provided that the inner rim is close to the star and the scale height of the disc is significant [e.g. @deroo07b]. The low observed $L_{\rm IR}/L_*$ values of both stars point to a small disc scale height and/or a much larger inner gap, as it is unlikely that a disc is optically thin in the radial direction. Not only do they have the lowest $L_{\rm IR}/L_*$ values, both stars show unique spectral signatures in comparison to the larger sample. We therefore selected these objects for a more detailed analysis.
--------- ------------------ ------------------ --------------- ---------- ---------- --------------- ------ -------------------- ------------------ ------------- -- --
Name $\alpha$ (J2000) $\delta$ (J2000) $T_{\rm eff}$ $\log g$ \[Fe/H\] P$_{\rm orb}$ $e$ $E(B-V)_{\rm tot}$ $L_{\rm IR}/L_*$ $d$
(h m s) ($^\circ$ ’ ”) (K) (cgs) (days) (%) (kpc)
EPLyr 19 18 17.5 $+$27 50 38 7000 2.0 -1.5 $0.51\pm0.01$ $3\pm0$ $3.2\pm1.0$
HD52961 07 03 39.6 $+$10 46 13 6000 0.5 -4.8 1310 0.21 $0.06\pm0.02$ $12\pm1$ $1.6\pm0.5$
--------- ------------------ ------------------ --------------- ---------- ---------- --------------- ------ -------------------- ------------------ ------------- -- --
Note: Listed are the name, equatorial coordinates $\alpha$ and $\delta$(J2000), effective temperature $T_{\rm eff}$, surface gravity $\log g$, and metallicity \[Fe/H\] of our sample stars. For the model parameters we refer to @deruyter06. Also given are the orbital period and the eccentricity [see references in @deruyter06; @gielen07]. The total reddening $E(B-V)_{\rm tot}$, the energy ratio $L_{\rm IR}/L_*$ and the calculated distance, assuming a luminosity of $L_*=3000\pm2000$L$_{\odot}$.
EPLyr
-----
@schneller31 discovered the variability of EPLyr and classified it as an RVb star. RVb stars are objects with a variable mean magnitude, in the General Catalogue of Variable Stars [@kholopov99]. Other studies [@zsoldos95; @gonzalez97a] classify the light curve as an RVa photometric variable, having a constant mean magnitude, with a period of $P=83.46$ days. @preston63 classify it as an RVB spectroscopic variable. @gonzalez97a performed an abundance analysis on EPLyr where they deduced stellar parameters (see Table \[sterren\]) and found the star to be metal-poor, oxygen-rich and severely depleted. Using the molecular lines found in the spectra, they also quantified the $^{12}$C/$^{13}$C ratio to be $9\pm1$. In the radial velocity data there is also evidence that EPLyr must have a stellar mass companion, but additional observations are necessary to determine the orbit.
HD52961
-------
HD52961 is an RVTauri like object, similar to class RVb objects [@waelkens91b], with a photometric variability of 72 days due to clear radial pulsations [@waelkens91b]. The binarity of HD52961 was first reported by @vanwinckel95 and further refined in @vanwinckel99 and @deroo06, where an orbital period of $P_{\rm orb}=1297\pm7$ days and an eccentricity of $e=0.22\pm0.05$ was found. On top of the stable photometric variation due to the pulsation, another long-term photometric variation was detected, correlated with the orbital period. @vanwinckel99 conclude that this can be understood as caused by variable circumstellar extinction during the orbital motion.
The star is a highly metal-poor object with \[Fe/H\]$=$$-4.8$ [@waelkens91a] and has an extremely high zinc to iron ratio of \[Zn/Fe\]$=$$+3.1$ [@vanwinckel92]. The star is one of the most extremely depleted objects known.
HD52961 has been studied with mid-IR long-baseline interferometry using the VLTI/MIDI instrument [@deroo06]. They find that the dust emission originates from a very small but resolved region, estimated to be $\sim35$mas at 8$\mu$m and $\sim55$mas at 13$\mu$m, likely trapped in a stable disc. The dust distribution through the disc is not homogeneous: the crystallinity is higher in the hotter inner region.
Observations
============
High- and low-resolution spectra of 21 post-AGB stars were obtained using the Infrared Spectrograph (IRS; [@houck04]) aboard the Spitzer Space Telescope [@werner04] in February 2005. The spectra were observed using combinations of the short-low (SL), short-high (SH) and long-high (LH) modules. SL ($\lambda$=$5.3-14.5$$\mu$m) spectra have a resolving power of R=$\lambda/\bigtriangleup\lambda
\sim$ 100, SH ($\lambda$=$10.0-19.5$$\mu$m) and LH ($\lambda$=$19.3-37.0$$\mu$m) spectra have a resolving power of $\sim$ 600. Exposure times were chosen to achieve an S/N ratio of around 400 for the high-resolution modes, which we complemented with short exposures in low-resolution mode with an S/N ratio around 100, using the first generation of the exposure time calculator of the call for proposals.
The spectra were extracted from the SSC data pipeline version S13.2.0 products, using the c2d Interactive Analysis reduction software package [@kessler06; @lahuis06]. This data processing includes bad-pixel correction, extraction, defringing and order matching. To match the different orders, we applied small scaling corrections.
Spectral analysis {#spectralanalysis}
=================
General
-------
A look at the spectra of EPLyr and HD52961 (Fig. \[eplyr\_hd52961\]) show very rich spectra with quite different continuum slopes. EPLyr shows strong emission features at longer wavelengths, with peak emission in the 20$\mu$m region, whereas HD52961 is characterised by a strong 10$\mu$m emission feature on top of a much steeper continuum.
Common dust species found in oxygen-rich post-AGB stars are amorphous silicates, namely olivine and pyroxene. Amorphous olivine (Mg$_{2x}$Fe$_{2(1-x)}$SiO$_4$, where $0 \leq x \leq
1$ denotes the magnesium content) has very prominent broad features around 9.8$\mu$m and 18$\mu$m. Amorphous pyroxene (Mg$_{x}$Fe$_{1-x}$SiO$_3$) shows a 10$\mu$m feature similar to that of amorphous olivine, but shifted towards shorter wavelengths. Also the shape of the 18$\mu$m feature is slightly different. For EPLyr it is unclear whether there is a significant contribution of amorphous silicates. Small amorphous silicates could contribute to the observed strong emission bump at 20$\mu$m in EPLyr, but as there does not seem to be a 10$\mu$m amorphous feature, the 20$\mu$m bump could be purely continuum dominated. HD52961 has clear strong emission of amorphous silicates at 10$\mu$m, but the profile shows complex narrow subfeatures. Very little contribution at 20$\mu$m is seen.
Both stars show strong narrow emission features which can be identified as being due to crystalline silicates. The Mg-rich end members of crystalline olivine and pyroxene, forsterite (Mg$_2$SiO$_4$) and enstatite (MgSiO$_3$), show strong but narrow features at distinct wavelengths around $11.3 - 16.2 - 19.7 - 23.7 -
28$ and 33.6$\mu$m. For EPLyr the silicate emission only clearly starts longward of 18$\mu$m, where strong emission features around $19-23-27$ and 33$\mu$m can be seen. HD52961 has strong narrow features at $9.8-11.3$$\mu$m and a remarkably strong 16$\mu$m feature. If this strong 16$\mu$m band is only due to forsterite it has shifted considerably to shorter wavelengths. A significant 16$\mu$m feature is seen in several evolved disc sources but it is never as strong as in HD52961 [@gielen08].
EPLyr shows evidence of the presence of carbon-rich dust species with probable PAH identifications at 8.1 and 11.3$\mu$m. The detection of PAH emission together with silicates is surprising and only observed in a few other post-AGB sources. The analysis of the PAH features is given in Sect. \[pah\].
The spectrum of EPLyr shows a strong resemblance to that of IRAS09245-6040 (Fig. \[eplyr-iras09425\]), a silicate J-type carbon AGB star [@molster01; @garciahernandez06]. Silicate J-type carbon stars have surprisingly low $^{12}$C/$^{13}$C ratios and do not show the typical s-process overabundances seen in N-type carbon stars [@abia00]. The infrared spectrum of these stars shows features of both carbon- and oxygen-rich dust species.
Of the silicate J-type carbon stars, only 10% show emission bands due to crystalline material [@lloydevans91; @ohnaka99]. The formation history of these stars is still unclear, but the most promising scenario for the presence of silicates in these stars, is that they are binaries with an undetected companion [@lloydevans90; @yamamura00]. A disc is supposed to be formed when the primary was still an O-rich giant. After that the star underwent thermal pulses and evolved into a carbon star. The silicate disc could be either captured from the wind [@mastrodemos99] or the result of a phase of strong binary interaction in a narrow system. Systems with strong crystalline features in their spectra, such as IRAS09425-6040 or IRAS18006-3213 [@deroo07a], would then be a result of mass-transfer into a circumbinary system, whereas sources dominated by amorphous silicates, such as V778Cygni [@yamamura00] or BHGem [@ohnaka08], consist of a wide binary with a circumcompanion disc. To date, no orbits are known, however, and direct evidence of binarity is found in a few objects only [@izumiura08].
For IRAS09245-6040, the $^{12}$C/$^{13}$C ratio is calculated to be $15\pm6$ [@garciahernandez06]. In the ISO-SWS spectrum features of C$_2$H$_2$, HCN, CO, C$_3$ and SiC are seen shortward of 15$\mu$m; after 15$\mu$m the spectrum is dominated by strong emission features of Mg-rich crystalline silicates [@molster01]. As in EPLyr, there is no evidence of a strong contribution of amorphous silicates.
Finally in both EPLyr and HD52961, clear CO$_2$ gas emission features are detected the $13-18$$\mu$m region. This is discussed in Sect. \[co2\].
Silicate dust emission {#silicates}
----------------------
We optimised the fitting procedure as discussed in @gielen08 for these two outliers, where we modelled the full Spitzer sample, consisting of 21 stars. In short we assume the flux to be originating from an optically thin region, so we can make linear combinations of the absorption profiles to calculate the model spectrum. In our previous modelling we found that, on average for the full sample, the best fit was obtained using relatively large grains ($\geq 2$$\mu$m) in an irregular Gaussian Random Fields (GRF) dust model. For EPLyr and HD52961 however, we already found that using smaller grain sizes ($\leq 2$$\mu$m) improved the fit considerably.
So we repeated the analysis for EPLyr and HD52961, allowing for different dust shapes, grain sizes and Mg/Fe content in the amorphous grains. We tested Mie, GRF and DHS (Distribution of Hollow Spheres) dust models in grain sizes ranging from 0.1 to 4.0$\mu$m. In order to test for the presence of Fe-poor amorphous dust, we perform the modelling both with pure Mg-rich amorphous silicates ($x=1$) and with the more standard Mg-Fe amorphous silicate dust ($x=0.5$). For a detailed description of the fitting routine we refer to @gielen08. The results of the fitting can be found in Table \[chi2\]. As for EPLyr the silicate signatures only appear after 18$\mu$m; we only fit this part of the Spitzer spectrum.
-------- ------------ ---------- ---------------------------------
EPLyr HD52961 model description
$ \chi^2$ $\chi^2$
model1 21.7 129.4 Mie - $0.1-2.0$$\mu$m - $x=0.5$
model2 6.2 67.5 DHS - $0.1-1.5$$\mu$m - $x=0.5$
model3 6.2 63.8 DHS - $0.1-1.5$$\mu$m - $x=1.0$
model4 8.4 101.4 DHS - $1.5-3.0$$\mu$m - $x=0.5$
model5 8.6 140.8 DHS - $1.5-3.0$$\mu$m - $x=1.0$
model6 5.9 64.2 GRF - $0.1-2.0$$\mu$m - $x=0.5$
model7 6.3 50.0 GRF - $0.1-2.0$$\mu$m - $x=1.0$
model8 5.8 96.5 GRF - $2.0-4.0$$\mu$m - $x=0.5$
model9 5.4 72.2 GRF - $2.0-4.0$$\mu$m - $x=1.0$
-------- ------------ ---------- ---------------------------------
: $\chi^2$ values for different models used in our full spectral fitting.[]{data-label="chi2"}
Note: For each model we give the used dust approximation, grain size and Mg-Fe content in the amorphous grains. $x=1.0$ denotes pure Mg-rich amorphous dust, $x=0.5$ the more standard Mg-Fe amorphous silicates.
The $\chi^2$ values of our fitting (Table \[chi2\]) are still quite high for HD52961 but, confirming the result of @gielen08, we can already tell that for both stars Mie theory is not a good dust approximation. For EPLyr the GRF grains prove the best match, but the difference in $\chi^2$ with the small DHS grain approximation is only minimal. The best fit to EPLyr is obtained using both small (0.1$\mu$m) and larger (2.0$\mu$m) silicate grains. The small difference in calculated $\chi^2$ values for EPLyr is due to the low signal-to-noise ratio, making it hard to distinguish between different synthetic emission profiles. For HD52961 small grains in Mg-rich silicates give the best $\chi^2$. Plots of our best fitting models can be found in Fig. \[silicate\_fit\]. Table \[fitresults1\] gives the resulting parameters.
The large $\chi^2$ value of HD52961 quantifies that this star has a very peculiar, unique chemistry, and we did not succeed in explaining all of the observed features. The strong forsterite 11.3$\mu$m feature in the GRF dust approximation is clearly too broad. DHS grains fit the feature better, but other feature profiles are fitted less well with this approximation. There also appears to be a short wavelength shoulder on the amorphous 9.8$\mu$m feature, which is not explained in the modelling. The strong 16.5$\mu$m feature is not reproduced in central wavelength by any of the different models. We already observed this trend in our full sample fitting [@gielen08], where the feature seemed to be shifted bluewards in comparison with the mean spectrum of the full sample. The two narrow features around 19$\mu$m could be an artifact of the data reduction, since in this region there can be a bad overlap between the SH and LH Spitzer-IRS high-resolution bands.
For EPLyr we fit the spectrum longwards of 18$\mu$m, where the silicate features are seen. This gives dust temperatures between 100 and 230K. This model, however, does not fit the spectrum before 18$\mu$m, since the continuum does not follow the observed strong downward slope before 20$\mu$m. If we try to fit the full Spitzer wavelength range we find we can get a better fit to the underlying continuum but then the features at 27 and 33$\mu$m are much stronger in the observed spectrum than in our best model. Unlike in other sources, a two temperature approach fails to model both the observed continuum and the coolest features for the full Spitzer wavelength spectrum of EPLyr. Irrespective of the derived continuum temperature, all the tested models give estimates of the dust temperatures between $100-300$K, which agrees with the temperatures derived in the SED modelling (Sect. \[sed\]). Clearly, the crystalline dust particles must be quite cold.
--------- ---------- ----------------------- ---------------------- ----------------------------------------- ----------------------- ----------------------- -----------------------------------------------
Name $\chi^2$ $T_{dust1}$ $T_{dust2}$ Fraction $T_{cont1}$ $T_{cont2}$ Fraction
(K) (K) $T_{dust1}$- $T_{dust2}$ (K) (K) $T_{cont1}$-$T_{cont2}$
EPLyr 5.4 $ 114_{ 14}^{ 122}$ $ 228_{ 89}^{ 488}$ $ 0.9_{ 0.6}^{ 0.1}- 0.1_{ 0.1}^{ 0.6}$ $ 205_{ 103}^{ 702}$ $ 641_{ 301}^{ 331}$ $ 0.96_{ 0.06}^{ 0.02}- 0.04_{ 0.02}^{ 0.06}$
HD52961 50.0 $ 200_{ 0}^{ 10}$ $ 724_{ 96}^{ 186}$ $ 0.9_{ 0.1}^{ 0.0}- 0.1_{ 0.0}^{ 0.1}$ $ 111_{ 11}^{ 356}$ $ 996_{ 111}^{ 4}$ $ 0.99_{ 0.03}^{ 0.00}- 0.01_{ 0.00}^{ 0.03}$
--------- ---------- ----------------------- ---------------------- ----------------------------------------- ----------------------- ----------------------- -----------------------------------------------
--------- ------------------------------------ --------------------------------- ----------------------------------- ----------------------------------- -----------------
Name Olivine Pyroxene Forsterite Enstatite Continuum
Small - Large Small - Large Small - Large Small - Large
EPLyr $ 6_{ 6}^{ 41} - 5_{ 5}^{25}$ $8_{7}^{34} - 6_{6}^{23}$ $34_{15}^{19} - 9_{ 8}^{33}$ $ 5_{ 5}^{20} - 28_{21}^{17}$ $53_{ 20}^{11}$
HD52961 $ 0_{ 0}^{ 13} - 1_{ 1}^{41}$ $55_{18}^{12} - 2_{2}^{43}$ $6_{5}^{18} - 33_{ 18}^{13}$ $ 1_{ 1}^{8} - 3_{3}^{26}$ $69_{ 4}^{ 4}$
--------- ------------------------------------ --------------------------------- ----------------------------------- ----------------------------------- -----------------
Note: Top part: The $\chi^2$, dust and continuum temperatures and their relative fractions. Bottom part: The abundances of small (0.1$\mu$m) and large (2.0$\mu$m) grains of the various dust species are given as fractions of the total mass, excluding the dust responsible for the continuum emission. The last column gives the continuum flux contribution, listed as a percentage of the total integrated flux over the full wavelength range. The errors were obtained using a Monte-Carlo simulation based on 100 equivalent spectra. Details of the modelling method are explained in @gielen08.
CO$_2$ emission {#co2}
---------------
### Introduction {#introduction}
CO$_2$ emission has been found in approximately 30% of all O-rich AGB stars [@justtanont98; @ryde99; @sloan03], but CO$_2$ detections in post-AGB stars are rare. To our knowledge CO$_2$ gas has been found in only two post-AGB stars, the Red Rectangle and HR4049 [@waters98; @cami01], which are also binaries surrounded by a stable circumstellar disc. HR4049 is the only example of a post-AGB star showing CO$_2$ in emission in the $13-16$$\mu$m region. @cami01 argued that the isotopic distribution of oxygen in HR4049 is abnormal, based on the isotope ratio analysis of the CO$_2$ emission features. This was not confirmed by @hinkle07, who use high-resolution spectra of the fundamental and first overtone CO vibro-rotational transition in the near-IR.
Both EPLyr and HD52961 show clear gas phase emission lines of $^{12}$CO$_2$ and $^{13}$CO$_2$. These emission lines were also seen in only one other source in our Spitzer sample [@gielen08], namely in IRAS10174-5704.
The CO$_2$ emission of EPLyr seems to be lying on top of a “plateau” that extends from 13 to 17$\mu$m. A similar plateau in this region is observed in PAH-rich sources [@peeters04], but this plateau is much broader and ranges from 15 to 20$\mu$m and is often characterised by strong emission features at 16.4$\mu$m (and less prominent at 15.8, 17.4 and 19$\mu$m), and thus quite different to the one seen in EPLyr.
### Analysis
To retrieve the very rich spectral information of the CO$_2$ emission bands, we calculate spectra of CO$_2$, using HITRAN line lists [@rothman05] and a circular slab model for the radiative-transfer [@matsuura02]. The model has four parameters: the excitation temperature ($T_{\rm ex}$), the total CO$_2$ column density ($N$), the radius of CO$_2$ gas ($r$) and the isotope ratio. The radius of the CO$_2$ layer is given relative to the radius of the background continuum source at 13$\mu$m. The dependence of the CO$_2$ model spectra on these parameters are described by @cami01. We estimate a pseudo-continuum by using a spline fit and a linear fit for HD52961 and EPLyr, respectively. For HD52961, we interpolate the spectrum at the spectral range where CO$_2$ bands have little influence on the observed spectrum. The continuum was also chosen so that the forsterite feature at 16$\mu$m would be removed. A spline fit was tested for EPLyr but failed because of the richness of CO$_2$ features in the mid-infrared range, so we simply use a linear interpolation between 13.4 and 17.8$\mu$m. Estimated continua are displayed as dotted lines in the top panels of Figs. \[Fig-EPLyr\] and \[Fig-HD52961\]. The resulted parameters for the model calculations are summarised in Table \[parameters\], and the resulted spectra show the identifications of the different CO$_2$ bands (bottom panels of Figs. \[Fig-EPLyr\] and \[Fig-HD52961\]).
Many small features in EPLyr in the $13.5-17$$\mu$m region are due to CO$_2$: features at 13.5, 13.9, 14.7, 14.9, 16.2$\mu$m are attributed to the CO$_2$ main isotope $^{12}$C$^{16}$O$_2$. The main isotopic $^{12}$C$^{16}$O$_2$ bands are probably optically thick, surpressing the line intensities. $^{13}$C$^{16}$O$_2$ and $^{16}$O$^{12}$C$^{18}$O bands are found at 15.3$\mu$m and 15.1$\mu$m, respectively. The prominent $^{16}$O$^{12}$C$^{18}$O feature is surprising. This feature was also found in the other binary post-AGB star HR4049 [@cami01].
The model uses a high fraction of isotopes, but actual abundance ratios remain largely uncertain, mainly because of the uncertainty of the interpolated continuum spectrum and the optical thickness of the main isotope. Nevertheless, these two features are particularly prominent in the spectrum of EPLyr, more than in HD52961, suggesting different isotope ratios for EPLyr. We see that the observed “plateau” in EPLyr can be explained by the richness of the $^{12}$C$^{16}$O$_2$ features, but the $^{12}$C/ $^{13}$C ratio is confirmed to be low. The strength of the isotopes, including the very rare $^{18}$O (in the Sun $^{16}$O/$^{18}$O is $\sim$500), is an exclusive feature of post-AGB stars.
--------- -------------- ------------------ --------- -------------------- -- -- -- -- -- -- -- -- -- -- -- -- --
$T_{\rm ex}$ $N$ $r$ isotope ratio
(K) (cm$^{-2}$) ($R_*$)
EPLyr 900 $8\times10^{18}$ 4.7 0.7 : 0.2 : 0.1
HD52961 800 $5\times10^{18}$ 4.7 0.93 : 0.05 : 0.02
--------- -------------- ------------------ --------- -------------------- -- -- -- -- -- -- -- -- -- -- -- -- --
: Resulting parameters for the model calculations. The isotope ratio is as follows: $^{12}$C$^{16}$O$_2$:$^{13}$C$^{16}$O$_2$:$^{16}$O$^{12}$C$^{18}$O. \[parameters\]
PAH features {#pah}
------------
Polycyclic aromatic hydrocarbons are found in a large variety of objects, including the diffuse ISM, HII regions, young stellar objects, post-AGB stars and planetary nebulae. They have strong emission features in the 3-13$\mu$m region [e.g. @tielens08]. The feature at 3.3$\mu$m is arises from the C-H stretching mode of neutral PAHs. The C-C modes produce features with typical central wavelengths at 6.2 and 7.7$\mu$m. The 8.6$\mu$m feature is due to C-H in-plane bending modes and features longward of 10$\mu$m can be attributed to C-H out-of-plane bending modes.
@peeters02 defined three groups of PAH spectra based on their emission profiles and peak positions. The “class A” sources have features at 6.22, 7.6 and 8.6$\mu$m. “Class B” sources show the same features but shifted to the red, peaking at 6.27, 7.8 and $>8.6$$\mu$m. They also identified two “class C” sources, the Egg Nebula (AFGL2688) and IRAS13416-6243, both post-AGB objects. These rare “class C” sources show emission features at 6.3$\mu$m, no emission near 7.6$\mu$m, and a broad feature centred around 8.2$\mu$m, extending beyond 9$\mu$m.
With the release of the IRS aboard the Spitzer Space Telescope, a limited number of additional class-C sources were discovered (Fig. \[splinesub\]). MSX SMC 029, a class-C post-AGB star in the SMC, was detected by @kraemer06. @sloan07 report on the detection of class-C PAH features in HD100764, a carbon-rich red giant with evidence of a circumstellar disc. @jura06 also report on the detection of class-C PAH features in a circumstellar disc around the oxygen-rich K-giant HD233517. Two young objects, the TTauri star SUAur [@Furlan06] and the Herbig Ae/Be source HD135344 [@sloan05], also show PAH spectra of class C, although in HD135344 the PAH features seem to be somewhat more in between B and C. This source is also slightly hotter than other class C sources. A comparison of the PAH features in all these sources is given by @sloan07. They find that all the known class-C spectra are excited by relatively cool stars of spectral type F or later and argue that the hydrocarbons in these sources have not been exposed to much ultraviolet radiation. The class-C PAHs are then relatively protected and unprocessed, while class A and B PAHs have been exposed to more energetic photons and are hence more processed.
EPLyr shows PAH emission bands at 8.1, 11.3 and 12.6$\mu$m. Other features can be seen at 13.25$\mu$m. Whether these can be attributed to PAHs remains uncertain. The PAH spectrum of EPLyr can be classified as class C. EPLyr is a high amplitude-variable with an effective temperature around 7000K, which is on the hot end of the other class-C emitters [see @sloan07].
Sofar, it is not clear whether the PAH carriers reside in the circumbinary disc, or in bipolar lobes created by a more recent mass-loss event, as observed in HR4049 [@johnson99; @dominik03; @antoniucci05; @hinkle07; @menut09] and in the Red Rectangle [@menshchikov02; @cohen04]. For HR4049, @johnson99 found that the optical polarisation seems to vary with orbital phase, suggesting that the polarisation in the optical is due to scattering in the circumbinary disc. In the UV, the polarisation is caused by scattering in the bipolar lobes, which should contain a population of small grains, including the PAH carriers. HR4049 and HD44179, as well as EPLyr and HD52961, are strongly depleted, and molecules or the formation of dust in these very depleted environments is likely very different from solar condenstation. As the CO molecule is abundant in the circumstellar environment, accretion of circumstellar gas will likely result in a C$\sim$O photosphere. HR4049 and HD44179 are stars with PAH emission belonging to the more standard class B.
Unlike what is detected around HD44179 (the ERE nebula) and HR 4049 (nano-diamonds), for EPLyr the PAHs are the only carbon-rich component observed in the circumstellar spectra. There is no evidence that the photosphere of EPLyr is in, or went through, a carbon-rich phase. The photosphere is depleted so that nucleosynthetic yields are very hard to recognise, but the photosphere is clearly oxygen-rich [@gonzalez97a]. A scenario involving hot-bottom burning to return to an oxygen-rich condition after a carbon-rich phase in the stellar evolution along the AGB is very unlikely: it would imply the object is of a more massive origin [@lattanzio96; @mcsaveney07], which is in contradiction with its Galactic coordinates of $b = 6.9^{\circ}$. Assuming 10000L$_\odot$ for a putative massive progenitor, the distance above the Galactic plane would be about 660pc, which is very high for an object of $5-6$M$_\odot$. Moreover, the initial metallicity is likely subsolar as indicated by the solfur and zinc abundances. Although, with the depletion, it is unclear whether these abunances of the volatiles do indeed represent the initial conditions. Also the pulsation period of EPLyr is similar to other RVTauri objects which are of low initial mass. We conclude that the photosphere of EPLyr is now O-rich, and we argue it is very unlikely that it was ever in its history in a carbon-rich phase. The PAH synthesis likely occurred in O-rich conditions.
As Figure \[hd233517\] shows, the spectrum of EPLyr has a striking resemblance to that of HD233517, shortward of 13$\mu$m [@jura06]. @jura03 hypothesises that HD233517 was a short-period binary on the main sequence. A circumstellar disc was then formed when the companion star was engulfed by the more massive star when it entered giant evolution, followed by a phase of strong mass ejection in the equatorial plane. Since HD233517 is an oxygen-rich star, it is remarkable that the disc shows features of carbon-rich chemistry. @jura06 propose a scenario in which the PAHs could be formed inside the disc due to Fischer-Tropsch (FT) catalysis on the surface of solid iron grains. These FT reactions can convert CO and H$_2$ into water and hydrocarbons [@willacy04], these hydrocarbons could then be converted in into PAHs.
So far it remains unclear whether the shape of the observed emission features can detect if the PAH carriers reside in the disc or an outflow.
Spectral energy distribution {#sed}
============================
2D disc modelling
-----------------
For both stars extensive photometric data are available. This, together with the Spitzer infrared spectral information, allows us to constrain some of the physical characteristics of the circumbinary disc.
We fit the SED using a Monte Carlo code, assuming 2D-radiative-transfer in a passive disc model [@dullemond01; @dullemond04]. This code computes the temperature structure and density of the disc. The vertical scale height of the disc is computed by an iteration process, demanding vertical hydrostatic equilibrium. The distribution of dust grain properties is fully homogeneous and, although this model can reproduce the SED, dust settling timescales indicate that settling of large grains to the midplane occurs. Using the dust settling timescale $$t_{set}=\frac{\pi}{2}\frac{\Sigma_0}{\rho_d a}\frac{1}{\Omega_k}\ln\frac{z}{z_0}$$ with $\Sigma_0$ the surface density, $\rho_d$ the particle density, $a$ the grain size and $\Omega_k=\sqrt{\frac{GM_*}{r^3}}$ the Keplerian rotation rate [@miyake95], we find that grains with sizes between 500$\mu$m and 0.1cm can descend 50AU on timescales similar to the estimated lifetime of the disc. An inhomogeneous disc model with a vertical gradient in grain-size distribution is thus necessary [@gielen07]. These large and cooler grains in the disc midplane mainly contribute to the far-IR part of the SED and constitute the main fraction of the total dust mass. The disc structure and near- and mid-IR flux are almost fully determined by small grains. So we use a homogeneous 2D disc model to fit the near- and mid-IR part of the SED and add a single blackbody temperature to represent the cooler midplane made up of large grains. The lack of observational constraints on the temperature structure of this component does not allow us to constrain this extra parameter.
Stellar input parameters of the model are the luminosity, the total mass (we assume the total gravitational potential to be spherically symmetric with a total mass of $M=1\,$M$_{\odot}$), and $T_{\rm eff}$. The luminosity (and thus the distance) for these sources is not well constrained so we use values between $L=1000-5000\,$L$_{\odot}$, typical values for post-AGB sources. Input disc parameters are $R_{\rm in}$ and $R_{\rm out}$, the different dust components, the total disc mass and the power law for the surface-density distribution. Since we are not dealing with outflow sources a power law $\alpha >-2$ is used. The gas-to-dust ratio is kept fixed at 100.
The modelling is still degenerate, especially in parameters like the outer radius and the total disc mass which can be easily interchanged, without strongly influencing the SED. We use a dust mixture of amorphous and crystalline silicates in grain sizes ranging of $0.1-20$$\mu$m, with a power law distribution of $-3.0$, for the homogeneous disc. For HD52961 the 850$\mu$m submillimetre data points to the presence of extremely large grains in the disc, but these large grains are assumed not to influence the near- and mid-IR part of the SED, and will only be important for the blackbody component. We use a value of 300AU for $R_{\rm out}$. For the inner rim we use the radius at which the temperature of the inner rim equals about 1500K. This is a typical value for the dust sublimation temperature of silicates, although values as low as 1200K are sometimes also used. The total SED energetics are then calculated, given a specific inclination angle of the system.
We do not aim to reproduce the observed narrow features in the Spitzer spectrum, since these features originate from an optically thin upper layer of small grains at the disc surface. They can only be fitted well using an inhomogeneous disc model with grain settling. Instead, we want to model the observed general energetics of the SED spectrum, thus the observed continuum and amorphous dust features.
Comparison with interferometric data
------------------------------------
The circumstellar environment of HD52961 has been resolved using the VLTI/MIDI instrument, with angular sizes in the N-band between 35mas and 55mas in a uniform disc approximation [@deroo06]. EPLyr is too faint for current interferometric capabilities.
To compare our disc model of HD52961 with the MIDI data we made model images from which we extracted visibilities, using the same projected baseline lengths (40m/46m) and angles ($45^{\circ}$/$46^{\circ}$) as the observations. The only free parameters here are the inclination of the disc and the orientation angle of the system on the sky. A range of inclinations which still fit the observed SED was tested, in steps of 15$^{\circ}$. The orientation angle is varied 1$^{\circ}$ at a time. The result of this comparison can be found in Sect. \[resultshd52961\].
Results
-------
When modelling the near- and mid-IR part of the SEDs, the feature-to-continuum ratio of the silicate features is too strong in comparison with the observed spectra for both stars. Moreover, the near-IR flux is often underestimated. This was also observed in @gielen07, where we fitted the SED similarly to two post-AGB stars, RUCen and ACHer. Including an extra continuum opacity source is needed to reduce the strength of the features (see Fig. \[comparison\_iron\]) and to increase the near-IR contribution. From our previous work [@gielen08] and Section \[silicates\], we found evidence that (a fraction of) the silicates might be iron poor. So we use metallic iron as a potential opacity source: its near-IR opacity is large, but the absorption coefficient is unfortunately featureless so direct detection is difficult. Inclusion of free metallic iron has a strong impact on the modelling because the near-IR excess increases significantly with a given inner radius. Another possible opacity source is the inclusion of hot large grains in the homogeneous disc model.
### HD52961 {#resultshd52961}
------------ ---------------------------- ---------------------- ---------- ------ -------------- ------------
$R_{\rm in} - R_{\rm out}$ $m$ $\alpha$ iron $T_{\rm bb}$ $i$
AU $10^{-5}$M$_{\odot}$ % K $^{\circ}$
HD52961: A $10 - 500$ $1.7$ -1.5 0 160 $0-90$
HD52961: B $10 - 500$ $0.7$ -1.5 10 160 $< 65$
EPLyr: A $40 - 300$ $6$ -1.0 0 - $0-90$
------------ ---------------------------- ---------------------- ---------- ------ -------------- ------------
: Results of our SED disc modelling. []{data-label="sedresults"}
Note: Given are the inner and outer radius ($R_{\rm in}$-$R_{\rm out}$), the total disc mass $m$ for the homogeneous disc model, the surface-density distribution power law $\alpha$, the percentage of iron in the homogeneous disc model, the blackbody temperature and the inclination of the system.
We can use the interferometric data for HD52961 to constrain the distance to the system (Fig. \[midimodel\]). When we compare the modelled visibilities with the observations for HD52961 we find that the visibilities of our model are too high, when using standard models of $L=5000\,$L$_{\odot}$ which fit the SED and impose the inner radius to be at sublimation radius. This means we need to increase the angular size of the N-band emission region, either by increasing the physical size or by decreasing the distance. Increasing the disc size to outer radii $>
500$AU does not influence the N-band emission so we need to increase the inner radius, flatten the density distribution power law and/or decrease the luminosity in the disc model. Changing the surface-density distribution power law to values $>-1.5$ proved incompatible with the observed SED. We therefore use an average luminosity of 3000L$_{\odot}$ and move the inner radius to larger distances. A good fit to the SED was obtained using an inner radius of 10AU. This assumed luminosity gives a distance to the system of about 1700pc. At 10AU the temperature of the inner rim is around 1100K, which is slightly below the canonical dust sublimation temperature for silicates.
For HD52961 we tested models with and without the inclusion of metallic iron. The resulting fits can be seen in Figure \[HD52961\_sedfit\], parameters can be found in Table \[sedresults\]. Since this star shows a rather strong 10$\mu$m feature, pointing to relatively large amounts of hot dust in the disc, we find we need a rather steep surface-density distribution, $\alpha < -1$. Since we only have one flux point at long wavelengths, we add a simple 160K blackbody model to fit the observed submillimetre emission.
When no iron is included (model A) we see that the flux around from 2 to 8$\mu$m is strongly underestimated. Including about 10$\%$ metallic iron (model B) increased this flux significantly. The inclusion of metallic iron has only a minor influence on the N-band interferometric measurements. It decreases the modelled visibilities by about 10%, which is still consistent with the observed visibilities.
The modelled visibilities (Fig. \[midimodel\]) lie within the observed visibility range but a remarkable detection is that the variation in visibilities between the two observations is quite large, despite the very limited difference in lengths ($41.3-46.3$m), as well as in projected angle ($45.6-46.3^\circ$). This is illustrated when we plot the visibility versus the spatial frequency for a given wavelength, as seen in Figure \[spatfreq\]. In this figure we illustrate that when using a uniform disc for the intensity distribution, the steep visibility drop can be accounted for. The physical disc model is, however, much smoother and does not contain the very sharp edge characteristic of the uniform disc. The model also do not reproduce the observed ‘bump’ in visibility between 9 and 12$\mu$m. @deroo06 explain this observed increase in visibility as being due to a non-homogeneous distribution of the silicates, which contribute most to the inner regions of the disc. The current disc model does not include the physics to be able to reproduce this radial distribution of dust species.
The submillimetre 850$\mu$m flux for HD52961 and the derived blackbody temperature of 160K can be used to estimate the dust mass of large grains in the disc. In the optically thin approach the disc mass can be estimated using [@hildebrand83] $$M_d=\frac{F_{850}\,D^2}{\kappa_{850}\,B_{850}(T)}.$$ Assuming a cross section of large spherical grains, the mass absorption coefficient of 850$\mu$m grains in blackbody approximation is about 2.4cm$^2$g$^{-1}$. The mass absorption coefficient is given by $\kappa=\frac{\pi a^2}{\frac{4}{3}\pi a^3 \rho}$, with $a$ the grain size, and $\rho$ typically 3.3gcm$^{-3}$ for astronomical silicates. This results in dust-mass estimates in large grains of $3.2\times
10^{-6}\,$M$_{\odot}$ for HD52961.
### EPLyr
The SED-fitting gives an estimate of the distance to the system of $d=3.4$kpc, assuming a luminosity $L=3000\,$L$_{\odot}$. This estimation is not only dependent on the assumed luminosity of the star, but also on the adopted inclination of the system. Other derived disc parameters can be found in Table \[sedresults\].
For EPLyr it proved very problematic to get a good fit to the observed strong 20$\mu$m feature, without introducing a strong 10$\mu$m silicate feature. The mixed chemistry adds to the complexity of the object as the PAH emission and underlying continuum in the near-IR may very well come from a distinct structural component, in for example the polar direction. This is already seen in HD44179, were the observed PAH emission comes from a bipolar outflow [@bregman93; @menshchikov02; @cohen04]. The lack of data shortwards of 7$\mu$m makes it hard to get a good continuum estimate of the near- and mid-IR energetics. Cool dust clearly dominates the SED, but without additional information from interferometry, we cannot constrain parameters like the inner radius or the surface-density distribution. We opted to keep a rather flat density distribution of $\alpha = -1.0$ and do not force the model to fit the spectrum shortward of 15$\mu$m.
To get a good fit to the SED a very large inner radius of about 40AU is necessary. At this distance the inner rim reaches temperatures of $\sim300$K. This temperature agrees with the temperatures found in the spectral modelling (Sect. \[silicates\]). If we include metallic iron in the model we find we need inner radii even larger than 200AU, which seems physically implausible. Unfortunately we do not posses submillimetre data for this star so we cannot determine the blackbody temperature associated with the midplane. The small $L_{\rm IR}/L_*$ as well as the lack of a near-IR dust excess shows that the inner rim is quite far from the sublimation radius.
Conclusions
===========
HD52961 and EPLyr both have rich infrared spectra, and the assembled multi-wavelength data show that these evolved objects are surrounded by a stable circumstellar disc. While the binary nature of HD52961 is well established, the binarity of EPLyr is suspected but not yet firmly proven. The discs must be circumbinary as the sublimation radius of the dust is larger than the determined (HD52961) and suspected (EPLyr) binary orbit.
Recent studies have shown that many of these binary post-AGB systems are already detected [@vanwinckel03; @deruyter06; @deroo06; @deroo07b; @gielen08], but EPLyr and HD52961 both show quite distinct characteristics in dust and gas chemistry as well as in physical properties of their discs.
EPLyr and HD52961 are the only stars from the larger Spitzer sample that have clear CO$_2$ gas emission lines in the mid-IR. Our modelling shows that the emission in both stars can be well fitted and is dominated by $^{12}$C$^{16}$O$_2$ features, but clear detections of other isotopes are present as well. Similar excitation temperatures and column densities are found in both objects, but with different ratios for $^{12}$C$^{16}$O$_2$ and $^{13}$C$^{16}$O$_2$. Why these two stars are the only ones from the larger sample showing strong CO$_2$ features, and if there is any relation with the low observed infrared flux remains unclear. Similar feature strengths observed in the other stars would have been easily detected. One possibility is that the low dust emission in the two sample stars, reflected in the low $L_{\rm IR}/L_{star}$, makes it easier for the CO$_2$ gas to become visible. This effect is also seen in AGB stars, where CO$_2$ gas emission is strongest in sources with the lowest mass-loss rates [@sloan96; @cami02; @sloan03].
One of the most remarkable features is the clear detection of $^{18}$O isotopes of CO$_2$ in both objects. Together with HR 4049 [@cami01], the strong $^{16}$O$^{12}$C$^{18}$O band is a systematic feature of the gas emission in the discs of post-AGB binaries when CO$_2$ emission is detected. The high $^{18}$O abundance of HR4049 derived in an optically thin approximation of a putative nucleosynthetic origin [@lugaro05] was not confirmed by the analysis of CO first overtone absorption [@hinkle07] in the same object. It is likely that the CO$_2$ gas is strongly optically thick, also in EPLyr and HD52961, so that very rare isotopes are detectable.
The high-resolution Spitzer spectra also reveal unique solid state features. As in the bulk of the disc sources [@gielen08], crystalline silicate features prevail in both stars, but unlike what we found for the larger sample, they proved very hard to model. In HD52961 we observe some unique strong crystalline features at 11.3 and 16$\mu$m, which could not be reproduced in the modelling, irrespective of the grain size used in the models, shape or assumed grain model. In the 2D disc modelling we could not fit the steep rise around 10$\mu$m, without the inclusion of metallic iron. Combining our physical model constrained by the SED, together with our interferometric data, we concluded that the inner dust rim is slightly beyond the dust sublimation radius. This is in contrast to similar binary objects like IRAS08544-4431 where the interferometric data shows that the dusty disc has to start very near to the dust sublimation radius [@deroo07b]. Assuming a luminosity of 3000L$_{\odot}$, we find that the inner disc radius of HD52961 is rather large, around 10AU. The strong 850$\mu$m flux shows that this object has a component of very large grains. This contribution was added to the SED fitting by an additional colder Planck curve.
EPLyr has only a very small infrared excess, but the Spitzer spectrum is very rich in spectral details. The most remarkable characteristic is the clear PAH emission, in combination with the strong crystalline features at longer wavelengths. There is no evidence that the central star evolved into a carbon star when on the AGB, yet unprocessed class-C PAH features are clearly detected. Whether these PAH species are formed in the circumbinary disc or in a recent, likely polar outflow of the depleted central star, remains unclear. An extra component of cold dust is necessary in this object as well, to fit the entire Spitzer spectrum. Unfortunately, EPLyr is too faint for the current interferometric possibilities.
The mixed chemistry, the strongly processed cold crystalline silicates and low $^{12}$C/$^{13}$C ratio are in common with the subgroup of silicate J-type carbon stars, which can also display strong crystalline material. This corroborates the conclusion that in the latter, the disc is circumbinary. The abundance studies of J-type carbon stars are not complete enough to probe whether photospheric depletion affected these objects as well.
Both objects are extreme examples of post-AGB binary stars, with characteristics dominated by the presence of a stable circumbinary disc. This disc environment is, to first order, well modelled by assuming a passive, irradiated stable disc. In this paper we corroborate that this geometry is ideal to induce strong grain processing and a rich, even mixed chemistry. We conclude also that a homogeneous disc model is too primitive to model the spectral details as evidence of grain settling is strong. The route to PAH formation (and excitation) in the O-rich EPLyr remains to be studied in detail as PAH emission is only observed in a very limited number of such sources.
This detailed study of HD52961 and EPLyr shows that many questions still remain in our current understanding of the evolution of a significant number of post-AGB binary stars, and the impact of the circumbinary discs on the entire system.
CG acknowledges support of the Fund for Scientific Research of Flanders (FWO) under the grant G.0178.02. and G.0470.07.
[^1]: Based on observations made with the 1.2m Flemish Mercator telescope at Roque de los Muchachos, Spain, the 1.2m Swiss Euler telescope at La Silla, Chile and on observations made with the Spitzer Space Telescope (program id 3274), which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. $^{**}$ NASA Postdoctoral Fellow
|
---
abstract: 'We analyze two volume limited galaxy samples from the SDSS photometric and spectroscopic data to test the isotropy in the local Universe. We use information entropy to quantify the global anisotropy in the galaxy distribution at different length scales and find that the galaxy distribution is highly anisotropic on small scales. The observed anisotropy diminishes with increasing length scales and nearly plateaus out beyond a length scale of $200 {{\, h^{-1}\, {\rm Mpc}}}$ in both the datasets. We compare these anisotropies with those predicted by the mock catalogues from the N-body simulations of the $\Lambda$CDM model and find a fairly good agreement with the observations. We find a small residual anisotropy on large scales which decays in a way that is consistent with the linear perturbation theory. The slopes of the observed anisotropy converge to the slopes predicted by the linear theory beyond a length scale of $\sim 200 {{\, h^{-1}\, {\rm Mpc}}}$ indicating a transition to isotropy. We separately compare the anisotropies observed across the different parts of the sky and find no evidence for a preferred direction in the galaxy distribution.'
title: Testing isotropy in the Universe using photometric and spectroscopic data from the SDSS
---
\[firstpage\]
methods: numerical - galaxies: statistics - cosmology: theory - large scale structure of the Universe.
Introduction
============
The statistical homogeneity and isotropy of the Universe is a fundamental premise of modern cosmology. The assumption of homogeneity and isotropy of the Universe is known as the Cosmological principle. The isotropy of the Universe demands that at any given instant of time, there are no special directions in the Universe. This assumption, considered to be a cornerstone of the standard cosmological model, can be tested with the plethora of data from the modern cosmological observations. The discovery of the Cosmic Microwave Background radiation (CMB) in 1964 [@penzias] proved to be a milestone in the establishment of the standard model of cosmology. The Cosmic Background Explorer (COBE) mission launched in 1990 revealed the uniformity of the CMB temperature across the entire sky [@smoot; @fixsen] and provided one of the most powerful evidences in favour of isotropy. COBE and two subsequent space missions, Wilkinson Microwave Anisotropy Probe (WMAP) and Planck, along with numerous ground and balloon based experiments [@wmap; @adeplanck3; @boomerang; @act; @spt] measured the small anisotropies in the CMB sky revealing a wealth of cosmological information and ushering in the era of precision cosmology. The CMB anisotropies are small fluctuations in the CMB temperature or intensity as a function of position on the sky.
We should clarify here that even though there are small anisotropies in the CMB, we can still have *statistical isotropy*. Statistical isotropy implies that the small fluctuations in the CMB should look same in every direction in the sky. In other words, the angular power spectrum of anisotropies should not depend on which part of the sky we choose to study. Even on the largest scales observable today and super horizon scales, we should expect small anisotropies and inhomogeneities of 1 part in $10^4-10^5$ seeded during inflation. We are however interested in *anomalous anisotropy*, defined as the anisotropy that is much larger than $10^{-4}-10^{-5}$ expected from the simplest models of inflation. In this paper we will use word isotropy to mean isotropy in this statistical sense on small scales and absence of anomalously large anisotropy on large scales and not absolute isotropy. The assumption of isotropy and homogeneity in this sense in the cosmological model allows us to treat the small anisotropies and inhomogeneities perturbatively on an absolute isotropic and homogeneous FRW background Universe.
A large number of studies from WMAP and PLANCK [@eriksen; @hoftuft; @akrami; @adeplanck1; @adeplanck2; @schwarz1; @land; @hanlewis; @moss; @grupp; @dai] reveal subtle anomalies in the CMB anisotropies that might challenge the assumption of *statistical isotropy*. Several unexpected features at large angular scales such as a hemispherical power asymmetry and an abnormally large cold spot, although modestly significant, suggest a critical examination of the assumption of isotropy. The assumption of isotropy needs to be tested with independent datasets and with diverse statistical measures. This has been done with various other observations such as the X-ray background [@wu; @scharf], radio sources [@wilson; @blake], Gamma-ray bursts [@meegan; @briggs], supernovae [@gupta; @lin], galaxies [@marinoni; @gibel; @yoon; @alonso; @pandey17], galaxy clusters [@bengaly17] and neutral hydrogen [@hazra]. All these observations are consistent with the assumption of statistical isotropy. Contrary to these findings, there are also other studies with Type-Ia supernovae [@schwarz2; @campanelli; @kalus; @javanmardi; @bengaly], radio sources [@jackson], galaxies [@javanmardi17], galaxy luminosity function [@appleby] and large scale bulk flows [@watkins; @kashlinsky1; @kashlinsky2] which reported statistically significant deviation from isotropy. The current observational status does not provide a clear consensus on the issue of isotropy of the Universe on large scales and further investigations are necessary to either establish or refute it.
The large-scale structures in the Universe emerge from the gravitational collapse of the primordial density fluctuations. The gravitational collapse is anisotropic in nature. In the Zeldovich approximation, an overdense region in the dark matter distribution first collapses along its shortest axis leading to a sheet-like structure [@zeldovich1970]. The subsequent collapse along the medium and the longest axis results in an elongated filament and a dense compact cluster respectively [@sz1989]. Galaxies are a biased tracer of the underlying mass distribution and any anisotropy in the distribution of the dark matter is also expected to be present in the galaxy distribution. The modern redshift surveys like 2dFGRS [@colles] and SDSS [@york] have now mapped a large number galaxies in the local Universe providing an unprecedented view of the galaxy distribution in our neighbourhood. The galaxies are found to be distributed in an interconnected network of filaments, sheets and clusters which are surrounded by large empty regions. The filaments, which acts as interconnecting bridges between the clusters are known to be statistically significant up to $\sim 70-80 {{\, h^{-1}\, {\rm Mpc}}}$ [@bharad04; @pandey05]. The Sloan Great Wall discovered in the SDSS [@gott05] is one of the richest galaxy system in the nearby Universe and appears to be contiguous over length scales of more than 400 Mpc. Observations suggests that there are voids of enormous sizes such as the Bootes void with radius of 62 Mpc [@kirshner] and the Eridanus supervoid extending up to 300 Mpc [@szapudi]. The Eridanus void is also known to be aligned with the CMB cold spot and believed to be associated with it. Observational detection of all these gigantic cosmic structures re-emphasize the necessity of testing the isotropy in the galaxy distribution.
The SDSS is a multiband photometric and spectroscopic redshift survey which covers one quarter of the celestial sphere in the Northern Galactic Cap. The photometric and spectroscopic catalogues of the SDSS now provide redshifts of millions of galaxies making them suitable for testing isotropy in the galaxy distribution. The spectroscopic redshifts are estimated from the spectra of galaxies and hence they are more reliable but difficult and costly to measure for a very large number of galaxies. On the other hand, there is a larger uncertainty in the estimate of the photometric redshifts but photometric data is easier to obtain for a large number of galaxies. Keeping this in mind, we consider both the spectroscopic and photometric catalogues from the SDSS for the present analysis.
Information entropy can be used to test the homogeneity and isotropy of the Universe [@pandey13; @pandey16a]. In this work, we use an information theory based method [@pandey16a] to test the isotropy of the local Universe using the spectroscopic and photometric redshift catalogues of the Sloan Digital Sky Survey (SDSS).
A brief outline of the paper follows. We describe the method of analysis in Section 2, the data in Section 3 and present the results and conclusions in Section 4 and Section 5 respectively.
We have used a $\Lambda$CDM cosmological model with matter density parameter today, the $\Omega_{m0}=0.31$, dark energy density parameter $\Omega_{\Lambda0}=0.69$ and Hubble parameter $h=1$ for converting redshifts to distances throughout the analysis.
METHOD OF ANALYSIS
==================
We use the anisotropy parameter defined in @pandey16a to quantify the anisotropy in the galaxy distribution. This anisotropy parameter uses the information entropy [@shannon48] to measure the non-uniformity in the distribution of galaxies. In this method, we first need to divide the entire sky into pixels of equal area and similar shape. The equal area of the pixels ensures that the solid angle subtended by each of these pixels on the observer is the same whereas the similar shapes ensure the same geometry for each of the angular bins. We use the Hierarchical Equal Area isoLatitude Pixelization (HEALPix) software [@gorski1; @gorski2] for this purpose. We use the HEALPix resolution parameter $N_{\rm side}$ to pixelate the sky into $N_{\rm pix}=12 \times N_{\rm side}^2$ pixels of equal area. The angular size of each pixel for a specific choice of $N_{\rm side}$ is $\sqrt{\frac{41253}{N_{\rm pix}}}$ degree where 41253 square degree is the total area of the sky. The angular bins subtended by each of these pixels have the same volumes but may contain different number of galaxies within them. We choose an upper limit $r_{\rm max}$ for the radial distance as the galaxies are mapped only within a finite region. Furthermore, the galaxy surveys very often do not provide a full sky coverage. The fact that only a part of the sky is mapped by the survey needs to be taken into account through a mask specific to the survey. The effective number of pixels ${N_{\rm effective}}$ available for the analysis is smaller than $N_{\rm pix}$ and depends on the size and geometry of the mask or the sky coverage of the survey.
We vary the radial distance within the limit $r_{\rm max}$ with uniform steps and cumulatively count the number of galaxies within each angular bin. We use an uniform step size of $10 {{\, h^{-1}\, {\rm Mpc}}}$ throughout this analysis. We consider a randomly selected galaxy lying within a radial distance $r$ from the observer. This galaxy resides in any one of the ${N_{\rm effective}}$ angular bins and the probability of finding the galaxy in any particular bin is proportional to the total number of galaxies residing in that bin. If $n_{i}$ is the number of galaxies located in the $i^{th}$ angular bin then the probability of finding the galaxy in the $i^{th}$ bin is given by, $f_{i}=\frac{n_{i}}{\sum^{{N_{\rm effective}}}_{i=1}
\, n_{i}}$ and $\sum^{{N_{\rm effective}}}_{i=1} \, f_{i}=1$ by definition. So the event of randomly selecting a galaxy has ${N_{\rm effective}}$ outcomes each with a different probability $f_{i}$. The information entropy associated with this event for a specific radial distance $r$ can be written as, $$\begin{aligned}
H_{lb}(r)& = &- \sum^{{N_{\rm effective}}}_{i=1} \, f_{i}\, \log\, f_{i} \nonumber\\ &=&
\log N - \frac {\sum^{{N_{\rm effective}}}_{i=1} \, n_i \, \log n_i}{N},
\label{eq:shannon2}\end{aligned}$$ where $N$ is the total number of galaxies located within a radial distance $r$ from the observer. The subscript $lb$ in $H_{lb}$ implies that we consider the pixels in the longitude-latitude $(l,b)$ space and study it as a function of the radial distance $r$ up to which the number counts are integrated. The base of the logarithm is arbitrary and only decides the unit of information. We use the base of $10$ for the present work. If the probabilities $f_{i}$ are identical for each of the angular bins then the information entropy will be maximum, $(H_{lb})_{max}=\log \, {N_{\rm effective}}$. This would only occur if each of the angular bins hosts exactly the same number of galaxies within a distance $r$. This corresponds to the situation when there is maximum uncertainty about the location of the randomly selected galaxy. We define the anisotropy parameter, $$\begin{aligned}
a_{lb}(r)=1-\frac{H_{lb}(r)}{\left(H_{lb}\right)_{max}},\end{aligned}$$ to measure the degree of radial anisotropy present in any distribution. It may be noted that for a completely isotropic distribution, the probability distribution is uniform leading to $H_{lb}(r)=\left(H_{lb}\right)_{max}$ and $a_{lb}(r)=0$. On the other hand, if all the galaxies are located only in one specific bin out of the ${N_{\rm effective}}$ angular bins then the galaxy distribution is maximally anisotropic. In this case, there is no uncertainty about the location of the randomly selected galaxy and consequently we have $H_{lb}(r)=0$ and $a_{lb}(r)=1$.
The galaxies are not distributed randomly but in a web-like network. The presence of the coherent patterns in the galaxy distribution like filaments, sheets, clusters and voids cause the distribution to be highly anisotropic on small scales. So the probabilities of locating randomly selected galaxies in different angular bins are not the same. The anisotropy parameter $a_{lb}(r)$ will measure non-uniformity in the distribution of galaxies as a function of the length scale $r$. We will measure the anisotropy parameter $a_{lb}(r)$ in the SDSS data as a function of $r$ up to the maximum radial distance $r_{\rm max}$. If the assumption of isotropy on large scales holds in the real Universe then we expect the anisotropy parameter to decrease with the increasing length scale $r$ and should become negligibly small on the scales where the Universe becomes isotropic. It should be noted that the value of $a_{lb}(r)$ is also expected to be sensitive to the choice of $N_{\rm side}$ as it decides the total number of pixels $N_{\rm pix}$. The pixel sizes will be larger for a smaller $N_{\rm side}$. As a result the volume covered by each angular bin will be also larger. This would increase the galaxy counts and reduce the Poisson noise leading to a decrease in the anisotropy. Even an isotropic distribution of finite size will exhibit some anisotropy due to the Poisson noise on small scales. To asses this, we compare our results to that obtained from a homogeneous and isotropic Poisson distribution which has the same geometry and sampling density as the actual data.
The anisotropy parameter measured at each length scale for the entire survey region provides the degree of global anisotropy present in the galaxy distribution. Besides the global isotropy, it is also important to compare the degree of anisotropy observed along the different directions in the sky. For this, we will be dividing the SDSS survey area into a small number of regions and separately measure the entropy and anisotropy in each of these regions. This would allow us to test the statistical isotropy of the galaxy distribution and identify the existence of any preferred directions.
The information entropy is related to the higher order moments of a distribution [@pandey16b], in addition to the second order moment or the power spectrum. The higher order moments of the galaxy density field are expected to be non zero as the present day galaxy distribution is known to be highly non-Gaussian. This provides the argument for using the information entropy as an effective measure of the anisotropy present in the galaxy distribution, since it captures information beyond the 2-point correlation function or the power spectrum.
\
DATA
====
We use the data from the twelfth data release of the Sloan Digital Sky Survey (SDSS, DR12) [@alam] which is the final data release of the SDSS-III. A description of the telescope used in the SDSS is provided in @gunn2 and the SDSS camera and filters are discussed in @gunn1 and @fukugita respectively. The target selection algorithm of the Main Galaxy Sample is described in @strauss.
THE SDSS PHOTOMETRIC DATA
-------------------------
We use the photometric redshift catalogue for the SDSS data prepared by @beck. We select only the galaxies with the photometric error classes $1$ and $-1$ which have accurate redshift error estimates. For other error classes, the redshift estimation errors are dependent on the position in color and magnitude space and hence requires additional statistical errors to be taken into account. We apply a cut in the redshift error estimate $\delta_{z}^{\rm
photo}<0.03$ and consider all the galaxies in the redshift range $0<z<0.3$ with r-band extinction $A_r<0.18$ which provides us with a sample containing $2827248$ galaxies. Further cuts in the r-band apparent magnitude $13<m_r<19$ and r-band absolute magnitude $M_{r}<-20.4$ are applied to construct a volume limited sample of galaxies. The resulting volume limited sample of galaxies extends up to redshift $z<0.2143$ or comoving distance $<609.197 {{\, h^{-1}\, {\rm Mpc}}}$ and consists of $1022630$ galaxies. The volume limited sample prepared from the SDSS photometric data does not start from $z=0$ but from $z=0.017$ which corresponds to a distance of $51.4 {{\, h^{-1}\, {\rm Mpc}}}$ due the apparent magnitude cut adopted here. We consider only the galaxies in the Northern Galactic Cap to get a contiguous sky coverage which reduces the available galaxies to $784329$ in our volume limited sample.
THE SDSS SPECTROSCOPIC DATA
---------------------------
We use publicly available spectroscopic data from the SDSS CasJobs[^1]. We retrieve the spectroscopic information of all the galaxies in the redshift range $0<z<0.3$ yielding $916633$ galaxies. We then construct a volume limited sample of galaxies by restricting the r-band Petrosian magnitude in the range $13<m_{r}<17.77$ and r-band absolute magnitude to $M_{r}<-20.4$. The values of k-corrections are also obtained from the SDSS CasJobs. These cuts yield a volume limited sample of $180181$ galaxies distributed within redshift $z<0.1341$ or comoving distance $<391.8 {{\, h^{-1}\, {\rm Mpc}}}$. The resulting volume limited sample does not start from $z=0$ but from $z=0.0167$ due the apparent magnitude range chosen. So the volume limited sample prepared from the SDSS spectroscopic data starts at $50 {{\, h^{-1}\, {\rm Mpc}}}$ and extends upto $391.8 {{\, h^{-1}\, {\rm Mpc}}}$. We require a contiguous region of the sky for our analysis. So we only consider the galaxies in the Northern Galactic Cap which finally leaves us with a volume limited galaxy sample comprised of $152860$ galaxies.
PREPARING THE MASKS
-------------------
We use HEALPix to divide the entire sky into pixels of equal sizes and count the number of galaxies inside each pixel. The galaxy number counts in different pixels for the volume limited samples of galaxies constructed from the spectroscopic and photometric redshift catalogues are shown in . The top left and bottom left panels of show the number counts in the SDSS photometric data with $N_{\rm side}=4$ and $N_{\rm side}=32$ respectively. The top right and bottom right panels show the same but for the SDSS spectroscopic data. The galaxy distributions are 3-dimensional and these counts are the integrated galaxy counts inside each pixel up to the maximum radial extent of the corresponding galaxy samples. $N_{\rm side}$ sets the resolution of the map. As a result we see relatively smaller number counts inside the pixels for $N_{\rm
side}=32$ than $N_{\rm side}=4$. It may be also noted that the pixels near the boundary of the survey regions preferentially show lower number counts. This is related to the fact that only parts of the boundary pixels lie inside the survey region. These pixels need to be discarded from any analysis of isotropy and we take this into account by preparing a mask for each dataset and for each resolution.
To prepare the mask we sub-pixelate each pixel into smaller sub-pixels using a new variable $N_{\rm side}^{\rm sub}$ which we have taken to be $N_{\rm side}^{\rm sub}=64$ for the present work. We count galaxies inside each of the sub-pixels and create a mask map by assigning a value 1 to the non-empty sub-pixels and 0 to the empty sub-pixels. We then degrade the mask map to a new mask map for $N_{\rm side}$. The pixels for which all the sub-pixels are empty have value $0$ and are masked. The values for pixels at the boundary indicate the fraction of non-empty sub-pixels in that pixel. We identify and mask the sparsely populated pixels near the boundary by using a threshold value $m_{\rm th}$ for this map. If a pixel in the $N_{\rm side}$ map has a value $< m_{\rm th} $ then we mask the corresponding pixel. In our analysis we have used $m_{\rm th} = 0.75$.
RANDOM MOCK CATALOGUES
----------------------
We generate $10$ Poisson random distributions for both the volume limited samples constructed from the spectroscopic and photometric data. The mean density of these samples is chosen to be same as the respective volume limited samples under consideration. We then apply the respective masks to generate $10$ random mock catalogues each for the SDSS spectroscopic and photometric samples.
MOCK CATALOGUES FROM N-BODY SIMULATIONS
---------------------------------------
We use a Particle-Mesh (PM) N-Body code to simulate the present day distributions of dark matter in the $\Lambda$CDM model. We use $\Omega_{m0}=0.31$, $\Omega_{\Lambda0}=0.69$, $h=0.68$, $\sigma_{8}=0.81$ and $n_{s}=0.96$ [@adeplanck3] as the values of the cosmological parameters. We simulate the distributions using $256^{3}$ particles on a $512^{3}$ grid in a comoving volume of $(
1433.6 \, h^{-1}\, {\rm Mpc})^3$. We run the simulations for three different realizations of the initial density fluctuations. We place an observer inside the centre of each simulation box and map the distributions to redshift space using the peculiar velocities of particles. The individual particles are treated as galaxies and we construct $3$ mock catalogues from each of the three boxes for each of the two volume limited SDSS samples (photometric and spectroscopic) by sampling particles with the same mean density as the corresponding SDSS sample and applying the respective masks. This provides us with $9$ mock galaxy catalogues for the $\Lambda$CDM model for each of the volume limited SDSS samples.
RESULTS
=======
THE GLOBAL ANISOTROPY IN THE PHOTOMETRIC AND SPECTROSCOPIC GALAXY SAMPLES
-------------------------------------------------------------------------
\
\
\
\
\
\
\
\
\
\
\
. \[fig:entropymap\_p\]
\
\
\
We measure the anisotropy parameter $a_{lb}(r)$ defined in using all the pixels inside the unmasked region of the sky. We separately measure the anisotropy in the photometric and the spectroscopic datasets from the SDSS at four different resolutions with HEALPix $N_{\rm side}=4,8,16$ and $32$. The corresponding mock galaxy catalogues from the random distributions and the N-body simulations of the $\Lambda$CDM model are also analyzed at the same resolution for comparison. The results for the spectroscopic and photometric data are shown in and respectively. The top left panel of shows the variation of the anisotropy parameter $a_{lb}(r)$ as a function of distance for the SDSS spectroscopic data for $N_{\rm side}=4$. The results for the mock galaxy catalogues from the random distributions and the N-body simulations of the $\Lambda$CDM model are also shown together in this panel for comparison with the observations. The $1-\sigma$ errorbars for the SDSS data are estimated from the $10$ Jackknife samples drawn from the data. The $1-\sigma$ errorbars for the mock catalogues from the random distributions and the $\Lambda$CDM model are estimated from $10$ and $9$ independent realizations respectively. The top right, bottom left and bottom right panels of show the same quantities but for $N_{\rm side}=8,16$ and $32$ respectively. In each of these panels we see that the galaxy distribution in the SDSS is highly anisotropic on small scales. The observed anisotropy in the galaxy distribution decreases with increasing length scales. It is interesting to note that the mock galaxy catalogues from the $\Lambda$CDM N-body simulations reproduce the observed anisotropy in the SDSS data remarkably well for each value of $N_{\rm side}$. The anisotropy observed in the mock catalogues from the random distributions also decreases with increasing length scales but the degree of anisotropy observed in these cases are noticeably smaller than the SDSS and $\Lambda$CDM. The anisotropy observed in the random distributions are sourced by only the Poisson noise which naturally decreases with increasing length scales due to the increase in galaxy counts at larger scales. On the other hand the anisotropies in the galaxy distribution in the SDSS and the $\Lambda$CDM model are sourced by both the Poisson noise and anisotropic gravitational clustering. The effect of Poisson noise on the anisotropy is expected to diminish with increasing galaxy counts in similar manner in all 3 datasets but the additional anisotropy present in the SDSS and $\Lambda$CDM model due to the gravitational clustering would change differently depending on the nature and strength of clustering present in them. We find that the anisotropies in the SDSS and the $\Lambda$CDM model decreases to a small value and plateaus out beyond a length scale of $200 {{\, h^{-1}\, {\rm Mpc}}}$. Since the $\Lambda$CDM simulations assume an isotropic background and statistically isotropic initial perturbations, the comparison of the data with simulations tests the validity of these assumptions. The difference between the simulations and data in particular becomes negligible at scales $r \gtrsim 200
~{\rm Mpc/h}$.
There is excess residual anisotropy, over the isotropic Poisson samples, present in the SDSS data and the $\Lambda$CDM simulations. The two contributions to the anisotropy in the observed data come from clustering in real space and from the effect of peculiar velocities on the redshift measurements, the so called redshift space distortions. The SDSS galaxies are mapped in redshift space where the peculiar velocities perturb the redshifts and distort the galaxy distribution. The large scale coherent inflow towards the overdense regions and outflow from the underdense regions introduce specific anisotropic features in clustering pattern of the galaxies. Also, the random peculiar velocities inside the virialized bound structures elongate the structures along the line of sight giving rise to what is popularly known as the “Fingers of God (FOG)” effect. Since we integrate along the radial direction, the errors in distance measurements coming from peculiar velocities should get averaged out. We therefore do not expect the redshift space distortions to contribute significantly to our anisotropy parameter $a_{lb}$. For the same reason, our results are also insensitive to the errors in redshifts, especially in the photometric sample. We can explicitly check this using the mock catalogues from the N-body simulations. We separately construct $10$ mock galaxy catalogues for the SDSS spectroscopic sample from the $\Lambda$CDM N-body simulations without taking into account the effect of peculiar velocities, i.e. without the redshift space distortions. We measure the anisotropy parameter $a_{lb}(r)$ in these samples and compare the results with that for the mock galaxy samples prepared by taking into account the effect of peculiar velocities, as in real observations. The results are shown in . We find that the anisotropy parameter $a_{lb}(r)$ is insensitive to the redshift space distortions. So the small residual anisotropy on large scales are not sourced by the random peculiar velocities inside FOGs. The remaining small difference in anisotropy between the galaxy distribution and the Poisson distribution indicates that the galaxy distribution can not be represented by a Poisson distribution on any length scale. We should expect this just from linear growth. The scales of 200 Mpc entered horizon around $z\approx 1000$ with an amplitude of $\sim 10^{-5}$ and would have grown by a factor of $\sim 1000$ today to an amplitude of $\sim 10^{-2}$. This is also the level of anisotropy that we see in $a_{lb}$. We will show explicitly below that this residual anisotropy is nothing but the linear large scale structure in the $\Lambda$CDM universe and comparison with linear theory defines an unambiguous scale of isotropy.
We also analyze the SDSS photometric data exactly in the same way and present our results in . The SDSS photometric sample covers a much larger volume and hence contains a significantly larger number of galaxies. Interestingly, we find that the results are very similar to those found with the SDSS spectroscopic sample. The anisotropy in the galaxy distribution decreases with length scale in each case and the galaxy distribution appears to be nearly isotropic beyond a length scale of $200 {{\, h^{-1}\, {\rm Mpc}}}$. We note that the anisotropies observed in the SDSS photometric sample are relatively smaller than what is predicted by the dark matter only $\Lambda$CDM simulations on scales below $200 {{\, h^{-1}\, {\rm Mpc}}}$ i.e the distribution of the galaxies in the local Universe is more uniform than the expectations from the $\Lambda$CDM model. But interestingly the anisotropy curves for both the distributions flatten out on the same scales. The differences between the observed anisotropy in the SDSS photometric data and the $\Lambda$CDM model may arise due to the larger uncertainties associated with the photometric redshifts in the SDSS. We test this possibility using the redshift estimation errors provided by @beck. @beck use a linear fit in colour magnitude space to describe the photometric redshift and estimate the rms error $\delta z_{phot,i}$ in photometric redshift measurement for each galaxies. We draw the errors on the photometric redshift for each galaxy from Gaussians with standrd deviations $\delta z_{phot,i}$ and simulate a set of galaxy catalogues from the primary SDSS photometric redshift catalogue. We construct $10$ volume limited galaxy samples from these datasets and analyze them separately. The results are presented in which indicates that the uncertainty in the measurement of photometric redshift is unlikely to change the anisotropy. As stated above, this is expected since we integrate along the line of sight and any errors in redshift estimates would average out, similar to the effect of the peculiar velocities. The difference in dark matter only simulations and SDSS data is therefore mostly because of the baryonic astrophysics. However, further investigations are necessary to either support or refute this claim. One may also note an apparent increase in the anisotropy at $\sim 300 {{\, h^{-1}\, {\rm Mpc}}}$ for $N_{\rm side}=4$ and $N_{\rm side}=8$ which may arise due to the presence of a large supercluster or void at this distance.
In the top left and right panels of , we compare the anisotropy in the SDSS photometric and spectroscopic galaxy samples respectively for different $N_{\rm side}$. We observe a larger degree of anisotropy in the galaxy distribution at higher resolution due to a larger contribution from the Poisson noise. We show the rate of change of anisotropy in each case for the photometric and spectroscopic data in the bottom left and right panels of respectively. We find that for both the SDSS photometric and spectroscopic data, the observed slopes flattens out nearly to zero on scales of $\sim 100 {{\, h^{-1}\, {\rm Mpc}}}$ for $N_{\rm side}=4$ and $8$. The same trend is observed on $\sim 200 {{\, h^{-1}\, {\rm Mpc}}}$ for $N_{\rm
side}=16$ and $32$ for both the datasets.
We linearly evolve the initial power spectrum upto present day and then use it to generate particle distributions at present within the N-body cube. These distributions represent the linear version of the present day mass distribution. We generate $3$ such distributions and construct $10$ independent mock samples for the spectroscopic and photometric sample from these distributions. We analyzed them in exactly same manner and find that a nearly constant residual anisotropy of the order of $10^{-2}$ persists on large scales irrespective of the chosen value of $N_{side}$. The top two panels of show that there is a small offset between the residual anisotropy observed in the SDSS and that predicted by linear theory. The residual anisotropy in the $\Lambda$CDM mock catalogues are consistent with the SDSS observations.
We will use the slope of the anisotropy parameter $a_{lb}$, which is a local quantity and insensitive to the overall normalization and therefore cosmological parameters, to define the scale of isotropy.
In , we see that for any $N_{side}$, the anisotropy parameter eventually plateaus out to a small value ($\sim
10^{-2}$) on a certain scale and the corresponding slope of the anisotropy parameter tends to zero. However it may be noted that the slope of the anisotropy parameter does not become exactly zero on any length scales. In the two bottom panels of , we blow up the y-axes representing the slopes of the anisotropy parameter and find that the slopes are of the order of $10^{-4}~{\rm h/Mpc}$ beyond a length scale of $200 {{\, h^{-1}\, {\rm Mpc}}}$ for both the photometric and spectroscopic sample. We calculate the slopes of the anisotropy curves expected in linear theory and compare them to the observed values in for $N_{side}=32$. We find that the slopes of the anisotropy curves for both the photometric and spectroscopic samples come within $1-\sigma$ errorbars of the slopes expected from the linear theory beyond a length scale of $\sim 200 {{\, h^{-1}\, {\rm Mpc}}}$. We thus conclude that the anisotropy decays in the same way as predicted by the linear theory beyond $r\sim 200 {{\, h^{-1}\, {\rm Mpc}}}$. We note that the anisotropy parameter $a_{lb}$ has contributions from all scales while the slope is a local quantity and is only sensitive to the scale at which it is being calculated. We therefore use the slope as the criteria to define the scale of isotropy yielding a scale of isotropy of $\sim 200 {{\, h^{-1}\, {\rm Mpc}}}$.
THE STATISTICAL ISOTROPY IN THE PHOTOMETRIC AND SPECTROSCOPIC GALAXY SAMPLES
----------------------------------------------------------------------------
Besides measuring the global isotropy across the entire contiguous region of the SDSS photometric and spectroscopic data, we also quantify local anisotropy in smaller regions of the sky and study how it varies over the survey area. We divide the survey area into smaller regions with each *region* defined as a pixel of HEALPix resolution $N_{\rm side}^{\rm region}=4$ and use resolution of $N_{\rm
side}=32$ to study isotropy in each region individually. We obtain $23$ and $22$ regions in the SDSS photometric and spectroscopic survey areas respectively. The galactic co-ordinates of these pixels are tabulated in in the Appendix. We then compute the degree of anisotropy in each region by calculating the entropy associated with the counts in all $N_{\rm side}=32$ pixels inside each region. We show the maps of $a_{lb}$ for the SDSS photometric and spectroscopic data at different radial distances in and respectively. The radial distance provided in each panel of these figures indicates the maximum radial distance up to which the galaxy counts are integrated to calculate the anisotropy inside each region. It may be noted that at a given length scale, these anisotropies are relatively higher as compared to the global anisotropies observed in the bottom right panels of and . This is simply due to the smaller sample size in each region compared to the full survey area and therefore larger Poisson noise. We find that both the photometric and spectroscopic SDSS sky show significant variations in the measured anisotropy across different regions of the sky at a radial distance of $100 {{\, h^{-1}\, {\rm Mpc}}}$. The variations in the local anisotropy across regions decrease with increasing length scales and nearly cease to exist at $\sim 200 {{\, h^{-1}\, {\rm Mpc}}}$.
In , we also show the variation in the anisotropy parameter $a_{lb}(r)$ with distance $r$ for each of the individual 23 and 22 regions in the photometric and spectroscopic data respectively. The observed anisotropy in each region is higher compared to the full survey due to the larger Poisson noise in each smaller regions. The bottom panels shows the rate of change of anisotropy in each of these $23/22$ regions. The results shown in indicate that different parts of the SDSS photometric and spectroscopic sky exhibit similar degree of anisotropy beyond a length scale of $\sim
200 {{\, h^{-1}\, {\rm Mpc}}}$ after which the anisotropy parameter $a_{lb}$ is approximately constant. We do not find any significantly divergent pixel beyond a length scale of $200 {{\, h^{-1}\, {\rm Mpc}}}$. This suggests that the galaxy distribution mapped by the SDSS in both the photometric and spectroscopic redshift survey is isotropic on a length scale of $200
{{\, h^{-1}\, {\rm Mpc}}}$ and there are no preferred directions in the SDSS survey volume beyond this length scale.
CONCLUSIONS
===========
In this work, we have analyzed the SDSS photometric and spectroscopic data with information entropy to test the isotropy of the galaxy distribution in the local Universe. We find that the galaxy distribution is highly anisotropic on small scales but these anisotropies decrease with increasing radial distances. The anisotropies predicted by the $\Lambda$CDM N-Body simulations are in fairly good agreement with the observed anisotropies with the SDSS. Both photometric and spectroscopic data from SDSS exhibit a small residual anisotropy on large scales. The $\Lambda$CDM simulations accurately reproduce the small residual anisotropy observed on large scales. These residual anisotropies on large scales are expected just from the linearly evolved primordial anisotropies. To verify this, we also study the anisotropies in the distributions generated from a linearly evolved $\Lambda$CDM power spectrum and find a residual anisotropy of the same order as observed in the SDSS and $\Lambda$CDM N-body simulations. We find a small offset in the magnitude of the residual anisotropy observed in the $\Lambda$CDM N-Body simulations and the linear theory. We show that this offset originates from the differences in the mass variance in the two distributions on the corresponding length scales. To avoid the complications associated with the offsets and normalization of the fluctuations which would be a function of cosmological parameters, we use the slope of the anisotropy, a local quantity, to define the scale of isotropy. We study the slopes of the anisotropy parameter as a function of length scales in the SDSS and linear theory and find that the slopes agree with each other at $1-\sigma$ level beyond a length scale of $200
{{\, h^{-1}\, {\rm Mpc}}}$ indicating the onset of isotropy. To be precise, the decay in anisotropy parameter $a_{lb}$ is consistent beyond this scale with the linear perturbation theory expectation.
It may be noted that the degree of residual anisotropy depends on the value of $N_{side}$ () which decides the size of the angular patches or the volumes subtended by them at the observer. The value of $N_{side}$ controls the magnitude of the shot noise and hence the anisotropy. We find that irrespective of the choice of $N_{side}$, the galaxy distribution shows a transition to isotropy on large scales. We would like to clarify here that the *length scale* throughout our analysis refers to the farthest distance along the radial direction up to which the number counts are integrated.
An analysis of the Luminous Red Galaxies (LRG) from the SDSS DR7 by @marinoni find the scale of isotropy to be $150 {{\, h^{-1}\, {\rm Mpc}}}$ which is somewhat different than the value obtained in this work. The LRG sample analyzed by @marinoni span a redshift range $0.22<z<5$ whereas the SDSS galaxy samples analyzed here are limited to redshift $z<0.21$. So the difference in the result arises both due to different magnitude limits of the samples and different methods of analysis.
We test for the statistical isotropy by separately measuring anisotropy in different sub-samples/regions of the SDSS survey and find no evidence for a preferred direction beyond the scale $200
{{\, h^{-1}\, {\rm Mpc}}}$.
Our analysis indicates that the galaxy distribution shows a transition to isotropy on a scale $\sim 200 {{\, h^{-1}\, {\rm Mpc}}}$ for both the SDSS photometric and spectroscopic data. We conclude from the present analysis that the galaxy distribution in the local Universe is indeed isotropic on a scale of $200 {{\, h^{-1}\, {\rm Mpc}}}$ which reaffirms the validity of the assumption of isotropy of the Universe on large scales.
ACKNOWLEDGEMENT
===============
The authors thank an anonymous reviewer for useful comments and suggestions. The authors would like to thank the SDSS team for making the data public. The authors would like to thank Maciej Bilicki for providing the SDSS photometric catalogue. BP would also like to thank Maciej Bilicki for helpful comments and suggestions on the draft. SS would like to thank UGC, Government of India for providing financial support through a Rajiv Gandhi National Fellowship. B.P. would like to acknowledge financial support from the SERB, DST, Government of India through the project EMR/2015/001037. B.P. would also like to acknowledge IUCAA, Pune and CTS, IIT, Kharagpur for providing support through associateship and visitors programme respectively. RK was supported by SERB grant number ECR/2015/000078 from Science and Engineering Research Board, Department of Science and Technology, Govt. of India and by MPG-DST partner group between Max Planck Institute for Astrophysics and Tata Institute of Fundamental Research, Mumbai funded by Max Planck Gesellschaft, Germany.
[99]{}
Akrami, Y., Fantaye, Y., Shafieloo, A., et al. 2014, , 784, L42
Alam, S., Albareti, F. D., Allende Prieto, C., et al. 2015, , 219, 12
Alonso, D., Salvador, A. I., S[á]{}nchez, F. J., et al. 2015, , 449, 670
Appleby, S., & Shafieloo, A. 2014, , 10, 070
Beck, R., Dobos, L., Budav[á]{}ri, T., Szalay, A. S., & Csabai, I. 2016, , 460, 1371
Bengaly, C. A. P., Jr., Bernui, A., & Alcaniz, J. S. 2015, , 808, 39
Bengaly, C. A. P., Bernui, A., Ferreira, I. S., & Alcaniz, J. S. 2017, , 466, 2799
Bharadwaj, S., Bhavsar, S. P., & Sheth, J. V. 2004, , 606, 25
Blake, C., & Wall, J. 2002, , 416, 150
Briggs, M. S., Paciesas, W. S., Pendleton, G. N., et al. 1996, , 459, 40
Campanelli, L., Cea, P., Fogli, G. L., & Marrone, A. 2011, , 83, 103503
Carlstrom, J. E., Ade, P. A. R., Aird, K. A., et al. 2011, , 123, 568
Crill, B. P., Ade, P. A. R., Artusa, D. R., et al. 2003, , 148, 527
Colles, M. et al.(for 2dFGRS team) 2001,,328,1039
Dai, L., Jeong, D., Kamionkowski, M., & Chluba, J. 2013, , 87, 123005
Eriksen, H. K., Banday, A. J., G[ó]{}rski, K. M., Hansen, F. K., & Lilje, P. B. 2007, , 660, L81 Fixsen, D. J., Cheng, E. S., Gales, J. M., et al. 1996, , 473, 576
Fowler, J. W., Niemack, M. D., Dicker, S. R., et al. 2007, , 46, 3444
Fukugita, M., Ichikawa, T., Gunn, J. E., et al. 1996, , 111, 1748
Gibelyou, C., & Huterer, D. 2012, , 427, 1994
Gorski, K. M., Wandelt, B. D., Hansen, F. K., Hivon, E., & Banday, A. J. 1999, arXiv:astro-ph/9905275
G[ó]{}rski, K. M., Hivon, E., Banday, A. J., et al. 2005, , 622, 759
Gott, J. R., III, Juri[ć]{}, M., Schlegel, D., et al. 2005, , 624, 463
Gruppuso, A., Natoli, P., Paci, F., et al. 2013, , 7, 047
Gupta, S., & Saini, T. D. 2010, , 407, 651
Gunn, J. E., Carr, M., Rockosi, C., et al. 1998, , 116, 3040
Gunn, J. E., Siegmund, W. A., Mannery, E. J., et al. 2006, , 131, 2332
Hanson, D., & Lewis, A. 2009, , 80, 063004
Hazra, D. K., & Shafieloo, A. 2015, , 11, 012
Hoftuft, J., Eriksen, H. K., Banday, A. J., et al. 2009, , 699, 985
Jackson, J. C. 2012, , 426, 779
Javanmardi, B., Porciani, C., Kroupa, P., & Pflamm-Altenburg, J. 2015, , 810, 47
Javanmardi, B., & Kroupa, P. 2017, , 597, A120
Kalus, B., Schwarz, D. J., Seikel, M., & Wiegand, A. 2013, , 553, A56
Kashlinsky, A., Atrio-Barandela, F., Kocevski, D., & Ebeling, H. 2008, , 686, L49
Kashlinsky, A., Atrio-Barandela, F., Ebeling, H., Edge, A., & Kocevski, D. 2010, , 712, L81
Kirshner, R. P., Oemler, A., Jr., Schechter, P. L., & Shectman, S. A. 1987, , 314, 493
E., [Smith]{} K. M., [Dunkley]{} J., [Bennett]{} C. L., [Gold]{} B., [Hinshaw]{} G., [Jarosik]{} N., [Larson]{} D., [Nolta]{} M. R., [Page]{} L., [Spergel]{} D. N., [Halpern]{} M., [Hill]{} R. S., [Kogut]{} A., [Limon]{} M., [Meyer]{} S. S., [Odegard]{} N., [Tucker]{} G. S., 2011, , 192, 18
Land, K., & Magueijo, J 2005, , 95, 071301
Lin, H.-N., Wang, S., Chang, Z., & Li, X. 2016, , 456, 1881
Marinoni, C., Bel, J., & Buzzi, A. 2012, , 10, 036
Meegan, C. A., Fishman, G. J., Wilson, R. B., et al. 1992, , 355, 143
Moss, A., Scott, D., Zibin, J. P., & Battye, R. 2011, , 84, 023014
Pandey, B., & Bharadwaj, S. 2005, , 357, 1068
Pandey, B. 2013, , 430, 3376 Pandey, B. 2016, , 462, 1630
Pandey, B. 2016, , 463, 4239
Pandey, B. 2017, , 468, 1953
Penzias, A. A., & Wilson, R. W. 1965, , 142, 419
Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2014, , 571, A23
Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2016, , 594, A16
Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2016, , 594, A13
Schwarz, D. J., Starkman, G. D., Huterer, D., & Copi, C. J. 2004, , 93, 221301
Schwarz, D. J., & Weinhorst, B. 2007, , 474, 717 Shandarin, S. F., & Zeldovich, Y. B. 1989, Reviews of Modern Physics, 61, 185 Shannon, C. E. 1948, Bell System Technical Journal, 27, 379-423, 623-656
Scharf, C. A., Jahoda, K., Treyer, M., et al. 2000, , 544, 49
Smoot, G. F., Bennett, C. L., Kogut, A., et al. 1992, , 396, L1
Strauss, M. A., Weinberg, D. H., Lupton, R. H., et al. 2002, , 124, 1810
Szapudi, I., Kov[á]{}cs, A., Granett, B. R., et al. 2015, , 450, 288
Watkins, R., Feldman, H. A., & Hudson, M. J. 2009, , 392, 743
Wilson, R. W., & Penzias, A. A. 1967, Science, 156, 1100
Wu, K. K. S., Lahav, O., & Rees, M. J. 1999, , 397, 225
Yoon, M., Huterer, D., Gibelyou, C., Kov[á]{}cs, A., & Szapudi, I. 2014, , 445, L60
York, D. G., et al. 2000, , 120, 1579 Zeldovich, Y. B. 1970, , 5, 84
[ccc]{}
\
Region & galactic longitude & galactic latitude\
1 & 56.25000 & 41.81031\
2 & 75.00000 & 54.34091\
3 & 15.00000 & 54.34091\
4 & 45.00000 & 54.34091\
5 & 67.50000 & 66.44354\
6 & 22.50000 & 66.44354\
7 & 45.00000 & 78.28415\
8 & 168.75000 & 41.81031\
9 & 165.00000 & 54.34091\
10 & 157.50000 & 66.44354\
11 & 112.50000 & 66.44354\
12 & 135.00000 & 78.28415\
13 & 202.50000 & 30.00000\
14 & 213.75000 & 41.81031\
15 & 191.25000 & 41.81031\
16 & 195.00000 & 54.34091\
17 & 225.00000 & 54.34091\
18 & 247.50000 & 66.44354\
19 & 202.50000 & 66.44354\
20 & 225.00000 & 78.28415\
21 & 337.50000 & 66.44354\
22 & 315.00000 & 78.28415\
23 & 180.00000 & 30.00000\
\[lastpage\]
[^1]: http://skyserver.sdss.org/casjobs/
|
---
abstract: 'We have systematically studied the nematic susceptibility in non-superconducting Ba(Fe$_{1-x}$TM$_{x}$)$_2$As$_2$ (TM = Cr, Mn, V and Cu) by measuring the uniaxial pressure dependence of the resistivity along the Fe-As-Fe direction. The nematic susceptibilities in all samples show the Curie-Weiss-like behavior at high temperatures, where the nematic Curie constant $A_n$ can be derived, similar to the Curie constant in a paramagnetism. While all these dopants do not introduce superconductivity in BaFe$_2$As$_2$, their effects on nematic fluctuations are different. In Mn, Cr and V doped samples, $|A_n|$ decreases significantly with the increasing doping level. On the other hand, $|A_n|$ increases dramatically with Cu doping, similar to the superconducting Ni-doped BaFe$_2$As$_2$. However, the nematic susceptibility is suppressed at low temperatures for $x$ larger than $0.04$, which may be related to the short-range antiferromagnetic order that survives up to very high doping level. Doping Mn, Cr and Cu into the optimally-doped superconducting BaFe$_2$(As$_{0.69}$P$_{0.31}$)$_2$ also strongly reduces $|A_n|$. Compared with those systems that clearly exhibit superconductivity, such as Ni, K or P doped samples, our results suggest a strong connection between the nematic and spin degrees of freedom. Moreover, the reason of the suppression of superconductivity by dopants such as Cr, Mn, V and Cu may be correlated with the suppression of nematic fluctuations.'
author:
- Yanhong Gu
- Yuan Wei
- Dongliang Gong
- Wenliang Zhang
- Wenshan Hong
- Xiaoyan Ma
- Xingguang Li
- Congkuan Tian
- Peng Cheng
- Hongxia Zhang
- Wei Bao
- Guochu Deng
- Xin Li
- Jianming Song
- 'Yi-feng Yang'
- Huiqian Luo
- Shiliang Li
title: 'The effect of non-superconducting dopants (Mn, V, Cr and Cu) on the nematic fluctuations in iron-based superconductors'
---
introduction
============
Iron-based superconductors have attracted many interests due to their high-temperature superconductivity, but the underlying mechanism is still unclear [@SiQ16; @DaiP15; @ScalapinoDJ12]. Compared to cuprates and many other unconventional superconductors, a particular interesting fact is that iron-based superconductivity can be achieved by various dopants at different sites [@HosonoH15]. For example, BaFe$_2$As$_2$ (Ba-122) shows both the antiferromagnetic (AFM) and tetragonal-to-orthorhombic structural transitions with a collinear or stripe magnetic structure [@HuangQ08]. Taking it as the parent compound, superconductivity can be achieved by substituting Fe by Ni, Co, Ru and Rh, or Ba by K and Na, or As by P, and so on [@SefatAS08; @LiLJ09; @NiN09; @SharmaS10; @RotterM08; @CortesGilR10; @JiangS09; @LuoH12]. However, one cannot always obtain superconductivity by the substitution, as shown in Mn, Cr, V and Cu doped systems [@NiN10; @ThalerA11; @LiX18]. In both Mn and Cr doped Ba-122, the stripe AFM structure changes to G-type structure with the competition between stripe-type and G-type spin fluctuations [@KimMG10; @MartyK11; @TuckerGS12; @InosovDS13], most likely because the magnetic structures in both BaMn$_2$As$_2$ and BaCr$_2$As$_2$ are G-type [@YogeshS09; @FilsingerKA17]. In V and Cu doped Ba-122, the AFM orders are gradually suppressed to short-range or spin-glass-like state without the appearance of superconductivity [@KimMG12; @KimMG15; @TakedaH14; @WangW17; @LiX18]. It is not only that doping these dopants cannot lead to superconductivity but also that they can quickly suppress superconductivity in superconducting samples [@NiN09; @ChengP10; @LiJ12; @ZhangR14; @ZhangW19]. It has been suggested that the above results are due to their local effects on the electronic and magnetic properties [@ChengP10; @TuckerGS12; @OnariS09; @TakedaH14; @TexierY12; @SuzukiH13; @IdetaS13; @WangX14; @UrataT15; @KobayashiT16; @GastiasoroMN14]. Since these explanations depend on the detailed effects of particular dopants, there is a lack of comprehensive understanding on the physics that is directly associated with superconductivity. This is not surprising because we have not found a general picture for the superconductivity itself yet.
Dopants affect not only antiferromagnetism and superconductivity, but also nematicity. The nematic order in iron-based superconductors breaks the rotational symmetry of the tetragonal lattice and is believed to result in the tetragonal-to-orthorhombic structural transition at $T_s$ [@FernandesRM14]. While its mechanism is still under debates [@FernandesRM14], the nematic order and its fluctuations can be observed by the anisotropic properties revealed by many different techniques [@ChuJH10; @ChuJH12; @YiM11; @NakajimaM11; @LuX14; @BohmerAE14; @ZhangW16; @LiuZ16]. There are increasing evidences both theoretically and experimentally that nematic fluctuations may be important for superconductivity [@FernandesRM13; @LedererS15; @KleinA18; @kushnirenkoYS18; @KuoHH16; @GuY17]. We have shown in a previous study that the enhancement of nematic fluctuations seems to be associated with both the suppression of the AFM ordered moment and the appearance of superconductivity [@GuY17]. These observations are made in superconducting systems, here we further study the correlation between nematic fluctuations and the AFM order in non-superconducting materials and hope to gain further insights into the role of nematic fluctuations.
The materials studied in this work are Mn, Cr, V and Cu doped Ba-122. The high-temperature nematic susceptibilities can be all fitted by the Curie-Weiss-like function as described before [@LiuZ16; @GuY17], where a nematic Curie constant $A_n$ can be obtained. The value of $A_n$ is quickly suppressed in Mn, Cr and V doped samples. In Ba(Fe$_{1-x}$Cu$_x$)$_2$As$_2$, although the mean-field nematic transition temperature $T'$ becomes zero around 0.045, no nematic quantum critical point (QCP) exists since the nematic fluctuations at low temperatures are significantly suppressed. This is consistent with the fact that the AFM order becomes short-range above $x$ = 0.04 and persists up to $x$ = 0.08 as shown by the neutron diffraction measurements. Moreover, doping Mn, Cr and Cu into optimally doped BaFe$_2$(As$_{0.69}$P$_{0.31}$)$_2$ suppresses both superconductivity and nematic fluctuations. These results suggest the nematic and spin degrees of freedom are strongly coupled. Moreover, despite of different macroscopic effects of these dopants on the electronic and magnetic properties, the suppression of nematic fluctuations may be correlated with the disappearance of superconductivity.
experiments
===========
Single crystals of Ba(Fe$_{1-x}$$TM$$_x$)$_2$As$_2$ ($TM$ = Mn, Cr, V and Cu) and Ba(Fe$_{1-x}$$TM$$_{x}$)$_2$(As$_{0.69}$P$_{0.31}$)$_2$ ($TM$ = Mn and Cu) were grown by flux-method as reported elsewhere[@ChenY11; @ZhangW19]. For the sake of simplicity, we will label them as $TM$-Ba122 and $TM$-BFAP, respectively. The doping levels of $TM$-Ba122 are actual doping levels determined from the inductively coupled plasma technique. The doping levels of $TM$-BFAP are nominal. The nematic susceptibility is obtained by measuring the resistance change under uniaxial pressure, which was carried out on a physical property measurement system (PPMS, Quantum Design). The uniaxial pressure is applied by a piezoelectric device as discussed previously [@LiuZ16]. The sign of the pressure is consistent with the experiments in hydrostaic pressure, i.e., positive and negative pressures correspond to compress and tensile a sample, respectively. The samples were cut along the tetragonal (110) direction by a diamond wire saw and then glued on the uniaxial pressure device. Neutron diffraction experiments were carried out at Kunpeng triple-axis spectrometer (TAS) at Key Laboratory of Neutron Physics and Institute of Nuclear Physics and Chemistry, Mianyang, China, SIKA TAS at Open Pool Australian Lightwater reactor (OPAL) [@sika], Australia and Xingzhi TAS at China Advanced Research Reactor (CARR), Beijing, China, with the final energy E$_f$ = 5 meV. Cooled Be filters were used after the sample. The samples were aligned in the \[H,H,L\] scattering plane in the tetragonal notation.
results
=======
$TM$-Ba122 ( $TM$ = Mn, Cr and V )
----------------------------------
![(a) Temperature dependence of the normalized resistivity $R_N$ = $R/R_{300K}$ in Mn-Ba122. (b) Temperature dependence of $\zeta_{(110)}$ in Mn-Ba122. The inset shows the resistivity change under pressure for the $x$ = 0.05 sample at several temperatures. (c) Temperature dependence of $R_N$ in Cr-Ba122. (d) Temperature dependence of $\zeta_{(110)}$ in Cr-Ba122. (e) Temperature dependence of $R_N$ in V-Ba122. (f) Temperature dependence of $\zeta_{(110)}$ in V-Ba122. The arrows in (b), (d) and (f) indicate $T_N$ obtained from the resistivity measurements that are consistent with previous reports [@KimMG10; @MartyK11; @LiX18]. []{data-label="fig1"}](fig1){width="\columnwidth"}
We first provides results of Mn-, Cr- and V-Ba122. One of the common features of the magnetic properties in these systems is that the G-type AFM order strongly competes with the stripe AFM order [@KimMG10; @MartyK11; @TuckerGS12; @InosovDS13; @LiX18]. It is thus interesting to see whether their nematic fluctuations also share some similarities.
Figure 1(a) shows the temperature dependence of the normalized resistivity $R_N$ for Mn-Ba122. For the $x$ = 0.05 sample, $R_N$ exhibits a sharp upturn at $T_N$ with decreasing temperature, which is a typical behavior for samples with the substitution of Fe by 3d transition-metal elements [@NiN09]. It should be noted that the structural transition temperature $T_s$ is usually the same or slightly above $T_N$ in most of the samples studied here, so we will not distinguish them unless ortherwise specified. This upturn disappears in the $x$ = 0.14 sample although $T_N$ changes little, which is consistent with the suppression of the stripe AFM order [@KimMG10; @InosovDS13]. A kink feature is found in the $x$ = 0.28 sample, and we attribute it to the AFM transition [@KimMG10; @TuckerGS12; @InosovDS13; @FilsingerKA17].
To study the nematic susceptibility, we have measured the resistance change under the uniaxial pressure to obtain $\zeta = d(\Delta R / R _0) / dP $, where $P$ and $R_0$ are the pressure and the resistance at zero pressure, respectively, and $\Delta R = R(P)-R_0$. The subscript means that the pressure is along the tetragonal (110) direction. As discussed previously [@LiuZ16; @GuY17], $\zeta_{(110)}$ can be treated as the nematic susceptibility, which is analogous to the magnetic susceptibility in a paramagnetism. The inset of Fig. 1(b) gives some examples of the pressure dependence of $\Delta R / R _0$ for the $x$ = 0.05 Mn-Ba122 sample. Well above $T_N$, the resistance changes linearly with pressure, so the slope is used to calculate $\zeta_{(110)}$. When the temperature is close to $T_N$, nonlinear pressure dependence of resistance appears, which is due to the effect of large pressure [@MaoH18]. Since this nonlinear effect is weak, one can still roughly make a linear fit to the data. Below $T_N$, clear hysteresis behavior appears because of magnetic or nematic domains [@GongD17]. In this case, $\zeta_{(110)}$ is not well defined anymore and the linear fit is forced to obtain its value, which will not affect our analyses that focus on the nematic susceptibility above $T_N$.
Figure 1(b) shows the temperature dependence of $\zeta_{(110)}$ for Mn-Ba122. For the $x$ = 0.05 and 0.14 samples, $\zeta_{(110)}$ above $T_N$ can be fitted by the Curie-Weiss-like function, $A/(T-T')+y_0$, where $A$, $T'$ and $y_0$ are temperature-independent parameters [@LiuZ16; @GuY17]. The fittings are good until the temperature is below $T_N$, which is not just due to the ill-defined $\zeta_{(110)}$ below $T_N$ but also because the nematic susceptibility cannot go infinite with decreasing temperature and should change below the phase transition as the magnetic susceptibility in a typical magnetically ordered system. For the $x$ = 0.28 sample, where the magnetic structure would have changed, $\zeta_{(110)}$ starts deviating from the Curie-Weiss-like behavior well above $T_N$.
Figure 1(c) shows the temperature dependence of $R_N$ for Cr-Ba122. Sharp upturns are found in the $x$ = 0.07 and 0.22 samples where the AFM order is still stripe-type. For the $x$ = 0.3 sample which is near the crossover from the stripe to G-type AFM order, the upturn of $R_N$ below $T_N$ becomes smooth. Figure 1(d) shows the temperature dependence of $\zeta_{(110)}$ in these samples. The nematic susceptibility in the $x$ = 0.07 behaves similar to that in the $x$ = 0.05 and 0.14 Mn-Ba122 samples. $\zeta_{(110)}$ becomes negative in the $x$ = 0.22 and 0.3 samples but can be still fitted by the Curie-Weiss-like function with negative value of $A$. This sign change has also been found in the hole-doped Ba$_{1-x}$K$_x$Fe$_2$As$_2$ system [@GuY17].
Figure 1(e) and 1(f) show the results of V-Ba122. The major features are similar to those observed in Mn- and Cr-Ba122, including the sharp upturn of $R_N$ associated with $T_N$ and the Curie-Weiss-like behavior of $\zeta_{(110)}$ above $T_N$. We will further analysis the nematic susceptibility in these materials later and here we would like to point it out that there is an intimate relationship between the magnetic and nematic systems.
Ba(Fe$_{1-x}$Cu$_{x}$)$_2$As$_2$
--------------------------------
![(a) Q-scans along the \[H,H,3\] direction for Ba(Fe$_{1-x}$Cu$_x$)$_2$As$_2$ at 2 K. (b) Q-scans along the \[0.5,0.5,L\] direction for Ba(Fe$_{1-x}$Cu$_x$)$_2$As$_2$ at 2 K. The solid lines in (a) and (b) are fitted by the Guassian function. (c) Doping dependence of the magnetic correlation length along HH and L directions. (d) Doping dependence of the peak intensity at (0.5, 0.5, 3). (e) & (f) Temperature dependence of the peak intensity at (0.5, 0.5, 3). The labels in (a), (b), (e) and (f) represent the doping levels and the scale factors of the intensities. []{data-label="fig2"}](fig2){width="\columnwidth"}
The Cu doping is unique in the Ba-122 system in that the AFM order is continuously suppressed but no superconductivity is found [@KimMG12; @TakedaH14]. Moreover, it seems that the AFM and structural transitions are separated and may disappear at different doping levels. In other words, we may expect to observe the magnetic and nematic QCPs individually without the presence of superconductivity. However, there are some discrepancies in phase diagram [@KimMG12; @TakedaH14]. Therefore, we provide our elastic neutron scattering studies on the AFM order here, which will help us to understand the nematic fluctuations in this system.
Figure 2(a) and 2(b) shows the HH-scans and L-scans at the magnetic peak (0.5, 0.5, 3) at 2 K for Cu-Ba122. For $x \leq $ 0.04, the magnetic peak is very sharp and the width is essentially resolution-limited. With further increasing $x$, the peak widths become much broader. This is because the AFM order becomes short-ranged as reported previously [@KimMG12]. To quantitatively describe the change from long-range to short-range AFM order, the doping dependence of the magnetic correlation lengths is shown in Fig. 2(c). Here the correlation $\xi$ is calculated as $\xi$ = $2\pi/FWHM$, where FWHM is the full width at half maximum obtained by the Guassian fit to the peak. For $x \leq$ 0.04, the correlation lengths along both directions are larger than 200 Å. Since the resolution effect has not been considered, these large values of $\xi$ suggest that the AFM order is still long-range. Above $x$ = 0.04, $\xi$ quickly drops in both directions, indicating the magnetic system becomes short-range. The change from long-range to short-range AFM order is also evidenced by the doping dependence of the peak intensity as shown in Fig. 2(d), which also drastically decreases above $x$ = 0.04. Figure 2(e) and 2(f) shows the temperature dependence of the magnetic peak intensity at (0.5, 0.5, 3), which are used to determine $T_N$. We will compare the results of $T_N$ with the mean-field nematic transition temperature $T'$ later.
![(a) Temperature dependence of $R_N$ for Cu-Ba122. (b) Temperature dependence of $dR_N/dT$ for Cu-Ba122. (c) Temperature dependence of $\zeta_{(110)}$ for Cu-Ba122. The solid lines are the fitted results by the Curie-Weiss-like function. The arrows in (b) and (c) indicate $T_N$. (d) Doping dependence of $T_N$ and $T'$. The labels LR, SR and SG represent long-range, short-range and spin-glass, respectively. The dashed lines are guides to the eye.[]{data-label="fig3"}](fig3){width="\columnwidth"}
Figure 3(a) shows the temperature dependence of $R_N$ for Cu-Ba122, where clear sharp upturns can be found for $x <$ 0.04. This is consistent with the above observation that the AFM order is long-range in these samples. Above $x$ = 0.04, the upturn in $R_N$ becomes smooth, which makes it hard to judge whether there is an AFM transition or not. The temperature dependence of $dR_N/dT$ shown in Fig. 3(b) also makes it clear that only the long-range AFM order can result in a dip at $T_N$. Since we have already obtained $T_N$ for $x >$ 0.04 from the neutron diffraction experiment, it seems that the short-range AFM order can still give rise to a kink feature in $dR_N/dT$.
Figure 3(c) shows the temperature dependence of $\zeta_{(110)}$ for Cu-Ba122. For $x \leq$ 0.048, the high-temperature data can be fitted by the Curie-Weiss-like function. With further increasing doping, $\zeta_{(110)}$ shows a broad hump, similar to that in overdoped BaFe$_{2-x}$Ni$_x$As$_2$ [@LiuZ16]. From the fittings, we can obtain the mean-field nematic transition temperature $T'$, whose doping dependence is plotted together with $T_N$ in Fig. 3(d). The short-range AFM order survives up to 0.08. At doping level higher than 0.14, the system changes into a spin-glass-like state as reported previously [@WangW17]. For the nematic order, $T'$ continuously decreases with increasing $x$ and becomes zero at about 0.046.
$TM$-BFAP ( $TM$ = Mn, Cr and Cu ) and summarized phase diagrams
----------------------------------------------------------------
![(a) Temperature dependence of $R_N$ for Mn-BFAP. (b) Temperature dependence of $\zeta_{(110)}$ for Mn-BFAP. (c) Temperature dependence of $R_N$ for Cu-BFAP. (b) Temperature dependence of $\zeta_{(110)}$ for Cu-BFAP. The arrows in (b) and (d) indicate assumed $T_N$ from the resistivity.[]{data-label="fig4"}](fig4){width="\columnwidth"}
In previous subsections, the effects of nonsuperconducting dopants on the nematic susceptibility in BaFe$_2$As$_2$ have been shown. In this subsection, we further show how the nematic susceptibility in optimally doped BaFe$_2$(As$_{0.69}$P$_{0.31}$)$_2$ (BFAP) changes with these dopants when the superconductivity is completely suppressed. The BFAP is chosen here as a starting material because $P$ and the $TM$ elements sit at different sites in the unit cell and their effects on superconductivity, magnetism and nematicity may be relatively easier to be separated.
While the AFM order is nearly completely suppressed in BaFe$_2$(As$_{0.69}$P$_{0.31}$)$_2$ [@HuD15], upturns appear in the temperature dependence of $R_N$ for nonsuperconducting Mn-BFAP, as shown in Fig. 4(a). The uptrun temperature decreases with increasing Mn doping level. As shown in Cr-BFAP [@ZhangW19], the phosphorus doping level where the AFM order disappears changes with Cr doping. Therefore, it is assumed that the upturns are associated with the AFM transitions. This assumption is also evidenced by the temperature dependence of $\zeta_{(110)}$, which shows kinks at the same temperatures as shown in Fig. 4(b). In both the $x$ = 0.1 and 0.2 samples, $\zeta_{(110)}$ above $T_N$ can be well fitted by the Curie-Weiss-like function, but the signs are opposite.
Figure 4(c) and 4(d) shows the temperature dependence of $R_N$ and $\zeta_{(110)}$ for Cu-BFAP, respectively. An upturn appears in $R_N$ for the $x$ = 0.1 sample, suggesting the presence of the AFM order, and disappears for the $x$ = 0.2 sample. Correspondingly, $\zeta_{(110)}$ in the $x$ = 0.1 sample can be fitted by the Curie-Weiss-like function above $T_N$ and shows a kink at it, while that in the $x$ = 0.2 sample shows a broad hump. Both signs of $\zeta_{(110)}$ are negative. For Cr-BFAP ($x$ = 0.03), the nematic susceptibility has already been studied previously [@ZhangW19].
Figure 5(a) shows the doping dependence of $|A_n|^{-1}$, where $A_n$ = $\kappa A$ with $A$ as the nematic Curie constant from the Curie-Weiss-like fit of $\zeta_{(110)}$. The $\kappa$ is a phenomenological parameter associated with the effect of Fermi velocities [@ZhangW19]. For the system studied here, we have assumed that $\kappa$ = 1. In other words, the changes of Fermi velocities with doping are supposed to be small. This assumption will not affect the main results here. As discussed previously, the magnitude of $|A_n|$ corresponds to the strength of nematic fluctuations [@ZhangW19]. It has been shown in a previous study that $|A_n|^{-1}$ decreases with increasing doping level in Ni-Ba122 [@LiuZ16]. Similar behavior is found in Cu-Ba122. For Mn-, V- and Cr-Ba122, $|A_n|^{-1}$ increases with increasing doping level. In other words, the value of $|A_n|$ becomes much smaller than that in BaFe$_2$As$_2$, suggesting the suppression of nematic fluctuations. While the effect of Cu on $|A_n|$ is different from those of other nonsuperconducting dopants, their effects are similar in BFAP, i.e., the value of $|A_n|^{-1}$ increases significantly with all kinds of non-superconducting dopants.
Figure 5(b) plot the relationship between the AFM ordered moment $M$ and $|A_n|^{-1}$ of these nonsuperconducting systems together with the other systems that have been reported previously [@GuY17]. The data for Cu-Ba122 still falls onto the same linear relationship as shown by the dashed brown line. For Mn-, Cr- and V-Ba122, a new relationship is shown by the red dashed line in Fig. 5(b), which shows that $|A_n|^{-1}$ increases with decreasing $M$. Finally, starting from the optimally doped BFAP, $|A_n|^{-1}$ also increases with Mn, Cr and Cu doping as shown by the blue dashed line in Fig. 5(b).
discussions
===========
![(a) Doping dependence of $|A_n|^{-1}$ for $TM$-Ba122 and $TM$-BFAP ( TM = Mn, Cr, V and Cu ). The data for Ba(Fe$_{1-x}$Ni$_{x}$)$_2$As$_2$ and Cr-BFAP are from Ref. [@LiuZ16] and [@ZhangW19], respectively. (b) The relationship between the AFM ordered moment $M$ and $|A_n|^{-1}$. The compounds listed in the right and the dashed straight brown line are from [@GuY17]. The values of $M$ for Cu-Ba122 are determined by comparing the intensities between the magnetic and nuclear peaks [@LuoH12]. The values of $M$ for Mn, Cr and V doped Ba-122 are estimated from Ref. [@KimMG10], [@MartyK11] and [@LiX18], respectively. Both $|A_n|^{-1}$ and $M$ of Cr-BFAP are from Ref. [@ZhangW19]. Since no measurements on the values of $M$ in Mn and Cr doped BFAP have been done yet, they are set as zero. The red and blue dashed lines are guides to the eye. The arrows in the red and blue dashed lines indicate increasing doping in $TM$-Ba122 ( Mn, Cr and V ) and $TM$-BFAP ( Mn, Cr and Cu ), respectively. []{data-label="fig5"}](fig5)
We start the discussions from Mn, Cr and V dopants. In both Mn- and Cr-Ba122, the suppression of the stripe AFM order is caused by the competition from the G-type AFM order [@KimMG10; @MartyK11; @TuckerGS12; @InosovDS13; @YogeshS09; @FilsingerKA17]. This kind of competition is also shown to exist in V-Ba122, where a spin-glass state appears first after the stripe AFM order is completely suppressed above $x$ = 0.25 [@LiX18]. Our results in Fig. 5(a) show that Mn, Cr and V also have the similar effects on the nematic susceptibility, i.e., they all result in the increase of $|A_n|^{-1}$. Moreover, the rate of this increase is largest for Cr doping, which accords with that the suppression of $T_N$ of the stripe AFM order is also fastest in Cr-Ba122. Since the G-type AFM order does not break the $C_4$ rotational symmetry, it is reasonable to conclude that the suppression of nematic fluctuations in these systems is due to the competition from the G-type AFM spin fluctuations. It is thus not surprising that in both Mn- and Cr-BFAP, $|A_n|^{-1}$ also increases significantly with increasing doping levels as shown in Fig. 5(a).
The Cu doping has different effects on the magnetic system in BaFe$_2$As$_2$ from the above non-superconducting dopants. First, the stripe AFM order is suppressed but is not replaced by the G-type order. In heavily overdoped Cu-Ba122 ( $x \geq$ 0.145 ), the magnetic system is spin-glass-like and the AFM order is short-range and stripe-type [@WangW17]. At first glance, this is the same as what we observed here for the samples with 0.044 $\leq x \leq$ 0.08. However, $T_N$ decreases and increases with increasing doping for $x \leq$ 0.08 and $x \geq$ 0.145, respectively. The difference of magnetic orders in these two doping regions can also be found from the doping dependence of the integrated magnetic intensity, which increases with doping for $x \geq$ 0.145 but decreases for $x \leq$ 0.08. It has been suggested that the certain arrangements of Fe and Cu may favor a magnetically ordered state [@WangW17], as also seen in heavily overdoped NaFe$_{1-x}$Cu$_x$As [@SongY16]. In our case, the content of Cu may be too little to introduce any kind of particular Fe-Cu arrangement. Therefore, the short-range AFM order below $x$ = 0.08 should be the result of randomly distributed Cu dopants, which may locally disturb the long-range stripe AFM order and make it short-range. Another possibility is that the electron doping from Cu dopants may result in short-range AFM order. We note that the short-range AFM order has also been found in the electron-doped systems of Co- and Ni-Ba122 [@LuoH12; @LuX14; @PrattDK11], although it is incommensurate.
The effects of Cu doping on nematic fluctuations in Cu-Ba122 can also be understood by the local effects of Cu dopants. At one hand, the Cu doping results in the increase of $A_n$ ( Fig. 5(a) ), suggesting the enhancement of nematic fluctuations. On the other hand, the nematic susceptibility does not follow the Curie-Weiss-like behavior for samples with $T'$ close to zero at low temperatures ( Fig. 3(c) ), which suggests that the nematic fluctuations are significantly suppressed by Cu dopants. This seemingly contradictory results may be explained as two different effects of Cu dopants at different temperature regions. First, the nematic fluctuations will be enhanced if the magnetic order is suppressed with the stripe-type AF fluctuations maintained, as shown by the very similar doping dependence of $A_n$ and $T'$ in Cu- and Ni-Ba122 [@LiuZ16; @GuY17]. This effect may be treated as a global effect of dopants to the nematic system. Second, the strong local effects of Cu dopants may prevent the nematic susceptibility from further increasing at low temperatures. Since the nematic order is directly coupled to the lattice, it is not surprising that $\zeta_{(110)}$ cannot be infinite when $T'$ is close to zero because it would mean that the lattice will become unstable. Naively, it may suggest that Cu dopants may limit the nematic correlations and keep the nematic system from long-range ordering. Further high-resolution structural studies may give an answer to this hypothesis. It is interesting to note that many superconducting systems may have an avoided nematic QCP near optimally doping levels where $T'$ becomes zero [@LiuZ16; @KuoHH16; @GuY17]. Similarly, an avoided nematic QCP may also present in Cu-Ba122, i.e., quantum nematic fluctuations could dominate at high temperatures when $T'$ = 0 but there is no actual nematic QCP.
Our results suggest a close relationship between the stripe AFM order and the nematicity. It has been shown that the ordered moment has a roughly linear relationship with $|A_n|^{-1}$ for many parent compounds and doped superconducting samples, as shown in Fig. 5(b). It suggests that the nematic fluctuations are enhanced when the stripe AFM order is suppressed. For Cu-Ba122 which does not induce any other type of magnetic order competing with the stripe AFM order, this relationship still holds. However, for Mn-, Cr- and V-Ba122, $|A_n|^{-1}$ increases drastically with the suppression of the stripe AFM order as shown by the red dashed line in Fig. 5(b), which clearly suggests that the competition from G-type AFM order suppresses the nematic fluctuations. This results, together with the linear relationship between $|A_n|^{-1}$ and $M$ in other systems, indicate that the stripe AFM order is directly associated with nematic fluctuations.
Our results also suggest a close relationship between the nematic fluctuations and superconductivity. We have suggested in a previous work that superconductivity appears only when $|A_n|^{-1}$ is small enough [@GuY17]. This is consistent with the results in this work that $|A_n|^{-1}$ is drastically increased with non-superconducting dopants doping into the superconducting samples. Moreover, whether a dopant can lead to superconductivity in BaFe$_2$As$_2$ seems associated with whether it can enhance nematic fluctuations. A particular case is Cu-doped Ba-122, where Cu doping only suppress low-temperature nematic fluctuations. Of course, it is not to say that the superconductivity must come from nematic fluctuations since one mechanism may affect both superconductivity and nematicity simultaneously. Still, our results put nematicity as one of the key aspects to understand superconductivity in iron-based superconductors.
conclusions
===========
We have systematically studied the doping effects of Mn, Cr, V and Cu on the nematic susceptibility in parent compound Ba-122 and optimally-doped superconducting BFAP. The nematic Curie constant $|A_n|$ are drastically decreased in all cases except in Cu-Ba122. In the latter, nematic fluctuations are only suppressed at low temperatures. Combining our previous studies, our results suggest that the stripe AFM order and superconductivity may both have intimate relationship with nematic fluctuations. In other words, nematicity may play one of the key roles in the low-energy physics of iron-based superconductors.
This work was supported by the National Key R&D Program of China (Grants No. 2017YFA0302900 and No. 2016YFA0300502,2016YFA0300604), the National Natural Science Foundation of China (Grants No. 11874401 and No. 11674406, No. 11374346, No. 11774399, No. 11474330, No. 11421092, No. 11574359 and No. 11674370), the Strategic Priority Research Program(B) of the Chinese Academy of Sciences (Grants No. XDB25000000 and No. XDB07020000, No. XDB28000000), China Academy of Engineering Physics (Grant No. 2015AB03) and the National Thousand-Young Talents Program of China. H. L. is grateful for the support from the Youth Innovation Promotion Association of CAS (2016004).
[60]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, , , ****, ().
, ****, ().
, ****, ().
, ****, ().
, , , , , , , , , ****, ().
, , , , , , ****, ().
, , , , , , , , , , , ****, ().
, , , , , , ****, ().
, , , , , , , , , ****, ().
, , , ****, ().
, , , , , ****, ().
, , , , , , , , , ****, ().
, , , , , , , , , , , ****, ().
, , , , , , , , ****, ().
, , , , , , , , , ****, ().
, , , , , , , , , , , ****, ().
, , , , , , , , , , , ****, ().
, , , , , , , , , , ****, ().
, , , , , , , , , , , ****, ().
, , , , , , , , , , , ****, ().
, , , , , , , ****, ().
, , , , , , , , , , ****, ().
, , , , , , , , , , , ****, ().
, , , , , , , , , , , ****, ().
, , , , , , , ****, ().
, , , , , , , , , ****, ().
, , , , ****, ().
, , , , , , , , , , , ****, ().
, , , , , , ****, ().
, , , , , , , , , , , ****, ().
, ****, ().
, , , , , , , , , ****, ().
, , , , , , , , , , , ****, ().
, , , , , , , , , , , ****, ().
, ****, ().
, , , , , , , ****, ().
, , , , ****, ().
, ****, ().
, , , ****, ().
, , , , , , , ****, ().
, , , , ****, ().
, , , , , , , , , , , ****, ().
, , , , , , , , , , , ****, ().
, , , , , , , ****, ().
, , , , , , , , , ****, ().
, , , , , , , , , , , ****, ().
, , , , , , , , , , , ****, ().
, ****, ().
, , , , ****, ().
, , , , , ****, ().
, , , , , , , , , ().
, , , , , ****, ().
, , , , , , , , , , , ****, ().
, , , , , ****, ().
, , , , , , , , ****, (????).
, , , , , , , ****, ().
, , , , , , , , ****, ().
, , , , , , , , , , , ****, ().
, , , , , , , , , , , ****, ().
, , , , , , , , , , , ****, ().
|
---
abstract: 'Variational inference is a scalable technique for approximate Bayesian inference. Deriving variational inference algorithms requires tedious model-specific calculations; this makes it difficult to automate. We propose an automatic variational inference algorithm, . The user only provides a Bayesian model and a dataset; nothing else. We make no conjugacy assumptions and support a broad class of models. The algorithm automatically determines an appropriate variational family and optimizes the variational objective. We implement in Stan (code available now), a probabilistic programming framework. We compare to <span style="font-variant:small-caps;">mcmc</span> sampling across hierarchical generalized linear models, nonconjugate matrix factorization, and a mixture model. We train the mixture model on a quarter million images. With we can use variational inference on any model we write in Stan.'
author:
- |
**Alp Kucukelbir**\
Data Science Institute\
Department of Computer Science\
Columbia University\
`alp@cs.columbia.edu`\
\
**Rajesh Ranganath**\
Department of Computer Science\
Princeton University\
`rajeshr@cs.princeton.edu`\
\
**Andrew Gelman**\
Data Science Institute\
Depts. of Political Science, Statistics\
Columbia University\
`gelman@stat.columbia.edu`\
\
**David M. Blei**\
Data Science Institute\
Depts. of Computer Science, Statistics\
Columbia University\
`david.blei@columbia.edu`
bibliography:
- 'advi.bib'
title: '**Automatic Variational Inference in Stan**'
---
=1
Introduction
============
Bayesian inference is a powerful framework for analyzing data. We design a model for data using latent variables; we then analyze data by calculating the posterior density of the latent variables. For machine learning models, calculating the posterior is often difficult; we resort to approximation.
approximates the posterior with a simpler density [@jordan1999introduction; @wainwright2008graphical]. We search over a family of simple densities and find the member closest to the posterior. This turns approximate inference into optimization. has had a tremendous impact on machine learning; it is typically faster than sampling (as we show here too) and has recently scaled up to massive data [@hoffman2013stochastic].
Unfortunately, algorithms are difficult to derive. We must first define the family of approximating densities, and then calculate model-specific quantities relative to that family to solve the variational optimization problem. Both steps require expert knowledge. The resulting algorithm is tied to both the model and the chosen approximation.
In this paper we develop a method for automating variational inference, . Given any model from a wide class (specifically, differentiable probability models), determines an appropriate variational family and an algorithm for optimizing the corresponding variational objective. We implement in Stan [@stan-manual:2015], a flexible probabilistic programming framework originally designed for sampling-based inference. Stan describes a high-level language to define probabilistic models (e.g., Figure \[fig:example\_poisson\]) as well as a model compiler, a library of transformations, and an efficient automatic differentiation toolbox. With we can now use variational inference on any model we can express in Stan.[^1] (See Appendices \[app:linreg\_ard\] to \[app:gmm\].)
[2.6in]{} ![ Held-out predictive accuracy results | of the image<span style="font-variant:small-caps;">clef</span> image histogram dataset. **(a)** outperforms the , the default sampling method in Stan . **(b)** scales to large datasets by subsampling minibatches of size $B$ from the dataset at each iteration [@hoffman2013stochastic]. We present more details in Section \[sub:scaling\_gmm\] and Appendix \[app:gmm\]. []{data-label="fig:gmm_plots"}](advi-figure0.pdf "fig:"){width="2.6in"}
[2.6in]{} ![ Held-out predictive accuracy results | of the image<span style="font-variant:small-caps;">clef</span> image histogram dataset. **(a)** outperforms the , the default sampling method in Stan . **(b)** scales to large datasets by subsampling minibatches of size $B$ from the dataset at each iteration [@hoffman2013stochastic]. We present more details in Section \[sub:scaling\_gmm\] and Appendix \[app:gmm\]. []{data-label="fig:gmm_plots"}](advi-figure1.pdf "fig:"){width="2.6in"}
Figure \[fig:gmm\_plots\] illustrates the advantages of our method. We present a nonconjugate Gaussian mixture model for analyzing natural images; this is 40 lines in Stan (Figure \[fig:code\_gmm\_diag\]). Section \[sub:gmm\_1000\] illustrates Bayesian inference on $1000$ images. The $y$-axis is held-out likelihood, a measure of model fitness; the $x$-axis is time (on a log scale). is orders of magnitude faster than , a state-of-the-art algorithm (and Stan’s default inference technique) [@hoffman2014nuts]. We also study nonconjugate factorization models and hierarchical generalized linear models; we consistently observe speed-up against .
Section \[sub:gmm\_adsvi\] illustrates Bayesian inference on $250\,000$ images, the size of data we more commonly find in machine learning. Here we use with stochastic variational inference [@hoffman2013stochastic], giving an approximate posterior in under two hours. For data like these, techniques cannot even practically begin analysis, a motivating case for approximate inference.
automates variational inference within the Stan probabilistic programming framework [@stan-manual:2015]. This draws on two major themes.
The first is a body of work that aims to generalize . @kingma2013auto and @rezende2014stochastic describe a reparameterization of the variational problem that simplifies optimization. @ranganath2014black and @salimans2014using propose a black-box technique that only uses the gradient of the approximating family for optimization. @titsias2014doubly leverage the gradient of the model for a small class of models. We build on and extend these ideas to automate variational inference; we highlight technical connections as we develop our method.
The second theme is probabilistic programming. @wingate2013automated study in general probabilistic programs, as supported by languages like Church [@goodman2008church], Venture [@mansinghka2014venture], and Anglican [@wood2014new]. Another probabilistic programming framework is infer.NET, which implements variational message passing [@winn2005variational], an efficient algorithm for conditionally conjugate graphical models. Stan supports a more comprehensive class of models that we describe in Section \[sub:diff\_prob\_models\].
Automatic Differentiation Variational Inference
===============================================
follows a straightforward recipe. First, we transform the space of the latent variables in our model to the real coordinate space. For example, the logarithm transforms a positively constrained variable, such as a standard deviation, to the real line. Then, we posit a Gaussian variational distribution. This induces a non-Gaussian approximation in the original variable space. Last, we combine automatic differentiation with stochastic optimization to maximize the variational objective. We begin by defining the class of models we support.
Differentiable Probability Models {#sub:diff_prob_models}
---------------------------------
Consider a dataset $\mbX = \mbx_{1:N}$ with $N$ observations. Each $\mbx_n$ is a discrete or continuous random vector. The likelihood $p (\mbX\mid\mbtheta)$ relates the observations to a set of latent random variables $\mbtheta$. Bayesian analysis posits a prior density $p(\mbtheta)$ on the latent variables. Combining the likelihood with the prior gives the joint density $p(\mbX,\mbtheta) = p(\mbX\mid\mbtheta)\,p(\mbtheta)$.
We focus on approximate inference for differentiable probability models. These models have continuous latent variables $\mbtheta$. They also have a gradient of the log-joint with respect to the latent variables $\nabla_\mbtheta
\log p(\mbX,\mbtheta)$. The gradient is valid within the support of the prior $ \supp(p(\mbtheta)) = \big\{\, \mbtheta \mid \mbtheta \in \bbR^K
\text{ and } p(\mbtheta) > 0 \,\big\} \subseteq \bbR^{K}$, where $K$ is the dimension of the latent variable space. This support set is important: it determines the support of the posterior density and will play an important role later in the paper. Note that we make no assumptions about conjugacy, either full[^2] or conditional.[^3]
Consider a model that contains a Poisson likelihood with unknown rate, $p(x\mid\lambda)$. The observed variable $x$ is discrete; the latent rate $\lambda$ is continuous and positive. Place an exponential prior for $\lambda$, defined over the positive real numbers. The resulting joint density describes a nonconjugate differentiable probability model. (See Figure \[fig:example\_poisson\].) Its partial derivative $\partial/\partial\lambda \, p (x,\lambda)$ is valid within the support of the exponential distribution, $\supp(p(\lambda)) = \bbR^+ \subset \bbR$. Since this model is nonconjugate, the posterior is not an exponential distribution. This presents a challenge for classical variational inference. We will see how handles this model later in the paper.
Many machine learning models are differentiable probability models. Linear and logistic regression, matrix factorization with continuous or discrete measurements, linear dynamical systems, and Gaussian processes are prime examples. In machine learning, we usually describe mixture models, hidden Markov models, and topic models with discrete random variables. Marginalizing out the discrete variables reveals that these are also differentiable probability models. (We show an example in Section \[sub:scaling\_gmm\].) Only fully discrete models, such as the Ising model, fall outside of this category.
Variational Inference {#sub:variational_inference}
---------------------
In Bayesian inference, we seek the posterior density $p (\mbtheta\mid\mbX)$, which describes how the latent variables vary, conditioned on a set of observations $\mbX$. Many posterior densities are intractable because they lack analytic (closed-form) solutions. Thus, we seek to approximate the posterior.
Consider an approximating density $q (\mbtheta\,;\,\mbphi)$ parameterized by $\mbphi$. We make no assumptions about its shape or support. We want to find the parameters of $q(\mbtheta\,;\,\mbphi)$ to best match the posterior according to some loss function. minimizes the divergence, $$\begin{aligned}
\min_\mbphi \KL{q (\mbtheta\,;\,\mbphi)}{p(\mbtheta\mid\mbX)},
\label{eq:min_KL}\end{aligned}$$ from the approximation to the posterior [@wainwright2008graphical]. Typically the divergence also lacks an analytic form. Instead we maximize a proxy to the divergence, the $$\begin{aligned}
\cL (\mbphi)
&=
\E_{q (\mbtheta)} \big[ \log p (\mbX,\mbtheta) \big]
-
\E_{q (\mbtheta)} \big[ \log q (\mbtheta\,;\,\mbphi) \big].\end{aligned}$$ The first term is an expectation of the joint density under the approximation, and the second is the entropy of the variational density. Maximizing the minimizes the divergence [@jordan1999introduction; @bishop2006pattern].
The minimization problem from Equation \[eq:min\_KL\] becomes $$\begin{aligned}
\mbphi^*
&=
\argmax_\mbphi \cL(\mbphi)
\quad\text{such that}\quad
\supp(q(\mbtheta\,;\,\mbphi)) \subseteq \supp(p(\mbtheta \mid \mbX)),
\label{eq:max_elbo}\end{aligned}$$ where we explicitly specify the support matching constraint implied in the divergence.[^4] We highlight this constraint, as we do not specify the form of the variational approximation; thus we must ensure that $q(\mbtheta\,;\,\mbphi)$ stays within the support of the posterior, which is equal to the support of the prior.
In classical variational inference, we typically design a conditionally conjugate model; the optimal approximating family matches the prior, which satisfies the support constraint by definition [@bishop2006pattern]. In other models, we carefully study the model and design custom approximations. These depend on the model and on the choice of the approximating density.
One way to automate is to use black-box variational inference [@ranganath2014black; @salimans2014using]. If we select a density whose support matches the posterior, then we can directly maximize the using integration and stochastic optimization. Another strategy is to restrict the class of models and use a fixed variational approximation [@titsias2014doubly]. For instance, we may use a Gaussian density for inference in unrestrained differentiable probability models, i.e. where $\supp (p (\mbtheta)) = \bbR^K$.
We adopt a transformation-based approach. First, we automatically transform the support of the latent variables in our model to the real coordinate space. Then, we posit a Gaussian variational density. The inverse of our transform induces a non-Gaussian variational approximation in the original variable space. The transformation guarantees that the non-Gaussian approximation stays within the support of the posterior. Here is how it works.
Automatic Transformation of Constrained Variables
-------------------------------------------------
Begin by transforming the support of the latent variables $\mbtheta$ such that they live in the real coordinate space $\bbR^K$. Define a one-to-one differentiable function $$\begin{aligned}
T &: \text{supp}(p(\mbtheta)) \rightarrow \bbR^K,
\label{eq:transformation}\end{aligned}$$ and identify the transformed variables as $\mbzeta = T(\mbtheta)$. The transformed joint density $g(\mbX,\mbzeta)$ is a function of $\mbzeta$; it has the representation $$\begin{aligned}
g(\mbX,\mbzeta)
&=
p
(\mbX,T^{-1}(\mbzeta))
\big|
\det J_{T^{-1}}(\mbzeta)
\big|,\end{aligned}$$ where $p$ is the joint density in the original latent variable space, and $J_{T^{-1}}(\mbzeta)$ is the Jacobian of the inverse of $T$. Transformations of continuous probability densities require a Jacobian; it accounts for how the transformation warps unit volumes [@olive2014statistical]. (See Appendix \[app:jacobian\].)
Consider again our running example. The rate $\lambda$ lives in $\bbR^+$. The logarithm $\zeta = T(\lambda) = \log(\lambda)$ transforms $\bbR^+$ to the real line $\bbR$. Its Jacobian adjustment is the derivative of the inverse of the logarithm, $|\det J_{T^{-1}(\zeta)}| = \exp(\zeta)$. The transformed density is $
g(x,\zeta) =
\text{Poisson}(x\mid\exp(\zeta))
\,
\text{Exponential}(\exp(\zeta))
\,
\exp(\zeta)
$. Figures \[sub:A\] and \[sub:B\] depict this transformation.
As we describe in the introduction, we implement our algorithm in Stan to enable generic inference. Stan implements a model compiler that automatically handles transformations. It works by applying a library of transformations and their corresponding Jacobians to the joint model density.[^5] This transforms the joint density of any differentiable probability model to the real coordinate space. Now, we can choose a variational distribution independent from the model.
Implicit Non-Gaussian Variational Approximation
-----------------------------------------------
After the transformation, the latent variables $\mbzeta$ have support on $\bbR^K$. We posit a diagonal (mean-field) Gaussian variational approximation $$\begin{aligned}
q(\mbzeta \,;\, \mbphi)
&=
\cN(\mbzeta \,;\, \mbmu, \mbsigma^2)
=
\prod_{k=1}^K
\cN
(\zeta_k \,;\, \mu_k, \sigma^2_k),\end{aligned}$$ where the vector $\mbphi = (\mu_{1},\cdots,\mu_{K}, \sigma^2_ {1},\cdots,\sigma^2_{K})$ concatenates the mean and variance of each Gaussian factor. This defines our variational approximation in the real coordinate space. (Figure \[sub:B\].)
The transformation $T$ from Equation \[eq:transformation\] maps the support of the latent variables to the real coordinate space. Thus, its inverse $T^{-1}$ maps back to the support of the latent variables. This implicitly defines the variational approximation in the original latent variable space as $%
% q(\mbtheta)%\,;\, \mbphi)
% =
\cN (T^{-1}(\mbzeta)\,;\, \mbmu, \mbsigma^2)
\big| \det J_{T^{-1}}(\mbzeta) \big|.
$ The transformation ensures that the support of this approximation is always bounded by that of the true posterior in the original latent variable space (Figure \[sub:A\]). Thus we can freely optimize the in the real coordinate space (Figure \[sub:B\]) without worrying about the support matching constraint.
The in the real coordinate space is $$\begin{aligned}
\cL(\mbmu,\mbsigma^2)
&=
\E_{q(\mbzeta)}
\bigg[
\log p (\mbX, T^{-1}(\mbzeta))
+
\log \big| \det J_{T^{-1}}(\mbzeta) \big|
\bigg]
+
\frac{K}{2}\left(1+\log(2\pi)\right)
+
\sum_{k=1}^K \log \sigma_k,\end{aligned}$$ where we plug in the analytic form for the Gaussian entropy. (Derivation in Appendix \[app:elbo\_unconstrained\].)
We choose a diagonal Gaussian for its efficiency and analytic entropy. This choice may call to mind the Laplace approximation technique, where a second-order Taylor expansion around the maximum-a-posteriori estimate gives a Gaussian approximation to the posterior. However, using a Gaussian variational approximation is not equivalent to the Laplace approximation [@opper2009variational]. Our approach is distinct in another way: the posterior approximation in the original latent variable space (Figure \[sub:A\]) is non-Gaussian, because of the inverse transformation $T^{-1}$ and its Jacobian.
Automatic Differentiation for Stochastic Optimization
-----------------------------------------------------
We now seek to maximize the in real coordinate space, $$\begin{aligned}
\mbmu^*,{\mbsigma^2}^*
&=
\argmax_{\mbmu,\mbsigma^2} \cL(\mbmu,\mbsigma^2)
\quad\text{such that}\quad
\mbsigma^2 \succ 0.
\label{eq:elbo_unconstrained}\end{aligned}$$ We can use gradient ascent to reach a local maximum of the . Unfortunately, we cannot apply automatic differentiation to the in this form. This is because the expectation defines an intractable integral that depends on $\mbmu$ and $\mbsigma^2$; we cannot directly represent it as a computer program. Moreover, the variance vector $\mbsigma^2$ must remain positive. Thus, we employ one final transformation: elliptical standardization[^6] [@hardle2012applied], shown in Figures \[sub:B\] and \[sub:C\].
First, re-parameterize the Gaussian distribution with the log of the standard deviation, $\mbomega = \log(\mbsigma)$, applied element-wise. The support of $\mbomega$ is now the real coordinate space and $\mbsigma$ is always positive. Then, define the standardization $\mbeta = S_{\mbmu,\mbomega}(\mbzeta) =
\diag(\exp(\mbomega^{-1})) (\mbzeta - \mbmu)$. The standardization encapsulates the variational parameters; in return it gives a fixed variational density $$\begin{aligned}
q (\mbeta \,;\,
\mb{0}, \mbI) &= \cN
(\mbeta \,;\, \mb{0}, \mbI)
=
\prod_{k=1}^K
\cN
(\eta_k \,;\, 0, 1).\end{aligned}$$ The standardization transforms the variational problem from Equation \[eq:elbo\_unconstrained\] into $$\begin{aligned}
\mbmu^*,{\mbomega}^*
&=
\argmax_{\mbmu,\mbomega} \cL (\mbmu, \mbomega)\\
&=
\argmax_{\mbmu,\mbomega}
\E_{\cN(\mbeta\,;\, \mb{0}, \mbI)}
\bigg[
\log p
(\mbX,T^{-1}(S_{\mbmu,\mbomega}^{-1}(\mbeta)))
+
\log \big| \det J_{T^{-1}}(S_{\mbmu,\mbomega}^{-1}(\mbeta)) \big|
\bigg] + \sum_{k=1}^K \omega_k,\end{aligned}$$ where we drop independent term from the calculation. The expectation is now in terms of the standard Gaussian, and both parameters $\mbmu$ and $\mbomega$ are unconstrained. (Figure \[sub:C\].) We push the gradient inside the expectations and apply the chain rule to get $$\begin{aligned}
\nabla_\mbmu \cL
&=
\E_{\cN(\mbeta)}
\left[
\nabla_\mbtheta \log p(\mbX,\mbtheta)
\nabla_{\mbzeta} T^{-1}(\mbzeta)
+
\nabla_{\mbzeta}
\log \big| \det J_{T^{-1}}(\mbzeta) \big|
\right],
\label{eq:elbo_us_grad_mu}
\\
\nabla_{\omega_k} \cL
&=
\E_{\cN(\eta_k)}
\left[
\left(
\nabla_{\theta_k} \log p(\mbX,\mbtheta)
\nabla_{\zeta_k} T^{-1}(\mbzeta)
+
\nabla_{\zeta_k}
\log \big| \det J_{T^{-1}}(\mbzeta) \big|
\right)
\eta_k
\exp(\omega_k)
\right]
+ 1.
\label{eq:elbo_us_grad_L}\end{aligned}$$ (Derivations in Appendix \[app:gradient\_elbo\].)
We can now compute the gradients inside the expectation with automatic differentiation. This leaves only the expectation. integration provides a simple approximation: draw $M$ samples from the standard Gaussian and evaluate the empirical mean of the gradients within the expectation [@robert1999monte]. This gives unbiased noisy estimates of gradients of the .
Scalable Automatic Variational Inference
----------------------------------------
Equipped with unbiased noisy gradients of the , implements stochastic gradient ascent. (Algorithm \[alg:ADVI\].) We ensure convergence by choosing a decreasing step-size schedule. In practice, we use an adaptive schedule [@duchi2011adaptive] with finite memory. (See Appendix \[app:adaGrad\] for details.)
has complexity $\mathcal{O}(2NMK)$ per iteration, where $M$ is the number of samples (typically between 1 and 10). Coordinate ascent has complexity $\mathcal{O}(2NK)$ per pass over the dataset. We scale to large datasets using stochastic optimization [@hoffman2013stochastic; @titsias2014doubly]. The adjustment to Algorithm \[alg:ADVI\] is simple: sample a minibatch of size $B \ll N$ from the dataset and scale the likelihood of the model by $N/B$ [@hoffman2013stochastic]. The stochastic extension of has a per-iteration complexity $\mathcal
{O}(2BMK)$.
Set iteration counter $i = 0$ and choose a stepsize sequence $\mbrho^{(i)}$. Initialize $\mbmu^{(0)} = \mb{0}$ and $\mbomega^{(0)} = \mb{0}$. Return $\mbmu^* \longleftarrow \mbmu^{(i)}
\text{ and }\:
\mbomega^* \longleftarrow \mbomega^{(i)}$. \[alg:ADVI\]
Empirical Study {#sec:empirical}
===============
We now study across a variety of models. We compare its speed and accuracy to two sampling algorithms: [@girolami2011riemann] and the [^7] [@hoffman2014nuts]. We assess convergence by tracking the ; assessing convergence with techniques is less straightforward. To place and on a common scale, we report predictive accuracy on held-out data as a function of time. We approximate the Bayesian posterior predictive using integration. For the techniques, we plug in posterior samples into the likelihood. For , we do the same by drawing a sample from the posterior approximation at fixed intervals during the optimization. We initialize with a draw from a standard Gaussian.
We explore two hierarchical regression models, two matrix factorization models, and a mixture model. All of these models have nonconjugate prior structures. We conclude by analyzing a dataset of $250\,000$ images, where we report results across a range of minibatch sizes $B$.
A Comparison to Sampling: Hierarchical Regression Models
--------------------------------------------------------
We begin with two nonconjugate regression models: linear regression with [@bishop2006pattern] and hierarchical logistic regression [@gelman2006data].
This is a sparse linear regression model with a hierarchical prior structure. (Details in Appendix \[app:linreg\_ard\].) We simulate a dataset with $250$ regressors such that half of the regressors have no predictive power. We use $10\,000$ training samples and hold out $1000$ for testing.
This is a hierarchical logistic regression model from political science. The prior captures dependencies, such as states and regions, in a polling dataset from the United States 1988 presidential election [@gelman2006data]. The model is nonconjugate and would require some form of approximation to derive a algorithm. (Details in Appendix \[app:logreg\].)
We train using $10\,000$ data point and withhold $1536$ for evaluation. The regressors contain age, education, and state and region indicators. The dimension of the regression problem is $145$.
Figure \[fig:regression\_models\] plots average log predictive accuracy as a function of time. For these simple models, all methods reach the same predictive accuracy. We study with two settings of $M$, the number of samples used to estimate gradients. A single sample per iteration is sufficient; it also is the fastest. (We set $M=1$ from here on.)
[2.6in]{} ![Hierarchical Generalized Linear Models.[]{data-label="fig:regression_models"}](advi-figure8.pdf "fig:"){width="2.4in"}
[2.6in]{} ![Hierarchical Generalized Linear Models.[]{data-label="fig:regression_models"}](advi-figure9.pdf "fig:"){width="2.6in"}
Exploring nonconjugate Models: Non-negative Matrix Factorization {#sub:NMF}
----------------------------------------------------------------
[2.6in]{} ![Non-negative matrix factorization of the Frey Faces dataset.[]{data-label="fig:nmf"}](advi-figure10.pdf "fig:"){width="2.6in"}
\[sub:mf\_gap\_pred\]
[2.6in]{} ![Non-negative matrix factorization of the Frey Faces dataset.[]{data-label="fig:nmf"}](advi-figure11.pdf "fig:"){width="2.6in"}
\[sub:mf\_dirichlet\_exp\_pred\]
[2.6in]{} ![Non-negative matrix factorization of the Frey Faces dataset.[]{data-label="fig:nmf"}](img/MF_gap.png "fig:"){width="1.6in"}
[2.6in]{} ![Non-negative matrix factorization of the Frey Faces dataset.[]{data-label="fig:nmf"}](img/MF_dirichlet_exp.png "fig:"){width="1.6in"}
We continue by exploring two nonconjugate non-negative matrix factorization models: a constrained Gamma Poisson model [@canny2004gap] and a Dirichlet Exponential model. Here, we show how easy it is to explore new models using . In both models, we use the Frey Face dataset, which contains $1956$ frames ($28\times20$ pixels) of facial expressions extracted from a video sequence.
This is a Gamma Poisson factorization model with an ordering constraint: each row of the Gamma matrix goes from small to large values. (Details in Appendix \[app:gap\].)
This is a nonconjugate Dirichlet Exponential factorization model with a Poisson likelihood. (Details in Appendix \[app:dir\_exp\].) Figure \[fig:nmf\] shows average log predictive accuracy as well as ten factors recovered from both models. provides an order of magnitude speed improvement over (Figure \[sub:mf\_gap\_pred\]). struggles with the Dirichlet Exponential model (Figure \[sub:mf\_dirichlet\_exp\_pred\]). In both cases, does not produce any useful samples within a budget of one hour; we omit from the plots.
The Gamma Poisson model (Figure \[sub:mf\_gap\_factors\]) appears to pick significant frames out of the dataset. The Dirichlet Exponential factors (Figure \[sub:mf\_dirichlet\_exp\_factors\]) are sparse and indicate components of the face that move, such as eyebrows, cheeks, and the mouth.
Scaling to Large Datasets: Gaussian Mixture Model {#sub:scaling_gmm}
-------------------------------------------------
We conclude with the example we highlighted earlier. This is a nonconjugate applied to color image histograms. We place a Dirichlet prior on the mixture proportions, a Gaussian prior on the component means, and a lognormal prior on the standard deviations. (Details in Appendix \[app:gmm\].) We explore the image<span style="font-variant:small-caps;">clef</span> dataset, which has $250\,000$ images [@villegas13_CLEF]. We withhold $10\,000$ images for evaluation.
In Figure \[sub:gmm\_1000\] we randomly select $1000$ images and train a model with $10$ mixture components. struggles to find an adequate solution and fails altogether. This is likely due to label switching, which can affect -based techniques in mixture models [@stan-manual:2015].
Figure \[sub:gmm\_adsvi\] shows results on the full dataset. Here we use with stochastic subsampling of minibatches from the dataset [@hoffman2013stochastic]. We increase the number of mixture components to $30$. With a minibatch size of $500$ or larger, reaches high predictive accuracy. Smaller minibatch sizes lead to suboptimal solutions, an effect also observed in [@hoffman2013stochastic]. converges in about two hours.
Conclusion
==========
We develop in Stan. leverages automatic transformations, an implicit non-Gaussian variational approximation, and automatic differentiation. This is a valuable tool. We can explore many models, and analyze large datasets with ease. We emphasize that is currently available as part of Stan; it is ready for anyone to use.
### Acknowledgments {#acknowledgments .unnumbered}
We acknowledge our amazing colleagues and funding sources here.
Transformation of the Evidence Lower Bound {#app:elbo_unconstrained}
==========================================
Recall that $\mbzeta = T(\mbtheta)$ and that the variational approximation in the real coordinate space is $q(\mbzeta \,;\, \mbmu, \mbsigma^2)$.
We begin with the in the original latent variable space. We then transform the latent variable space of to the real coordinate space. $$\begin{aligned}
\cL
&=
\int
q(\mbtheta\,;\,\mbphi)
\log \left[\frac{p(\mbX,\mbtheta)}{q(\mbtheta\,;\,\mbphi)}\right]
\dif\mbtheta
\\
&=
\int
q(\mbzeta \,;\, \mbmu, \mbsigma^2)
\log
\left[
\frac
{p (\mbX,T^{-1}(\mbzeta)) \big| \det J_{T^{-1}}(\mbzeta) \big|}
{q(\mbzeta \,;\, \mbmu, \mbsigma^2)}
\right]
\dif \mbzeta
\\
&=
\int q(\mbzeta \,;\, \mbmu, \mbsigma^2)
\log
\left[
p (\mbX,T^{-1}(\mbzeta)) \big| \det J_{T^{-1}}(\mbzeta) \big|
\right]
\dif \mbzeta
-
\int q(\mbzeta \,;\, \mbmu, \mbsigma^2)
\log
\left[
q(\mbzeta \,;\, \mbmu, \mbsigma^2)
\right]
\dif \mbzeta
\\
&= \E_{q(\mbzeta)} \left[ \log p (\mbX,T^{-1}(\mbzeta))
+
\log \big| \det J_{T^{-1}}(\mbzeta) \big| \right]
-
\E_{q(\mbzeta)} \left[ \log q(\mbzeta \,;\, \mbmu, \mbsigma^2) \right]\end{aligned}$$ The variational approximation in the real coordinate space is a Gaussian. Plugging in its entropy gives the in the real coordinate space $$\begin{aligned}
\cL
&=
\E_{q(\mbzeta) } \left[ \log p (\mbX,T^{-1}(\mbzeta))
+
\log \big| \det J_{T^{-1}}(\mbzeta) \big| \right]
+
\frac{1}{2}K\left(1+\log(2\pi)\right)
+
\sum_{k=1}^K \log \sigma_k.\end{aligned}$$
Gradients of the Evidence Lower Bound {#app:gradient_elbo}
=====================================
First, consider the gradient with respect to the $\mbmu$ parameter of the standardization. We exchange the order of the gradient and the integration through the dominated convergence theorem [@cinlar2011probability]. The rest is the chain rule for differentiation. $$\begin{aligned}
\nabla_\mbmu \cL
&=
\nabla_\mbmu
\Big\{
\E_{\cN(\mbeta\,;\, \mb{0}, \mbI)}
\left[
\log p
(\mbX,T^{-1}(S_{\mbmu,\mbomega}^{-1}(\mbeta)))
+
\log \big| \det J_{T^{-1}}(S_{\mbmu,\mbomega}^{-1}(\mbeta)) \big|
\right]\\
&\qquad\quad+ \frac{K}{2}(1+\log(2\pi)) + \sum_{k=1}^K \log \sigma_k
\Big\}\\
&=
\E_{\cN(\mbeta\,;\, \mb{0}, \mbI)}
\left[
\nabla_\mbmu
\left\{
\log p
(\mbX,T^{-1}(S^{-1}(\mbeta))
+
\log \big| \det J_{T^{-1}}(S^{-1}(\mbeta)) \big|
\right\}
\right] \\
&=
\E_{\cN(\mbeta\,;\, \mb{0}, \mbI)}
\left[
\nabla_\mbtheta \log p(\mbX,\mbtheta)
\nabla_{\mbzeta} T^{-1}(\mbzeta)
\nabla_{\mbmu} S_{\mbmu,\mbomega}^{-1} (\mbeta)
+
\nabla_{\mbzeta}
\log \big| \det J_{T^{-1}}(\mbzeta) \big|
\nabla_{\mbmu} S_{\mbmu,\mbomega}^{-1} (\mbeta)
\right] \\
&=
\E_{\cN(\mbeta\,;\, \mb{0}, \mbI)}
\left[
\nabla_\mbtheta \log p(\mbX,\mbtheta)
\nabla_{\mbzeta} T^{-1}(\mbzeta)
+
\nabla_{\mbzeta}
\log \big| \det J_{T^{-1}}(\mbzeta) \big|
\right]\end{aligned}$$
Similarly, consider the gradient with respect to the $\mbomega$ parameter of the standardization. The gradient with respect to a single component, $\omega_k$, has a clean form. We abuse the $\nabla$ notation to maintain consistency with the rest of the text (instead of switching to $\partial$). $$\begin{aligned}
\nabla_{\omega_k} \cL
&=
\nabla_{\omega_k}
\Big\{
\E_{\cN(\mbeta\,;\, \mb{0}, \mbI)}
\left[
\log p
(\mbX,T^{-1}(S_{\mbmu,\mbomega}^{-1}(\mbeta))
+
\log \big| \det J_{T^{-1}}(S_{\mbmu,\mbomega}^{-1}(\mbeta)) \big|
\right]\\
&\qquad\quad+ \frac{K}{2}(1+\log(2\pi)) + \sum_{k=1}^K \log( \exp
(\omega_k))
\Big\}\\
&=
\E_{\cN(\eta_k)}
\left[
\nabla_{\omega_k}
\big\{
\log p
(\mbX,T^{-1}(S_{\mbmu,\mbomega}^{-1}(\mbeta)))
+
\log \big| \det J_{T^{-1}}(S_{\mbmu,\mbomega}^{-1}(\mbeta)) \big|
\big\}
\right] + 1\\
&=
\E_{\cN(\eta_k)}
\left[
\left(
\nabla_{\theta_k} \log p(\mbX,\mbtheta)
\nabla_{\zeta_k} T^{-1}(\mbzeta)
+
\nabla_{\zeta_k}
\log \big| \det J_{T^{-1}}(\mbzeta) \big|
\right)
\nabla_{\omega_k} S_{\mbmu,\mbomega}^{-1}(\mbeta))
\right]
+ 1.\\
&=
\E_{\cN(\eta_k)}
\left[
\left(
\nabla_{\theta_k} \log p(\mbX,\mbtheta)
\nabla_{\zeta_k} T^{-1}(\mbzeta)
+
\nabla_{\zeta_k}
\log \big| \det J_{T^{-1}}(\mbzeta) \big|
\right)
\eta_k
\exp(\omega_k)
\right]
+ 1.\end{aligned}$$
Running in Stan {#app:stan}
===============
Use `git` to checkout the `feature/bbvb` branch from `https://github.com/stan-dev/stan`. Follow instructions to build Stan. Then download `cmdStan` from `https://github.com/stan-dev/cmdstan`. Follow instructions to build `cmdStan` and compile your model. You are then ready to run .
The syntax is
----------- ---------------------------------------------------------
./myModel experimental variational
grad\_samples=M $\qquad\qquad\qquad$( $M = 1$ default )
data file=myData.data.R
output file=output\_advi.csv
diagnostic\_file=elbo\_advi.csv
----------- ---------------------------------------------------------
where `myData.data.R` is the dataset in the `R` language `dump` format. `output_advi.csv` contains samples from the posterior and `elbo_advi.csv` reports the .
Transformations of Continuous Probability Densities {#app:jacobian}
===================================================
We present a brief summary of transformations, largely based on [@olive2014statistical].
Consider a univariate (scalar) random variable $X$ with probability density function $f_X(x)$. Let $\mathcal{X} = \supp(f_X(x))$ be the support of $X$. Now consider another random variable $Y$ defined as $Y = T(X)$. Let $\mathcal{Y} = \supp(f_Y(y))$ be the support of $Y$.
If $T$ is a one-to-one and differentiable function from $\mathcal{X}$ to $\mathcal{Y}$, then $Y$ has probability density function $$\begin{aligned}
f_Y(y)
&=
f_X(T^{-1}(y))
\left|
\frac{\dif T^{-1}(y)}{\dif y}
\right|.\end{aligned}$$ Let us sketch a proof. Consider the cumulative density function $Y$. If the transformation $T$ is increasing, we directly apply its inverse to the cdf of $Y$. If the transformation $T$ is decreasing, we apply its inverse to one minus the cdf of $Y$. The probability density function is the derivative of the cumulative density function. These things combined give the absolute value of the derivative above.
The extension to multivariate variables $\mb{X}$ and $\mb{Y}$ requires a multivariate version of the absolute value of the derivative of the inverse transformation. This is the the absolute determinant of the Jacobian, $|\det J_{T^{-1}}(\mb{Y})|$ where the Jacobian is $$\begin{aligned}
J_{T^{-1}}(\mb{Y})
&=
\left(
\begin{matrix}
\frac{\partial T_1^{-1}}{\partial y_1}
&
\cdots
&
\frac{\partial T_1^{-1}}{\partial y_K}\\
\vdots & & \vdots\\
\frac{\partial T_K^{-1}}{\partial y_1}
&
\cdots
&
\frac{\partial T_K^{-1}}{\partial y_K}\\
\end{matrix}
\right).\end{aligned}$$
Intuitively, the Jacobian describes how a transformation warps unit volumes across spaces. This matters for transformations of random variables, since probability density functions must always integrate to one. If the transformation is linear, then we can drop the Jacobian adjustment; it evaluates to one. Similarly, affine transformations, like elliptical standardizations, do not require Jacobian adjustments; they preserve unit volumes.
Setting a Stepsize Sequence for {#app:adaGrad}
================================
We use adaGrad [@duchi2011adaptive] to adaptively set the stepsize sequence in . While adaGrad offers attractive convergence properties, in practice it can be slow because it has infinite memory. (It tracks the norm of the gradient starting from the beginning of the optimization.) In we randomly initialize the variational approximation, which can be far from the true posterior. This makes adaGrad take very small steps for the rest of the optimization, thus slowing convergence. Limiting adaGrad’s memory speeds up convergence in practice, an effect also observed in training neural networks [@rmsprop]. (See [@kingma2014adam] for an analysis of these trade-offs and a method that combines benefits from both.)
Consider the stepsize $\mbrho^{(i)}$ and a gradient vector $\mb{g}^{(i)}$ at iteration $i$. The $k$th element of $\mbrho^{(i)}$ is $$\begin{aligned}
\rho_k^{(i)}
&=
\frac{\eta}
{\tau + \sqrt{s^{(i)}_k}}\end{aligned}$$ where, in adaGrad, $\mb{s}$ is the gradient vector squared, summed over all times steps since the start of the optimization. Instead, we limit this to the past ten iterations and compute $\mb{s}$ as $$\begin{aligned}
s^{(i)}_k
&=
{g^2_k}^{(i-10)} + {g^2_k}^{(i-9)} + \cdots + {g^2_k}^{(i)}.\end{aligned}$$ (In practice, we implement this recursively to save memory.) We set $\eta=0.1$ and $\tau=1$ as the default values we use in Stan.
Linear Regression with Automatic Relevance Determination {#app:linreg_ard}
========================================================
Linear regression with is a high-dimensional sparse regression model [@bishop2006pattern; @drugowitsch2013variational]. We describe the model below. Stan code is in Figure \[fig:code\_linreg\].
The inputs are $\mbX=\mbx_{1:N}$ where each $\mbx_n$ is $D$-dimensional. The outputs are $\mb{y}=y_{1:N}$ where each $y_n$ is $1$-dimensional. The weights vector $\mb{w}$ is $D$-dimensional. The likelihood $$\begin{aligned}
p(\mb{y} \mid \mbX, \mb{w}, \tau)
&=
\prod_{n=1}^N \cN \left( y_n \mid \mb{w}^\top \mbx_n \;,\; \tau^{-1}\right)\end{aligned}$$ describes measurements corrupted by iid Gaussian noise with unknown variance $\tau^{-1}$.
The prior and hyper-prior structure is as follows $$\begin{aligned}
p(\mb{w},\tau,\mb{\alpha})
&=
p(\mb{w},\tau \mid \mb{\alpha})
p(\mb{\alpha})\\
&=
\cN \left( \mb{w} \mid 0 \,,\, (\tau\diag[\mb{\alpha}])^{-1} \right)
\Gam (\tau \mid a_0, b_0)
\prod_{i=1}^D \Gam({\alpha}_i \mid c_0, d_0)\end{aligned}$$ where $\mb{\alpha}$ is a $D$-dimensional hyper-prior on the weights, where each component gets its own independent Gamma prior.
We simulate data such that only half the regressions have predictive power. The results in Figure \[sub:linreg\] use $a_0 = b_0 = c_0 = d_0 = 1$ as hyper-parameters for the Gamma priors.
Hierarchical Logistic Regression {#app:logreg}
================================
Hierarchical logistic regression models dependencies in an intuitive and powerful way. We study a model of voting preferences from the 1988 United States presidential election. Chapter 14.1 of [@gelman2006data] motivates the model and explains the dataset. We also describe the model below. Stan code is in Figure \[fig:code\_election88\], based on [@stan-manual:2015]. $$\begin{aligned}
\Pr(y_n=1)
&=
\sigma
\bigg(
\beta^0
+ \beta^\text{female}\cdot\text{female}_n
+ \beta^\text{black}\cdot\text{black}_n
+ \beta^\text{female.black}\cdot\text{female.black}_n \\
&\qquad\quad + \alpha^\text{age}_{k[n]}
+ \alpha^\text{edu}_{l[n]}
+ \alpha^\text{age.edu}_{k[n],l[n]}
+ \alpha^\text{state}_{j[n]}
\bigg)\\
\alpha^\text{state}_j
&\sim
\cN
\left(
\alpha^\text{region}_{m[j]}
+ \beta^\text{v.prev}\cdot\text{v.prev}_j
\,,\, \sigma^2_\text{state}
\right)\end{aligned}$$ where $\sigma(\cdot)$ is the sigmoid function (also know as the logistic function).
The hierarchical variables are $$\begin{aligned}
\alpha^\text{age}_k
&\sim
\cN \left(0\,,\, \sigma^2_\text{age} \right)
\text{ for } k = 1,\ldots,K\\
\alpha^\text{edu}_l
&\sim
\cN \left(0\,,\, \sigma^2_\text{edu} \right)
\text{ for } l = 1,\ldots,L\\
\alpha^\text{age.edu}_{k,l}
&\sim
\cN \left(0\,,\, \sigma^2_\text{age.edu} \right)
\text{ for } k = 1,\ldots,K, l = 1,\ldots,L\\
\alpha^\text{region}_m
&\sim
\cN \left(0\,,\, \sigma^2_\text{region} \right)
\text{ for } m = 1,\ldots,M.\end{aligned}$$
The variance terms all have uniform hyper-priors, constrained between 0 and 100.
Non-negative Matrix Factorization: Constrained Gamma Poisson Model {#app:gap}
==================================================================
The Gamma Poisson factorization model is a powerful way to analyze discrete data matrices [@canny2004gap; @cemgil2009bayesian].
Consider a $U \times I$ matrix of observations. We find it helpful to think of $u=\{1,\cdots,U\}$ as users and $i=\{1,\cdots,I\}$ as items, as in a recommendation system setting. The generative process for a Gamma Poisson model with $K$ factors is
1. For each user $u$ in $\{1,\cdots,U\}$:
- For each component $k$, draw $\theta_{uk} \sim \Gam(a_0, b_0)$.
2. For each item $i$ in $\{1,\cdots,I\}$:
- For each component $k$, draw $\beta_{ik} \sim \Gam(c_0, d_0)$.
3. For each user and item:
- Draw the observation $y_{ui} \sim
\text{Poisson}(\mbtheta_u^\top\mbbeta_i)$.
A potential downfall of this model is that it is not uniquely identifiable: scaling $\mbtheta_u$ by $\alpha$ and $\mbbeta_i$ by $\alpha^{-1}$ gives the same likelihood. One way to contend with this is to constrain either vector to be a positive, ordered vector during inference. We constrain each $\mbtheta_u$ vector in our model in this fashion. Stan code is in Figure \[fig:code\_nmf\_pf\]. We set $K=10$ and all the Gamma hyper-parameters to 1 in our experiments.
Non-negative Matrix Factorization: Dirichlet Exponential Model {#app:dir_exp}
==============================================================
Another model for discrete data is a Dirichlet Exponential model. The Dirichlet enforces uniqueness while the exponential promotes sparsity. This is a non-conjugate model that does not appear to have been studied in the literature.
The generative process for a Dirichlet Exponential model with $K$ factors is
1. For each user $u$ in $\{1,\cdots,U\}$:
- Draw the $K$-vector $\mbtheta_{u} \sim \text{Dir}(\mb{\alpha}_0)$.
2. For each item $i$ in $\{1,\cdots,I\}$:
- For each component $k$, draw $\beta_{ik} \sim \text{Exponential}
(\lambda_0)$.
3. For each user and item:
- Draw the observation $y_{ui} \sim
\text{Poisson}(\mbtheta_u^\top\mbbeta_i)$.
Stan code is in Figure \[fig:code\_nmf\_dir\_exp\]. We set $K=10$, $\alpha_0 = 1000$ for each component, and $\lambda_0 = 0.1$. With this configuration of hyper-parameters, the factors $\mbbeta_i$ are sparse and appear interpretable.
Gaussian Mixture Model {#app:gmm}
======================
The is a powerful probability model. We use it to group a dataset of natural images based on their color histograms. We build a high-dimensional with a Gaussian prior for the mixture means, a lognormal prior for the mixture standard deviations, and a Dirichlet prior for the mixture components.
The images are in $\mbX = \mbx_{1:N}$ where each $\mbx_n$ is $D$-dimensional and there are $N$ observations. The likelihood for the images is $$\begin{aligned}
p(\mbX \mid \mbtheta, \mbmu, \mbsigma)
&=
\prod_{n=1}^N
\prod_{k=1}^K
\theta_k
\prod_{d=1}^D
\cN(x_{nd}\mid\mu_{kd},\sigma_{kd})\end{aligned}$$ with a Dirichlet prior for the mixture proportions $$\begin{aligned}
p(\mbtheta)
&=
\text{Dir}(\mbtheta\,;\, \mb{\alpha}_0),\end{aligned}$$ a Gaussian prior for the mixture means $$\begin{aligned}
p(\mbmu)
&=
\prod_{k=1}^D
\prod_{d=1}^D
\cN(\mu_{kd}\,;\, 0, \sigma_\mu)\end{aligned}$$ and a lognormal prior for the mixture standard deviations $$\begin{aligned}
p(\mbsigma)
&=
\prod_{k=1}^D
\prod_{d=1}^D
\text{logNormal}(\sigma_{kd}\,;\, 0, \sigma_\sigma)\end{aligned}$$
The dimension of the color histograms in the image<span style="font-variant:small-caps;">clef</span> dataset is $D =
576$. These a concatenation of three $192$-length histograms, one for each color channel (red, green, blue) of the images.
We scale the image histograms to have zero mean and unit variance and set $\alpha_0 = 10\,000$, $\sigma_\mu=0.1$ and $\sigma_\mu$. code is in Figure \[fig:code\_gmm\_diag\]. The stochastic data subsampling version of the code is in Figure \[fig:code\_gmm\_diag\_adsvi\].
[^1]: is available in Stan 2.7 (development branch). It will appear in Stan 2.8. See Appendix \[app:stan\].
[^2]: The posterior of a *fully* conjugate model is in the same family as the prior.
[^3]: A *conditionally* conjugate model has this property within the complete conditionals of the model [@hoffman2013stochastic].
[^4]: If $\supp(q) \not\subseteq \supp (p)$ then outside the support of $p$ we have $\KL {q} {p} = \E_q [\log q] -
\E_q[\log p] = -\infty$.
[^5]: Stan provides transformations for upper and lower bounds, simplex and ordered vectors, and structured matrices such as covariance matrices and Cholesky factors [@stan-manual:2015].
[^6]: Also known as a “co-ordinate transformation” [@rezende2014stochastic], an “invertible transformation” [@titsias2014doubly], and the “re-parameterization trick” [@kingma2013auto].
[^7]: is an adaptive extension of . It is the default sampler in Stan.
|
---
abstract: 'A number of techniques in Lorentzian geometry, such as those used in the proofs of singularity theorems, depend on certain smooth coverings retaining interesting global geometric properties, including causal ones. In this note we give explicit examples showing that, unlike some of the more commonly adopted rungs of the causal ladder such as strong causality or global hyperbolicity, less-utilized conditions such as causal continuity or causal simplicity [*do not*]{} in general pass to coverings, as already speculated by one of the authors (EM). As a consequence, any result which relies on these causality conditions transferring to coverings must be revised accordingly. In particular, some amendments in the statement and proof of a version of the Gannon-Lee singularity theorem previously given by one of us (IPCS) are also presented here that address a gap in its original proof.'
address:
- 'Dipartimento di Matematica e Informatica “U. Dini”, Università degli Studi di Firenze, Via S. Marta 3, I-50139, Firenze, Italy'
- 'Department of Mathematics, Universidade Federal de Santa Catarina, 88.040-900 Florianópolis-SC, Brazil.'
author:
- Ettore Minguzzi
- 'Ivan P. Costa e Silva'
title: A note on causality conditions on covering spacetimes
---
\[section\] \[thm\][Proposition]{} \[thm\][Lemma]{} \[thm\][Corollary]{} \[thm\][Definition]{} \[thm\][Notation]{} \[thm\][Example]{} \[thm\][Conjecture]{} \[thm\][Problem]{} \[thm\][Remark]{} \[thm\][Convention]{} \[thm\][Criterion]{} \[thm\][Claim]{}
Introduction {#section1}
============
Let $(M^n,g)$ be a [*spacetime*]{}, i.e., a pair consisting of a connected $C^{\infty}$ smooth manifold (Hausdorff and secound-countable) $M$ of dimension $n\geq 2$ and a time-oriented $C^\infty$ Lorentz metric $g$. Let $\pi:\tilde{M}\rightarrow M$ be a smooth covering map, and endow $\tilde{M}$ with the pullback metric $\tilde{g}:= \pi ^{\ast}g$ and the induced time-orientation. Since $\pi$ is a local isometry, any [*local*]{} geometric condition that might hold on $(M,g)$, such as (say) $Ric_g(v,v)\geq 0$ for all lightlike vectors $v \in TM$, must hold on $(\tilde{M},\tilde{g})$ as well.
Geodesic (lightlike, or timelike, or spacelike) completeness, on the other hand, is a key example of a [*global*]{} geometric feature which holds on $(M,g)$ if and only if it holds on $(\tilde{M},\tilde{g})$. This is of substantial technical importance in the proofs of singularity theorems, since some constructions are carried out on (a suitable choice of) $(\tilde{M},\tilde{g})$ [@C; @G; @Haag; @H; @HE]. For example, (cf. [@oneill Prop. 14.2]) if $(M,g)$ admits as (topologically) closed connected spacelike hypersurface $\Sigma \subset M$, then $(\tilde{M},\tilde{g})$ can be chosen so that it possesses a diffeomorphic copy $\tilde{\Sigma}$ of $\Sigma$ which is in addition [*acausal*]{}, so that its Cauchy development can be suitably analyzed for the existence of certain maximal geodesics normal to $\tilde{\Sigma}$, an important step in some proofs. Of course, such constructions can only be meaningfully carried out provided there is some control on whether the required properties still hold on covering manifolds. An important hypothesis in a number of theorems, and in singularity theorems in particular, is on which rung of the so-called [*causal ladder*]{} [@minguzzi_living_reviews; @causalladder] of spacetimes $(M,g)$ sits. In view of the remarks in the previous paragraph, it is therefore of interest to know whether that rung is shared with $(\tilde{M},\tilde{g})$. A positive statement in this regard has been summarized by one of us (EM) as follows. (See [@minguzzi_living_reviews Thm. 2.99] for an extended discussion and a proof.)
\[thm1\] Let $\pi : (\tilde{M},\tilde{g})\rightarrow (M,g)$ be a Lorentzian covering. If $(M,g)$ is chronological \[resp. causal, non-totally imprisoning, future/past distinguishing, strongly causal, stably causal, globally hyperbolic\] then $(\tilde{M},\tilde{g})$ has the same property.
Just after the statement of that theorem, the author mentions in passing that [*reflectivity*]{} and [*closure of causal futures/pasts*]{} of points in $(M,g)$ do not seem to pass to coverings. It is the purpose of this note to both confirm the latter claim by means of concrete (counter)examples, as well as to modify the statement and proof of [@costa_e_silva_gannon Thm. 2.1] to incorporate this discovery. As it stands, the latter proof has a gap if the underlying spacetime is not globally hyperbolic, precisely because it assumes without further discussion that simple causality also applies to a certain Lorentzian covering thereof.[^1] The version we present here, however, is [*not*]{} a mere amendment. Recently it has been shown that the assumptions of Penrose’s theorem can be improved by weakening global hyperbolicity to past reflectivity [@minguzzi_new]. Adapting some arguments to the Gannon-Lee case we are able to accomplish a similarly interesting result, that is, we can dramatically decrease the causality requirements of the Gannon-Lee theorem by demanding $(M,g)$ [*and its coverings*]{} to be just past reflecting.
This paper is organized as follows. In section \[s1\], we discuss two examples, one of which is adapted from the spacetime constructed by Hedicke and Suhr in [@hedicke19 Thm. 2.7] (with the entirely different purpose of providing an example of a causally simple spacetime for which the space of null geodesics is not Hausdorff). These examples show that $(i)$ the closure of the causal relation, and $(ii)$ past reflectivity, do not pass to $(\tilde{M},\tilde{g})$ while holding on $(M,g)$. In section \[s2\] we give an alternative statement and proof of the Gannon-Lee theorem presented in [@costa_e_silva_gannon Thm. 2.1] it terms of past reflectivity of Lorentzian coverings.
We shall assume throughout that the reader is familiar with the elements of causal theory in the core references [@beem; @oneill], up to and including the best-known singularity theorems originally proven by R. Penrose and S.W, Hawking described in those references. We also assume the reader is acquainted with the basic structure of the causal ladder, that is, the basic hierarchy of causal conditions listed from the weakest - [*non-totally viciousness*]{} - at the bottom, through the strongest - [*global hyperbolicity*]{} at the top, which can be found, e.g., in Ch. 2 of [@beem]. However, since the notions of past/future reflectivity are somewhat less known, we briefly recall these in section \[s1\].
The results in this paper are purely geometric in that no field equations are assumed, and they hold for any spacetime dimension $\geq 3$. Our conventions for the signs of spacetime curvature are those of [@beem], but for the mean curvature vector of submanifolds within it we use those of [@oneill]. In particular, we borrow the definition of [*convergence*]{} of a semi-Riemannian submanifold from the latter reference (cf. [@oneill Def. 10.36]). Unless otherwise explicitly stated, all maps and (sub)manifols are assumed to be $C^{\infty}$, and submanifolds to be embedded.
(Counter)examples of causality conditions on coverings {#s1}
======================================================
As announced in the Introduction, we present in this section examples showing that certain causal conditions which hold on a spacetime $(M,g)$ may fail to hold on one (or more) of its coverings, including the universal covering.
First, we recall some terminology.
\[reflectdef\] A spacetime $(M,g)$ is said to be [*past reflecting*]{} \[resp. [*future reflecting*]{}\] if one (and hence both) of the two equivalent statements hold for any two $p,q\in M$:
- $I^+(q)\subset I^+(p) \Rightarrow I^-(p)\subset I^-(q)$ \[resp. $I^-(q)\subset I^-(p) \Rightarrow I^+(p)\subset I^+(q)$\];
- $q \in \overline{I^+(p)} \Rightarrow p \in \overline{I^-(q)} $ \[resp. $q \in \overline{I^-(p)} \Rightarrow p \in \overline{I^+(q)} $\].
If $(M,g)$ is [*both*]{} past and future reflecting, it is simply said to be [*reflecting*]{}.
For further discussion on reflectivity, and a proof of the equivalence of $(i)$ and $(ii)$ in Def. \[reflectdef\] (as well as other equivalent statements) see, e.g., [@minguzzi_living_reviews Section 4.1].
We just comment here that unlike usual causality conditions, reflectivity [*by itself*]{} lies outside the causal ladder, and is known as a [*transversal*]{} condition. (See, for example, the discussion around [@k_causalityminguzzi Fig. 2] or [@minguzzi_living_reviews].) In particular, it may well be present even in non-chronological spacetimes (e.g. Gödel spacetime).
However, if combined with other, even quite mild, causality conditions, reflectivity does imply - and is implied by - some rather strong causality requirements [@minguzzi_living_reviews]:
- $(M,g)$ is distinguishing $+$ reflecting $\Leftrightarrow$ $(M,g)$ is causally continuous;
- $(M,g)$ is causal $+$ closure of causal relation $\Leftrightarrow$ $(M,g)$ is causally simple;
- closure of causal relation $\Rightarrow$ $(M,g)$ is reflecting;
- $(M,g)$ is non-totally vicious + reflecting $\Rightarrow$ $(M,g)$ is chronological.
We are ready to discuss the examples.
\[exe1\] We construct a causally simple spacetime $(M^3,g)$ in dimension 3 for which the [*universal covering*]{} $(\tilde{M}, \tilde{g})$ is such that $\tilde J$, the causal relation on $\tilde M$, is not closed. Thus, $(\tilde{M}, \tilde{g})$ cannot be causally simple. The spacetime $(M^3,g)$ was introduced, with an entirely different purpose, in [@hedicke19 Thm. 2.7].
Consider the static spacetime $$(M,g)=(\mathbb{R}\times \Sigma, -d t^2+ k)$$ where $\partial _t$ is taken to be future-directed and $(\Sigma,k)$ is a suitable surface of revolution given as follows. Consider the map $$\varphi : (0,1)\times \mathbb{R} \to \mathbb{R}^3, \quad (u,v) \mapsto
(u, f(u)\cos\ v, f(u) \sin \ v),$$ with $f :[0,1] \to [0,\infty)$ being any concave function with $f(0)=f(1)=0$ and $0<f'(0)=-f'(1)<\tfrac{1}{2\pi}$. $(\Sigma, k)$ is then the image of $\varphi$ endowed with the (Riemanniann) metric induced by the Euclidean metric on $\mathbb{R}^3$. Its topology is that of a sphere minus the two points intersecting the $x$ axis, but its shape resembles that of a lemon, or an ellipsoid with a deficit angle at the removed points. As a result $(\Sigma, k)$ is [*geodesically convex*]{}, meaning that the distance between any two points is realized by a minimizing geodesic segment $\sigma$ connecting them, which ultimately is what guarantees the causal simplicity property [@hedicke19 Cor. 4.1]. Let $a=\varphi((1/2,0))$, $b=\varphi((1/2,\pi))$ be two points that belong to the surface of revolution and to the same plane $xy$ passing through the axis. Consider the fixed-endpoint homotopy class of a curve $\gamma_1$ starting from $a$ and reaching $b$, that revolves once in the positive direction over the singularity at $x=0$. Denote this class by $[\gamma_1]$, which is obviously distinct from that of $\sigma$. We have $$\ell _0:= \textrm{inf}_{\gamma\in [\gamma_1]}
\ell^\kappa(\gamma)=\ell^\kappa(\eta_1)+\ell^\kappa(\eta_2)>
\ell^\kappa(\sigma)$$ where $\ell^\kappa$ is the length functional in $(\Sigma,\kappa)$ and where $\eta_1$ and $\eta_2$ are incomplete geodesics that connect $a$ to the singularity at $0$ and the singularity at $0$ to $b$, respectively. Notice that the infimum of length $\ell _0$ is not realized since the putative ‘curve’ $\eta_2 \circ \eta_1$ would pass through the singularity at $x=0$ if connected. But we can consider a sequence of $\ell ^\kappa$-parametrized curves $(\gamma_k)$ in $[\gamma_1]$ starting at $a$ and ending at $b$ such that $\ell^\kappa(\gamma _k)\rightarrow \ell _0$. Thus, $\eta_1$ and $\eta_2$ are, separately, limit curves of this sequence, but no limit curve can connect $a$ and $b$.
In $(M,g)$ we find that the causal (indeed lightlike) curves $t
\mapsto (t,\gamma_k(t))$ have final endpoints $(\ell^\kappa(\gamma _k),b) \in J^+(0,a)$; the latter set is closed since $(M,g)$ is causally simple. This means that $(
\ell_0,b)$ also belongs to $J^+((0,a))$. However, in the covering spacetime $\tilde M=\mathbb{R}\times \tilde
\Sigma \simeq \mathbb{R}^3$, the lifts $t \mapsto (t,\tilde \gamma _k(t))$ of the curves $t \mapsto (t,\gamma _k(t))$ that start at the same representative $(0,\tilde a)$ are causal curves that have final endpoints $(\ell^\kappa(\gamma _k),\tilde{b}) \in \tilde{J}^+(0,\tilde{a})$ converging to a point $(\ell _0,\tilde b)$, which now, however, can [*not*]{} belong to $\tilde J^+((0,\tilde a))$, precisely because the above-mentioned infimum is not attained. We conclude that in $(\tilde M,\tilde g)$, the causal relation $\tilde J$ is not closed.
We now present an example of causally continuous (and in particular reflecting) spacetime $(M,g)$ for which the universal covering $(\tilde M, \tilde g)$ is not past reflecting.
\[exe2\] Start with Minkowski 2d spacetime $\mathbb{R}^2$ of coordinates $(t,x)$, endowed with the metric $-d t^2+d x^2$. Let $r=(0,0)$ and $r_k=(0,-1/k)$ ($k\in \mathbb{N}$). Consider now the spacetime given as the set $M=\mathbb{R}^2 \backslash \{r,r_1,r_2, \cdots \}$ endowed with the restricted metric and time-orientation. This spacetime is causally continuous, as can be easily established by looking at the continuity of the volume functions [@minguzzi_living_reviews Def.4.6(vii)]. Let $p=(-1,1)$ and $q=(1,-1)$, so that $q\in
\overline{I^+(p)}$ and $p\in \overline{I^-(q)}$.
Consider the universal covering $\pi:(\tilde M,\tilde{g}) \rightarrow (M,g)$. Let $\gamma:[0,1]\rightarrow M$ be any curve such that $\gamma(0)=p$, $\gamma(1)=q$, and $\gamma (0,1) \subset I^+(p)$, and consider its lift $\tilde{\gamma}$ starting at a fixed representative $\tilde p \in \pi ^{-1}(p)$. Then, for each $k\in \mathbb{N}$ with $k\geq 2$, $$\tilde{q}_k:=\tilde{\gamma}(1-1/k) \in \pi^{-1}(I^+(p))\equiv \tilde{I}^+(\pi^{-1}(p));$$ but since $\gamma |_{[0,1-1/k]}$ is clearly endpoint homotopic to any future-directed timelike curve from $p$ to $\gamma (1-1/k)$ we conclude that $\tilde{q}_k \in \tilde{I}^+(\tilde{p})$, so $\tilde{q}:= \tilde{\gamma}(1) \in \pi ^{-1}(q)\cap \overline{\tilde I^+(\tilde p)}$.
We claim, however, that $\tilde p \notin \overline{\tilde I^-(\tilde q)}$, i.e., past reflectivity is violated on $(\tilde M,\tilde{g})$. For suppose there is a sequence $(\tilde{p}_k)\subset \tilde {I}^-(\tilde q)$ converging to $\tilde{p}$. Pick any future-directed timelike curves $\tilde{\sigma}_k:[0,1]\rightarrow \tilde{M}$ starting at $\tilde{p}_k$ and ending at $\tilde{q}$, so that $p_k:=\pi(\tilde{p}_k) \rightarrow p$. The curves $\sigma_k:= \pi\circ \tilde{\sigma}_k$ are timelike, so $(p_k)\subset I^-(q)$. Moreover, the intersections of the images of the $\sigma_k$’s with the axis $t=0$ clearly must occur at points $c_k \rightarrow (0,0)$ (in $\mathbb{R}^2$). Therefore, for some subsequence $(c_{k_i})_{i\in \mathbb{N}}$ we can assume that each $c_{k_i}$ belongs to a different segment $\{0\}\times (-\frac{1}{k_i}, -\frac{1}{k_i+1})$. Pick any distinguished small open disc $U\ni p$ in $M$ not intersecting the $t=0$ axis. We can assume, without loss of generality, that $p_{k_i} \in U$ for every $i\in \mathbb{N}$. But then the points $\tilde p_{k_i}$ must be in different connected components of $\pi^{-1}(U)$, and thus we cannot have $\tilde p_{k_i}\rightarrow \tilde{p}$, a contradiction.
Finally, we mention that not only the properties ‘reflectivity’ and ‘closure of the causal relation’ do not pass to coverings; these properties are also known to be distinguished among all causality properties for not being preserved by isocausal mappings, see [@garciaparrado05b].
A new version of the Gannon-Lee theorem in low causality {#s2}
========================================================
The [*Gannon-Lee singularity theorem*]{} was independently discovered by D. Gannon and C.W. Lee in 1975/1976 [@gannon1; @gannon2; @lee]. Its importance lies in its application to general relativity, wherein it suggests that certain “localized non-trivial topological structures” in spacetime (meaning non-trivial fundamental group of certain spacelike hypersurfaces), such as wormholes, are [*gravitationally unstable*]{}, at least if one neglects quantum effects.
To make precise statements, we again fix a spacetime $(M^n,g)$, but throughout this section we assume $n\geq 3$. We shall recall some terminology which we believe is unfamiliar for many readers, largely following [@costa_e_silva_gannon].
Fix a smooth, connected, spacelike partial Cauchy hypersurface (i.e, a submanifold of codimension one) $\Sigma^{n-1} \subset M$[^2], and a smooth, connected, compact spacelike submanifold $S^{n-2} \subset M$ of codimension two.
Suppose $S$ [*separates*]{} $\Sigma$, i.e., $S \subset \Sigma$ and $\Sigma \setminus S$ is not connected. This means, in particular, that $\Sigma \setminus S$ is a disjoint union $\Sigma_{+}\dot{\cup} \Sigma_{-}$ of open submanifolds of $\Sigma$ having $S$ as a common boundary. We shall loosely call $\Sigma_{+}$ \[resp. $\Sigma_{-}$\] the [*outside*]{} \[resp. [*inside*]{}\] of $S$ in $\Sigma$. (In most interesting examples there is a natural choice for these.) It also means that there are unique unit spacelike vector fields $N_{\pm}$ on $S$ normal to $S$ in $\Sigma$, such that $N_{+}$ \[resp. $N_{-}$\] is outward-pointing \[resp. inward-pointing\]. Let $U$ be the unique timelike, future-directed, unit normal vector field on $\Sigma$. Then $K_{\pm} := U|_{S}+N_{\pm}$ are future-directed null vector fields on $S$ normal to $S$ in $M$ spanning the normal bundle $NS \subset TM$ (which is in particular trivial). The [*outward* ]{}\[resp. [*inward*]{}\] [*null convergence*]{} of $S$ in $M$ is the smooth function $k_{+}: S \rightarrow \mathbb{R}$ \[resp. $k_{-}: S \rightarrow \mathbb{R}$\] given by $$\label{maineq1}
k_{+}(p) = \langle H_p, K_{+}(p)\rangle _{p} \mbox{ [resp. $k_{-}(p) = \langle H_p, K_{-}(p) \rangle_p $]},$$ for each $p \in S$, where $H_p$ denotes the mean curvature vector of $S$ in $M$ at $p$ [@oneill], and we denote $g$ as $\langle \, , \, \rangle$ here and hereafter, if there is no risk of confusion. Under the sign conventions we adopt here, if $S$ is a round sphere in a Euclidean slice of Minkowski spacetime with the obvious choices of inside and outside, then $k_{+}<0$ and $k_{-}>0$. One also expects this to be the case if $S$ is a “large" sphere in an asymptotically flat spatial slice.
We say that a smooth future-directed timelike vector field $X:M \rightarrow TM$ is a [*piercing*]{} of $\Sigma$ (or [*pierces*]{} $\Sigma$) if every maximally extended integral curve of $X$ intersects $\Sigma$ exactly once. There is no loss of generality in assuming that $X$ is a complete vector field, and we shall do so in what follows. Using its flow, it is not difficult to show that if such a piercing exists, then $M$ is diffeomorphic to $\mathbb{R}\times \Sigma$.
Of course, such a partial Cauchy hypersurface $\Sigma$ and/or a piercing of $\Sigma$ may not exist for general spacetimes. On the other hand, if $(M,g)$ is globally hyperbolic and $\Sigma$ is a Cauchy hypersurface, then [*every*]{} smooth future-directed timelike vector field in $M$ pierces $\Sigma$. [*However, the existence of a piercing for a partial Cauchy hypersurface $\Sigma$ is strictly weaker than the requirement that $\Sigma$ be Cauchy*]{}. In [@costa_e_silva_gannon], anti-de Sitter spacetime is given as an example, and we may add here any standard (conforma)stationary spacetime with respect to a fiducial timelike (conformal) Killing vector field defining the standard splitting.
Using the terminology above, we shall adopt the following useful definition:
\[asympticallyregular\] A smooth, connected, spacelike partial Cauchy hypersurface $\Sigma \subset M$ is [*asymptotically regular*]{} if there exists a smooth, connected, compact submanifold $S \subset \Sigma$ of dimension $n-2$ such that
- $S$ separates $\Sigma$, and $\overline{{\Sigma}}_{+} \equiv S \cup \Sigma_{+}$ is non-compact;
- The map $h_{\#}: \pi_1(S) \rightarrow \pi_1(\overline{\Sigma}_{+})$ induced by the inclusion $h: S \hookrightarrow \overline{\Sigma}_{+}$ is onto;
- $S$ is [*inner trapped*]{}, i.e., $k_{-} >0$ everywhere on $S$.
we shall call such an $S$ an [*enclosing surface*]{} in $\Sigma$.
Let us briefly pause to explain the motivation behind the clauses $(i)-(iii)$ of this definition. First, it is meant as a convenient adaptation of Gannon’s definition of a [*regular near infinity*]{} hypersurface, so item $(i)$ presents no novelty. Clause $(ii)$, however, might look somewhat opaque. But it simply means that the (closure of the) outside of $S$ has only topological (or more precisely path-homotopic) complexities arising from having $S$ itself as a boundary. Specifically, [*it means that every loop in the exterior of $S$ in $\Sigma$ is homotopic to a loop on $S$*]{}. Note that this is certainly the case if $\overline{\Sigma}_{+} \equiv S \cup \Sigma_{+}$ is homeomorphic to $S \times [0, +\infty)$, as in the original Gannon-Lee theorem, but the condition as stated gives rise to the much wider set of topological possibilities which are likely to arise in higher dimensions. Note that $(iii)$ refers only to the inward-pointing family of null geodesics normal to $S$, namely that they converge “on average”.
We are finally ready to state our main result.
\[maintheorem\] Let $(M,g)$ be an $n$-dimensional (with $n \geq 3$) null geodesically complete spacetime, which satisfies the null energy condition (i.e., $Ric(v,v)\geq 0$, for any null vector $v\in TM$), and possesses an asymptotically regular hypersurface $\Sigma \subset M$ pierced by some timelike vector field $X\in \mathfrak{X}(M)$. Let an enclosing surface $S \subset \Sigma$ be given, and assume, in addition, that at least one of the following conditions holds:
- $(M,g)$ and each one of its covering spacetimes $(\tilde{M},\tilde{g})$ are past reflecting, or else
- $S$ is simply connected and both $(M,g)$ and its [*universal covering*]{} are past reflecting.
Then, the group homomorphism $i_{\#}: \pi_1(S) \rightarrow \pi_1(\Sigma)$ induced by the inclusion $i: S \hookrightarrow \Sigma$ is surjective. In particular, if $S$ is simply connected, then so is $\Sigma$.
In its version in [@costa_e_silva_gannon], clauses $(i)$ and $(ii)$ were simply replaced by the condition that $(M,g)$ be [*causally simple*]{}. They obviously hold if $(M,g)$ is globally hyperbolic and $\Sigma$ is a Cauchy hypersurface, as in the original Gannon-Lee theorem. However, the proof in [@costa_e_silva_gannon] implicitly took for granted that causal simplicity holds on a certain Lorentzian [*covering*]{} of $(M,g)$, which as discussed in Example \[exe1\] need not be the case. Now, on the one hand this is [*always*]{} the case if $\Sigma$ is a Cauchy hypersurface as in the original Gannon-Lee theorems. On the other hand, in the present version, [*provided one imposes past reflectivity on covering spacetimes as well as on $(M,g)$, these spacetimes do not even need to be chronological*]{}. [*Proof of Thm. \[maintheorem\]*]{}.\
The key to the proof is establishing the following\
[*Claim:*]{} [*the interior region $\overline{\Sigma}_{-}$ in $\Sigma$ is compact.*]{} (cf. [@costa_e_silva_gannon Prop. 4.1]).\
In order to prove this Claim we begin by considering the set $\mathcal{H}^+$ of all the points $p$ of $E^+(S)(:= J^+(S)\setminus I^+(S))$ such that either $p \in S$ or else $p$ can be reached from $S$ by a future-directed null geodesic $\eta:[0,b]\rightarrow M$ with $\eta(0)\in S$, $\eta(b)=p$ and $\eta'(0) =K_{-}(\eta(0))$. By a standard limit curve argument, either $\mathcal{H}^+$ is compact or else there exists a future-directed null geodesic $S$-ray $\gamma:[0,+\infty)\rightarrow M$, with $\gamma'(0) = K_{-}(\gamma(0))$. But the latter alternative is impossible due to the null convergence condition[^3] and the fact that $k_{-}>0$, which together imply the appearance of a focal point to $S$ along $\gamma$ incompatible with its maximal status [@oneill Prop. 10.43]. We conclude that $\mathcal{H}^+$ has to be compact.
Consider the closed set $T:=\partial I^+(\Sigma _+)\setminus \Sigma_+$. If we can show that $T\subset \mathcal{H}^+$, then it follows that $T$ is compact; in that case, arguing exactly as in the proof of Claim 3 in [@costa_e_silva_gannon Prop. 4.1]) we conclude that $\rho_X(T)\equiv \overline{\Sigma}_{-}$, where $\rho_X:M\rightarrow \Sigma$ is the retract associated with the piercing $X$ as discussed therein, and our Claim follows.
Suppose, then, by way of contradiction, that there is some $q \in T\setminus \mathcal{H}^+$. As mentioned in the Introduction, our strategy here is to adapt the proof of [@minguzzi_new Thm. 3.3] for the present context (see especially Fig. 3 in that reference). Let $(q_k)\subset I^+(q)$ be a sequence of points such that $q_k\rightarrow q$. Thus, $(q_k)\subset I^+(\Sigma_+)$. The maximal integral curve $\alpha$ of $X$ through $q$ must intersect $\Sigma$; it cannot do so to the future of $q$, or else this would violate the acausality of $\Sigma$. Thus, it either intersects $\Sigma$ at $q$ itself or to the past thereof. In the first case, since $q\notin \overline{\Sigma}_+$ we would have to have $q\in \Sigma_{-}$. In the second case, since $q\notin I^+(\overline{\Sigma} _+)$, we must have $q\in I^+(\Sigma_{-})$. In any case, $(q_k)\subset I^+(\Sigma_{-})$. Therefore, for each $k \in \mathbb{N}$ we have $I^-(q_k)\cap \Sigma_{\pm}\neq \emptyset$, whence we conclude that $I^-(q_k)\cap S\neq \emptyset$ since $I^-(q_k)\cap \Sigma$ is connected[^4]. In other words, $(q_k)\subset I^+(S)$.
Fix a background complete Riemannian metric $h$ on $M$ with associated distance function $d_h$, and let $\sigma_k:[0,+\infty)\rightarrow M$ be a sequence of future-directed, future-inextendible timelike, $h$-arc-length-parametrized curves starting at $S$ and such that $\sigma_k(t_k)=q_k$ for some $t_k\in (0,+\infty)$. By the compactness of $S$ and the Limit Curve Lemma, we can, up to passing to subsequences, assume that $\sigma_k(0)\rightarrow r\in S$, and that there exists a future-directed, future inextendible $C^0$ causal curve $\sigma:[0,+\infty)\rightarrow M$ with $\sigma(0)=r$ such that $$\sigma_k|_C \rightarrow \sigma|_C$$ $d_h$-uniformly for each compact set $C\subset [0,+\infty)$.
Suppose the sequence $(t_k)$ is bounded. Then we can assume, again up to passing to a subsequence, that $t_k\rightarrow t_0 \Rightarrow q=\sigma(t_0)$. Now, $q\notin S$, so $t_0>0$. By the achronality of $\partial I^+(\Sigma_+)$, the causal curve segment $\sigma |_{[0,t_0]}$ can be reparametrized as a future-directed null geodesic segment $\eta:[0,b]\rightarrow M$ without focal points to $S$ before $q=\eta(b)$. In particular, the null vector $\eta'(0)$ is normal to $S$, and hence it is either parallel to $K_{+}(r)$ or to $K_{-}(r)$. In the latter case, however, we’d have $q \in \mathcal{H}^+$, a contradiction. Thus we can assume, affinely reparametrizing $\eta$ if needed, that $\eta'(0)=K_{+}(r)$. But the acausality of $\Sigma$ implies that $\eta(0,b]$ cannot intersect $\Sigma$, in which case, as we discussed above, $q\in I^+(\Sigma_{-})$, and indeed the maximal integral curve $\alpha$ through $q$ intersects $\Sigma_{-}$. The continuous curve $\rho_X\circ \eta:[0,b]\rightarrow \Sigma$ enters initially in $\Sigma _{+}$, but $\rho_X\circ \eta(b) \in \Sigma_{-}$, so there exists some $s_0\in (0,b)$ for which $\rho_X\circ \eta(s_0) \in \Sigma _{+}$. But then $\eta(s_0) \in I^+(\Sigma _{+})$, so that $q\in I^+(\Sigma _{+})\cap \partial I^+(\Sigma_{+})$, again a contradiction. We conclude that $(t_k)$ must be unbounded. We can assume $t_k\rightarrow +\infty$. If $\sigma:[0,+\infty)\rightarrow M$ never left $\partial I^+(S)$, it could again be reparametrized as a (future-complete) null geodesic $S$-ray $\gamma:[0,+\infty)\rightarrow M$ initially parallel to $K_{-}(\gamma(0))$, a contradiction. Thus, for some $b\in (0,+\infty)$ $\sigma(b)\in I^+(S)$ and we can pick an open set $U\ni \sigma(b)$ such that $U\subset I^+(S)$, and also pick $p \in I^-(\sigma(b),U)$. Since $\sigma_k(b)\rightarrow \sigma(b)$, and eventually $t_k>b$, eventually $p\ll \sigma_k(b)\leq \sigma _k(t_k)=q_k$, and we conclude that $q\in \overline{I^+(p)}$. But then, past reflectivity implies that $p\in \overline{I^-(q)}$. Since $p \in I^+(S)\subset I^+(\overline{\Sigma}_{+}) (\equiv I^+(\Sigma_{+}) )$, we thus have $q \in I^+(\Sigma_{+})\cap \partial I^+(\Sigma_{+})$. This final contradiction thus establishes the Claim.
The rest of the proof of Thm. \[maintheorem\] now proceeds exactly as the proof of [@costa_e_silva_gannon Thm. 2.1], only with the caveat that if clause $(ii)$ in the statement holds, then we may use the universal covering instead of the more elaborate one therein. Therefore, we omit further details here.
$\Box$
Acknowledgments {#acknowledgments .unnumbered}
===============
IPCS is partially supported by the project MTM2016-78807-C2-2-P (Spanish MINECO with FEDER funds). Both authors wish to express their deep gratitude for the warm hospitality we received at the Dept. of Geometry and Topology of the University of Malaga, Spain, where this work was initiated. They extend their special thanks to José Luis Flores and Miguel Sánchez for helpful discussions.
[99]{}
J.K. Beem, P. Ehrlich, P. and K. Easley, [*Global Lorentzian geometry*]{}, Marcel Dekker, NY, 1996. B. Carter, [*Causal structure in space-time*]{}, Gen. Relativ. Gravit. 1:349–391 (1971). I.P. Costa e Silva, [*On the Gannon–Lee singularity theorem in higher dimensions*]{}, Class. Quantum Grav. 27:155016 (2010). https://doi.org/10.1088/0264-9381/27/15/155016. D. Gannon, [*Singularities in nonsimply connected space-times*]{}, J. Math. Phys. 16(12):2364–2367 (1975). https:// doi.org/10.1063/1.522498. D. Gannon, [*On the topology of spacelike hypersurfaces, singularities, and black holes*]{}, Gen. Relativ. Gravit. 7:219–232 (1976). https://doi.org/10.1007/BF00763437. A. Garc[í]{}a-Parrado and M. S[á]{}nchez, [*Further properties of causal relationship: causal structure stability, new criteria for isocausality and counterexamples*]{}, Class. Quantum Grav. 22:4589–4619 (2005). R. Geroch, [*Topology in general relativity*]{}, J. Math. Phys. 8:782–786 (1967). B.C. Haggman, G.W. Horndeski and G. Mess, [*Properties of a covering space defined by Hawking*]{}, J. Math. Phys. 21:2412–2416 (1980). S.W. Hawking, [*The occurrence of singularities in cosmology. III. Causality and singularities*]{}, Proc. R. Soc. London, Ser. A 300(1461):187–201 (1967). S.W. Hawking and G.F.R. Ellis, [*The large scale structure of space-time*]{}, Cambridge Monographs on Mathematical Physics, No. 1. Cambridge University Press, London-N.Y, 1973. J. Hedicke and S. Suhr, [*Conformally embedded spacetimes and the space of null geodesics*]{}, Commun. Math. Phys., 375:1561–1577 (2020). C.W. Lee, [*A restriction on the topology of Cauchy surfaces in general relativity*]{}, Commun. Math. Phys., 51:157–162 (1976). E. Minguzzi, [*K-causality coincides with stable causality*]{}, Commun. Math. Phys., 290:239–248 (2009), ArXiv:0809.1214. E. Minguzzi, [*Lorentzian causality theory*]{}, Living Rev. Relativ., 22:3 (2019). https://doi.org/10.1007/s41114-019-0019-x. E. Minguzzi, [*A gravitational collapse singularity theorem consistent with black hole evaporation*]{}, ArXiv: 1909.07348 (2019). E. Minguzzi and M. Sánchez, [*The causal hierarchy of spacetimes*]{}, in [*Recent developments in pseudo-Riemannian geometry*]{}, ESI Lect. Math. Phys, 299-358 (2008). B. O’Neill, [*Semi-Riemannian geometry with applications to relativity*]{}, Pure and Applied Mathematics 103, Academic Press Inc., Harcourt Brace Jovanovich, Publishers, NY, 1983.
[^1]: IPCS wishes to thank Roland Steinbauer for calling his attention to the fact that that assumption is made in [@costa_e_silva_gannon] without proper justification.
[^2]: Recall that a [*partial Cauchy hypersurface*]{} is by definition an acausal edgeless subset of a spacetime, which means in particular that it is a topological (i.e. $C^0$) hypersurface [@oneill]. In this paper, however, we always deal with smooth hypersurfaces.
[^3]: Note that in this argument we can weaken our convergence assumption to an [*averaged*]{} null convergence condition in the following form: $\int_0^{+\infty}Ric(\gamma'(t),\gamma'(t))dt \geq 0$ along any null geodesic $\gamma:[0,+\infty)\rightarrow M$, with $\gamma'(0) = K_{-}(\gamma(0))$.
[^4]: To see this, just project a continuous curve in $I^-(q_k)$ between two points of $I^-(q_k)\cap \Sigma$ onto $\Sigma$ using $\rho_X$
|
---
abstract: 'We show that time translation symmetry of a ring system with a macroscopic quantum ground state is broken by decoherence. In particular, we consider a ring-shaped incommensurate charge density wave (ICDW ring) threaded by a fluctuating magnetic flux: the Caldeira-Leggett model is used to model the fluctuating flux as a bath of harmonic oscillators. We show that the charge density expectation value of a quantized ICDW ring coupled to its environment oscillates periodically. The Hamiltonians considered in this model are time independent unlike “Floquet time crystals" considered recently. Our model forms a metastable quantum time crystal with a finite length in space and in time.'
author:
- 'K. Nakatsugawa$^1$, T. Fujii$^3$, S. Tanda$^{1,2}$'
bibliography:
- '20170829APS\_With\_Referee\_Reply.bib'
title: |
Quantum Time Crystal By Decoherence:\
Proposal With Incommensurate Charge Density Wave Ring
---
INTRODUCTION
============
The original proposal of a quantum time crystal (QTC) given by Wilczek [@QTC] and Li *et al.* [@Wigner] is a quantum mechanical ground state which breaks time translation symmetry. In this QTC ground state there exists an operator $\hat Q$ whose expectation value oscillates permanently with a well-defined “lattice constant" $P$, that is, with a well-defined period.\
Volovik [@Volovic] relaxed the condition of permanent oscillation and proposed the possibility of effective QTC, that is, a periodic oscillation in a metastable state such that the oscillation will persist for a finite duration $\tau_Q\gg P$ in the time domain and will eventually decay.\
In this paper we promote Volovik’s line and consider the possibility of a metastable QTC state without spontaneous symmetry breaking: We consider symmetry breaking by *decoherence* of a macroscopic quantum ground state (FIG. \[Symm.Break by Decoh.\]). Decoherence is defined as the loss of quantum coherence of a system coupled to its environment. Coupling to environment will inevitably introduce friction to the system such that the oscillation will eventually decay at $t=\tau_\text{damp}$. However, for $t<\tau_Q\ll\tau_\text{damp}$ the oscillation period $P$ is well defined. If friction is sufficiently weak such that $P\ll\tau_Q$, then we have a model of effective QTC with life time $\tau_Q$.
![The concept of translation symmetry breaking by decoherence is illustrated. (a) A simplified version of the two-state system considered by Leggett *et al.*[@LeggettTwoState]. A particle can tunnel through a potential barrier and exist at two states simultaneously. However, if this system starts to interact with its surrounding environment, then the particle will localize at one of the states. (b) Similarly, the ground state of a free particle confined on a ring is a plane wave state. Coupling to environment will localize the particle and break rotational symmetry. This “particle" corresponds to the phase of an incommensurate charge density wave ring (ICDW ring) in our model (FIG. \[FluctuatingB\]).[]{data-label="Symm.Break by Decoh."}](Symm_Break_by_Decoh.eps){width="\linewidth"}
\
Our model consists of a ring-shaped incommensurate charge density wave (ICDW ring) threaded by a fluctuating magnetic flux (FIG. \[FluctuatingB\](a)). A charge density wave (CDW) is a periodic (spatial) modulation of electric charge density which occurs in quasi-one-dimensional [crystals]{} [@Gruner2; @*Sambongi; @*Monceau]: the periodic modulation of the electric charge density occurs due to electron-phonon interaction. [If the ratio between the CDW wavelength $\lambda$ and the lattice constant $a$ of the crystal is a simple fraction like $2$, $5/2$ etc, then the CDW is commensurate with the underlying lattice. A commensurate CDW cannot move freely because of commensurability pinning, i.e. the CDW phase is pinned by ions’ position in the crystal. On the other hand, if $\lambda/a$]{} is effectively an irrational number [@Sambongi], then the CDW is incommensurate with the underlying lattice. An ICDW ring with a radius of $10\mu$m, for instance, contains approximately $10^5$ wavelengths [@Zettl], so it is possible that $a/\lambda$ is very close to an irrational number. [See the discussion section for an elaboration of this assumption.]{}
![(a) We consider an incommensurate CDW ring threaded by a fluctuating magnetic flux. This figure shows a commensurate CDW with $\lambda/a=2$ because it is easier to visualise. The wave (typically $\sim 10^5$ wavelengths) represents the charge density and the dots represent the atoms of a quasi-one dimensional crystal.(b) ICDW ring crystals [such as monoclinic TaS$_3$ ring crystals and NbSe$_3$ ring crystals ]{}can be produced experimentally [@ring; @Matsuura; @ABCDW]. Our model can be tested provided clean ring crystals with almost no defects and impurities can be produced. (c)The ground state of an isolated quantized ICDW ring is a superposition of periodically oscillating ICDWs, hence the oscillation is unobservable. Coupling to environment (fluctuating magnetic flux) will break the superposition and the oscillation becomes apparent.[]{data-label="FluctuatingB"}](FluctuatingB.eps){width="\linewidth"}
The sliding of an ICDW [without pinning]{} is described by a gapless Nambu-Goldstone (phason) mode [@Gruner2] and the energy of an ICDW is independent of its phase (i.e. position), which implies that the expected ground state of an ICDW ring is a superposition of ICDWs with different phases. Ring-shaped crystals and ring-shaped (I)CDWs have been produced [@ring] (FIG. \[FluctuatingB\] (b)). The presence of circulating CDW current [@Matsuura] and Aharonov-Bohm oscillation (evidence of macroscopic wave function) [@ABCDW] are verified experimentally.\
We show in section \[Without\_Dissipation\] that the charge density expectation value of an isolated ICDW ring [with moment of inertia $I$ is periodic in time with period $P=4\pi I/\hbar$]{}. This periodicity is a consequence of the uncertainty relation on $S^1$ (ring). However, this oscillation becomes unobservable at ground state because the ground state of an isolated ICDW ring is a plane wave state, *i.e.* a coherent superposition of ICDWs with different phases. Therefore, in section \[With\_Dissipation\] we use the Caldeira-Leggett model [@Caldeira; @Weiss] to show that time translation symmetry is broken by decoherence. More precisely, the [superposition is broken by decoherence]{} and the amplitude of the ICDW oscillates periodically (FIG. \[FluctuatingB\] (c)). If the ICDW ring weakly couples to its environment then this state is a metastable ground state. Therefore, our model forms an effective QTC with a finite length in space and in time.\
Before developing our main arguments, we compare our work to recent developments of QTC. In analogy with spatial crystal, the original proposal of QTC is based on the spontaneous breaking of time translation symmetry. However, Bruno [@Bruno3] and Watanabe and Oshikawa [@Watanabe] theoretically proved that spontaneous breaking of time translation symmetry cannot occur at ground state. Recently, it was shown that there is a notion of spontaneous breaking of time translation symmetry in periodically driven (Floquet) states [@FTCTheory; @*DTCYao; @*Prethermal] and this idea was proved experimentally[@FTCChoi; @*FTCZhang]. On the other hand, the periodic oscillation we consider in this paper is inherent to ring systems with a macroscopic wave function.
Ground State of an Isolated ICDW Ring {#Without_Dissipation}
=====================================
Classical Theory of ICDW Ring
-----------------------------
It is well known that the electric charge density of a quasi-one-dimensional crystal becomes periodic by opening a gap at the [Fermi wave number $k_\mathrm{F}$]{} and form a charge density wave (CDW) [ground]{} state with a wavelength $\lambda=\pi/k_\mathrm{F}$ [@Gruner2]. Consider a CDW formed on a ring-shaped quasi-one-dimensional crystal with radius $R$. The order parameter of this CDW ring is a complex scalar $\Delta(x,t)=|\Delta(x,t)|\exp[i\theta(x,t)]$, where $|\Delta(x,t)|$ is the size of the energy gap at $\pm k_\mathrm{F}$, $\theta(x,t)$ is the phase of the CDW, $x\in[0,2\pi R)$ is the coordinate on the crystal, and $t$ is the time coordinate. The charge density is given by $$n(x,t)=n_0+n_1\cos[2k_\text Fx+\theta(x,t)]
\label{chargedensity}$$ where $n_0$ is the average charge density and $n_1$ is the amplitude of the wave. Bogachek *et al.* [@Bogachek1] derived the following Lagrangian density of the phase of a ring-shaped incommensurate CDW (ICDW ring) threaded by a magnetic flux $${\mathscr{L}_0\left(\frac{\partial\theta}{\partial t},\frac{\partial\theta}{\partial x}\right)}=\frac{N_0}{2}\left[\left(\frac{\partial \theta}{\partial t}\right)^2-c_0^2\left(\frac{\partial \theta}{\partial x}\right)^2\right]+\frac{eA}{\pi}\frac{\partial \theta}{\partial t}$$ where $A$ is the magnetic vector potential, $N_0=v_\text F^2\hbar^2N(\varepsilon_\mathrm{F})/(2c_0^2)$, $N(\varepsilon_\mathrm{F})$ is the density of states of electrons at the Fermi level per unit length and per spin direction, $v_\text F$ is the Fermi velocity of the crystal, $c_0
=\sqrt{m/m^\ast}v_\text F$ is the phason velocity, $m^\ast$ is the effective mass of electrons and $\hbar$ is the reduced Planck constant. We first consider an isolated ICDW ring with $A=0$. Assuming that $N(\varepsilon_\mathrm{F})$ is equivalent to the density of states of electrons on a one dimensional line, that is, $N(\varepsilon_\mathrm{F})=1/(\pi\hbar v_\mathrm{F})$, we have $
N_0=\hbar v_\mathrm F/(2\pi c_0^2)$. An incommensurate CDW (ICDW) can slide freely because of spatial translation symmetry, so the dynamics of an ICDW is understood by its phase $\theta$. We further assume the rigid-body model of ICDW, i.e. the ICDW ring does not deform locally and the phase $\theta(x,t)=\theta(t)$ is independent of position. Then, the Lagrangian, the canonical angular momentum and the Hamiltonian of the ICDW ring are, respectively $$\begin{aligned}
L_0(\dot\theta)&=\int_0^{2\pi R}dx\mathscr L_0(\dot\theta)=\frac{I}{2}\dot\theta^2,
\label{Classical_Phase_Lagrangian}
\\
\pi_\theta(\dot\theta)&=\frac{\partial L_0(\dot\theta)}{\partial \dot\theta}=I\dot\theta,
\label{momentum}
\\
H_0(\pi_\theta)&=\pi_\theta\dot\theta-L_0(\dot\theta)=\frac{\pi_\theta^2}{2I}
\label{Classical_Phase_Hamiltonian}\end{aligned}$$ where $\dot\theta=d\theta/dt$ and $I=\hbar R v_\mathrm{F}/ c_0^2$ is the moment of inertia. We note that , , and are time independent.
Quantization of [an Isolated]{} ICDW Ring
-----------------------------------------
Next, we quantize the ICDW ring system. We show that a quantized ICDW ring possesses an inherent oscillation which originates from the uncertainty principle. Let $\hat H_0=\hat\pi_\theta^2/(2I)$ and $\hat\pi_\theta$ be the Hamiltonian operator and angular momentum operator of the ICDW ring, respectively. The macroscopic quantum state $\psi\in\mathscr H$ is defined in the Hilbert space $\mathscr H$ of positive square-integrable functions with the periodic boundary condition $\psi(\theta+2\pi)=\psi(\theta)$. The canonical commutation relation $[\hat\theta,\hat\pi_\theta]=i\hbar$ is not satisfactory because $\hat\theta$ is a multi-valued operator and is not well-defined. Ohnuki and Kitakado [@Ohnuki] resolved this difficulty by using the unitary operator $\hat W$ and the self-adjoint angular momentum operator $\hat \pi_\theta$ defined by $$\braket{\theta|\hat W|\psi}=e^{i\theta}\psi(\theta),\qquad
\braket{\theta|\hat \pi_\theta|\psi}=-i\hbar\frac{\partial\psi(\theta)}{\partial\theta}$$ which satisfy the commutation relation on $\mathscr H$ $$[\hat \pi_\theta,\hat W]=\hbar\hat W.\label{Algebra_Ohnuki}$$ $\hat H_0$ is a function of $\hat\pi_\theta$ only, hence the complete orthonormal set $\{\psi_l\}_{l=-\infty}^\infty$ of momentum eigenstates spans $\mathscr H$ and satisfy $\psi_l(\theta)=e^{il\theta}/\sqrt{2\pi}$. The eigenvalues of $\hat\pi_\theta$ are quantized with $\braket{\psi_l|\hat\pi_\theta|\psi_l}=l\hbar,l\in\mathbb Z$. $\hat W$ and $\hat W^\dagger$ are ladder operators which satisfy $\hat W\psi_l=\psi_{l+1}$ and $\hat W\psi_l=\psi_{l-1}$. Therefore, is the one dimensional version of the well known angular momentum algebra [@Sakurai]. Time evolution is introduced via the Heisenberg picture: $\hat \pi_\theta(t)=e^{i\hat H_0t/\hbar}\hat \pi_\theta e^{-i\hat H_0t/\hbar}$ and $\hat W(t)=e^{i\hat H_0t/\hbar}\hat W e^{-i\hat H_0t/\hbar}$. $\hat \pi_\theta$ commutes with $\hat H_0$, so $\hat \pi_\theta(t)=\hat \pi_\theta$. From the commutation relation we obtain the following solutions of $\hat W(t)$: $$\hat W(t)=e^{it\hat\pi_\theta/I}\hat We^{-\frac{it}{2\mu}}=\hat We^{it\hat\pi_\theta/I}e^{\frac{it}{2\mu}}\label{TimeDependentW}$$ where $\mu=I/\hbar$. [The two different expressions in arise from the noncommutativity between $\hat W$ and $e^{it\hat\pi_\theta/I}$. ]{}For a QTC we need a periodic expectation value at the ground state. So, we define the time dependent charge density operator $${\hat n(x,t)=n_0+\frac{n_1}{2}\left(e^{2ik_\mathrm F x}\hat W(t)+\mathrm{h.c.}\right)}\label{chargedensityoperator}$$ and replace the classical charge density by the expectation value $$n(x,t)=\braket{\hat n(x,t)}.$$ Any states in $\mathscr H$ must be a linear superposition of $\{\psi_l\}$, that is $\psi=\sum_{l\in\mathbb Z}c_l\psi_l$ provided $\sum_{l}|c_l|^2=1$. Therefore, the expectation values $\braket{\hat W(t)}$ and $\braket{\hat n(x,t)}$ [are periodic with]{} period $P=4\pi\mu$ for any state $\psi$: $$\begin{aligned}
\begin{split}
&\braket{\hat W(t)}{=\frac{1}{2}\text{tr}[\hat W(t)\hat\rho+\hat\rho\hat W(t)]}
\\
&=\frac{1}{2}\!\int_{-\pi}^\pi\! d\theta e^{i\theta}[e^{\frac{it}{2\mu}}\rho(\theta+ t/\mu,\theta)\!+\!\rho(\theta,\theta- t/\mu)e^{-\frac{it}{2\mu}}].\end{split}
\label{WExpectationValue}\end{aligned}$$ From the two different expressions of $W(t)$ in we can define the Weyl form of the commutation relation [@WeylCCR] $$\begin{aligned}
\hat We^{it\hat\pi_\theta/I}=e^{it\hat\pi_\theta/I}\hat We^{-\frac{it}{\mu}}\label{WeylCCR}\end{aligned}$$ hence the phase $\frac{t}{2\mu}$ and the periodic oscillation with period $4\pi\mu$ is a manifestation of the uncertainty principle. For an alternative explanation, let us consider an electron with effective mass $m^\ast$ confined in a finite space with volume $L\sim 2R$. From the uncertainty principle, the momentum [uncertainty]{} of this particle is $\Delta p\sim\hbar/(4R)$. This means that the particle’s wave packet expands with velocity $v=\Delta p/m^\ast\sim\hbar/(4m^\ast R)$. Then, because of the periodic boundary condition, the physical quantity $W=e^{i x/\lambda}$ is periodic with period $P=\lambda/v\sim4\pi m^\ast R/(\hbar k_\text F)=4\pi m^\ast R/mv_\text F=4\pi Rv_\text F/c_0^2=4\pi\mu$. So, the origin of the periodicity is (i) the macroscopic wave function of the ICDW ring diffuses due to the uncertainty principle then (ii) $W=e^{i\theta}$ oscillates periodically. However, the oscillation in is not observable at the ground state $\hat\rho_0\equiv\ket{\psi_0}\bra{\psi_0}$ because $\braket{\theta|\hat\rho_0|\phi}=\frac{1}{2\pi}$ and the $\theta$ integral vanishes. [Therefore, the ground state of an isolated ICDW ring is not yet a time crystal because of superposition.]{}
COUPLING TO ENVIRONMENT {#With_Dissipation}
=======================
Now, suppose that the ICDW ring starts to interact with its surrounding enviromnent at $t=0$. Then, we expect decoherence of the phase $\theta$. This interaction is modeled using the Caldeira-Leggett model [@Caldeira] which is a model quantum Brownian motion. It describes a particle coupled to its environment. This environment is described as a set of non-interacting harmonic oscillators. [First, the classical solution $\theta(t)$ is calculated to study the dynamics of the ICDW ring. Next, this system is quantized to calculate the amplitude of the charge density expectation value $\braket{\hat n(x,t)}$]{}.
Classical Theory of ICDW Ring With Environment
----------------------------------------------
Let us consider the following Lagrangian of an ICDW ring threaded by a fluctuating magnetic flux $$\tilde L(\dot\theta,\mathbf q,\dot{\mathbf q})=\frac12I\dot\theta^2+A(\mathbf q)\dot\theta+\sum_{j=1}^{\mathcal N}\left(\frac12m\dot q_j^2-\frac12 m\omega_j^2q_j^2\right)
\label{ClassicalLagrangian1}$$ where $q_j$ are the normal coordinates of the fluctuation and $\tilde\pi_\theta=\partial \tilde L/\partial\dot\theta$ and $p_j=\partial \tilde L/\partial\dot q_j$ are the canonical momenta of the ICDW ring and the environment, respectively. The fluctuating magnetic flux is given by $$A(\mathbf q)=\sum_{j=1}^\mathcal N cq_j.$$ Classically, this magnetic flux will randomly changes the phase and the *mechanical* angular momentum $I\dot\theta$ of the ICDW ring due to electromotive force. An equivalent Lagrangian obtained by a Canonical transformation is $$\begin{aligned}
\begin{split}
L(\dot\theta,\mathbf R,\dot{\mathbf R})&=\frac12I\dot\theta^2\!+\!\frac{m}{2}\sum_j^\mathcal N\!\left[\!\dot R_j^2(\theta)\!-\!\omega_j^2\left(\!\!R_j(\theta)-\frac{C_j\theta}{m\omega_j^2}\! \right)^2\right]
\label{ClassicalLagrangian2}
\end{split}\end{aligned}$$ which is the Lagrangian of a bath of field particles $R_j$ coupled to the phase $\theta$ by springs. $R_j(p_j,\theta)=-\frac{p_j-c\theta}{m\omega_j}$, $P_j(q_j)=m\omega_jq_j$ and $C_j=c\omega_j$. and are precisely the kinds of Lagrangian considered by Caldeira and Leggett, so we can use the results in [@Caldeira] but with slight modifications due to the periodicity of the ring.
[Classical Solution]{}
----------------------
The equation of motion of $\theta$ obtained from the Lagrangian is the generalized Langevin equation[@Hanggi1997] $$I\ddot\theta(t)+2\int_0^td\tau\alpha_\text I(t-\tau)\theta(\tau)=\xi(t)\label{GLEModified}$$ with the dissipation kernel $\alpha_\text I(t-\tau)$, the memory function $\gamma(t-\tau)$ and the classical fluctuating force $\xi(t)$ defined by $$\begin{aligned}
\alpha_{\text I}(t-\tau)&=I\gamma(0)\delta(t-\tau)+\frac{I}{2}\frac{d}{dt}\gamma(t-\tau),\label{alphaIDef}
\\
\gamma(t-\tau)&=\sum_{j=1}^{\mathcal N}\frac{C_j^2}{Im\omega_j^2}\cos\omega_j(t-\tau),
\\
\xi(t)&=\sum_{j=1}^{\mathcal N}C_j\left[R_j(0)\cos\omega_jt+\frac{P_j(0)}{m\omega_j}\sin\omega_jt\right].\end{aligned}$$ The correlation function [of the classical force is]{} given by the noise kernel $$\begin{aligned}
\braket{\xi(t)\xi(\tau)}_\text{env}&=\hbar\alpha_{\text R}(t-\tau)
\\
\alpha_\text R(t-\tau)&=\sum_j^\mathcal N\frac{C_j^2}{2m\omega_j}\coth\left(\frac{\hbar\omega_j}{2k_\mathrm{B}T}\right)\cos\omega_j(t-\tau)
$$ where the average $\braket{\cdot}_\text{env}$ is taken with respect to the environment coordinate at equilibrium. It is convenient to define the spectral density function $$\mathcal J(\omega)=\frac{\pi}{2}\sum_j^\mathcal N\frac{C_j^2}{m_j\omega_j}\delta(\omega-\omega_j)
\label{J_def}$$ and assume the power law spectrum $\mathcal J(\omega)=I g_s\omega^s$ [@Grabert1988115; @*Schramm1987] with a cutoff frequency $\Omega$ and $0<s<2$. Then, $\alpha_\text R(t-\tau)$ can be written $$\begin{aligned}
\alpha_\text R(t-\tau)&=\frac{Ig_s}{\pi}\int_0^\Omega \omega^s\coth\left(\frac{\hbar \omega}{2k_\mathrm BT}\right)\cos\omega(t-\tau)d\omega.\label{alphaRIntegral}\end{aligned}$$ The classical solution of $\theta$ is $$\theta(t)=G(t)\dot\theta(0)+\dot G(t)\theta(0)+\frac{1}{I}\int_0^td\tau G(t-\tau)\xi(t)$$ with the fundamental solution $$G(t)=\mathcal L^{-1}\left[\frac{1}{z^2+z\hat\gamma(z)}\right](t).$$ where $\mathcal L^{-1}$ is the inverse Laplace transform. The Laplace transform of the memory function $\gamma(t)$ can be written [@Weiss] $$\hat\gamma(z)=\omega_s^{2-s}z^{s-1},\qquad \omega_s=\left(\frac{g_s}{\sin\frac{\pi s}{2}}\right)^{1/(2-s)}$$ and $G(t)$ takes the form of a generalized Mittag-Leffler function $E_{\alpha,\beta}(x)=\sum_{k=0}^\infty\frac{x^k}{\Gamma(\alpha k+\beta)}$: $$\begin{aligned}
G(t)=tE_{2-s,2}[-(\omega_s t)^{2-s}].\end{aligned}$$ For ohmic damping with $s=1$ and $\hat \gamma(z)=g_1\equiv2\gamma$, we obtain $$G(t)=\frac{1-e^{-2\gamma t}}{2\gamma}.\label{ClassicalMotionOhmic}$$ [$G(t)$ and $\dot G(t)$ are shown in FIG. \[ClassicalMotion\]. We note that $\dot G(t)\approx 1$ for $t$ less than some damping time scale $\tau_{\text{damp},s}$. In other words, the fluctuating magnetic flux does not affect the dynamics of an ICDW ring for $t<\tau_{\text{damp},s}$ and $e^{i\theta(t)}$ oscillates periodically with period $P\approx 4\pi\mu$. Next, we quantize the ICDW ring + environment system to show that this oscillation is observable for a finite time $\tau_Q$ and form an effective QTC as a metastable state.]{}
![These plots are shown with $g_s=1$ Hz$^{2-s}$ and $\mu= 10^{-8}$ sec. (a)The fundamental solution $G(t)$ of a classical ICDW ring coupled to its environment is shown for sub-ohmic ($s<1$), ohmic $(s=1)$ and super-ohmic ($s>1$) damping. Note that $G(t)\approx t$ for $t$ less than some time scale $\tau_{\text{damp},s}$. The time $t$ is normalized by $4\pi\mu$, so the horizontal axis gives the number of “lattice points". (b)The derivative of the fundamental solution, $\dot G(t)$, heuristically describes the velocity of the ICDW ring. Note that $\dot G(t)\approx 1$ for $t\ll \tau_{\text{damp,s}}$.[]{data-label="ClassicalMotion"}](ClassicalMotion.eps){width="0.8\linewidth"}
Quantization [of ICDW Ring Coupled to Environment]{}
----------------------------------------------------
[The ICDW ring $+$ environment system is]{} quantized using the commutation relations $[\hat{\tilde\pi}_\theta,\hat W]=\hbar\hat W$ and $[\hat q_j,\hat p_k]=i\hbar\delta_{jk}$. Define the orthonormal position state $\ket{\mathbf q}=\prod_{i=1}^\mathcal{N}\ket{q_i}$ and the orthonormal momentum state $\ket{\mathbf p}=\prod_{i=1}^\mathcal{N}\ket{p_i}$ such that $$\braket{\mathbf q|\hat q_j|\psi}=q_j\braket{\mathbf q|\psi},\qquad \braket{\mathbf q|\hat p_j|\psi}=-i\hbar\frac{\partial}{\partial q_j}\braket{\mathbf q|\psi},$$ and the inner product of $\ket{\mathbf q}$ and $\ket{\mathbf p}$ is defined as $\braket{\mathbf q|\mathbf p}=\frac{1}{\sqrt{2\pi\hbar}^\mathcal{N}}\exp\left(\frac{i}{\hbar}\mathbf q\cdot\mathbf p\right)$. The periodic boundary condition of the ring implies that $\braket{\theta+2\pi n|\tilde\pi_\theta}=\braket{\theta|\tilde\pi_\theta}$ for some integer $n$, hence the angular momentum eigenstates are quantized: $\braket{\psi_l|\hat{\tilde\pi}_\theta|\psi_l}=l\hbar, l=0,\pm,\pm2,\dots$, $\ket{\tilde\pi_\theta}=\hbar^{-1/2}\ket{\psi_l}$, $\braket{\theta|\psi_l}=\frac{1}{\sqrt{2\pi}}e^{il\theta}$. Moreover, one can easily show that $$\braket{\theta,\mathbf p|\tilde\pi_\theta,\mathbf q}=\braket{\theta,\mathbf R(\theta)|\pi_\theta,\mathbf P}.$$ $\hat W$ is independent of the environmental coordinate. So, the expectation value of $\hat W$ is $$\begin{aligned}
\braket{\hat W(t)}
&=\frac{1}{2}\int_{-\pi}^\pi d\theta_\text f\int_{-\pi}^\pi d\phi_\text f \rho(\theta_\text f,\phi_\text f,t)\braket{\phi_\text f|\hat W|\theta_\text f}
\\
&+\frac{1}{2}\int_{-\pi}^\pi d\theta_\text f\int_{-\pi}^\pi d\phi_\text f\braket{\theta_\text f|\hat W|\phi_\text f}\rho(\phi_\text f,\theta_\text f,t).\end{aligned}$$ where the reduced density matrix of the ICDW ring is (see *Appendix* \[CLS1\]) $$\begin{aligned}
\begin{split}
\rho(\theta_\text f,\phi_\text f,t)&=\int_{-\pi}^{\pi}d\theta_\text i\int_{-\pi}^{\pi}d\phi_\text i\sum_{l_1,l_2\in\mathbb Z}\rho(\theta_\text i,\phi_\text i,0)
\\
&\times
J(\theta_\text f+2\pi l_1,\phi_\text f+2\pi l_2,t;\theta_\text i,\phi_\text i,0).
\label{rho}
\end{split}\end{aligned}$$ The exact form of $J(\theta_\text f,\phi_\text f,t;\theta_\text i,\phi_\text i,0)$ for ohmic dissipation $s=1$ was calculated in [@Caldeira]. For general damping with arbitrary $s$ the computation of the reduced density matrix is essentially equivalent to [@Caldeira] and we obtain $$\begin{aligned}
J(\theta_\text f,\phi_\text f,t;\theta_\text i,\phi_\text i,0)&=F^2(t)\exp\left(\frac{i}{\hbar}S[\varphi^+_\text{cl},\varphi^-_\text{cl}]-\Gamma[\varphi^-_\text{cl}]\right).\label{JClassical}\end{aligned}$$ $\varphi^+_\text{cl}=\frac12(\theta_\text{cl}+\phi_\text{cl})$ and $\varphi^-_\text{cl}=\phi_\text{cl}-\theta_\text{cl}$ are the classical coordinates obtained from the Euler-Lagrange equation $$\begin{aligned}
I\ddot\varphi_\text{cl}^-(u)+2\int_u^td\tau\varphi_\text{cl}^-(\tau)\alpha_\text I(\tau-u)=0,\label{eqn1}
\\
I\ddot\varphi^+_\text{cl}(u)+2\int_0^ud\tau\varphi^+_\text{cl}(\tau)\alpha_\text I(u-\tau)=0\label{eqn2}\end{aligned}$$ whose solution are given in terms of boundary conditions $\varphi^\pm_\text i=\varphi_\text{cl}^\pm(0),\varphi^\pm_\text f=\varphi_\text{cl}^\pm(t)$: $$\begin{aligned}
\varphi^+_\text{cl}(u)&=\kappa_i(u;t)\varphi^+_\text i+\kappa_f(u;t)\varphi^+_\text f,
\label{varphisolfinal}
\\
\varphi^-_\text{cl}(u)&=\kappa_i(t-u;t)\varphi^-_\text f+\kappa_f(t-u;t)\varphi^-_\text i,
\label{varphiprimesolfinal}
\\
\kappa_i(u;t)&=\dot G(u)-\frac{\dot G(t)}{G(t)}G(u),
\qquad
\kappa_f(u;t)=\frac{G(u)}{G(t)}.\end{aligned}$$ The classical action and the noise action are given by $$\begin{aligned}
S[\varphi^+_\text{cl},\varphi^-_\text{cl}]
&=S_\text{cl}(\varphi^+_\text f,\varphi^-_\text f,t;\varphi^+_\text i,\varphi^-_\text i,0)
\\
&=-I[\dot\varphi^+_\text{cl}(t)\varphi^-_\text f-\dot\varphi^+_\text{cl}(0)\varphi^-_\text i],
\\
\Gamma[\varphi_\text{cl}^-]&=\Gamma_\text{cl}(\varphi_f^-,t;\varphi_i^-,0)
\\
&=\frac{1}{2\hbar}\int_0^td\tau\int_0^td\tau'\varphi^-(\tau)\alpha_\text R(\tau-\tau')\varphi^-(s).\end{aligned}$$ $F^2(t)$ is a normalization function such that $\text{tr}[\rho(t)]=\braket{1}=1$. The winding numbers $l_1$ and $l_2$ can be absorbed into the $\theta_\text f$ and $\phi_\text f$ integrals, respectively, by changing the domain of $\theta_\text f$ and $\phi_\text f$ from $S^1$ to $\mathbb R^1$. Then, taking care of the non-Hermiticity of $\hat W$, we obtain $$\begin{aligned}
\braket{\hat W(t)}&=\frac{r_1^+(t)+r_1^-(t)}{r_2^+(t)+r_2^-(t)},\label{WExpGeneral}
\\
r_1^+&=\int_{-\pi}^{\pi}d\theta_\text i\sum_{n\in\mathbb S_1(\theta,t)}\rho(\theta_\text i-f_1(t),\theta_\text i,0)\nonumber
\\
&\times e^{-in\pi-i\mu \theta \dot f_1(t)+\frac{i \mu}{2} f_1(t) \dot f_1(t)-\Gamma_\text{cl}(2\pi n,t;f_1(t),0)},\nonumber
\\
r_1^-&=\int_{-\pi}^{\pi}d\theta_\text i\sum_{n\in\mathbb S_1(\theta,t)}\rho(\theta_\text i,\theta_\text i+f_1(t),0)\nonumber
\\
&\times e^{-in\pi-i\mu \theta \dot f_1(t)-\frac{i \mu}{2} f_1(t) \dot f_1(t)-\Gamma_\text{cl}(2\pi n,t;f_1(t),0)},\nonumber
\\
r_2^+&=\int_{-\pi}^{\pi}d\theta_\text i\sum_{n\in\mathbb S_2(\theta,t)}\rho(\theta_\text i-f_2(t),\theta_\text i,0)\nonumber
\\
&\times e^{-i\mu \theta \dot f_2(t)+\frac{i\mu}{2} f_2(t)\dot f_2(t)-\Gamma_\text{cl}(2\pi n,t;f_2(t),0)},\nonumber
\\
r_2^-&=\int_{-\pi}^{\pi}d\theta_\text i\sum_{n\in\mathbb S_2(\theta,t)}\rho(\theta_\text i,\theta_\text i+f_2(t),0)\nonumber
\\
&\times e^{-i\mu \theta \dot f_2(t)-\frac{i\mu}{2} f_2(t)\dot f_2(t)-\Gamma_\text{cl}(2\pi n,t;f_2(t),0)}.\nonumber\end{aligned}$$ with $f_1(t)=2\pi n \dot G(t)-G(t)/\mu$, $f_2(t)=2\pi n \dot G(t)$, $\mathbb S_1(\theta,t)=\{n\in\mathbb Z|-\pi<\theta+f_1(t)<\pi\}$, and $\mathbb S_2(\theta,t)=\{n\in\mathbb Z|-\pi<\theta+f_2(t)<\pi\}$. This is the most general form of the expectation value of $\hat W$ for an ICDW ring coupled to its environment. Although the derivation of is exact, it is not very insightful, so we make some approximations.
Early Time [Approximation]{}
----------------------------
Let us consider the classical solution (22) with $\varphi^+_\text{cl}=\theta_\text f$ and ohmic damping , then we see immediately that $\dot\theta(t)\sim \dot\theta(0)e^{-2\gamma t}$. Therefore, we are interested in the range $t\ll1/(2\gamma)\equiv\tau_{\text{damp},1}$. For general dissipation, we can see from FIG. \[ClassicalMotion\] that there exist a time scale $\tau_{\text{damp},s}$ such that $G(t)\approx t$, $\dot G(t)\approx 1$ for $t\ll\tau_{\text{damp},s}$. Then, writing $t=2\pi I(m+a)/\hbar$ for an integer $m$ and $0<a<1$, we have $\mathbb S_1(\theta<-2a\pi,t)=\{m+1\}$, $\mathbb S_1(\theta>-2a\pi,t)=\{m\}$, and $\mathbb S_2(\theta,t)=\{0\}$. Therefore, we conclude that $\mathbb S_1(\theta,t)$ is approximately the $(m/2)^{\text{th}}$ lattice point. Then, using $\dot f_1(t)\approx -\dot G(t)\hbar/I$ we obtain the approximate form $$\begin{aligned}
\braket{\hat W(t)}&\approx\int_{-\pi}^{\pi}d\theta_\text i\sum_{n\in\mathbb S_1(\theta,t)}\rho(\theta_\text i,\theta_\text i-G(t)/\mu,0)
\\
&\times\exp\left\{i\theta_\text i\dot G(t)-i\frac{G(t)\dot G(t)}{2\mu}-\Gamma_\text{cl}(2\pi n,t;f_1(t),0)\right\}
\\
&+\int_{-\pi}^{\pi}d\theta_\text i\sum_{n\in\mathbb S_1(\theta,t)}\rho(\theta_\text i+G(t)/\mu,\theta_\text i,0)
\\
&\times\exp\left\{i\theta_\text i\dot G(t)+i\frac{G(t)\dot G(t)}{2\mu}-\Gamma_\text{cl}(2\pi n,t;f_1(t),0)\right\}.\end{aligned}$$ For $t\ll\tau_{\text{damp},s}$ we have $\varphi^-_\text{cl}(\tau)\approx \tau/\mu$ and the noise action can be written $$\begin{aligned}
&\Gamma_\text{cl}(2\pi n,t;f_1(t),0)\approx \Gamma_{T,s}(t)
\\
&\qquad=\frac{g_s}{2\pi \mu}\int_0^\Omega d\omega\coth\left(\frac{\hbar \omega}{2k_\text BT}\right)\Upsilon(\omega),
\\
&\Upsilon(\omega)=\omega^{s-4}(2+\omega^2t^2-2\cos\omega t-2\omega t\sin\omega t)\end{aligned}$$
![(a) The amplitude of the charge density oscillation is shown for $g_s=1$ Hz$^{2-s}$, $\mu=10^{-8}$sec. and $\Omega=1/\mu$. This oscillation is an effective QTC. In general, the oscillation amplitude is larger for super-ohmic($s>1$) damping but has a small life time $\tau_{Q}$. On the other hand, the oscillation amplitude is small for sub-ohmic ($s<1$) damping but has a long $\tau_Q$. (b) The charge density oscillation is shown for super-ohmic damping with $s=1.2$, $g_{1.2}=1$ Hz$^{0.8}$, $\mu=10^{-8}$sec. and $\Omega=1/\mu$. The time $t$ is normalized such that the horizontal axis gives the number of “lattice points".[]{data-label="GSOsc"}](n1plots.eps){width="0.8\linewidth"}
Taking the low temperature limit $\frac{\hbar \Omega}{2k_\text BT}\to\infty$ such that $\coth\frac{\hbar \Omega}{2k_\text BT}\to1$ we obtain $$\begin{aligned}
\Gamma_{T,s}(t)
=&-\frac{g_s}{\pi \mu}\frac{\Omega ^{s-3} \left(\,
_1F_2\left(\frac{s-3}{2};\frac{1}{2},\frac{s-1}{2};-\frac{1}
{4} t^2 \Omega ^2\right)-1\right)}{s-3}
\\
&-\frac{g_st^2}{2\pi \mu}\frac{\Omega ^{s-1} \left(\,
_1F_2\left(\frac{s-1}{2};\frac{1}{2},\frac{s+1}{2};-\frac{1}
{4} t^2 \Omega ^2\right)-1\right)}{s-1}\end{aligned}$$ where $\,_1F_2\left(a_1;b_1,b_2;z\right)$ is the generalized hypergeometric function. Define the decoherence time $\tau_{\text{decoh},s}$ such that $\Gamma_{T,s}(\tau_{\text{decoh},s})=1$. Numerical analysis shows that the order of $\tau_{\text{decoh},s}$ does not change with $\Omega<1/(2\mu)$ and decreases rapidly for $\Omega>1/(2\mu)$. If we set $\Omega\sim1/\mu$ [and]{} use the ground state $\rho_0$, then for $t\ll\tau_Q\equiv\min\{\tau_{\text{damp,s}},\tau_{\text{decoh},s}\}$ we have $$\begin{aligned}
n(x,t)&\approx n_0+n^\text{osc.}_1(t)\cos(2k_\text F x),\label{CDWResult}\\
n^\text{osc.}_1(t)&=n_1\text{sinc}[\pi\dot G(t)]\cos\left[\frac{\dot G(t)G(t)}{2\mu}\right]e^{-\Gamma_{T,s}(t)}.\end{aligned}$$ Equation is the main result of this paper. It shows that the amplitude of an ICDW ring threaded by a (time independent) fluctuating magnetic flux oscillates for a finite time $\tau_Q$ and form an effective QTC. In the no-damping limit $\hat\gamma(z)\to0$ we have $G(t)\to t,G(t)\to 1$ and recover $\braket{\hat W(t)}=0$. This charge density oscillation is shown in FIG. \[GSOsc\].
DISCUSSION {#Conclusion}
==========
[First, we elaborate our assumtion of ICDW ring. A mathematical definition of ICDW is that $\lambda/a$ is an irrational number. We note that a CDW formed on a macroscopic crystal is basically incommensurate because the wavelength of a CDW is given by $\lambda=\pi/k_\text{F}$, where the Fermi wave number $k_\text{F}$ is usually an irrational number for an arbitrary band filling. However, strictly speaking, $\lambda/a$ of a finite-size system can never be irrational. A physical condition is that the commensurability pinning energy is negligible, which is possible if $\lambda/a$ cannot be expressed as a simple fraction like 2, 5/2, etc. In order to explain this, suppose that for some integer $M\geq2$, $Ma/\lambda$ is an integer. In other words, the same atom-electron configuration is obtained if we move the CDW by $M$ wavelengths and $\epsilon_{k+2Mk_\text{F} }=\epsilon_k$ ($\epsilon_k$ is the energy of an electron with momentum $\hbar k$). Then, the energy required to move a CDW by a small phase $\phi$ from its equilibrium is [@LRA] $$\epsilon(\phi;M)=\frac{|\Delta|^2}{\epsilon_\text{F}} \left(\frac{e|\Delta|}{W}\right)^{M-2}\frac{M\phi^2}{2}$$ where $\epsilon_\text{F}$ is the Fermi energy, $W$ is the band width, $|\Delta|$ is the CDW gap width and $e$ is the elementary charge in natural units. We see that $\epsilon(\phi;M)$ approaches zero rapidly for large $M$ as the distinction between rational and irrational numbers becomes academic. For example, for a ring crystal with $N_a$ atoms and $N_\lambda$ CDW wavelengths, we obtain $\lambda/a=N_a/N_\lambda$. Therefore, for a large $N_a$ and a large $N_\lambda$, $M$ can always be arbitrary large (order of $N_a$), hence the commensurability energy is completely negligible. In fact, we can experimentally make sub-micrometer scale ICDW rings such that, we expect, $N_a$ and $N_\lambda$ are not so large but $M$ is very large such that commensurability energy is negligible. ]{}
[ Usually, superposition of ICDWs with different phases is not observed because of impurity pinning. However, the probability of impurity decreases with decreasing radius. Commensurability effect may become significant for small radius (more precisely, for some small $M$). But, the origin of the “time crystal periodicity” in our model is the uncertainty principle, which appears as a collective fluctuation of the ICDW phase. If commensurability effect becomes significant, then the ICDW phase is expected to fluctuate periodically around some phase $\theta=\theta_0$ determined by the ions’ position. This fluctuation is expected to become apparent and oscillate periodically by coupling to environment (fluctuating magnetic flux).]{}
[Next, we discuss the presence of an upper bound and a lower bound for the radius of the ICDW ring in our model. Decoherence induced breaking of time translation symmetry occurs, in principle, only in mesoscopic systems: There is an upper bound for the CDW radius determined by the coupling strength $\gamma=\omega_s/2$ and a lower bound given by the CDW wavelength $\lambda$. We will first calculate the upper bound by replacing the environment with an equivalent LC circuit. Here, we focus on Ohmic damping because an approximate form of $\gamma$ can be calculated explicitly. In general, the coupling strength $\gamma$ depends on the CDW radius $R$. In order to explicitly see the $R$ dependence, suppose that the fluctuating magnetic flux in our model comes from an external coil connected to a series of parallel capacitors. Then, the Lagrangian of the CDW + environment system is $$L_\text{circuit}=L_0+MI_{CDW} \sum_{j=1}^\mathcal{N}I_j +\sum_{j=1}^\mathcal{N}\frac m2(I_j^2-\omega_j^2 Q_j^2 ).$$ $L_0$ is the Lagrangian of an isolated ICDW ring, $I_{CDW}=e\dot\theta/\pi$ is the CDW current induced by the fluctuating flux, $Q_j$ is the net charge on the capacitor $j$ with capacitance $\mathscr C_j$, $m$ is the inductance of the coil, $\omega_j=1/\sqrt{m\mathscr C_j}$, and $M$ is the mutual inductance between the coil and the CDW. We immediately see that $L_\text{circuit}$ is equivalent to the Lagrangian in equation but with the interaction Lagrangian replaced by $\dot \theta\sum_{j=1}^\mathcal N \frac{Me}{\pi} \dot Q$. Therefore, define $p_j=mI_j$, $C_j=Me\omega_j^2/\pi$ and we obtain the Lagrangian in equation after the canonical transformation $$L=L_\text{circuit}-\frac{d}{dt}\sum_j^\mathcal{N}(MI_{CDW}+p_j) Q_j$$ with $P_j=m\dot R_j=m\omega_j Q_j$ and $R_j=-(p_j-MI_{CDW})/(m\omega_j)$. Assume that the radius of the coil $r_\text{coil}$ is much larger than the CDW radius $R$ and that the coil and the CDW are concentric. Then, we obtain $M=(\mu_0 \pi R^2)/(2r_\text{coil} )$. Next, integrate with respect to $\omega$ from $\omega=0$ to $\omega=\Omega=1/\mu$ and obtain $\gamma=\beta R$, $\beta=(\pi \mu_0^2 e^2 \rho c_0^6 )/(32\hbar r_\text{coil}^2 m v_F^3 )$. Here, $\mu_0$ is the permeability of free space and $\rho$ is the density of states defined by $\sum_j\to\int d\omega\rho$. If we assume that $\rho\sim\Omega$ like in the Caldeira-Leggett model, then the order of $\beta$ may change depending on the parameters of the CDW and the parameters of the coil, but $\gamma=\beta R$ is usually smaller than 1Hz. Now, if the radius of the CDW ring is too large, then periodicity does not appear because the oscillation period exceeds the lifetime $\tau_Q$ of the time crystal. The upper bound of the CDW radius R is given by the condition $N>1$, where $N=\min\{\tau_{\text{damp},s},\tau_{\text{deph},s}\}/(4\pi\mu)$ is the number of oscillations. For Ohmic damping ($s=1$) we have the approximate form $\tau_{\text{damp},1}=\gamma^{-1}$ and $\tau_{\text{deph},1}=\sqrt{\mu\gamma^{-1}}=\sqrt{\mu\gamma}\tau_{\text{damp},1}$. Let $\mu\gamma<1$, i.e. $\tau_{\text{damp},1}>\tau_{\text{deph},1}$ (which is a valid assumption because a typical value for $\mu$ with radius $10^{-6}$m is $10^{-6}$s and $\gamma$ is usually smaller than 1Hz), then, the upper bound to observe more than one oscillation is $R<\frac{c_0}{4\pi\sqrt{v_F\beta}}$ which is typically 1mm. There is also a lower bound determined by the CDW wavelength $\lambda$: The radius should be large enough to define a Fermi surface. This condition is given by $p_\text F=\hbar\pi/\lambda\gg\hbar/R$. In other words, $R$ should be much larger than $\lambda$.]{}
Finally, we discuss how our model can be tested experimentally and discuss how our results may be applicable to other physical systems. Ring-shaped crystals and ICDW ring crystals (such as monoclinic TaS$_3$ ring crystals and NbSe$_3$ ring crystals) have been produced and studied by the Hokkaido group [@ring; @Matsuura; @ABCDW]. Therefore, our model can be tested provided ring crystals with almost no defects and impurities can be produced. The oscillation in implies that the local charge density of the ICDW ring oscillates with frequency $\omega=\hbar/2I$. For a ring with diameter $2R=1\mu$m, $v_\text F/c_0= 10^3$ and $v_\mathrm{F}=10^5$m/s, we have $\omega=10^8Hz.$ The time dependence of the charge density modulation can be measured using scanning tunneling microscopy (STM) [@Ichimura] and/or using narrow-band noise with vanishing threshhold voltage [@IDO].
We recall that the origin of the quantum oscillation in our model is the uncertainty principle. If we were to consider a single particle with mass $m^\ast$ confined on a ring with radius $R$, then the particle’s wave packet expands with a velocity $v\sim \hbar/(4m^\ast R)$ and $e^{i\theta}$ oscillates with period $P=2\pi R/v$. However, a charge density wave has an internal periodicity given by the wavelength $\lambda$. Then the period of oscillation is $P\sim\lambda/v$. Therefore, because an ICDW ring is described by a macroscopic wave function with internal periodicity, the number of lattice points $N$ in our model is numerous. And yet, the periodicity of $\hat W(t)$ seems to be universal for any wave functions on $S^1$ (ring system). Therefore, our results may be applicable to earlier models of QTC such as [@QTC; @Wigner; @Josephson] and annular Josephson junctions [@AnnularJJ]. Moreover, it was shown vely recently in [@Haffner] that the ground state of a $^{40}$Ca$^+$ ring trap possesses rotational symmetry as the number of ions is decreased. Our results predict that quantum oscillations may appear in such ring traps with the appropriate set up. We also recall that Volovik’s proposal of metastable effective QTC [@Volovic] is not restricted to ring systems. Therefore, time translation symmetry breaking by *decoherence* may occur in other systems coupled to time-independent environment, and without a periodic driving field. We also expect that many other incommensurate systems such as incommensurate spin density waves [@Gruner2], incommensurate mass density waves [@PhysRevLett.40.1507; @PhysRevB.20.751; @PhysRevB.71.104508], or possibly some dielectrics that exhibit incommensurate phases [@blinc1986incommensurate], may be used to test our results and to model QTC without spontaneous symmetry breaking. We thank Kohichi Ichimura, Toru Matsuura, Noriyuki Hatakenaka, Avadh Saxena, Yuji Hasegawa, Kousuke Yakubo and Tatsuya Honma for stimulating and valuable discussions.
QUANTUM BROWNIAN MOTION ON $S^1$ {#CLS1}
================================
Reduced Density Matrix On $S^1$
-------------------------------
Consider an ICDW ring $A$ coupled to its environment $B$. Let $\rho_A$ and $\rho_B$ denote the density operators of $A$ and $B$, respectively. Let $\theta$ and $\phi$ be angular coordinates on the ring system $A$ and let $\mathbf p=\{p_k:k=1,\mathcal N\}$ and $\mathbf s=\{s_k:k=1,\mathcal N\}$ be momentum coordinates of the bath $B$. Then, the density matrix of the coupled system can be written $$\begin{aligned}
\rho(\theta,\mathbf p,\phi,\mathbf s)&=
\braket{\theta , \mathbf p|\rho_{AB}(t)|\phi ,\mathbf s}
\\
&=\int_0^{2\pi}d\theta'd\phi'\int_{-\infty}^{\infty}d\mathbf p'd\mathbf s'K(\theta ,\mathbf p,t;\theta',\mathbf p',0)K^\ast(\phi,\mathbf s,t;\phi',\mathbf s',0)\\
&\quad\times\braket{\theta',\mathbf p'|\rho_{AB}(0)|\phi',\mathbf s'}\end{aligned}$$ where $$K(\theta,\mathbf p,t;\theta',\mathbf p',0)=\braket{\theta,\mathbf p|e^{-\frac{i}{\hbar}\hat H t}|\theta',\mathbf p'}
\label{PropagatorTheta}$$ and $$K^\ast(\phi,\mathbf s,t;\phi',\mathbf s',0)=\braket{\phi',\mathbf s'|e^{\frac{i}{\hbar}\hat H t}|\phi,\mathbf s}
\label{PropagatorPhi}$$ are recognized as Feynman propagators if we notice that $\ket{\mathbf p}$ and $\ket{\mathbf s}$ are actually position states after canonical transformation. The propagators and can be written using path integrals by dividing the time $t$ into $N$ time steps of length $\epsilon=t/(N+1)$, $\mathbf p=\mathbf p_N$, $\mathbf p'=\mathbf p_0$, $\theta=\theta_N$ and $\theta'=\theta_0$. For $N\to\infty$ the propagator becomes $$\begin{aligned}
K(\theta,\mathbf p,t;\theta',\mathbf p',0)&=\Braket{\theta,\mathbf p|\lim_{\epsilon\to 0}\left(\exp-\frac{i\epsilon}{\hbar}\hat H\right)^N|\theta',\mathbf p'}\\
&=\lim_{\epsilon\to 0}\int_0^{2\pi}\left(\prod_{n=1}^{N-1}d\theta_n\right)\int_{-\infty}^\infty\left(\prod_{n=1}^{N-1}d\mathbf p_n\right)\prod_{n=1}^NK_n(\theta_n,\mathbf p_n,\epsilon;\theta_{n-1},\mathbf p_{n-1},0).\end{aligned}$$ The Hamiltonian operator $\hat H$ can be decomposed into the kinetic part $\hat{\mathcal K}$ and the potential part $\hat{\mathcal V}$, i.e. $\hat H=\hat{\mathcal K}+\hat{\mathcal V}$, which satisfy the eigenvalue equations $$\hat{\mathcal K}\ket{\psi_l,\mathbf q}=\mathcal K(l,\mathbf P)\ket{\psi_l,\mathbf q},\quad \hat{\mathcal V}\ket{\theta,\mathbf p}=\mathcal V(\theta,\mathbf R(\theta))\ket{\theta,\mathbf p}.$$ Then we obtain $$\begin{aligned}
K&(\theta_n,\mathbf p_n,\epsilon;\theta_{n-1},\mathbf p_{n-1},0)\\
&=\sum_{l_n}\int_{-\infty}^\infty d\mathbf q_n\Braket{\theta_n,\mathbf p_n|\exp\left(-\frac{i\epsilon}{\hbar}\hat{\mathcal K}\right)|\psi_{l_n},\mathbf q_n}\Braket{\psi_{l_n},\mathbf q_n|\exp\left(-\frac{i\epsilon}{\hbar}\hat{\mathcal V}\right)|\theta_{n-1},\mathbf p_{n-1}}\\
&=\sum_{l_n=-\infty}^\infty\frac{1}{2\pi}\int_{-\infty}^\infty \frac{d\mathbf q_n}{(2\pi \hbar)^{\mathcal N}}\exp\left\{-\frac{i\epsilon}{\hbar}\mathcal K(l_n,\mathbf P_n)-\frac{i\epsilon}{\hbar}\mathcal V(\theta_{n-1},\mathbf R_{n-1}(\theta_{n-1}))\right\}\\
&\quad\times\exp\left\{il_n(\theta_n-\theta_{n-1})-\frac{i}{\hbar}\mathbf q_n\cdot(\mathbf p_n-\mathbf p_{n-1}))\right\}.\end{aligned}$$ Next, define $A_n=\sum_jcq_{j,n}=\sum_j\frac{C_j}{m\omega_j^2}P_{j,n}$ and obtain $$\begin{aligned}
K&(\theta_n,\mathbf p_n,\epsilon;\theta_{n-1},\mathbf p_{n-1},0)\\
&=\sum_{l_n=-\infty}^\infty\frac{1}{2\pi}\int_{-\infty}^\infty \frac{d\mathbf q_n}{(2\pi \hbar)^{\mathcal N}}\exp\left\{-\frac{i\epsilon}{\hbar}\mathcal K(l_n,\mathbf P_n)-\frac{i\epsilon}{\hbar}\mathcal V(\theta_{n-1},\mathbf R_{n-1}(\theta_{n-1}))\right\}\\
&\quad\times\exp\left\{i(l_n-A_n/\hbar)(\theta_n-\theta_{n-1})+\frac{i}{\hbar}\mathbf P_n\cdot(\mathbf R_n(\theta_n)-\mathbf R_{n-1}(\theta_{n-1}))\right\}.\end{aligned}$$ The sum over $l_n$ can be replaced by a sum of integrals using the Poisson resummation formula $$\sum_{l\in\mathbb Z}f(l)=\sum_{l\in\mathbb Z}\int_{-\infty}^\infty f(\zeta)\exp(2\pi l \zeta)d\zeta.$$ Then, $$\begin{aligned}
K&(\theta_n,\mathbf p_n,\epsilon;\theta_{n-1},\mathbf p_{n-1},0)\\
&=\sum_{l_n=-\infty}^\infty\int_{-\infty}^\infty\frac{d\zeta_n}{2\pi}\int_{-\infty}^\infty \frac{d\mathbf q_n}{(2\pi \hbar)^{\mathcal N}}\exp\left\{-\frac{i\epsilon}{\hbar}\mathcal K(\zeta_n,\mathbf P_n)-\frac{i\epsilon}{\hbar}\mathcal V(\theta_{n-1},\mathbf R_{n-1}(\theta_{n-1}))\right\}\\
&\quad\times\exp\left\{i(\zeta_n-A_n/\hbar)(\theta_n-\theta_{n-1}+2\pi l_n)+\frac{i}{\hbar}\mathbf P_n\cdot(\mathbf R_n(\theta_n+2\pi l_n)-\mathbf R_{n-1}(\theta_{n-1}))\right\}.\end{aligned}$$ For our Hamiltonian of an ICDW ring threaded by a fluctuating magnetic flux we have $$\begin{aligned}
\mathcal K(\zeta_n,\mathbf P_n)&=\frac{\left(\zeta_n\hbar-A_n\right)^2}{2I}+\sum_j\frac{1}{2m}P_{j,n}^2\\
\mathcal V(\theta_{n-1},\mathbf R_{n-1}(\theta_{n-1}))&=\sum_j\frac{1}{2}m\omega_j^2 \left(R_{j,n-1}(\theta_{n-1})-\frac{C_j\theta_{n-1}}{m\omega_j^2}\right)^2.\end{aligned}$$ We note that that the potential $\mathcal V(\theta,\mathbf R(\theta))$ is rotationally invariant. That is, for an arbitrary rotation $\theta\to\theta+\delta$, the potential is $\mathcal V(\theta+\delta,\mathbf R(\theta+\delta))=\mathcal V(\theta,\mathbf R(\theta))$. Let $\tilde\zeta_n=\zeta_n-A_n/\hbar$. We use the fact that the phase space volume is conserved under canonical transformation. Then, $$\begin{aligned}
K&(\theta,\mathbf p,t;\theta',\mathbf p',0)
\\
&=\left(\prod_{j=1}^\mathcal{N}\frac{1}{m\omega_j}\right)\lim_{\epsilon\to 0}\left(\prod_{n=1}^{N-1}\int_0^{2\pi}d\theta_n\int_{-\infty}^\infty d\mathbf R_{n}(\theta_n)\right)\left(\prod_{n=1}^N\sum_{l_n=-\infty}^\infty\int_{-\infty}^\infty\frac{d\mathbf P_n}{(2\pi \hbar)^{\mathcal N}}\frac{d\tilde \zeta_n}{2\pi}\right)
\\
&\times\exp\sum_{n=1}^N\left\{-\frac{i\epsilon}{\hbar}\left(\frac{\hbar^2}{2I}\tilde \zeta_n^2+\sum_j\frac{1}{2m}P_{j,n}^2\right)-\frac{i\epsilon}{\hbar}\mathcal V(\theta_{n-1},\mathbf R_{n-1}(\theta_{n-1}))\right\}
\\
&\times\exp\sum_{n=1}^N\left\{i\tilde\zeta_n(\theta_n-\theta_{n-1}+2\pi l_n)+\frac{i}{\hbar}\mathbf P_n\cdot(\mathbf R_n(\theta_n+2\pi l_n)-\mathbf R_{n-1}(\theta_{n-1}))\right\}\\
$$ Solve the $\tilde \zeta_n$ and $P_{j,n}$ integrals to obtain $$\begin{aligned}
K&(\theta,\mathbf p,t;\theta',\mathbf p',0)\\
&=\left(\prod_{j=1}^\mathcal{N}\frac{1}{m\omega_j}\right)\lim_{\epsilon\to 0}\left(\prod_{n=1}^{N-1}\int_0^{2\pi}d\theta_n\int_{-\infty}^\infty d\mathbf R_{n}(\theta_n)\right)\left(\prod_{n=1}^N\sum_{l_n=-\infty}^\infty\sqrt{\frac{I}{2\pi i\epsilon\hbar}}\sqrt{\frac{m}{2\pi i\epsilon \hbar}}^{\mathcal N}\right)\\
&\times\exp\sum_{n=1}^N\left\{\frac{i}{\hbar}\frac{I}{2\epsilon}(\theta_n-\theta_{n-1}+2\pi l_n)^2+\frac{i}{\hbar}\frac{m}{2\epsilon}(\mathbf R_n(\theta_n+2\pi l_n)-\mathbf R_{n-1}(\theta_{n-1}))^2\right\}\\
&\times\exp\sum_{n=1}^N\left\{-\frac{i\epsilon}{\hbar}\mathcal V(\theta_{n-1},\mathbf R_{n-1}(\theta_{n-1}))\right\}.\end{aligned}$$ The integral including $\theta_{N-1}$ is $$\begin{aligned}
I_{N-1}&=\sum_{l_N=-\infty}^\infty\sum_{l_{N-1}=-\infty}^\infty\sqrt{\frac{I}{2\pi i\epsilon\hbar}}\sqrt{\frac{m}{2\pi i\epsilon \hbar}}^{\mathcal N}\int_0^{2\pi}d\theta_{N-1}\int_{-\infty}^\infty d\mathbf R_{N-1}(\theta_{N-1})\\
&\times\exp\left\{\frac{i}{\hbar}\frac{I}{2\epsilon}(\theta_N-\theta_{N-1}+2\pi l_N)^2+\frac{i}{\hbar}\frac{I}{2\epsilon}(\theta_{N-1}-\theta_{N-2}+2\pi l_{N-1})^2\right\}\\
&\times\exp\left\{\frac{i}{\hbar}\frac{m}{2\epsilon}[\mathbf R_N(\theta_N+2\pi l_N)-\mathbf R_{N-1}(\theta_{N-1})]^2\right\}\\
&\times\exp\left\{\frac{i}{\hbar}\frac{m}{2\epsilon}[\mathbf R_{N-1}(\theta_{N-1}+2\pi l_{N-1})-\mathbf R_{N-2}(\theta_{N-2})]^2\right\}\\
&\times\exp\left\{-\frac{i\epsilon}{\hbar}\mathcal V(\theta_{N-1},\mathbf R_{N-1}(\theta_{N-1}))\right\}.\end{aligned}$$ The sum over $l_{N-1}$ can be absorbed into the $\theta_{N-1}$ integral by changing the domain of $\theta_{N-1}$ integration from $[0,2\pi)$ to $(-\infty,\infty)$. This is done by the following procedure:
1. transform $l_{N}$ to $\tilde l_N=l_N+l_{N-1}$ and write $\tilde \theta_{N-1}=\theta_{N-1}+2\pi l_{N-1}$ to obtain $$\begin{aligned}
I_{N-1}&=\sum_{l_N=-\infty}^\infty\sum_{l_{N-1}=-\infty}^\infty\sqrt{\frac{I}{2\pi i\epsilon\hbar}}\sqrt{\frac{m}{2\pi i\epsilon \hbar}}^{\mathcal N}\int_0^{2\pi}d\theta_{N-1}\int_{-\infty}^\infty d\mathbf R_{N-1}(\theta_{N-1})
\\
&\times\exp\left\{\frac{i}{\hbar}\frac{I}{2\epsilon}(\tilde\theta_N-\theta_{N-1}+2\pi \tilde l_N)^2+\frac{i}{\hbar}\frac{I}{2\epsilon}(\tilde \theta_{N-1}-\theta_{N-2})^2\right\}
\\
&\times\exp\left\{\frac{i}{\hbar}\frac{m}{2\epsilon}[\mathbf R_N(\theta_N+2\pi \tilde l_N)-\mathbf R_{N-1}(\tilde\theta_{N-1})]^2\right\}
\\
&\times\exp\left\{\frac{i}{\hbar}\frac{m}{2\epsilon}[\mathbf R_{N-1}(\tilde\theta_{N-1})-\mathbf R_{N-2}(\theta_{N-2})]^2\right\}
\\
&\times\exp\left\{-\frac{i\epsilon}{\hbar}\mathcal V(\tilde\theta_{N-1},\mathbf R_{N-1}(\tilde\theta_{N-1}))\right\}.\end{aligned}$$
2. Change the variable of integration $\int_0^{2\pi}d\theta_n\to\int_{2\pi n}^{2\pi(n+1)}d\tilde \theta_n$ and change the domain of $\theta_{N-1}$ integration from $[0,2\pi)$ to $(-\infty,\infty)$ and use the periodicity of $\mathcal V(\theta_{N-1},\mathbf R_{N-1}(\theta_{N-1}))$ to obtain $$\begin{aligned}
I_{N-1}&=\sum_{l_N=-\infty}^\infty\sqrt{\frac{I}{2\pi i\epsilon\hbar}}\sqrt{\frac{m}{2\pi i\epsilon \hbar}}^{\mathcal N}\int_{-\infty}^\infty d\theta_{N-1}\int_{-\infty}^\infty d\mathbf R_{N-1}(\theta_{N-1})\\
&\times\exp\left\{\frac{i}{\hbar}\frac{I}{2\epsilon}(\theta_N-\theta_{N-1}+2\pi l_N)^2+\frac{i}{\hbar}\frac{I}{2\epsilon}(\theta_{N-1}-\theta_{N-2})^2\right\}\\
&\times\exp\left\{\frac{i}{\hbar}\frac{m}{2\epsilon}[\mathbf R_N(\theta_N+2\pi l_N)-\mathbf R_{N-1}(\theta_{N-1})]^2\right\}\\
&\times\exp\left\{\frac{i}{\hbar}\frac{m}{2\epsilon}[\mathbf R_{N-1}(\theta_{N-1})-\mathbf R_{N-2}(\theta_{N-2})]^2\right\}\\
&\times\exp\left\{-\frac{i\epsilon}{\hbar}\mathcal V(\theta_{N-1},\mathbf R_{N-1}(\theta_{N-1}))\right\}.\end{aligned}$$
Repeat this procedures for the integrals involving $\theta_{n}$ from $n=N-2$ to $n=1$, and obtain $$\begin{aligned}
K&(\theta,\mathbf q,t;\theta',\mathbf q',0)\\
&=\sum_{l=-\infty}^\infty\lim_{\epsilon\to 0}\int_{-\infty}^{\infty}\left(\prod_{n=1}^{N-1}d\theta_n\right)\int_{-\infty}^\infty\left(\prod_{n=1}^{N-1}d\mathbf R_{n-1}(\theta_{n-1})\right)\prod_{n=1}^N\sqrt{\frac{I}{2\pi i\epsilon\hbar}}\sqrt{\frac{m}{2\pi i\epsilon \hbar}}^{\mathcal N}\\
&\times\exp\left\{\frac{i}{\hbar}\frac{I}{2\epsilon}(\theta_n-\theta_{n-1}+2\pi l\delta_{n,N})^2\right\}\\
&\times\exp\left\{\frac{i}{\hbar}\frac{m}{2\epsilon}[\mathbf R_n(\theta_n+2\pi l\delta_{n,N})-\mathbf R_{n-1}(\theta_{n-1})]^2-\frac{i\epsilon}{\hbar}\mathcal V(\theta_{n-1},\mathbf R_{n-1}(\theta_{n-1}))\right\}.\end{aligned}$$ Define $$\begin{aligned}
\mathcal A_A&=\sqrt{\frac{2\pi i\epsilon\hbar}{I}},\qquad \mathcal A_B=\sqrt{\frac{2\pi i\epsilon\hbar}{m}}^\mathcal{N},
\\
\int \mathcal{D}\theta&=\frac{1}{\mathcal A_A}\int_{-\infty}^\infty\left(\prod_{n=1}^{N-1}\frac{d\theta_n}{\mathcal A_A}\right),
\qquad
\int \mathcal{D}\mathbf R(\theta)=\frac{1}{\mathcal A_B}\int_{-\infty}^\infty\left(\prod_{n=1}^{N-1}\frac{d\mathbf R_n(\theta_n)}{\mathcal A_B}\right),\end{aligned}$$ $$\begin{aligned}
L_n(\theta_n,\mathbf R_n(\theta_n))&=\frac{I}{2}\left(\frac{\theta_n-\theta_{n-1}+2\pi l\delta_{n,N}}{\epsilon}\right)^2\nonumber
\\
&+\frac{m}{2}\left(\frac{\mathbf R_n(\theta_n+2\pi l\delta_{n,N})-\mathbf R_{n-1}(\theta_{n-1})}{\epsilon}\right)^2-\mathcal V(\theta_{n-1},\mathbf R_{n-1}(\theta_{n-1}))\end{aligned}$$ where $$\begin{aligned}
L[\theta,\mathbf R(\theta)]&=\lim_{\epsilon\to 0}L_n[\theta_n,\mathbf R_n(\theta_n)]=L_0[\theta]+L_B[\theta,\mathbf R(\theta)]
\\
L_0[\theta]&=\frac{I}{2}\dot\theta^2
\\
L_B[\theta,\mathbf R(\theta)]&=\sum_{j=1}^\mathcal N\frac{m}{2}\dot R_j(\theta)^2-\sum_{j=1}^\mathcal N\frac{1}{2}m\omega_j^2\left(R_j(\theta)-\frac{C_j\theta}{m\omega_j^2}\right)^2\end{aligned}$$ is the Lagrangian of the coupled system. The action $S[\theta,\mathbf R(\theta)]$ is given by $$\begin{aligned}
S[\theta,\mathbf R(\theta)]=\int_0^tL[\theta,\mathbf R(\theta)]ds&=S_0[\theta]+S_B[\theta,\mathbf R(\theta)]
\\
S_0[\theta]&=\int_0^td\tau L[\theta]
\\
S_B[\theta,\mathbf R(\theta)]&=\int_0^td\tau L_B[\theta,\mathbf R(\theta)].
\label{S_0}\end{aligned}$$ Now, suppose that we initially have the total density operator given by $$\hat\rho_{AB}(0)=\hat\rho_A(0)\hat\rho_B(0).$$ Then, the reduced density matrix is $$\begin{aligned}
\rho(\theta,\phi,t)&=\int_{-\infty}^\infty d\mathbf R(\theta)d\mathbf Q(\theta)\delta(\mathbf R(\theta)-\mathbf Q(\theta))\rho(\theta,\mathbf p,\phi,\mathbf s,t)
\nonumber\\
&=\int_0^{2\pi}d\theta' d\phi' J(\theta,\phi,t;\theta',\phi',0)\rho_A(\theta',\phi',0)
\label{Reduced_Density_Matrix}\end{aligned}$$ where $$J(\theta,\phi,t;\theta',\phi',0)=\sum_{l=-\infty}^\infty\sum_{l'=-\infty}^\infty\int_{\theta'}^{\theta+2\pi l}\mathcal D\theta\int_{\phi'}^{\phi+2\pi l'}\mathcal D\phi^\ast\exp\frac{i}{\hbar}\left(S_0[\theta]-S_0[\phi]\right)\mathcal F[\theta,\phi]$$ is the propagator of the density matrix, $$\begin{aligned}
\mathcal F[\theta,\phi]&=\int_{-\infty}^\infty d\mathbf R(\theta)d\mathbf Q(\phi) d\mathbf R'(\theta') d\mathbf Q'(\phi')\delta(\mathbf R(\theta)-\mathbf Q(\phi))\rho_B(\mathbf p',\mathbf s',0)
\\
&\times \int_{\mathbf R'(\theta')}^{\mathbf R(\theta)}\mathcal D\mathbf R(\theta)\int_{\mathbf Q'(\phi')}^{\mathbf Q(\phi)}\mathcal D\mathbf Q(\phi)\exp\frac{i}{\hbar}\left(S_B[\theta,\mathbf R(\theta)]-S_B[\phi,\mathbf Q(\phi)]\right)\end{aligned}$$ is the influence functional and $Q(\phi)=-(s_j-c\phi)/m\omega_j$. Supposed that the density operator $\hat \rho_B$ can be written as a canonical ensemble at $t=0$. That is, $$\rho_B=\frac{\exp(-\beta\hat H_B)}{\mathrm{tr}_B[\exp(-\beta\hat H_B)]}$$ Introduce the imaginary time $\tau=-i\hbar\beta$, then the density matrix of the environment at $t=0$ is $$\begin{aligned}
\rho_B(\mathbf p',\mathbf s',0)=\frac{\Braket{\mathbf p'|\exp\left(-\frac{i\tau}{\hbar}\hat H_B\right)|\mathbf s'}}{\mathrm{tr}_B\left[\exp\left(-\frac{i\tau}{\hbar}\hat H_B\right)\right]}=\frac{K_B(\mathbf p',\tau;\mathbf s',0)}{\int d\mathbf qK_B(\mathbf q,\tau;\mathbf q,0)}.\end{aligned}$$
\[F\_Derivation\]
Momentum Representation of the Density matrix of a Harmonic Oscillator
----------------------------------------------------------------------
The propagator of a harmonic oscillator with the Lagrangian $$L=\frac{m}{2}\dot x^2-\frac{1}{2}m\omega x^2$$ has a well known solution $$\begin{aligned}
K(x,t;x',0)&=\int \mathcal Dx\exp\frac{i}{\hbar}\int_0^tLds
\\
&=\sqrt{\frac{m\omega}{2\pi i\hbar\sin\omega t}}\exp\left\{\frac{i}{\hbar}\frac{m\omega}{2\sin\omega t}[(x^2+x'^2)\cos\omega t-2xx']\right\}\end{aligned}$$ $K(p,t;p',0)$ is obtained by a Fourier transformation $$K(p,t:p',0)=\int_{-\infty}^\infty\frac{dxdx'}{2\pi\hbar}K(x,t;x',0)\exp\frac{i}{\hbar}\left(px-p'x'\right).$$ Let $A=\frac{m\omega\cos\omega t}{2\sin\omega t}$ and $B=\frac{m\omega}{2\sin\omega t}$, then $$\begin{aligned}
K(p,t;p',0)=\frac{F(t)}{\sqrt{4B^2-4A^2}}\exp\frac{i}{\hbar}\frac{Ap^2+Ap'^2-2Bpp'}{4B^2-4A^2}\end{aligned}$$ where $$4B^2-4A^2=\frac{m^2\omega^2(1-\cos^2\omega t)}{\sin^2\omega t}=m^2\omega^2,$$ so, $$K(p,t;p',0)=\frac{F(t)}{m\omega}\exp\frac{i}{\hbar}S[p(t)/m\omega].$$ Therefore, using $\tau=-i\hbar\beta$, $\beta=1/k_BT$, $\nu_j=\hbar\omega_j\beta$ and $i\sinh x=\sin ix$ we have $$\begin{aligned}
\braket{\mathbf p'|\exp(-i\tau\hat H_B/\hbar)|\mathbf s'}=\prod_{j=1}^\mathcal{N}\frac{1}{m\omega_j}\sqrt{\frac{m\omega_j}{2\pi \hbar\sinh\nu_j}}\exp-\frac{m\omega_j}{2\hbar\sinh\nu_j}\left[\frac{{p'}_j^2+{s'}_j^2}{m^2\omega_j^2}\cosh\nu-2\frac{{p'}_j{s'}_j}{m^2\omega_j^2}\right],\end{aligned}$$ and $$\begin{aligned}
\mathrm{tr}_B[\exp(-i\tau\hat H_B/\hbar)]=\int_{-\infty}^\infty d\mathbf q\braket{\mathbf q|\exp(-i\tau\hat H_B/\hbar)|\mathbf q}=\prod_{j=1}^\mathcal{N}\frac{1}{2\sinh\frac{\nu_j}{2}}.\end{aligned}$$ We are assuming that the environment is not coupled to the system at $t=0$, so $R'_j=p'_j/m\omega_j$ and $Q'_j=s'_j/m\omega_j$. Therefore, $$\rho_B(\mathbf p',\mathbf s',0)=\left(\prod_{j=1}^\mathcal{N}\frac{1}{m\omega_j}\right)\rho_B(\mathbf R',\mathbf Q',0)$$ where $$\rho_B(\mathbf R',\mathbf Q',0)=\prod_{j=1}^\mathcal{N}2\sinh\frac{\nu_j}{2}\sqrt{\frac{m\omega_j}{2\pi \hbar\sinh\nu_j}}\exp-\frac{m\omega_j}{2\hbar\sinh\nu_j}\left[({R'}_j^2+{Q'}_j^2)\cosh\nu_j-2{R'}_j{Q'}_j\right].$$ Let us rewrite this density matrix into a simpler form. Define $x_{0j}=R'_j+Q'_j$ and $x'_{0j}=R'_j-Q'_j$. Then, using $$({R'_j}^2+{Q'_j}^2)\cosh\nu_j-2{R'}_j{Q'}_j=\frac12(\cosh\nu_j-1)x_{0j}^2+\frac12(\cosh\nu_j+1){x'_{0j}}^2$$ and the identity $$\frac{\cosh\nu_j-1}{\sinh\nu_j}=\frac{\sinh\nu_j}{\cosh\nu_j+1}=\tanh\frac{\nu_j}{2},$$ we obtain $$\rho_B(\mathbf R',\mathbf Q',0)=\prod_{j=1}^\mathcal{N}\sqrt{\frac{m\omega_j}{\pi \hbar\mu_j}}\exp-\frac{m\omega_j}{4\hbar}\left[\frac{x_{0j}^2}{\mu_j}+{x'_{0j}}^2\mu_j\right]. \label{densitmatrix RQ}$$ $$\mu_j=\coth\frac{\nu_j}{2}.$$
The Influence Functional $\mathcal F[\theta,\phi]$
--------------------------------------------------
Now, we calculate the density functional $\mathcal F[\theta,\phi]$ explicitly. We will write $\mathbf R$ and $\mathbf Q$ instead of $\mathbf R[\theta]$ and $\mathbf Q[\phi]$ for simplicity, but keep in mind that they depend on $\theta$ and $\phi$, respectively. Then, the influence functional can be written as $$\begin{aligned}
\mathcal F[\theta,\phi]&=\int_{-\infty}^\infty d\mathbf x_t d\mathbf x'_td\mathbf x_0d\mathbf x'_0(1/2)^{\mathcal{N}}\delta(\mathbf x'_t)\rho_B(\mathbf R',\mathbf Q',0)\int_{\mathbf R'}^{\mathbf R}\mathcal D\mathbf R\int_{\mathbf Q'}^{\mathbf Q}\mathcal D\mathbf Q^\ast\nonumber\\
&\times\exp\left\{\frac{i}{\hbar}\int_0^tds\sum_j\left[\frac{m}{2}(\dot R_j^2-\dot Q_j^2)-\frac{1}{2}m\omega_j^2(R_j^2-Q_j^2)+C_j(\theta R_j-\phi Q_j)-\frac{C_j^2}{2m\omega_j^2}(\theta^2-\phi^2)\right]\right\}.\end{aligned}$$ To evaluate this we introduce the variables $$\begin{aligned}
\begin{split}
\varphi&=\theta+\phi,\qquad \varphi'=\theta-\phi,
\\
x_j&=R_j+Q_j,\qquad x'_j=R_j-Q_j.
\end{split}\end{aligned}$$ Then, the influence functional becomes $$\begin{aligned}
\mathcal F[\theta,\phi]&=\int_{-\infty}^\infty d\mathbf x_td\mathbf x'_td\mathbf x_0d\mathbf x'_0(1/2)^{\mathcal{N}}\delta(\mathbf x'_t)\rho_B(\mathbf R',\mathbf Q',0)\int\mathcal D\mathbf x\mathcal D \mathbf x'
\\
&\times\exp\left\{\frac{i}{\hbar}\int_0^tds\sum_j\left[\frac{m}{2}\dot x_j\dot x'_j-\frac{1}{2}m\omega_j^2x_jx'_j+\frac{C_j}{2}(\varphi x'_j+\varphi'x_j)-\frac{C_j^2}{2m\omega_j^2}\varphi'\varphi\right]\right\}
\\
&=\int_{-\infty}^\infty d\mathbf x_td\mathbf x'_td\mathbf x_0d\mathbf x'_0(1/2)^{\mathcal{N}}\delta(\mathbf x'_t)\rho_B(\mathbf R',\mathbf Q',0)
\\
&\times\exp\left\{\frac{im}{2\hbar}\sum_j(x_{tj}\dot x'_{tj}-x_{0j}\dot x'_{0j})-\frac{i}{\hbar}\int_0^tds\sum_j\frac{C_j^2}{2m\omega_j^2}\varphi'\varphi\right\}
\\
&\times\int\mathcal D\mathbf x\mathcal D \mathbf x'\exp\left\{\frac{i}{\hbar}\int_0^tds\sum_j\left[-g(x'_j)x_j+\frac{C_j}{2}\varphi x_j'\right]\right\}\end{aligned}$$ where $$g(x_j')=\frac{m}{2}\ddot x'_j+\frac{1}{2}m\omega_j^2x'_j-\frac{C_j}{2}\varphi'.$$ We note that the classical solution $x_{j,cl}'$ satisfies the Euler-Lagrange equation $$g(x_{j,cl}')=0,\quad g(x_{j}'(0))=0,\quad g(x_{j}'(t))=0.$$ The path integral over $\mathbf x$ can be done first. Calling this integral $I_{\mathbf x,\mathbf x'}$ we obtain $$\begin{aligned}
I_{\mathbf x,\mathbf x'}&=\int\mathcal D\mathbf x\mathcal D \mathbf x'\exp\left\{\frac{i}{\hbar}\int_0^tds\sum_j\left[-g(x'_{j})x_{j}+\frac{C_j}{2}\varphi x_{j}'\right]\right\}
\\
&=\frac{1}{|\mathcal A_B|^2}\int_{-\infty}^\infty\left(\prod_{n=1}^{N-1}\frac{d\mathbf xd\mathbf x'}{|\mathcal A_B|^2}\right)\prod_{n=1}^N\prod_j\exp\frac{i\epsilon}{\hbar}\left[-g(x_{n,j}')x_{n,j}+\frac{C_j}{2}\varphi x_j'\right]
\\
&=\frac{1}{|\mathcal A_B|^2}\int_{-\infty}^\infty\left(\prod_{n=1}^{N-1}\frac{d\mathbf x'}{|\mathcal A_B|^2}\right)\prod_j\left(\prod_{n=1}^{N-1}\delta\left[\frac{\epsilon}{2\pi\hbar}g(x'_{n,j})\right]\right)\prod_{n=1}^N\exp\frac{i\epsilon}{\hbar}\frac{C_j}{2}\varphi x_{n,j}'.\end{aligned}$$ $|\mathcal A|^{-2}$ is absorbed into the delta functions. The interpretation of the delta function is to insert the classical solution $x'_{j,cl}$ into $x'_j$ which satisfy $g(x'_{j,cl})=0$. The outcome is, $$I_{\mathbf x,\mathbf x'}=\frac{1}{|\mathcal A_B|^2}\exp\left(\frac{i}{\hbar}\int_0^tds\sum_j\frac{C_j}{2}\varphi x'_{cl}\right).$$ Therefore, the influence functional is $$\begin{aligned}
\mathcal F[\theta,\phi]&=\int_{-\infty}^\infty d\mathbf x_td\mathbf x'_td\mathbf x_0d\mathbf x'_0(1/2)^{\mathcal{N}}\delta(\mathbf x'_t)\rho_B(\mathbf R',\mathbf Q',0)\exp\frac{i}{\hbar}\frac{m}{2}\sum_j(x_{tj}\dot x'_{tj}-x_{0j}\dot x'_{0j})
\\
&\times\frac{1}{|\mathcal A_B|^2}\exp\left(\frac{i}{\hbar}\int_0^tds\sum_j\left[-\frac{C_j^2}{2m\omega_j^2}\varphi'\varphi+\frac{C_j}{2}\varphi x_{cl}'\right]\right).\end{aligned}$$ The $\mathbf x_t$ integral gives a delta function of $\dot{\mathbf x}'_t$ $$\begin{aligned}
\mathcal F[\theta,\phi]&=\int_{-\infty}^\infty d\mathbf x'_td\mathbf x_0d\mathbf x'_0(1/2)^{\mathcal{N}}\delta(\mathbf x'_t)\delta(\dot{\mathbf x}'_t)\rho_B(\mathbf R',\mathbf Q',0)
\\
&\times\exp\left(\frac{i}{\hbar}\int_0^tds\sum_j\left[-\frac{C_j^2}{2m\omega_j^2}\varphi'\varphi+\frac{C_j}{2}\varphi x_{cl}'\right]+\frac{i}{\hbar}\frac{m}{2}\sum_jx_{0j}\dot x'_{0j}\right).\end{aligned}$$ Insert equation for the density matrix and do the $\mathbf x_0$ integral, then we obtain $$\begin{aligned}
\mathcal F[\theta,\phi]&=\int_{-\infty}^\infty d\mathbf x'_td\mathbf x'_0\delta(\mathbf x'_t)\delta(\dot{\mathbf x}'_t)
\\
&\times\prod_j\exp\left(\frac{i}{\hbar}\int_0^tds\left[-\frac{C_j^2}{2m\omega_j^2}\varphi'\varphi+\frac{C_j}{2}\varphi x_{cl}'\right]-\frac{m\mu_j\omega_j}{4\hbar}\left(\frac{\dot{x}{'_{0j}}^2}{\omega_j^2}+{x'_{0j}}^2\right)\right).\end{aligned}$$ The two delta functions $\delta(\mathbf x'_t)$ and $\delta(\dot{\mathbf x}'_t)$ can be interpreted as boundary conditions on the classical solution of $g(\mathbf x'(s))=0$. The result of doing the remaining integrals would be to substitute the classical solution of $x'_{cl}$, $\dot{x}{'_{0j}}$ and $x'_{0j}$ which satisfy these boundary conditions. The solution to the classical solution of $g(x)=0$, that is $$\ddot x'_j+\omega_j^2x'_j-\frac{C_j}{m}\varphi'=0.$$ is $$\begin{aligned}
x_j(\tau)&=-\int_\tau^t\frac{C_j^2\varphi'(s)}{m\omega_j}\sin\omega_j(\tau-s)ds+x'_{tj}\cos\omega_j(t-\tau)-\frac{\dot x'_{tj}}{\omega_j}\sin\omega_j(t-\tau).\end{aligned}$$ For our boundary condition this solution reduces to $$\begin{aligned}
x'_j(\tau)&=-\int_\tau^t\frac{C_j\varphi'(s)}{m\omega_j}\sin\omega_j(\tau-s)ds\\
\dot{x}'_j(\tau)&=-\int_\tau^t\frac{C_j\varphi'(s)}{m}\cos\omega_j(\tau-s)ds$$ Therefore, the result of integration is $$\begin{aligned}
\mathcal F[\theta,\phi]=\exp\left(-\sum_j\frac{i}{\hbar}\frac{C_j^2}{2m\omega_j}\int_0^t\int_\tau^t\varphi(\tau)\left(\frac{2}{\omega_j}\delta(\tau-s)-\sin\omega_j(\tau-s)\right)\varphi'(s)d\tau ds\right)
\\
\times\exp\left(-\sum_j\frac{\mu_j}{4\hbar}\int_0^t\int_0^t\frac{C_j^2}{m\omega_j}\varphi'(\tau)\cos\omega_j(\tau-s)\varphi'(s)dsd\tau\right).\end{aligned}$$ Since $s$ and $\tau$ enter into the second integral symmetrically it can be rewritten $$\int_0^t\int_0^tE(\tau,s)dsd\tau=2\int_0^t\int_0^\tau E(\tau,s)dsd\tau.$$ The first integral can be written $$\int_0^t\int_\tau^tE(\tau,s)dsd\tau=\int_0^t\int_0^\tau E(s,\tau)dsd\tau.$$ Then, the influence functional becomes $$\mathcal F[\theta,\phi]=\exp i\Phi[\theta,\phi]$$ where the influence phase $\Phi[\theta,\phi]$ can be written as $$\begin{aligned}
i\Phi[\theta,\phi]&=-\frac{i}{\hbar}\int_0^t\int_0^\tau d\tau ds\varphi(s)\alpha_I(\tau-s)\varphi'(\tau)-\frac{1}{\hbar}\int_0^t\int_0^\tau dsd\tau\varphi'(\tau)\alpha_R(\tau-s)\varphi'(s)
\\
\alpha_R(\tau-s)&=\sum_j\frac{C_j^2}{2m\omega_j}\coth\frac{\hbar\omega_j}{2k_BT}\cos\omega_j(\tau-s)
\\
\alpha_I(\tau-s)&=\sum_j\frac{C_j^2}{2m\omega_j}\left(\frac{2}{\omega_j}\delta(\tau-s)+\sin\omega_j(\tau-s)\right).\end{aligned}$$ Therefore, we obtain the density matrix propagator $J(\theta,\phi,t;\theta',\phi',0)$ $$\begin{aligned}
&J(\theta_\text f,\phi_\text f,t;\theta\text i,\phi_\text i,0)
\\
&=\left.
\int_{\theta_\text i}^{\theta_\text f}D\theta\int_{\phi_\text i}^{\phi_\text f}D\phi^\ast\exp\left\{\frac{i}{\hbar}S[\varphi^+,\varphi^-]-\Gamma[\varphi^-]\right\}
\right|_{\substack{\varphi^+=(\theta+\phi)/2\\\varphi^-=\phi-\theta\quad}},
\label{JDef}\end{aligned}$$ the classical action $S[\varphi^+,\varphi^-]$ is $$\begin{aligned}
S[\varphi,\varphi']&=-\int_0^td\tau I\dot\varphi^+(\tau)\dot\varphi^-(\tau)
\\
&+2\int_0^td\tau\int_0^s ds\varphi^-(\tau)\alpha_\text I(\tau-s)\varphi^+(s)
$$ and the $\Gamma[\varphi^-]$ is given by $$\begin{aligned}
\Gamma[\varphi^-]&=\frac{1}{2\hbar}\int_0^td\tau\int_0^tds\varphi^-(\tau)\alpha_\text R(\tau-s)\varphi^-(s).\end{aligned}$$ The path integral in can be done exactly and we obtain $$\begin{aligned}
J(\theta_\text f,\phi_\text f,t;\theta_\text i,\phi_\text i,0)&=F^2(t)\exp\left(\frac{i}{\hbar}S[\varphi^+_\text{cl},\varphi^-_\text{cl}]-\Gamma[\varphi^-_\text{cl}]\right)\label{JClassical}\end{aligned}$$ $\varphi^\pm_\text{cl}$ are the classical coordinates obtained from the Euler-Lagrange equation $$\begin{aligned}
I\ddot\varphi_\text{cl}^-(u)+2\int_u^td\tau\varphi_\text{cl}^-(\tau)\alpha_\text I(\tau-u)=0\label{eqn1}
\\
I\ddot\varphi^+_\text{cl}(u)+2\int_0^ud\tau\varphi^+_\text{cl}(\tau)\alpha_\text I(u-\tau)=0\label{eqn2}\end{aligned}$$ whose solution are given in terms of boundary conditions $\varphi^\pm_\text i=\varphi^\pm(0),\varphi^\pm_\text f=\varphi^\pm(t)$: $$\begin{aligned}
\varphi^+_\text{cl}(u)&=\kappa_i(u;t)\varphi^+_\text i+\kappa_f(u;t)\varphi^+_\text f
\label{varphisolfinal}
\\
\varphi^-_\text{cl}(u)&=\kappa_i(t-u;t)\varphi^-_\text f+\kappa_f(t-u;t)\varphi^-_\text i
\label{varphiprimesolfinal}
\\
\kappa_i(u;t)&=\dot G(u)-\frac{\dot G(t)}{G(t)}G(u),
\qquad
\kappa_f(u;t)=\frac{G(u)}{G(t)}.\end{aligned}$$ Then, the classical action reduces to $$\begin{aligned}
S[\varphi^+_\text{cl},\varphi^-_\text{cl}]&=-I[\dot\varphi^+_\text{cl}(t)\varphi^-_\text f-\dot\varphi^-_\text{cl}(0)\varphi^-_\text i].
\label{ClassicalAction}\end{aligned}$$
|
---
abstract: 'Supermassive black holes (SMBHs) have been found to be ubiquitous in the nuclei of early-type galaxies and of bulges of spirals. There are evidences of a tight correlation between the SMBH masses, the velocity dispersions of stars in the spheroidal components galaxies and other galaxy properties. Also the evolution of the luminosity density due to nuclear activity is similar to that due to star formation. All that suggests an evolutionary connection between Active Galactic Nuclei (AGNs) and their host galaxies. After a review of these evidences this lecture discusses how AGNs can affect the host galaxies. Other feedback processes advocated to account for the differences between the halo and the stellar mass functions are also briefly introduced.'
author:
- Gianfranco De Zotti
bibliography:
- 'ptapapdoc.bib'
title: 'Co-evolution of galaxies and Active Galactic Nuclei'
---
Introduction
============
The mutual interactions between super-massive black holes (SMBHs) and host galaxies are a key ingredient to understand the evolution of both source populations. There are now clear evidences that SMBHs and host galaxies evolve in a coordinated way over cosmic history. The understanding of the processes that drive the SMBH-galaxy co-evolution is a central topic in current extragalactic studies.
The starting point was the discovery, via stellar/gas dynamics and photometric observations, that nearby galaxies possessing massive spheroidal components (bulges) host, at their centers, a massive dark object (MDO) endowed with a mass proportional to the mass in old stars or to the K-band luminosity [@KormendyRichstone1995; @Magorrian1998]. The MDOs were generally interpreted as the black hole (BH) remnants of a past nuclear activity. @Salucci1999 demonstrated that the mass function of MDOs, derived on the basis of the observed correlation between their masses and the stellar masses in galactic bulges, is consistent with the SMBH mass function derived from the redshift-dependent luminosity function of Active Galactic Nuclei (AGNs), for standard values of the radiative efficiency of BH accretion.
The SMBH interpretation of MDOs was strongly confirmed by the analysis of the orbits for a number of individual stars in the central region of our Galaxy [@Ghez2008; @Genzel2010], ruling out alternative possibilities. A further quite unambiguous evidence of a central SMBH was provided by VLBI measurements at milli-arcsecond (mas) resolution of the $H_2 O$ mega-maser at 22GHz of the galaxy NGC4258 [@Miyoshi1995].
The luminosity/mass of the stellar component is not the only global property of the local ETGs that correlates with the central BH mass, $M_{\rm BH}$. In fact, a tighter correlation was found between $M_{\rm BH}$ and the stellar velocity dispersion [@FerrareseMerritt2000; @Gebhardt2000]. The connection between SMBH mass and galaxy properties, linking scales differing by as much as nine orders of magnitude is likely imprinted by the huge amount of energy released by AGNs, a small fraction of which may come out in a mechanical form (AGN feedback). However, many important details of the processes governing the AGN-galaxy co-evolution are still to be clarified.
In this paper, after a short introduction to AGNs (Sect. \[sect:AGNintro\]) I will touch on their demography (Sect. \[sect:BHdemography\]). Section \[sect:Soltan\] deals the main mechanism of SMBH growth: radiative accretion and/or non-radiative processes such as BH mergers? The argument put forward long ago by @Soltan1982 has still an important bearing on this matter. Section \[sect:interaction\] is about relationships between galaxy properties and the central SMBH. Section \[sect:coev\] concerns the AGN impact on the host galaxy; the two main mechanisms corrently discussed in the literature, energy- and momentum-driven winds, are illustrated. In Section \[sect:feedback\] winds powered by AGNs are put in the more general context of feedback processes invoked to account for the different shapes of the mass functions of dark matter halos and of galaxies. Finally, the main conclusions are summarized in Sect. \[sect:conclusions\].
Introduction to AGNs {#sect:AGNintro}
====================
The AGN (or quasar) discovery dates back to the @Schmidt1963 paper in which the redshift measurement of a bright radio source, 3C273, was reported: “The stellar object is the nuclear region of a galaxy with a cosmological redshift of 0.158, corresponding to an apparent velocity of 47,400 km/s. The distance would be around 500 megaparsecs, and the diameter of the nuclear region would have to be less than 1 kiloparsec. This nuclear region would be about 100 times brighter than the luminous galaxies which have been identified with radio sources so far...”.
The extreme luminosity of these sources, coming from very compact regions, called for a new energy source. @HoyleFowler1963b were the first to argue that such energy was of gravitational origin. @Salpeter1964 and @Zeldovich1964 showed that accretion into a SMBH can indeed account for the quasar luminosity. The idea was further elaborated by several authors [e.g., @LyndenBell1969; @LyndenBell1978; @LyndenBellRees1971] and gained widespread acceptance.
The argument went as follows. The total energy output from a quasar is at least the energy stored in its radio halo ($\sim 10^{54}\,\hbox{J}=10^{61}\,$erg); via $E=mc^2$ this corresponds to $10^7\,M_\odot$. Nuclear reactions have at best an efficiency of 0.7% (H burning). So the mass undergoing nuclear reactions capable of powering a quasar is $> 10^9\,M_\odot$. Rapid variability implies that a typical quasar is no bigger than a few light-hours. But the gravitational energy of $10^9\,M_\odot$ compressed within this size is $10^{55}\,$J, i.e. 10 times larger than the fusion energy. In Lynden-Bell’s words: “Evidently, although our aim was to produce a model based on nuclear fuel, we have ended up with a model which has produced more than enough energy by gravitational contraction. The nuclear fuel has ended as an irrelevance.”
@Salpeter1964 considered the radiative efficiency of accretion onto a “Schwarzschild singularity”, showing that it can release an energy of $0.057\,c^2$ per unit mass. @Bardeen1970 showed that for a rotating (Kerr) singularity up to nearly 42% of the accreted rest mass energy is emitted.
BH demography {#sect:BHdemography}
=============
Why did it take so long to realize the AGN role in galaxy evolution?
--------------------------------------------------------------------
Although it was clear from the beginning that quasars are located in the nuclei of galaxies, their presence was considered for decades just as an incidental diversion, an ornament irrelevant for galaxy formation and evolution. There are reasons for that:
- The physical scale of the AGNs is incomparably smaller than that of galaxies: typical radii of the stellar distribution of galaxies are of several kpc, to be compared with the Schwarzschild radius $$r_S = \frac{2 G\,M_{\rm BH}}{c^2} \simeq 9.56\times 10^{-6}\,\frac{M_{\rm
BH}}{10^8\,M_\odot}\ \hbox{pc},$$ i.e. $r_{\rm BH} \sim 10^{-9}\,r_{\rm gal}$.
- The radius of the “sphere of influence” of the SMBH (the distance at which its potential significantly affects the motion of the stars or of the interstellar medium) is also small: $$r_{\rm inf} = \frac{G\,M_{\rm BH}}{\sigma_\star^2} \simeq 11 \frac{M_{\rm BH}}{10^8\,M_\odot}\,
\left(\frac{\sigma_\star}{200\,\hbox{km}\,\hbox{s}^{-1}}\right)^{-2}\ \hbox{pc},$$ $\sigma_\star$ being the velocity dispersion of stars in the host galaxies (SMBHs are generally associated to spheroidal components of galaxies, whose stellar dynamics is dominated by random motions, not by rotation). Hence SMBHs have a negligible impact on the global stellar and interstellar medium (ISM) dynamics.
- AGNs and galaxies have very different evolutionary properties. AGNs evolve much faster: they are much rarer than galaxies at low redshifts, where galaxies were most extensively studied, and become much more numerous at $z \ge 2$.
Although powerful AGNs are rare locally, the SMBHs powering the quasars at high $z$ do not disappear. After they stop accreting, they should live essentially forever as dark remnants. So dead quasar engines should hide in many nearby galaxies. This was pointed out early on [@LyndenBell1969; @Schmidt1978], but a direct test of this idea had to wait for decades.
How solid is the evidence of SMBHs in galactic nuclei?
------------------------------------------------------
The difficulty to reveal inactive SMBHs in galactic nuclei stems from the fact that the SMBH masses required to power the AGNs are a tiny fraction of the stellar mass (let alone the total mass, including dark matter!) of galaxies[^1], and therefore their radius of influence is very small. Even in nearby galaxies the angular scale associated to $r_{\rm inf}$ is sub-arcsecond: $$\theta_{\rm inf} \sim 0.2\frac{M_{\rm BH}}{10^8\,M
_\odot} \left(\frac{\sigma_\star}{200\,\hbox{km}\,\hbox{s}}\right)^{-2}
\left(\frac{D}{10\,\hbox{Mpc}}\right)^{-1}\ \hbox{arcsec},$$ where $D$ is the distance. Thus dynamical evidence for SMBHs is hard to find.
The first stellar dynamical SMBH detections followed in the mid- to late-1980s, when CCDs became available on spectrographs and required the excellent seeing of observatories like Palomar and Mauna Kea [for reviews see @KormendyRichstone1995; @FerrareseFord2005]. The Hubble Space Telescope (HST), by delivering five-times-better resolution than ground-based optical spectroscopy, made it possible to find SMBHs in many more galaxies. This led to the convincing conclusion that *SMBHs are present in essentially every galaxy that has a bulge component*.
Because of the smallness of their sphere of influence, resolving it is possible only for relatively nearby, very massive SMBHs. Dynamical estimates based on line widths may not be reliable because of contributions to the mass from other components (dense star clusters, dark matter, ...).
Completely reliable estimates would require resolved proper motions of stars surrounding the SMBH, but so far this could be achieved only for the Milky Way, for which orbits have been determined for about 30 stars within $\simlt
0.5\,$arcsec from the SMBH and distances $\simlt 0.1\,$arcsec at periastron. The SMBH mass can be derived for each of them. However a complete orbit is well measured only for the star S2 which then provides the most accurate determination of the SMBH mass: $M_{\rm BH}=4.30\pm 0.20\, ({\rm
statistical})\pm 0.30\,({\rm systematic})\times 10^6\,M_\odot$ [@KormendyHo2013].
The pericenter radius of S2 is 0.0146arcsec. At the distance $R_0=8.28\pm0.15\,({\rm statistical}) \pm 0.29\,({\rm systematic})\,{\rm kpc}$ [@Genzel2010], this angular radius corresponds to 0.00059pc or $1,400\,
r_S$. The mass density within this radius is $\simeq 5\times
10^{15}\,M_\odot\,\hbox{pc}^{-3}$. The extended mass component within the orbit of S2 (visible stars, stellar remnants and possible diffuse dark matter) contributes less than 4 to 6.6% of this central mass [$2\,\sigma$; @Gillessen2009].
@Maoz1998 investigated the dynamical constraints on alternatives to the SMBH interpretation. Can the dark mass density be accounted for not by a point mass but by an ultra-dense cluster of any plausible form of nonluminous objects, such as brown dwarfs or stellar remnants? To answer this question he has investigated the maximum possible lifetime of such dark cluster against the processes of evaporation and physical collisions. It is highly improbable that clusters with a lifetime much shorter than the age of a galaxy ($\sim
10^{10}\,$yr, i.e. 10Gyr) survives till the present epoch. Thus short cluster lifetimes lend strong support to the SMBH interpretation.
The mass and the lower limit to the mass density of the Milky Way central object imply a cluster lifetime of only $\hbox{a\,few}\times 10^5\,$yr, implying that the case for a SMBH is quite strong. The Milky Way situation is however still unique. @Maoz1998 found a cluster lifetime shorter than $\simeq 10\,$Gyr for only another galaxy, NGC4258, for which VLBI measurements at mas resolution (i.e. about 100 times better than the HST) of the $H_2 O$ mega-maser at 22GHz from the circum-BH molecular gas disk were available [@Miyoshi1995].
The observations of @Miyoshi1995 established the $H_2 O$ mega-masers as one of the most powerful tools for measuring SMBH masses. At the distance of NGC4258 1 mas corresponds to 0.035pc. The masers have almost exactly a Keplerian velocity ($V\propto r^{-1/2}$), as expected in the case of a point source gravitational field. So it is likely that the disk mass is negligible so that the SMBH mass can be straightforwardly derived as $M_{\rm BH}= V^2 r/G$.
Useful maser disks are however rare, not least because they must be edge-on, and the orientation of the host galaxy gives no clue about when this is the case. Also in several cases the disk mass was found to be comparable to the SMBH mass, a major complication for SMBH mass measurements. However, progress on this subject has been accelerating in recent years [@Kuo2017a; @Kuo2017b; @Gao2017; @Henkel2018].
Information on the immediate environment, within a few gravitational radii, of the SMBH is provided by the asymmetric K$\alpha$ iron line due to fluorescence [e.g., @Fabian2013]. X-ray observations have shown that AGNs possess a strong, broad Fe K$\alpha$ line at 6.4keV (rest-frame). Spectral-timing studies, such as reverberation and broad iron line fitting, of these sources yield coronal sizes, often showing them to be small and in the range of 3 to 10 gravitational radii in size [@Fabian2015]. The line then experiences general-relativistic effects, such as strong light bending and large gravitational redshift. If the SMBH is rotating, its angular momentum manifests through the Lense-Thirring precession which occurs only in the innermost part of the accretion disk where the space-time becomes twisted in the same direction that the SMBH is rotating.
The BH spin is measured by the parameter $a=c\,J/G\,M_{\rm BH}^2$, $J$ being the angular momentum. Negative spin values represent retrograde configurations in which the BH spins in the opposite direction to the disk, positive values denote prograde spin configurations, and $a = 0$ implies a non-spinning BH. The relativistic effects allow the BH spin to be measured. In particular, the Fe K$\alpha$ line shape changes, as a function of BH spin; the breadth of the red wing of the line is enhanced as the BH spin increases [@Brenneman2013].
The So[ł]{}tan argument {#sect:Soltan}
=======================
A still not completely settled issue is: which is the main mechanism of SMBH growth? Radiative accretion certainly contributes. Actually, most of the accreted material is incorporated by the BH: the standard radiative efficiency adopted by AGN evolutionary models is $\simeq 10\%$, implying that $\sim 90\%$ of the mass adds to the BH. On the other hand, models of merger-driven galaxy evolution envisage that also the central SMBHs grow by merging. In this case the angular momentum is dissipated by gravitational radiation, i.e. happens without electromagnetic emission.
@Soltan1982 pointed out that, if the SMBH growth is mostly due to radiative accretion, the luminosity function of QSOs as a function of redshift traces the accretion history of the SMBHs: for an assumed mass-to-energy conversion efficiency, the luminosity function at any given redshift directly translates into an accreted mass density at that redshift. Integrating such mass density over redshift, gives a present day accreted mass density, which is a lower limit to the present day SMBH mass density since the SMBH mass can have also increased non-radiatively (e.g. via mergers).
Thus a comparison of the accreted mass density with the local SMBH mass density provides considerable insight into the formation and growth of massive SMBHs. If a BH is accreting at a rate [$\dot M$]{}, its emitted luminosity is $$L={\ensuremath{\epsilon}}{\ensuremath{\dot M}}_\mathrm{acc} c^2$$ where ${\ensuremath{\epsilon}}$ is the radiation efficiency, i.e. the fraction of the accreted mass which is converted into radiation and thus escapes the BH.
The growth rate of the BH, [$\dot M$]{}, is thus given by $${\ensuremath{\dot M}}=(1-{\ensuremath{\epsilon}}) {\ensuremath{\dot M}}_\mathrm{acc}.$$ Let’s neglect any process which, at time $t$, might ‘create’ or ‘destroy’ a BH with mass $M$. In particular, *this means neglecting BH merging*. Indeed, the merging process of two BHs, $M_1+M_2\rightarrow M_{12}$, means that BHs with $M_1$ and $M_2$ are destroyed while a BH with $M_{12}$ is created. Then, if ${\ensuremath{\epsilon}}$ is constant we have: $${\ensuremath{\rho_\mathrm{BH}}}= \frac{1-{\ensuremath{\epsilon}}}{{\ensuremath{\epsilon}}c^2} \, U_{T}$$ where $U_{T}$ is the total [*comoving*]{} energy density from AGNs (not to be confused with the total [*observed*]{} energy density), given by $$U_T = \int_0^{{{\ensuremath{z_s}}}} dz \frac{dt}{dz} \int_{L_1}^{L_2} L \phi(L,z)\,d L\,.$$ Here $\phi(L,z)\,d L$ is the *bolometric* AGN luminosity function. Note the factor $(1-{\ensuremath{\epsilon}})$ which is needed to account for the part of the accreting matter which is radiated away during the accretion process. If BH mergers, which don’t yield electromagnetic radiation but only gravitational waves, are important, the derived ${\ensuremath{\rho_\mathrm{BH}}}$ is a lower limit.
@Marconi2004 used luminosity functions in the optical B-band, soft X-ray (0.5–2 keV) and hard X-ray (2–10 keV) band [@Ueda2003] transformed into a bolometric luminosity function using bolometric corrections obtained from template spectral energy distributions (SEDs). In addition, they constrained the redshift-dependent X-ray luminosity function to reproduce the hard X-ray background, which can be considered an integral constraint on the total mass accreted over the cosmic time and locked in SMBHs. They obtained: $${\ensuremath{\rho_\mathrm{BH}}}=(4.7-10.6) \frac{(1-{\ensuremath{\epsilon}})}{9{\ensuremath{\epsilon}}}{\ensuremath{\,\times 10^5{\ensuremath{~\mathrm{M}_\odot}}{\ensuremath{\mathrm{\,Mpc}}}{\ensuremath{^{-3}}}}},$$ consistent with their own estimate from the local SMBH mass function, ${\ensuremath{\rho_\mathrm{BH}}}=(3.2-6.5){\ensuremath{\,\times 10^5{\ensuremath{~\mathrm{M}_\odot}}{\ensuremath{\mathrm{\,Mpc}}}{\ensuremath{^{-3}}}}}$, for the ‘canonical’ value ${\ensuremath{\epsilon}}=0.1$.
A similar conclusion was reached by Ueda et al. (2014) using updated X-ray luminosity functions, still constrained to reproduce the X-ray background, and comparing with the local SMBH mass density by @Vika2009, ${\ensuremath{\rho_\mathrm{BH}}}=(4.9
\pm 0.7) {\ensuremath{\,\times 10^5{\ensuremath{~\mathrm{M}_\odot}}{\ensuremath{\mathrm{\,Mpc}}}{\ensuremath{^{-3}}}}}$, derived from the empirical relation between SMBH mass and host-spheroid luminosity (or mass).
However, recent analyses of SMBH mass measurements and scaling relations concluded that the BH-to-bulge mass ratio shows a mass dependence and varies from 0.1–0.2% at $M_{\rm bulge} \simeq 10^9\,M_\odot$ to $\simeq 0.5\%$ at $M_{\rm bulge} \simeq 10^{11}\,M_\odot$ [@GrahamScott2013; @KormendyHo2013]. A similarly large median $M_{\rm BH}/M_{\rm bulge}$ ratio for *early type galaxies* was found by @Savorgnan2016 who however reported a substantial decrease of the ratio with decreasing stellar mass of the bulges of late-type galaxies.
The revised normalization is a factor of 2 to 5 larger than previous estimates ranging from $\simeq 0.10\%$ [@MerrittFerrarese2001; @McLureDunlop2002; @Sani2011] to $\simeq 0.23\%$ [@MarconiHunt2003], therefore resulting in an overall increase in the effective ratio, which is dominated by massive bulges.
This would imply either a lower mean radiative efficiency or an important non-radiative (e.g. merging) contribution to the BH growth. Radiatively inefficient processes (i.e. slim accretion disks) have been independently advocated to explain the fast growth of SMBHs in the early Universe [e.g., @Madau2014].
On the other hand, it was pointed out that the bulge-disk decomposition can lead to a considerable underestimate of the spheroid luminosity and stellar mass, $M_\star$ [@SavorgnanGraham2016] and the selection bias (the AGNs with higher luminosity, i.e., generally, with higher SMBH mass for given stellar mass, are more easily detected) can lead to an overestimate of $M_{\rm
BH}$. Both effects lead to an overestimate of the $M_{\rm BH}/M_\star$ ratio. According to @Shankar2016 the selection bias leads to an overestimate of $M_{\rm BH}$ by at least a factor of 3.
Based on a comprehensive analysis of the co-evolution of galaxies and SMBHs throughout the history of the universe by a statistical approach using the continuity equation and the abundance matching technique, @Aversa2015 found a mean $M_{\rm BH}$–$M_\star$ relation systematically lower by a factor of $\simeq 2.5$ than that proposed by @KormendyHo2013. The SMBH-to-stellar mass ratio was found to evolve mildly at least up to $z\simlt
3$, indicating that the SMBH and stellar mass growth occurs in parallel by in situ accretion and star formation processes, with dry mergers playing a marginal role at least for the stellar and SMBH mass ranges for which the observations are more secure.
Although the issue is still debated, there are physical arguments indicating that the AGN radiative efficiency, $\epsilon$, varies with the system age, although how this happens is not yet clear. The global consistency between the SMBH mass density inferred from the So[ł]{}tan approach and from the local SMBH mass function constrains predictions on the gravitational wave signals expected from SMBH mergers.
In the case of thin-disk accretion, $\epsilon$ may range from 0.057 for a non-rotating BH to 0.32 for a rotating Kerr BH with spin parameter $a=0.998$ [@Thorne1974]. During a coherent disk accretion, the SMBH is expected to spin up very rapidly, and correspondingly the efficiency is expected to increase up to $\simeq 0.3$. On the other hand, when the mass is flowing towards the SMBH at high rates (super-Eddington accretion), the matter accumulates in the vicinity of the BH and the accretion may happen via the radiatively-inefficient ‘slim-disk’ solution [@Abramowicz1988; @Begelman2012; @Madau2014; @AbramowiczStraub2014; @Volonteri2015] that speeds up the SMBH growth.
This would relieve the challenge set by the existence of billion-solar-mass black holes at the end of the reionization epoch [see, e.g., @Haiman2013 for a review]. The most distant quasar discovered to date, ULAS J$1120+0641$ at a redshift $z = 7.084$, is believed to host a black hole with a mass of $2.0(+1.5, -0.7)\times 10^9\,M_\odot$, only 0.78Gyr after the big bang [@Mortlock2011]. The need of a radiatively inefficient phase is however debated. For example, @Trakhtenbrot2017 argue that “the available luminosities and masses for the highest-redshift quasars can be explained self-consistently within the thin, radiatively efficient accretion disk paradigm”.
@Trakhtenbrot2018 have observed with ALMA six luminous quasars at $z\sim
4.8$ finding spectroscopically confirmed companion sub-millimeter galaxies for three of them. The companions are separated by $\sim 14-45\,$kpc from the quasar, supporting the idea that major mergers may be important drivers for rapid, early BH growth. However, the fact that not all quasar hosts with intense star formation are accompanied by interacting sub-mm galaxies, and their ordered gas kinematics observed by ALMA, suggest that other processes may be fueling these systems. They then conclude that data demonstrate the diversity of host galaxy properties and gas accretion mechanisms associated with early and rapid SMBH growth.
Relationships between galaxies and central SMBHs {#sect:interaction}
================================================
A tight correlation between $M_{\rm BH}$ and the galaxy velocity dispersion, $\sigma$, was reported, independently, by @FerrareseMerritt2000 and by @Gebhardt2000. Both papers claimed that the scatter was only 0.30 dex over almost 3 orders of magnitude in $M_{\rm BH}$ and no larger than expected on the basis of measurement errors alone. This suggested that the most fundamental relationship between SMBHs and host galaxies had been found and that it implies a close link between SMBH growth and bulge formation.
Many papers have expanded on this result with bigger samples [for a review, see @KormendyHo2013]. The $M_{\rm BH}$–$\sigma$ correlation has been confirmed to be the strongest, with the lowest measured and intrinsic scatter [@Saglia2016]. The correlations with the bulge mass, $M_{\rm bulge}$, is also strong, except for the pseudo-bulge[^2] subsample .
@Saglia2016 find, for ellipticals and classical bulges: $$\begin{aligned}
\log(M_{\rm BH}/M_\odot)&=&(4.868\pm 0.32)\log(\sigma/\hbox{km}\,\hbox{s}^{-1})-(2.827\pm 0.75) \\
\log(M_{\rm BH}/M_\odot)&=&(0.846\pm 0.064)\log(M_{\rm bulge}/M_\odot)-(0.713 \pm 0.697)\end{aligned}$$ Both relationships agree, within the errors, with those derived by @KormendyHo2013; however, while @Saglia2016 find a mean $M_{\rm
BH}/M_{\rm bulge}$ ratio slowly decreasing with increasing $M_{\rm bulge}$, @KormendyHo2013 find a slowly increasing trend.
The normalization of the $M_{\rm BH}$–$M_{\rm bulge}$ relation is currently debated. As mentioned above, the main issues are the difficulties of removing the disk component to derive $M_{\rm bulge}$ and, even more, the bias on $M_{\rm BH}$ estimates.
An inspection of the @Saglia2016 results shows that:
- “core” ellipticals[^3] have more massive BHs than other classical bulges, at a given $\sigma$ or bulge mass;
- the smallest intrinsic and measured scatters of the $M_{\rm
BH}$–$\sigma$ and $M_{\rm BH}$–$M_{\rm bulge}$ relations are measured for the sample of “core” ellipticals;
- “power-law” early-type galaxies and classical bulges follow similar $M_{\rm BH}$–$\sigma$ and $M_{\rm BH}$–$M_{\rm bulge}$ relations;
- pseudo-bulges have smaller SMBH masses than the rest of the sample at a given $\sigma$ or $M_{\rm bulge}$;
- disks do not correlate with $M_{\rm BH}$ and it is unclear whether pseudo-bulges do: SMBH masses do not “know about” galaxy disks [@Kormendy2011].
These relations suggest a tight link between star-formation activity in the spheroidal components of galaxies (not in the disks) and SMBH growth. Such link is strongly confirmed by the striking similarity of the evolution of the SMBH accretion rate and of the star formation rate or of the AGN and galaxy luminosity densities, especially at substantial redshifts where the star formation mostly occurs in the spheroidal components [@Shankar2009; @Fiore2017].
The growth histories of the stellar mass and of the AGN luminosity (hence of SMBH mass) share further similarities. The most massive early-type galaxies form in short (duration $\simlt 1\,$Gyr), intense starbursts at high redshift while less massive galaxies have more extended star formation histories that peak later with decreasing mass [e.g., Fig.9 of @Thomas2010]. This “anti-hierarchical” nature (called “downsizing”) is mirrored in the SMBH growth: the most massive SMBHs likely grow in intense quasar phases which peak in the early universe, while less massive SMBHs have more extended, less intense growth histories that peak at lower redshift [e.g., @Ueda2014]. All that suggests a strong galaxy-AGN co-evolution.
AGN impact on galaxy evolution {#sect:coev}
==============================
AGN-driven winds
----------------
As we have seen, the smallness of the radius of influence means that the SMBH’s gravity has a completely negligible effect on its host galaxy. On the other hand, the energy released by the AGN $$E_{\rm BH} \simeq \epsilon Mc^2 \sim 2\times 10^{61}{\epsilon \over 0.1}{M_{\rm BH}\over 10^8\, M_\odot}\,{\rm erg},$$ where $\epsilon$ is the mass to radiation conversion efficiency, is far larger than the gas binding energy. Setting $M_{\rm gas} = f M_{\rm bulge}$, with $f<1$, we have $$E_{\rm gas} \sim {3\over 2} f M_{\rm bulge}\sigma^2 \sim 1.2 \times 10^{58}f {M_{\rm bulge}/M_{\rm BH}\over 10^3}{M_{\rm BH}\over 10^8\, M_\odot}\left({\sigma\over 200\,\hbox{km/s}}\right)^{-2}\,{\rm erg},$$ where $\sigma$ is the line-of-sight velocity dispersion (the corresponding 3D velocity is $v=\sqrt{3}\sigma$). This means that only a few percent of the SMBH energy output may have a strong influence on the gas in the host galaxy, potentially expelling it and, at the same time, limiting the SMBH own growth.
The SMBH energy release can potentially affect its surroundings in two main ways. By far the stronger one (in principle) is through direct radiation. This is particularly effective during heavily dust-obscured phases of AGN evolution [@Fabian2002].
The second form of coupling the SMBH energy release to a host bulge is mechanical. The huge accretion luminosity of SMBH’s may drive powerful gas flows into the host, impacting into its interstellar medium. A well know form of flow is jets, highly collimated flows driven from the immediate vicinity of the SMBH (“radio-mode” feedback). However to affect most of the bulge requires a way of making the interaction relatively isotropic, perhaps with changes of the jet direction over time. Moreover, radio observations show that the jet energy is dissipated on scales from several kpc to Mpc, i.e. on scales larger than that of a galaxy. Therefore they are more relevant for heating the intergalactic medium (IGM) in galaxy clusters.
A form of mechanical interaction that has automatically the right property are near-isotropic winds carrying large momentum fluxes (“quasar mode” feedback). Such winds are indeed observed in many AGNs, as we will see.
@SilkRees1998 pointed out that an AGN emitting at close to the Eddington rate could expel gas completely from its host galaxy provided that $$M_{\rm BH} > {{f\sigma^5\sigma_{\rm T}}\over{4\pi G^2 m_{\rm p} c}},$$ where $\sigma_{\rm T}$, $G$, $m_p$, $c$ and $f$ are the Thomson cross section for electron scattering, the gravitational constant, the proton mass, the speed of light and the gas mass fraction, respectively. The galaxy bulge is assumed to be isothermal with the radius $r$, so that its mass is $M_{\rm
bulge}=2\sigma^2 r/G$. The gas mass can be written as $M_{\rm gas} = f M_{\rm
bulge}$ with $f<1$. The maximum collapse rate of the gas is $\dot{M}_{\rm gas}=
{M_{\rm gas}/t_{\rm free-fall}}$ with $t_{\rm free-fall}=r/\sigma$. The corresponding power is $$\dot{E}_{\rm gas}= {1/2}({M_{\rm gas}/t_{\rm free-fall}})v^2 = 3{f\sigma^5/G},$$ ($v=\sqrt{3}\sigma$). The relation $M_{\rm BH}$–$\sigma$ then follows equating $\dot{E}_{\rm gas}$ to the Eddington luminosity $$L_{\rm Edd}={4\pi G M_{\rm BH}m_p c\over \sigma_T}.$$ Plugging in the numbers and considering that only a fraction, $f_{\rm Edd}$, of the Eddington luminosity can be used to throw the gas out we get: $$M_{\rm BH}={3.6\times 10^5 \over f_{\rm Edd}}\left({\sigma\over 100\,\hbox{km/s}}\right)^5
\,M_\odot.$$ A comparison with the empirical $M_{\rm BH}$–$\sigma$ relation shows that $f_{\rm Edd}$ at the few/several percent level is enough to stop the accretion and to expel the gas from the host galaxy.
Alternatively, we may have momentum or force (instead of energy) balance [@Fabian1999; @Fabian2002; @King2003; @King2005; @Murray2005]. Balancing the outward radiation force with the inward one due to gravity gives $${{4\pi GM_{\rm BH}m_{\rm p}}\over \sigma_{\rm T}}=
{L_{\rm Edd}\over c}={{GM_{\rm gal}M_{\rm gas}}\over
r^2}={{fGM_{\rm gal}^2}\over r^2}={{fG}\over r^2}{\left({2\sigma^2
r}\over G\right)^2}$$ i.e., $${{4\pi G M_{\rm BH}m_{\rm p} }\over{\sigma_{\rm T}
}}={{4f\sigma^4}\over G},$$ from which we get $$M_{\rm BH}={{f\sigma^4\sigma_{\rm T}}\over{\pi G^2 m_{\rm p}}}=
{1.43\times 10^9}f \left({\sigma\over 100\,\hbox{km/s}}\right)^4\,M_\odot.$$ Thus in the case of momentum-driven flows, the SMBH mass required to stop the accretion is a factor $\sim c/\sigma$ larger than in the energy-driven case. A comparison with the empirical $M_{\rm BH}$–$\sigma$ relation shows that in this case, even the full Eddington luminosity is not enough to unbind the gas unless the gas fraction is small ($f < 0.1$) or the AGN has a strongly super-Eddington luminosity. That the full AGN luminosity goes into radiation pressure is expected in the case of “Compton-thick” objects, most of whose radiation is absorbed by the gas.
In the radiation-driven case it is implicitly assumed that the cooling of the SMBH wind is negligible. Under what conditions does this apply? We can crudely model the outflows as quasi-spherical winds from SMBHs accreting at about the Eddington rate $$\dot{M}_w\simeq \dot{M}_{\rm Edd}= {L_{\rm Edd}\over \epsilon c^2}\simeq 0.22
{M_{\rm BH}\over 10^8}\,M_\odot\,\hbox{yr}^{-1}.$$ Winds like this have electron scattering optical depth $\tau\sim 1$, measured inward from infinity to a distance of order the Schwarzschild radius $R_{\rm
S}=2 G M/ c^2$ [@KingPounds2015]. So on average every photon emitted by the AGN scatters about once before escaping to infinity.
Because electron scattering is front-back symmetric, each photon on average gives up all its momentum to the wind, and so the total (scalar) wind momentum should be of order the photon momentum, or $$\dot{M}_w v\sim {L_{\rm Edd}\over c}\simeq \dot{M}_{\rm Edd} \epsilon c,$$ where $v$ is the wind’s terminal velocity. Since $\dot{M}_w\simeq \dot{M}_{\rm
Edd}$ we get [@KingPounds2015]: $$v\simeq 0.1 {\epsilon\over 0.1} c\, .$$ The instantaneous wind mechanical luminosity is then $$L_{\rm BHwind}={1\over 2}v^2\dot{M}_w\simeq {1\over 2}{v\over c} L_{\rm Edd}
\simeq 0.05 {\epsilon\over 0.1} L_{\rm Edd},$$ in good agreement with the earlier estimate for the energy-driven winds.
A self-consistent model is described by @KingPounds2015. The black hole wind is abruptly slowed in an inner (within the SMBH sphere of influence) shock, in which the temperature approaches $\sim 10^{11}\,$K. The shocked wind gas acts like a piston, sweeping up the host ISM at a contact discontinuity moving ahead of it. Because this swept-up gas moves supersonically into the ambient ISM, it drives an outer (forward) shock into it. The dominant interaction here is the reverse shock slowing the black hole wind, which injects energy into the host ISM. The nature of this shock differs sharply depending on whether some form of cooling (typically radiation) removes significant energy from the hot shocked gas on a timescale shorter than its flow time.
If the cooling is strong (momentum-driven flow), most of the pre-shock kinetic energy is lost (usually to radiation). As momentum must be conserved, the post-shock gas transmits just its ram pressure to the host ISM. As we have seen, this amounts to transfer of only a fraction $\sim \sigma/c\sim 10^{-3}$ of the mechanical luminosity $L_{\rm BH,wind}\simeq 0.05\,L_{\rm Edd}$ to the ISM. In other words, for SMBHs close to the $M_{\rm BH}$–$\sigma$ relation, in the momentum-driven limit only $\sim 10\%$ of the gas binding energy is injected into the bulge ISM, which is therefore stable.
In the opposite limit in which cooling is negligible, the post-shock gas retains all the mechanical luminosity and expands adiabatically into the ISM. The post-shock gas is now geometrically extended. The mechanical energy, thermalized in the shock and released to the ISM, now equals the gas binding energy. The energy-driven flow is much more violent than the momentum-driven flow and can unbind the bulge.
[Note that if the SMBH and galaxy evolve strictly in parallel, preserving the $M_{\rm BH}$–$\sigma$ relation, a SMBH in an energy-driven environment is unlikely to reach the observed SMBH masses because it would stop its gas accretion]{}. We will come back to this point in the following.
To sum up, for both momentum- and energy-driven outflows a fast wind (velocity $\sim 0.1c$) impacts the interstellar gas of the host galaxy, producing an inner reverse shock that slows the wind and an outer forward shock that accelerates the swept-up gas. In the momentum-driven case, the shocks are very narrow and rapidly cool to become effectively isothermal; only the ram pressure is communicated to the outflow, leading to very low kinetic energy, $\sim
(\sigma/c)\,L_{\rm Edd}$. In an energy-driven outflow, the shocked regions are much wider and do not cool; they expand adiabatically, transferring most of the kinetic energy of the wind to the outflow.
Observed wind properties
------------------------
A recent study of relations between AGN properties, host galaxy properties, and AGN winds has been carried out by @Fiore2017. This paper also contains an exhaustive list of references to observations of massive outflows of ionised, neutral and molecular gas, extended on kpc scales, with velocities of order of $1000\,\hbox{km}\,\hbox{s}^{-1}$. Three main techniques have been used to detect such outflows [see @Fiore2017 for references]: deep optical/near-infrared spectroscopy, mainly from integral field observations; interferometric observations in the (sub)millimetre domain; far-infrared spectroscopy from *Herschel*. In addition, AGN-driven winds on sub-parsec scales, from the accretion disk scale up to the dusty torus, are now detected routinely up to $z>2$ as blue-shifted absorption lines in the X-ray spectra of a substantial fraction of AGNs [e.g.. @Kaastra2014]. The most powerful of these winds have extreme velocities (ultrafast outflows, UFOs, with $v\sim 0.1$–$0.3c$) and are made by highly ionised gas which can be detected only at X-ray energies.
The main conclusions of the @Fiore2017 study are:
- the mass outflow rate is correlated with the AGN bolometric luminosity;
- the fraction of outflowing gas in the ionised phase increases with the bolometric luminosity;
- the wind kinetic energy rate (kinetic power) $\dot{E}_{\rm kin}$ is correlated with the AGN bolometric luminosity, $L_{\rm bol}$, for both molecular and ionized outflows: we have $\dot{E}_{\rm kin}/L_{\rm
bol}\sim 1 - 10\%$ for molecular winds and $\dot{E}_{\rm kin}/L_{\rm
bol}\sim 0.1 - 10\%$ for ionised winds;
- About half X-ray absorbers and broad absorption line (BAL) winds have $\dot{E}_{\rm kin}/L_{\rm bol}\sim 0.1 - 1\%$ with another half having $\dot{E}_{\rm kin}/L_{\rm bol}\sim 1 - 10\%$.
- Most molecular winds and the majority of ionised winds have kinetic power in excess to what would be predicted if they were driven by supernovae, based on the SFRs measured in the AGN host galaxies. The straightforward conclusion is that the most powerful winds are AGN driven.
- The average AGN wind mass-loading factor[^4], $\langle \eta \rangle$, is between 0.2 and 0.3 for the full galaxy population while $\langle \eta
\rangle \gg 1$ for massive galaxies at $z \simlt 2$. We may then tentatively conclude that AGN winds are, on average, powerful enough to clean galaxies from their molecular gas (either expelling it from the galaxy or by destroying the molecules) in massive systems only, and at $z
\simlt 2$.
- What happens at $z> 2$ is still unclear.
AGN winds may then be the manifestation of AGN feedback linking nuclear and galactic processes in massive galaxies and accounting for the correlation of SMBH masses and properties of host bulges.
Do SFR and BH accretion rate simply track each other?
-----------------------------------------------------
A widely cited galaxy-AGN co-evolution model [@Hopkins2008 see, in particular, their Fig. 1] envisage the following scenario for galaxy evolution. Early galaxies have a disk morphology and grow mainly in quiescence until the onset of a major merger. During the early stages of the merger, tidal torques excite some enhanced star formation and SMBH accretion. During the final coalescence of the galaxies, massive inflows of gas trigger strong starbursts. The high gas densities feed a rapid SMBH growth.
In this scenario, star formation and SMBH accretion evolve strictly in parallel. If so we expect a direct proportionality between the corresponding luminosities, i.e., in practice between infrared (IR)[^5] and X-ray luminosities, $L_{\rm X}$. In fact, since star formation at substantial redshifts is dust enshrouded, the light emitted by young stars is absorbed by dust and re-emitted at far-IR wavelengths; so $L_{\rm IR}$ is a measure of the luminosity produced by star formation. In turn, $L_{\rm X}$ is the best indicator of accretion luminosity.
The connection between star formation and SMBH accretion has been investigated by several authors using samples of far-IR selected galaxies followed up in X-rays and of X-ray/optically selected AGNs followed up in the far-IR band [e.g., @Lapi2014 and references therein]. Most recently, @Lanzuisi2017 looked for correlations between the average properties of X-ray detected AGNs and their far-IR detected star forming host galaxies, using a large sample of X-ray and far-IR detected objects in the COSMOS field. The sample covered the redshift range $0.1< z <4$ and about 4 orders of magnitude in X-ray and far-IR luminosity, and in stellar mass. They found that $L_{\rm
X}$ and $L_{\rm IR}$ are significantly correlated. However, splitting the sample into five redshift bins, they found that for every redshift bin both luminosities have a broad distribution, with weak or no signs of a correlation (see their Fig.4).
@Lanzuisi2017 further investigated, for each redshift bin, the relationships between $L_{\rm IR}$ and $L_{\rm X}$ in $L_{\rm X}$ bins and between $L_{\rm X}$ and $L_{\rm IR}$ in $L_{\rm IR}$ bins (see their Fig.5). The two relationships can be very different if the link between the two luminosities is complex, as in the case of a weak correlation for the bulk of the population and a strong correlation only for the most extreme objects. This turns out to be case. The average host $L_{\rm IR}$ has a quite flat distribution in bins of $L_{\rm X}$ (i.e. the two quantities are weakly, if at all, correlated), while the average $L_{\rm X}$ somewhat increases in bins of $L_{\rm IR}$ with logarithmic slope of $\simeq 0.7$ in the redshift range $0.4
< z <1.2$. At higher redshifts the slopes become flatter.
The simplest interpretation of these data goes as follows. At high-$z$ the gas is very abundant in galactic halos and therefore both the star formation and the SMBH accretion can proceed vigorously. However the timescales are widely different. In the case of star formation, the relevant timescale is the minimum between the dynamical time, which is $\sim 0.1\,$Gyr for massive galaxies, and the cooling timescale, that can be much longer. Also, since there is an abundant amount of gas available, the star formation can proceed until the gas is swept outside the galaxy by some feedback process (see Sect. \[sect:feedback\]). As argued by several authors [@Granato2004; @Thomas2010; @Lapi2014] the star formation timescale is likely of at least 0.5–0.7Gyr for massive spheroidal galaxies and longer for less massive galaxies.
The SMBH accretion timescale is much shorter. During the early evolutionary phases the AGN accretion rate likely occurs at about the Eddington limit: $$L_{\rm AGN}=\epsilon c^2 \dot{M}_{\rm BH}=\lambda L_{\rm Edd}$$ where $\lambda$ is the Eddington ratio and $$L_{\rm Edd}= \frac{4\pi c G M_{\rm BH} \mu_e}{\sigma_T} =
1.51\times 10^{38}\frac{M_{\rm BH}}{M_\odot}\ \hbox{erg}\,\hbox{s}^{-1}$$ $\mu_e\simeq 1.2 m_p$ being the mass per unit electron and $\sigma_T$ the Thomson cross-section.
If $L_{\rm AGN}= L_{\rm Edd}$, i.e. $\lambda=1$, $M_{\rm BH}$ grows exponentially: $$M_{\rm BH}=M_{\rm seed} e^{t/\tau_{\rm S}}$$ where $\tau_{\rm S}$ is the Salpeter time $$\tau_{\rm S}=\frac{\epsilon c \sigma_T}{4\pi G \mu_e}= 3.7\times 10^7 \frac{\epsilon}{0.1}\, \hbox{yr}.$$ Because of the large difference between the accretion and the star-formation timescale, during *the Eddington-limited regime* the star-formation and the AGN luminosities cannot be proportional to each other: the AGN luminosity increases exponentially while that due to star formation is approximately constant.
On the other hand, at later times, the accretion is strongly sub-Eddington and the accretion timescale can match the star-formation timescale, as in the case of re-activation of star-formation and nuclear activity (“rejuvenation”) by interactions or mergers.
Feedback and galaxy evolution {#sect:feedback}
=============================
The AGN feedback avoids the formation of too many massive galaxies and the presence of too many baryons in their galactic halos. In fact, while the mean cosmic baryon density in units of the critical density is $\Omega_{\rm b}
=0.0486$ [@PlanckCollaborationXIII2016], the baryon density in galaxies is $\Omega_{\rm b, gal} \simeq 0.00345$ [@FukugitaPeebles2004], i.e. only $\sim 7\%$ of baryons are in galaxies. Thus a process capable of sweeping out more that 90% of baryons initially present in galactic halos is necessary.
There are however other structural problems that cannot be solved by the AGN feedback alone. The naïve assumption that the stellar mass follows the halo mass leads also to too many small galaxies [see Fig. 1 of @SilkMamon2012]. For halo masses $\simlt 10^{11}\,M_\odot$ the energy input into the interstellar medium is dominated by supernovae [@Granato2004] that become the key player in determining the slope of the low-mass portion of the galaxy stellar mass function as well as the shape of the Faber-Jackson and of the $M_{\rm BH}$–$\sigma$ for less massive bulges.
Why is the energy injection by supernovae insufficient for the most massive galaxies? Although the *total energy* released by supernovae, integrated over the galaxy lifetime, may be large, the mean *power* (energy released per unit time) is not enough to expel the gas from large galaxies. According to @DekelSilk1986 supernovae can cause gas disruption and dispersal in intermediate mass and massive dwarf galaxies (halo mass $\sim 10^8
- 10^{10}\,M_\odot$) and can expel the remaining baryons in systems of halo mass up to $\sim 10^8\,M_\odot$, leaving behind dim dwarf galaxy remnants. According to @Granato2004, if 5% of the energy released by supernova explosions goes into heating of the gas, supernova-driven winds can eject large gas fractions from halos of up to $\sim 10^{11}\,M_\odot$.
In very low-mass halos gas cannot even fall in, because its specific entropy is too high [@Rees1986]. Only halos of mass $> 10^5\,M_\odot$ can trap baryons that are able to undergo early $\hbox{H}_2$ cooling and eventually form stars. Heating by re-ionization increases this mass limit. The abrupt increase of the sound speed to $10 - 20\,\hbox{km}\,\hbox{s}^{-1}$ at $z \sim 8$ means that dwarfs of halo mass $\sim 10^6 - 10^7\,M_\odot$, which have not yet collapsed and fragmented into stars, will be disrupted [@SilkMamon2012].
Conclusions {#sect:conclusions}
===========
SMBH masses are tightly correlated with properties of the spheroidal component (classical bulges) of their host galaxies, such as the stellar velocity dispersion (that shows the tightest correlation), the stellar mass, the luminosity. In contrast, they do not correlate with disk properties.
Since the spheroidal components of galaxies possess the older stellar populations, this suggests that the SMBH growth happens in the context of dissipative baryon collapse at substantial redshifts. The evolution of luminosity densities due to star-formation in spheroidal galaxies and to AGN activity proceed in parallel, i.e. the SMBH growth and galaxy build up follow a similar evolution through cosmic history, with a peak at $z\sim 2 - 3$ and a sharp decline toward the present age.
This implies a mutual relationship beween star formation and SMBH accretion rates, which however do not simply track each other. Most growth of large SMBHs happens by radiatively efficient gas accretion (So[ł]{}tan argument). The energy radiated by a SMBH ($\sim 0.1\,M_{\rm BH} c^2$, assuming a radiative efficiency of 10%) is much larger than the binding energy of its host bulge ($\sim M_{\rm
bulge} \sigma^2$, $\sigma$ being the stellar velocity dispersion). If only $\sim 5\%$ of the AGN energy output couples to gas in the forming galaxy, then all of the gas can be blown. Thus, SMBH growth may be self-limiting, and AGNs may quench star formation.
The substantial stellar masses and star-formation rates of sub-millimeter galaxies (SMGs) and the evidence for subdominant AGN activity and moderate SMBH masses in these objects imply that most of the star formation occurs before the SMBH reach large masses. This is easily understood since the early SMBH growth is Eddington limited.
When the SMBHs reach a critical threshold, their “quasar-mode energy feedback” balances outward radiation or mechanical pressure against gravity. Then the AGNs blow away the interstellar gas, quenching star formation and leaving the galaxies red and dead. AGNs become visible and continue to shine for a few Salpeter times, accreting the “reservoir” (torus) mass. This convincingly solves some problems of galaxy formation, such as expelling a large fraction of initial gas to account for the present day baryon to dark matter ratio in galaxies and preventing excessive star formation (the stellar mass function of galaxies sinks down, at large masses, much faster than the halo mass function).
However, observations of winds capable of removing the interstellar gas from $z> 2$ galaxies are still missing, and a large fraction of massive early type galaxies formed most of their stars at these redshifts. To remove the gas we need a kinetic power of $\sim 5\%$ of the AGN bolometric luminosity, but the scanty observations of high $z$ winds generally indicate lower kinetic powers.
Two kinds of AGN feedback have been considered. Radio-mode feedback (powerful jets of radio sources) is a well known phenomenon but can hardly substantially affect the galaxy evolution: it is very hard to confine well-collimated jets within their galaxies. As [@KormendyHo2013] put it: “Firing a rifle in a room does not much heat the air in the room”. It is much more plausible that the jet energy is dissipated by interactions with the intergalactic gas, especially in galaxy clusters.
To efficiently operate on galaxy scales, the feedback must be relatively isotropic (quasar-model feedback). But the physics is not well understood. The correlations between SMBH masses and the galaxy velocity dispersions are consistent with both energy- and momentum-driven feedback. But only energy-driven feedback can efficiently quench star formation.
Outflows with kinetic power sufficient to clean the galaxy of cold gas were found for only a few high-$z$ galaxies. They are more common at $z \simlt 2$. But most local AGNs accrete at very sub-Eddington rates. Very few galaxies are still growing their SMBHs at a significant level. Rapid SMBH growth by radiatively efficient accretion took place mostly in more massive galaxies that are largely quenched today. That is, the era of SMBH growth by radiatively efficient accretion is now mostly over: co-evolution happened at high $z$.
[^1]: Estimates of $M_{\rm BH}/M_\star$ range from $\simeq 0.1\%$ [@Sani2011] to $\simeq 0.49\%$ [@KormendyHo2013].
[^2]: While bulge properties are indistinguishable from those of elliptical galaxies, except that they are embedded in disks, pseudo-bulges have more disk-like properties and are though to be made by slow (“secular”) evolution internal to isolated galaxy disks.
[^3]: Detailed studies of the central regions of early-type galaxies [@Ferrarese1994; @Faber1997] have identified two distinct classes of galaxy centers: “power-law” galaxies, whose central surface brightness shows a steep power-law profile; and “core” galaxies, where the luminosity profile turns over at a fairly sharp “break radius” into a shallower power-law. @Ferrarese1994 and @Faber1997 found evidence that global parameters of early-type galaxies correlated with their nuclear profiles: “core” galaxies tend to have high luminosities, boxy isophotes, and pressure-supported kinematics, while “power-law” galaxies are typically lower-luminosity and often have disky isophotes and rotationally supported kinematics.
[^4]: The mass-loading factor, $\eta$, is the ratio between the mass outflow rate, $\dot{M}_w$, and the SFR: $\eta=\dot{M}_w/\hbox{SFR}$.
[^5]: It has become common practice to define the IR luminosity, $L_{\rm IR}$, as the integrated emission between 8 and $1000\,\mu$m. In this paper we adopt this convention.
|
---
abstract: 'For any non-uniform lattice ${\Gamma}$ in ${\operatorname{SL}}_2({\mathbb{R}})$, we describe the limit distribution of orthogonal translates of a [*divergent*]{} geodesic in ${\Gamma}{\backslash}{\operatorname{SL}}_2({\mathbb{R}})$. As an application, for a quadratic form $Q$ of signature $(2,1)$, a lattice ${\Gamma}$ in its isometry group, and $v_0\in {\mathbb{R}}^3$ with $Q(v_0)>0$, we compute the asymptotic (with a logarithmic error term) of the number of points in a discrete orbit $v_0{\Gamma}$ of norm at most $T$, when the stabilizer of $v_0$ in ${\Gamma}$ is finite. Our result in particular implies that for any non-zero integer $d$, the smoothed count for number of integral binary quadratic forms with discriminant $d^2$ and with coefficients bounded by $T$ is asymptotic to $c\cdot T \log T +O(T)$.'
address:
- 'Mathematics department, Brown university, Providence, RI, U.S.A and Korea Institute for Advanced Study, Seoul, Korea'
- 'Department of Mathematics, The Ohio State University, Columbus, OH 43210, U.S.A'
author:
- Hee Oh
- 'Nimish A. Shah'
title: 'Limits of translates of divergent geodesics and Integral points on one-sheeted hyperboloids'
---
[^1]
[^2]
Introduction
============
Motivation
----------
Let $Q\in {\mathbb{Z}}[x_1, \cdots, x_n]$ be a homogeneous polynomial and set $V_m:=\{x\in {\mathbb{R}}^n: Q(x)=m\}$ for an integer $m$. It is a fundamental problem to understand the set $V_m({\mathbb{Z}})=\{x\in {\mathbb{Z}}^n: Q(x)=m\}$ of integral solutions.
In particular, we are interested in the asymptotic of the number $N(T):=\#\{x\in V_m({\mathbb{Z}}): \|x\|<T\}$ as $T\to \infty$, where $\|\cdot \|$ is a fixed norm on ${\mathbb{R}}^n$.
The answer to this question depends quite heavily on the geometry of the ambient space $V_m$. We suppose that the variety $V_m$ is homogeneous, i.e., there exist a connected semisimple real algebraic group $G$ defined over ${\mathbb{Q}}$ and a ${\mathbb{Q}}$-rational representation $\iota: G\to {\operatorname{SL}}_n$ such that $V_m=v_0. \iota(G)$ for some non-zero $v_0\in {\mathbb{Q}}^n$.
Let ${\Gamma}<G({\mathbb{Q}})$ be an arithmetic subgroup preserving $V_m({\mathbb{Z}})$. By a theorem of Borel and Harish-Chandra [@BH], the co-volume of ${\Gamma}$ in $G$ is finite and there are only finitely many ${\Gamma}$-orbits in $V_m({\mathbb{Z}})$. Hence understanding the asymptotic of $N(T)$ is reduced to the orbital counting problem on $\#(v_0\Gamma \cap B_T)$ for $B_T=\{x\in V_m: \|x\|<T\}$ and $v_0\in V_m({\mathbb{Z}})$.
\[drs\] Set $H$ to be the stabilizer subgroup of $v_0$ in $G$. Suppose that $H$ is either a symmetric subgroup or a maximal ${\mathbb{Q}}$-subgroup of $G$. If the volume of $(H\cap {\Gamma}){\backslash}H$ is finite, i.e., if $H\cap {\Gamma}$ is a lattice in $H$, we have $$\#(v_0\Gamma \cap B_T) \sim \frac{{\operatorname}{vol}_{H}(H\cap {\Gamma}{\backslash}H)}{{\operatorname}{vol}_G({\Gamma}{\backslash}G)}{\operatorname}{vol}_{H{\backslash}G} (B_T)$$ where the volumes on $H, G$ and $v_0G\simeq H{\backslash}G$ are computed with respect to invariant measures chosen compatibly; that is, $d{\operatorname}{vol}_G=d{\operatorname}{vol}_H\times d{\operatorname}{vol}_{H{\backslash}G}$ locally.
This theorem was first proved by Duke, Rudnick, Sarnak [@DukeRudnickSarnak1993] when $H$ is symmetric and Eskin and McMullen gave a simplified proof in [@EskinMcMullen1993]. When $H$ is a maximal ${\mathbb{Q}}$-subgroup, it is proved by Eskin, Mozes and Shah in [@EskinMozesShah1996].
As apparent from the main term of the asymptotic, it is crucial to assume ${\operatorname}{vol}(H\cap {\Gamma}{\backslash}H)<\infty$ in Theorem \[drs\]. The main aim of this paper is to break this barrier; to investigate the counting problem in the case when ${\operatorname}{vol}(H\cap {\Gamma}{\backslash}H)=\infty$.
We focus on the case when $Q$ is a quadratic form of signature $(n-1,1)$ with $n\ge 3$ and $G$ is the special orthogonal group of $Q$. In this situation, the case of ${\operatorname}{vol}(H\cap {\Gamma}{\backslash}H)=\infty$ for $H={\operatorname}{Stab}_G(v_0)$ arises only when $n=3$ and $Q(v_0)=m>0$, that is, when the variety $V_m=\{x\in {\mathbb{R}}^3: Q(x)=m\}$ is a one-sheeted hyperboloid. To prove this claim, note first that if $H$ is a non-compact simple Lie group, then any closed ${\Gamma}{\backslash}{\Gamma}H$ in ${\Gamma}{\backslash}G$ must be of finite volume by Dani [@Dani1979] and Margulis [@MargulisICM1991] (see also [@Shah1991]). Any non-compact stabilizer $H$ of $v_0\in {\mathbb{R}}^n$ in $G$ is either locally isomorphic to ${\operatorname{SO}}(n-2,1)$ (which is a simple Lie group except for $n=3$) or a compact extension of a horospherical subgroup. Since any orbit of a horospherical subgroup is either compact or dense in ${\Gamma}{\backslash}G$ (cf. [@Dani1986]), it follows that the case of ${\operatorname}{vol}(H\cap {\Gamma}{\backslash}H)=\infty $ arises only when $H\simeq {\operatorname{SO}}(1,1)$; hence $n=3$ and $Q(v_0)>0$.
In the next subsection, we state our main theorem in a greater generality, not necessarily in the arithmetic situation.
Counting integral points on a one-sheeted hyperboloid {#sec:1.2}
-----------------------------------------------------
Let $Q(x_1, x_2, x_3)$ be an real quadratic form of signature $(2,1)$. Denote by $G$ the identity component of the special orthogonal group ${\operatorname{SO}}_Q({\mathbb{R}})$. Let ${\Gamma}<G$ be a lattice and $v_0\in {\mathbb{R}}^3$ be such that $Q(v_0)>0$ and the orbit $v_0{\Gamma}$ is discrete. As before, we fix a norm $\|\cdot \|$ on ${\mathbb{R}}^3$ and set $B_T:=\{x\in v_0G:\|w\|<T\}$.
To present our theorem with a best possible error term, we consider the following smoothed counting function: fixing a non-negative function $\psi\in C_c^\infty(G)$ with integral one, let $$\tilde N_T:=\sum_{v\in v_0{\Gamma}} (\chi_{B_T}* \psi )(v)$$ where $\chi_{B_T}*\psi (x)=\int_{G}\chi_{B_T}(x g)\psi(g) \;dg$, $x\in v_0G$, is the convolution of the characteristic function of $B_T$ and $\psi$. Note that $\tilde N_T \asymp \# (v_0{\Gamma}\cap B_T)$ in the sense that their ratio is in between two uniform constants for all $T> 1$.
Denoting by $H\simeq {\operatorname{SO}}(1,1)^\circ$ the one-dimensional stabilizer subgroup of $v_0$ in $G$, note that ${\operatorname}{vol}(H\cap {\Gamma}{\backslash}H)<\infty$ if and only if $H\cap {\Gamma}$ is infinite. In order to state our theorem, we write $H$ as a one-parameter subgroup $\{h(s):s\in {\mathbb{R}}\}$ so that the Lebesgue measure $ds$ defines a Haar measure on $H$: $\int_{-\log T}^{\log T} ds={\operatorname{vol}}_H(\{h(s):|s|<\log T\})$.
\[main\] If the volume of $(H\cap {\Gamma}){\backslash}H$ is infinite, we have the following:
1. As $T\to \infty$, $$N_T\sim
\frac{\int_{-\log T}^{\log T} ds}{{\operatorname}{vol}_G({\Gamma}{\backslash}G)}{\operatorname}{vol}_{H{\backslash}G}(B_T)$$ where $d{\operatorname}{vol}_G=ds\times d{\operatorname}{vol}_{H{\backslash}G}$ locally.
2. for $T \gg 1$, $$\tilde N_T =c\cdot T\log T +O(T)$$ where $c=\lim_{T\to \infty}\frac{2{\operatorname}{vol}_{H{\backslash}G}(B_T)}{T{\operatorname}{vol}_G({\Gamma}{\backslash}G)}$.
We note that when ${\operatorname}{vol}(H\cap {\Gamma}{\backslash}H)<\infty $, $\tilde N_T =c\cdot T + O(T^{\alpha})$ for $0<\alpha <1$ is obtained in [@DukeRudnickSarnak1993]. We believe, as suggested by Z. Rudnick to us, that $\tilde N_T = c \cdot T\log T +c' \cdot T +O(T^\alpha) $ for some $c'>0$ and $0<\alpha<1$ and hence the order of the second term for $\tilde N_T$ cannot be improved.
Theorem \[main\] can be generalized to the orbital counting for more general representations of ${\operatorname{SL}}_2({\mathbb{R}})$ (see section \[sec:SL2-gen-rep\]).
In the case when $Q=x_1^2+x_2^2-d^2x_3^2$ for $d\in {\mathbb{Z}}$, $v_0=(1,0,0)$, and ${\Gamma}={\operatorname{SO}}_Q({\mathbb{Z}})$, it was pointed out in [@DukeRudnickSarnak1993] that an elementary number theoretic computation of [@Scourfield1961] leads to the asymptotic $$\#\{(x_1, x_2, x_3)\in v_0\Gamma: \sqrt{x_1^2+x_2^2+d^2x_3^2}< T\} =
c \cdot T \log T +O(T\log(\log T)) .$$ However this deduction seems to work only for this very special case; for instance, we are not aware of any other approach than ours which can deal with non-arithmetic situtations.
Arithmetic case and Integral binary quadratic forms
---------------------------------------------------
In the arithmetic case, Theorem \[main\] together with Theorem \[drs\] implies the following:
\[arith\] Let $Q(x_1, x_2, x_3)$ be an integral quadratic form with signature $(2,1)$. Suppose that for some $v_0\in {\mathbb{Z}}^3$ with $Q(v_0)>0$, the stabilizer subgroup of $v_0$ is isotropic over ${\mathbb{Q}}$.
Then there exists $c=c(\|\cdot \|) >0$ such that as $T\to \infty$, $$\# \{x\in {\mathbb{Z}}^3: Q(x)=Q(v_0), \|x\|<T\}\sim c\cdot T \log T .$$
For a binary quadratic form $q(x,y)=ax^2+bxy+cy^2$, its discriminant ${\operatorname}{disc}(q)$ is defined to be $b^2-4ac$. The group ${\operatorname{SL}}_2({\mathbb{R}})$ acts on the space of binary quadratic forms by $(g. q)(x, y)= q(g^{-1}(x,y))$ and preserves the discriminant. For $d\in {\mathbb{Z}}$, denote by ${\mathcal{B}}_d({\mathbb{Z}})$ the space of integral binary quadratic forms with discriminant $d$. Note that ${\mathcal{B}}_d({\mathbb{Z}})\ne \emptyset$ if and only if $d$ congruent to $0$ or $1 \mod 4$. Now $d$ is a square [*if and only if*]{} the stabilizer of every $q\in {\mathcal{B}}_d({\mathbb{Z}})$ in ${\operatorname{SL}}_2({\mathbb{Z}})$ is infinite [*if and only if*]{} every $q\in {\mathcal{B}}_d({\mathbb{Z}})$ is decomposable over ${\mathbb{Z}}$. (cf. [@BV]).
Therefore Corollary \[arith\] implies the following:
. For any non-zero square $d\in {\mathbb{Z}}$, there exists $c_0>0$ such that $$\#\{q\in {\mathcal{B}}_d({\mathbb{Z}}): {\operatorname}{disc}(q)=d, \|q\|<T\} \sim c_0 \cdot T\log T$$ where $\|ax^2+bxy+cy^2\|=\|(a,b,c)\|$.
Orthogonal translates of a divergent geodesic
---------------------------------------------
Let $G={\operatorname{SL}}_2({{\mathbb{R}}})$ and $\Gamma$ be a non-uniform lattice in $G$. For $s\in{{\mathbb{R}}}$, define $$\label{ha}
h(s)=\Bigl[\begin{smallmatrix} \cosh(s/2) & \sinh(s/2) \\
\sinh(s/2) & \cosh(s/2)\end{smallmatrix} \Bigr], \
a(s)=\Bigl[\begin{smallmatrix} e^{s/2} & 0 \\
0 & e^{-s/2}\end{smallmatrix}\Bigr]$$ and set $H=\{h(s):s\in {\mathbb{R}}\}$.
In the case when the orbit ${\Gamma}{\backslash}{\Gamma}H$ is closed and of finite length, the limiting distribution of the translates ${\Gamma}{\backslash}{\Gamma}H a(T)$ as $T\to\infty$ is described by the unique $G$-invariant probability measure $d\mu(g)=dg$ on ${\Gamma}{\backslash}G$ [@DukeRudnickSarnak1993], that is, if $s_0$ is the period of ${\Gamma}\cap H{\backslash}H$, then for any $\psi \in C_c({\Gamma}{\backslash}G)$, $$\lim_{T\to \pm \infty} \frac{1}{s_0} \int_{s=0}^{s_0} \psi (h(s)a(T))ds =\int_{{\Gamma}{\backslash}G}\psi \; dg.$$
Similarly, understanding the limit of the translates ${\Gamma}{\backslash}{\Gamma}H a(T)$ when ${\Gamma}{\backslash}{\Gamma}H$ is divergent (and of infinite length) is the main new ingredient in our proofs of Theorem \[main\].
\[thm:main\] Let $x_0\in \Gamma{\backslash}G$ and suppose that $x_0h(s)$ diverges as $s\to+\infty$, that is, $x_0h(s)$ leaves every compact subset for all sufficiently large $s\gg 1$. For a given compact subset ${\mathcal{K}}\subset {\Gamma}{\backslash}G$, there exist $c=c({\mathcal{K}})>0$ and $M=M({\mathcal{K}})>0$ such that for any $\psi \in C^\infty({\Gamma}{\backslash}G)$ with support in ${\mathcal{K}}$, we have, as $|T|\to \infty$, $$\int_{0}^{\infty}\psi (x_0h(s)a(T))ds= \int_{0}^{T+M}\psi (x_0h(s)a(T))ds = |T|\int \psi\; d\mu +O ( 1)$$ where the implied constant depends only on ${\mathcal{K}}$ and a Sobolev norm of $\psi$.
Consider the hyperbolic plane $\mathbb H^2$. A parabolic fixed point for ${\Gamma}$ is a point in the geometric boundary $\partial_\infty(\mathbb H^2)$ fixed by a parabolic element of ${\Gamma}$. If $\mathcal F\subset \mathbb H^2$ is a finite sided Dirichlet region for ${\Gamma}$, then the parabolic fixed points of ${\Gamma}$ are precisely the ${\Gamma}$-orbits of vertices of $\overline{\mathcal F}$ lying in $\partial_\infty(\mathbb H^2)$. Let $\pi: G\to \mathbb H^2$ denote the orbit map $g\mapsto g(i)$. For $x_0={\Gamma}g_0\in {\Gamma}{\backslash}G$, the image $\pi(g_0 H)$ is a geodesic in $\mathbb H^2$ with two endpoints $g_0H(+\infty):=\lim_{s\to \infty} \pi(g_0h(s))$ and $g_0H(-\infty):=
\lim_{s\to -\infty} \pi(g_0h(s))$ in $\partial_\infty(\mathbb H^2)$. We remark that $x_0h(s)$ diverges as $s\to + \infty$ (resp. $s\to -\infty$) if and only if $g_0H(+\infty)$ (resp. $g_0H(-\infty)$) is a parabolic fixed point for ${\Gamma}$ (cf. Theorem \[thm:cusps\]).
\[corodiv\] Suppose that $x_0 H$ is closed and non-compact. For any $\psi\in
C_c(\Gamma{\backslash}G)$, $$\lim_{T\to\pm\infty} \frac{1}{2{\lvertT\rvert}} \int_{
-\infty}^\infty
\psi(x_0h(s)a(T))\,ds=\int_{\Gamma{\backslash}G} \psi\,d\mu.$$
Acknowledgements {#acknowledgements .unnumbered}
----------------
We thank Zeev Rudnick for insightful comments.
Structure of cusps in $\Gamma{\backslash}G$ and divergent trajectory {#sec2}
====================================================================
Let $G={\operatorname{SL}}_2({{\mathbb{R}}})$ and $\Gamma$ be a non-uniform lattice in $G$. We will keep the notation for $h(s)$ and $a(s)$ from in the introduction. Let $$N=\{\left(\begin{smallmatrix}1 & s \\ 0 &
1\end{smallmatrix}\right):s\in{{\mathbb{R}}}\}\quad\text{ and } \quad U=wNw^{-1}$$ where $w=\bigl[\begin{smallmatrix} \cos(\pi/4) & \sin(\pi/4) \\
-\sin(\pi/4) & \cos(\pi/4) \end{smallmatrix}\bigr]$. Note that $h(s)=w a(s)w^{-1}$ for all $s\in {\mathbb{R}}$. For $\eta>0$, let $$H_\eta=\{h(s):s/2 >-\log \eta\} .$$
Let $K={\operatorname{SO}}(2)=\{g\in G: g g^t=I\}$. Then the multiplication map $U\times H\times K\to G$: $(u,h,k)\mapsto uhk$ is a diffeomorphism.
The following classical result may be found at [@GarlandRaghunathan1970 Thm. 0.6] or [@DaniSmillie1984]:
\[thm:cusps\] There exists a finite set $\Sigma\subset G$ such that the following holds:
1. $\Gamma{\backslash}\Gamma \sigma U$ is compact for every $\sigma\in\Sigma$.
2. For any $\eta>0$, the set $${\mathcal{K}}_\eta:=\Gamma{\backslash}G {\smallsetminus}\bigcup_{\sigma\in\Sigma} {\Gamma}{\backslash}{\Gamma}\sigma UH_\eta K$$ is compact; and any compact subset of $\Gamma{\backslash}G$ is contained in ${\mathcal{K}}_\eta$ for some $\eta>0$.
3. There exists $\eta_0>0$ such that for $i=1,2$, if $\sigma_i\in \Sigma$, $u_i\in U$, $h_i\in H_{\eta_0}$, and $\Gamma \sigma_1 u_1 h_1 k_1 = \Gamma \sigma_2 u_2 h_2 k_2 $, then $\sigma_1=\sigma_2$, $k_1=\pm k_2$ and $h_1=h_2$.
Consider the standard representation of $G={\operatorname{SL}}_2({{\mathbb{R}}})$ on ${{\mathbb{R}}}^2$: $((v_1, v_2), g)\mapsto (v_1,v_2)g$. Let ${\lVert\cdot\rVert}$ denote the Euclidean norm on ${{\mathbb{R}}}^2$. Let $$p=(0,1) w^{-1} =(-\sin(\pi/4),
\cos(\pi/4)).$$ Then $pU=p$, and $ph(s)=(0,1) a(s) w^{-1} = e^{-s/2}p$ for all $s\in {\mathbb{R}}$. Also $$\label{eq:eta}
g\in UH_\eta K \Leftrightarrow {\lVertpg\rVert}<\eta.$$
\[prop:divergent\] Let $x_0\in \Gamma{\backslash}G$ be such that the trajectory $\{x_0 h(s):s\geq 0\}$ is divergent. Then there exist $\sigma_0\in {\pm I}\Sigma$, $s_0\in {{\mathbb{R}}}$ and $u\in U$ such that $x_0 ={\Gamma}\sigma_0 uh(s_0)$.
By Theorem \[thm:cusps\], there exists $s_1>0$ and $\sigma\in\Sigma$ such that $x_0h(s) =\Gamma \sigma UH_{\eta_0/2}K$ for all $s\geq s_1$. Let $g_1\in UH_{\eta_0/2}K$ be such that $x_0h(s_1)=\Gamma\sigma g_1$. We claim that $pg_1\in {{\mathbb{R}}}p$. If not, then ${\lVertph(s)\rVert}\to \infty$ as $s\to\infty$, and hence there exists $s>0$ such that $\eta_0/2 {\lVertpg_1h(s)\rVert} < \eta_0$. By , $g_1h(s)\in uhk$ for some $u\in U$, $h\in H_{\eta_0}$ and $k\in K$. Therefore $$\Gamma \sigma uhk=\Gamma \sigma g_1h(s)=x_0h(s_1+s)\in \Gamma \sigma UH_{\eta_0/2}K.$$ By Theorem \[thm:cusps\](3), we have that $h\in H_{\eta_0/2}$. But then ${\lVertpg_1h(s)\rVert}={\lVertpuhk\rVert}<\eta_0/2$, a contradiction. Therefore our claim that $pg_1\in {{\mathbb{R}}}p$ is valid. Hence $g_1= u_1h(s)\{\pm I\}$ for some $u_1\in U$ and $s/2\geq -\log(\eta_0/2)$. Thus $x_0h(s_1)=\Gamma \sigma u_1 h(s) \{\pm I\}$, and hence $x_0=\Gamma\sigma_0 u_1 h(s-s_1)$, where $\sigma_0={\pm I}\sigma$.
\[prop:zero\] Let $x_0\in \Gamma{\backslash}G$ be such that the trajectory $\{x_0 h(s):s\geq 0\}$ is divergent. Let ${\mathcal{K}}\subset {\Gamma}{\backslash}G$ be a compact subset. There exists $M_1=M_1({\mathcal{K}})> 0$ such that $$x_0 h(s) a(T)\not\in {\mathcal{K}}$$ for any $T\in{{\mathbb{R}}}$ and $s>0$ satisfying $s>|T|+M_1$. In particular, for any $f\in C({\Gamma}{\backslash}G)$ with support inside ${\mathcal{K}}$, $$\int_{0}^\infty f(x_0h(s) a(T))\,ds=\int_0^{{\lvertT\rvert}+M_1}
f(x_0 h(s)a(T) )\,ds.$$
By Proposition \[prop:divergent\], $x_0=\Gamma \sigma_0 uh(s_0)$ for some $\sigma_0\in {\pm \Sigma}, u\in U, s_0\in {\mathbb{R}}$. By Theorem \[thm:cusps\](2), let $\eta>0$ be such that ${\mathcal{K}}\subset {\mathcal{K}}_\eta$. Let $M_1=-s_0-2\log(\eta)$. Since $s-|T|>-s_0-2\log \eta$, we have $$\label{eq:sT}
\begin{array}{ll}
{\lVert p uh(s_0)h(s)a(T)\rVert}
&={\lVert ph(s+s_0)a(T) \rVert}\\
&=e^{-(s+s_0)/2}{\lVertpa(T)\rVert} \\
& < e^{-(s+s_0)/2}e^{{{\lvertT\rvert}/2}}\\
&= e^{-(s+s_0-{\lvertT\rvert})/2}<\eta.
\end{array}$$ Therefore by , $uh(s_0)h(s)a(T)\in UH_{\eta}K$, and hence $$x_0h(s)a(T)\in \Gamma \sigma_0UH_\eta K\subset \Gamma{\backslash}G{\smallsetminus}K_\eta.$$
Uniform mixing on compact sets
==============================
Let $G={\operatorname{SL}}_2({\mathbb{R}})$ and ${\Gamma}<G$ be a lattice. Let $\mu$ denote the $G$-invariant probability measure on ${\Gamma}{\backslash}G$. For an orthonormal basis $X_1, X_2, X_3$ of $\mathfrak{sl}(2, {{\mathbb{R}}})$ with respect to an $Ad$-invariant scalar product, and $\psi \in C^\infty({\Gamma}{\backslash}G)$, we consider the Sobolev norm $${\mathcal S}_m(\psi )=\max\{\|X_{i_1}\cdots X_{i_j}(\psi )\|_2:1\le i_j \le 3 ,0\le j\le m\} .$$
The well-known spectral gap property for $L^2(\Gamma{\backslash}G)$ says that the trivial representation is isolated (see [@Bekka1998 Lemma 3]) in the Fell topology of the unitary dual of $G$. It follows that there exist $\theta>0$ and $c>0$ such that for any $\psi_1, \psi_2\in C^\infty(\Gamma{\backslash}G)$ with $\int\psi_i d\mu=0$, ${\mathcal S}_1(\psi_i)<\infty$ and for any $T>0$, $$\label{theta}
|\langle a(T)\psi_1, \psi_2\rangle | \le c e^{-\theta |T|}{\mathcal S}_1(\psi_1){\mathcal S}_1(\psi_2)$$ (cf. [@CowlingHaggerupHowe], [@Venkatesh2007])
Write ${\mathcal O}_{\epsilon}=\{g\in G: \|g- I\|_\infty\le {\epsilon}\}$. For a compact subset ${\mathcal{K}}\subset\Gamma{\backslash}G$, let $0<
{\epsilon}_0({\mathcal{K}})\le 1 $ be the injectivity radius of ${\mathcal{K}}$, that is, ${\epsilon}_0({\mathcal{K}})$ is the supremum of $0< {\epsilon}\le 1$ such that the multiplication map $ {\mathcal{K}}\times {\mathcal O}_{{\epsilon}} \to \Gamma{\backslash}G$ is injective.
For $s\in {\mathbb{R}}$, let $$n_+(s)=\left(\begin{smallmatrix} 1 & 0 \\ s & 1 \end{smallmatrix}\right)
\quad \text{ and } \quad n_-(s)=\left(\begin{smallmatrix}1 & s \\ 0 &
1\end{smallmatrix}\right).$$
\[thm:mixing\] Let ${\mathcal{K}}\subset{\Gamma}{\backslash}G$ be a compact subset and $\eta>0$. There exists $c=c({\mathcal{K}})>0$ such that for any $\psi \in C^\infty(\Gamma{\backslash}G)$ with support in ${\mathcal{K}}$, for any $|T|\geq 1$, $x\in {\mathcal{K}}$, and $0<r_0<{\epsilon}_0({\mathcal{K}})$, we have $${\Bigl\lvert \int_0^{r_0} \psi (xn_\nu(s)a(T) )\,ds- r_0 \int \psi \,d\mu \Bigr\rvert}
\leq c ({\mathcal S}_3(\psi) +1) e^{-\theta_0 |T|}$$ for some $\theta_0$ depending only on the spectral gap for $L^2({\Gamma}{\backslash}G)$. Here and in what follows, the sign $\nu=+$ if $T>0$ and $\nu=-$ if $T<0$.
Consider the case when $T>0$ and hence $\nu=+$. The other case can be proved similarly.
Let ${\epsilon}>0$. Fix a non-negative function $\rho_{\epsilon}\in C^\infty_c(N^+)$ which is $1$ on $n_+[0,r_0]$ and $0$ outside $n_+[-{\epsilon}, r_0+ {\epsilon}]$. Let $N^{\pm}=\{n_{\pm}(s):s\in {\mathbb{R}}\}$ and $ W_{\epsilon}:=AN^-\cap {\mathcal O}_{\epsilon}$. Let $\mu_0$ denote the right invariant measure on $AN^-$ such that $d\mu_0 \otimes dn=d\mu$. We choose a non-negative function $\phi_{\epsilon}\in C^\infty(AN^-)$ supported inside $W_{\epsilon}$ and $\int \phi_{\epsilon}d\mu_0=1$.
If we a consider a function $\tau_{x,{\epsilon}}$ on ${\Gamma}{\backslash}G$ which is defined to be $\tau_{x,{\epsilon}}(y):= \rho_{\epsilon}(n_+(s) )\phi_{\epsilon}(w) \in C^\infty(\Gamma{\backslash}G)$ if $y=xn_+(s)w\in x \;\text{supp}(\rho_{\epsilon}) W_{\epsilon}$ and $0$ otherwise, then ${\mathcal S}_1(\tau_{x,{\epsilon}})\ll {\epsilon}^{-3}$ where the implied constant is independent of $x$ and $$\begin{gathered}
\label{idd}\langle a(T)\psi, \tau_{x,{\epsilon}}\rangle
=\int_{y\in \Gamma{\backslash}G } \psi(ya(T)) \tau_{x,{\epsilon}}(y) d\mu(y)
\\= \int_{u\in W_{\epsilon}, s\in {{\mathbb{R}}}} \psi( xn_+(s)wa(T)) \phi_{\epsilon}(w) \rho_{\epsilon}(n_+(s))d\mu_0(w)ds .\end{gathered}$$
As ${\mathcal{K}}$ is compact, the $C^1$-norm of $f$ supported inside ${\mathcal{K}}$ is bounded above by a uniform multiple of ${\mathcal S}_3(\psi )$ (cf. [@Aubinbook Thm 2.20]) and hence for some $c_1>0$, $$\label{sup}
\max\{\|\psi\|_\infty, C_\psi\}<c_1 {\mathcal S}_3(\psi)$$ where $C_\psi$ is the Lipschitz constant of $\psi$.
Since for all $T>0$, $W_{\epsilon}a(T)\subset a(T){\mathcal O}_{2{\epsilon}} , $ we have for all $w\in W_{\epsilon}$ and $T\gg 1$, $${\Bigl\lvert \psi(x n_+(s)wa(T)) -\psi(x n_+(s)a(T)) \Bigr\rvert}\le 2 c_1 {\mathcal S}_3(\psi) {\epsilon}.$$
Hence by , $$\begin{aligned}
&{\Bigl\lvert \langle a(T)\psi, \tau_{x,{\epsilon}}\rangle -\int_{s\in {{\mathbb{R}}}} \psi(x n_+(s) a(T)) \rho_{\epsilon}(n_+(s)) ds \Bigr\rvert}\\
&=\Bigl| \int_{w,s} \psi(x n_+(s) w a(T)) \phi_{\epsilon}(w) \rho_{\epsilon}(n_+(s))d\mu_0(w)ds\\
& \qquad\qquad\qquad \qquad -\int_{s\in {{\mathbb{R}}}} \psi(x n_+(s) a(T)) \rho_{\epsilon}(n_+(s)) ds \Bigr| \\
&\ll 2 c_1 {\mathcal S}_3(\psi) {\epsilon}\|\rho_{\epsilon}\|_1\le 2 c_1 {\mathcal S}_3(\psi) {\epsilon}(r_0+2{\epsilon}).\end{aligned}$$
Since $${\Bigl\lvert \langle a(T)\psi, \tau_{x,{\epsilon}}\rangle-\int \psi d\mu\cdot
\|\rho_{\epsilon}\|_1 \Bigr\rvert} \ll e^{-\theta T} {\epsilon}^{-3}{\mathcal S}_1(\psi),$$ we deduce $$\begin{aligned}
& {\Bigl\lvert \int_{0}^{r_0} \psi(xn_+(s) a(T)) ds -r_0\int \psi d\mu \Bigr\rvert}
\\ &\le {\Bigl\lvert \int_{s\in {\mathbb{R}}} \psi(xn_+(s) a(T))\rho_{\epsilon}(n_+(s)) ds -r_0\int \psi d\mu \|\rho_{\epsilon}\|_1 \Bigr\rvert} +4c_1 {\epsilon}{\mathcal S}_3(\psi)\\
&\le {\Bigl\lvert \langle a(T)\psi, \tau_{x,{\epsilon}}\rangle
- r_0\int \psi d\mu\cdot \|\rho_{\epsilon}\|_1 \Bigr\rvert} + 6 c_1 {\epsilon}{\mathcal S}_3(\psi) \\
&\le 6c_1 {\epsilon}{\mathcal S}_3(\psi) +c\cdot e^{-\theta T} {\epsilon}^{-3} {\mathcal S}_3(\psi)\end{aligned}$$ for some $c>0$. Hence for ${\epsilon}=e^{-\theta T/4}$ and some $c_2>0$, $${\Bigl\lvert \int_{0}^{r_0} \psi(xn_+(s)a(T)) ds - r_0 \int \psi d\mu \Bigr\rvert}\le c_2 ({\mathcal S}_3(\psi) +1) e^{-\theta T/4} .$$
Translates of divergent orbits
==============================
Let $x_0\in {\Gamma}{\backslash}G$ be such that $x_0h(s)$ diverge as $s\to \infty$.
\[thm:combine2\] For any $|T|>1$ and any $\psi\in C_c^\infty({\Gamma}{\backslash}G)$ $$\int_0^{|T|} \psi(x_0h(s) a(T) )\,ds =| T| \int
\psi\,d\mu +O(1){\mathcal S}_3(\psi).$$
Let $R_0=-\log \eta_0$. Due to Proposition \[prop:divergent\], replacing $x_0$ by another point in $x_0H$, we may assume that $x_0=\Gamma \sigma_0 h(R_0)$. For any $S>0$, ${\lVertph(R_0)h(S)a(S)\rVert}\in [\eta_0/\sqrt{2}, \eta_0]$. Hence $x_0h(R_0)h(S)a(S)\in K_{\eta_0/\sqrt{2}}$.
Let $r_0$ be the injectivity radius of $K_{\eta_0/\sqrt{2}}$, that is, $r_0={\epsilon}_0( K_{\eta_0/\sqrt{2}})$. Let $S_0=0$, and choose $S_i$ such that $ r_0e^{-S_i}\le \delta_i:=S_{i+1}-S_i\le 2r_0e^{-S_i}$ for each $i$. We will choose $S_i=\log(2r_0 i +1)$ for each $i$. Then $x_0h(S_i)a(S_i)\in K_{\eta_0/\sqrt{2}}$. We put $R_i=T-S_i$.
We will express $x_0h([S_i,S_{i+1}])a(T)=x_ih^{a(S_i)}([0,\delta_i])a(R_i)$, where $x_i=x_0h(S_i)a(S_i)$ and $h^{a(S_i)}(s)=a(-S_i)h(s)a(S_i)=n(e^{S_i}s/2)w_i(s)$, and ${\lvertw_i(s)\rvert}=O(e^{-2S_i})$. Note that $r_0/2\leq e^{S_i}\delta_i/2 \leq r_0$.
By Theorem \[thm:mixing\], we have $$\int_0^{r_0} \psi(x_i n(s) a(R_i)) ds - r_0\int\psi d\mu = {\mathcal S}_3(\psi)\cdot O(e^{-\theta_0R_i})$$ and hence $$\int_{S_i}^{S_{i+1}} \psi(x_0h(s)a(T)) ds = \frac{\delta_i}{r_0}\int_0^{r_0}\psi(x_in(s)a(R_i))ds +
{\mathcal S}_3(\psi)\cdot O(e^{-2S_i}\delta_i).$$
Let $k=k(T)$ be such that $S_k\le T<S_k +r_0e^{-S_k}$. Therefore, since $\delta_i r_0^{-1} \leq 2e^{-S_i}$,
$$\begin{aligned}
&\int_0^T \psi(xh(s)a(T)) ds =\sum_{i=0}^{k-1}\int_{S_i}^{S_{i+1}} \psi(xh(s)a(T)) ds +O(e^{-S_{k}})
\\ & = \sum_{i=0}^{k-1} \delta_i \frac{1}{r_0} \int_0^{r_0}\psi(x_in(s)a(T))ds +
{\mathcal S}_3(\psi)\cdot O(e^{-2S_i}\delta_i)+O(1)
\\&= \sum_{i=0}^{k-1} \delta_i\mu(\psi) + \sum_{i=0}^{k-1} \delta_i r_0^{-1}{\mathcal S}_3(\psi)\cdot O(e^{-\theta_0R_i})
+{\mathcal S}_3(\psi)\cdot O(e^{-2S_i}\delta_i) +O(1)
\\
&= T \mu (\psi) + O( \sum_{i=1}^{k-1} e^{-S_i}e^{-\theta_0 R_i} +\sum_{i=1}^k e^{-3S_i} ){\mathcal S}_3(\psi) +O(1)
\\&= T \mu (\psi) + O( e^{-\theta_0 T}\sum_{i=0}^{k-1} e^{(1-\theta_0) S_i} +\sum_{i=0}^{k-1} e^{-3S_i} ){\mathcal S}_3(\psi) +O(1).\end{aligned}$$
Since $S_i=\log(2r_0 i +1)$, $0<T-S_k<2e^{-T}$ implies that $k<\frac{e^T-1}{2r_0}<k+1$, and hence $$\sum_{i=0}^{k-1} e^{-3S_i}\ll \sum_{i=1}^{k-1} \frac{1}{(2r_0 i+1)^3}=O(k^{-2}+1)=O(e^{-2T}+1)<\infty$$ and $$\sum_{i=0}^{k-1} e^{(1-\theta_0) S_i } \ll \int_{0}^{e^T} \frac{1}{(2r_0 x+1)^{1-\theta_0}} dx=O(e^{\theta_0 T}).$$ Hence $$e^{-\theta_0 T}\sum_{i=0}^{k-1} e^{(1-\theta_0) S_i} +\sum_{i=0}^{k-1} e^{-3S_i} =O(1).$$ Therefore $$\int_0^T \psi(xh(s)a(T)) ds = T \mu (\psi) + O(1){\mathcal S}_3(\psi) .$$
Theorem \[thm:main\] follows from the following:
\[mct2\] Let $x_0h(s)$ diverge as $s\to \infty$. For a given compact subset ${\mathcal{K}}\subset {\Gamma}{\backslash}G$, and $\psi\in C^\infty({\Gamma}{\backslash}G)$ with support in ${\mathcal{K}}$, we have $$\int_{0}^{\infty}\psi(x_0h(s)a(T))ds=|T|\cdot \int \psi\;d\mu+O(1){\mathcal S}_3(\psi).$$
Since $x_0 h(s)$ diverges as $s\to\infty$, by Proposition \[prop:zero\], there exists $M_1=M_1({\mathcal{K}}) >0$ such that $$\begin{aligned}
& \int_{0}^{\infty}\psi (x_0h(s)a(T))ds=
\int_{0}^{|T|+M_1}\psi (x_0h(s)a(T))ds \\
& = (|T|+M_1) \int \psi \; d\mu + O (1){\mathcal S}_3(\psi)\\
&= |T| \int \psi \; d\mu + O (1){\mathcal S}_3(\psi).\end{aligned}$$
By a similar argument, we also deduce the following:
\[cor:negativeside\] If $x_0h(s)$ diverges as $s\to -\infty$, then $$\int_{-\infty}^{0}\psi(x_0h(s)a(T))ds=|T| \int \psi d\mu+O(1){\mathcal S}_3(\psi) .$$
\[div\] If $x_0h({\mathbb{R}})$ is closed and non-compact, then $x_0h(s)$ diverges as $s\to \pm \infty$.
We use a well-known fact that for a closed subgroup $H$ of a locally compact second countable group $G$ and a discrete subgroup ${\Gamma}$ of $G$, if ${\Gamma}H$ is closed in $G$, then the canonical projection map $H\cap {\Gamma}{\backslash}H\to {\Gamma}{\backslash}G$ is a proper map (cf. [@OhShahGFH]). Since $x_0h({\mathbb{R}})$ is non-compact and $h({\mathbb{R}})$ is one-dimensional with no non-trivial finite subgroups, the stabilizer of $x_0$ in $h({\mathbb{R}})$ is trivial. Therefore the map $h({\mathbb{R}})\to {\Gamma}{\backslash}G$ given by $h\to x_0h$ is a proper injective map. This implies that $x_0h(s)$ diverges as $s\to \pm \infty$.
As the set $C_c^\infty({\Gamma}{\backslash}G)$ is dense in $C_c({\Gamma}{\backslash}G)$, the claim follows from Lemma \[div\], Theorem \[thm:combine2\], and Corollary \[cor:negativeside\].
Counting: Proof of Theorem \[main\]
===================================
Let $Q$ be a real quadratic form in $3$ variables of signature $(2,1)$ and ${\Gamma}_0$ a lattice in the identity component $G_0$ of ${\operatorname{SO}}_Q({\mathbb{R}})$. We assume that $v_0{\Gamma}_0$ is discrete for some vector $v_0\in {\mathbb{R}}^3$ with $Q(v_0)=d>0$ and that the stabilizer $H_0$ of $v_0$ in $G_0$ is finite.
It suffices to prove Theorem \[main\] in the case when $Q=x^2+y^2-z^2$ and $v_0=(\sqrt d, 0,0)$ by the virtue of Witt’s theorem.
Consider the spin double cover map $\iota: G:={\operatorname{SL}}_2({\mathbb{R}})\to G_0$ given by $$\left(\begin{smallmatrix} a&b\\c&d\end{smallmatrix}\right)\mapsto
\left(\begin{smallmatrix}
\frac{a^2-b^2-c^2+d^2}{2}& {ac-bd} &\frac{a^2-b^2+c^2-d^2}{2}\\
{ab-cd}& {bc+ad}& {ab+cd}\\
\frac{a^2+b^2-c^2-d^2}{2}& {ac+bd} &\frac{a^2+b^2+c^2+d^2}{2}
\end{smallmatrix}\right).$$
For $s\in {\mathbb{R}}$, we set $$h(s)=\left(\begin{smallmatrix} \cosh(s/2) & \sinh(s/2) \\
\sinh(s/2) & \cosh(s/2)\end{smallmatrix}\right);\quad\text{and} \quad a(s)=\left(\begin{smallmatrix} e^{s/2} & 0 \\
0 & e^{-s/2}\end{smallmatrix}\right) .$$
Recall that $H:=\{h(s):s\in {\mathbb{R}}\}$, $A:=\{a(t):t\in {\mathbb{R}}\}$ and $K_1:=\{k(\theta):\theta\in[0,2\pi]\}$, here $K_1$ is half of the circle group. Observing that $$\iota(h(s))=\left(\begin{smallmatrix} 1&0&0\\
0&\cosh s &\sinh s\\ 0&\sinh s&\cosh s\end{smallmatrix}\right) \quad\text{ and}\quad
\iota(a(t))=\left(\begin{smallmatrix} \cosh t& 0 & \sinh t\\
0& 1& 0\\ \sinh t&0 &\cosh t\end{smallmatrix}\right),$$ the subgroup $\tilde H:=\pm H$ is the stabilizer of $v_0$ in $G$. We have a generalized Cartan decomposition $G=\tilde HAK_1$ in the sense that every $g$ is of the form $hak$ for unique $h\in \tilde H, a\in A, k\in K_1$. And for $g=h(s)a(t)k$, $d\mu(g)=\sinh(t)dsdtdk$ defines a Haar measure on $G$, where $dk=(1/2\pi) dk(\theta)$, and $ds$, $ dt$ and $d\theta$ are Lebesgue measures. As $v_0G=\pm H{\backslash}G \simeq A\times K_1$, $\sinh(t)dtdk$ defines an invariant measure on $v_0G$. We consider the volume forms on $G$ and $v_0G$ with respect to these measures. Via the map $\iota$, these define invariant measures on $G_0$ and $v_0G_0$ as well.
Denote by ${\Gamma}$ the pre-image of ${\Gamma}_0$ under $\iota$. Then ${\operatorname}{Stab}_{\Gamma}(v_0)=
\tilde H\cap {\Gamma}=\{\pm I\}$.
For each $T>1$, define a function on ${\Gamma}{\backslash}G$: $$F_T(g):=\sum_{\gamma\in {\pm I}{\backslash}{\Gamma}}\chi_{B_T}(v_0 \gamma g) .$$
\[inner\] For any $\Psi\in C_c^\infty({\Gamma}{\backslash}G)$, $${\langle}F_{ T}, \Psi{\rangle}=\frac{T\log T\mu(\Psi) } {{\operatorname}{vol}({\Gamma}{\backslash}G)} \cdot
2\int_{K_1} \frac{1}{\|v^+k\|}dk
+O(T){\mathcal S}_3(\psi)$$ where $v^{\pm}=\frac{\sqrt d}{2} (e_1\pm e_3) $. Here the implied constant depends only on $\mathcal S_3(\Psi)$ and the support of $\Psi$.
Then $v_0= v^++v^-$ and $v_0 a(t)= e^tv^++e^{-t}v^-$. Since $B_T=\{ v_0a(t)k: \|v_0a(t)k\|<T\,t\in{{\mathbb{R}}},\,k\in K_1\}$, we have $$\begin{aligned}
&{\langle}F_{ T}, \Psi{\rangle}=\int_{{\Gamma}{\backslash}G}\sum_{\gamma\in {\pm I}{\backslash}{\Gamma}}\chi_{B_T}(v_0\gamma g)
\Psi(g)d\mu(g)\\ & =
\int_{k\in K_1} \int_{\|v_0a(t) k\|< T }
\left( \int_{h(s) \in \pm I {\backslash}\tilde H }\Psi(h(s) a(t) k)ds\right) \sinh(t) dt dk \\
&= \int_{k\in K_1}\int_{\|v_0a(t)k\|< T }
\left( \int_{s\in {\mathbb{R}}}\Psi (h(s)a(t) k)ds\right) \sinh(t) dt dk .\end{aligned}$$
Since $v_0{\Gamma}$ is discrete and $H\cap {\Gamma}$ is trivial, it follows that ${\Gamma}{\backslash}{\Gamma}H$ is closed and non-compact in ${\Gamma}{\backslash}G$. Now fix any $k\in K_1$. Hence by Theorem \[mct2\] and Lemma \[div\], $$\begin{aligned}
\label{coo}
& \int_{t\gg 1, \|v_0a(t)k\|< T}\left( \int_{s\in {\mathbb{R}}}\Psi(h(s)a(t) k)ds\right) \sinh(t) dt\\
&= \frac{1}{{\operatorname}{vol}({\Gamma}{\backslash}G)} \int_{t\gg 1, e^t \|v^+ k \|< T +O(1) }
(2t \mu(\psi) +O(1){\mathcal S}_3(\psi)) (e^t/2+O(1)) dt\\
& = \frac{T\log T \mu(\Psi)}{{\operatorname}{vol}({\Gamma}{\backslash}G)\cdot \|v^+k\|} +O(T){\mathcal S}_3(\psi).
\end{aligned}$$
Similarly, $$\begin{aligned}
& \int_{t\ll -1,
\|v_0a(t)k\|< T}\left( \int_{s\in {\mathbb{R}}}\Psi(h(s)a(t) k)ds\right)
\sinh(t) dt\\&=\int_{t\gg 1,
\|v_0a(-t)k\|< T}\left( \int_{s\in {\mathbb{R}}}\Psi(h(s)a(-t) k)ds\right)
\sinh(t) dt
\\ & =\frac{1}{{\operatorname}{vol}({\Gamma}{\backslash}G)} \int_{t\gg 1, e^t \|v^- k \|< T + O(1) }
(2t\mu(\psi) +O(1){\mathcal S}_3(\psi)) (e^t/2+O(1)) dt\\
&= \frac{T\log T\mu(\Psi) }{{\operatorname}{vol}({\Gamma}{\backslash}G) \|v^-k\|} +O(T){\mathcal S}_3(\psi). \end{aligned}$$ Since $v^-k(\pi)=-v^+$, $$\int_{k\in K_1} {\lVertv^-k\rVert}{^{-1}}dk = \int_{k\in K_1} {\lVertv^+k(\pi)k\rVert}{^{-1}}dk=\int_{K_1}{\lVertv^+k\rVert}{^{-1}}dk.$$ The required formula can be deduced in a straightforward manner from this.
Fix a non-negative function $\psi\in C_c^\infty(G)$ whose support injects to ${\Gamma}{\backslash}G$ and with integral $\int \psi(g)\;d\mu(g)=1$. Consider a function $\xi_T$ on ${\mathbb{R}}^3$ defined by $$\xi_T(x)=\int_{g\in G}\chi_{B_T}(xg)\psi(g) d\mu(g) .$$ Then the sum $\sum_{\gamma\in \pm I{\backslash}{\Gamma}}\xi_T(v_0\gamma)$ is a smoothed over counting satisfying $$\sum_{\gamma\in \pm I{\backslash}{\Gamma}}\xi_T(v_0\gamma)\asymp \#v_0{\Gamma}\cap B_T .$$
As $T\to \infty$, $$\sum_{\gamma\in \pm I{\backslash}{\Gamma}}\xi_T(v_0 \gamma) =
\frac{2 T\log T } {{\operatorname}{vol}({\Gamma}{\backslash}G)} \cdot \int_{k\in K_1} \frac{1}{\|v^+k\|}dk
+O( T){\mathcal S}_3(\psi).$$
It is not hard to verify that $$\sum_{\gamma\in \pm I{\backslash}{\Gamma}}\xi_T(v_0\gamma)={\langle}F_T, \Psi{\rangle}$$ where $\Psi({\Gamma}g)=\sum_{\gamma\in {\Gamma}}\psi(\gamma g)$. Therefore the claim follows from Proposition \[inner\].
\[mctwo\] For $T\gg 1$, we have $$\#\{w\in v_0{\Gamma}: \|w\|<T\}=
\frac{2 T\log T }{{\operatorname}{vol}({\Gamma}{\backslash}G)} \int_{K_1}\frac{1}{ \|w^+k\|} dk (1+(\log T)^{-\alpha})$$ where $\alpha=-1/5.5$.
Note that $F_T(I)=\#\{w\in v_0{\Gamma}: \|w\|<T\}$. For each ${\epsilon}>0$, let ${\mathcal O}_{\epsilon}=\{g\in G:\|g-I\|_\infty\le {\epsilon}\}$. There exists $0< \ell \le 1 $ such that for all small ${\epsilon}>0$,$$\label{oe} {\mathcal O}_{\ell {\epsilon}} B_T\subset B_{(1+{\epsilon})T},\quad B_{(1-{\epsilon})T}\subset \cap_{u\in {\mathcal O}_{\ell {\epsilon}} }u B_T .$$ Let $\psi^{\epsilon}$ be a non-negative smooth function on $G$ supported in ${\mathcal O}_{\ell {\epsilon}}$ and with integral $\int \psi^{\epsilon}d\mu=1$ and define $\Psi^{\epsilon}\in C_c^\infty({\Gamma}{\backslash}G)$ by $\Psi^{\epsilon}({\Gamma}g):=\sum_{\gamma\in {\Gamma}}\psi^{\epsilon}(\gamma g)$.
Using , we have $${\langle}F_{(1-{\epsilon})T}, \Psi^{\epsilon}{\rangle}\le F_T(I)\le
{\langle}F_{(1+{\epsilon})T}, \Psi^{\epsilon}{\rangle}.$$
Therefore by Proposition \[inner\] $$\begin{aligned}
{\langle}F_{(1\pm {\epsilon}) T}, \Psi^{\epsilon}{\rangle}&= \frac{2 T\log T }{{\operatorname}{vol}({\Gamma}{\backslash}G)} \int_{K_1}\frac{1}{ \|w^+k\|} dk
+O({\epsilon}T\log T) + O(\mathcal S_3 (\Psi^{\epsilon}) T)\\
&= \frac{2 T\log T }{{\operatorname}{vol}({\Gamma}{\backslash}G)} \int_{K_1}\frac{1}{ \|w^+k\|} dk (1+(\log T)^{-1/5.5}, \end{aligned}$$ where the last equality follows because $\mathcal S_3 (\Psi^{\epsilon})=O({\epsilon}^{-4.5})$, and if we put $\epsilon=(\log T)^{-1/5.5}$ then $$O(\mathcal S_3(\Psi^{\epsilon})T)=O(\epsilon T\log T)= (T\log T)(\log T)^{-1/5.5}.$$
The above computation in the proof of Proposition \[inner\] also shows that $${\operatorname}{vol}(B_T)=\int_{k\in K_1} \int_{\|v_0a(t)k\|<T }
\sinh(t) dt dk
=T \int_{k\in K}\frac{1}{\|v^+k\|} dk +O(\log T).$$ From Theorem \[mctwo\], it follows that $$\label{f2}
F_T(I)= \frac{2\log T {\operatorname}{vol}(B_T)} {{\operatorname}{vol}({\Gamma}{\backslash}G)} (1+O(\log T)^{-\alpha})).$$ Since $F_T(I)=\#(v_0{\Gamma}\cap B_T)$, this completes the proof.
Orbital counting for general representations of ${\operatorname{SL}}_2({{\mathbb{R}}})$ {#sec:SL2-gen-rep}
=======================================================================================
Let $G={\operatorname{SL}}_2({{\mathbb{R}}})$ and $\Gamma$ be a non-uniform lattice in $G$. For $s\in{{\mathbb{R}}}$, define $$h(s)=\Bigl[\begin{smallmatrix} \cosh(s/2) & \sinh(s/2) \\
\sinh(s/2) & \cosh(s/2)\end{smallmatrix} \Bigr], \
a(s)=\Bigl[\begin{smallmatrix} e^{s/2} & 0 \\
0 & e^{-s/2}\end{smallmatrix}\Bigr], \ k(\theta)=\Bigl[\begin{smallmatrix} \cos(\theta/2) & -\sin(\theta/2) \\ \sin(\theta/2) & \cos(\theta/2)\end{smallmatrix}\Bigr]$$ Put $H=\{h(s):s\in{{\mathbb{R}}}\}$, $A^+=\{a(t):t>0\}$, and $K_1=\{k(\theta):\theta\in [0,2\pi]\}$, here $K_1$ is half of the circle group. Put $w_0=k(\pi)$. Then $\{\pm I\}{\backslash}G=HA^+K_1\cup Hw_0A^+K_1$, $w_0{^{-1}}h(s)w_0 = h(-s)$ and $w_0{^{-1}}a(t)w_0= a(-t)$.
Let $V$ be any finite dimensional representation of $G$ and $v_0\in G$ be such that $H$ is the stabilizer subgroup of $v_0$ in $G$, i.e., $H=G_{v_0}$ where $G_{v_0}=\{g\in G: v_0g=v_0\}$. Assume that $V$ is linearly spanned by $v_0G$. Then if $e^{mt}$ is the highest eigenvalue for $a(t)$-action on $V$, then $m\in {{\mathbb{N}}}$, and the $G$ action factors through $\{\pm I\}{\backslash}G={{\operatorname}{PSL}}_2({{\mathbb{R}}})\cong {\operatorname{SO}}(2,1)^0$.
For example, let $V_m$ denote the $(2m+1)$-dimensional space of real homogeneous polynomials of degree $2m$ in two variables, and consider the standard right action of $g\in {\operatorname{SL}}(2,{{\mathbb{R}}})$ on $P(x,y) \in V_m$ by $(Pg)(x,y)=P((x,y)g)$, where $(x,y)\bigl[\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\bigr]=(ax+cy,bx+dy)$. Let $v_0(x,y)=(x^2-y^2)^m$. Then $G_{v_0}=H{\mathcal{W}}$, where ${\mathcal{W}}=\{\pm I\}$ if $m$ is odd and ${\mathcal{W}}=\{\pm I, \pm w_0\}$ if $m$ is even. Moreover, $\{P\in V_m:Ph=P,\,\text{ for all }h\in H\}={{\mathbb{R}}}P_0$. A general finite dimensional representation of $G$ with a nonzero $H$-fixed vector is a direct sum of such irreducible representations, and $v_0$ is a sum of one nonzero $H$-fixed vector from each of the irreducible representations; we assume that $V$ is a span of $v_0G$.
\[thm:gen-count\] Let $V$, $v_0$ and $m$ be as above. Suppose that $\Gamma$ is a lattice in $G$, $v_0\Gamma$ is discrete, and $\Gamma_{v_0}:=\Gamma\cap G_{v_0}$ is finite. Let ${\lVert\cdot\rVert}$ be any norm on $V$, and $v_0^+=\lim_{t\to\infty} v_0a_t/{\lVertv_0 a_t\rVert}$. Let $C$ be an open subset of $\{v\in V:{\lVertv\rVert}=1\}$ such that $\Theta=\{\theta\in[0,2\pi]: v_0^+k(\theta)\in {{\mathbb{R}}}C\}$ has positive Lebesgue measure, and $\{\theta\in [0,2\pi]: v_0^+ k(\theta)\in {{\mathbb{R}}}({\overline{C}}{\smallsetminus}C)\}$ has zero Lebesgue measure. Then for $T\gg 1$, $$\begin{aligned}
\# (v_0\Gamma &\cap [0,T]C )\\
&=\frac{4 (2\pi){^{-1}}\int_{\Theta}{\lVertv_0^+ k(\theta)\rVert}^{-1/m}\,d\theta} {{\lvert\Gamma_{v_0}\rvert}\cdot {\operatorname{vol}}_G(\Gamma{\backslash}G)}\times \frac{\log T}{m}T^{1/m} (1+(\log T)^{-\alpha})
\nonumber\end{aligned}$$ where ${\operatorname{vol}}_G$ is given by the Haar integral $dg=\sinh(t)dtdsd\theta$ on $G$, where $g=h(s)a(t)k(\theta)$, and $\alpha=\frac{1}{5.5}$.
Moreover, if $C\subset V$ satisfies ${{\mathbb{R}}}{\overline{C}}\cap v_0^+K_1=\emptyset$, then $\# (v_0\Gamma\cap {{\mathbb{R}}}C )<\infty$.
The result can be deduced by the arguments as in the proof of Theorem \[mctwo\]; one may also use the basic ideas from [@OhShahGFH] about using the highest weight.
[10]{}
Thierry Aubin. , volume 252 of [*Grundlehren der Mathematischen Wissenschaften \[Fundamental Principles of Mathematical Sciences\]*]{}. Springer-Verlag, New York, 1982.
M. B. Bekka. On uniqueness of invariant means. , 126(2):507–514, 1998.
Borel, Armand and Harish-Chandra , , vol 75, 485–535, 1962
Buchmann, Johannes and Vollmer, Ulrich , , Algorithms and Computation in Mathematics, Vol 20.
M. Cowling, U. Haagerup, and R. Howe. Almost [$L\sp 2$]{} matrix coefficients. , 387:97–110, 1988.
S. G. Dani. On invariant measures, minimal sets and a lemma of [M]{}argulis. , 51(3):239–260, 1979.
S. G. Dani. Divergent trajectories of flows on homogeneous spaces and Diophantine approximation. , 359:55–89, 1985.
S. G. Dani. Orbits of horospherical flows. , 53(1):177–188, 1986.
S. G. Dani and John Smillie. Uniform distribution of horocycle orbits for [F]{}uchsian groups. , 51(1):185–194, 1984.
W. Duke, Z. Rudnick, and P. Sarnak. Density of integer points on affine homogeneous varieties. , 71(1):143–179, 1993.
Alex Eskin and C. T. McMullen. Mixing, counting, and equidistribution in [L]{}ie groups. , 71(1):181–209, 1993.
Alex Eskin, Shahar Mozes, and Nimish Shah. Unipotent flows and counting lattice points on homogeneous varieties. , 143(2):253–299, 1996.
H. Garland and M. S. Raghunathan. Fundamental domains for lattices in [R]{}-rank [$1$]{} semisimple [L]{}ie groups. , 92:279–326, 1970.
Alex Gorodnik, Hee Oh, and Nimish Shah. Integral points on symmetric varieties and [S]{}atake compactifications. , 131(1):1–57, 2009.
G.A. Margulis. Dynamical and ergodic properties of subgroup actions on homogeneous spaces with applications to number theory. In [*Proceedings of the [I]{}nternational [C]{}ongress of [M]{}athematicians, [V]{}ol. [I]{}, [II]{} ([K]{}yoto, 1990)*]{}, pages 193–215, Tokyo, 1991. Math. Soc. Japan.
Hee Oh and Nimish Shah. Equidistribution and counting for orbits of geometrically finite hyperbolic groups. .
E.J. Scourfield. The divisors of a quadratic polynomial. , 5:8–20, 1961.
N.A. Shah. Uniformly distributed orbits of certain flows on homogeneous spaces. , 289:315–334, 1991.
A. Venkatesh. Sparse equidistribution problem, period bounds, and subconvexity, 2007. , 172:989-1094, 2010.
[^1]: Oh was partially supported by NSF Grant \#0629322.
[^2]: Shah was partially supported by NSF Grant \#1001654.
|
---
abstract: 'We give a new, simplified and detailed account of the correspondence between levels of the Sherali–Adams relaxation of graph isomorphism and levels of pebble-game equivalence with counting (higher-dimensional Weisfeiler–Lehman colour refinement). The correspondence between basic colour refinement and fractional isomorphism, due to Ramana, Scheinerman and Ullman [@RamanaScheinermanUllman], is re-interpreted as the base level of Sherali–Adams and generalised to higher levels in this sense by Atserias and Maneva [@AtseriasManeva], who prove that the two resulting hierarchies interleave. In carrying this analysis further, we here give (a) a precise characterisation of the level $k$ Sherali–Adams relaxation in terms of a modified counting pebble game; (b) a variant of the Sherali–Adams levels that precisely match the $k$-pebble counting game; (c) a proof that the interleaving between these two hierarchies is strict. We also investigate the variation based on boolean arithmetic instead of real/rational arithmetic and obtain analogous correspondences and separations for plain $k$-pebble equivalence (without counting). Our results are driven by considerably simplified accounts of the underlying combinatorics and linear algebra.'
author:
- |
Martin Grohe\
RWTH Aachen University
- |
Martin Otto\
Technische Universität Darmstadt
bibliography:
- 'cklp.bib'
title: Pebble Games and Linear Equations
---
Introduction
============
We study a surprising connection between equivalence in finite variable logics and a linear programming approach to the graph isomorphism problem. This connection has recently been uncovered by Atserias and Maneva [@AtseriasManeva], building on earlier work of Ramana, Scheinerman and Ullman [@RamanaScheinermanUllman] that just concerns the 2-variable case.
Finite variable logics play a central role in finite model theory. Most important for this paper are finite variable logics with counting, which have been specifically studied in connection with the question for a logical characterisation of polynomial time and in connection with the graph isomorphism problem (e.g.[@caifurimm92; @graott93; @gro10; @immlan90; @lau10; @ott97]). Equivalence in finite variable logics can be characterised in terms of simple combinatorial games known as pebble games. Specifically, $\LC^k$-equivalence can be characterised by the bijective $k$-pebble game introduced by Hella [@hel92]. Cai, Fürer and Immerman [@caifurimm92] observed that $\LC^k$-equivalence exactly corresponds to indistinguishability by the $k$-dimensional Weisfeiler-Lehman (WL) algorithm,[^1] a combinatorial graph isomorphism algorithm introduced by Babai, who attributed it to work of Weisfeiler and Lehman in the 1970s. The 2-dimensional version of the WL algorithm precisely corresponds to an even simpler isomorphism algorithm known as colour refinement.
The isomorphisms between two graphs can be described by the integral solutions of a system of linear equations. If we have two graphs with adjacency matrices $A$ and $B$, then each isomorphism from the first to the second corresponds to a permutation matrix $X$ such that $X^tAX=B$, or equivalently $$\label{eq:iso-lp}
AX=XB.$$ If we view the entries of $X$ as variables, this equation corresponds to a system of linear equations. We can add inequalities that force $X$ to be a permutation matrix and obtain a system $\ISO$ of linear equations and inequalities whose integral solutions correspond to the isomorphisms between the two graphs. In particular, the system $\ISO$ has an integral solution if any only if the two graphs are isomorphic.
What happens if we drop the integrality constraints, that is, we admit arbitrary real solutions of the system $\ISO$? We can ask for doubly stochastic matrices $X$ satisfying equation . (A real matrix is *doubly stochastic* if its entries are non-negative and all row sums and column sums are one.) Ramana, Scheinerman and Ullman [@RamanaScheinermanUllman] proved a beautiful result that establishes a connection between linear algebra and logic: the system $\ISO$ has a real solution if, and only if, the colour refinement algorithm does not distinguish the two graphs with adjacency matrices $A$ and $B$. Recall that the latter is equivalent to the two graphs being $\LC^2$-equivalent.
To bridge the gap between integer linear programs and their LP-relaxations, researchers in combinatorial optimisation often add additional constraints to the linear programs to bring them closer to their integer counterparts. The Sherali–Adams hierarchy [@sheada90] of relaxations gives a systematic way of doing this. For every integer linear program $\IL$ in $n$ variables and every positive integer $k$, there is a *rank-$k$ Sherali–Adams relaxation* $\IL[k]$ of $\IL$, such that $\IL[1]$ is the standard LP-relaxation of $\IL$ where all integrality constraints are dropped and $\IL[n]$ is equivalent to $\IL$. There is a considerable body of research studying the strength of the various levels of this and related hierarchies (e.g. [@bieozb04; @burgalhoo+03; @charmakmak09; @matsin09; @schtretul07; @sch08]).
Quite surprisingly, Atserias and Maneva [@AtseriasManeva] were able to lift the Ramana–Scheinerman–Ullman result, which we may now restate as an equivalence between $\ISO[1]$ and $\LC^2$-equivalence, to a close correspondence between the higher levels of the Sherali–Adams hierarchy for $\ISO$ and the logics $\LC^k$. They proved for every $k\ge 2$:
1. if $\ISO[k]$ has a (real) solution, then the two graphs are $\LC^k$-equivalent;
2. if the two graphs are $\LC^k$-equivalent, then $\ISO[k-1]$ has a solution.
Atserias and Maneva used this results to transfer results about the logics $\LC^k$ to the world of polyhedral combinatorics and combinatorial optimisation, and conversely, results about the Sherali–Adams hierarchy to logic.
Atserias and Maneva [@AtseriasManeva] left open the question whether the interleaving between the levels of the Sherali–Adams hierarchy and the finite-variable-logic hierarchy is strict or whether either the correspondence between $\LC^k$-equivalence and $\ISO[k]$ or the correspondence between $\LC^{k}$-equivalence and $\ISO[k-1]$ is exact. Note that for $k=2$ the correspondence between $\LC^{k}$-equivalence and $\ISO[k-1]$ is exact by the Ramana–Scheinerman–Ullman theorem. We prove that for all $k\ge3$ the interleaving is strict. However, we can prove an exact correspondence between $\ISO[k-1]$ and a variant of the bijective $k$-pebble game that characterises $\LC^k$-equivalence. This variant, which we call the weak bijective $k$-pebble game, is actually equivalent to a game called $(k-1)$-sliding game by Atserias and Maneva.
Maybe most importantly, we prove that a natural combination of equalities from $\ISO[k]$ and $\ISO[k-1]$ gives a linear program $\ISO[k-1/2]$ that characterises $\LC^k$-equivalence exactly.
To obtain these results, we give simple new proofs of the theorems of Ramana, Scheinerman and Ullman and of Atserias and Maneva. Whereas the previous proofs use two non-trivial results from linear algebra, the Perron–Frobenius Theorem (about the eigenvalues of positive matrices) and the Birkhoff–von Neumann Theorem (stating that every doubly stochastic matrix is a convex combination of permutation matrices), our proofs only use elementary linear algebra. This makes them more transparent and less mysterious (at least to us).
In fact, the linear algebra we use is so simple that much of it can be carried out not only over the field of real numbers, but over arbitrary semirings. By using similar algebraic arguments over the boolean semiring (with disjunction as addition and conjunction as multiplication), we obtain analogous results to those for $\LC^k$-equivalence for the ordinary $k$-variable logic $\LL^k$, characterising $\LL^k$-equivalence, i.e., $k$-pebble game equivalence without counting, by systems of ‘linear’ equations over the boolean semiring.
For the ease of presentation, we have decided to present our results only for undirected simple graphs. It is easy to extend all results to relational structures with at most binary relations. Atserias and Maneva did this for their results, and for ours the extension works analogously. An extension to structures with relations of higher arities also seems possible, but is more complicated and comes at the price of loosing some of the elegance of the results.
Finite variable logics and pebble games {#sec:logic}
=======================================
We assume the reader is familiar with the basics of first-order logic . We almost exclusively consider first-order logic over finite graphs, which we view as finite relational structures with one binary relation, and in a few places over other finite relational structures. We assume graphs to be undirected and loop-free. For every positive integer $k$, we let $\LL^k$ be the fragment of $\FOL$ consisting of all formulae that contain at most $k$ distinct variables.
We write $\str{A}\equiv_\LL^k \str{B}$ to denote that two structures $\str{A},\str{B}$ are *$\LL^k$-equivalent*, that is, satisfy the same $\LL^k$-sentences. $\LL^k$-equivalence can be characterised in terms of the *$k$-pebble game*, played by two players on a pair $\str{A},\str{B}$ of structures. A *play* of the game consists of a (possibly infinite) sequence of *rounds*. In each round, player picks up one of his pebbles and places it on an element of one of the structures $\str{A},\str{B}$. Player answers by picking up her pebble with the same label and placing it on an element of the other structure.
Note that after each round $r$ there is a subset $p\subseteq \str{A}\times
\str{B}$ consisting of the at most $k$ pairs of elements on which the pairs of corresponding pebbles are placed. We call $p$ the *position* after round $r$. Player wins the play if every position that occurs is a local isomorphism, that is, a local mapping from $\str{A}$ to $\str{B}$ that is injective and preserves membership and non-membership in all relations (adjacency and non-adjacecny if $\str{A}$ and $\str{B}$ are graphs).
$\str{A}\equiv_\LL^k \str{B}$ if, and only if, player has a winning strategy for the $k$-pebble game on $\str{A},\str{B}$.
We extend $\LL^k$-equivalence to structures with distinguished elements. For tuples ${\mathbf{a}}$ and ${\mathbf{b}}$ of the same length $\ell\le
k$ we let $\str{A},{\mathbf{a}}\equiv_\LL^k\str{B},{\mathbf{b}}$ if $\str A,{\mathbf{a}}$ and $\str{B},{\mathbf{b}}$ satisfy the same $\LL^k$-formulae ${\varphi}({\mathbf{x}})$ with $\ell$ free variables ${\mathbf{x}}$. The pebble game characterisation extends: $\str{A},{\mathbf{a}}\equiv_\LL^k\str{B},{\mathbf{b}}$ if, and only if, player has a winning strategy for the $k$-pebble game on $\str{A},\str{B}$ starting with pebbles on ${\mathbf{a}}$ and the corresponding pebbles on ${\mathbf{b}}$. The *$\LL^k$-type* of a tuple ${\mathbf{a}}$ in a structure $\str A$ is the $\equiv_\LL^k$-equivalence class of $\str A,{\mathbf{a}}$. More syntactically, we may also view the $\LL^k$-type of ${\mathbf{a}}$ as the set of all $\LL^k$-formulae ${\varphi}({\mathbf{x}})$ satisfied by $\str A,{\mathbf{a}}$.
Let us turn to the $k$-variable counting logics. It is convenient to start with the (syntactical) extension $\LC$ of $\FOL$ by *counting quantifiers* $\exists^{\ge n}$. The semantics of these counting quantifiers is the obvious one: $\exists^{\ge n}x\,{\varphi}$ means that there are at least $n$ elements $x$ such that ${\varphi}$ is satisfied. Of course this can be expressed in $\FOL$, but only by a formula that uses at least $n$ variables. For all positive integers $k$, we let $\LC^k$ denote the $k$-variable fragment of $\LC$. Whereas $\LC$ and $\FOL$ have the same expressive power, $\LC^k$ is strictly more expressive than $\LL^k$.
We write $\str{A}\equiv_\LC^k \str{B}$ to indicate that structures $\str{A}$ and $\str{B}$ are $\LC^k$-equivalent. $\LC^k$-equivalence can be characterised in terms of the *bijective $k$-pebble game*, which, like the $k$-pebble game, is played by two players by placing $k$ pairs of pebbles on a pair of structures $\str{A},\str{B}$. The rounds of the bijective game are as follows. Player picks up one of his pebble, and player picks up her corresponding pebble. Then player chooses a bijection $f$ between $\str{A}$ and $\str{B}$ (if no such bijection exists, that is, if the structures have different cardinalities, player immediately looses). Then player places his pebble on an element $a$ of $\str{A}$, and player places her pebble on $f(a)$. Again, player wins a play if all positions are local isomorphisms.
\[theo:hel\] $\str{A}\equiv_\LC^k \str{B}$ if, and only if, player has a winning strategy for the bijective $k$-pebble game on $\str{A},\str{B}$.
As $\LL^k$-equivalence, we extend $\LC^k$-equivalence to structures with distinguished elements, writing $\str A,{\mathbf{a}}\equiv_\LC^k\str
B,{\mathbf{b}}$. Again, the pebble-game characterisation of the equivalence extends. We define $\LC^k$-types analogously to $\LL^k$-types.
The *colour refinement* algorithm is a simple combinatorial heuristics for testing whether two graphs are isomorphic: Given two graphs $\str{A}$ and $\str{B}$, which we assume to be disjoint, it computes a colouring of their vertices by the following iterative procedure: Initially, all vertices have the same colour. Then in each round, the colouring is refined by assigning different colours to vertices that have a different number of neighbours of at least one colour assigned in the previous round. Thus after the first round, two vertices have the same colour if, and only if, they have the same degree. After the second round, two vertices have the same colour if, and only if, they have the same degree and for each $d$ the same number of neighbours of degree $d$. The algorithm stops if no further refinement is achieved; this happens after at most $|\str{A}|+|\str{B}|$ rounds. We call the resulting colouring of $\str{A}\cup \str{B}$ the *stable colouring* of $\str{A},\str{B}$. If the stable colouring differs on the two graphs, that is, for some colour $c$ the graphs have a different number of vertices of colour $c$, then we say that colour refinement *distinguishes* the graphs.
\[theo:immlan\] $\str{A}\equiv_\LC^2 \str{B}$ if, any only if, colour refinement does not distinguish $\str{A}$ and $\str{B}$.
The *$k$-dimensional Weisfeiler-Lehman algorithm* (for short: *$k$-WL*) is a generalisation of the colour refinement algorithm, which instead of vertices colours $k$-tuples of vertices. Given two structures $\str{A}$ and $\str{B}$, which we assume to be disjoint, $k$-WL iteratively computes a colouring of $\str{A}^k\cup \str{B}^k$. Initially, two tuples ${\mathbf{a}}=(a_1,\ldots,a_k),{\mathbf{b}}=(b_1,\ldots,b_k)\in \str{A}^k\cup \str{B}^k$ get the same colour if the mapping defined by $p(a_i)=b_i$ is a local isomorphism. In each round of the algorithm, the colouring is refined by assigning different colours to tuples that for some $j\in[k]$ and some colour $c$ have different numbers of $j$-neighbours of colour $c$ in their respective graphs. Here we call two $k$-tuples *$j$-neighbours* if they differ only in their $j$th component. The algorithm stops if no further refinement is achieved; this happens after at most $|\str{A}|^k+|\str{B}|^k$ rounds. If after the refinement process the colourings of the two graphs differ, that is, for some colour $c$ the graphs have a different number of $k$-tuples of colour $c$, then we say that $k$-WL *distinguishes* the graphs.
$\str{A}\equiv_\LC^k \str{B}$ if, and only if, $k$-WL does not distinguish $\str{A}$ and $\str{B}$.
More significantly, Cai, Fürer, and Immerman [@caifurimm92] proved that for all $k$ there are nonisomorphic graphs $\str{A}_k,\str{B}_k$ of size $O(k)$ such that $\str{A}\equiv_\LC^k \str{B}$.
Note that the previous two theorems imply that colour refinement and $2$-WL distinguish the same graphs.
There are also ‘boolean’ versions of the two algorithms characterising $\LL^k$-equivalence (see [@ott97]).
Basic combinatorics and linear algebra
======================================
We consider matrices with entries in $\B = \{0,1\}$, $\Q$ or $\R$. A matrix $X \in \R^{m,n}$ with $m$ rows and $n$ columns has entry $X_{ij}$ in row $i \in [m] = \{ 1,\ldots, m\}$ and column $j \in [n] = \{ 1,\ldots, n \}$. We write $E_n$ for the $n$-dimensional unit matrix.
We write $X {\geqslant}0$ to say that (the real or rational) matrix $X$ has only non-negative entries, and $X > 0$ to say that all entries are strictly positive. We also speak of *non-negative* or *strictly positive matrices* in this sense. For a boolean matrix, strict positivity, $X > 0$ means that all entries are $1$.
A square $n\!\times\!n$-matrix is *doubly stochastic* if its entries are non-negative and if the sum of entries across every row and column is $1$. *Permutation matrices* are doubly stochastic matrices over $\{0,1\}$, with precisely one $1$ in every row and in every column. Permutation matrices are orthogonal, i.e., $P P^t = P^t P = E_n$ for every permutation matrix $P$. The permutation $p \in S_n$ associated with a permutation matrix $P \in \R^{n,n}$ is such that $P \mathbf{e}_j = \mathbf{e}_{p(j)}$, i.e., it describes the permutation of the standard basis vectors that is effected by $P$. We also say that $P$ represents $p$. The permutation matrices form a subgroup of the general linear groups. The doubly stochastic matrices do not form a subgroup, but are closed under transpose and product.
It will be useful to have the shorthand notation $$X_{D_1D_2} = 0$$ for the assertion that $X_{d_1d_2} = 0$ for all $d_1 \in D_1$, $d_2 \in D_2$. If $p$ and $q$ are permutations in $S_n$ represented by permutation matrices $P$ and $Q$, then $$(P^tXQ)_{D_1D_2} = 0 \quad\mbox{ iff }\quad
X_{p(D_1)q(D_2)} = 0.$$ So, if $X_{D_1D_2} = 0$ and $P$ and $Q$ are chosen such that $p^{-1}(D_1)$ and $q^{-1}(D_2)$ are final and initial segments of $[n]$, respectively, then $P^t X Q$ has a null block of dimensions $|D_1| \times |D_2|$ in the upper right-hand corner.
Decomposition into irreducible blocks
-------------------------------------
\[irreddef\] With $X \in \R^{n,n}$ associate the directed graph $$\mathrm{G}(X) := ([n], \{ (i,j) \colon X_{ij} \not= 0\}).$$ The strongly connected components of $\mathrm{G}(X)$ induce a partition of the set $[n] = \{ 1,\ldots, n\}$ of rows/columns of $X$. $X$ is called *irreducible* if this partition has just the set $[n]$ itself.
Note that $X$ is irreducible iff $P^t X P$ is irreducible for every permutation matrix $P$.
\[posobs\] Let $X \in \R^{n,n} {\geqslant}0$ with strictly positive diagonal entries. If $X$ is irreducible, then all powers $X^\ell$ for $\ell {\geqslant}n-1$ have non-zero entries throughout. Moreover, if $X$ is irreducible, then so is $X^\ell$ for all $\ell {\geqslant}1$.
It is easily proved by induction on $\ell {\geqslant}1$ that $(X^\ell)_{ij} \not= 0$ if, and only if there is a directed path of length $\ell$ from vertex $i$ to vertex $j$ in $G(X)$. For $X$ with positive diagonal entries, $G(X)$ has loops in every vertex, and therefore there is a path of length $\ell$ from vertex $i$ to vertex $j$ if, and only if, there is path of length $M'$ for every $\ell' {\geqslant}\ell$ from $i$ to $j$. If $G(X)$ is also strongly connected, then any two vertices are linked by a path of length up to $n-1$.
Let us call two matrices $Z,Z' \in \R^{n,n}$ *permutation-similar* or *$S_n$-similar*, $Z \sim_{S_n} Z'$, if $Z' = P^t Z P$ for some permutation matrix $P$, i.e., if one is obtained from the other by a coherent permutation of rows and columns.
\[symdecomplem\] Every symmetric $Z \in \R^{n,n} {\geqslant}0$ is permutation-similar to some block diagonal matrix $\mathrm{diag}(Z_1,\ldots,Z_s)$ with irreducible blocks $Z_i \in \R^{n_i,n_i}$.
The permutation matrix $P$ corresponding to the row- and column-permutation $p \in S_n$ that puts $Z$ into block diagonal form $P^t Z P = \mathrm{diag}(Z_1,\ldots,Z_s)$ with irreducible blocks, is unique up to an outer permutation that re-arranges the block intervals $([k_i+1,k_i+n_i])_{1{\leqslant}i {\leqslant}s}$ where $k_i = \sum_{j < i} n_j$, and a product of inner permutations within each one of these $s$ blocks.
The underlying partition $[n] = \dot{\bigcup}_{1 {\leqslant}i {\leqslant}s} D_i$ where $D_i :=
p([k_i+1,k_i+n_i])$ for $k_i = \sum_{j < i} n_j$, is uniquely determined by $Z$.[^2]
In the following we refer to the *partition induced by a symmetric matrix $Z$*.
Obvious, based on the partition of the vertex set $[n]$ of $G(Z)$ into connected components (note that symmetry of $Z$ is preserved under similarity, and strong connectivity is plain connectivity in $G(Z)$ for symmetric $Z$).
\[decomppartobs\] In the situation of Lemma \[symdecomplem\], the partition $[n] = \dot{\bigcup}_i D_i$ induced by the symmetric matrix $Z$ is the partition of $[n]$ into the vertex sets of the connected components of $G(Z)$. Then, for every pair $i \not= j$, $Z_{D_i D_{j}} = 0$, while all the minors $Z_{D_iD_i}$ are irreducible.[^3]
If, moreover, $Z$ has strictly positive diagonal entries, then the partition induced by $Z$ is the same as that induced by $Z^\ell$, for any $\ell {\geqslant}1$; for $\ell {\geqslant}n-1$, the diagonal blocks $(Z^\ell)_{D_iD_i}$ have non-zero entries throughout: $(Z^\ell)_{D_iD_i} > 0$ .
The last assertion says that for a symmetric $n\!\times\!n$ matrix $Z$ with non-negative entries and no zeroes on the diagonal, all powers $Z^\ell$ for $\ell {\geqslant}n-1$ are *good symmetric* in the sense of the following definition.
\[goodsymmatrixdef\] Let $Z {\geqslant}0$ be symmetric with strictly positive diagonal. Then $Z$ is called *good symmetric* if w.r.t. the partition $[n] = \dot{\bigcup}_i D_i$ induced by $Z$, all $Z_{D_iD_i} > 0$.
More generally, a not necessarily symmetric matrix $X {\geqslant}0$ without null rows or columns is *good* if $Z = XX^t$ and $Z' = X^tX$ are good in the above sense.
The importance of this notion lies in the fact that, as observed above, for an arbitrary symmetric $n\!\times\!n$ matrix $Z {\geqslant}0$ without zeroes on the diagonal, the partition induced by $Z$ is the same as that induced by the good symmetric matrix $\hat{Z} := Z^{n-1}$; and, as for any good matrix, this partition can simply be read off from $\hat{Z}$: $i,j \in [n]$ are in the same partition set if, and only if, $\hat{Z}_{ij} \not= 0$.
\[relpartdef\] Consider partitions $[n] = \dot{\bigcup}_{i \in I} D_i$ and $[m] = \dot{\bigcup}_{i \in I} D_i'$ of the sets $[n]$ and $[m]$ with the same number of partition sets. We say that these two partitions are *$X$-related* for some matrix $X \in \R^{n,m}$ if
$X {\geqslant}0$ has no null rows or columns, and
$X_{D_i {D_{j}}\!\!\nt'} = 0$ for every pair of distinct indices $i,j \in I$.
Note that partitions that are $X$-related are $X^t$-related in the opposite direction. More importantly, each one of the $X/X^t$-related partitions can be recovered from the other one through $X$ according to $$\barr{rcl}
D_i' &=& \{ d' \in [m] \colon X_{dd'} > 0 \mbox{ for some } d \in D_i
\},
\\
\hnt
D_i &=& \{ d \in [n] \colon X_{dd'} > 0
\mbox{ for some } d' \in D_i'\}.
\earr$$
For a more algebraic treatment, we associate with the partition sets $D_i$ of a partition $[n] = \dot{\bigcup}_{i \in I} D_i$ the *characteristic vectors* $\mathbf{d}_i$ with entries $1$ and $0$ according to whether the corresponding component belongs to $D_i$: $$\textstyle \mathbf{d}_i = \sum_{d \in D_i} \mathbf{e}_d,$$ where $\mathbf{e}_d$ is the $d$-th standard basis vector. In terms of these characteristic vectors $\mathbf{d}_i$ for $[n] = \dot{\bigcup}_{i \in I} D_i$ and $\mathbf{d}_i'$ for $[m] = \dot{\bigcup}_{i \in I} D_i'$, the $X/X^t$-relatedness of these partitions means that $$\barr{rcl}
D_i'
&=&
\{ d' \in [m] \colon ( X^t \mathbf{d}_i)_{d'} > 0 \},
\\
\hnt
D_i
&=&
\{ d \in [n] \colon (X \mathbf{d}_i')_{d} > 0 \}.
\earr$$
\[stochrelatedpartlem\] If two partitions $[n] = \dot{\bigcup}_{i \in I} D_i$ and $[n] = \dot{\bigcup}_{i \in I} D_i'$ of the same set $[n]$ are $X$-related for some doubly stochastic matrix $X \in \R^{n,n}$, then $|D_i| = |D_i'|$ for all $i \in I$, and for the characteristic vectors $\mathbf{d}_i$ and $\mathbf{d}_i'$ of the partition sets $D_i$ and $D_i'$ even $$\mathbf{d}_i = X \mathbf{d}_i'
\quad
\mbox{ and }
\quad
\mathbf{d}_i' = X^t \mathbf{d}_i.$$
Observe that for all $d\in[n]$ we have $0\le (X{\mathbf{d}}_i')_d=\sum_{d'\in
D_i'}X_{dd'}\le 1$. It follows immediately from the definition of $X$-relatedness that $(X{\mathbf{d}}_i')_d =0$ for all $d\not\in D_i$. Therefore, $$|D_i|{\geqslant}\sum_{d\in D_i}(X{\mathbf{d}}_i')_d=\sum_{d\in[n]}(X{\mathbf{d}}_i')_d
=\sum_{d'\in D_i'}\sum_{d\in[n]}X_{dd'}=|D_i'|.$$
Similarly, $0\le (X^t{\mathbf{d}}_i)_{d'}\le 1$ for $d'\in[n]$, and $|D_i'|\ge \sum_{d'\in D_i'}(X^t{\mathbf{d}}_i)_{d'}=|D_i|$. Together, we obtain $$|D_i|=\sum_{d\in D_i}(X{\mathbf{d}}_i')_d=|D_i'|=\sum_{d'\in D_i'}(X^t{\mathbf{d}}_i)_{d'}.$$
As all summands are bounded by $1$, this implies $(X{\mathbf{d}}_i')_d=1$ for all $d\in D_i$ and $(X^t{\mathbf{d}}_i)_{d'}=1$ for all $d'\in D_i$.
\[newdecomplem\] Let $X {\geqslant}0$ be an $m\!\times\!n$ matrix without null rows or columns. Then the $m\!\times\!m$ matrix $Z := X X^t$ and the $n\!\times\!n$ matrix $Z' := X^t X$ are symmetric with positive entries on their diagonals. Moreover, the (unique) partitions of $[m]$ and $[n]$ that are induced by $Z$ and $Z'$, respectively, are $X/X^t$-related.[^4]
It is obvious that $Z$ and $Z'$ are symmetric with positive diagonal entries. Let partitions $[m] = \dot{\bigcup}_{i\in I} D_i$ and $[n] = \dot{\bigcup}_{i\in I'} D_i'$ be obtained from decompositions of $Z$ and $Z'$ into irreducible blocks. We need to show that the non-zero entries in $X$ give rise to a coherent bijection between the index sets $I$ and $I'$ of the two partitions, in the sense that partition sets $D_i$ and $D_j'$ are related if, and only if, some pair of members $d \in D_i$ and $d' \in D_j'$ have a positive entry $X_{dd'}$. Then a re-numbering of one of these partitions will make them $X$-related in the sense of Definition \[relpartdef\]. Recall from Observation \[decomppartobs\] that the $D_i$ are the vertex sets of the connected components of $G(XX^t)$ on $[m]$, while the $D_i'$ the are the vertex sets of the connected components of $G(X^tX)$ on $[n]$.
Consider the uniformly directed bipartite graph $G(X)$ on $[m] \,\dot{\cup}\, [n]$ with an edge from $i \in [m]$ to $j \in [n]$ if $X_{ij} > 0$. In light of the symmetry of the whole situation w.r.t.$X$ and $X^t$, it just remains to argue for instance that no $i \in [m]$ can have edges into two distinct sets of the partition $[n] = \dot{\bigcup}_{i\in I'} D_i'$. But any two target nodes of edges from one and the same $i \in [n]$ are in the same connected component of $G(X^t X)$, hence in the same partition set.
In the situation of Lemma \[newdecomplem\], powers of $Z$ induce the same partitions as $Z$, and the partitions induced by $(Z^\ell X)(Z^\ell X)^t = Z^{2\ell+1}$ are $X/X^t$-related as well as $Z^\ell X/X^tZ^\ell$-related, for all $\ell {\geqslant}1$.
For $\ell {\geqslant}n/2 - 1$, the matrix $Z^\ell X$ has no null rows or columns: else $Z^\ell X (Z^\ell X)^t = Z^{2\ell+1}$ would have to have a zero entry on the diagonal, contradicting the fact that this symmetric matrix is good symmetric in the sense of Definition \[goodsymmatrixdef\]. The same reasoning shows that $Z^\ell X$ is itself good in the sense of Definition \[goodsymmatrixdef\].
\[goodlinkcor\] Let $X {\geqslant}0$ be an $m\!\times\!n$ matrix without null rows or columns, $Z = XX^t$, $Z' = X^tX$ the associated symmetric matrices with non-zero entries on the diagonal. Then for $\ell {\geqslant}m-1$, the matrix $\hat{X} := Z^\ell X = X (Z')^\ell$ and its transpose $\hat{X}^t = X^t Z^\ell = (Z')^\ell X^t$ are good and relate the partitions $[m] = \dot{\bigcup}_i D_i$ and $[n] = \dot{\bigcup}_i D_i'$ induced by $Z$ and $Z'$, respectively.
Moreover,
$\hat{X}_{D_iD_i'}>0$ for all $i$, and
$\hat{X}_{D_iD_j'} = 0$ for all $i \not= j$.
$Z^\ell X$ is good symmetric by the above reasoning. So $(Z^\ell)_{D_iD_i} > 0$ for all $i$, while $(Z^m)_{D_i D_j} = 0$ for all $j \not= i$. It follows that $(Z^\ell X)_{D_i D_i'} = (Z^\ell)_{D_iD_i} X_{D_i D_i'}$ has only non-zero entries because $X_{D_i D_i'}$ does not have null columns. This proves (i). Assertion (ii) is clear as, for $i \not= j$, $(Z^\ell X)_{D_i D_j'} = (Z^\ell)_{D_iD_j} X_{D_j D_j'} = 0 \,
X_{D_jD_j'} = 0$.
### Aside: boolean vs. real arithmetic {#aside-boolean-vs.real-arithmetic .unnumbered}
Looking at matrices with $\{0,1\}$-entries, we may not only treat them as matrices over $\R$ as we have done so far, but also over other fields, or as matrices over the boolean semiring $\B = \{ 0,1\}$ with the logical operations of $\vee$ for addition and $\wedge$ for multiplication. Though not even forming a ring, boolean arithmetic yields a very natural interpretation in the context where we associate non-negative entries with edges, as we did in passage from $X$ to $G(X)$ (cf. Definition \[irreddef\] and Observation \[posobs\]). The ‘normalisation map’ $\chi \colon \R_{{\geqslant}0} \rightarrow \{ 0, 1\}$, $x \mapsto 1$ iff $x > 0$, relates the arithmetic of reals $x,y {\geqslant}0$ to boolean arithmetic in $$\chi(x + y) = \chi(x) \vee \chi(y)
\quad
\mbox{ and }
\quad
\chi(x y) = \chi(x) \wedge \chi(y).$$
This is the ‘logical’ arithmetic that supports, for instance, arguments used in Observation \[posobs\]: for any real $n\!\times\!n$ matrix $X {\geqslant}0$, $(XX)_{ij} = \sum_{k} X_{ik} X_{kj} \not= 0$ iff there is at least one $k \in [n]$ for which $X_{ik} \not=0$ and $X_{kj} \not= 0$ iff $\bigvee_{k\in[n]} (\chi(X_{ik}) \wedge \chi(X_{kj})) = 1$. It is no surprise, therefore, that several of the considerations apparently presented for real non-negative matrices above, have immediate analogues for boolean arithmetic – in fact, one could argue, that the boolean interpretation is closer to the combinatorial essence. We briefly sum up these analogues with a view to their use in the analysis of $\LL^k$-equivalence, while the real versions are related to $\LC^k$-equivalence. Note also that the boolean analogue of a doubly stochastic matrix with non-negative real entries is a matrix without null rows or columns.
Also note that Definitions \[irreddef\] (irreducibility) and \[relpartdef\] ($X$-relatedness) are applicable to boolean matrices without any changes. Observations \[posobs\] and \[decomppartobs\] go through (as just indicated), and so does Lemma \[symdecomplem\]. For Lemma \[stochrelatedpartlem\], one may look at $X$-related partitions of sets $[m]$ and $[n]$, where not necessarily $n=m$, by any boolean matrix $X$ without null rows or columns and obtains the relationship between the characteristic vectors as stated there, now in terms of boolean arithmetic – but of course we do not get any numerical equalities between the sizes of the partition sets. Lemma \[newdecomplem\], finally, applies to boolean arithmetic, exactly as stated.
\[summaryboollem\] In the sense of boolean arithmetic for matrices with entries in $\B = \{ 0,1 \}$:
Any symmetric $Z \in \B^{n,n}$ induces a unique partition of $[n]$ for which the diagonal minors induced by the partition sets are irreducible and the remaining blocks null; $d,d' \in [n]$ are in the same partition set if, and only if, in the sense of boolean arithmetic $(Z^\ell)_{dd'} = 1$ for any/all $\ell {\geqslant}n-1$.
If two partitions (not necessarily of the same set) with the same number of partition sets are related by some boolean matrix $X \in \B^{m,n}$, then the characteristic vectors $(\mathbf{d}_i)_{i \in I}$ and $(\mathbf{d}_i')_{i \in I}$ of the partitions are related by $\mathbf{d}_i = X \mathbf{d}_i'$ and $\mathbf{d}_i' = X^t \mathbf{d}_i$ in the sense of boolean arithmetic.
For any matrix $X \in \B^{m,n}$ without null rows or columns, the symmetric boolean matrices $Z = XX^t$ and $Z' = X^tX$ have diagonal entries $1$ and induce partitions that are $X/X^t$-related, and agree with the partitions induced by higher powers of $Z$ and $Z'$ or on the basis of $Z^\ell X$ and $X(Z')^\ell$ for any $\ell \in \N$. For $\ell {\geqslant}m-1,n-1$, the partition blocks in $Z$ and $Z'$ have entries $1$ throughout, and $Z^\ell X$ and $X (Z')^\ell$ have entries $1$ in all positions relating elements from matching partition sets.
\[inducedpartboolobs\] For a symmetric boolean matrix $Z \in \B^{n,n}$ with $Z_{dd} = 1$ for all $d \in [n]$, the characteristic vectors $\mathbf{d}_i$ of the partition $[n] = \dot{\bigcup}_{i\in I} D_i$ induced by $Z$ satisfy the following ‘eigenvector’ equation in terms of boolean arithmetic: $$Z \mathbf{d}_i = \mathbf{d}_i \quad \mbox{\rm (boolean), \; for all $i \in I$.}$$
Eigenvalues and -vectors
------------------------
\[evallem\] If $Z \in \R^{n,n}$ is doubly stochastic, then it has eigenvalue $1$. If $Z$ is doubly stochastic and irreducible with strictly positive diagonal entries, then the eigenspace for eigenvalue $1$ has dimension $1$ and is spanned by the vector $\mathbf{d} := (1,\ldots,1)^t$.
Clearly $Z \mathbf{d} = \mathbf{d}$ for any stochastic matrix $Z$. If $Z$ is moreover irreducible with positive diagonal entries, then by Observation \[posobs\], $Z^\ast := Z^{n-1}$ has strictly positive entries and, being doubly stochastic, therefore entries strictly between $0$ and $1$.
If $\mathbf{v}$ is an eigenvector for eigenvalue $1$ of $Z$, then also of $Z^\ast$. If $\mathbf{v} = (v_1,\ldots,v_n)$, this is equivalent to $$\textstyle
v_i = \sum_j Z^\ast_{ij} v_j\quad \mbox{ for all } i \in [n].$$
Looking at an index $i$ for which $v_j {\leqslant}v_i$ for all $j$, we see that the maximal $v_i$ is a convex combination of the $v_j$ to which every $v_j$ contributes. This implies that all $v_j = v_i$, so that $\mathbf{v}$ is a scalar multiple of $\mathbf{d}$ as claimed.
\[evalcor\]
Let $Z \in \R^{n,n}$ be doubly stochastic with positive diagonal, and $[n] = \dot{\bigcup}_i D_i$ a partition with $Z_{D_iD_j} = 0$ for $i \not= j$ and such that the minors $Z_{D_iD_i}$ are irreducible for all $i$. Then the eigenspace for eigenvalue $1$ of $Z$ is the direct sum of the $1$-dimensional subspaces spanned by the characteristic vectors $\mathbf{d}_i$ of the partition sets $D_i$.
If $Z = X^tX \in \R^{n,n}$ for some doubly stochastic matrix $X$, then the eigenspace for eigenvalue $1$ is the direct sum of the spans of the characteristic vectors $\mathbf{d}_i$ from the unique partition $[n] = \dot{\bigcup}_i D_i$ of $[n]$ induced by $Z$ according to Lemma \[symdecomplem\].
Towards (a), it is clear that $Z \mathbf{d}_i= \mathbf{d}_i$, so that each $\mathbf{d}_i$ is an eigenvector with eigenvalue $1$. Let $V_i := \mathrm{span}(\mathbf{e}_d \colon d
\in D_i)$; then $\R^n = \bigoplus_i V_i$ is a direct sum decomposition, and $Z_{D_jD_i} = 0$ for $j \not= i$ implies that $Z$ maps $V_i$ to itself. Therefore any eigenvector $\mathbf{v}$ with eigenvalue $1$ decomposes as $\mathbf{v} = \sum_i \mathbf{v}_i$, where $\mathbf{v}_i \in V_i$, in such manner that $Z \mathbf{v}_i =
\mathbf{v}_i$. Since the restriction of $Z$ to $V_i$ is irreducible with positive diagonal, $\mathbf{v}_i \in
\mathrm{span}(\mathbf{d}_i)$ by Lemma \[evallem\], as claimed.
Statement (b) is s direct consequence, since $Z$ is symmetric with positive diagonal.
Stable partitions
-----------------
\[stablepartdef\] Let $A \in \R^{n,n}$, $[n] = \dot{\bigcup}_{i\in I} D_i$ be a partition. We call this partition a *stable partition for $A$* if there are numbers $(s_{ij})_{i,j \in I}$ and $(t_{ij})_{i,j \in I}$ such that for all $i,j \in I$: $$d \in D_i \quad \Rightarrow \quad \sum_{d' \in D_j} A_{dd'} = s_{ij}
\quad \mbox{ and }\quad
\sum_{d' \in D_j} A_{d'\!d} = t_{ij}.$$
If there are $s_{ij}$ such that $\sum_{d' \in D_j} A_{dd'} = s_{ij}$ for all $d \in D_i$, we call the partition *row-stable*; similarly, for $t_{ij}$ such that $\sum_{d' \in D_j} A_{d'\!d} = t_{ij}$ for all $d \in D_i$, *column-stable*.
For symmetric $A$, column- and row-stability are equivalent (with $t_{ij} = s_{ij}$).
Note that the row and column sums in the definition are the $D_i$-components of $A \mathbf{d}_j$ and of $\mathbf{d}_j^t A = (A^t \mathbf{d}_j)^t$, respectively. So, for instance, row stability precisely says that $$A \mathbf{d}_j
= \sum_i s_{ij} \mathbf{d}_i \in \bigoplus_i \mathrm{span}(\mathbf{d}_i).$$
\[commstablelem\] Let $A \in \R^{n,n}$ commute with some symmetric matrix of the form $Z = XX^t \in \R^{n,n}$ for some doubly stochastic $X \in \R^{n,n}$. Then the partition $[n] = \dot{\bigcup}_i D_i$ of $[n]$ induced by $Z$ according to Lemma \[symdecomplem\] is stable for $A$.
We use the characteristic vectors $\mathbf{d}_i$ of the partition sets. By Corollary \[evalcor\], the eigenspace for eigenvalue $1$ of $Z$ is the direct sum of the spans of the vectors $\mathbf{d}_i$.
Now $Z A \mathbf{d}_i = AZ \mathbf{d}_i = A \mathbf{d}_i$ shows that $A \mathbf{d}_i$ is an eigenvector of $Z$ with eigenvalue $1$, whence$$A \mathbf{d}_i
\in \bigoplus_i \mathrm{span}(\mathbf{d}_i).
$$
It follows that the partition $[n] = \dot{\bigcup}_i D_i$ is row-stable.
Note again that $(A \mathbf{d}_j)_d = \sum_{d' \in D_j} A_{d d'}$ and $A \mathbf{d}_j \in \bigoplus_i \mathrm{span}(\mathbf{d}_i)$ precisely means that this value $(A \mathbf{d}_j)_d$ only depends on the partition set $D_i$ to which $d$ belongs. I.e., $\sum_{d'\in D_j} A_{d d'} = s_{ij}$ for all $d \in D_i$.
As $Z = XX^t = Z^t$, $A^t$ commutes with $Z$ if $A$ does: $A^t Z = A^t Z^t = (ZA)^t = (AZ)^t = Z^t A^t = Z A^t$. The above reasoning therefore shows that the partition into the $D_i$ is row-stable for $A^t$ as well, hence column stable for $A$. Hence it is stable for $A$.
NB: symmetry of $A$ is not required here. It is essential for deriving commutation of $A$ (and $A^t$) with $Z = XX^t$ from an equation of the form $AX = XB$, as we shall see below. But first a corollary from the argument just given.
\[innersymcor\] Let $A$ commute with $Z = XX^t$ and $B$ commute with $Z' = X^t X$, where $X$ is doubly stochastic (cf. Lemma \[commstablelem\]). Then the partitions induced by $Z$ and $Z'$, which are $X$-related by Lemma \[newdecomplem\], are stable for $A$ and $B$, respectively.
### Aside: boolean arithmetic {#aside-boolean-arithmetic .unnumbered}
We give a separate elementary proof of the analogue of Lemma \[commstablelem\] for boolean arithmetic. Here the definition of a *boolean* stable partition is this natural analogue of Definition \[stablepartdef\].
\[stablepartbooldef\] A partition $[n] = \dot{\bigcup}_{i \in I} D_i$ is *boolean stable* for $A \in \B^{n,n}$ if, in the sense of boolean arithmetic, $\sum_{d' \in D_j} A_{dd'}$ and $\sum_{d' \in D_j} A_{d'd}$ only depend on the partition set $i$ for which $d \in D_i$.
Note that boolean stability implies that, for the characteristic vectors $\mathbf{d}_i$ of the partition, $(A \mathbf{d}_j)_d = \sum_{d' \in D_j} A_{dd'}$ is the same for all $d \in D_i$, so that also here $A \mathbf{d}_j$ is a boolean linear combination of the characteristic vectors $\mathbf{d}_i$.
\[commstableboollem\] Let $A \in \B^{n,n}$ commute, in the sense of boolean arithmetic, with some symmetric matrix of the form $Z = XX^t \in \B^{n,n}$ with entries $Z_{dd} = 1$ for all $d \in [n]$. Then the partition $[n] = \dot{\bigcup}_i D_i$ induced by $Z$ according to Lemma \[summaryboollem\] is boolean stable for $A$.
Recall from Observation \[inducedpartboolobs\] that the characteristic vectors $\mathbf{d}_i$ of the induced partition behave like eigenvector with eigenvalue $1$ for boolean arithmetic: $Z \mathbf{d}_i = \mathbf{d}_i$. Moreover, we may assume that $Z_{dd'} = 1$ iff $d$ and $d'$ are in the same partition set (after passage to $Z^{n-1}$ if necessary). Let us write $\brck{\ell \in D_j}$ for the boolean truth value of the assertion $\ell \in D_j$. Then, for $d \in D_i$, $$\barr{rcl}
\sum_{d' \in D_j} A_{dd'} &=& (A \mathbf{d}_j)_d
= (A Z \mathbf{d}_j)_d
\\
&=& (Z A \mathbf{d}_j)_d =
\sum_{k,\ell} Z_{dk} A_{k\ell} \, \brck{\ell \in D_j}
=
\sum_{k \in D_i,\ell \in D_j} A_{k\ell}
\earr$$ does indeed not depend on $d \in D_i$, whence the partition is boolean row-stable. Column-stability again follows from similar considerations based on commutation of $Z = Z^t$ with $A^t$.
Fractional isomorphism
======================
$\LC^2$-equivalence and linear equations
----------------------------------------
The *adjacency matrix* of graph $\str A$ is the square matrix $A$ with rows and columns indexed by vertices of $\mathcal A$ and entries $A_{aa'}=1$ if $aa'$ is an edge of $\str A$ and $A_{aa'}=0$ otherwise. By our assumption that graphs are undirected and simple, $A$ is a symmetric square matrix with null diagonal. It will be convenient to assume that our graphs always have an initial segment $[n]$ of the positive integers as their vertex set. Then the adjacency matrices are in $\mathbb B^{n,n}\subseteq\mathbb R^{n,n}$. Throughout this subsection, we assume that $\str A$ and $\str B$ are graphs with vertex set $[n]$ and with adjacency matrices $A,B$, respectively. It will be notationally suggestive, to denote typical indices of matrices $a,a',\ldots \in [n]$ when they are to be interpreted as vertices of $\str{A}$, and $b,b',\ldots \in n]$ when they are to be interpreted as vertices of $\str{B}$.
Recall (from the discussion in the introduction) that two graphs $\str A,\str
B$ are isomorphic if, and only if, there is a permutation matrix $X$ such that $AX=XB$. We can rewrite this as the following integer linear program in the variables $X_{ab}$ for $a,b\in[n]$.
Then $\str A$ and $\str B$ are isomorphic if, and only if, $\ISO$ has an integer solution.
Two graphs $\str A,\str B$ are *fractionally isomorphic*, $\str
A\approx\str B$, if, and only if, the system $\ISO$ has a real solution.
Observe that graphs are fractionally isomorphic if, and only if, there is a doubly stochastic matrix $X$ such that $AX=XA$.
Note that fractionally isomorphic graphs necessarily have the same number of vertices (this will be different for the boolean analogue, which cannot count).
The established theorem on fractional isomorphism, by Ramana, Scheinerman and Ullman from [@RamanaScheinermanUllman; @ScheinermanUllman], relates fractional isomorphis to the colour refinement algorithm (‘iterated degree sequences’ in [@ScheinermanUllman]) introduced in Section \[sec:logic\] and stable partitions (‘equitable partitions’ in [@ScheinermanUllman]).
A *stable partition* of the vertex set of an undirected graph is a stable partition $[n] = \dot{\bigcup}_{i \in I} D_i$ for its adjacency matrix in the sense of Definition \[stablepartdef\]. Reading that definition for the (symmetric) adjacency matrix $A$ of a graph on $[n]$, and thinking of the partition sets $D_i$ as vertex colours, stability means that the colour of any vertex determines the number of its neighbours in every one of the colours. This is stability in the sense of colour refinement; it means that the colour refinement algorithm produces the coarsest stable partition.
The characteristic parameters for a stable partition $[n] = \dot{\bigcup}_{i \in I} D_i$ for $A$ are the numbers $s_{ij}=s_{ij}^A$ such that $s_{ij}=\sum_{d' \in D_j} A_{dd'}$ for all $d\in D_i$. (As $A$ is symmetric, the parameters $t_{ij}$ of Definition \[stablepartdef\] are equal to the $s_{ij}$.) We call two stable partitions $\dot{\bigcup}_{i \in I} D_i$ for a matrix $A$ and $\dot{\bigcup}_{i
\in J} D_i'$ for a matrix $B$ *equivalent* if $I=J$ and $|D_i|=|D_i'|$ for all $i\in I$ and $s_{ij}^A=s_{ij}^B$ and for all $i,j\in I$.
\[lem:c2stable\] $\str A$ and $\str B$ are $\LC^2$-equivalent if, and only if, there are equivalent stable partitions $\dot{\bigcup}_{i \in I} D_i$ for $A$ and $\dot{\bigcup}_{i \in I} D_i'$ for $B$.
The forward direction follows from Theorem \[theo:immlan\], because the colour refinement algorithm computes equivalent stable partitions of $\mathcal A$ and $\mathcal B$.
To establish the converse implication, we use the bijective $2$-pebble game, which characterises $\LC^2$-equivalence by Theorem \[theo:hel\]. Suppose we have equivalent stable partitions $\dot{\bigcup}_{i \in I} D_i$ of $A$ and $\dot{\bigcup}_{i \in J} D_i'$ of $B$. Then it is a winning strategy for player to maintain the following invariant for every position $p$ of the game: $p$ is a local isomorphism (that is, if $\dom(p)=\{a,a'\}$ then $a=a'$ if, and only if, $p(a)=p(a')$, and $a$ and $a'$ are adjacent in $\str A$ if, and only if, $p(a)$ and $p(a')$ are adjacent in $\str B$), and if $a\in\dom(p)\cap D_i$ then $p(a)\in D_i'$. It follows easily from the definition of stable partitions that player can indeed maintain this invariant.
\[simplegraphthm\] Two graphs are $\LC^2$-equivalent if, and only if, they are fractionally isomorphic.
In view of Lemma \[lem:c2stable\], it suffices to prove that $\str A$ and $\str B$ have equivalent stable partitions if, and only if, they are fractionally isomorphic.
For the forward direction, suppose that we have equivalent stable partitions $\dot{\bigcup}_{i \in I} D_i$ for $A$ and $\dot{\bigcup}_{i \in J} D_i'$ for $B$. For all $a\in D_i,b\in D_j'$ we let $$X_{ab}:=\delta(i,j)/n_i,$$ where $n_i:=|D_i|=|D_i'|$. (Here and elsewhere we use Kronecker’s $\delta$ function defined by $\delta(i,j)=1$ if $i=j$ and $\delta(i,j)=0$ otherwise.) An easy calculation shows that this defines a doubly stochastic matrix $X$ with $AX=XB$, that is, a solution for $\ISO$.
For the converse direction, suppose that $X$ is a doubly stochastic matrix such that $AX = XB$. Since $A$ and $B$ are symmetric, also $X^t A = B X^t$, and $$A XX^t = XB X^t = XX^t A \quad \mbox{ and } \quad
B X^tX = X^tA X = X^t X B,$$ show that $A$ commutes with $Z := XX^t$ and $B$ with $Z' := X^tX$.
From Lemma \[commstablelem\] and Corollary \[innersymcor\], the partitions $[n] = \dot{\bigcup}_{i \in I} D_i$ and $[n] = \dot{\bigcup}_{i \in I} D_i'$ that are induced by the symmetric matrices $Z$ and $Z'$ are $X$-related and stable for $A$ and for $B$, respectively. We need to show that $|D_i| = |D_i'|$ and that the partitions also agree w.r.t. the parameters $s_{ij}$.
By Lemma \[stochrelatedpartlem\] we have $|D_i| = |D_i'|$ and $$\label{parttransleqn}
\mathbf{d}_i = X \mathbf{d}_i'
\quad \mbox{ and } \quad
\mathbf{d}_i' = X^t \mathbf{d}_i,$$ where ${\mathbf{d}}_i$ and ${\mathbf{d}}_i'$ for $i\in I$ are the characteristic vectors of the two partitions. Thus for all $i,j\in I$, $$(\mathbf{d}_i')^t B \mathbf{d}_j'
=
(X^t \mathbf{d}_i)^t B X^t \mathbf{d}_j
=
\mathbf{d}_i^t X B X^t \mathbf{d}_j
=
\mathbf{d}_i^t A X X^t \mathbf{d}_j
=
\mathbf{d}_i^t A Z \mathbf{d}_j
=
\mathbf{d}_i^t A \mathbf{d}_j,$$ where the last equality follows from the fact that ${\mathbf{d}}_j$ is an eigenvector of $Z$ with eigenvalue $1$ by Corollary \[evalcor\].
Note that $\mathbf{d}_i^t A \mathbf{d}_j$ is the number of edges of $\str A$ from $D_i$ to $D_j$. By stability of the partition, we have $s_{ij}^A=\mathbf{d}_i^t A \mathbf{d}_j/|D_i|$ and similarly $s_{ij}^B=(\mathbf{d}_i')^t B \mathbf{d}_j'/|D_i'|$, so that $s_{ij}^A=s_{ij}^B$.
$\LL^2$-equivalence and boolean linear equations {#boolfracisosec}
------------------------------------------------
W.r.t. an adjacency matrix $A \in \B^{n,n}$, a boolean stable partition $[n] = \dot{\bigcup}_{i \in I} D_i$ has as parameters just the boolean values $$\iota_{ij}^A = \left\{
\barr{ll}
0 & \mbox{if } A_{D_iD_j} = 0,
\\
\hnt
1 & \mbox{else.}
\earr
\right.$$ Boolean (row-)stability of the partition for $A$ implies that $\iota_{ij}^A = 1$ if, and only if, for each individual $d \in D_i$ there is at least one $d' \in D_j$ such that $A_{dd'} = 1$, and similarly for column stability.
To capture the situation of $2$-pebble game equivalence, though, we now need to work with similar partitions that are stable both w.r.t. $A$ and w.r.t. to the adjacency matrix $A^c$ of the complement of the graph with adjacency matrix $A$. Here the complement of a graph $\str A$ is the graph $\str A^c$ with the same vertex set as $\str A$ obtained by replacing edges by non-edges and vice versa. Hence $A^c_{aa'}=1$ if $A_{aa'}=0$ and $a\neq a'$, and $A^c_{aa'}=0$ otherwise. While a partition in the sense of real arithmetic is stable for $A$ if, and only if, it is stable for $A^c$, this is no longer the case for boolean arithmetic. Let us call a partition that is boolean stable for both $A$ and $A^c$, *boolean bi-stable* for $A$.
Then the following captures the situation of two graphs that are $2$-pebble game equivalent. We note that $2$-pebble equivalence is a very rough notion of equivalence, if we look at just simple undirected graphs – but the concepts explored here do have natural extensions to coloured, directed graphs, and form the basis for the analysis of $k$-pebble equivalence, which is non-trivial even for simple undirected graphs.
$\LL^2$-equivalence of two graphs does not imply that the graphs have the same size. In the following, we always assume that $\mathcal
A,\mathcal B$ are graphs with vertex sets $[m],[n]$ respectively and that $A\in\mathbb B^{m,m}$ and $b\in\mathbb B^{n,n}$ are their adjacency matrices. We call two bi-stable partitions $[m]=\dot{\bigcup}_{i \in I} D_i$ for $A$ (and $A^c$) and $[n]=\dot{\bigcup}_{i
\in J} D_i'$ for $B$ (and $B^c$) *b-equivalent* if $I=J$ and $\iota_{ij}^A=\iota_{ij}^B$ and $\iota_{ij}^{A^c}=\iota_{ij}^{B^c}$ and for all $i,j\in I$. Note that b-equivalence does not imply that $|D_i|=|D_i'|$.
\[lem:l2stable\] $\str A$ and $\str B$ are $\LL^2$-equivalent if, and only if, there are b-equivalent bi-stable partitions $[m]=\dot{\bigcup}_{i \in I}
D_i$ for $A$ and $[n]=\dot{\bigcup}_{i \in J} D_i'$ for $B$.
The proof is analogous to the proof of Lemma \[lem:l2stable\]. For the backward direction, we need bistability to guarantee that player can maintain position $p$ that preserve adjacency, non-adjacency, and (in)equality. Stability alone would only enable her to maintain adjacency and equality.
\[fracisobooldef\] $\str{A}$ and $\str{B}$ are *boolean isomorphic*, $\str{A} \fracsimeqbool \str{B}$, if there is some boolean matrix $X$ without null rows or columns such that $A X = X B$ and $A^c X = X B^c$.
\[simplegraphboolthm\] Two graphs are $\LL^2$-equivalent if, and only if, they are boolean isomorphic.
For the forward direction, suppose that $A \equiv_\LL^2 B$, and let $[m] = \dot{\bigcup}_{1 {\leqslant}i {\leqslant}s} D_i$ and $[n] =
\dot{\bigcup}_{1 {\leqslant}i {\leqslant}s} D_i'$ be the similar boolean bi-stable partitions. For all $a\in D_i, b \in D_j'$ we let $X_{ab}:=\delta(i,j)$. This defines a boolean matrix $X\in \B^{m,n}$ without null rows or columns. One checks that $AX = XB$, in boolean arithmetic: for $a \in D_i$ and $b \in D_j'$, and for the characteristic vectors $\mathbf{d}_i$ and $\mathbf{d}_j'$ for the partitions, $$\barr{rcccl}
(AX)_{ab} &=&
\sum_{k} A_{ak}X_{kb}
=
(A \mathbf{d}_j)_a
= \iota_{ij}^A &&
\\
\hnt
&=& \iota_{ij}^B
=
(B \mathbf{d}_i')_{b}
=
((\mathbf{d}_i')^t B^t)_{b}
&=&
\sum_{k} X_{ak} B_{kb} = (XB)_{ab}.
\earr$$
The argument for $A^c X = X B^c$ is completely analogous.
For the converse, suppose that $A \fracsimeqbool B$, and let $X$ be a boolean matrix without null rows or columns such that $AX = XB$ and $A^c X = X B^c$. Since $A$ and $B$ are symmetric, also $X^t A = B X^t$ $X^t A^c = B^c X^t$, and $$A XX^t = XB X^t = XX^t A \quad \mbox{ and } \quad
B X^tX = X^tA X = X^t X B,$$ and the analogue for the complements, show that both $A$ and $A^c$ commute with $Z := XX^t$ and both $B$ and $B^c$ commute with $Z' := X^tX$. Moreover, the matrices $Z$ and $Z'$ have entries $1$ on the diagonal.
From Lemma \[commstableboollem\] and the straightforward analogue of Corollary \[innersymcor\], the partitions $[m] = \dot{\bigcup}_{i \in I} D_i$ and $[n] = \dot{\bigcup}_{i \in I} D_i'$ induced by the symmetric matrices $Z$ and $Z'$ are $X$-related and boolean bi-stable for $A$ and for $B$, respectively. We need to show that these partitions also agree w.r.t. the characteristic $\iota_{ij}$. By Lemma \[summaryboollem\], the characteristic vectors $\mathbf{d}_i'$ and $\mathbf{d}_i'$ of the partitions are related by $\mathbf{d}_i = X \mathbf{d}_i'$ and $\mathbf{d}_i' = X^t \mathbf{d}_i$ in the sense of boolean arithmetic.
Since $AX=XB$ and as the $\mathbf{d}_j$ are boolean eigenvectors of $Z = XX^t$ with eigenvalue $1$ by Observation \[inducedpartboolobs\], $$\barr{rcl}
\iota_{ij}^B &=&
(\mathbf{d}_i')^t B \mathbf{d}_j'
=
(X^t \mathbf{d}_i)^t B X^t \mathbf{d}_j
=
\mathbf{d}_i^t X B X^t \mathbf{d}_j
\\
&=&
\mathbf{d}_i^t A X X^t \mathbf{d}_j
=
\mathbf{d}_i^t A Z \mathbf{d}_j
=
\mathbf{d}_i^t A \mathbf{d}_j
\;=\; \iota^A_{ij}.
\earr$$
The argument for $\iota_{ij}^{B^c} = \iota_{ij}^{A^c}$ is strictly analogous.
Relaxations in the style of Sherali–Adams
=========================================
In this section we refine the connection between the Sherali–Adams hierarchy of LP relaxation of the integer linear program $\ISO$ to equivalence in the finite variable counting logics or the higher-dimensional Lehman–Weisfeiler equivalence.
NB: our parameter $k {\geqslant}2$ is the number of pebbles, or the variables available in the $k$-variable logics $\LC^k$ or $\LL^k$.
As before, $\str A$ and $\str B$ are graphs with vertex sets $[m]$ and $[n]$, respectively, and $A$ and $B$ are their adjacency matrices. We denote typical elements and tuples of elements from $\str{A}$ and $\str{B}$ as $\abar = (a_1,\ldots, a_r)$ or $\bbar = (b_1,\ldots, b_r)$, for $0 {\leqslant}r {\leqslant}k$; correspondingly, we typically denote entries of the adjacency matrices as, e.g., $A_{aa'}$. This device will help in an intuitive consistency check also in matrix compositions like $AX$ with entries $(AX)_{ab}$ if $A$ is an $m\!\times\!m$ matrix over $[m]$ and $X$, as an $m\!\times\!n$ matrix, relates $[m]$ and $[n]$ through entries $X_{ab}$: $(AX)_{ab} = \sum_{a'} A_{aa'}X_{a'b}$ (which rightly suggests paths of length two in a suitable composition of graphs $\str{A}$ and $G(X)$).
#### Types. {#types. .unnumbered}
Let $\etp(\abar)$ denote the equality type of tuple $\abar$ in $\str{A}$, $\atp(\abar)$ its quantifier-free type, and $\tp(\abar)$ its complete type in the logic $\LC^k$, that is, the set of all $\LC^k$-formulae ${\varphi}({\mathbf{x}})$ such that $\str A$ satisfies ${\varphi}({\mathbf{a}})$. Note that $\abar \mapsto \bbar$ constitutes a local bijection if, and only if, $\etp(\abar) = \etp(\bbar)$, and a local isomorphism if, and only if $\atp(\abar) = \atp(\bbar)$.
Types with parameters are denoted as in $\tp_\abar(a)$ for the type of $a$ w.r.t. to the fixed parameter tuple $\abar$; we may treat types as if they were formulae, with free variables for the main argument (as opposed to the parameter tuple), as in $(\tp_\abar(a))$ viewed as a formula with parameters for $\abar$ and a free variable $x$ assuming the role of $a$. By logics with restrictions on the number of variables, however, these parameters are accessible as assignments to variables; they thus count towards the variables in formulae, and are not treated as constants. The distinction between plain types and types with parameters, therefore, is one of semantic intention rather than syntactic. It is often suggestive when it comes to counting realisations. E.g., we let $$\#_\xbar^\str{A} (\tp(\xbar) \!=\! \tp(\abar))$$ denote the number of tuples $\abar'$ in $\str{A}$ that realise the type of $\abar$, i.e., those $\abar'$ for which $\str{A},\abar' \equiv_\LC^k
\str{A},\abar$. If the structure in which realisations are counted is obvious or does not matter because of $\LC^k$-equivalence, we drop the superscript and write e.g. just $\#_\xbar$ instead of $\#_\xbar^\str{A}$ or $\#_\xbar^\str{B}$. The number of realisations of the $1$-type of $a$ over parameters $\abar$ (in $\str{A}$) is denoted by $$\#_{x}(\tp_\abar(x) \!=\!\tp_\abar(a))
\quad\bigl(= \#_{x}(\tp(\abar x) \!=\! \tp(\abar a))\bigr).$$ Regarding the counting of realisations we note that generally $$\label{quotienteqn}
\#_{\xbar x} (\tp(\xbar x) \!=\! \tp(\abar a))
\quad=\quad
\#_\xbar (\tp(\xbar) \!=\! \tp(\abar))
\cdot \#_x (\tp_\abar(x) \!=\! \tp_\abar(a)).$$
#### The equations. {#the-equations. .unnumbered}
In the following we use variables $X_p$ indexed by subsets $p {\subseteq}[m]\times[n]$ of size up to $k-1$; we may think of such $p$ as being specified by two tuples $\abar$ and $\bbar$ of length $|p|$ that enumerate the first and second components of the pairs in $p$ in any coherent order. In this sense we write $p = \abar\bbar$. As remarked before, $p$ is a local bijection between $\str A$ and $\str B$ iff $\etp(\abar) = \etp(\bbar)$, and a local isomorphism iff $\atp(\abar) = \atp(\bbar)$ (and neither of these conditions depends on the chosen enumeration of tuples in $p$, which gives rise to the order of components in both $\abar$ and $\bbar$).
The augmentation of $p {\subseteq}[m]\times[n]$ by some pair $ab \in [m]\times[n]$ is simply denoted $p \,\widehat{\ }\,ab$. It is crucial that the notation $p \,\widehat{\ }\,ab$ does not refer to a *tuple* of pairs but to a *set* of pairs, in which the pair $(a,b)$ is not distinguished.
For further reference we isolate and name equation types as follows. For given $n,m {\geqslant}1$ and matrices $A \in \B^{n,n}$ and $B \in \B^{m,m}$: $$\barr{l}
\,X_\emptyset = 1
\hfill \CONT0
\\
\\
\left.\barr{@{}l@{}}
X_p = \sum_{b'} X_{p\,\widehat{\ }\, ab'} = \sum_{a'} X_{p\,\widehat{\ }\, a'b}
\\
\hnt
\mbox{for } |p| = \ell-1, a \in [m], b \in [n]
\earr
\qquad\;
\right\}
\hfill \CONT{\ell}
\\
\\
\left.\barr{@{}l@{}}
\sum_{a'} A_{aa'} X_{p\,\widehat{\ }\, a'b} =
\sum_{b'} X_{p\,\widehat{\ }\, ab'} B_{b'b}
\\
\hnt
\mbox{for } |p| = \ell-1, a \in [m], b \in [n]
\earr
\quad\;\,
\right\}
\qquad \COMP{\ell}
\earr$$
Here *level $\ell$* refers to $\ell$ as the size of the pairings $\abar \bbar$ in the typical variables $X_{\abar \bbar}$ involved; note that the size of $p$ mentioned in $X_{p\,\widehat{\ }\,ab}$ therefore remains one below this $\ell$. In the generic formats for $\CONT{\ell}$ and $\COMP{\ell}$ above, we assume $\ell {\geqslant}1$. Note that the combination of $\CONT{\ell}$ and $\COMP{\ell}$ for $\ell =1$ precisely corresponds to the equations for fractional isomorphism.
If we think of the matrix entries $(X_{\abar\bbar\,\widehat{\ }\,ab})_{a \in [n], b \in [m]}$ as specifying extensions of $\abar \mapsto \bbar$ in form of a distribution on possible pairings $a \mapsto b$, then equations $\CONT{\ell}$ may be seen as *continuity conditions*, while equations $\COMP{\ell}$ specify *compatibility conditions* with the edge relations encoded in $A$ and $B$. Variants of the compatibility conditions can be expressed for matrices other than the adjacency matrices $A$ and $B$ that we primarily think of. We saw one such variation in the discussion on boolean isomorphisms above, where $\COMP1$ was postulated for both $A,B$ and $A^c,B^c$. Further variants will play a role in Section \[boolLksec\].
Sherali–Adams of level $k-1$ {#sec:sa}
----------------------------
For $k {\geqslant}2$, the *level-$(k-1)$ Sherali–Adams relaxation* of the integer linear program $\ISO$ consists of the collection of the equations $\CONT{\ell}$ and $\COMP{\ell}$ for $\ell < k$:
### From $\LC^k$-equivalence to a level $k-1$ solution
Assume that $\str{A} \equiv_\LC^k \str{B}$. This implies that $\str{A}$ and $\str{B}$ realise exactly the same types of $r$-tuples for $r {\leqslant}k$ and with the same number of realisations: $$\#_\xbar^\str{A} (\tp(\xbar) \!=\! \tp(\abar))
=
\#_\xbar^\str{B} (\tp(\xbar) \!=\! \tp(\abar))$$ and similarly for all types $\tp(\bbar)$ of $r$-tuples in $\str{B}$ for $r {\leqslant}k$. In particular $m=|\str{A}| = |\str{B}|=n$ so that both structures have domain $[n]$.
If $\tp(\abar) = \tp(\bbar)$, where $\abar$ and $\bbar$ are $r$-tuples for $r {\leqslant}k-1$, then, for any $a \in [n]$, there are $\hat{b} \in [n]$ such that $\tp(\bbar \hat{b}) = \tp(\abar a)$; and for any such choice of $\hat{b}$ we find (cf. equation (\[quotienteqn\])): $$\label{quotienteqntwo}
\barr{r@{\;=\;}l}
\#_x^\str{A}(\tp_\abar(x) \!=\! \tp_\abar(a)) &
\#_{\xbar x}^\str{A}(\tp (\xbar x) \!=\! \tp(\abar a))
\bigm/
\#_\xbar^\str{A} (\tp(\xbar) \!=\! \tp(\abar))
\\
\hnt
&
\#_{\xbar x}^\str{B}(\tp(\xbar x) \!=\! \tp(\abar a))
\bigm/
\#_\xbar^\str{B} (\tp(\xbar) \!=\! \tp(\abar))
\\
\hnt
&
\#_{\xbar x}^\str{B}(\tp(\xbar x) \!=\! \tp(\bbar \hat{b}))
\bigm/
\#_\xbar^\str{B} (\tp(\xbar) \!=\! \tp(\bbar))
\\
\hnt
&
\#_x^\str{B} (\tp_\bbar(x) \!=\! \tp_\bbar(\hat{b})).
\earr$$
Similarly, for $r$-tuples $\abar$ and $\bbar$ such that $\tp(\abar) = \tp(\bbar)$, where $r {\leqslant}k-2$ (!), and for any $a$ and $b$, there are $\hat{a}$ and $\hat{b}$ such that $\tp(\bbar \hat{b} b) = \tp(\abar a \hat{a})$ and $$\label{edgecount}
\#_{xy}^\str{A} (\tp_\abar(xy) \!=\! \tp_\abar(a\hat{a})) =
\#_{xy}^\str{B} (\tp_\bbar(xy) \!=\! \tp_\bbar(\hat{b}b)).$$ For the desired solution put $$\label{soldef}
\barr{l}
X_\emptyset := 1,
\\
X_p := \delta(\tp(\abar),\tp(\bbar)) \bigm/ \#_\xbar
(\tp(\xbar) \!=\! \tp(\abar)),
\\
\nt\hfill
\mbox{ for $p = \abar\bbar$, $0 < |p| <k$.}
\earr$$
For the denominator note that $\#_\xbar (\tp(\xbar) \!=\! \tp(\abar))
=
\#_\xbar (\tp(\xbar) \!=\! \tp(\bbar))$ whenever $\tp(\abar) = \tp(\bbar)$. Clearly $X_p {\geqslant}0$. Note that $X_p \not= 0$ implies $\tp(\abar) =\tp(\bbar)$, which implies that $\atp(\abar) =\atp(\bbar)$ whereby $\abar \mapsto \bbar$ is a local isomorphism.
We check that the given assignment to the variables $X_p$ satisfies all instances of the equations $\CONT{\ell}$ and $\COMP{\ell}$ in Sherali–Adams of level $k-1$. In fact, the $X_p$ as specified by (\[soldef\]), satisfy all instances of the continuity equations $\CONT{\ell}$ of levels $\ell {\leqslant}k$ (!) and all instances of the compatibility equations $\COMP{\ell}$ of levels $\ell < k$, while level $k-1$ Sherali–Adams just requires both equation types for levels $\ell < k$.
Consider an instance of $\CONT{\ell}$ of level $\ell {\leqslant}k$, i.e., for $|p| < k$, with $p = \abar \bbar$, $a \in [n]$. If $\tp(\abar) \not= \tp(\bbar)$, then both sides of the equation are zero. In case $\tp(\abar) = \tp(\bbar)$, let $\hat{b} \in [n]$ be such that $\tp(\bbar \hat{b}) = \tp (\abar a)$ (such $\hat{b}$ exist as $\tp(\abar) = \tp(\bbar)$ and since $|p| < k$). Then $$\barr{r@{\;=\;}l}
\sum_{b'} X_{p\,\widehat{\ }\, ab'} &
\sum_{b'} \; \delta(\tp(\abar a),\tp(\bbar b'))
\bigm/
\#_{\xbar x}(\tp(\xbar x) \!=\! \tp(\abar a))
\\
\vhnt
&
\#_{x} (\tp_\bbar(x) \!=\! \tp_\abar (a))
\bigm/
\#_{\xbar x}(\tp(\xbar x) \!=\! \tp(\abar a))
\\
\vhnt
&
\#_{x} (\tp_\bbar(x) \!=\! \tp_\bbar (\hat{b}))
\bigm/
\#_{\xbar x}(\tp(\xbar x) \!=\! \tp(\bbar \hat{b}))
\\
\vhnt
&
\#_\xbar (\tp(\xbar) \!=\! \tp(\bbar))^{-1} = X_p,
\earr$$ where the crucial equality leading to the last line is from equation (\[quotienteqn\]).
Consider now an instance of equation $\COMP{\ell}$ of level $\ell < k$, i.e., with $|p| < k - 1$, with $p = \abar \bbar$, $a \in [n]$, $b \in [n]$. Again, the case of $\tp(\abar) \not= \tp(\bbar)$ is trivial. So we are left with the case of $p = \abar \bbar$ with $\tp(\abar) = \tp(\bbar)$ and $|p| {\leqslant}k-2$. These imply that there are $\hat{a}$ and $\hat{b}$ such that $\tp(\abar a \hat{a}) = \tp (\bbar \hat{b} b)$. Then $$\label{evallhs}
\barr{rl}
&
\sum_{a'} A_{aa'} X_{p\,\widehat{\ }\, a'b}
\\
=&
\vhnt
\sum_{a'} A_{aa'} \, \delta(\tp(\abar a'),\tp(\bbar b))
\bigm/ \#_{\xbar x}(\tp(\xbar x) \!=\! \tp(\bbar b))
\\
=&
\vvhnt
\frac{\displaystyle
\#_{y}
\bigl( \mathrm{edge}(ay) \wedge
\tp_\abar(y) \!=\! \tp_\abar(\hat{a}) \bigr)}{\displaystyle
\#_{\xbar x}(\tp(\xbar x) \!=\! \tp(\bbar b))}
\\
=&
\vvhnt
\frac{\displaystyle
\#_{xy}
\bigl( \mathrm{edge}(xy) \wedge \tp_\abar(x) \!=\! \tp_\abar(a) \wedge
\tp_\abar(y) = \tp_\abar(\hat{a}) \bigr)}{\displaystyle
\#_{\xbar x}(\tp(\xbar x) \!=\! \tp(\bbar b))
\cdot \#_x(\tp_\abar(x) \!=\! \tp_\abar(a))}
\\
=&
\vvhnt
\frac{\displaystyle
\#_{xy}
\bigl( \mathrm{edge}(xy) \wedge \tp_\abar(x) \!=\! \tp_\abar(a) \wedge
\tp_\abar(y) \!=\! \tp_\abar(\hat{a}) \bigr)}{\displaystyle
\#_{\xbar}(\tp(\xbar) \!=\! \tp(\bbar))
\cdot
\#_{x}(\tp_\bbar(x) \!=\! \tp_\bbar(b))
\cdot
\#_x(\tp_\abar(x) \!=\! \tp_\abar(a))}\;,
\earr$$ where we use instances of equation (\[quotienteqn\]), and, in the passage form the third to the forth line, artificially count over all realisations of $\tp_\abar(a)$ instead of just the fixed parameter $a$, and compensate for that in the denominator.
The counting term in the enumerator of this expression, $$\#_{xy}
\bigl( \mathrm{edge}(xy) \wedge \tp_\abar(x) \!=\! \tp_\abar(a)\wedge
\tp_\abar(y) \!=\! \tp_\abar(\hat{a})\bigr),$$ is the sum of the number of realisations of all those types $(\tp_\abar(a'',a'))$ that simultaneously extend $\tp_\abar(a)$, $\tp_\abar(\hat{a})$ and contain the formula $\mathrm{edge}(xy)$. Each one of these types has exactly the same number of realisations in $\str{A}$ as the corresponding type that simultaneously extends $\tp_\bbar(\hat{b})$, $\tp_\bbar(b)$ and contains the formula $\mathrm{edge}(xy)$. By symmetry of the graphs under consideration, $\mathrm{edge}(xy) $ is equivalent with $\mathrm{edge}(yx)$ and what we obtained in (\[evallhs\]) coincides with the corresponding evaluation of the right-hand side of this instance of equation $\COMP{\ell}$ as desired:
$$\label{evalrhs}
\barr{rl}
&\sum_{b'} B_{b'b} X_{p\,\widehat{\ }\, ab'}
\\
\vhnt
=&
\sum_{b'} B_{b'b} \, \delta(\tp(\abar a),\tp(\bbar b'))
\bigm/ \#_{\xbar x}(\tp(\xbar x) \!=\! \tp(\abar a))
\\
\vhnt
=&
\frac{\displaystyle
\#_{xy}
\bigl( \mathrm{edge}(yx) \wedge \tp_\bbar(x) \!=\! \tp_\bbar(b) \wedge
\tp_\abar(y) \!=\! \tp_\abar(a) \bigr)}{\displaystyle
\#_{\xbar x}(\tp(\xbar x) \!=\! \tp(\abar a))
\cdot \#_x(\tp_\bbar(x) \!=\! \tp_\bbar(b))}
\\
\vhnt
=&
\frac{\displaystyle
\#_{xy}
\bigl( \mathrm{edge}(yx) \wedge \tp_\bbar(x) \!=\! \tp_\bbar(b) \wedge
\tp_\bbar(y) \!=\! \tp_\bbar(\hat{b}) \bigr)}{\displaystyle
\#_{\xbar}(\tp(\xbar) \!=\! \tp(\abar))
\cdot
\#_{x}(\tp_\abar(x) \!=\! \tp_\abar(a))
\cdot
\#_x(\tp_\bbar(x) \!=\! \tp_\bbar(b))}\;.
\earr$$
\[mixedlevelcor\] If $\str{A} \equiv_\LC^k \str{B}$, then there is a solution $(X_p)$ for the combination of the continuity equations $\CONT{\ell}$ of levels $\ell {\leqslant}k$ (!) with the compatibility equations $\CONT{\ell}$ of levels $\ell < k$.
### From a level $k-1$ solution to $\LC^{<k}$-equivalence
In the following we discuss what it means that some admissible real or rational non-negative assignment to the variables $X_p$ for all $|p| < k$ satisfies the equations $\CONT{\ell}$ and $\COMP{\ell}$ for $\ell < k$, i.e., .
\[locisolem\] If $(X_p)_{|p| < k}$ is a solution to $\ISO[k-1]$, then $X_p \not= 0$ implies that
$p$ is a local bijection; and, strengthening this, that
$p$ is a local isomorphism.
For (a) we only need to use instances of the continuity equations $\CONT{\ell}$. Suppose that $p = \abar\bbar$ is not a local bijection, w.l.o.g. (by symmetry) assume that there are $a$ and $b_1 \not= b_2$ such that $(a,b_1), (a,b_2) \in p$. For $p_0 := p \setminus (a,b_2)$ we clearly have $\ell := |p_0| < k-1$, and looking at the instance of $\CONT{\ell}$ for this $p_0$ and $a$, we find that the two summands for $b' = b_i$, $i = 1,2$, both contribute to the left-hand side. So the equation and non-negativity of all $X$-assignments imply that $X_{p_0} + X_p {\leqslant}X_{p_0}$, whence $X_p =0$.
For (b) we use (a) and instances of the equations $\COMP{\ell}$ . The claim is true for $p = \emptyset$, and we lift it to all other $p$ by induction. Note that for $|p|=1$, $p = ab$ cannot fail to be a local isomorphism (in undirected, loop-free graphs).
Assume then that $X_{p \,\widehat{\ }\, ab} > 0$, that $0 < |p| < k-1$ and that (b) holds true for $p$. Note that equations $\CONT{|p|+1}$ and $X_{p \,\widehat{\ }\, ab} > 0$ imply that $X_p >0$. Hence, by the inductive hypothesis, $p$ is a local isomorphism.
By (a), $p \,\widehat{\ }\, ab$ is a local bijection, and we check that it also must be a local isomorphism. For this it remains to show that, for instance $(a,a_1)$ is an edge of $\str{A}$ iff $(b,b_1)$ is an edge of $\str{B}$. Since the edge relations are symmetric, it suffices to show that
$A_{a_1,a} = 1 \;\Rightarrow\; B_{b_1b} \not=0$, and
$B_{b,b_1} = 1 \;\Rightarrow\; A_{aa_1} \not=0$.
For (i) we use this instance of equation $\COMP{\ell}$ for $\ell := |p|+1$: $$\sum_{a'} A_{a_1a'} X_{p\,\widehat{\ }\,a'b} =
\sum_{b'} X_{p\,\widehat{\ }\,a_1b'} B_{b'b}.$$
The right-hand side reduces to the single term $X_{p} B_{b_1b}$ because $a_1 \in \dom(p)$. As the left-hand side is positive if $A_{a_1,a} = 1$ and $X_{p \,\widehat{\ }\, ab} > 0$, (i) follows.
An analogous argument for (ii) is based on the instance $\sum_{a'} A_{aa'} X_{p\,\widehat{\ }\,a'b_1} =
\sum_{b'} X_{p\,\widehat{\ }\,ab'} B_{b'b_1}$ of equation $\COMP{\ell}$.
### [$\LC^{<k}$]{}-equivalence {#lck-equivalence .unnumbered}
A solution for the variables $X_{p}$ satisfying $\ISO(k-1)$ is in fact not strong enough to guarantee $\LC^k$-equivalence, but a slightly lesser equivalence, $$\str{A}\equiv_\LC^{<k} \str{B},$$ which we characterise in terms of a modified, *weak bijective $k$-pebble game* over $\str{A},\str{B}$. The game is played by two players. If $m\neq n$, player loses immediately. Otherwise, a play of the game proceeds in a sequence of rounds. Positions of the game are sets $p\subseteq [m]\times [n]$ of size $|p|\le k-1$. Normally, the initial position is $\emptyset$, but we will also consider plays of the game starting from other initial positions. A single round of the game, starting in position $p$, is played as follows.
1. If $|p|=k-1$, player selects a pair $ab\in p$. If $|p|<k-1$, he omits this step.
2. Player selects a bijection between $[m]$ and $[n]$ (recall that $m=n$).
3. Player chooses a pair $a'b'$ from this bijection.
4. If $p^+:=p\,\widehat{\ }\,a'b'$ is a local isomorphism then the new position is $$p':=
\begin{cases}
\;\; (p\!\setminus\! ab)\,\widehat{\ }\,a'b'&\text{ if }|p|=k-1,\\
\hnt
\;\;\, p \,\widehat{\ }\,a'b'&\text{ if }|p|<k-1.
\end{cases}$$ Otherwise, the play ends and player loses.
Player wins a play if it lasts forever, i.e., if $m=n$ and she never loses in step 4 of a round.
Note that the weak bijective $k$-pebble game requires more of the second player than the bijective $(k-1)$-pebble game, because $p^+$ rather than just $p'$ is required to be a local isomorphism. On the other hand, it requires less than the bijective $k$-pebble game: the bijective $k$-pebble game precisely requires the second player to choose the bijection without prior knowledge of the pair $ab$ that will be removed from the position. A strategy for player in the weak version is good for the usual version if it is fully symmetric or uniform w.r.t. the pebble pair that is going to be removed.
However, this is only relevant if $k\ge 3$. The weak bijective $2$-pebble game and the bijective $2$-pebble game are essentially the same.
$\str{A}$ and $\str{B}$ are $\LC^{<k}$-equivalent, $\str{A}\equiv_\LC^{<k} \str{B}$, if the second player has a winning strategy in the weak bijective $k$-pebble game on $\str A$, $\str B$.
Furthermore, for tuples ${\mathbf{a}}$ and ${\mathbf{b}}$ of the same length $\ell<k$ we let $\str{A},{\mathbf{a}}\equiv_\LC^{<k} \str{B},{\mathbf{b}}$ if the second player has a winning strategy in the weak bijective $k$-pebble game on $\str A$, $\str B$ with initial position $\abar
{\mathbf{b}}$.
\[sandwichobs\] $
\str A\equiv_\LC^2\str B\;\Leftrightarrow\;\str A\equiv_\LC^{<2}\str B,
$ and for all $k {\geqslant}3$: $$\str{A}\equiv_\LC^k \str{B}
\quad \Rightarrow \quad
\str{A}\equiv_\LC^{<k} \str{B}
\quad \Rightarrow \quad
\str{A}\equiv_\LC^{k-1} \str{B}.$$
\[edgemoverem\] The weak bijective $k$-pebble game is equivalent to a bisimulation-like game with $k-1$ pebbles where in each round the first player may slide a pebble along an edge of one of the graphs and the dublicator has to answer by sliding the corresponding pebble along an edge of the other graph. In this version, the game coresponds to the $(k-1)$-pebble sliding game introduced by Atserias and Maneva [@AtseriasManeva]. They prove that equivalence of two graphs with respect to the $(k-1)$-pebble sliding game implies that $\ISO[k-1]$ has a solution. In view of the equivalence of the sliding game with our weak bijective $k$-pebble game, this implies the backward direction of Theorem \[SAkminus1prop\] below.
Let $I_{n}$ be the set of positions of the weak bijective $k$-pebble game over $\str{A},\str{B}$ in which the second player has a strategy to survive through $n$ rounds. Let $\sim^n$ stand for the equivalence relation induced by $I_n$, i.e., the transitive closure of the relation that puts $\abar \sim^n \bbar$ if $p = \abar \bbar \in I_n$. Note that $\sim^n$ is compatible with permutations in the sense that, for instance, $\abar \sim \bbar$ iff $\pi(\abar) \sim \pi(\bbar)$ for any $\pi \in S_n$. We write $\pi(\abar)$ for the application of the permutation $\pi \in S_n$ to the components of $\abar = (a_1,\ldots, a_n)$, which results in $\pi(\abar) = (a_{\pi(1)},\ldots, a_{\pi(n)})$.
For $n = 0$, $I_0$ consists of all size $k-1$ local isomorphisms. We characterise $I_{n+1}$ and $\sim^{n+1}$ in terms of $I_n$ by means of back&forth conditions for a single round: $\abar \sim^{n+1} \bbar$ ($p = \abar\bbar \in I_{n+1}$) iff position $\abar\bbar$ is good in the following sense: for $1 {\leqslant}j {\leqslant}k-1$, the second player has a response that guarantees a target position in $I_n$ if the first player chooses index $j$.
I.e., for each $1 {\leqslant}j{\leqslant}k-1$, the second player needs to have a between $[m]$ and $[n]$ such that for every $(a,b) \in \rho_j$ $$\atp(\abar a) = \atp(\bbar b)
\quad\mbox{ and }\quad
\textstyle \abar\frac{a}{j} \bbar\frac{b}{j} \in I_n.$$
The first condition says that $p\,\widehat{\ }\,ab$ is a local isomorphism, the second that the new position is good for $n$ further rounds.
Note that, since $\str A$ is a graph, the quantifier-free type $\atp(\abar a)$ is fully determined by $\atp(\abar)$ and the $\atp(a_i a)$ for $1 {\leqslant}i {\leqslant}k-1$, which in turn are determined by $\atp(\abar\frac{a}{j} a)$. The condition that $\atp(\abar) = \atp(\bbar)$ is a pre-condition for the round to be played; the condition that $\atp(a_i a) = \atp(b_i b)$ is part of the post-condition that $p\!\setminus\! (j) \,\widehat{\ }\,ab$ is a local isomorphism, for all $i$ apart from $i = j$.
Let $(\alpha_i)_{i\in I}$ be an enumeration of the $\sim^n$-classes over $\str{A}$ and $\str{B}$. Then the above conditions on membership of $p = \abar\bbar$ in $I_{n+1}$ are equivalent to the following:
$$\barr{l}
\mbox{\btfll for each $1 {\leqslant}j {\leqslant}k-1$,
\\
for every $\sim^n$-class $\alpha$, and
\\
for every quantifier-free type $\eta(x,y)$:
\etfll}
\\
\#_a^\str{A}
\bigl(
\abar {\textstyle \frac{a}{j}} \in \alpha
\wedge
\atp(a_ja) \!=\! \eta
\bigr)
\;\;=\;\;
\#_b^\str{B}
\bigl(
\bbar {\textstyle \frac{b}{j}} \in \alpha
\wedge
\atp(b_jb) \!=\! \eta
\bigr).
\earr$$ Note, towards the claimed equivalence, that these numerical equalities allow the second player to piece together a bijection that respects the partition of $[m]$ and $[n]$ according to different combinations of $\alpha$ and $\eta$, which in turn guarantees that any pair $(a,b)$ drawn from the bijection respects this partition and hence leads to a position $\abar\frac{a}{j}\bbar\frac{b}{j} \in I_n$ as required.
Conversely, if one of these equalities were violated, then any bijection will have to have at least one pair that does not respect the partition of $[m]$ and $[n]$ w.r.t. the $\alpha$ and $\eta$. If the first player picks such a bad pair $(a,b)$, then the second player loses during this round because $\atp(\abar a) \not= \atp(\bbar b)$, or the resulting new position $\abar\frac{a}{j}\bbar\frac{b}{j}$ is not in $I_n$.
For later use we state the condition on full $\LC^{<k}$-equivalence, corresponding to the stable limit of the above refinement step. For $\abar \in [m]^{k-1}$ and $\bbar \in [n]^{k-1}$, $\str{A},\abar \equiv_\LC^{<k} \str{B},\bbar$ iff $$\label{equivlimitcond}
\barr{l}
\mbox{\btfll for each $1 {\leqslant}j {\leqslant}k-1$,
\\
for all $\LC^{<k}$-equivalence classes $\alpha$, and
\\
for every quantifier-free type $\eta(x,y)$:
\etfll}
\\
\#_a^\str{A}
\bigl(
\abar {\textstyle \frac{a}{j}} \in \alpha
\wedge
\atp(a_ja) \!=\! \eta
\bigr)
\;\;=\;\;
\#_b^\str{B}
\bigl(
\bbar {\textstyle \frac{b}{j}} \in \alpha
\wedge
\atp(b_jb) \!=\! \eta
\bigr).
\earr$$
#### From local isomorphisms to [$\LC^{<k}$]{}-equivalence. {#from-local-isomorphisms-to-lck-equivalence. .unnumbered}
We observe that for any solution $(X_p)$ of the level $k-1$ equations, and for every $|p| {\leqslant}k-2$ such that $X_p \not= 0$, the matrix $$\left(\frac{\displaystyle X_{p\,\widehat{\ }\,ab}}
{\displaystyle X_p}
\right)_{a \in [m], b \in [n]} $$ is doubly stochastic by equations $\CONT{\ell}$. In particular, the continuity equations, even for $\ell = 0$, also enforce that $\str{A}$ and $\str{B}$ have the same number of vertices, and we may assume that $m=n$.
\[solutiontoskequivlem\] For $k {\geqslant}3$, if $(X_{p\,\widehat{\ }\,ab})_{|p| {\leqslant}k-2, a\in [n],b\in [n]}$ is a solution to $\ISO[k-1]$ then for all $\abar \in [n]^{k-1}$ and $\bbar \in [n]^{k-1}$: $$X_{\abar\bbar} > 0 \;
\quad \Longrightarrow \quad
\str{A},\abar \equiv_\LC^{<k} \str{B},\bbar.$$
Consider tuples $\abar$ and $\bbar$ of length $k-1$ such that $X_{\abar\bbar} > 0$. It suffices to show that, for each $1 {\leqslant}j {\leqslant}k-1$, the second player has a response $\rho_j$ in a single round played in component $j$ that guarantees a resulting position $\abar'\bbar' = \abar\frac{a'}{j}\bbar\frac{b'}{j}$ for which again $X_{\abar'\bbar'} > 0$.
Fix some $j$. Let $a := a_j$, $b := b_j$ and $p$ such that $\abar\bbar = p\,\widehat{\ }\, ab$.[^5] Since $X_{\abar\bbar} >0$, equations $\CONT{\ell}$ imply that also $X_p >0$. Note that $|p| {\leqslant}k-2$. The matrix $Y$ defined by $Y_{ab} := X_{p\,\widehat{\ }\,ab}/X_p$ is doubly stochastic, with a strictly positive entry in the position $ab$ under consideration.
By Corollary \[innersymcor\], $Z = YY^t$ and $Z' = Y^tY$ are symmetric and doubly stochastic with positive entries on the diagonal and induce $Y$-related partitions $[n] = \dot{\bigcup}_{i \in I} D_i$ and $[n] = \dot{\bigcup}_{i \in I} D_i'$ that are stable w.r.t. $A$ and $B$, respectively. As discussed in the proof of Theorem \[simplegraphthm\], cf. equation (\[parttransleqn\]), the characteristic vectors $\mathbf{d}_i$ and $\mathbf{d}_i'$ of these stable partitions are related according to $\mathbf{d}_i' = Y^t \mathbf{d}_i$ and $\mathbf{d}_i = Y \mathbf{d}_i'$. Similarity of the partitions as stated in the proof of Theorem \[simplegraphthm\] implies that the partitions agree w.r.t. sizes of the partition sets, $|D_i| = |D_i'| = n_i$ and w.r.t. the characteristic counts $\nu_{ij}^\str{A} = \nu_{ij}^\str{B}$, where $\nu_{ij}^\str{A} = \mathbf{d}_i^t A \mathbf{d}_j$ and $\nu_{ij}^\str{B} = (\mathbf{d}_i')^t B \mathbf{d}_j'$.
Because $X_{\abar\bbar} = X_{p\,\widehat{\ }\,ab} = Y_{ab} >0$, $a \in D_k$ and $b \in D_k$ for the same $k \in I$. It follows that, for every $i \in I$, $$| \{ a' \in D_i \colon A_{aa'} = 1 \}|
=
| \{ b' \in D_i' \colon B_{bb'} = 1 \}|.$$ For this observe that these counts are $\nu_{ki}/n_k $. The second player may therefore choose a bijection that bijectively maps $D_i$ to $D_i'$ for $i \in I$ in such a manner that for every pair $(a',b')$ in this bijection, $(a,a')$ is an edge in $\str{A}$ iff $(b,b')$ is an edge in $\str{B}$.
If the first player now picks any pair $(a',b')$ from this bijection, then $a' \in D_i$ and $b' \in D_i$ for some $i \in I$, and the $Y$-relatedness of the partitions implies that $Y_{a'b'} > 0$ and hence $Y_{a'b'} X_p = X_{p\,\widehat{\ }\,a'b'} > 0$. By Lemma \[locisolem\] (b), $p\,\widehat{\ }\,a'b'$ is a local isomorphism; and so is, by assumption, $p\,\widehat{\ }\,ab$. As also $(a,a')$ is an edge in $\str{A}$ iff $(b,b')$ is an edge in $\str{B}$ by choice of the bijection, $p^+ := p\,\widehat{\ }\,ab\,\widehat{\ }\, a'b' =
\abar\bbar\,\widehat{\ }\, a'b'$ is a local isomorphism, too. The second player is thus guaranteed to reach a resulting position $p'= \abar'\bbar'= p\,\widehat{\ }\,a'b'$ for which $X_{p'} >0$.
\[SAkminus1prop\] $\ISO[k-1]$ has a solution if, and only if, $\str A\equiv_\LC^{<k}\str B$.
The last lemma settles one implication. For the converse implication, it remains to argue that $\LC^{<k}$-equivalence suffices in place of $\LC^k$-equivalence to provide a solution to the Sherali–Adams relaxation of level $k-1$. We now let $\sktp(\abar)$ stand for the $\LC^{<k}$-type, or the $\LC^{<k}$-equivalence class of the tuple $\abar$. We may look at just tuples of length $k-1$, by trivial padding through repetition of the last component say. Put $$\barr{l}
X_\emptyset := 1
\\
X_p := \delta(\sktp(\abar),\sktp(\bbar))
\bigm/ \#_\xbar (\sktp(\xbar) \!=\!
\sktp(\abar))
\\
\nt\hfill
\mbox{ for $p = \abar\bbar$, $0 < |p| {\leqslant}k-1$.}
\earr$$
We know that $\LC^{<k}$-equivalence refines $\LC^{k-1}$-equivalence, and that an assignment to $X_p$ according to $\LC^{k-1}$-types of $(k-1)$-tuples was shown above to satisfy the continuity equations $\CONT{\ell}$ of levels $\ell < k$, cf. Corollary \[mixedlevelcor\]. One can infer from this that also the refinement used here satisfies these equations.
For satisfaction of equations $\COMP{\ell}$ of level $\ell < k$, however, we need to appeal to something less than the extension property that boosts $\abar$ and $\bbar$ to $k$-tuples $\abar a \hat{a}$ and $\bbar \hat{b} b$ of the same $\LC^k$-type, as we used in connection with (\[evallhs\]) above.
Here as there, however, we only need to look at $p = \abar \bbar$ of size (up to) $k-2$ for which $\str{A},\abar \equiv_\LC^{<k} \str{B},\bbar$, because all other instances of the equation are trivially true with $0$ on both sides. We fix such $p$.
Now, for any combination of $\LC^{<k}$-types $\alpha$ and $\beta$ of $(k-1)$-tuples and quantifier-free type $\eta$ of a pair, $$\label{matchnoeqn}
\barr{rl}
&\#_{aa'}^\str{A}
\bigl( \sktp (\abar a) \!=\! \alpha \wedge
\sktp(\abar a') \!=\! \beta \wedge \atp(aa') \!=\! \eta \bigr)
\\
\hnt
=&
\#_{bb'}^\str{B}
\bigl( \sktp (\bbar b) \!=\! \alpha \wedge
\sktp(\bbar b') \!=\! \beta \wedge \atp(bb') \!=\! \eta \bigr).
\earr$$
This follows from an analysis of the $\LC^{<k}$-game from position $p = \abar\bbar$ through two rounds, in which the first player first gets the last pebble placed on any one of the possible choices for $a$, with responses $b$ as provided by the second player’s bijection (in exactly the same number); then the first player plays on that last component again, and replaces it with any one of the choices he may have for $a'$ and its match $b'$ according to the second player’s bijection (again, the same number of positive choices).
For given $a$ and $b$, let now $\alpha := \sktp(\abar a)$ and $\beta := \sktp(\bbar b)$. Then $$\barr{rl}
&
\sum_{a'} A_{aa'} X_{p\,\widehat{\ }\, a'b}
\\
=&
\vvhnt
\sum_{a'} A_{aa'} \, \delta(\sktp(\abar a'),\sktp(\bbar b))
\bigm/ \#_{\xbar x}(\sktp(\xbar x) \!=\! \sktp(\bbar b))
\\
=&
\vvhnt
\frac{\displaystyle
\#_{a'}^\str{A}
\bigl( \sktp(\abar a') \!=\! \beta \wedge \mathrm{edge}(aa') \bigr)}{\displaystyle
\#_{\xbar a'}(\sktp(\xbar a') \!=\! \beta)}
\\
=&
\vvhnt
\frac{\displaystyle
\#_{aa'}^\str{A}
\bigl( \sktp (\abar a) \!=\! \alpha \wedge
\sktp(\abar a') \!=\! \beta \wedge \mathrm{edge}(aa') \bigr)}{\displaystyle
\#_{\xbar a'}(\sktp(\xbar a') \!=\! \beta) \cdot
\#_{a}(\sktp(\abar a) \!=\! \alpha)}
\\
=&
\vvhnt
\frac{\displaystyle
\#_{aa'}^\str{A}
\bigl( \sktp (\abar a) \!=\! \alpha \wedge
\sktp(\abar a') \!=\! \beta \wedge \mathrm{edge}(aa') \bigr)}{\displaystyle
\#_{a'}(\sktp(\abar a') \!=\! \beta)
\cdot
\#_{\xbar}(\sktp(\xbar) \!=\! \sktp(\abar))
\cdot
\#_{a}(\sktp(\abar a) \!=\! \alpha)}.
\earr$$ We transform this term further, using (\[matchnoeqn\]), a renaming of dummy variables in counting terms and the symmetry of the unique quantifier-free type $\eta$ determined by $\mathrm{edge}(xy)$ in simple undirected graphs. The goal is to show equality with the corresponding term obtained for $\sum_{b'} X_{p\,\widehat{\ }\, ab'} B_{b'b}$. Equality (\[matchnoeqn\]) is used in the first step of these transformations, starting from the term just obtained: $$\barr{rl}
&
\frac{\displaystyle
\#_{aa'}^\str{A}
\bigl( \sktp (\abar a) \!=\! \alpha \wedge
\sktp(\abar a') \!=\! \beta \wedge \mathrm{edge}(aa') \bigr)}{\displaystyle
\#_{a'}(\sktp(\abar a') \!=\! \beta)
\cdot
\#_{\xbar}(\sktp(\xbar) \!=\! \sktp(\abar))
\cdot
\#_{a}(\sktp(\abar a) \!=\! \alpha)}
\\
\vvhnt
=&
\frac{\displaystyle
\#_{bb'}^\str{B}
\bigl( \sktp (\bbar b) \!=\! \alpha \wedge
\sktp(\bbar b') \!=\! \beta \wedge \mathrm{edge}(bb') \bigr)}{\displaystyle
\#_{b'}(\sktp(\bbar b') \!=\! \beta)
\cdot
\#_{\xbar}(\sktp(\xbar) \!=\! \sktp(\bbar))
\cdot
\#_{b}(\sktp(\bbar b) \!=\! \alpha)}
\\
\vvhnt
=&
\frac{\displaystyle
\#_{bb'}^\str{B}
\bigl( \sktp (\bbar b') \!=\! \alpha \wedge
\sktp(\bbar b) \!=\! \beta \wedge \mathrm{edge}(bb') \bigr)}{\displaystyle
\#_{b}(\sktp(\bbar b) \!=\! \beta)
\cdot
\#_{\xbar}(\sktp(\xbar) \!=\! \sktp(\bbar))
\cdot
\#_{b'}(\sktp(\bbar b') \!=\! \alpha)}
\\
\vvhnt
=&
\frac{\displaystyle
\#_{b'}^\str{B}
\bigl( \sktp (\bbar b') \!=\! \alpha
\wedge \mathrm{edge}(bb') \bigr)}{\displaystyle
\#_{\xbar}(\sktp(\xbar) \!=\! \sktp(\bbar))
\cdot
\#_{b'}(\sktp(\bbar b') \!=\! \alpha)}
\\
\vvhnt
=&
\sum_{b'}
B_{b'b} \;
\delta(\tp^{<k}(\bbar b'),\tp^{<k}(\abar a)) \bigm/
\#_{\xbar x}(\sktp(\xbar x) \!=\! \tp^{<k}(\abar a))
\\
\vvhnt
=&
\sum_{b'}
X_{p\,\widehat{\ }\,ab'} B_{b'b}.
\earr$$
The gap {#gapsec}
-------
Based on a construction due to Cai, Fürer, and Immerman [@caifurimm92], for $k{\geqslant}3$ we construct graphs showing that $\str{A} \equiv_\LC^{<k} \str{B}
\not\Rightarrow \str{A} \equiv_\LC^{k} \str{B}$, and that $\str{A} \equiv_\LC^{k-1} \str{B}
\not\Rightarrow \str{A} \equiv_\LC^{<k} \str{B}$. So the implications in Observation \[sandwichobs\] are strict for $k {\geqslant}3$.
For every $k {\geqslant}3$, there are graphs $\str{A}$ and $\str{B}$ such that $\str{A} \equiv_\LC^{k-1} \str{B}$ but $\str{A} \not\equiv_\LC^{<k} \str{B}$.
We present the argument explicitly for $k=4$; its variant for $k=3$ and the generalisation to higher $k$ are straightforward. For $\str{A}$ and $\str{B}$ we use the straight and the twisted version of the Cai–Fürer–Immerman companions of the $4$-clique.
We use copies of the standard degree $3$ gadget $$\barr{rcl}
H_3 &:=& \bigl( O \cup I, E \bigr), \mbox{ where }
\\
O &:=& \{ 1,2,3\} \cup \{ \bar{1},\bar{2},\bar{3} \} \quad \mbox{ (outer nodes)}
\\
I &:=& \mathcal{P}_{\ssc \mathrm{odd}}(\{1,2,3\}) \quad \mbox{ (inner nodes: odd subsets)}
\\
E &:=&
\{
\{i,s\} \colon i \in O, s \in I, i \in s
\}
\cup
\{
(\bar{i},s) \colon i \in O, s \in I, i \not\in s
\}
\earr$$ and its dual $$\barr{rcl}
\bar{H}_3 &:=& \bigl( O \cup I, E \bigr), \mbox{ where }
\\
O &:=& \{ 1,2,3\} \cup \{ \bar{1},\bar{2},\bar{3} \} \quad \mbox{ (outer nodes)}
\\
I &:=& \mathcal{P}_{\ssc \mathrm{even}}(\{1,2,3\}) \quad \mbox{ (inner nodes: even subsets)}
\\
E &:=&
\{
\{ i,s\} \colon i \in O, S \in I, i \in s
\}
\cup
\{
(\bar{i},s) \colon i \in O, s \in I, \colon i \not\in s
\}.
\earr$$
In each of these, we think of the three pairs of outer nodes as positive markers ($1,2,3$) and negative markers ($\bar{1},\bar{2},\bar{3}$) for the elements of the set $[3] = \{ 1,2,3 \}$, and of the four inner nodes as subsets $s {\subseteq}[3]$ (of odd cardinalities in the case of $H_3$, and of even cardinalities in the case of $\bar{H}_3$); the edges incident with a particular $s \in I$ encode elementhood of $i = 1,2,3$ in $s$ by linking $s$ to $i$ if $i \in s$ and to $\bar{i}$ if $i \not\in s$. Both graphs are bipartite with inner nodes of degree $3$ and outer nodes of degree $2$, which serve as ports to link copies of $H_3$ and $\bar{H}_3$. We really use coloured variants of $H_3$ and $\bar{H}_3$ that distinguish the vertices of the distinct copies of $H_3$ and $\bar{H}_3$, and inner and outer as well as the three groups of port vertices (i.e., $\{ i, \bar{i} \}$ for $i = 1,2,3$) within each of them. This is without loss of generality, since we may eliminate colours, e.g., by attaching simple, disjoint paths of different lengths to the members of each group of vertices.
The non-trivial automorphisms of this decorated variant of $H_3$ and $\bar{H}_3$ precisely allow for simultaneous swaps within exactly two pairs of port vertices. In the sketch we distinguish positive element markers from negative ones by using filled and open circles; this is just to highlight the combinatorial source of the graph structure, and in (the decorated versions of) $H_3$ and $\bar{H}_3$ *this* distinction is *not* present.
$$\begin{xy}0;<1.5cm,0cm>:
0,
+/d.5cm/,
*[F]{\rule{0ex}{1ex}\rule{1ex}{0ex}} ="000",
+/r.5cm/,
*[F]{\rule{0ex}{1ex}\rule{1ex}{0ex}} ="011",
+/r.5cm/,
*[F]{\rule{0ex}{1ex}\rule{1ex}{0ex}} ="101",
+/r.5cm/,
*[F]{\rule{0ex}{1ex}\rule{1ex}{0ex}} ="110",
+/d1.5cm/,
+/l2cm/,
*{\bullet} ="1",
+/r.3cm/,
*{\circ} ="1bar",
+/r.8cm/,
*{\bullet} ="2",
+/r.3cm/,
*{\circ} ="2bar",
+/r.8cm/,
*{\bullet} ="3",
+/r.3cm/,
*{\circ} ="3bar",
"1", +/d.3cm/,*{\ssc 1},
"1bar", +/d.3cm/,*{\ssc \bar{1}},
"2", +/d.3cm/,*{\ssc 2},
"2bar", +/d.3cm/,*{\ssc \bar{2}},
"3", +/d.3cm/,*{\ssc 3},
"3bar", +/d.3cm/,*{\ssc \bar{3}},
"000", +/u.3cm/,*{\ssc [3]},
"011", +/u.3cm/,*{\ssc \{1\}},
"101", +/u.3cm/,*{\ssc \{2\}},
"110", +/u.3cm/,*{\ssc \{3\}},
"110", +/r1.5cm/,*{I},
"000", +/l1.5cm/, +/d.5cm/, *{H_3},
"3bar", +/r1cm/,*{O},
+/d1cm/,
*{\ },
\ar@{-} "000" ; "1",
\ar@{-} "000" ; "2",
\ar@{-} "000" ; "3",
\ar@{-} "011" ; "1",
\ar@{-} "011" ; "2bar",
\ar@{-} "011" ; "3bar",
\ar@{-} "101" ; "1bar",
\ar@{-} "101" ; "2",
\ar@{-} "101" ; "3bar",
\ar@{-} "110" ; "1bar",
\ar@{-} "110" ; "2bar",
\ar@{-} "110" ; "3",
\end{xy}$$
Let $\str{A}$ consist of four decorated copies of $H_3$, copies $a,b,c,d$ say, that are linked by edges in corresponding outer nodes:
$(a,1)$ is linked to $(b,1)$ and $(a,\bar{1})$ to $(b,\bar{1})$,
$(a,2)$ is linked to $(c,1)$ and $(a,\bar{2})$ to $(c,\bar{1})$,
$(a,3)$ is linked to $(d,1)$ and $(a,\bar{3})$ to $(d,\bar{1})$,
$(b,2)$ is linked to $(c,2)$ and $(b,\bar{2})$ to $(c,\bar{2})$,
$(d,2)$ is linked to $(b,3)$ and $(d,\bar{2})$ to $(b,\bar{3})$,
$(d,3)$ is linked to $(c,3)$ and $(d,\bar{3})$ to $(c,\bar{3})$.
In short: bridges are built between the $1$-ports of $b/c/d$ with the $1/2/3$-ports of $a$; between the $2$-ports of $b$ and $c$; and between the $2/3$-ports of $d$ and the $3$-ports of $b/c$. $$\begin{xy}0;<1.5cm,0cm>:
0,
*+{a} ="a",
+/d3cm/, +/l1.7cm/,
*+{b} ="b",
+/r3.4cm/,
*+{c} ="c",
+/u1cm/, +/l1.7cm/,
*+{d} ="d",
\ar@{-} |{\rotatebox{60}{\makebox(10,10)[c]{$\ssc 1|1$}}} "a" ; "b",
\ar@{-} |{\rotatebox{-60}{\makebox(10,10)[c]{$\ssc 2|1$}}} "a" ; "c",
\ar@{-} |{\rotatebox{-90}{\makebox(12,10)[c]{$\ssc 3|1$}}} "a" ; "d",
\ar@{-} |{\rotatebox{0}{\makebox(12,10)[c]{$\ssc 2|2$}}} "b" ; "c",
\ar@{-} |{\rotatebox{30}{\makebox(10,10)[c]{$\ssc 3|2$}}} "b" ; "d",
\ar@{-} |{\rotatebox{-30}{\makebox(10,10)[c]{$\ssc 3|3$}}} "c" ; "d",
\end{xy}$$ $\str{B}$ consists of three decorated copies of $H_3$ (labelled $a,b,c$) and one of $\bar{H}_3$ (labelled $d$), and linked in the same manner.
We speak of the vertex sets in the individual decorated copies of $H_3$ and $\bar{H}_3$ in $\str{A}$ and $\str{B}$ as *regions $a$, $b$, $c$ and $d$.*
\[symgadgetKfourobs\]
The automorphism groups of both $\str{A}$ and $\str{B}$ are such that only the trivial automorphism fixes any three inner vertices from three distinct regions. Fixing one inner vertex each in two different regions leaves precisely one non-trivial automorphism, which swaps the two bridges between the outer vertices connecting the remaining two regions.
Any three corresponding regions in $\str{A}$ and $\str{B}$ are linked by a local isomorphism. Any local isomorphism between two corresponding regions of $\str{A}$ and $\str{B}$ has precisely two extensions to each one of the remaining two regions, which are related by a swap of the bridges between those two regions. Extensions of these assertions to structures similarly obtained from $k$-cliques for $k > 4$ are straightforward.
Property (b) can be used to show that $\str{A} \equiv_\LC^3 \str{B}$, by exhibiting a strategy for the second player that maintains a local isomorphism between the unions of the pebbled regions.
We now argue that $\str{A} \not\equiv_\LC^{<4} \str{B}$. The first player can initially force a position in which pebble $i$ in $\str{A}$ is on the (positive) port node $(a,i)$ for $i=1,2,3$; w.l.o.g. (viz., up to an automorphism of $\str{B}$) the pebbles in $\str{B}$ are on $(a,i)$ as well.
Next we let the first player move pebbles $1$ and $2$ along paths of length 2 to the ports in copies $b$ and $c$: pebble $1$ to $(b,1)$, pebble $2$ to $(c,1)$. Necessarily, over $\str{B}$, these pebbles will then also be on the nodes with these labels.
In the next two steps, we let the first player move these two pebbles along edges to the inner nodes $(b,[3])$ and $(c,[3])$ in copies $b$ and $c$. The forced responses in $\str{B}$ will put pebble $1$ to one of $(b,\{1\})$ or $(b,[3])$, and pebble $2$ to one of $(c,\{1\})$ or $(c,[3])$. The two skew combinations $((b,\{1\}),(c,[3]))$ or $((b,[3]),(c,\{1\}))$ for pebbles $1$ and $2$ are bad for the second player, because the two pebbles end up at distance greater than $3$ in $\str{B}$ while their distance in $\str{A}$ is $3$ – the first player can force a win in two rounds involving pebble $3$. Up to an automorphism of $\str{B}$ that fixes the location of the third pebble, we may therefore assume that the pebble configuration is $((b,[3]), (c,[3]), (a,3))$ in both $\str{A}$ and $\str{B}$.
In further moves along edges that take all three pebbles towards the port nodes of the $d$-regions in both $\str{A}$ and $\str{B}$, the first player forces the configuration $((d,2), (d,3), (d,1))$ in both $\str{A}$ and $\str{B}$.
Since the $d$-region of $\str{A}$ is a copy of $H_3$ while that of $\str{B}$ is a copy of $\bar{H}_3$, however, all three pebbles have edges to the inner node $(d,[3])$ in $\str{A}$, but there is no such node in $\str{B}$. So a single $\LC^{k}$-round in which the fourth pebble in $\str{A}$ is put on $(d,[3])$, lets the first player win the game (the first player can also win by moving one of the $3$ pebbles along an edge in $\str{A}$ to $(d,[3])$).
For every $k {\geqslant}3$, there are graphs $\str{A}$ and $\str{B}$ such that $\str{A} \equiv_\LC^{<k} \str{B}$ but $\str{A} \not\equiv_\LC^{k} \str{B}$.
We describe the argument for the case $k =3$, where we use variants of $\str{A}$ and $\str{B}$ as in the last proof, but with one marked inner node: in both $\str{A}$ and $\str{B}$ we mark the inner node $(a,[3])$ by a new colour (which can be eliminated by attaching a path of some characteristic length, as observed above). We denote these modified structures as $\str{A}_\ast$ and $\str{B}_\ast$. In effect this means that the second player needs to respect moves to these marked vertices as if they were coloured in both the $\LC^3$- and the $\LC^{<3}$-game: if the second player fails to pair $(a,[3])^{\str{A}_\ast}$ with $(a,[3])^{\str{B}_\ast}$, this mismatched pair would allow the first player an easy win. We claim that $\str{A}_\ast \not\equiv_\LC^3 \str{B}_\ast$ while $\str{A}_\ast \equiv_\LC^{<3} \str{B}_\ast$.
First, for $\str{A}_\ast \not\equiv_\LC^3 \str{B}_\ast$, we let the first player play the first three rounds so that the inner nodes $(b,[3])$, $(c,[3])$ and $(d,[3])$ are pebbled in $\str{A}_\ast$, and (unless the second player has lost already or will lose for trivial reasons involving the decorations) matched with inner nodes $(b,s_b)$, $(c,s_c)$ and some $(d,s_d)$ in $\str{B}_\ast$, which means that $s_b, s_c {\subseteq}[3]$ are of odd size while $s_d {\subseteq}[3]$ is of even size. It follows that between at least two pairs of nodes from $((a,[3]),(b,[3]),(c,[3]),(d,[3]))^{\str{A}_\ast}$ and $((a,[3]),(b,s_b),(c,s_c),(d,s_d))^{\str{B}_\ast}$, distances in $\str{A}_\ast$ and $\str{B}_\ast$ are different: the nodes in $\str{A}_\ast$ are at pairwise distance $3$; for at least one pair of the nodes in $\str{B}_\ast$ the distance is greater than $3$. This is easily turned into a strategy for the first player to pebble along one of these short connecting paths in $\str{A}_\ast$, alternating between two pebbles; this forces a mismatch w.r.t. to the target node, which is either pebbled (if in regions $b,c,d$) or coloured with the special marker path (if in region $a$), and in either situation the second player can be made to lose.
For $\str{A}_\ast \equiv_\LC^{<3} \str{B}_\ast$ we claim that the second player can maintain the condition that the current configuration $(a_1,a_2;b_1,b_2) \in \dom(\str{A}_\ast)^2
\times \dom(\str{B}_\ast)^2$ extends to a local isomorphism on the union of the $a$-region with those regions to which $a_1$ and $a_2$ (and thus $b_1$ and $b_2$) belong. In fact we may add an extra virtual pebble pair on $a_0 := (a,[3])^{\str{A}_\ast}$ and $b_0 := (a,[3])^{\str{B}_\ast}$ and maintain a local isomorphism $\xi$ whose domain consists of up to three out of the four regions and such that $\xi(a_i) = b_i$ for $i = 0,1,2$. Suppose, w.l.o.g., that in such a situation the first player announces play on pebble $2$, i.e., that the pair $(a_2,b_2)$ will be withdrawn at the end of the round. Let $\xi_0$ be the restriction of $\xi$ to the $a$-region and the region of $a_1/b_1$. Since $\xi_0$ covers at most two of the four regions, Observation \[symgadgetKfourobs\] guarantees extensions to local isomorphisms that
cover any one of the remaining (two or three) regions, and
at the same time respect a given pairing between the outer nodes of that new region in the direction of one of the (one or two) further remaining regions.
From such extensions, the second player can piece together a bijection that allows her to maintain the desired condition, as follows. The chosen bijection extends $\xi$ to the (one or two) regions not covered by $\xi$ as follows.
*Case 1:* $a_2/b_2$ are inner nodes or outer nodes in the direction of a region covered by $\xi_0$. Extend $\xi$ by bijections obtained from extensions of $\xi_0$ to local isomorphisms involving one extra region at a time, as in (a).
*Case 2:* $a_2/b_2$ are outer nodes in the direction of a further region not covered by $\xi_0$. Extend $\xi$ to that further region according to an extension of $\xi_0$ as a local isomorphism that also respects the neighbours of $a_2/b_2$ in this region, in the sense of (b); the extension to a remaining region (if there is such) can be completed as in Case 1.
Any bijection pieced together like this is a good choice:
- the second player does not lose during this round: any pair selected from the bijection belongs to a local isomorphism extending $\xi_0$ and thus comprising $a_0/b_0$ and $a_1/b_1$ and respects $a_2/b_2$ as well as $N(a_2)/N(b_2)$;
- the second player maintains the local isomorphism condition: the extension of $\xi_0$ by that part of the bijection that covers the region of the new pair may serve as the new $\xi'$.
It follows that the second player has a strategy to respond indefinitely, so that $\str{A}_\ast \equiv_\LC^{<3} \str{B}_\ast$.
The level of equivalence captured by the Sherali–Adams relaxation of fractional graph isomorphism of level $k-1$ is strictly between $\LC^{k-1}$-equivalence and $\LC^k$-equivalence, for every $k {\geqslant}3$.
Closing the gap {#sec:close}
---------------
As before, we let $\str A$, $\str B$ be graphs with vertex sets $[m],[n]$, respectively, and we let $A,B$ be their adjacency matrices. We use variables $X_p$ for set $p\subseteq[m]\times[n]$ of size $|p|\le
k$ For each $1 {\leqslant}j {\leqslant}k$ let $J \in \{0,1\}^{[n]^k \times [n^k]}$ be the adjacency matrix of the $j$-th accessibility relation, which relates two $k$-tuples if the coincided in all but their $j$th component. We would like to put the extra condition that $J X = X J$, where we think of the family $(X_p)_{|p|=k}$ as one square matrix with rows and columns indexed by $[m]^k$ and $[n]^k$, respectively. For a particular choice of $j$, this condition $J X = X J$ translates into $$\sum_a X_{\abar{\ssc \frac{a}{\j}} \bbar}
=
\sum_b X_{\abar \bbar{\ssc \frac{b}{\j}}},$$ for all $\abar\in [m]^k$, $\bbar \in [n]^k$, $a \in [m]$ and $b \in [n]$. For this we just observe that $J$ has entries $1$ precisely for all tuples $\abar\abar'$ and $\bbar\bbar'$ where $\abar'= \abar\frac{a}{j}$ or $\bbar'= \bbar\frac{b}{j}$ for some $a \in [m]$ or $b \in [n]$. Note also that $J$ is fully invariant under the action of permutations (of $[m]$ or $[n]$) on the $k$-tuples indexing its rows and columns.
If we look at the indices $p$ as *sets* of pairs over $[m] \times [n]$ again, we want to impose the additional requirement of permutation-invariance on the original $X$, which we had thought of as in terms of variables $X_{\abar\bbar}$ indexed by tuples of pairs over $[m] \times [n]$. Let us adopt momentarily the notation $p\!\setminus\!(j)$ for the result of removing the $j$-th pair $(a_j,b_j)$ from $p = \abar \bbar$. Then, up to the obvious permutation of the resulting tuple, $p'= \abar\frac{a}{j}\bbar\frac{b}{j} =
\abar\bbar\!\setminus\!(j)\,\widehat{\ }\,ab$. Under the natural assumption of permutation invariance, the above can now be re-written into $$\sum_a X_{p \setminus(j)\,\widehat{\ }\,ab}
=
\sum_b X_{p \setminus(j)\,\widehat{\ }\,ab},$$ for $|p| = k$, $a \in [m]$, $b \in [n]$, $1{\leqslant}j {\leqslant}k$, and hence to the familiar format $$\sum_a X_{p\,\widehat{\ }\,ab}
=
\sum_b X_{p\,\widehat{\ }\,ab},$$ for $|p| = k-1$, $a \in [m]$, $b \in [n]$, which is a consequence of Sherali–Adams equations $\CONT{\ell}$ of level $\ell = k$ (one up!).
Note that, as is always the case for the continuity equations, these conditions make no reference to the edge relations of $\str{A}$ and $\str{B}$.
We now combine the equations $\COMP{\ell}$, concerning compatibility with the edge relations, of level $\ell < k$ with the continuity equations $\CONT{\ell}$ of levels $\ell {\leqslant}k$.
We know from Corollary \[mixedlevelcor\] that $\LC^k$-equivalence of simple graphs implies the existence of a solution for exactly this combination of equations. We now want to show that conversely, a solution to $\ISO[k-1/2]$ yields a strategy for the second player in the bijective $k$-pebble game, i.e., implies $\LC^k$-equivalence.
First some preparation concerning the solution space of the equations.
\[modifysolutionlem\] Let $(X_p)_{|p| {\leqslant}k}$ be a solution and $\ell {\geqslant}1$ a natural number. Then $(\hat{X}_p)_{|p| {\leqslant}k}$ is also a solution where $\hat{X}_p$ is defined by induction on $|p|$ according to $$\barr{r@{\;:=\;}l@{\quad}l}
\hat{X}_\emptyset & X_\emptyset = 1
\\
\hnt
\hat{X}_{p\,\widehat{\ }\,ab} & X_{p\,\widehat{\ }\,ab} = 0 &
\mbox{ if $X_p =0$,}
\\
\hnt
\hat{X}_{p\,\widehat{\ }\,ab} &
\hat{X}_p \bigl((YY^t)^\ell Y)\bigr)_{ab}
&
\mbox{ if $X_p > 0$ and $Y_{ab} := 1/X_p \; X_{p\,\widehat{\ }\,ab}$.}
\earr$$
W.l.o.g. $[m] = [n] = [n]$. The continuity equations involving the matrix $X = (X_{p\,\widehat{\ }\,ab})_{a,b \in [n]}$ are preserved under multiplication from the left by arbitrary doubly stochastic matrices of the form $Z = (Z_{aa'})_{a,a' \in [n]}$ (just as under multiplication from the right by some doubly stochastic $Z' = (Z_{bb'})$). We note that $Z = YY^t = XX^t/X_p^2$ is doubly stochastic by the given continuity equations for $X = (X_{p\,\widehat{\ }\,ab})_{a,b \in [n]}$. It follows that $\hat{X} =
(\hat{X}_{p\,\widehat{\ }\,ab})_{a,b\in[n]}$ satisfies the new continuity equations which stipulate row- and column-sums $\hat{X}_p$ for this matrix.
For consistency with the view of the $\hat{X}_p$ as being indexed by *sets* of pairs, we need to verify that $\hat{X}_{p\widehat{\
}ab} = \hat{X}_p$ whenever $ab \in p$. But in that case, already $X_{p\,\widehat{\ }\,ab} = X_p$ was the only non-zero entry in the row of $a$ as well as in the column of $b$, by the given continuity equations. It follows that $(YY^t)_{aa'}
= 1/X_p^2 \, \sum_{b'} X_{p\,\widehat{\ }\,ab'} X_{p\,\widehat{\ }\,a'b'} =
(1/X_p) X_{p\,\widehat{\ }\,a'b} = \delta(a',a)$ so that also $\hat{X}_{p\,\widehat{\ }\,ab} = \hat{X_p}/X_p X_{p\,\widehat{\ }\,ab} =
\hat{X}_p$ as desired.
Compatibility equations of the form $A X = XB$ are compatible with multiplication of $X$ by $Z = XX^t$ from the left, whenever $A$ and $B$ are symmetric. We note that the adjacency matrices $A$ and $B$ commute with $Z = XX^t$ and $Z' = X^tX$ by the given compatibility equations.
Passage from $X = X_{p\,\widehat{\ }\,ab}$ and $X_p$ to $\hat{X}_{p\,\widehat{\ }\,ab}$ and $\hat{X}_p$ therefore preserves all equations of level $k-1/2$.
For the following recall the definition of a *good* matrix without null rows or columns from Definition \[goodsymmatrixdef\].
\[goodsoldef\] Call a solution $(X_p)_{|p| {\leqslant}k}$ a *good solution* if, whenever $X_p >0$ for $|p| < k$, the associated doubly stochastic matrix $Y = (1/X_p) (X_{p\,\widehat{\ }\,ab})_{a, b \in [n]}$ is good in the sense of Definition \[goodsymmatrixdef\].
If there is any solution $(X_p)_{|p| {\leqslant}k}$, then there is a good solution. For a good solution $(X_p)_{|p| {\leqslant}k}$ and $|p| < k$ with $X_p \not= 0$, let $Y$ be the doubly stochastic matrix $Y := (X_{p\,\widehat{\ }\,ab})_{a,b\in [n]}$. Then the symmetric matrices $Z = YY^t$ and $Z' = Y^tY$ are good symmetric and induce $Y/Y^t$-related partitions $[n] = \dot{\bigcup}_i D_i$ and $[n] = \dot{\bigcup}_i D_i'$, and satisfy
$Y_{D_iD_i'} >0$ for all $i$, and
$Y_{D_iD_j'} = 0$ for all $i \not= j$.
An application of Lemma \[modifysolutionlem\] with $\ell {\geqslant}n-1$ produces a good solution from any given solution. The stated property for doubly stochastic matrices $Y$ induced by a good solution follows directly from Corollary \[goodlinkcor\].
\[solutiontokequivlem\] If $(X_{p})_{|p| {\leqslant}k}$ is a good solution to $\ISO[k-1/2]$, then for all $\abar \in [n]^{k}$ and $\bbar \in [n]^{k}$: $$X_{\abar\bbar} > 0 \;
\quad \Longrightarrow \quad
\str{A},\abar \equiv_\LC^{k} \str{B},\bbar.$$
For $k=2$, we know that even the level $1$ equations suffice. We therefore assume that $k {\geqslant}3$ and fix a good solution $(X_{p})_{|p| {\leqslant}k}$. From the proof of Lemma \[locisolem\] (a) we know that $X_p > 0$ implies that $p = \abar \bbar$ is a local bijection, while (b) from Lemma \[locisolem\] tells us that at least every $p = \abar\bbar$ of size up to $k-1$ with $X_p>0$ must be a local isomorphism. But for $k {\geqslant}3$ and simple graphs the local isomorphism property for all $p$ of size $k-1$ with $X_p >0$ implies the same for $p = \abar\bbar$ of size $k$, essentially by monotonicity as imposed by the continuity equations:
Let $p = \abar\bbar$, $|p| = k$, $X_p > 0$. Then the continuity equations imply that $X_{q} > 0$ for every restriction $q {\subseteq}p$. E.g., if $p = q\,\widehat{\ }\,ab$, then $X_{q} = \sum_{a'} X_{q\,\widehat{\ }\,a'b} {\geqslant}X_{q\,\widehat{\ }\,ab}
= X_p$ shows that $X_{q} > 0$. But if every size $k-1$ restriction of $p$ is a local isomorphism, then so is $p$ itself. Here it is important that $k {\geqslant}3$ is strictly larger than the arity of the relations in $\str{A}$ and $\str{B}$, the edge relation in this case.
It remains to argue that the second player has a strategy to maintain the condition that $X_p > 0$ in the bijective $k$-pebble game on positions $p = \abar\bbar$. I.e., for a single round played from a position $p\,\widehat{\ }\,ab$, where $p = \abar\bbar$ is of size up to $k-1$:
> if $X_{p\,\widehat{\ }\,ab} > 0$, then there is a bijection $\rho$ between $[m]$ and $[n]$\
> such that for every pair $(a',b') \in \rho$, also $X_{p\,\widehat{\ }\,a'b'} > 0$.
Fix $p$ of size up to $k-1$ with $X_p >0$, and let $Y$ be the doubly stochastic matrix with entries $Y_{ab} := (1/X_p) X_{p\,\widehat{\ }\,ab}$. Let $Z := YY^t$ and $Z' := Y^tY$ be the induced symmetric doubly stochastic matrices with positive entries on the diagonal as considered in Lemma \[newdecomplem\]. Let $[n] = \dot{\bigcup}_{i \in I} D_i$ and $[n] = \dot{\bigcup}_{i\in I} D_i'$ be the $Y$-related partitions of $[n]=[m]$ induced by $Z$ and $Z'$, respectively. Then $|D_i| = |D_i'|$ for all $i \in I$ by Lemma \[stochrelatedpartlem\]. So there is a bijection that is compatible with these two partitions in the sense that it associates $D_i$ bijectively to $D_i'$. Any choice of a pair $(a',b')$ from such a bijection corresponds to an entry $X_{p\,\widehat{\ }\,a'b'}$ with $a'\in D_i$ and $b'\in D_i'$ for the same $i \in I$; because $(X_p)$ is a good solution, this means that the corresponding entry $Y_{a'b'}$ is positive, i.e., that $X_{p\,\widehat{\ }\,a'b'}>0$, as desired. So, if the second player chooses such a bijection, the resulting position $p'$ is guaranteed to have positive $X_{p'}$ again.
The following theorem sums up the results of the previous lemmas and should be compared to Theorem \[SAkminus1prop\] for $\ISO[k-1]$.
\[SAkminus1/2prop\] $\ISO[k-1/2]$ has a solution if, and only if, $\str A\equiv_\LC^{k}\str B$.
Boolean arithmetic and $\LL^k$-equivalence {#boolLksec}
------------------------------------------
We saw in Section \[boolfracisosec\] that equations, which are direct consequences of the basic continuity and compatibility equations w.r.t. the adjacency matrices $A$ and $B$, may carry independent weight in their boolean interpretation. This is no surprise, because the boolean reading is much weaker, especially due to the absorptive nature of $\vee$, which unlike $+$ does not allow for inversion. $AX = XB$ for doubly stochastic $X$ and $A,B \in \B^{n,n}$ implies $A^c X = X B^c$. Similarly, we found in part (a) of Lemma \[locisolem\] that the continuity equations imply that $p$ is a local bijection whenever $X_p>0$, under real arithmetic; this also fails for boolean arithmetic.
We now augment the boolean requirements by corresponding boolean equations that express
compatibility also w.r.t. $A^c$ and $B^c$, as in boolean fractional isomorphism,
the new constraint $X_p = 0$ whenever $p$ is not a local bijection.
In the presence of the continuity equations, which force monotonicity, it suffices for (b) to force $X_{aa'bb'} = 0$ for all $a,a'\in [m]$, $b,b' \in [n]$ such that *not* $a=a' \;\Leftrightarrow\; b=b'$. This is captured by the constraint $\MATCH2$ below. Together with the continuity and compatibility equations, $\MATCH2$ then implies that $X_p = 0$ unless $p$ is a local isomorphism, just as in the proof of part (b) of Lemma \[locisolem\], also in terms of boolean arithmetic.
So we now use the following boolean version of the Sherali–Adams hierarchy $\ISO[k-1]$ and $\ISO[k-1/2]$ for $k\ge 2$.
For $\BISO[k-1/2]$ we just require $\CONT{\ell}$ for all $\ell {\leqslant}k$, i.e., additionally for $\ell = k$.
$\BISO[k-1]$ and $\BISO[k-1/2]$ are systems of boolean equations, and the reader may wonder whether they can be solved efficiently. At first sight, it may seem -complete to solve such systems (just like boolean satisfiability). However, our systems consist of “linear” equations of the following forms: $$\begin{aligned}
\label{eq:be1}
\sum_{i\in I}X_i=\sum_{j\in J}X_j,\\
\label{eq:be2}
\sum_{i\in I}X_i=0,\\
\label{eq:be3}
\sum_{i\in I}X_i=1.\end{aligned}$$ (The equations of the form are actually subsumed by those of the form with $J=\emptyset$.) It is an easy exercise to prove that such systems of linear boolean equations can be solved in polynomial time.
The *weak* $k$-pebble game is the straightforward adaptation of the weak bijective $k$-pebble game to the setting without counting. A single round of the game is played as follows.
1. If $|p|=k-1$, player selects a pair $ab\in p$. If $|p|<k-1$, he omits this step.
2. Player chooses an element $a'$ of $\str A$ or $b'$ of $\str B$.
3. Player answers with an element $b'$ of $\str B$ or $a'$ of $\str A$, respectively.
4. If $p^+:=p\,\widehat{\ }\,a'b'$ is a local isomorphism then the new position is $$p':=
\begin{cases}
\;\; (p\!\setminus\! ab)\,\widehat{\ }\,a'b'&\text{ if }|p|=k-1,\\
\hnt
\;\;\, p \,\widehat{\ }\,a'b'&\text{ if }|p|<k-1.
\end{cases}$$ Otherwise, the play ends and player loses.
We denote weak $k$-pebble equivalence as in $\str{A}\equiv_\LL^{<k}
\str{B}$ and extend this to $\str{A},\abar \equiv_\LL^{<k} \str{B},\bbar$ for tuples ${\mathbf{a}},{\mathbf{b}}$ of the same length $<k$. We sketch a proof of the following, which is a boolean analogue of the correspondences between half-step levels of Sherali–Adams and $\LC^k$- and $\LC^{<k}$-equivalence established in the last section.
\[boolthm\]
$\BISO[k-1]$ has a solution (w.r.t. boolean arithmetic) if, and only if, $\str A\equiv_\LL^{<k}\str B$.
$\BISO[k-1/2]$ has a solution (w.r.t. boolean arithmetic) if, and only if, $\str A\equiv_\LL^{k}\str B$.
We start with the second part of the theorem. For the backward direction, suppose that $\str{A} \equiv_\LL^k \str{B}$ and let, for $p
= \abar \bbar$ of size $|p| {\leqslant}k$, $$X_p := \left\{ \barr{ll}
1 & \mbox{if } \str{A},\abar \equiv_\LL^k \str{B},\bbar,
\\
\hnt
0 &\mbox{else.}
\earr\right.$$
Clearly this assignment satisfies $\MATCH2$, and one easily checks that it also satisfies the boolean continuity equations $\CONT{\ell}$ for $\ell
{\leqslant}k$. For the boolean compatibility equations $\COMP{\ell}$ and $\COMP{\ell}^c$ for $\ell < k$, let us check, for instance, an equation $\COMP{k-1}$. The non-trivial case is that of $p = \abar\bbar$ where $\abar \in [n]^{k-2}$ and $\bbar \in [m]^{k-2}$ are such that $\str{A},\abar \equiv_\LL^k
\str{B},\bbar$ so that $X_p =1$. Consider the instance of equation $\CONT{k-1}$ for $a\in[m],b\in[n]$: $$\sum_{a'} A_{aa'} X_{p\,\widehat{\ }\, a'b} =
\sum_{b'} X_{p\,\widehat{\ }\, ab'} B_{b'b}.$$ Since $\str{A},\abar \equiv_\LL^k \str{B},\bbar$, there is a $\hat{b} \in [m]$ such that $\str{A}, \abar a \equiv_\LL^k \str{B},\bbar \hat{b}$. Suppose the left-hand side of the equation evaluates to $1$. This means that there is an edge in $\str{A}$ from $a$ to some $a'$ for which $X_{p\,\widehat{\ }\,a'b} = 1$, i.e., for which $\str{A},\abar a' \equiv_\LL^k \str{B},\bbar b$.
In other words, there is an edge in $\str{A}$ from some $a'$ for which $\str{A},\abar a' \equiv_\LL^k \str{B},\bbar b$ to some $a$ for which $\str{A},\abar a \equiv \str{B},\bbar \hat{b}$. So every realisation of the $\LL^k$-type of $\str{B},\bbar b$ has an edge between $b$ and some $b'$ where $\str{B},\bbar b' \equiv_\LL^k \str{A},\abar a$, which implies that the right-hand side of the equation evaluates to $1$, too.
For the forward direction, we extract from a good solution to $\ISO[k-1/2]$ a strategy for the second player to maintain pebbles in configurations $\abar\bbar$ for which $X_{\abar\bbar} =1$. Let $p = \abar\bbar$ of size $|p| < k$ be such that the current position is $p\;\,\widehat{\ }\,ab$, where $X_{p\,\widehat{\ }\,ab} = 1$, and let the first player move the pebble from $a$ to $a'$ in $\str{A}$, say. The second player needs to find some $b'$ in $\str{B}$ such that again $X_{p\,\widehat{\ }\,a'b'} = 1$.
Analogous to the real case, a *good solution* is one where all the matrices $Y_{ab} = X_{p\,\widehat{\ }\,ab}$ for $X_p \not= 0$ (which are necessarily without null rows or columns), are such that the $Y$-related partitions of $[n]$ and $[m]$ induced by $Z = YY^t$ and $Z'=Y^tY$ put $a$ and $b$ in related partition sets if, and only if, $Y_{ab}=1$. Here again, it suffices to postulate that $Z$ and $Z'$ are good symmetric, and an arbitrary solution can be turned into a good solution on the basis of Lemma \[summaryboollem\].
So, if the second player picks any $b' \in [m]$ for which $Y_{a'b'} = 1$, this makes sure that $X_{p\,\widehat{\ }\,a'b'} = 1$ is maintained, and that $p\;\,\widehat{\ }\,a'b'$ is a local isomorphism.
We turn to part (a) of the theorem.
For the forward direction, we want to extract from a good (!) solution a strategy for the player in the weak $k$-pebble game that maintains positions $p$ s.t. $X_p = 1$. Recall that positions are now of size $<k$. Consider a position $p=\abar a \bbar b$ of size $|p|=k-1$ with $X_p = 1$. Suppose that player selects $ab$ in the first step of the next round (“he announces to withdraw the pebble pair on $ab$”) and selects $a'$ in the second step (“he places a pebble on $a'$”). W.l.o.g. suppose that $a' \not= a$, because $a' = a$ has the obvious response $b' := b$. Player needs to find some $b' \in [m]$ such that $p^+ = p\,\widehat{\ }\, a'b'$ is a local isomorphism and for $p'=(p\setminus ab)\,\widehat{\ }\, ab$ also $X_{p'} = 1$. It suffices to make sure that $X_{p'} = 1$, that $b' \not= b$, and that $bb'$ is an edge in $\str{B}$ iff $aa'$ is an edge in $\str{A}$. Suppose, for instance, that $aa'$ is not an edge, i.e., that $A_{aa'} = 0$.
Let $p^-=p\setminus ab$, and let $Y$ be the matrix with entries $Y_{a''b''} = X_{p^-\,\widehat{\ }\,a''b''}$. Let $[m] = \dot{\bigcup}_i D_i$ and $[n] = \dot{\bigcup}_i D_i'$ the partitions induced by $YY^t$ and $Y^tY$, which are boolean bi-stable for $A$ and $B$, respectively, and $Y$-related. Suppose $a \in D_i$ and $a' \in D_j$, so that $A_{aa'} = 0$ implies that $\iota_{ij}^{A^c} =1$. Since $Y_{ab} = 1$, we have $b \in D_i'$; and as $\iota_{ij}^{B^c} = \iota_{ij}^{A^c} = 1$, there is also some $b' \in D_j'$ for which $B^c_{b,b'} = 1$, so that $bb'$ is a not an edge in $\str{B}$ and $b' \not= b$. As we are dealing with a good solution, $a' \in D_j$ and $b' \in D_j'$ imply that $Y_{a'b'} = 1$, so that $X_{p'} =X_{p^-\,\widehat{\ }\,a'b'}= 1$.
For the backward direction, suppose that $\str{A} \equiv_{<k}
\str{B}$ and let $X_\emptyset := 1$ and, for $\abar \in [n]^{k-1}$ and $\bbar \in [m]^{k-1}$, $X_{\abar \bbar} := 1$ iff $\str{A},\abar \equiv_\LL^{<k} \str{B},\bbar$. It is clear that this assignment satisfies $\MATCH2$ and $\CONT{\ell}$ for $\ell < k$. Consider then an instance of $\COMP{\ell}$ for $\ell < k$, $$\label{eq:bcompl}
\sum_{a'} A_{aa'} X_{p\,\widehat{\ }\, a'b} =
\sum_{b'} X_{p\,\widehat{\ }\, ab'} B_{b'b},$$ where $a \in [n], b \in [m]$, $|p| < k-1$, $p = \abar \bbar$. Let us assume that $|p|=k-2$; this is the most difficult case. If $X_p=0$ then $X_{p\,\widehat{\ }\,a'b'}=0$ for all $a'b'$, and thus equation is trivially satisfied. So assume $X_p=1$, that is, $\str{A},\abar \equiv_\LL^{<k} \str{B},\bbar$. Suppose for instance that the left-hand side of equation evaluates to $1$, i.e., that there is some $a'$ adjacent to $a$ in $\str A$ for which $\str{A}, \abar a' \equiv_\LL^{<k} \str{B},\bbar b$. Consider the weak $k$-pebble game in position ${\mathbf{a}}a'{\mathbf{b}}b$. Assume player selects the pair $a'b$ in the first step of the next round and selects $a$ in the second step. Let $b'$ be the answer of when she plays according to her winning strategy. Then ${\mathbf{a}}a'a\mapsto{\mathbf{b}}bb'$ is a local isomorphism and the new position ${\mathbf{a}}a{\mathbf{b}}b'$ is a winning position for player , that is, $\str A,{\mathbf{a}}a\equiv_\LL^{<k}\str B,{\mathbf{b}}b'$. Since ${\mathbf{a}}a'a\mapsto{\mathbf{b}}bb'$ is a local isomorphism and $aa'$ is an edge of $\str A$, the pair $bb'$ is an edge of $\str B$ and thus $B_{b'b}=1$. Since $\str A,{\mathbf{a}}a\equiv_\LL^{<k}\str B,{\mathbf{b}}b'$, we have $X_{p\widehat{\
}ab'}=1$. Thus the right-hand side of equation evaluates to $1$ as well.
\[seprem\] For all $k {\geqslant}3$, $\equiv_\LL^{k-1}$, $\equiv_\LL^{<k}$, $\equiv_\LL^{k}$ form a strictly increasing hierarchy of discriminating power.
The examples for the gaps between $\equiv_\LC^{k-1}$, $\equiv_\LC^{<k}$, $\equiv_\LC^{k}$ given in Section \[gapsec\], are in fact good in the setting without counting. The strategy analysis given there does not involve counting in any non-trivial manner.
#### Acknowledgements {#acknowledgements .unnumbered}
We are most grateful to Albert Atserias for his valuable comments on an earlier draft of this exposition.\
The second author gratefully acknowledges the academic hospitality in the first author’s group at HU Berlin during his recent sabbatical.
[^1]: The dimensions of the WL algorithm are counted differently in the literature; what we call “$k$-dimensional” here is sometimes called “$(k-1)$-dimensional”.
[^2]: Here we regard two partitions as identical if they have the same partition sets, i.e., we ignore their indexing/enumeration.
[^3]: Note that this does not depend on the enumeration of the partition set $D_i$, because irreducibility is invariant under permutation-similarity.
[^4]: As $X/X^t$-relatedness refers to partitions presented with an indexing of the partition sets, we need to allow a suitable re-indexing for at least one of them, so as to match the other one.
[^5]: Note that $\abar\bbar = p\,\widehat{\ }\, ab$ just specifies the sub*set* $p$ consisting of pairs in $\abar \mapsto \bbar$ other than $(a,b)$; our notation should not wrongly suggest an appeal to ordered tuples.
|
---
abstract: 'We discuss power corrections to infrared safe cross sections and event shapes, and identify a nonperturbative function that governs $1/Q$ corrections to these quantities.'
---
ø
0.0in 0.0in 6.in 9.in -1.9 cm
ITP-SB-97-41\
LPTHE-ORSAY 97/40
[**Power Corrections\
and Nonlocal Operators[^1]**]{}
Gregory P. Korchemsky\
\
Gianluca Oderda and George Sterman\
\
April 1997
Introduction
============
Nonperturbative corrections to infrared safe jet cross sections and event shapes are an important issue in the study of QCD. They are the natural starting point for a unified perturbative-nonperturbative treatment of high energy cross sections, because they enter at the level of nonleading powers of the hard momentum scale, $Q$.
For example, a classic analysis [@Mueller] of the total cross section for ${\rm e}^+{\rm e}^-$ annihilation to hadrons identifies terms $\alpha_s^n(Q^2)b_2^n\; n!$ at $n$th order, which may be attributed to the infrared (IR) behavior of the QCD running coupling, $\alpha_s(k^2)=4\pi/[b_2\ln(k^2/\Lambda^2)]$. Borel analysis shows that this nonconvergence is due to an ambiguity in the cross section at $Q^{-4}$ relative to leading behavior. This ambiguity is in direct correspondence with the contribution of the gluon condensate to the operator product expansion (OPE). The OPE, however, is available in only a few cases. Nevertheless, we would like to abstract from these considerations a method of “substitution". That is, we shall assume that the contribution of [*any*]{} region of momentum space where the contour integrals of perturbation theory (PT) are trapped by mass-shell and IR singularities is a source of nonperturbative corrections, whose power suppression may be estimated from PT itself. We outline a general procedure, which we shall briefly illustrate below.
Let $\sigma$ an be an IR safe cross section at large scale $Q$: [(i)]{} Identify regions $R$ in momentum space where lines are pinched on-shell in $\sigma$, by use of Landau equations or an equivalent method. [(ii)]{} Organize logarithms of momenta $k$ that occur in $R$ into: (a) $\alpha_s\left(f(k)\right)$, with $f(k)$ a characteristic momentum scale, and/or (b) explicit kinematic integrals. [(iii)]{} Introduce a cutoff $\kappa$ on (some) components: $k^\nu<\kappa$, to define the contribution $\sigma_R$ from region $R$, such that $\alpha_s\left(f(k)\right)>
\alpha_0$, $\alpha_0$ fixed. [(iv)]{} With the coupling fixed, evaluate the power behavior $\sigma_R \sim Q^{-2-m}\kappa^m$. [(v)]{} Find a “universal" matrix element $\langle {\cal O} \rangle$, of dimension $m$, whose perturbative expansion is identical to that for $R$. [(vi)]{} Remove $\sigma_R$ from $\sigma$, replacing it with $\langle {\cal O} \rangle$, $\sigma= \sigma^{({\rm reg})}(\kappa)
+{\langle {\cal O} \rangle(\kappa)/ Q^{-2-m}}\, .
$Of these steps, items (ii) and (v) require spectial treatment on a case-by-case basis. Nevertheless, this approach includes the analysis of infrared renormalons [@universal; @KoSt], and represents, we believe, a somewhat more general viewpoint.
Power Corrections in Event Shapes
=================================
As an application, we consider the perturbative expansion for infrared safe event shapes, such as the thrust $T$, close to $T=1$, the limit of two perfectly narrow jets. The leading behavior for a large class of such event shapes $w$ is $1/w$ (times logs of $w$) in this limit. Keeping only the $1/w$ terms, the program outlined above may be carried out explicitly. To be specific, we shall assume that to the power $1/w$, the weight factorizes into contributions from individual particles, w(k) = \_[particles [*i*]{}]{} w(k\_i/Q)=\_[particles [*i*]{}]{} [k\_i\^0Q]{}f\_w(cos \_i) , \[wsum\] for some function $f_w$ of the cosine of the angle of particle $i$ to the two-jet axis. In the case of thrust, $f_{1-T}=(1-|cos\; \theta_i|)$.
In the two-jet limit, the differential cross section $d\sigma/dw$ factorizes into functions describing the internal evolutions of the jets convoluted in $w$ with a “two-jet" soft-gluon function $\sigma_2(Q,w)$. $\sigma_2$ is the eikonal approximation to the cross section at fixed weight $w$, in which the jets are replaced by oppositely-moving, lightlike Wilson lines [@KoSt]. For the $1/w$ contributions, the directions of the Wilson lines may be considered as fixed. We now apply our reasoning to $\sigma_2$. Suitably defined, the jet functions give nonleading power corrections.
Our first observation is that $\sigma_2(Q,w)$ is a convolution, \_[2]{}(Q,w) = \_[n=0]{}\^ dw\_1…dw\_n (w-\_[i=1]{}\^n w\_i) \_[i=1]{}\^n S(Q,w\_i) , \[sig2Jconv\] in terms of a kernel, $S(Q,w)$. The convolution fixes the weights of final states that contribute to $\sigma_2(Q,w)$. Then, if the weight function satisfies Eq. (\[wsum\]), the Laplace transform of $\sigma_2(Q,w)$ exponentiates (up to small corrections, which we suppress), \_[2]{}(Q,N) = \_0\^[w\_[max]{}]{} dw e\^[-Nw]{}[\_[2]{}(Q,w)]{}= . \[sigexp\] By construction of $w$, this transform is infrared finite. The perturbative expansion of the kernel $S$ in Eq. (\[sig2Jconv\]) is identical to a sum of two-particle irreducible diagrams called “webs" long ago [@GFT]. Webs have a number of important properties. First, they require only a single, additive UV renormalization, corresponding to multiplicative renormalization for $\sigma_{2}$. Second, they give rise to only a single overall collinear and infrared logarithm each, aside from logarithms which may be organized into the running of the coupling. These conditions are summarized in the integral representation, $$\begin{aligned}
{\tilde S}(Q,N)
&=&
\int_0^{Q^2}{dk^2\over k^2}
\int_0^{Q^2-k^2} {dk_T^2\over k^2+k_T^2}\;
\nonumber \\
&\ & \quad \times
\int_{\sqrt{k^2+k_T^2}}^Q{dk_0\over \sqrt{k_0^2-k^2-k_T^2}}\;
\gamma_w \left ( {k\over \mu},{\mu\over Q},\alpha_s(\mu),N\right )\, ,
\label{stilde}\end{aligned}$$ where $k$ represents $k_0$, $k_T$ and $\sqrt{k^2}$. The combination ${\gamma_w(k,N)/[k^2(k^2+k_T^2)]}$ is an integrable distribution for $k^2,k_T^2\rightarrow 0$. The two overall logarithms of $N$ are generated from the $k$ integrals. In Eq. (\[stilde\]), we have implemented items (i) and (ii) in the method of substitution above. To identify the explicit forms of power corrections in $Q$, we expand $\gamma_w$ in $Q$ at fixed values of $N$. The precise form of dependence on soft regions in (\[stilde\]), and the corresponding substitutions (see above), depend on the weight, but are simplified by the web structure, leading to clear sources of power corrections.
For $w=1-T$, the expansion of $\gamma_{1-T}$ gives = [NQ]{} (k\^2) 2C\_F [\_s(k\_T)]{} (k\_0-) +[O]{}(N\^2/Q\^2) , \[gammaexpand\] where we have used the renormalization-group invariance of $\gamma_{1-T}$ to set $\mu=k_T$. After the $k_0$ integral in (\[stilde\]), we find an exponentiating $1/Q$ correction, which multiplies the (infrared regulated) perturbative result, \_[2]{}\^[(1-T,[corr]{})]{}(Q,N) \~ , \[exp1overQ\] where we have introduced, as above, a new scale $\kappa$ to isolate the infrared-sensitive region around $k_T=0$. This, of course, is a typical infrared renormalon, now in the exponent [@universal; @KoSt].
We now ask if it is possible to give an operator interpretation to this result, and thus to “substitute" a nonperturbative matrix element for the IR region of PT, leading to a universal quantity that controls $1/Q$ behavior [@universal; @KoSt]. To construct this matrix element, we define operators that measure the energy that arrives over time at a sphere “at infinity", () = \_[|[y]{}|]{}\_0\^ [dy\_0(2)\^2]{} ||\^2 \_i \_[0i]{}(y\^) , \[thetadef\] with $\theta_{\mu\nu}$ the energy-momentum tensor, and $\hat{y}$ a unit vector. The energy density that flows in direction $\hat{y}$ for $\sigma_{2}(Q,w)$ is () = 0W\^\_[v\_1v\_2]{}(0) () W\_[v\_1v\_2]{}(0)0 , \[calEdef\] where $W_{v_1v_2}(0)$ is the product of outgoing Wilson lines in the $v_i$ directions, joined by a color singlet vertex at the origin. In these terms, the leading power correction, Eq. (\[exp1overQ\]), is of the general form \_[2]{}\^[(w,[corr]{})]{}(Q,N) = [NQ]{} f\_w()() , with $f_w$ the function in Eq. (\[wsum\]). $1/Q$ corrections for a wide class of event shapes are thus generated from the nonperturbative function ${\cal E}(\hat{y})$, the matrix element of a nonlocal operator, which describes the nonperturbative component of energy flow, associated with two-jet color flow. Generalizations to multijet cross sections are possible. We anticipate that this approach will help to unify the treatments of power corrections for a variety of infrared-safe quantities.
[*Acknowledgement*]{}. This work was supported in part by the National Science Foundation, under grant PHY9309888. We would like to thank R. Akhoury, Eric Laenen and Nikolaos Kidonakis for helpful conversations.
[999]{}
A.H. Mueller [*Nucl. Phys*]{}. [**B250**]{}, 327 (1985); in [*QCD 20 years later*]{}, Eds. P.M. Zerwas and H.A. Kastrup, World Scientific, Singapore, 1993 pp. 162-171.
A.V. Manohar and M.B. Wise, [*Phys. Lett*]{}. [**B344**]{}, 407 (1995); Yu.L. Dokshitser, G. Marchesini and B.R. Webber, hep-ph/9512336; R. Akhoury and V.I. Zakharov, [*Phys. Rev. Lett*]{}. [**76**]{}, 2238 (1996); P. Nason and M.H. Seymour, [*Nucl. Phys*]{}. [**B454**]{}, 291 (1995); M. Beneke and V.M. Braun, [*Nucl. Phys*]{}. [**B454**]{}, 253 (1995); Yu.L. Dokshitser and B.R. Webber, hep/ph-9704298.
G.P. Korchemsky and G. Sterman, [*Nucl. Phys*]{}. [**B437**]{}, 415 (1995); and hep-ph/9505391, in [*QCD and High Energy Hadronic Interactions*]{}, Ed.J. Trân Thanh Vân, Editions Frontieres, Gif-sur-Yvette, 1995, pp. 383-392.
J.G.M. Gatheral, [*Phys. Lett*]{}. [**B133**]{}, 90 (1983); J. Frenkel and J.C. Taylor, [*Nucl. Phys*]{}. [**B246**]{}, 231 (1984).
[^1]: Presented at the Fifth International Workshop on Deep Inelastic Scattering and QCD, Chicago, IL, April 14-18, 1997.
|
---
abstract: 'We show how to associate an $\mathbb{R} $–tree to the set of cut points of a continuum. If $X$ is a continuum without cut points we show how to associate an $\mathbb{R} $–tree to the set of cut pairs of $X$.'
address:
- |
Mathematics Department\
University of Athens\
Athens 157 84\
Greece
- |
Mathematics Department\
Brigham Young University\
Provo UT 84602\
USA
author:
- Panos Papasoglu
- Eric Swenson
bibliography:
- 'link.bib'
title: 'From continua to $\mathbb{R}$–trees'
---
We show how to associate an R-tree to the set of cut points of a continuum. If X is a continuum without cut points we show how to associate an R-tree to the set of cut pairs of X.
We show how to associate an <b>R</b>–tree to the set of cut points of a continuum. If X is a continuum without cut points we show how to associate an <b>R</b>–tree to the set of cut pairs of X.
Introduction
============
The study of the structure of cut points of continua has a long history. Whyburn [@WHY] in 1928 showed that the set of cut points of a Peano continuum has the structure of a “dendrite”. This “dendritic” decomposition of continua has been extended and used to prove several results in continua theory.
We recall here that a continuum is a compact, connected metric space and a Peano continuum is a locally connected continuum. If $X$ is a continuum we say that a point $c$ is a cut point of $X$ if $X-\{c \}$ is not connected.
Continua theory became relevant for group theory after the introduction of hyperbolic groups by Gromov [@GRO]. The Cayley graph of a hyperbolic group $G$ can be “compactified” and if $G$ is one-ended its Gromov boundary $\partial G$ is a continuum. Moreover the group $G$ acts on $\partial G$ as a convergence group. It turns out that algebraic properties of $G$ are reflected in topological properties of $\partial G$. A fundamental contribution to the understanding of the relationship between $\partial G$ and algebraic properties of $G$ was made by Bowditch. In [@BOW1] Bowditch shows how to pass from the action of a hyperbolic group $G$ on its boundary $\partial G$ to an action on an $\mathbb{R}$–tree. The construction of the tree (under the hypothesis of the $G$–action) from the continuum is similar to the dendritic decomposition of Whyburn. The difficulty here comes from the fact that the continuum is not assumed to be locally connected.
The second author in [@SWE] (see also [@SWE3]) explained how to associate to any continuum a “regular big tree” $T$, and conjectured that $T$ is in fact an $\mathbb{R}$–tree. It is this conjecture that we prove in the first part of this paper.
Let $X$ be a continuum without cut points. If $a,b\in X$ we say that $a,b$ is a [*cut pair*]{} if $X-\{a,b \}$ is not connected. In the second part of this paper we show how to associate an $\mathbb{R}$–tree to the set of cut pairs of $X$ (compare [@BOW6]). We call this tree a JSJ-tree motivated by the fact that if $G$ is a one-ended hyperbolic group then the tree associated to $\partial G$ by this construction is the tree of the JSJ-decomposition of $G$ (in this case one obtains in fact a simplicial tree). Continua appear in group theory also as boundaries of ${\rm CAT}(0)$ groups. In [@P-S] we use the construction of $\mathbb{R}$–trees from cut pairs presented here to extend Bowditch’s results on splittings [@BOW1] to ${\rm CAT}(0)$ groups. We show in particular that if $G$ is a one-ended ${\rm CAT}(0)$ group such that $\partial G$ has a cut pair then $G$ splits over a virtually cyclic group.
### Acknowledgements {#acknowledgements .unnumbered}
We would like to thank the referee for many suggestions that improved the exposition and for correcting a mistake in the proof of .
This work is co-funded by European Social Fund and National Resources (EPEAEK II) PYTHAGORAS.
Preliminaries
=============
Let ${{\mathcal P}}$ be a set with a betweenness relation. If $y$ is between $x$ and $z$ we write $xyz$. ${{\mathcal P}}$ is called a pretree if the following hold:
1\. There is no $y$ such that $xyx$ for any $x\in {{\mathcal P}}$.
2\. $xzy \Leftrightarrow yzx$.
3\. For all $x,y,z$, if $y$ is between $x$ and $z$ then $z$ is not between $x$ and $y$.
4\. If $xzy$ and $z\ne w$ then either $xzw$ or $yzw$.
We say that a pretree ${{\mathcal P}}$ is discrete if for any $x,y\in {{\mathcal P}}$ there are finitely many $z\in {{\mathcal P}}$ such that $xzy$.
A compact connected metric space is called a continuum.
Let $X$ be a topological space. We say that a set $C$ *separates* the nonempty sets $A,B \subset X $ if there are disjoint open sets $U,V$ of $X-C$, such that $A\subset U$, $B \subset V$ and $U \cup V =X- C$. We say $C$ separates the points $a,b \in X$ if $C$ separates $\{a\}$ and $\{b\}$. We say that $C$ separates $X$ if $C$ separates two points of $X$. If $C=\{c\}$ then we call $c$ a cut point. If $C=
\{c,d\}$ where $c\neq d$ and *neither* $c$ nor $d$ is cut point, then we call $\{c,d\}$ a (unordered) cut pair.
The proof of the following Lemma is an elementary exercise in topology and will be left to the reader.
\[L:etop\] Let $A$ be a connected subset of the space $X$ and $B$ closed in $X$. If $A
\cap \mathrm{Int} B \neq \emptyset$, then either $A \subset B$ or $A \cap {\partial}B$ separates the subspace $A$.
\[L:eas\] Let $X$ be a continuum and $C \subset X$ be minimal with the property that $X-C$ is not connected. The set $C$ separates $A \subset X-C$ from $B \subset X-C$ if and only if there exist continua $Y,Z$ such that $A \subset Y$, $B \subset Z$, $Y \cup Z = X$ and $Y \cap Z = C$.
We first show that $C$ is closed in $X$. There are disjoint nonempty subsets $D$ and $E$ open in $X-C$ with $D \cup E = X-C$. By symmetry, it suffices to show that $D$ is open in $X$. Suppose that $d \in D \cap {\partial}D$. There is a neighborhood $G$ of $d$ in $X$ such that $\bar G \cap \bar E =
\emptyset$. Since $G \not \subset D$, there is $c \in C \cap G$. Notice that $D \cup
\{c\}$ and $E$ are disjoint open subsets of $X -(C-\{c\})$ with $(D\cup
\{c\}) \cup E = X-(C-\{c\})$, and $C$ is not minimal. Therefore $C$ must be closed.
Suppose now that $C$ separates $A$ from $B$. Thus there exist disjoint nonempty $U$ and $V$ open in $X-C$ (this implies open in $X$) such that $A
\subset U$, $B \subset V$, $U \cup V = X-C$. Since ${\partial}U$ separates $X$ , by the minimality of $C$, ${\partial}U = C = {\partial}V$. Suppose the closure $\bar U$ is not connected. Then $\bar U = P \cup Q$ where $P$ and $Q$ are disjoint nonempty clopen (closed and open) subsets of $\bar U$. Since $\bar U$ is closed, this implies that $P$ and $Q$ are closed subsets of $X$. Since $\bar U \not \subset C$, we may assume that $P\not \subset C$. The boundary of $P$ in $\bar U$ is empty, so ${\partial}P
\subset {\partial}\bar U = {\partial}U =C$. Again by minimality ${\partial}P = C$. Since $P$ is closed in $X$, $C \subset P$. Thus $Q \subset U$, and $Q$ is open in $U$ since it is open $\bar U$. Thus $Q$ is clopen in $X$ which contradicts $X$ being connected.
The implication in the other direction is trivial.
This next result is just an application of the previous result.
\[L:top\] Let $X$ be a continuum and $A,B \subset X$.
- The point $c \in X$ is a cut point of $X$ which separates $A$ from $B $ if and only if there exist continua $Y,Z \subset X$ such that $A \subset
Y-\{c\}$, $B \subset Z-\{c\}$, $Y\cup Z= X$ and $Y \cap Z =
\{c\}$.
- The pair of non-cut points $\{c,d\}$ is a cut pair separating $A$ from $B $ if and only if there exist continua $Y,Z
\subset X$ such that $A \subset Y-\{c,d\}$, $B\subset Z-\{c,d\}$, $Y\cup Z = X$, $Y \cap Z = \{c,d\}$.
Cutpoint trees
==============
Let $X$ be a metric continuum. In this section we show that the big tree constructed in [@SWE3] is always a real tree. For the reader’s convenience we recall briefly the construction here.
For the remainder of this section, $X$ will be a continuum.
If $a, b \in X$ we say that $c\in (a,b)$ if the cut point $c$ separates $a$ from $b$.
We call $(a,b)$ an interval and this relation an interval relation. We define closed and half open intervals in the obvious way, ie $[a,b)=\{a \}\cup (a,b)$, $[a,b]=\{a,b \}\cup (a,b)$ for $a\ne b$ and $[a,a)=\emptyset, \, [a,a]=\{a \} $.
We define an equivalence relation on $X$. Each cut point is equivalent only to itself and if $a, b\in X$ are not cut points we say that $a$ is equivalent to $b$ if $(a,b)=\emptyset $.
Let’s denote by ${{\mathcal P}}$ the set of equivalence classes for this relation. We can define an interval relation on ${{\mathcal P}}$ as follows:
If $x, y \in {{\mathcal P}}$ and $c$ is a cut point (so $c \in {{\mathcal P}}$) we say that $c \in (x,y)$ if for some (any) $a\in x$, $b \in y$, we have $c \in (a,b)$.
For $z\in {{\mathcal P}}$, $z$ not a cut point we say that $z\in (x,y)$ if for some (any) $a\in
x,b\in y, c\in z$ we have that $$[a,c)\cap (c,b]=\emptyset.$$
If $x, y,z\in {{\mathcal P}}$ we say that $z$ is between $x,y$ if $z\in
(x,y)$. We will show that ${{\mathcal P}}$ with this betweenness relation is a pretree. The first two axioms of the definition of pretree are satisfied by definition.
For the remaining two axioms we recall the following lemmas (for a proof see Bowditch [@BOW5] or Swenson [@SWE]).
\[L:ax3\] For any $x,y \in {{\mathcal P}}$, if $z\in (x,y)$ then $x\notin (y,z)$.
\[L:ax4\] For any $x,y,z\in P$, $(x,z)\subset (x,y]\cup [y,z)$.
Axiom 3 follows from and Axiom 4 from .
Now consider the following example where $X\subset \R^2$ is the union of a Topologist’s sine curve, two arcs, five circles and two disks:
$X$ at 256 43 
${{\mathcal P}}$ at 250 32 
$T$ at 249 32 
The tree $T$ is obtained from ${{\mathcal P}}$ by “connecting the dots" according to the pretree relation on ${{\mathcal P}}$. We will give the rigorous definition of $T$ later.
We have the following results about intervals in pretrees from [@BOW5]:
\[L:subset\] If $x,y,z \in {{\mathcal P}}$, with $y\in [x,z]$ then $[x,y]\subset [x,z]$.
\[L:order\] Let $[x,y]$ be an interval of ${{\mathcal P}}$. The interval structure induces two linear orderings on $[x,y]$, one being the opposite of the other, with the property that if $<$ is one of the orderings, then for any $z$ and $w\in [x,y]$ satisfying $z<w$, we have $(z,w)=\{ u\in [x,y]:z<u<w
\}$. In other words the interval structure defined by one of the orderings is the same as our original interval structure.
If $x$ and $y$ are distinct points of ${{\mathcal P}}$ we say that $x$ and $y$ are *adjacent* if $(x,y)=\emptyset $. We say $x \in {{\mathcal P}}$ is [*terminal*]{} if there is no pair $y,z \in {{\mathcal P}}$ with $x \in (y,z)$.
We recall the following lemma from [@SWE].
\[L:adjacent\] If $x,y \in {{\mathcal P}}$, are adjacent then exactly one of them is a cut point and the other is a nonsingleton equivalence class whose closure contains this cut point.
\[L:singing\] If $p \in {{\mathcal P}}$ is a singleton equivalence class and $p$ is not a cut point, then $p$ is terminal in ${{\mathcal P}}$.
Let $x \in X$ with $[x] =\{x\}$, and $x \in (a,b)$ for some $a,b \in {{\mathcal P}}$. Suppose that $x$ is not a cut point. By there is no point of ${{\mathcal P}}$ adjacent to $[x]$. Thus there are infinitely many cut points in $[a,x]$. For each such cut point $c \in (a,x) $ choose a continuum $A_c \ni a$ with ${\partial}A_c = \{c\}$. Considered the nested union $A = \bigcup A_c$. We will show that ${\partial}A = \{x\}$.
First consider $ y \in A$. There exists a cut point $c\in (a,x)$, $c
\neq y$, with $y \in A_c$. Thus $y \in \mathrm{Int}
A_c$ so $y \not \in {\partial}A$.
Now consider $z \in X-A$ with $z \neq x$. Since $z \not \in A$, by definition $x \in ([z], [y])$ for any $y \in A$. Since $x$ and $[z]$ are not adjacent there is a cut point $d \in ([z],x) $. There exist continua $Z, B$ such that $Z \cup B = X$, $z \in Z$, $x \in
B$ and $Z\cap B =\{d\}$. Since $x \in ([z], [y])$ for any $y \in A$, by definition $A \subset B$, and $z \not \in {\partial}A$.
The fact that $x\in {\partial}A$ follows since $b \not \subset A$, so $X \neq
A$, and so ${\partial}A \neq \emptyset$.
We have the theorem [@SWE Theorem 6]:
\[T:nested\] A nested union of intervals of ${{\mathcal P}}$ is an interval of ${{\mathcal P}}$.
\[C:supremum\] Any interval of ${{\mathcal P}}$ has the supremum property with respect to either of the linear orderings derived from the interval structure.
Let $[x,y]$ be an interval of ${{\mathcal P}}$ with the linear order $\leq $. Let $A\subset [x,y]$. The set $\{[x,a]:a \in A\}$ is a set of nested intervals so their union is an interval $[x,s]$ or $[x,s)$ and $s=\sup\,A$.
A [*big arc*]{} is the homeomorphic image of a compact connected nonsingleton linearly ordered topological space. A separable big arc is called an [*arc*]{}. A [*big tree*]{} is a uniquely big-arcwise connected topological space. If all the big arcs of a big tree are arcs, then the big tree is called a [*real tree*]{}. A metrizable real tree is called an $\R$–tree. An example of a real tree which is not an $\mathbb{R}$–tree is the long line [@HOC-YOU Section 2.5, page 56].
A pretree ${{\mathcal R}}$ is [*complete*]{} if every closed interval is complete as a linearly ordered topological space (this is slightly weaker than the definition given in [@BOW5]). Recall that a linearly ordered topological space is complete if every bounded set has a supremum.
Let ${{\mathcal R}}$ be a pretree. An interval $I\subset {{\mathcal R}}$ is called [*preseparable*]{} if there is a countable set $Q \subset I$ such that for every nonsingleton closed interval $J \subset I$, we have $J \cap Q \neq \emptyset$. A pretree is [*preseparable*]{} if every interval in it is preseparable.
We now give a slight generalization of a construction in [@SWE; @SWE3]. Let ${{\mathcal R}}$ be a complete pretree. Set $$T={{\mathcal R}}\cup \underset {{x,\,y\, \text{adjacent}}} { \bigsqcup } I_{x,y}$$ where $I_{x,y}$ is a copy of the real open interval $(0,1)$ glued in between $x$ and $y$. We extend the interval relation of ${{\mathcal R}}$ to $T$ in the obvious way (as in [@SWE; @SWE3]), so that in $T$, $(x,y) = I_{x,y}$. It is clear that $T$ is a complete pretree with no adjacent elements. When ${{\mathcal R}}={{\mathcal P}}$, we call the $T$ so constructed the cut point tree of $X$.
For $A$ finite subset of $T$ and $s\in T$ we define $$U(s,A)=\{ t\in T:[s,t]\cap A=\emptyset \}.$$
The following is what the proof of [@SWE Theorem 7] proves in this setting.
Let ${{\mathcal R}}$ be a complete pretree. The pretree $T$, defined above, with the topology defined by the basis $\{U(s,A)\}$ is a regular big tree. If in addition ${{\mathcal R}}$ is preseparable, then $T$ is a real tree.
We now prove the conjecture from [@SWE].
\[T:countable\] The pretree ${{\mathcal P}}$ is preseparable, so the cut point tree $T$ of $X$ is a real tree.
By the proof of [@SWE Theorem 7], it suffices to show that there are only countably many adjacent pairs in a closed interval $[a,b]$ of ${{\mathcal P}}$. By , adjacent elements of ${{\mathcal P}}$ are pairs $E,c$ where $E$ is a nonsingleton equivalence class, $c$ is a cut point and $c \in \bar E-E$. Let’s assume that there are uncountably many such pairs in $[a,b]$. By symmetry we may assume that $E \in (a,c)$ for uncountably many pairs $(E,c)$, and for each such pair we pick an $e\in E$.
Since $c$ separates $e$ from $b$ choose continua $A$, $B$ such that $X = A \cup B$, $\{c\} = A \cap B$, $e \in A$ and $b \in B$. Since $e \not \in B$ and $B$ is compact $d(e,B) >0$. Let $\epsilon_e=d(e,B)$.
In this way to each pair $E,c$ we associate $e\in E$ continua $A$, $B$ and $\epsilon_e >0$. Since there are uncountably many $e$, for some $n \in N$ there are uncountably many $e$ with $\epsilon_e>1/n$. Let’s denote by $S$ the set of all such $e$ with $\epsilon_e >1/n$. Consider a finite covering of $X$ by open balls of radius $\unpfrac{1}{2n}$. Since $S$ is infinite there are distinct elements $e_1,e_2,e_3\in S$ lying in the same ball. It follows that $d(e_i,e_j)<1/n$ for all $i,j$.
The points $e_1,e_2,e_3$ correspond to adjacent elements of ${{\mathcal P}}$, say $E_1,c_1$, $E_2,c_2,E_3,c_3$. Since all these lie in an interval of ${{\mathcal P}}$ they are linearly ordered and we may assume, without loss of generality, that $E_1\in [a,c_2)$ and $E_3\in
(c_2,b]$. Let $A_1$ and $B_1$ be the continua chosen for $E_1,c_1$ such that $A_1 \cap B_1 = \{c_1\}$, $A_1\cup B_1 = X$, $e_1 \in A_1$, $b
\in B_1$ and $d(e_1,B_1) =\epsilon_{e_1}>1/n$. It follows that $e_3 \in B_1$ and so $d(e_1,e_3)\ge d(e_1,B_1) > 1/n$, which is a contradiction.
The real tree $T$ is not always metrizable. Take for example $X$ to be the cone on a Cantor set $C$ (the so-called Cantor fan). Then $X$ has only one cut point, the cone point $p$, and ${{\mathcal P}}$ has uncountable many other elements $q_c$, one for each point $c \in C$. As a pretree, $T$ consists of uncountable many arcs $\{ [p,q_c]:\, c \in C\}$ radiating from a single central point $p$. However, in the topology defined from the basis $\{U(s,A)\}$, every open set containing $p$ contains the arc $[p,q_c]$ for all but finitely many $c \in C$. There can be no metric, $d$, giving this topology since $d(p,q_c)$ could only be nonzero for countably many $c \in C$.
It is possible however to equip $T$ with a metric that preserves the pretree structure of $T$. This metric is “canonical” in the sense that any homeomorphism of $X$ induces a homeomorphism of $T$. The idea is to metrize $T$ in two steps. In the first step one metrizes the subtree obtained by the span of cut points of ${{\mathcal P}}$. This can be written as a countable union of intervals and it is easy to equip with a metric.
$T$ is obtained from this tree by gluing intervals to some points of $T$. In this step one might glue uncountably many intervals but the situation is similar to the Cantor fan described above. The new intervals are metrized in the obvious way, eg one can give all of them length one.
\[T:Rtree\] There is a path metric $d$ on $T$, which preserves the pretree structure of $T$, such that $(T,d)$ is a metric $\mathbb{R}
$–tree. The topology so defined on $T$ is canonical (and may be different from the topology with basis $\{U(s,A)\}$). Any homeomorphism $\phi $ of $X$ induces a homeomorphism $\hatphi$ of $T$ equipped with this metric.
Let ${{\mathcal C}}$ be the set of cut points of $X$ and let $S$ be a countable dense subset of ${{\mathcal C}}$. Choose a base point $s\in S$. Denote by $T'$ the union of all intervals $[s,s']$ of $T$ with $s' \in S$. Now we remark that at most countably many cut points of $X$ are not contained in $T'$. Indeed if $c\in {{\mathcal C}}$ is a cut point not in $T'$ then $X-c=U\cup V$ where $U,V$ are disjoint open sets and one of the two (say $U$) contains no cut points. Let $\epsilon >0$ be such that a ball $B(c)$ in $X$ of radius $\epsilon $ is contained in $U$. So we associate to each $c$ not in $T'$ a ball $B(c)$ and we remark that to distinct $c$’s correspond disjoint balls. Clearly there can be at most countably many such disjoint balls in $X$. Thus by enlarging $S$ we may assume that $T'$ contains all cut points, so $T'$ is the convex hull of ${{\mathcal C}}$ in $T$, and so is canonical.
Since $S$ is countable we can write $S=\{s_1,s_2, \dots \}$ and we metrize $T'$ by an inductive procedure: we give $[s,s_1]$ length 1 (Choose $f\co [0,1]\to [s,s_1]$, a homeomorphism, and define $d(f(a),f(b))=|a-b|$). $[s,s_2]$ intersects $[s,s_1]$ along a closed interval $[s,a]$. If $[a,s_2]$ is non empty we give it length $1/2$ and we obtain a finite tree. At the $n$–th step of the procedure we add the interval $[s,s_{n+1}]$ to a finite tree $T_n$. If $[s,s_{n+1}]\cap T_n=[a,s_{n+1}]$ a nondegenerate interval we glue $[a,s_{n+1}]$ to $T_n$ and give it length $1/2^n$. Note that if $a,b\in T'$ then $a\in [s,s_n], b\in
[s,s_k]$ for some $k,n\in \Bbb N$. Without loss of generality, $k\le n$ and so $a,b\in T_n$ and $d(a,b)$ is determined at some finite stage of the above procedure.
We remark that each end of $ T'$ corresponds to an element of ${{\mathcal P}}$ and by adding these points to $ T'$ one obtains an $\Bbb
R$–tree that we still denote by $ T'$. Here by end of $T'$ we mean an ascending union $\bigcup [s,s_i]\ (i\in \Bbb N,\,s_i\in S)$ which is not contained in any interval of $ T'$. If $C_i$ is the closure of the union of all components of $X-s_i$ not containing $s$ we have that $C_i\subset C_{i-1}$ for all $i$ and $\bigcap C_i$ is an element of ${{\mathcal P}}$.
If $x\in {{\mathcal P}}$ does not lie in $T'$ then $x$ is adjacent to some cut point $c\in T'$. For each such adjacent pair $(c,x)$, by construction $(c,x)$ is a copy of the unit interval $(0,1)$ and this gives the path metric on $[c,x]$. In this way we equip $T$ with a path metric $d$.
Clearly a homeomorphism $\phi\co X\to X$ induces a pretree isomorphism $\hatphi\co {{\mathcal P}}\to {{\mathcal P}}$. By extending $\hatphi$ to the intervals corresponding to adjacent points of ${{\mathcal P}}$ (via the identity map on the unit interval, $(0,1) \to(0,1)$) we get a pretree isomorphism function $\hatphi\co T \to T$ which restricts to a pretree isomorphism $\hatphi\co T' \to T'$. For any (possibly singleton) arc $\alpha \in T'$ let ${ {\mathcal B}}_{\alpha }$ be the set of components of $T' -\alpha$. By the construction of $d$, for any $\epsilon >0$, the set $\{B \in { {\mathcal B}}_{\alpha }:\,
{\text{diam}}(B) > \epsilon\}$ is finite. It follows that $\hatphi\co T'
\to T'$ is continuous (using the metric $d$) and therefore a homeomorphism. We extend $\hatphi$ to $T$ by defining it to be an isometry on the disjoint union of intervals $T-T'$. Thus we get $\hatphi\co T \to T$ a homeomorphism.
JSJ-trees
=========
Let $X$ be a continuum without cut points. A finite set $S\subset X$ with $|S|>2$ is called [*cyclic subset*]{} if there is an ordering $S=\{x_1,\dots x_n\}$ and continua $M_1,\dots M_n$ with the following properties:
- $ M_n \cap M_1 = \{x_1\} $, and for $i>1$, $\{x_i\} = M_{i-1} \cap
M_{i}$
- $M_i \cap M_j = \emptyset$ for $i-j \neq \pm 1 \mod n$
- $\bigcup M_i = X$
The collection $M_1, \dots M_n$ is called the (a) [*cyclic decomposition*]{} of $X$ by $\{x_1,\dots x_n\}$. This decomposition is unique as we show in (for $n>3$).
We also define a cut pair to be [*cyclic*]{}.
Clearly every nonempty nonsingleton subset of a cyclic set is cyclic.
If $S$ is an infinite subset of $X$ and every finite subset $A \subset S$ with $|A|>1$ is cyclic, then we say $S$ is [*cyclic*]{}.
Clearly if $A$ is a subset of a cyclic set with $|A|>1$, then $A$ is cyclic.
\[L:cross\] Let $X$ be a connected metric space without cut points. If the cut pair ${a,b}$ separates the cut pair $c,d $ then $\{a,b,c,d\}$ is cyclic, so $\{c,d\}$ separates $a$ from $b$. Furthermore $X-\{c,d\}$ has exactly two components and $X-\{a,b\}$ has exactly two components.
By there exist continua $C,D$ with $C\cap D =
\{a,b\}$, $X= C \cup D$ , $c\in C$ and $d \in D$. Since $c,d$ is a cut pair there exist continuum $A,B$ such that $A\cup B = X$ and $A\cap B = \{c,d\}$. We may assume that $a \in A$. Since $B$ is connected, with $c,d \in B$ and $a\not \in B$, then $b$ must be a cut point of $B$ separating $c$ and $d$. Similarly $a$ is a cut point of $A$ separating $c$ and $d$. Thus by there exist continua $M_{a,c}$, $M_{a,d}$ $M_{b,c}$, $M_{b,d}$ with $c \in M_{a,c}$, $c \in M_{b,c}$, $d \in M_{a,d}$, and $d
\in M_{b,d}$, such that $M_{a,c}\cup M_{a,d} =A $, $M_{a,c}\cap
M_{a,d} =\{a\}$, $M_{b,c}\cup M_{b,d} =B $ and $M_{b,c}\cap
M_{b,d} =\{b\} $. It follows that $M_{a,c} \cap M_{b,c} =\{c\}$ and that $M_{a,d} \cap M_{b,d} =\{d\}$. Thus $\{a,b,c,d\}$ is cyclic.
Suppose that $\{c,d\}$ separated $A$, then there would be nonsingleton continua $F$,$G$ with $A = F\cup G$ and $\{c,d\}= F\cap G$. We may assume that $a \in F$. Since $a$ separates $c$ from $d$ in $A$, either $c \not \in
G$ or $d \not \in G$. With no loss of generality $d \not \in G$. Thus $F
\cup B$ and $G$ are continua with $X= (F\cup B) \cup G$ and $\{c\}= (F\cup
B) \cap G$, making $c$ a cut point of $X$. This is a contradiction, so $\{c,d\}$ doesn’t separate $A$ and similarly $\{c,d\}$ doesn’t separate $B$. Thus $X-\{c,d\}$ has exactly two components.
Let $X$ be a metric space without cut points. A nondegenerate nonempty set $A \subset X$ is called inseparable if no pair of points in $A$ can be separated by a cut pair.
Every inseparable set is contained in a maximal inseparable set. A maximal inseparable subset is closed (its complement is the union of open subsets).
A maximal inseparable set need not be connected, for example let $X$ be the complete graph on the vertex set $V$ with $3< |V|
<{\infty}$. The set $V$ is a maximal inseparable subset of $X$. $V$ also has the property that every pair in $V$ is a cut pair of $X$, but $V$ is not cyclic.
\[L:cyc\] Let $S$ be a subset of $X$ with $|S|>1$. If every pair of points in $S$ is a cut pair and $\{a,b\}$ is a cut pair separating points of $S$, then $S \cup \{a,b\}$ is cyclic.
It suffices to prove this when $S$ is finite. Let $c,d \in S$ be separated by $\{a,b\}$. By , $X- \{c,d\}$ has exactly two components, $X-\{a,b\}$ has exactly two components, and there are continua $M_{a,c}$, $M_{a,d}$ $M_{b,c}$, $M_{b,d}$ whose union is $X$ such that $M_{a,c}\cap M_{a,d} =\{a\}$, $M_{b,c}\cap
M_{b,d} =\{b\} $, $M_{a,c} \cap M_{b,c} =\{c\}$, $M_{a,d} \cap
M_{b,d} =\{d\}$.
Now let $e \in S -\{a,b,c,d\}$. We may assume $e \in M_{a,c}$. Now $\{a,b\}$ separates the cut pair $\{d,e\}$ and so by , $\{d,e\}$ separates $a$ from $b$. It follows that $e$ is a cut point of the continuum $M_{a,c}$. Thus there exist continua $M_{a,e}\ni a$ and $M_{e,c}\ni c$ such that $M_{a,e} \cup M_{e,c} = M_{a,c}$ and $M_{a,e} \cap M_{e,c} =\{e\}$. The set $\{a,b,c,d,e\}$ is now known to be cyclic. Continuing this process, we see that $S \cup \{a,b\}$ is cyclic.
If $S\subset X$ with $|S|>1$ and $S$ has the property that every pair of points in $S$ is a cut pair, then either $S$ is inseparable or $S$ is cyclic.
By Zorn’s Lemma, every cyclic subset of $X$ is contained in a maximal cyclic subset. A maximal cyclic subset with more than two elements is called a [*necklace*]{}. In particular, every separable cut pair is contained in a necklace.
\[L:twototango\] Let $S$ be a cyclic subset of $X$, a continuum without cut points. If $S$ separates the point $x$ from $y$ in $X$, then there is a cut pair in $ S$ separating $x$ from $y$.
Suppose not, then for any finite subset $\{x_1,\dots x_n\}
\subset S$ with $M_1,\dots M_n$ the cyclic decomposition of $X$ by $\{x_1,\dots x_n\}$, $x$ and $y$ are contained in the same element $M_i$ of this cyclic decomposition. There are two cases.
In the first case, we can find a strictly nested intersection of (cyclic decomposition) continua $C\ni x,y$, with the property that $|C \cap S|\leq 1$. Nested intersections of continua are continua, so $C$ connected, and similarly using , $C- (C \cap S)$ is connected, so $S$ doesn’t separate $x$ from $y$.
In the second case there is a cut pair $\{a,b\}\subset S$ and continua $M,N$ such that $ x,y\in M$, $N \cup M =X$, $N \cap M
=\{a,b\}$ and $ S \subset N$. It follows that $\{a,b\}$ separates $x$ from $y$ in $M$, so there exist continua $Y,Z$ with $M=Y\cup
Z$ where $Y\cap Z =\{a,b\}$, $y \in Y$ and $x \in Z$. Since $(N\cup
Y) \cap Z =\{a,b\}$ and $(N\cup Y) \cup Z =X$, it follows that $\{a,b\}$ separates $x$ from $y$ in $X$.
Let $S$ be a necklace of $X$. We say $y,z \in X-S$ are $S$ equivalent, denoted $y\sim_S z$, if for every cyclic decomposition $M_1, \dots M_n$ of $X$ by $\{x_1, \dots x_n \} \subset S$, both $y,z \in M_i$ for some $1\le i \le n$. The relation $\sim_S$ is clearly an equivalence relation on $X-S$. By , if $y,z$ are separated by $S$ then $y \not \sim_S z$, but the converse is false.
The closure (in $X$) of a ($\sim_S$)–equivalence class of $X-S$ is called a [*gap*]{} of $S$. Notice that every gap is a nested intersection of continua, and so is a continuum. Every inseparable cut pair in $S$ defines a unique gap. The converse is true if $X$ is locally connected, but false in the nonlocally connected case.
Let $s \in S$. Choose distinct $x,y \in S-\{s\}$, and take the cyclic decomposition $M_1,M_2,M_3$ of $X$ by $\{s,x,y\}$ with $M_1 \cap M_3 =\{s\}$. For each $i$, take a copy $\hat M_i$ of $M_i$. Let $\hat M$ be the disjoint union of the $\hat M_i$. For $i =1,2,3$ let $s_i, y_i, x_i$ be the points of $\hat M_i$ which correspond to $s,x,y$ respectively whenever they exist (for instance there is no $s_2$ since $s \not \in M_2$). Let $\hat X
$ be the quotient space of $\hat M$ under the identification $y_i =y_j$ and $x_i = x_j$ for all $i,j$. The metrizable continuum $\hat X$ is clearly independent of the choice of $x$ and $y$. The obvious map $q\co \hat X \to X$ is one to one except that $\{s_1,s_3\} =
q^{-1}(s)$. We will abuse notation and refer to points of $X-\{s\}$ as points of $\hat X -\{s_1,s_3\}$ and vice versa.
The cut points of the continuum $\hat X$ are exactly $S-\{s\}$. Consider the cut point pretree ${{\mathcal P}}$ for $\hat X$. By , the cut points of $\hat X$ will be exactly the singleton equivalence classes in ${{\mathcal P}}$ other than $\{s_1\}$ and $\{s_3\}$. The closures of nonsingleton equivalence classes in ${{\mathcal P}}$ are exactly the gaps of $S$. [*Thus every gap of $S$ has more than one point*]{}. The cut point real tree $T$ is in this case an arc (see ), so there is a linear order on ${{\mathcal P}}$ corresponding to the pretree structure. Let $A \in {{\mathcal P}}$ be a nonsingleton equivalence class (so $\bar A \subset X$ is a gap of $S$) with $s \not \in \bar A$. Let $U = \{ x \in \hat X:
[x]< A \}$ and let $B = q(\bar U \cap \bar A)$. Similarly let $O=
\{x \in \hat X:A < [x]\}$ and let $C= q(\bar O \cap \bar A)$. The two closed sets $B$ and $C$ are called the [*sides*]{} of the gap $\bar
A$. Notice that ${\partial}A = C \cup B$. Since $X$ has no cut points $B$ and $C$ are nonempty.
Let $D$ be a gap of $S$ with sides $B$ and $C$. If $B\cap C=\emptyset $, then we say $D$ is a [*fat*]{} gap of $S$. Each fat gap is a continuum whose boundary is the disjoint union of its sides. It follows that every fat gap has nonempty interior. Distinct fat gaps of $S$ will have disjoint interiors. Since the compact metric space $X$ is Lindelöf (every collection of nonempty disjoint open sets is countable), $S$ has only countably many fat gaps. If $X$ is locally connected then there are only fat gaps because the sides of a gap form (with local connectivity) an inseparable cut pair.
Consider the following example where $X$ is a continuum in $\R^2$ containing a single necklace $S$ and five gaps of $S$. The three solid rectangles are fat gaps, and the two thin gaps are limit arcs of Topologist’s sine curves.
$X$ at 71 48 
The union of the sides of a gap of $S$ is a nonsingleton inseparable set.
Take $A$, $U$,$O$, $B$, $C$, $s$ and $q\co \hat X \to X$ as above. We show that $B \cup C$ is a nonsingleton inseparable set. Suppose that $B\cup C = \{b\}$. Then ${\partial}A = \{b\}$ and since gaps are not singletons, $b$ is a cut point of $X$. Thus $B \cup C$ is not a singleton.
Now suppose that $d,e \in B\cup C$ and $\{r,t\}$ is a cut pair separating $d$ and $e$. Let $E=q(O) \cup q(U)$. Since $O$ and $U$ are nested unions of connected sets they are connected and since $s \in q(O) \cap q(U)$, $E$ is connected. Thus for any $P\subset X$, with $E \subseteq P \subseteq \bar E$, $P$ is connected. Since $d,e \in \bar E$ it follows that $\{r,t\}$ must separate $E$. Since the gap $\bar A\ni d,e$ is connected it follows that $\{r,t\}$ must separate $\bar A$. Notice that $E \cap \bar A$ consists of sides of $A$ which are points of $S$, so $|E\cap
\bar A|\le 2$ and if $|E\cap \bar A |=2$ then $E \cap \bar A =\{e,d\}$. It follows that $\{r,t \} \not \subset E\bar A$.
First consider the case where one of $\{r,t\}$, say $r$ is in $E\cap \bar
A$. It follows that $r \in S$ is one of the sides of $\bar A$, say $\{r\} =B$. Thus $d,e \in C$, the other side of $\bar A$. Since $q(O)$ is connected and its closure contains $C$, it follows that $t \in q(O) \cap S$. Since $B$ is not a point of $S$, there are infinitely many elements $u \in S$ such that $\{r,u\}$ separates $t$ from $C$. Replacing $t$ with such an $u$, we may assume that $r$ are $t$ are not inseparable and so $X- \{r,t\}$ has exactly two components by . One of these components will contain $s$ and the other will contain $\bar A -\{r\}$. Thus $\{r,t\}$ doesn’t separate $d$ from $e$. Contradiction.
Now we have the case where $\{r,t\} \cap (E\cap \bar A)= \emptyset$. It follows that one of them (say $r$) is a cut point of $E$ and the other $t$ is a cut point of $\smash{\bar A}$. Since $r$ is a cut point of $E$, it follows that $r \in S$, and since $r$ is not a side of $\bar A$, with no loss of generality $r=s$. Thus $t$ is a cut point in $\hat X$ and so $t \in S$. But $S \subset E$ and $t \not \in E$. Contradiction.
\[C:character\] Let $X$ be a continuum without cut points. Suppose that for every pair of points $c,d\in X$ there is a pair of points $a,b$ that separates $c,d$. Then $X$ is homeomorphic to the circle.
Let $S$ be a necklace of $X$. Using , we can show that $S$ is infinite and in fact any two points of $S$ are separated by a cut pair in $S$. If $X-S \neq \emptyset $ then there is a gap $A$ of $S$. The union of sides of the gap $A$ is a nonsingleton inseparable subset of $X$. There are no nonsingleton inseparable subsets of $X$, so $S=X$. Thus $X$ is homeomorphic to the circle. This follows from [@HOC-YOU Theorem 2-28, page 55] (our also proves this).
\[L:maptocircle\] Let $S$ be a necklace of $X$. There exists a continuous surjective function $f\co X \to S^1$, with the following properties:
1. The function $f$ is one to one on $S$.
2. The image of a fat gap of $S$ is a nondegenerate arc of $S^1$.
3. For $x,y\in
X$ and $a,b\in S$:
1. If $\{f(a), f(b)\}$ separates $`f(x)$ from $`f(y)$ then $\{a,b\}$ separates $x$ from $y$.
2. If $x \in S$ and $\{a,b\}\subset S$ separates $x$ from $y$, then $\{f(a),
f(b)\}$ separates $f(x)$ from $f(y)$.
The function $f$ is unique up to homotopy and reflection in $S^1$. In addition, if $G$ is a group acting by homeomorphisms on $X$ which stabilizes $S$, then the action of $G$ on $S$ extends to an action of $G$ on $S^1$.
We use the strong Urysohn Lemma [@MUN 4.4 Exercise 5] If $A$ and $B$ are disjoint closed $G_\delta$ subsets of a normal space $Y$, then there is a continuous $f\co Y \to [0,1]$ such that $f^{-1}(0) = A$ and $f^{-1}(1) = B$. In a metric space, all closed sets are $\smash{G_\delta}$. Since $X$ has a countable basis, the subspace $S$ has a countable dense subset $\smash{\hat S}$. Since the fat gaps of $S$ are countable, the collection $R$ of all sides of fat gaps of $S$ is countable. Let $\{s_n:n\in {{\mathbb N}}\}= \hat S \cup
R$. Notice that now some of the elements of $\{s_n: n \in {{\mathbb N}}\}$ are points (singleton sets) of $S$ and some of them are sides of gaps (and therefore closed sets of $X$). In particular all inseparable cut pairs of $S$ are in $\{s_n:n \in {{\mathbb N}}\}$.
For the remainder of this proof, we will maintain the useful fiction that each element of $\{s_n:n \in {{\mathbb N}}\}$ is a point (which would be true if $X$ were locally connected), and leave it to the reader (with some hints) to check the details for the nonsingleton sides of gaps.
Notice that the elements of $\{s_n: n \in {{\mathbb N}}\}$ are pairwise disjoint.
We inductively construct the map $f$. We take as $S^1$, quotient space of the interval $[0,1]/(0=1)$ with $0$ identified with $1$. Since $\{s_1,s_2,s_3\}$ is cyclic, there exist cyclic decomposition $M_1,M_2,M_3$ of $X$ with respect to $\{s_1,s_2,s_3\}$. We define the map $f_3 \co M_
1\to [0,\unfrac 13]$ by $f_3(s_1) =0$, $f_3(s_2) = \unfrac 1 3$ and then extend to $M_1$ using the strong Urysohn Lemma so that $f_3^{-1}(0)
=\{ s_1\}$ and $ f_3^{-1}(\unfrac 1 3) =\{s_2\}$. Similarly we define the continuous map $
f_3\co M_2 \to [\unfrac 1 3, \unfrac 2 3] $ such that $ \smash{f_3^{-1}}( \unfrac 1 3)
=\{s_2\}$ and $ \smash{f_3^{-1}}(\unfrac 2 3) = \{s_3\}$. Lastly we define $
f_3\co M_3 \to [\unfrac 2 3, 1] $ such that $ \smash{f_3^{-1}}( \unfrac 2 3) =\{s_3\}$ and $ \smash{f_3^{-1}}(1) = \{s_1\}$. Since $0=1$ we paste to get the function $f_3\co X \to S^1$.
Now inductively suppose that we have $N_1, \dots N_k$ a cyclic decomposition of $X$ with respect to $\{s_i : i \le k\}$ (when the $s_i$ are sides of gaps the definition of cyclic decomposition will be similar), and a map $f_k\co X \to S^1$ such that $f_k(N_j) = [f_k(s_p),f_k(s_q)]$ for each $1\le j
\le k$, where ${\partial}N_j =\{s_p,s_q\}$ and $q,p \le k$, satisfying $f_k^{-1}(f(s_j))=\{s_j\}$ for all $j\le k$. If $s_{k+1} \in N_j$ with ${\partial}N_j =\{s_p,s_q\} $ then there exists continua $A,B$ such that $A\cup B = N_j$, $A \cap B = \{s_{k+1}\}$, $s_p \in A$ and $s_q \in B$ (in the case where $s_{k+1}$ is the side of a gap, then one of $A$, $B$ will be a nested union of continua, and the other will be a nested intersection). Using the strong Urysohn Lemma, we define $f_{k+1}\co N_j \to [f_k(s_p),f_k(s_q)]$ such that $\smash{f_{k+1}^{-1}}(f_k(s_p))=\{s_p\}$, $\smash{f_{k+1}^{-1}}(f_k(s_q))=\{s_q\}$, $f_{k+1}^{-1}(
\upnfrac {s_q +s_p}2)=\{s_{k+1}\}$, $f_{k+1}(A) =[f_{k+1}(s_p),
f_{k+1}(s_{k+1})]$ and lastly $f_{k+1}(B) =[f_{k+1}(s_{k+1}), f_{k+1}(s_{q})]$. We define $f_{k+1}$ to be equal to $f_k$ on $X- N_j$ and by pasting we obtain $f_{k+1}\co X \to
S^1$. By construction, the sequence of functions $f_k$ converges uniformly to a continuous function $f\co X \to S^1$. Property (2) follows from the construction of $f$.
For uniqueness, consider $h\co X \to S^1$ satisfying these properties. Since the cyclic ordering on $S$ implies that $f\co S\to S^1$ is unique up to isotopy and reflection [@BOW1], we may assume that $h$ and $f$ agree on $S$. Thus for any fat gap $O$ of $S$, we have $h(O) =
f(O)=J$, an interval. Since $f$ and $h$ agree on the sides of $O$, which are sent to the endpoints of $J$, we simply straight line homotope $h$ to $f$ on each fat gap. Clearly after the homotopy they are the same.
The action of $G$ on $S$ gives an action on $\overline{f(S)} \subset S^1$, which preserves the cyclic order. Thus by extending linearly on the complementary intervals, we get an action of $G$ on $S^1$. This action has the property that for any $g \in G$, $f\circ g \simeq g \circ f$.
Let $X$ be a continuum without cut points. We define ${{\mathcal R}}\subset
2^X$ to be the collection of all necklaces of $X$, all maximal inseparable subsets of $X$, and all inseparable cut pairs of $X$. For the remainder of this section, $X$ is fixed.
\[L:cot\] Let $E$ be a nonsingleton subcontinuum of $X$. There exists $Q \in {{\mathcal R}}$ with $Q \cap E \neq \emptyset$.
Let $c,d\in E$ distinct. If $\{c,d\}$ is an inseparable set, then there is a maximal inseparable set $D \in {{\mathcal R}}$ with $c,d \in D$.
If not then there is a cut pair $\{a,b\}$ separating $c$ from $d$. It follows that $E\cap \{a,b\}\neq \emptyset$. Either $\{a,b\}$ is inseparable or there is a necklace $N \in {{\mathcal R}}$ with $\{a,b \} \subset N$, and so $E \cap N \neq \emptyset$.
\[T:cint\] Let $X$ be a continuum without cut points. If $S,
T\in {{\mathcal R}}$ are distinct then $|S\cap T|<3$ and if $|S\cap T| =2$, then $S
\cap T$ is an inseparable cut pair.
If $S$ or $T$ is an inseparable cut pair, then the result is trivial. We are left with three cases.
First consider the case where $S$ and $T$ are necklaces of $X$ Suppose there are distinct $a,b,c \in S \cap T$. Since $S,T$ are distinct necklaces, there exists $d\in S-T$. Since $\{a,b,c,d\}
\subset S$ is cyclic, renaming $a,b,c$ if needed, we have $X= A\cup B \cup
C \cup D$ where $A,B,C,D$ are continua and $A\cap B= \{b\}$, $B\cap C =
\{c\}$, $C\cap D =\{d\}$, and $D\cap A = \{a\}$, and all other pairwise intersections are empty. Thus $\{b,d\}$ separates $a$ and $c$, points of $T$. It follows by that $d \in T$. This contradicts the choice of $d$ so $|S
\cap T|<3$. Now suppose we have distinct $a,b \in S \cap T$. If $\{y,z\}$ is a cut pair separating $a$ from $b$ in $X$ then, by , $\{y,z\} \subset
T$ and $\{y,z \} \subset S$, so $|S\cap T| >3$. This is a contradiction, so $\{a,b\}$ is an inseparable cut pair.
Now consider the case where $S$ and $T$ are maximal inseparable subsets of $X$. Since $S$ and $T$ are distinct maximal inseparable sets, there exist $y \in S $, $z \in T$ and a cut pair $\{a,b\}$ separating $y$ from $z$. It follows that $y \not \in S$ and $z \not \in T$. Thus $X = C \cup D$ where $C$ and $D$ are continua, $y \in C$, $z \in D$ and $C \cap D = \{a,b\}$. By inseparability, $S \subset C$ and $T \subset D$. Clearly $S\cap T
\subset C \cap D =\{a,b\}$. If $S\cap T = \{a,b\}$ then $\{a,b\}$ is inseparable.
Finally consider the case where $S$ is a necklace of $X$, and $T$ is a maximal inseparable set of $X$. By definition, every cyclic subset with more than three elements is not inseparable. It follows that $|S\cap
T|
<4$. The only way that $|S \cap T| =3$ is if $S = T$ which is not allowed. If $|S\cap T|=2$ then $S\cap T$ is inseparable (since $T$ is) and cyclic (since $S$ is) and therefore $S\cap T$ is an inseparable cut pair.
\[L:nosep\] If $S,T \in {{\mathcal R}}$, then $S$ doesn’t separate points of $T$.
Suppose that $r,t \in T-S$ with $S$ separating $r$ and $t$. First suppose that $S$ is cyclic (so $S$ is a necklace or an inseparable cut pair). In this case by , there exists a cut pair $\{a,b\} \subset S$ such that $\{a,b\}$ separates $r$ from $t$.
If $\{r,t\}$ is a cut pair, then by , $a$ and $b$ are separated by $\{r,t\}$, so $S$ is not an inseparable pair. Thus $S$ is a necklace and it follows by that $r,t \in S$ (contradiction). Thus $\{r,t\}$ is not a cut pair, and so $T$ is a maximal inseparable set, but $\{a,b\}$ separates points of $T$ which is a contradiction.
We are left with the case where $S$ is a maximal inseparable set.
If $T$ is also a maximal inseparable set, then there is a cut pair $A$ separating a point of $S$ from a point of $T$. Thus there exist continua $N$ and $M$ such that $N\cup M =X$, $N \cap M =A$ and, since $S$ and $T$ are inseparable, we may assume $T\subset N$ and $S \subset M$. Since $A$ doesn’t separate points of $T$, and $S \cap N \subset A$, it follows that $S$ doesn’t separate points of $T$.
Lastly we have the case where $S$ is maximal inseparable, and $T$ is cyclic. Thus $\{r,t\}$ is a cut pair. So there exist continua $N,M$ such that $N\cup M =X$ and $N \cap M
=\{r,t\}$. Since $S$ is maximal inseparable, $S$ is contained in one of $N$ or $M$ (say $S \subset M$). However $r,t \subset N$ and since $r,t\not \in S$, we have $S \cap N = \emptyset $. Thus $S$ doesn’t separate $r$ from $t$.
We now define a symmetric betweenness relation on ${{\mathcal R}}$ under which ${{\mathcal R}}$ is a pretree. Let $R,S,T $ be distinct elements of ${{\mathcal R}}$. We say $S$ is between $R$ and $T$, denoted $RST$ or $TSR$, provided:
1. \[cutpair\] $S$ is an inseparable cut pair and $S$ separates a point of $R$ from a point of $T$.
2. $S$ is not an inseparable pair and:
1. $R \subset S$, so $R$ is an inseparable cut pair, and $R$ isn’t between $S$ and $T$ (see case ).
2. \[partof\] $S$ separates a point of $R$ from a point of $T$, and there is no cut pair $Q \in {{\mathcal R}}$ with $RQS$ and $TQS$ (see case ).
For $R,T \in {{\mathcal R}}$ we define the open interval $(R,T) = \{S\in
{{\mathcal R}}:\, RST\}$. We now defined the closed interval $[R,S]= (R,S)
\cup \{R,T\}$ and we define the half-open intervals analogously. We will show that ${{\mathcal R}}$ with this betweenness relation forms a pretree [@BOW5]. Clearly for any $R,S \in {{\mathcal R}}$, by definition $[R,S] = [S,R]$ and $R \not \in (R,S)$.
Consider the following example in where $X \subset \R^2$ is the union of 6 nonconvex quadrilaterals (meeting only at vertices) and a Topologist’s sine curve limiting up to one of them. There are two necklaces, one being the Topologist’s sine curve and the other consisting of the four green points. The cut pair tree $T$ is obtained from ${{\mathcal R}}$ by connecting the dots (definition to be given later).
$X$ at 200 39  at 195 35  at 195 35 
For any $R,S,T \in {{\mathcal R}}$, we have that $[R,T] \subset [R,S]\cup
[S,T]$.
We may assume $R,S,T$ are distinct. Let $ Q \in (R,T) $ with $Q \neq S$.
If $Q$ is an inseparable pair then $Q$ separates a point $r\in R$ from a point $t\in T$. Thus there exist continua $N$,$M$ such that $N\cup M =X$, $N\cap M = Q$, $r\in N$ and $t\in M$. Since $S \not \subset Q$, either $(S-Q) \cap N \neq
\emptyset$ implying $Q \in (S,T)$, or $(S-Q) \cap M \neq \emptyset $ implying $Q \in (R,S)$.
Now consider the case where $Q$ is not an inseparable pair.
Suppose that one of $R$,$T$ (say $R$) is contained in $Q$, so $R\subset Q$ is an inseparable cut pair and $R\not \in (Q, T)$. If $R \not \in (Q, S)$ then by definition $Q \in (R,S)$ as required. If on the other hand $R \in (Q,S)$ then there exist continua $N, M$ and $q\in Q-R$ and $s \in S-R$ such that $q\in N$, $s \in M$, $N\cup M =X$ and $N \cap M = R$. Since $R\not \in (Q, T)$, it follows that $(T-R) \subset
N$. If $T \subset Q$, then since $T\neq R$, it follows that $T \not \in
(Q,S)$, and so $Q \in (T,S)$. If $T\not \subset Q$, then there is $t \in
(T-Q)\subset (T-R) \subset N$ and it follows that $Q$ separates $s$ from $t$ since $R$ separated them, thus $Q \in (S,T)$ as required.
We are now left with the case where $Q$ is not an inseparable pair, $R
\not \subset Q$, and $T \not \subset Q$ (see Property (2) of the betweenness relation). Thus by definition there exists $r \in R-Q$, $t\in T-Q$ and disjoint continua $M,N$ with $r \in M $ and $t \in N$, $N\cup M = X$ and $N \cap M \subset Q$. If $S \not \subset Q$ then there exists $s\in S-Q$ and either $s \in M$ in which case $Q \in (S,T)$ or $s \in N$ in which case $Q \in (S,R)$. If on the other hand $S \subset Q$, then by Property (2), either $S \not
\in (Q,R)$ implying $Q \in (S,R)$ or $S \not \in (Q,T)$ implying $Q \in
(S,T)$.
For any $R,T \in {{\mathcal R}}$, if $S \in (R,T)$ then $R \not \in
(S,T)$.
First consider the case where $S$ is an inseparable cut pair. We have $r
\in R-S$, $t \in T-S$ and continua $N\ni r$ and $M\ni t $ such that $N\cup M
=X$, $N \cap M = S$. In fact by $R \subset N$ and $T \subset M$.
If $S \not \subset R$, then $|R\cap S|<2$. Since $X$ has no cut points, no point in $S$ is a cut point of $M$, so $M-R$ is connected. Thus $R$ doesn’t separate $S$ from $T$, so $R \not \in (S,T)$.
If $S \subset R$, then by definition since $S \in (R,T)$ then $R \not \in
(S,T)$.
Now consider the case where $S$ is not an inseparable pair. If $R \subset S$, then $R$ is an inseparable pair and $R \not \in (S,T)$ as required. We may now assume that $R \not \subset S$. If $T \subset S$, then $T$ is an inseparable pair and $T\not \in (R,S)$. By $R$ cannot separate a point of $T$ from a point of $S$, since $R\not \subset
S$, it follows that $R \not \in (S,T)$.
We are left with case (2,b) , so $S$ separates a point $r \in R-S$ from a point $t \in T-S$. Thus there exists continua $M,N$ with $r \in M$, $t\in N$, $N \cup M =X$ and $N \cap M \subset S$. In fact by $R \subset M$ and $T \subset N$. Since $X$ has no cut points and $|R \cap S|<2$, then $N-R$ is connected, and so $R \not \in (S,T)$.
We say distinct $R,S \in {{\mathcal R}}$ are adjacent if $(R,S) =
\emptyset$.
If $R,S \in {{\mathcal R}}$ are adjacent then $R \subset S$, $S \subset R$, or (interchanging if need be) $R$ is a necklace and $S$ is maximal inseparable with $[\bar R - R]\cap S \neq \emptyset$.
We need only consider the case where $R,S$ are adjacent and neither is a subset of the other.
First consider the case where one of $R$, $S$ (say $S$) is an inseparable set. There is no maximal inseparable set containing both $R$ and $S$, so there exists $r\in R-S$ and cut pair $A$ separating $r$ from a point of $S$. Notice that $A$ is contained in some necklace $T$. Since $A,T \not \in (R,S)$, it follows that $T=R$ and that $S$ is maximal inseparable.
Let $G$ be the gap of $R$ with $S\subset G$. Let $Q$ be a side of $G$ and $p \in Q$. If $p \not \in S$, then there exists a cut pair $B$ separating $p$ from $S$. Since $(R,S)= \emptyset$, $B$ doesn’t separate $R$ from $S$. It follows by definition of side, that $B$ separates points of $R$ which implies that $B
\subset R$. This contradicts the fact that $Q$ is a side of the gap $G\supset S$. If both sides of $G$ are points, then they form an inseparable cut pair in $(R,S)$. Thus they are not both points so $[\bar
R - R]\cap S \neq \emptyset$.
We are left with the case where $R$ and $S$ are each necklaces with more than $2$ elements. Again let $G$ be the gap of $R$ with $S \subset G$, and let $Q,P$ be sides of $G$. Since $Q \cup P$ is inseparable, there is a maximal inseparable set $A\supset Q \cup P$. It follows that $A \in (R,S)$ which is a contradiction.
Using the pretree structure on ${{\mathcal R}}$, we can put a linear order (two actually) on any interval of ${{\mathcal R}}$. We recall that the order topology on a linearly ordered set $I$ is the topology with basis $I_y=\{x:x>y \}, J_y=\{x:x<y \}$ and $K_{y,z}=\{x:z<x<y
\}$ where $y,z$ range over elements of $I$. The suspension of a Cantor set is a continuum with uncountably many maximal inseparable sets, but this doesn’t happen for inseparable cut pair and necklaces.
\[L:count\] Only countably many elements of ${{\mathcal R}}$ are inseparable pairs or necklaces.
We first show that any interval $I$ of ${{\mathcal R}}$ contains only countably many necklaces and inseparable pairs.
Let $Q$ be the set of all cut pairs in $I$ which have more than two complementary components, union the set of necklaces in $I$. Let $A \in Q$.
- If $A$ is a cut pair, then since $X-A$ has more than two components, and $\smash{\bigcup Q}$ will intersect two of the components, $A$ will separate $(\bigcup
Q)-A $ from some other point of $X$. Using Lemma 3, we find subcontinua $Y,Z$ of $X$ such that $ Y \cup Z=X$, $Y \cap Z = A$ where $(\bigcup Q)-A
\subset Y$. We define the open set $U_A= Z-A$
- If $A$ is a necklace, then $|A|>2$ and there is a cut pair $\{a,b \}
\subset A$ which doesn’t separate $(\bigcup Q)-A$. Using Lemma 3, we find subcontinua $Y,Z$ of $X$ such that $ Y \cup Z=X$, $Y \cap Z = A$ where $(\bigcup Q)- A\subset Y$. We define the open set $U_A= Z-\{a,b\}$
Notice that for any $A,B \in Q$, $U_A \cap U_B= \emptyset$. Since $X$ is Lindelöf, the collection $\{U_A: A \in Q\}$ is countable and therefore $Q$ is countable.
It is more involved to show that inseparable cut pairs $\{a,b\}$ of $I$ such that $X-\{a,b\}$ has 2 components are countable. Let $S$ be the set of inseparable cut pairs $\{a,b\}$ in $I$ such that $X-\{a,b \}$ has 2 components. We argue by contradiction, so we assume that $S$ is uncountable.
Let $\{a,b\}$ be a cut pair of $S$ and let $C_L,C_R$ be the components of $X-\{a,b\} $.
We say that $\{a,b\}$ is a limit pair if there are inseparable cut pairs $\{a_i,b_i\}$ and $\{a_i',b_i'\}$ in $S$ such that $\{a_i,b_i\}\subset C_L$, $\{a_i',b_i'\}\subset C_R$, and for each limit pair $\{ a',b' \}\ne \{a,b \}$ of $S$ one of the two components of $X-\{a',b'\} $ contains at most finitely many elements of the sequences $\{a_i,b_i\}$ and $\{a_i',b_i'\}$.
We claim that there are at most countable pairs in $S$ which are not limit pairs. Indeed if $\{
a,b \}$ is not a limit pair and $I=[x,y]$ let $C_L,C_R$ be the components of $X-\{ a,b \}$ containing, respectively, $x,y$ ($L,R$ stand for left, right). Since $\{ a,b \}$ is not a limit pair for some $\epsilon
>0$ one of the four sets $$C_L\cap B_{\epsilon }(a),\ C_R\cap B_{\epsilon
}(a),\ C_L\cap B_{\epsilon }(b,)\text{ or }C_R\cap B_{\epsilon }(b)$$ intersects the union of all cut pairs of $S$ at either $a$ or $b$.
We remark now that for fixed $\epsilon >0$ there are at most finitely many pairs $\{a,b \}$ in $S$ such that (say) $C_L\cap
B_{\epsilon }(a)$ intersects the union of all cut pairs of $S$ in a subset of $\{a,b \}$. Indeed if we take all pairs $\{ a,b \}
$ with this property the balls $B_{{\punfrac\epsilon 2} }(a)$ are mutually disjoint so there are finitely many such pairs. The same argument applies to each one of the three other sets $C_R\cap
B_{\epsilon }(a),\ C_L\cap B_{\epsilon }(b,)$ or $C_R\cap
B_{\epsilon }(b)$. This implies that non limit cut pairs are countable.
So we may assume $S$ has uncountably many limit pairs. Let $\{c,d \}$ be a limit pair in $S$, let $C_L,C_R$ be the components of $X-\{c,d \}$ and let $\{c_i,d_i \}\in \bar C_L$, $\{c_i',d_i' \}\in \bar C_R$ be sequences of distinct pairs in $S$ provided by the definition of limit pair.
Let $\smash{C_R^i}$ be the component of $X-\{c_i,d_i\}$ containing $c,d$. We claim that there is an $\epsilon $ such that for all $i$ there is some $x_i\in C_L\cap C_R^i$ with $d(x_i,\{ c,d \})> \epsilon $. Indeed this is clear if the accumulation points of the sequences $c_i$ and $d_i$ are not contained in the set $\{c,d \}$. Otherwise by passing to a subsequence and relabelling, if necessary, we may assume that either $c_i\to c,d_i\to d$ or both $c_i,d_i$ converge to, say, $c$.
In the first case we remark that there is a component $C_i$ of $X-\{c,d,c_i,d_i\}$, such that its closure contains both $c,d_i$ or both $d,c_i$. Indeed otherwise $\{c,d,c_i,d_i\}$ is a cyclic subset which is impossible since we assume that $\{ c_{j},d_{j}
\}\ (j>i)$ are all inseparable cut pairs.
Since $d_i\to d$ and $c_i\to c$ there exists $\epsilon >0$ and $x_i\in C_i$ such that $d(x_i,c)>\epsilon$ and $d(x_i,d)>\epsilon $ for all $i$.
In the second case we remark that since $c$ is not a cut point there is some $e >0$ such that for each $i$ there is a component $C_i$ of $C_L\cap \smash{C_R^i}$ with diameter bigger than $e $. It follows that there is an $\epsilon >0$ and $x_i\in C_i$ so that $d(x_i,c)>\epsilon ,d(x_i,d)>\epsilon $ for all $i$.
By passing to subsequence we may assume that $x_i$ converges to some $x_L\in C_L$. Clearly $d(x_L,c)\geq \epsilon
,d(x_L,d)\geq \epsilon $. It follows that $d(x_L,C_R)>0$.
We associate in this way to a limit pair $\{c,d
\}$ in $S$ a point $x_L$ and a $\delta >0$ such that:
1. $x_L\in C_L$
2. $d(x_L,C_R) >\delta $
Since there are uncountably many limit pairs in $S$ there are infinitely many such pairs for which $1,2$ above hold for some fixed $\delta >0$. But then the corresponding $x_L$’s are at distance greater than $\delta $ (by property 2 above). This is impossible since $X$ is compact. Thus $S$ is countable.
Thus for any interval $I$ of ${{\mathcal R}}$, the set of necklaces and inseparable cut pairs of $I$ is countable.
Let $E$ be a countable dense subset of $X$. For any $A$, a necklace with more than one gap or an inseparable pair, there exist $a,b \in E$ separated by $A$. Thus the interval $I=[[a],[b]]$ contains $A$. There are countably many such intervals, so the set of necklaces with more than one gap is countable, and the set of inseparable cut pairs of $X$ is countable.
If a necklace $N$ has less than two gaps, there is an open set $U \subset
N$. By Lindelöf, there are at most countably many such necklaces, and thus there are at most countably many inseparable cut pairs and necklaces in $X$.
The pretree ${{\mathcal R}}$ is preseparable and complete.
Let $[R,W]$ be a closed interval of ${{\mathcal R}}$.
We first show that any bounded strictly increasing sequence in $[R,W]$ converges. Let $(S_n)\subset [R,W]$ be strictly increasing. Let $C_n$ be the component of $X-S_n$ which contains $R$. Let $C= \overline{\bigcup C_n}$. Clearly $C$ is contained in the closure $Q$ of the component of $X-W$ which contains $R$, and so ${\partial}C \subset Q$. Clearly ${\partial}C$ is not a point (since it would by definition be a cut point separating $R$ from $W$). The set ${\partial}C$ is inseparable, and so ${\partial}C
\subset A$ is a maximal inseparable set. It follows that $A \in [R,W]$. If $S_n \not \to A$, then there is $B \in [R,A)$ with $S_n <B$ for all $n$. As before, we have $C$ contained in the closure $D$ of the component of $X-B$ containing $R$. This would imply that $A \in [R,B]$, a contradiction. Thus every strictly increasing sequence in $[R,W]$ converges.
We now show that there are only countably many adjacent pairs in $[R,W]$. We remark that if $A,B$ is an adjacent pair in ${{\mathcal R}}$ at most one of the sets $A,B$ is a maximal inseparable set. By there are only countably many inseparable cut pairs and necklaces in $X$. It follows that there are only countably many inseparable pairs in $[R,W]$.
We have shown thus that ${{\mathcal R}}$ is a complete preseparable pretree. By gluing in intervals to adjacent pairs of ${{\mathcal R}}$ we obtain a real tree $T$ as in .
There is a metric on $T$, which preserves the pretree structure of $T$, such that $T$ is an $\mathbb{R} $–tree. The topology so defined on $T$ is canonical.
We metrize $T$ as in . We metrize first the subtree spanned by the set of inseparable cut pairs and necklaces (which is countable) and then we glue intervals for the inseparable subsets of ${{\mathcal R}}$ which are not contained in this subtree.
We call this $\mathbb{R} $–tree the *JSJ- tree* of the continuum $X$ since in the case $X=\partial G$ with $G$ one-ended hyperbolic our construction produces a simplicial tree corresponding to the JSJ decomposition of $G$.
Combining the two trees
=======================
When $X$ is locally connected, one can combine the constructions of the previous 2 sections to obtain a tree for both the cut points and the cut pairs of a continuum $X$. The obvious application would be to relatively hyperbolic groups, and we should note that in that setting, the action of the tree may be nesting. We explain briefly how to construct this tree.
Let $X$ be a Peano continuum and let ${{\mathcal P}}$ be the cut point pretree.
Let $A\in {{\mathcal P}}$ be a nonsingleton equivalence class of ${{\mathcal P}}$. Then the closure $\bar A$ is a Peano continuum without cut points.
We first show that $\bar A$ is a Peano continuum. Since $\bar A$ is compact, and $X$ is (locally) arcwise connected, it suffices to show that $A$ is convex in the sense that every arc joining points of $\bar A$ is contained in $\bar A$.
Let $a,b$ be distinct points of $ \bar A$ and let $I$ be an arc from $a$ to $b$. Suppose $d \in I-A$. Thus either $c$ is a cut point adjacent to $A$, or there is a cut point $c
\in A$ separating $d$ from $A$, but then $I$ cannot be an arc since it must run through $d$ twice.
Let $a,b,e\in \bar A$. Since $e$ doesn’t separate $a$ from $b$ in $X$, there is an arc in $X$ from $a$ to $b$ missing $e$. By convexity, this arc is contained in $\bar A$, and so $e$ doesn’t separate $a$ from $b$ in $\bar
A$. It follows that the continuum $\bar A$ has no cut points.
Let $A$ be a nonsingleton equivalence class of the cut point pretree ${{\mathcal P}}$, and let $T_A$ be the ends compactification (well, it will not be compact, but we glue the ends to the tree anyway) of the cut pair tree for $\smash{\bar A}$. Since $X$ is locally connected, for any interval $(B,D)\ni A$ there are cut points $a_1, a_2\in \smash{\bar A}$ with $a_1,a_2 \in (B,D)$. Not every point of $\smash{\bar A}$ is contained in one of the defining sets of the cut pair pretree for $\smash{\bar A}$. Some of the points of $\smash{\bar A}$ are not contained in a cut pair, or in a maximal inseparable set with more than two elements, and these appear as ends of the cut pair tree ${{\mathcal R}}_{\bar A}$ for $\smash{\bar A}$.
For each nonsingleton class $A$ of a the cut point tree $T$ we replace $A$ by $T-A$. The end of the component of $T-A$ corresponding to a cut point $a_1 \in \smash{\bar A}$ is glued to the minimal point or end of $T_A$ containing $a_1$.
To see that this construction yields a tree, we use the following Lemma.
The set of classes of ${{\mathcal P}}$ with nontrivial relative JSJ-tree in any interval of ${{\mathcal P}}$ is countable.
Let $[u,v]$ be an interval of ${{\mathcal P}}$ and let $A$ be a class of $[u,v] $ with nontrivial JSJ-tree. Since $X$ is locally connected, $\bar A$ contains some cut point of $[u,v]$. If $c$ is a cut point of $[u,v]$ in $\bar A$ we have that $c,A$ are adjacent elements of $[u,v]$. But we have shown in that there are at most countable such pairs.
Clearly any group of homeomorphisms of $X$ acts on this combined tree.
Group actions
=============
The $\Bbb R$–trees we construct in the previous sections usually come from group boundaries and the group action on them is induced from the action on the boundary, so it’s an action by homeomorphisms. In this section we examine such actions and generalize some results from the more familiar setting of isometric actions.
We recall that the action of a group $G$ on an $\Bbb R$–tree $T$ is called non-nesting if there is no interval $[a,b]$ in $T$ and $g\in G$ such that $g([a,b])$ is properly contained in $[a,b]$. An element $g\in G$ is called elliptic if $gx=x$ for some $x\in T$. If $g$ is elliptic we denote by $\fix(g)$ the fixed set of $g$. An element which is not elliptic is called hyperbolic.
Let $G$ be a group acting on an $\Bbb R$–tree $T$ by homeomorphisms. Suppose that the action is non-nesting. Then if $g$ is elliptic $\fix(g)$ is connected. If $g$ is hyperbolic then $g$ has an “axis”, ie there is a subtree $L$ invariant by $g$ which is homeomorphic to $\Bbb R$.
Let $g$ be elliptic. We argue by contradiction. If $A,B$ are distinct connected components of $\fix(g)$ let $[a,b]$ be an interval joining them ($a\in A, b\in B$). Then $g([a,b])=[a,b]$. Since $[a,b]$ is not fixed pointwise there is a $c\in [a,b]$ such that $g(c)\ne c$. So $g(c)\in [a,c)$ or $g(c)\in (c,b]$. In the first case $g([a,c])\subset [a,c)$ and in the second $g([c,b])\subset (c,b]$. This is a contradiction since the action is non-nesting.
Let $g$ be hyperbolic. If $a\in T$ consider the interval $[a,g(a)]$. The set of all $x\in [a,g(a)]$ such that $g(x)\in
[a,g(a)]$ is a closed set. If $c$ is the supremum of this set then there is no $x\in [c,g(c)]$ such that $g(x)\in [c,g(c)]$. We take $L$ to be the union of all $g^n([c,g(c)])$ ($n\in \Bbb
Z$). Clearly $L$ is homeomorphic to $\Bbb R$ and is invariant by $g$.
Let $G$ be a finitely generated group acting on an $\Bbb R$–tree $T$ by homeomorphisms. Suppose that the action is non-nesting. Then if every element of $G$ is elliptic there is an $x\in T$ fixed by $G$.
We argue by contradiction. Let $G=\langle a_1,a_2,...,a_n \rangle$. If the intersection $\fix(a_1)\cap \fix(a_2)\cap ... \cap \fix(a_n)= \emptyset $ then $\fix(a_i)\cap \fix (a_j)=\emptyset $ for some $a_i,a_j$. We claim that $\smash{a_i^{-1}a_j^{-1}a_ia_j}$ is hyperbolic. Indeed if $\smash{a_i^{-1}a_j^{-1}a_ia_j(x)=x}$ then $a_ia_j(x)=a_ja_i(x)$. Let $A=\fix(a_i),B=\fix(a_j)$. We remark that the smallest interval joining $a_ia_j(x)$ to $A\cup B$ has one endpoint in $A$ while the smallest interval joining $a_ja_i(x)$ to $A\cup B$ has one endpoint in $B$ so these two points can not be equal. This is a contradiction.
Let $G$ be a group acting on an $\Bbb R$–tree $T$ by homeomorphisms. Suppose that the action is non-nesting. Then if every element of $G$ is elliptic $G$ fixes either an $x\in T$ or an end of $T$.
Suppose that $G$ does not fix any $x\in T$. Then there is a sequence $g_n \in G$ and $x_n\in T$ such that $x_n\in \fix(g_n)$, $x_n\notin \fix(g_{n-1})$ and $x_n$ goes to infinity. The sequence $x_n$ defines an end $e$ of $T$. If $r$ is a ray from $x_0\in T$ to $e$ then any $g\in G$ fixes a ray $r_g$ contained in $r$. Indeed if this is not the case, for some $n$, $\fix(g)$ and $\fix(g_n)$ are disjoint. It follows as in the previous proposition that $g^{-1}g_n^{-1}gg_n$ is hyperbolic, which is a contradiction.
|
---
abstract: 'When parameters are varied periodically, charge can be pumped through a mesoscopic conductor without applied bias. Here, we consider the inverse effect in which a transport current drives a periodic variation of an adiabatic degree of freedom. This provides a general operating principle for adiabatic quantum motors, for which we develop a comprehensive theory. We relate the work performed per cycle on the motor degree of freedom to characteristics of the underlying quantum pump and discuss the motors’ efficiency. Quantum motors based on chaotic quantum dots operate solely due to quantum interference, motors based on Thouless pumps have ideal efficiency.'
author:
- 'Raúl Bustos-Marún'
- Gil Refael
- Felix von Oppen
title: Adiabatic quantum motors
---
[*Introduction.—*]{}Popular culture has long been fascinated with microscopic and nanoscopic motors. Perhaps best known is the contest announced by Richard Feynman, who promised a $\$1000$ prize to the developer of an engine that fits a cube of sides 1/64” [@Feynman]. While this feat was carried out shortly thereafter, in 1960, and did not produce an intellectual breakthrough, Feynman’s contest has continued to provide tremendous inspiration to the field of nanotechnology. A prototypical nanomotor was unveiled in 2003, using tiny gold leaves mounted on multi-walled carbon nanotubes, with the carbon layers themselves carrying out the motion [@Zettl]. The motor was driven through AC actuation, and basically relied on classical physics for their operation. As the dimensions of motors are reduced, however, it is natural to expect that quantum mechanics could be used to operate and to optimize nanomotors. In fact, cold-atom-based AC-driven quantum motors have been explored in Refs. [@Haenggi; @Weitz].
Nanomotors can also be actuated by DC driving [@Bailey08; @Dundas09; @Qi09]. A general strategy towards realizing a DC nanoscale motor is based on operating an electron pump in reverse. Consider an electron pump in which the periodic variation of parameters (such as shape, gate voltage, or tunneling strength) originates from the adiabatic motion of, say, a mechanical rotor degree of freedom. To operate this pump as a motor, an applied bias voltage produces a charge current through the pump which, in turn, exerts a force on the mechanical rotor. The existence of quantum pumps [@Brouwer; @Switkes] suggests that by this operating principle, quantum mechanics can be put to work in DC-driven nanomotors. Here, we develop a theory of such adiabatic quantum motors, expressing the work performed per cycle in terms of characteristics of the pump on which the motor relies and discussing the efficiency of quantum motors in general terms.
Our theory relies on progress in the understanding of adiabatic reaction (or current-induced) forces [@Dundas09; @Lu10; @Bode11; @Bode12; @Thomas12] which applies when the mechanical motor degree of freedom is slow compared to electronic time scales and can be treated as classical. Conventionally, adiabatic reaction forces acting on the slow degree of freedom are considered for closed quantum systems [@Berry]. This has recently been extended to situations where the fast degrees of freedom constitute a quantum mechanical scattering problem and thus to mesoscopic conductors [@Bode11; @Bode12; @Thomas12]. The resulting expressions for the reaction forces in terms of the scattering matrix of the mesoscopic conductor allow one to explore the relations to quantum pumping in general terms.
![Generic adiabatic quantum motors building on (a) a quantum pump based on a chaotic quantum dot and (b) a Thouless pump. When a voltage is applied to the pump, the current ’turns the wheel’ and makes the phase angle $\theta$ wind. \[genericmotor\]](model3.pdf "fig:"){width="4cm"}![Generic adiabatic quantum motors building on (a) a quantum pump based on a chaotic quantum dot and (b) a Thouless pump. When a voltage is applied to the pump, the current ’turns the wheel’ and makes the phase angle $\theta$ wind. \[genericmotor\]](model5.pdf "fig:"){width="4.5cm"}
Before developing our general theory, we sketch two conceptual examples of adiabatic quantum motors in Fig. \[genericmotor\]. One motor is based on a chaotic quantum dot operated as a pump [@Brouwer; @Switkes], as illustrated in Fig. \[genericmotor\](a). In this motor, the time-dependent gate voltages varying the shape of the quantum dot are provided by a periodic set of charges situated around the rim of a wheel which approach and modify the quantum dot in two locations. A current flowing through the quantum dot will then produce a rotation of the wheel. Alternatively, we could base a quantum motor on a Thouless pump. A schematic of such a motor is shown in Fig. \[genericmotor\](b). A single-channel quantum wire is located next to a conveyor belt with periodic attached charges (alternatively, a cogwheel with periodically spaced and electrically charged teeth). The charges induce a periodic potential in the quantum wire which slides as the conveyor belt or cogwheel turns. It is well-known from the seminal works of Thouless [@Thouless] that when the Fermi energy lies in an energy gap, such pumps transport integer amounts of charge per cycle (i.e., when the periodic potential slides by one period). An alternative physical realization of a Thouless motor is based on a helical wire in an electric field [@Qi09].
[*Output power of adiabatic quantum motors.—*]{}We start by deriving a general expression for the output power of an adiabatic quantum motor. The motor consists of a mesoscopic conductor with left (L) and right (R) lead, described within the independent-electron approximation by an electronic scattering matrix. In the adiabatic quantum motors of Fig. \[genericmotor\], the mesoscopic conductor is coupled to a single (classical) angle degree of freedom $\theta$, as described through the dependence $S({\theta})$ of the $S$-matrix on the motor coordinate ${\theta}$. More generally, the mesoscopic conductor could be coupled to several mechanical motor degrees of freedom $X_\nu$ ($\nu=1,2\ldots N$) so that $S=S({\bf X})$.
Retaining the dependence on several mode coordinates ${\bf X}$ for generality, the adiabatic reaction force ${\bf F}({\bf X})$ on the motor degrees of freedom can be expressed in terms of the $S$-matrix of the mesoscopic conductor [@Bode11; @Bode12; @Thomas12], $$\label{force}
F_\nu({\bf X}) = \sum_\alpha \int \frac{d\epsilon}{2\pi i} f_\alpha
{\rm Tr} \left(\Pi_\alpha S^\dagger \frac{\partial S}{\partial X_\nu} \right).$$ Here, $f_\alpha(\epsilon)$ denotes the Fermi distribution function in lead $\alpha = {\rm L,R}$ with chemical potential $\mu_\alpha$ and $\Pi_\alpha$ is a projector onto the scattering channels in lead $\alpha$. It has been shown [@Dundas09; @Lu10; @Bode11; @Bode12; @Thomas12] that this adiabatic reaction force need not be conservative when the electronic conductor is out of equilibrium. Thus, the work per cycle performed by this force is nonzero and given by $$W_{\rm out} = \oint d{\bf X}\cdot {\bf F}({\bf X}) .
\label{OutWork}$$ Note that in the absence of an applied bias, $W_{\rm out}=0$ (i.e., the force is conservative). In this case, $f_\alpha(\epsilon)=f(\epsilon)$ and $\sum_\alpha \Pi_\alpha = {\bf 1}$, and inserting Eq. (\[force\]) into Eq. (\[OutWork\]) yields $$W_{\rm out} = \oint d{\bf X}\cdot \nabla_{\bf X} \int \frac{d\epsilon}{2\pi i}\, f(\epsilon)\,
{\rm Tr} \ln S(\epsilon) = 0 .$$ The work performed by the adiabatic quantum motor per cycle is nonzero when a finite bias $V$ is applied. In linear response, Eq. (\[force\]) yields $$\begin{aligned}
W_{\rm out} = \frac{ieV}{4\pi }\oint d{\bf X}\cdot \!\!\int {d\epsilon} f^\prime(\epsilon)
{\rm Tr} [ (\Pi_L - \Pi_R) S^\dagger \frac{\partial S}{\partial {\bf X}} ],
\label{Eq4}\end{aligned}$$ where we used that $W_{\rm out}=0$ in equilibrium and expanded Eq. (\[force\]) to linear order in the applied bias $V$.
Using Brouwer’s formula [@Brouwer] which expresses the charge $Q_p$ pumped during one cycle of ${\bf X}$ in terms of the electronic $S$-matrix $S({\bf X})$, the right-hand side of Eq. (\[Eq4\]) can be identified as $$W_{\rm out} = Q_p V.
\label{output}$$ Remarkably, the output of the nonequilibrium device is described by $Q_p$, which characterizes the underlying quantum pump in [*equilibrium*]{}. Eq. (\[output\]) shows that the mechanical output of the motor per cycle originates from the fact that a charge $Q_p$ is pumped through the system with every revolution of the motor, and that this pumped charge gains an electrical energy $Q_pV$ due to the applied bias. Thus, the average output power of the motor is $$P_{\rm out} = Q_p V/\tau,
\label{OutputP}$$ where $\tau$ denotes the motor’s cycle period. We also emphasize that Eq. (\[output\]) identifies quantum pumping as the physical origin of the nonconservative nature of the adiabatic reaction force in Eq. (\[force\]).
[*Efficiency of adiabatic quantum motors.—*]{}The applied bias $V$ induces a slowly-varying DC charge current $I$ in the adiabatic quantum motor. Thus, on average, operation of the motor requires an input power of $P_{\rm in} = \overline{I}V$. (The overline denotes an average over a single cycle). The efficiency $\eta $ of the adiabatic quantum motor is then naturally defined as the ratio of output to input power, $$\eta = P_{\rm out}/P_{\rm in} = Q_p/\overline{I}\tau.$$ Here, we have used Eq. (\[OutputP\]) in the second equality.
For adiabatic motor degrees of freedom, the current $I$ is made up of two contributions: the pumped charge and the transport current induced by the applied bias $V$. If $G({\bf X})$ denotes the conductance of the device for fixed ${\bf X}$, the linear-response current averaged over one cycle is $$\overline{I} = \overline{G({\bf X})} V + \frac{Q_p}{\tau}.
\label{current}$$ Note that the pumping current also depends on voltage through the motor’s operating frequency (as characterized by $\tau$). We note in passing that this expression can be obtained more formally, see Ref. [@Bode11].
With Eq. (\[current\]), the quantum motor’s efficiency becomes $$\eta = \frac{1}{1+ \overline{G} V\tau/Q_p }.
\label{Efficiency}$$ Interesting conclusions can be drawn directly from this expression: (i) Quantum motors can operate entirely on the basis of quantum interference and become ineffective due to phase-breaking processes, justifying the term [*quantum*]{} motor. A conceptually interesting example is the motor in Fig. \[genericmotor\](a) which is based on a chaotic quantum dot. It is well-known that the charge pumped through chaotic quantum dots (and hence the output power of the corresponding quantum motor) vanishes with increasing phase breaking. (ii) Quantum motors can have ideal efficiency $\eta=1$, implying perfect conversion of electrical into mechanical energy. Indeed, this can be realized by motors based on Thouless pumps; when the Fermi energy lies in the gap, the conductance vanishes while the pumped charge is quantized to integer multiples of $e$. Thus, Eq. (\[Efficiency\]) yields $\eta=1$, making Thouless pumps [*ideal*]{} adiabatic quantum motors.
[*Motor dynamics*]{}.—The output power of a quantum motor depends on its dynamics through the cycle period $\tau$. Here, we discuss this for the simplest case, in which both the driving force and the load $F_{\rm load}$ acting on the angular motor degree of freedom $\theta$ are independent of the state of the motor. (This is realized for Thouless motors, but typically not for motors based on chaotic quantum dots.) If the motor degree of freedom is subject to damping with damping coefficient $\gamma$, the steady-state velocity of the motor follows from the (classical) condition $$\gamma\dot \theta = \frac{Q_p V}{2\pi} - F_{\rm load}.
\label{EOM}$$ Thus, we obtain for the cycle period of the motor $\tau = {2\pi}/{|\dot \theta|} = {(2\pi)^2\gamma }/({Q_pV - 2\pi F_{\rm load}})$. We can use Eq. (\[current\]) to eliminate $V$ in favor of the current $\overline{I}$. This yields $$\frac{1}{\tau} = \frac{Q_p\overline{I} - 2\pi F_{\rm load} \overline{G}}{Q_p^2 + (2\pi)^2 \gamma \overline{G}}.
\label{period}$$ For an ideal Thouless motor with $\overline{G}=0$ [@Qi09], this yields the relation $1/\tau = \overline{I}/Q_p$. This is a direct consequence of the fact that in this case, the entire current passing the device must be due to pumping. More generally, this remains a good approximation as long as $\overline{G} \ll Q_p^2/(2\pi)^2\gamma $. This result also implies that the maximum load on the motor is given by $F_{\rm load}^{\rm max} = Q_p\overline{I}/2\pi\overline{G}$.
[*Thouless motor.—*]{}Thouless motors provide an instructive example not only because they realize ideal quantum motors but also because they allow for a thorough analytical discussion. Consider a single-channel quantum wire subject to a periodic potential of period $a$, as described by the Hamiltonian = [p]{}\^2/2m + 2(2x/a + ) (L/2-|x|) \[TH\] The periodic potential of strength $2\Delta$ acting for $-L/2 < x < L/2$ arises, e.g., from a periodic set of charges situated along a conveyor belt or cogwheel so that the nearby electrons in the wire experience an electrostatic potential \[cf. Fig. \[genericmotor\](b)\]. This potential slides as the cogwheel turns and the mechanical variable $\theta$ varies by $2\pi$ as the teeth of the cogwheel advance by one spacing $a$.
When the chemical potential $\mu$ is chosen such that the Fermi wavevector $k_F=({2m\mu/\hbar^2})^{1/2}$ is close to $k_0=\pi/a$, one can linearize the Hamiltonian for momenta close to $\pm k_0$. This results in an effective Hamiltonian $\H$ with counterpropagating linear channels and backscattering due to the periodic potential. Measuring momenta from $\pm k_0$ and energies from $\hbar^2 k_0^2/2m$, one has = v\_F p \^z + ł(\^x + \^y ) (L/2-|x|). \[TH2\] Here, the $\sigma^i$ denote the Pauli matrices in the space of the counterpropagating channels. We do not include the real electron spin for simplicity.
![Efficiency of the Thouless motor vs Fermi energy for $L=0.75\mu$m and $v=10^5$m/s. From top to bottom, the curves correspond to dissipative loads $\gamma/\hbar=1/2\pi,\,0.5,\,1$. The motor has ideal efficiency ($\eta = 1$) when the Fermi energy lies in the gap, $|E_F|<\Delta$, and the length is taken to infinity. The inset shows the cycle frequency for a current-biased Thouless motor vs Fermi energy. \[efficiency\]](eta-Ef-02.pdf){width="10.cm"}
Within the linearized model, the adiabatic $S$-matrix $S(\theta)$ can be readily obtained analytically. We start with the transfer matrix $M$ from $x=L/2$ to $x=-L/2$. Since the model is linear in momentum $p$, this can be done by analogy with the time-evolution operator in quantum mechanics which yields M ={ -\^z }. This can be rewritten as ${M}=\cos\lambda_L-i\sigma_{\rm eff}\sin\lambda_L$, where $\sigma_{\rm eff} = [{E \sigma^z -i\Delta\cos\theta\sigma^y+i\Delta\sin\theta\sigma^x}]/[{E^2-\Delta^2}]^{1/2}$ and $\lambda_L=(L/\hbar v_F) [E^2-\Delta^2]^{1/2}$. Note that $\sigma_{\rm eff}^2={\bf 1}$.
To obtain the $S$-matrix from the transfer matrix $M$, we first assume that there is only an outgoing wave on the right. Then, the wavefunction on the left is $(i_L,o_L)^T =M (o_R, i_R)^T=\l(M_{11} o_R,M_{21} o_R\rr)^T$, where $i$ and $o$ refer to the in- and outgoing waves, respectively. This immediately implies that the transmission $S_{21}$ is $1/M_{11}$, and the reflection $S_{11}$ is $M_{21}/M_{11}$. Repeating the same arguments with only an outgoing wave on the left, we also find $S_{22}= (M^{-1})_{12}/(M^{-1})_{22}$ and $S_{12} = 1/(M^{-1})_{22}$. With $M_{11}=(M^{-1})_{22}=\cos\lambda_L-i\frac{E}{\sqrt{E^2-\Delta^2}}\sin\lambda_L$, this yields S =ł( -i e\^[i]{} \_L & 1\
1 & -i e\^[-i]{} \_L ). We can now use this $S$-matrix to obtain explicit expressions for the efficiency of the Thouless motor.
Using the Landauer formula, the conductance for a Fermi energy $E_F$ takes the form G = = \[conductance\] In accord with the fact that the periodic potential opens a gap, the conductance is exponentially small in $L$ for $|E_F|<\Delta$ and becomes oscillatory and finite in $L$ for $|E_F| > \Delta$. Similarly, we can obtain the pumped charge in the standard way from Brouwer’s formula [@Brouwer] (evaluated at zero temperature and for an angular degree of freedom), Q\_p = . \[ThoulessCharge\] For Fermi energies in the gap, the charge pumped is quantized to $e$ with exponential precision. When the Fermi energy is outside the gap, $|E_F|>\Delta$, the charge is no longer quantized and smaller than $e$.
We can combine these results to obtain an explicit expression for the efficiency of the Thouless motor. To do so, we note that the force acting on the motor is independent of $\theta$. Thus, we can combine Eqs. (\[Efficiency\]), (\[period\]), (\[conductance\]), and (\[ThoulessCharge\]) to obtain (for zero load, $F_{\rm load}=0$) = \[eta1\] In Fig. \[efficiency\], we plot the efficiency of the Thouless motor as a function of the Fermi energy. As can be seen from Eq. (\[eta1\]), the efficiency is exponentially close to unity when the Fermi energy is within the gap. For this range of Fermi energies, the Thouless motor is an ideal adiabatic quantum motor. When the Fermi energy moves out of the energy gap, the efficiency is oscillatory with an algebraically dropping amplitude. In this regime, Fabry-Perot interference alone produces peaks in the efficiency, which appear when the reflection coefficients are maximal. The inset of Fig. \[efficiency\] also shows the cycle frequency of the Thouless motor, for a given current and zero load, as a function of Fermi energy, cf. Eq. (\[period\]).
[*Intrinsic damping.—*]{}So far, we have treated the damping coefficient $\gamma$ of the motor degree of freedom as phenomenological. However, in addition to extrinsic, purely mechanical friction, there is a contribution to $\gamma$ which arises intrinsically from the coupling to the electronic system. As shown recently, this intrinsic damping $\gamma_{\rm int}$ can also be obtained from the electronic $S$-matrix [@Bode11; @Bode12; @Thomas12]. Restricting attention to small bias voltages, we can approximate $\gamma_{\rm int}$ by its equilibrium value, $\gamma_{\rm int} = (\hbar/4\pi){\rm tr}[(\partial S^\dagger/\partial \theta)(\partial S/\partial \theta)]$. This is readily evaluated for the Thouless motor when the Fermi energy is in the vicinity of the fundamental gap. We find that the intrinsic damping can be expressed in terms of the pumped charge, $\gamma_{\rm int} = (\hbar/2\pi e) Q_p$. Quite surprisingly, the electronic system induces finite mechanical damping even when the Fermi energy lies in the gap (and $Q_p=e$). We interpret this damping as arising from forming plasmon excitations in the leads when pumped charge enters or leaves. When the Fermi energy of a current-biased Thouless motor lies inside the fundamental gap, the motor (without load) rotates at angular frequency $\omega = 2\pi I/e$, which, from Eq. (\[EOM\]), gives a friction-induced voltage drop of $V = I (2\pi)^2 \gamma /e^2 $. The existence of the intrinsic friction implies that for a given current, there is a minimal voltage of $V = (h/e^2)I$ at which the Thouless motor described by Eq. (\[TH2\]) can operate.
At first sight, the intrinsic damping may seem to negate the possibility of an ideal quantum motor when the motor is subject to a load. Indeed, the electrical input power is then split between the power consumed by the load, $P_{\rm load} = F_{\rm load}\dot\theta$, and the power dissipated by damping, $P_{\gamma} = \gamma {\dot \theta}^2$. Nevertheless, for a quantized Thouless pump, $\dot\theta = 2\pi I/Q_p$, so that $P_{\rm load} \propto I$ while $P_{\gamma} \propto I^2$. Hence, the power dissipated by damping becomes negligible at small currents, and the load efficiency $\eta_{\rm load} = P_{\rm load}/P_{\rm in}$ can be made arbitrarily close to unity by operating the motor at low currents.
[*Conclusions.—*]{}Motion at the nanoscale tends to be dominated by fluctuations. It is an important challenge to develop schemes to generate directed motion in nanoscale devices [@Kudernac; @Tierny; @Perera]. Here, we investigated a general strategy to this effect which is based on operating quantum pumps in reverse. We developed a corresponding theory which expresses the output power and the efficiency of such adiabatic quantum motors to characteristics of the pumps on which they are based. The concept of adiabatic quantum motors offers numerous possibilities for future research. Interesting directions include motors based on electron pumps which involve electron-electron interactions as well as systems in which the motor degree of freedom is itself quantum mechanical.
We acknowledge discussions with P. Brouwer as well as support by the Deutsche Forschungsgemeinschaft through SFB 658, the Humboldt Foundation through a Bessel Award, the Packard Foundation, and the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center with support of the Gordon and Betty Moore Foundation.
[12]{}
R.P. Feynman, Engineering and Science [**23**]{}, 22 (1960).
A.M. Fennimore, T.D. Yuzvinsky, W.-Q. Han, M.S. Fuhrer, J. Cumings, and A. Zettl, Nature [**424**]{}, 408 (2003).
A.V. Ponomarev, S. Denisov, and P. Hänggi, Phys. Rev. Lett. [**102**]{}, 230601 (2009).
T. Salger, S. Kling, T. Hecking, C. Geckeler, L. Morales-Molina, and M. Weitz, Science [**326**]{}, 1241 (2009).
S.W.D. Bailey, I. Amanatidis, and C.J. Lambert, Phys. Rev. Lett. [**100**]{}, 256802 (2008).
D. Dundas, E.J. McEniry, and T.N. Todorov, Nature Nanotech. [**4**]{}, 99 (2009).
X.-L. Qi and S.C. Zhang, Phys. Rev. B [**79**]{}, 235442 (2009).
P.W. Brouwer, Phys. Rev. B [**58**]{}, R10135 (1998).
M. Switkes, C.M. Marcus, K. Campman, A.C. Gossard, Science [**283**]{}, 1905 (1999).
J.T. Lü, M. Brandbyge, and P. Hedegard, Nano Lett. [**10**]{}, 1657 (2010).
N. Bode, S. Viola Kusminskiy, R. Egger, F. von Oppen, Phys. Rev. Lett. [**107**]{}, 036804 (2011).
N. Bode, S. Viola Kusminskiy, R. Egger, F. von Oppen, Beilstein J. Nanotechnol. [**3**]{}, 144 (2012).
M. Thomas, T. Karzig, S. Viola Kusminskiy, G. Zarand, F. von Oppen, Phys. Rev. B [**86**]{}, 195419 (2012).
M. V. Berry, in [*Geometric Phases in Physics*]{}, edited by A. Shapere and F. Wilczek (World Scientific, Singapore, 1989).
D.J. Thouless, Phys. Rev. B [**27**]{}, 6083 (1983).
T. Kudernac, N. Ruangsupapichat, M. Parschau, B. Macia, N. Katsonis, S.R. Harutyunyan, K.-H. Ernst, and B.L. Feringa, Nature [**479**]{}, 208 (2011).
H.L. Tierney, C.J. Murphy, A.D. Jewell, A.E. Baber, E.V. Iski, H.Y. Khodaverdian, A.F. McGuire, N. Klebanov, and E.C.H. Sykes, Nature Nanotech. [**6**]{}, 625 (2011).
U.G.E. Perera, F. Ample, H. Kersell, Y. Zhang, G. Vives, J. Echeverria, M. Grisolia, G. Rapenne, C. Joachim, and S.-W. Hla Nature Nanotech. [**8**]{}, 46 (2013).
|
---
abstract: 'Path integrals play a crucial role in describing the dynamics of physical systems subject to classical or quantum noise. In fact, when correctly normalized, they express the probability of transition between two states of the system. In this work, we show a consistent approach to solve conditional and unconditional Euclidean (Wiener) Gaussian path integrals that allow us to compute transition probabilities in the semi-classical approximation from the solutions of a system of linear differential equations. Our method is particularly useful for investigating Fokker-Planck dynamics, and the physics of string-like objects such as polymers. To give some examples, we derive the time evolution of the $d$-dimensional Ornstein-Uhlenbeck process, and of the Van der Pol oscillator driven by white noise. Moreover, we compute the end-to-end transition probability for a charged string at thermal equilibrium, when an external field is applied.'
author:
- Giulio Corazza
- Matteo Fadel
bibliography:
- 'paper.bib'
title: Normalized Gaussian Path Integrals
---
#### Introduction.
– Path integrals are an essential tool in many branches of physics and mathematics [@BookPapa; @BookSchulman; @BookChaichian; @BookZinn; @BookKleinert]. Originally introduced by Wiener as a method to study Brownian motion [@Wiener1; @Wiener2], their formalism was significantly developed by Feynman in the context of quantum mechanics [@FEY0; @FEY]. Since then, path integrals revealed themselves to be a powerful method for the investigation of systems subject to classical or quantum fluctuations, therefore finding a plethora of different applications.
In many relevant situations one is interested in evaluating transition ([*i.e.* ]{}conditional) probabilities. Namely, the probability for a system to be in a specific final state, given its initial state. Path integrals are precisely tailored to answer such questions, by expressing transition probabilities as an infinite weighted sum over all possible trajectories passing through both states. Typical examples where this formulation arises naturally include the stochastic motion of particles in diffusion processes, and the dynamics of quantum particles and fields.
It is worth emphasizing that the “paths” entering a path integral do not need to be the trajectories of a moving particle, but they can also be the stationary configurations of string-like objects [@Edwards1965; @Edwards1967; @Freed; @PapaPoly]. In this context, transition probabilities represent the probability of finding the string’s endpoints in specific positions. This observation turns out extremely useful for the study of organic and inorganic polymers at thermal equilibrium, such as chains of molecules ([*e.g.* ]{}DNA, actin filaments) and flexible rods [@Winkler94; @Winkler97; @Vilgis00; @LUD].
Despite their intuitive interpretation, path integrals are in general difficult to compute. Among several different strategies to circumvent this issue, the semiclassical (quadratic) approximation is one of the most adopted. In brief, the idea consists in approximating the weights for the paths so that a Gaussian integral is obtained. The solution is then straightforward for conditional path integrals (where both extremal points are fixed), while it often remains non-trivial for the unconditional case (where only the starting point is fixed). Addressing this remaining problem is of special interest for expressing transition probabilities that are properly normalized.
Here we focus on Euclidean (Wiener) path integrals, and propose a consistent method to compute from them transition probabilities in the semiclassical approximation. Our approach is based on the generalization of a result by Papadopulos [@PAP1], which allows us to evaluate both conditional and unconditional path integrals for general quadratic Lagrangians, from the solutions of the Euler-Lagrange equations and of a system of second-order nonlinear differential equations. Furthermore, we then show that the latter can be related to a simpler system of linear differential equations, by exploiting a link with the Jacobi equation. Interestingly, our study also sheds light on the relation between the choice for the discretization of continuous paths, and the path integral measure.
Our results are of interest for studying the dynamics of stochastic processes, such as the one described by the Fokker-Planck equation, and for investigating equilibrium configurations of string-like objects. This is illustrated here with three concrete examples. First, we show how to recover the transition probability for a $d$-dimensional Ornstein-Uhlenbeck process [@OU; @FALKOFF; @VATI]. Second, we investigate the non-linear Van der Pol oscillator driven by white noise, for which transition probabilities are not known analytically due to its chaotic dynamics [@NAESS]. Third, we compute in one spatial dimension the end-to-end transition probability for an elastic and electrically charged string at thermal equilibrium, when an external electric field is applied.
#### Statement of the problem and main result.
– Consider a system in configuration $q(\tau)\in\mathbb{R}^{d}$, whose dynamics is described by the Lagrangian $\mathcal{L}(q,\dot{q},\tau)$. Our goal is to calculate the transition probability for the system of being in final state $q(t)=q$, given its initial state $q(t_0)=q_0$, namely the conditional probability $\rho(q,t \vert q_0,t_0)$ satisfying $\rho(q,t_0 \vert q_0,t_0)=\delta(q-q_0)$. A prescription for this calculation is given by the path integral formalism, which allows us to write $$\label{pathint}
\rho(q,t \vert q_0,t_0) := \dfrac{\mathcal{K}}{\mathcal{N}} = \frac{1}{\mathcal{N}}\int\limits_{q(t_0)=q_0}^{q(t)=q}{\mathcal{D}q\,e^{-S(q,\dot{q})}} \;,$$where the (conditional) integration is taken over all path with fixed extremal points, weighted depending on the action $S(q,\dot{q}):=\int_{t_0}^t{d\tau\,\mathcal{L}(q,\dot{q},\tau)}$, and normalized by $\mathcal{N}$ to ensure $$\label{norm}
\int_{\mathbb{R}^{d}}{\rho(q,t \vert q_0,t_0)}\,{\text{d}}q = 1 \;, \qquad \forall t\geq t_0 \;.$$
From Eqs. (\[pathint\]) and (\[norm\]), we see that the normalization can be formally written as the (unconditional) path integral $$\label{normex}
\mathcal{N} = \int\limits_{q(t_0)=q_0}{\mathcal{D}q\,e^{-S(q,\dot{q})} } \;,$$where now the integral is over all paths satisfying only the initial condition $q(t_0)=q_0$.
For typical cases of interest, we are often in the situation where $\mathcal{L}$ is complicated enough that closed-form solutions for $\mathcal{K}$ and $\mathcal{N}$ do not exist. A standard technique to simplify part of the problem consist in taking the semi-classical approximation, where the action is expanded to second order around an isolated minimum. This allows us to approximate $\mathcal{K}$ by a solvable Gaussian integral, but the evaluation of $\mathcal{N}$ remains non-trivial because of the free boundary condition $q(t)$. The latter difficulty is often circumvented through demanding Monte-Carlo integrations, or by setting $\mathcal{N}=1$ and considering in $\mathcal{K}$ an effective (Onsager-Machlup) Lagrangian containing additional terms that ensure normalization [@GRAHAM; @FALKOFF; @HAKEN]. Our main result consists in solving this problem in a more general situation. In brief, we formulate a consistent approach to solve in the semi-classical approximations both path integrals appearing in Eq. , by relating their solutions to the solutions of a system of linear differential equations. The procedure we propose is the following.
As a starting point, in order to ensure the accuracy of the semi-classical approximation, let us restrict to Lagrangian functions where the leading order term for the second variation of the action, $G(\tau):= \frac{\partial^2 \mathcal{L}}{\partial \dot{q}^2}$, is independent on $q, \dot{q}$. This assumption is still general enough to include most cases of interest. On the other hand, we consider in the second variation arbitrary $V:= \frac{\partial^2 \mathcal{L}}{\partial {q}^2}$ and cross term matrix $A:= \frac{\partial^2 \mathcal{L}}{\partial \dot{q}\partial q}$.
Following the idea behind the semi-classical approximation, the first step of our method consists in deriving from the Euler-Lagrange equations for $\mathcal{L}$ two solutions: i) an isolated minimizer of the action $q^{\scaleto{D}{3.5pt}}(\tau)$, satisfying the Dirichlet boundary conditions $q^{\scaleto{D}{3.5pt}}(t_0)=q_0$ and $q^{\scaleto{D}{3.5pt}}(t)=q$, and ii) an isolated minimizer $q^{\scaleto{N}{3.5pt}}(\tau)$, satisfying $q^{\scaleto{N}{3.5pt}}(t_0)=q_0$ and the Neumann natural boundary condition $\frac{\partial \mathcal{L}}{\partial \dot{q}}(t)=0$. Then, the second step consists in deriving a set of solutions $W^{\scaleto{D}{3.5pt}(\scaleto{N}{3.5pt})}(\tau)$ of the Jacobi equation for the second variation of the action on the Dirichlet (Neumann) minimum. These can be obtained from the Hamiltonian formulation of the Jacobi equation, together with the necessary boundary conditions, as solutions of
$$\label{fin3}
\begin{cases}
\frac{{\text{d}}}{{\text{d}}\tau}\begin{pmatrix}
{{W^{\scaleto{D}{3.5pt}}}} \\
{{M^{\scaleto{D}{3.5pt}}}}
\end{pmatrix}
=J{{E^{\scaleto{D}{3.5pt}}}}\begin{pmatrix}
{{W^{\scaleto{D}{3.5pt}}}} \\
{{M^{\scaleto{D}{3.5pt}}}}
\end{pmatrix}
\\
\\
\begin{pmatrix}
{{W^{\scaleto{D}{3.5pt}}}} \\
{{M^{\scaleto{D}{3.5pt}}}}
\end{pmatrix}
(t)
=J\begin{pmatrix}
{\mathbb{1}} \\
\mathbb{0}
\end{pmatrix}
\end{cases} \,
\begin{cases}
\frac{{\text{d}}}{{\text{d}}\tau}\begin{pmatrix}
{{W^{\scaleto{N}{3.5pt}}}} \\
{{M^{\scaleto{N}{3.5pt}}}}
\end{pmatrix}
=J{{E^{\scaleto{N}{3.5pt}}}}\begin{pmatrix}
{{W^{\scaleto{N}{3.5pt}}}} \\
{{M^{\scaleto{N}{3.5pt}}}}
\end{pmatrix}
\\
\\
\begin{pmatrix}
{{W^{\scaleto{N}{3.5pt}}}} \\
{{M^{\scaleto{N}{3.5pt}}}}
\end{pmatrix}
(t)
=J\begin{pmatrix}
\mathbb{0} \\
{\mathbb{1}}
\end{pmatrix}
\end{cases}$$
where $M^{\scaleto{D}{3.5pt}(\scaleto{N}{3.5pt})}$ is the conjugate variable under the Legendre transform, $J=\bigl( \begin{smallmatrix} \mathbb{0} & \mathbb{1}\\ -\mathbb{1} & \mathbb{0}\end{smallmatrix}\bigr)$ is the symplectic matrix, and $E^{\scaleto{D}{3.5pt}(\scaleto{N}{3.5pt})}$ is the symmetric matrix driving the system, which reads $$E^{\scaleto{i}{5pt}} = \begin{pmatrix}
{A}^T{G}^{-1}{A}-{V}\,\,\, & -{A}^T{G}^{-1}\\
-{G}^{-1}{A}\,\,\, & {G}^{-1}
\end{pmatrix}\Bigg|_{q^{\scaleto{i}{5pt}}}\;, \quad {\scaleto{i}{5pt}}={\scaleto{D}{3.5pt},\scaleto{N}{3.5pt}} \;.$$
Finally, our main result consists in showing that we can write the semi-classical approximation of the transition probability Eq. as $$\label{fin2}
\rho^{\text{sc}}(q,t \vert q_0,t_0) = e^{S(q^{\scaleto{N}{3.5pt}})-S(q^{\scaleto{D}{3.5pt}})}\sqrt{\det\left[\frac{1}{2\pi}\frac{{W^{\scaleto{N}{3.5pt}}}}{{W^{\scaleto{D}{3.5pt}}}}(t_0)\right]} \;,$$ where $S(q^{\scaleto{D}{3.5pt}(\scaleto{N}{3.5pt})})$ is the action evaluated on the Dirichlet (Neumann) minimum, and $W^{\scaleto{D}{3.5pt}(\scaleto{N}{3.5pt})}$ are the solutions of Eq. .
In general, due to the semi-classical approximation, we have $\rho(q,t \vert q_0,t_0) \approx \rho^{\text{sc}}(q,t \vert q_0,t_0)$. However, let us mention that in the particular case where the Lagrangian is a quadratic function of $q$ and $\dot{q}$, then no error is introduced by the semi-classical approximation, and $\rho(q,t \vert q_0,t_0) = \rho^{\text{sc}}(q,t \vert q_0,t_0)$. In the latter case the matrix $E$ is now independent of the particular minimum, and Eq. simplifies further.
To summarize, our approach for computing the transition probability Eq. consists in taking the ratio of the conditional and unconditional Wiener path integrals in the semi-classical approximation, to then express their solutions in terms of the solutions of a set of ordinary differential equations. In the following we present the derivation of our results, and we apply them to three relevant examples.
#### Description of the method.
– The evaluation of $\mathcal{K}$ in the semi-classical approximation is a standard textbook technique, and for our Euclidean path integrals it is also known as Laplace asymptotic method [@PIT]. The idea is to first Taylor expand the action $S(q(\tau))$ to second order around the Dirichlet minimum $q^{\scaleto{D}{3.5pt}}(\tau)$, exploiting the fact that the first order variation on a minimum is zero. Here, the existence and stability of $q^{\scaleto{D}{3.5pt}}(\tau)$ are assumed. In particular, the second property involves [*e.g.* ]{}the conjugate point theory, as discussed in [@FOM]. Then, from the second variation of the action computed in $q^{\scaleto{D}{3.5pt}}(\tau)$, namely
$$\label{secv}
\delta^2 S(q^{\scaleto{D}{3.5pt}},y) = \int_{t_0}^t {\text{d}}\tau\, \bigg( \dot{y}^T{{G}}\dot{y}+2\dot{y}^T{{A^{\scaleto{D}{3.5pt}}}}{y}+{y}^T{{V^{\scaleto{D}{3.5pt}}}}{y}\bigg) \;,$$
and from the discretisation of $\tau\in[t_0,t]$ into $n$ intervals of length $\varepsilon=(t-t_0)/n$, the semi-classical approximation for $\mathcal{K}$ reads $$\label{appx}
\mathcal{K}^{\text{sc}} = e^{-S(q^{\scaleto{D}{3.5pt}})} \lim_{n\rightarrow\infty} I^{\scaleto{D}{3.5pt}}_n \;,$$
$$\label{num}
I^{\scaleto{D}{3.5pt}}_n=\displaystyle{\int\limits_{y(t_0)=0}^{y(t)=0}{\prod\limits_{j=1}^{n}\left[\frac{\det{(G_j)}}{(2\pi\varepsilon)^{d}}\right]^{\frac{1}{2}}\prod\limits_{j=1}^{n-1}{{\text{d}}y_j}\,\,\, e^{-\frac{\varepsilon}{2} \sum\limits_{j=0}^{n}\delta^2 S(q^{\scaleto{D}{3.5pt}},y)_j}}}.$$
The subscript $j$ indicates that the associated term is evaluated in $\tau_j=t_0+j\varepsilon$, [*e.g.* ]{}$y_j=y(\tau_j)$. Moreover, the integration boundaries come from the fact that $y(\tau)$ represents a perturbation around the minimum $q^{\scaleto{D}{3.5pt}}(\tau)$, and as such it must satisfy null Dirichlet boundary conditions. Let us mention that the products in Eq. give the integration measure for the integral, which is here a conditional Wiener measure [@BookChaichian].
At this point, it is straightforward to solve Eq. using the method presented by Papadopoulos in Ref. [@PAP1]. This results in the following Gelfand–Yaglom-type expression $$\label{pap}
\mathcal{K}^{\text{sc}} = e^{-S(q^{\scaleto{D}{3.5pt}})}\det\left[2\pi {D^{\scaleto{D}{3.5pt}}}(t)\right]^{-\frac{1}{2}} \;,$$ where ${D^{\scaleto{D}{3.5pt}}}(\tau)$ solves the second order nonlinear differential equation (omitting superscripts $\scaleto{D}{3.5pt}$ for $A$, $V$ and $D$)
$$\label{papeq}
\begin{split}
\frac{d}{d\tau}\left[\dot{D}{{G}}\right]+{{D}}\dot{A}^{(s)}-{{D}}\left[{{V}}+{{A}}^{(a)}{{G}}^{-1}{{A}}^{(a)}\right]=\\
\dot{D}{{A}}^{(a)}-{{D}}{{A}}^{(a)}{{G}}^{-1}{{D}}^{-1}\dot{D}{{G}} \;,
\end{split}$$
with ${A}^{(s)}$ $({A}^{(a)})$ the (anti-)symmetric part of $A^{\scaleto{D}{3.5pt}}$, and with initial conditions ${D^{\scaleto{D}{3.5pt}}}(t_0)=\mathbb{0}$, ${\dot{D}^{\scaleto{D}{3.5pt}}}(t_0)=G(t_0)^{-1}$.
We point out that the result Eq. (\[pap\]) is specific to the (Stratonovich-type) discretisation prescription adopted in Ref. [@PAP1] for the cross terms $2\dot{y}(\tau)^T{{A}}(\tau){y}(\tau)$, which gives $\frac{1}{\varepsilon}(y_{j+1}-y_j)^T(A_jy_j+A_{j+1}y_{j+1})$. In fact, there is in general a one-parameter family of discretisations $$\label{disc}
\frac{2}{\varepsilon}(y_{j+1}-y_j)^T((1-\gamma)A_jy_j+\gamma A_{j+1}y_{j+1}),\,\,\,\gamma\in[0,1]$$ leading to different results for Eq. (\[pap\]), [@LANG1; @LANG2]. Interestingly, we notice that the mid-point rule ($\gamma=1/2$) is the only one giving a finite result for Eq. (\[num\]) when $A(\tau)$ is not symmetric. For more details see Appendix \[appB\].
It is now easy to show that, even for simple quadratic Lagrangians, Eq. alone does not represent a transition probability satisfying the normalization condition Eq. . This can happen even if there are no cross terms ([*i.e.* ]{}$A=\mathbb{0}$), as we will see in the string example. To fix this issue, the condition Eq. is enforced by introducing the normalization factor $\mathcal{N}$, see Eq. . Unfortunately, computing $\mathcal{N}$ can be a non-trivial task, which we are now going to tackle.
Following the same approach as for $\mathcal{K}$, we compute the semi-classical approximation for $\mathcal{N}$ as defined in Eq. . This time we Taylor expand the action $S(q(\tau))$ to second order around the Neumann minimum $q^{\scaleto{N}{3.5pt}}(\tau)$, since the point $q(t)$ is unconstrained. Then, from the second variation of the action computed in $q^{\scaleto{N}{3.5pt}}(\tau)$, namely
$$\label{secv2}
\delta^2 S(q^{\scaleto{N}{3.5pt}},h) = \int_{t_0}^t {\text{d}}\tau\, \bigg( \dot{h}^T{{G}}\dot{h}+2\dot{h}^T{{A^{\scaleto{N}{3.5pt}}}}{h}+{h}^T{{V^{\scaleto{N}{3.5pt}}}}{h}\bigg) \;,$$
and the same discretisation as before, the semi-classical approximation for $\mathcal{N}$ reads $$\label{appx2}
\mathcal{N}^{\textit{sc}} = e^{-S(q^{\scaleto{N}{3.5pt}})} \lim_{n\rightarrow\infty} I^{\scaleto{N}{3.5pt}}_n \;,$$
$$\label{den}
I^{\scaleto{N}{3.5pt}}_n=\displaystyle{\int\limits_{h(t_0)=0}{\prod\limits_{j=1}^{n}\left[\frac{\det{(G_j)}}{(2\pi\varepsilon)^{d}}\right]^{\frac{1}{2}}\prod\limits_{j=1}^{n}{{\text{d}}h_j}\,\,\, e^{-\frac{\varepsilon}{2} \sum\limits_{j=0}^{n}\delta^2 S(q^{\scaleto{N}{3.5pt}},h)_j}}}.$$
Here, similarly to Eq. , the subscript $j$ indicates that the associated term is evaluated in $\tau_j$, and the integration boundaries come from the fact that $h(\tau)$ represents a perturbation around the minimum $q^{\scaleto{N}{3.5pt}}(\tau)$, and as such it must satisfy only the initial null Dirichlet boundary condition. The products in Eq. give the integration measure for the integral, which is here an unconditional Wiener measure [@BookChaichian]. In fact, note that contrary to Eq. the product of ${\text{d}}h_j$ runs here until $n$, which is what makes the integration of Eq. in general nontrivial.
One of our main results is now to show how to compute Eq. , from a modification of the method used by Papadopoulos in Ref. [@PAP1] to derive Eq. . The idea consists in performing a backward integration of Eq. , meaning that the standard direction of discretisation $(q_0,t_0)\rightarrow (q,t)$ is now replaced by $(q,t)\rightarrow (q_0,t_0)$. In summary, we are able to find a set of symmetric and positive definite matrices $F^{\scaleto{N}{3.5pt}}_j$, depending on $G$, $A^{\scaleto{N}{3.5pt}}$ and $V^{\scaleto{N}{3.5pt}}$, such that (see Appendix \[Ap1intN\]) $$\label{res1}
{I}^{\scaleto{N}{3.5pt}}_n=\det{\left[\prod\limits_{j=1}^{n} F^{\scaleto{N}{3.5pt}}_j\right]}^{-\frac{1}{2}} \;.$$ Using then the recursion relations for ${I}^{\scaleto{N}{3.5pt}}_n$ and $F^{\scaleto{N}{3.5pt}}_j$, we show that the limit in Eq. (\[res1\]) gives $$\label{pap2}
\mathcal{N}^{sc} = e^{-S(q^{\scaleto{N}{3.5pt}})}\det\left[{D^{\scaleto{N}{3.5pt}}}(t_0)\right]^{-\frac{1}{2}} \;,$$ where $D^{\scaleto{N}{3.5pt}}(\tau)$ solves Eq. , but this time with $A^{\scaleto{N}{3.5pt}}(\tau)$ and $V^{\scaleto{N}{3.5pt}}(\tau)$. As a consequence of the backward integration necessary for deriving the matrices $F^{\scaleto{N}{3.5pt}}_j$, also Eq. is now solved in the backward direction with boundary conditions ${D^{\scaleto{N}{3.5pt}}}(t)=\mathbb{1}$, ${\dot{D}^{\scaleto{N}{3.5pt}}}(t)=-{A^{\scaleto{N}{3.5pt}}}^{\small{(s)}}(t)G(t)^{-1}$. Detailed calculations to derive this result are given in Appendix \[Ap1limitN\].
Inspired by this strategy we compute again Eq. , but this time using the backward integration procedure. We find that the result coincides with Eq. , where now $D^{\scaleto{D}{3.5pt}}(\tau)$ solves Eq. in the backward direction, with boundary conditions ${D^{\scaleto{D}{3.5pt}}}(t)=\mathbb{0}$, ${\dot{D}^{\scaleto{D}{3.5pt}}}(t)=-G(t)^{-1}$.
In conclusion, the results obtained so far allow us to express the transition probability Eq. in the semi-classical approximation as $$\label{fin1}
\rho^{\text{sc}}(q,t \vert q_0,t_0):=\frac{\mathcal{K}^{\text{sc}}}{\mathcal{N}^{\text{sc}}} = e^{S(q^{\scaleto{N}{3.5pt}})-S(q^{\scaleto{D}{3.5pt}})}\sqrt{\det\left[\frac{1}{2\pi}\frac{{D^{\scaleto{N}{3.5pt}}}}{{D^{\scaleto{D}{3.5pt}}}}(t_0)\right]} \;.$$
Despite the simplicity of Eq. , let us remember that the $D$’s have to be found by solving two second order nonlinear differential equations of the form of Eq. , which can be a demanding task. Remarkably, we are able to simplify this problem significantly by relating Eq. to a system of linear differential equations.
As noted in Ref. [@LUD], it turns out that there is a relation between the matrix Eq. and the linear Jacobi equation for a vector field ${w}\in\mathbb{R}^d$ $$\label{jac}
\frac{d}{d\tau}\left[{G}\dot{w}+{A}{w}\right]-{A}^T \dot{w}-{V}{w}={{0}} \;.$$ In fact, if $W=L^T$ is a matrix whose columns $w$ are solutions of the Jacobi equation Eq. , then the solutions of Eq. and the ones of $$\label{jactr}
\frac{d}{d\tau}\left[\dot{L}{G}+{L}{A}^T\right]-\dot{L}{A}-{L}{V}={\mathbb{0}}$$ are related by the nonlinear transformation $$\label{tran}
L^{-1}\dot{L}=D^{-1}\dot{D}+{{A}}^{(a)}G^{-1} \;.$$
Then, it is possible to impose the condition $\det{(L)}=\det{(D)}$, which is also equal to $\det{(W)}$, to ensure the uniqueness of the change of variables and to find the associated boundary conditions for the Jacobi equation. These read ${W^{\scaleto{D}{3.5pt}}}(t)=\mathbb{0}$, $ {\dot{W}^{\scaleto{D}{3.5pt}}}(t)=-G(t)^{-1}$ for $\mathcal{K}^{\text{sc}}$, and ${W^{\scaleto{N}{3.5pt}}}(t)=\mathbb{1}$, ${\dot{W}^{\scaleto{N}{3.5pt}}}(t)=-G(t)^{-1}{A}(t)$ for $\mathcal{N}^{\text{sc}}$. More details on this transformation are given in Appendix \[appC\].
As a final step, since the Jacobi equation is the Euler-Lagrange equation for the second variation of the action, we can easily provide the more elegant Hamiltonian formulation given in Eq. . Namely, if $m=G\dot{w}+Aw$ is the conjugate variable under the Legendre transform of the second variation with respect to $\dot{w}$, and if we define $M=G\dot{W}+AW$, we can express Eq. in terms of the solutions $W^{\scaleto{D}{3.5pt}}$ and $W^{\scaleto{N}{3.5pt}}$ of the Jacobi equation in Hamiltonian form subjected to the transformed final boundary conditions, [*i.e.* ]{}Eq. (\[fin3\]). This results then in Eq. .
Let us emphasize that when the Lagrangian is quadratic in its variables we have $\rho(q,t \vert q_0,t_0)=\rho^{\text{sc}}(q,t \vert q_0,t_0)$, since the second order expansion used in the semi-classical approximation does not neglect any term of higher order.
In conclusion, we have shown how to compute the path integrals appearing in Eq. in the semi-classical approximation, from the solutions of the Euler-Lagrange equations and of the system of linear differential equations Eq. .
#### Examples.
– To show how our approach applies to a number of relevant problems, we present here three illustrative examples.
To begin, let us summarize briefly the relations between the Langevin and the Fokker-Planck equations, with the associated path integral formulation [@GRAHAM; @FALKOFF; @HAKEN].
We consider the Langevin equation $$\label{lang}
{\text{d}}Q(t)=\mu(Q(t),t){\text{d}}t+\sigma(t){\text{d}}B(t),$$ with $Q(t),\,\,\mu(Q(t),t)\in\mathbb{R}^d$, $\sigma(t)\in\mathbb{R}^{d\times l}$ and $B(t)$ an $l-$dimensional standard Wiener process. It is known that the transition probability $\rho(q,t \vert q_0,t_0)$ for the continuous Markovian process $Q(t)$ is the fundamental solution of the Fokker-Planck equation $$\label{fok}
\frac{\partial}{\partial t}\rho=-\frac{\partial}{\partial q_i}\left[\mu_i(q,t)\rho\right]+\frac{1}{2}\Sigma_{ij}(t)\frac{\partial^2}{\partial q_i\partial q_j}\rho$$ with $\rho(q,t_0 \vert q_0,t_0)=\delta(q-q_0)$, and where $\Sigma(t)=\sigma(t)\sigma(t)^T\in\mathbb{R}^{d\times d}$ is the diffusion matrix, $\mu(q,t)$ the drift vector, and the Einstein summation convention is adopted for $i,j=1,...,d$. In particular, if $\Sigma$ is constant and $\mu(q)$ is a function of the configuration, $\rho(q,t \vert q_0,t_0)$ has the path integral representation (\[pathint\]), where the Lagrangian is given by the Onsager-Machlup function [@OM] $$\label{OM}
\mathcal{L}(q(\tau),\dot{q}(\tau))=\frac{1}{2}\left[\left(\dot{q}-\mu(q)\right)^T\Sigma^{-1}\left(\dot{q}-\mu(q)\right)\right] \;.$$ In the literature, however, an additional factor $+\frac{1}{2}\text{div}\left({\mu(q)}\right)$ usually appears in Eq. (\[OM\]). As mentioned before, we observe that this correction is necessary for providing a normalized result when the path integral expression for the transition probability is only defined by $\mathcal{K}$. Our method offers an alternative approach, where we avoid the problem of finding an effective Lagrangian for every application by introducing the normalization constant $\mathcal{N}$.
In the situation where $\Sigma$ is not strictly positive definite, even if the Onsager-Machlup Lagrangian Eq. is ill-defined, its Hamiltonian form is well defined by $\mathcal{L}(p(\tau))=\frac{1}{2} p^T\Sigma\, p$, where $\dot{q}=\Sigma \,p+\mu(q)$, and $p$ is the conjugate variable of $\dot{q}$ under the Legendre transform. This procedure is justified by taking the limit for a sequence of strictly positive definite matrices converging to $\Sigma$. At the same time the Hamilton and Jacobi equations for the minima and the fluctuations are also well defined. In particular, the Hamilton equations $$\label{Ham}
\displaystyle{\frac{{\text{d}}}{{\text{d}}\tau}\begin{pmatrix}
{q} \\
{p}
\end{pmatrix}
=J\begin{pmatrix}
{\text{d}}[\mu(q)]^T\,p \\
\Sigma \,p+\mu(q)
\end{pmatrix}}$$ are subject to the boundary conditions $q^{\scaleto{D}{3.5pt}}(t_0)=q_0$, $q^{\scaleto{D}{3.5pt}}(t)=q$ for the Dirichlet minimum $q^{\scaleto{D}{3.5pt}}(\tau)$, and $q^{\scaleto{N}{3.5pt}}(t_0)=q_0$, $p^{\scaleto{N}{3.5pt}}(t)=0$ for the Neumann minimum $q^{\scaleto{N}{3.5pt}}(\tau)$. On the other hand, the Jacobi equation in Hamiltonian form Eq. (\[fin3\]) is driven by the matrix $$E^{\scaleto{i}{5pt}}=\begin{pmatrix}
{\text{d}}^2[\mu(q)]^T\,p\,\,\, \,\,& {\text{d}}[\mu(q)]^T\\
\\
{\text{d}}[\mu(q)]\,\,\,\,\, & \Sigma
\end{pmatrix}\Bigg\vert_{q^{\scaleto{i}{5pt}},p^{\scaleto{i}{5pt}}}\;, \quad \scaleto{i}{5pt}={\scaleto{D}{3.5pt},\scaleto{N}{3.5pt}}\;.$$ Here and in Eq. , ${\text{d}}[\mu(q)]$ and ${\text{d}}^2[\mu(q)]$ denote respectively the rank-2 and rank-3 tensors of the first and second derivatives in $q$ of the vector field $\mu(q)$.
As a first application of our method, we consider the d-dimensional Ornstein-Uhlenbeck process [@OU; @FALKOFF; @VATI], which is described by the Fokker-Planck equation where $\Sigma\in\mathbb{R}^{d\times d}$ is a constant symmetric diffusion matrix and $\mu(q)=-\Theta\, q$, $\Theta\in\mathbb{R}^{d\times d}$ defines the drift. It is easy to see that the system is exactly characterised by the same linear Hamilton and Jacobi equation $$\label{OUeq}
\displaystyle{\frac{{\text{d}}}{{\text{d}}\tau}\begin{pmatrix}
{w} \\
{m}
\end{pmatrix}
=\begin{pmatrix}
-\Theta &\,\,\,\Sigma \\
\mathbb{0} &\,\,\,\Theta^T
\end{pmatrix}}
\begin{pmatrix}
{w} \\
{m}
\end{pmatrix} \;.$$ The analytical solution is given by
$$\label{analy}
w(\tau)=e^{-\Theta\tau}\left[\int_0^{\tau}{e^{\Theta s}\,\Sigma\, e^{\Theta^T s}C_1}\,{\text{d}}s+C_2\right],\,\,m(\tau)=e^{\Theta^T\tau}C_1,$$
where $C_1$ and $C_2$ are determined from the appropriate boundary conditions. Setting $t_0=0$, we can use Eq. and recover the Gaussian transition probability
$$\rho(q,t \vert q_0,0)=\frac{\exp{[-\frac{1}{2}(q-\text{Av}(q_0,t))^T\text{Co}^{-1}(t)(q-\text{Av}(q_0,t))]}}{\sqrt{\det[2\pi\,\text{Co}(t)]}},$$
$$\label{gauss}
\text{Av}(q_0,t)=e^{-\Theta t}q_0,\,\,\,\text{Co}(t)=\int_0^{t}{e^{\Theta (s-t)}\,\Sigma\, e^{\Theta^T (s-t)}}\,{\text{d}}s \;.$$
As a second application, we consider the Van der Pol oscillator driven by white noise [@NAESS], that is described by the Langevin equation of motion for the coordinate $z$ as $$\label{vdp}
\ddot{z}(t)+2\xi[z(t)^2-1]\dot{z}(t)+z(t)=\sqrt{2 \lambda}f(t) \;,$$ where $f(t)$ denotes a standard stationary gaussian white noise, $\lambda>0$ represents the diffusion coefficient and $\xi>0$ the strength of the non-linearity. By defining the terms $$\Omega:=\begin{pmatrix}
0 &\,\,\,-1 \\
1 &\,\,\,-2\xi
\end{pmatrix}\,\,\,\,\,\text{and}\,\,\,\,\,\nu(q):=\begin{pmatrix}
0 \\
q_1^2q_2
\end{pmatrix} \;,$$ it is possible to write the stochastic equation of motion in phase space as a 2-dimensional Langevin equation in the form of Eq. (\[lang\]), with $\sigma=(0,\sqrt{2 \lambda})^T$. The associated Fokker-Planck equation has then coefficients $$\Sigma=
\begin{pmatrix}
0 &\,\,\,0 \\
0 &\,\,\,2\lambda
\end{pmatrix},\qquad\text{and}\qquad\mu(q)=
-\Omega q-2\xi \nu(q) \;.$$ Note that, in this example, the corresponding Onsager-Machlup function Eq. (\[OM\]) is no longer a quadratic function of $q$ and $\dot{q}$. Therefore, the semi-classical approximation will lead to a result that is *a priori* not exact. From the second order expansion we nevertheless expect the result to be accurate for small values of diffusion $\lambda$ and final time $t$. Applying the method we presented, we obtain that the Hamilton and Jacobi equations for the system are respectively of the form $$\label{Hamvdp}
\displaystyle{\frac{{\text{d}}}{{\text{d}}\tau}\begin{pmatrix}
{q} \\
{p}
\end{pmatrix}
=\begin{pmatrix}
-\Omega &\,\,\,\Sigma \\
\mathbb{0} &\,\,\,\Omega^T
\end{pmatrix}
\begin{pmatrix}
q \\
p
\end{pmatrix}}-2\xi\psi(q,p) \;,$$ $$\label{Jacvdp}
\frac{{\text{d}}}{{\text{d}}\tau}\begin{pmatrix}
{{W^{\scaleto{i}{5pt}}}} \\
{{M^{\scaleto{i}{5pt}}}}
\end{pmatrix}
=\begin{pmatrix}
-\Omega &\,\,\,\Sigma \\
\mathbb{0} &\,\,\,\Omega^T
\end{pmatrix}
\begin{pmatrix}
{{W^{\scaleto{i}{5pt}}}} \\
{{M^{\scaleto{i}{5pt}}}}
\end{pmatrix}
-2\xi\Psi^{\scaleto{i}{5pt}}\begin{pmatrix}
{{W^{\scaleto{i}{5pt}}}} \\
{{M^{\scaleto{i}{5pt}}}}
\end{pmatrix} \;,$$ with ${\scaleto{i}{5pt}}={\scaleto{D}{3.5pt},\scaleto{N}{3.5pt}}$, $\psi(q,p)=(0,\,q_1^2 q_2,\,-2q_1 q_2 p_2,\,-q_1^2 p_2)^T$, and $$\Psi^{\scaleto{i}{5pt}}=\begin{pmatrix}
0 &\,\,\,0 &\,\,\,0 &\,\,\,0 \\
2q_1^{\scaleto{i}{5pt}} q_2^{\scaleto{i}{5pt}} &\,\,\,(q_1^{\scaleto{i}{5pt}})^2 &\,\,\,0 &\,\,\,0 \\
-2q_2^{\scaleto{i}{5pt}} p_2^{\scaleto{i}{5pt}} &\,\,\,-2q_1^{\scaleto{i}{5pt}} p_2^{\scaleto{i}{5pt}} &\,\,\,0 &\,\,\,-2q_1^{\scaleto{i}{5pt}} q_2^{\scaleto{i}{5pt}} \\
-2q_1^{\scaleto{i}{5pt}} p_2^{\scaleto{i}{5pt}} &\,\,\,0 &\,\,\,0 &\,\,\,-(q_1^{\scaleto{i}{5pt}})^2
\end{pmatrix} \;.$$ Solving numerically Eqs. (\[Hamvdp\],\[Jacvdp\]), subject to the associated boundary conditions, we are able to obtain through Eq. (\[fin2\]) the semi-classical approximation of the (non-Gaussian) transition probability solving the Fokker-Planck equation for the stochastic Van der Pol oscillator, see Appendix \[appD\].
As a final example, let us investigate in one dimension the stationary configuration of a charged extensible string at thermal equilibrium, when an external field is applied. We consider the action [@Rudi04] $$S = \beta \int_0^L \left[ \dfrac{1}{2} \alpha \dot{q}(\tau)^2 - \sigma \phi(q(\tau)) \right] {\text{d}}\tau \;,$$ where $\beta$ is the inverse temperature, $\alpha$ the elastic constant, $\sigma$ the charge density per unit length, and $\phi(q(\tau))$ the electric potential. Note that here $\tau\in[0,L]$ is a parametrisation of the string, and not a time, so that $\dot{q}(\tau)$ represent the elongation. To be concrete, let us assume a potential of the form $\phi(q)=a q^2 + b q$, which could be the second order approximation of a more general potential. Let us define $\kappa:=\sqrt{2\sigma a / \alpha}$. From the Euler-Lagrange equations with the appropriate boundary conditions we obtain $q^{\scaleto{D}{3.5pt}}(\tau)$ and $q^{\scaleto{N}{3.5pt}}(\tau)$, while from the Jacobi equation we obtain ${W^{\scaleto{D}{3.5pt}}}(\tau)=(\beta\alpha\kappa)^{-1}\sin(\kappa(L-\tau))$ and ${W^{\scaleto{N}{3.5pt}}}(\tau)=\cos(\kappa(L-\tau))$. These results allow us to express the probability that $q(L)=q_L$, given that $q(0)=0$, as $ p(q_L,L\vert 0,0) = e^{-\frac{1}{2}\frac{(q_L-\text{Av})^2}{\text{Var}}} /\sqrt{2\pi \text{Var}}$, which is a Gaussian probability distribution with mean $\text{Av}=\frac{b}{2a}\left(\frac{1-\cos(\kappa L)}{\cos(\kappa L)}\right)$ and variance $\text{Var}=\frac{1}{\beta\alpha \kappa}\tan(\kappa L)$. Interestingly, this example illustrates how $\mathcal{N}\neq 1$ in general, even if $A=0$.
#### Conclusions.
– In this work we presented a consistent approach to compute transition probabilities in the semi-classical approximation, from a path integral formulation. Our method is based on the generalization of a work by Papaopulos [@PAP1], which allows us to express the solutions of both conditional and unconditional Gaussian path integrals from the solutions of the Euler-Lagrange equation and of s system linear differential equations. Remarkably, the accuracy of our method is only dependent on the accuracy of the semi-classical approximation. In particular, when the Lagrangian of the system is quadratic in position and velocity there is no approximation, and the results are exact. As a side note, we discussed what is the effect of choosing different discretisation prescriptions for continuous paths, and mention under which circumstances this can be arbitrary. To conclude, we applied our method to three examples of general interest. These illustrate how our results can be applied to a variety of problems in physics and mathematics, such as the study of stochastic processes or the analysis of equilibrium configurations of polymers.
#### Acknowledgments.
– We are grateful to Prof. John Maddocks for the fruitful discussions and insights, as well as to all the LCVMM group of Lausanne for the constant support. G.C. was supported by the Swiss National Science Foundation through Grant No. 163324. MF acknowledges support by the Swiss National Science Foundation.
SUPPLEMENTARY MATERIALS
=======================
Evaluation of $\mathcal{N}^{\text{sc}}$
=======================================
Derivation of Eq. (\[res1\]) {#Ap1intN}
----------------------------
In order to derive Eq. (\[res1\]), we first express the second variation Eq. (\[secv2\]) using the method of finite differences. Recalling that we discretised $\tau\in[t_0,t]$ into $n$ intervals of length $\varepsilon=(t-t_0)/n$, we obtain $$\label{start}
\begin{split}
\delta^2S(q^{\scaleto{N}{3.5pt}},h)\approx\varepsilon\sum\limits_{j=0}^{n}\delta^2S(q^{\scaleto{N}{3.5pt}},h)_j&:=\frac{1}{\varepsilon}\sum\limits_{j=1}^n\left[\Delta {h}_j^TG_j\Delta {h}_j+\varepsilon \Delta {h}_j^T\left({A^{\scaleto{N}{3.5pt}}_j} {h}_j+A^{\scaleto{N}{3.5pt}}_{j-1}{h}_{j-1}\right)+\varepsilon^2 {h}_j^TV^{\scaleto{N}{3.5pt}}_j{h}_j \right]\\
&=\frac{1}{\varepsilon}\sum\limits_{j=1}^n\big{[}{h}_j^T\left(G_j+\varepsilon {A^{\scaleto{N}{3.5pt}}_j}+\varepsilon^2V^{\scaleto{N}{3.5pt}}_j\right){h}_j+{h}_{j-1}^T\left(G_j-\varepsilon A^{\scaleto{N}{3.5pt}}_{j-1}\right){h}_{j-1}\\
&\qquad -{h}_{j}^T\left(G_j-\varepsilon A^{\scaleto{N}{3.5pt}}_{j-1}\right){h}_{j-1}-{h}_{j-1}^T\left(G_j+\varepsilon A^{\scaleto{N}{3.5pt}}_{j}\right){h}_{j}\big{]} \;,
\end{split}$$ where we defined $\Delta {h}_j:={h}_j-{h}_{j-1}$ for $j=1,...,n$, and the subscript $j$ indicates that the associated term is evaluated in $\tau_j=t_0+j\varepsilon$ for $j=0,1,...,n$.
Since $h(\tau)$ is a perturbation around the Neumann minimum, then ${h}_0={0}$ and we can rearrange the terms in the sum in order to isolate the slice for $j=n$: $$\label{fin_dif1.2}
\begin{split}
\varepsilon\sum\limits_{j=0}^n\delta^2S(q^{\scaleto{N}{3.5pt}},h)_j&=\frac{1}{\varepsilon}\sum\limits_{j=1}^{n-1}\left[h_j^T\left(G_j+G_{j+1}+\varepsilon^2V^{\scaleto{N}{3.5pt}}_j\right)h_j-h_{j}^T\left(G_j-\varepsilon A^{\scaleto{N}{3.5pt}}_{j-1}\right)h_{j-1}-h_{j-1}^T\left(G_j+\varepsilon A^{\scaleto{N}{3.5pt}}_{j}\right)h_{j}\right]\\
&\quad\quad +h_n^T\left(G_n+\varepsilon A^{\scaleto{N}{3.5pt}}_n+\varepsilon^2V^{\scaleto{N}{3.5pt}}_n\right)h_n-h_{n}^T\left(G_n-\varepsilon A^{\scaleto{N}{3.5pt}}_{n-1}\right)h_{n-1}-h_{n-1}^T\left(G_n+\varepsilon A^{\scaleto{N}{3.5pt}}_{n}\right)h_{n} \;.
\end{split}$$Introducing now the matrices $U^{\scaleto{N}{3.5pt}}_j:=G_j+\frac{\varepsilon}{2}[(A^{\scaleto{N}{3.5pt}}_j)^T-A^{\scaleto{N}{3.5pt}}_{j-1}]$ for $j=1,...,n$, we have that Eq. (\[fin\_dif1.2\]) can be written as
$$\label{fin_dif2.2}
\begin{split}
\frac{1}{\varepsilon}\sum\limits_{j=1}^{n-1}\left[h_j^T\left(G_j+G_{j+1}+\varepsilon^2V^{\scaleto{N}{3.5pt}}_j\right)h_j-h_j^T U^{\scaleto{N}{3.5pt}}_j h_{j-1}-h_{j-1}^T (U^{\scaleto{N}{3.5pt}}_j)^T h_{j}\right]+h_n^T\left(G_n+\varepsilon A^{\scaleto{N}{3.5pt}}_n+\varepsilon^2V^{\scaleto{N}{3.5pt}}_n\right)h_n-h_{n}^T U^{\scaleto{N}{3.5pt}}_n h_{n-1}-h_{n-1}^T (U^{\scaleto{N}{3.5pt}}_n)^Th_{n} \;.
\end{split}$$
At this point, we perform a change of variables. We define the transformation with unit Jacobian ${\phi}_j:=h_j-\beta^{\scaleto{N}{3.5pt}}_j h_{j-1}$ for $j=1,...,n$, where the matrices $\beta^{\scaleto{N}{3.5pt}}_j$ are given recursively by the following construction
\[cons\] $$\begin{aligned}
\alpha^{\scaleto{N}{3.5pt}}_{n} &:= G_{n}+\varepsilon\frac{A^{\scaleto{N}{3.5pt}}_n+(A^{\scaleto{N}{3.5pt}}_n)^T}{2}+\varepsilon^2V^{\scaleto{N}{3.5pt}}_{n} \;,&&\\
\alpha^{\scaleto{N}{3.5pt}}_{j} &:= G_{j}+G_{j+1}+\varepsilon^2 V^{\scaleto{N}{3.5pt}}_{j}-(\beta^{\scaleto{N}{3.5pt}}_{j+1})^T\alpha^{\scaleto{N}{3.5pt}}_{j+1}\beta^{\scaleto{N}{3.5pt}}_{j+1} \;&&\text{for }\; j=n-1,...,1 \;, \\
U^{\scaleto{N}{3.5pt}}_j&=\alpha^{\scaleto{N}{3.5pt}}_j\beta^{\scaleto{N}{3.5pt}}_j \;&&\text{for }\; j=1,...,n \;.\end{aligned}$$
These expressions are motivated by the fact that they allow to express Eq. as a quadratic form, which is desired in view of a Gaussian integration. In fact, ${\phi}_j^T\alpha^{\scaleto{N}{3.5pt}}_j{\phi}_j=h_j^T\alpha^{\scaleto{N}{3.5pt}}_j h_j-h_j^T U^{\scaleto{N}{3.5pt}}_j h_{j-1}-h_{j-1}^T(U^{\scaleto{N}{3.5pt}}_j)^T h_{j}+h_{j-1}^T(\beta^{\scaleto{N}{3.5pt}}_j)^T\alpha^{\scaleto{N}{3.5pt}}_j\beta^{\scaleto{N}{3.5pt}}_j h_{j-1}$, which gives $$\label{quad}
\varepsilon\sum\limits_{j=0}^{\scaleto{N}{3.5pt}}\delta^2S(q^{\scaleto{N}{3.5pt}},h)_j=\frac{1}{\varepsilon}\sum\limits_{j=1}^{n}{\phi}_j^T\alpha^{\scaleto{N}{3.5pt}}_j{\phi}_j \;.$$Finally, we define $F^{\scaleto{N}{3.5pt}}_j:=\alpha^{\scaleto{N}{3.5pt}}_jG_j^{-1}$ for $j=1,...,n$, to recover Eq. (\[res1\]) by computing the Gaussian integrals as: $$\begin{aligned}
I^{\scaleto{N}{3.5pt}}_n &= \displaystyle{\int\limits_{h(t_0)=0}{\prod\limits_{j=1}^{n}\left[\frac{\det{(G_j)}}{(2\pi\varepsilon)^{d}}\right]^{\frac{1}{2}}{{\text{d}}h_j}\,\,\, e^{-\frac{\varepsilon}{2} \sum\limits_{j=0}^{n}\delta^2 S(q^{\scaleto{N}{3.5pt}},h)_j}}} \nonumber\\
&= \displaystyle{\int{\prod\limits_{j=1}^{n}\left[\frac{\det{(G_j)}}{(2\pi\varepsilon)^{d}}\right]^{\frac{1}{2}}{{\text{d}}\phi_j}\,\,\, e^{-\frac{1}{2\varepsilon}\sum\limits_{j=1}^{n}{\phi}_j^T\alpha^{\scaleto{N}{3.5pt}}_j{\phi}_j}}} \nonumber\\
&=\det{\left[\prod\limits_{j=1}^{n} F^{\scaleto{N}{3.5pt}}_j\right]}^{-\frac{1}{2}} \;. \label{inttg} \end{aligned}$$
Derivation of Eq. (\[pap2\]) {#Ap1limitN}
----------------------------
In order to derive Eq. (\[pap2\]) we need to compute the limit in Eq. . To this end, we look for recurrence relations in order to express Eq. (\[inttg\]) through a difference equation. On the basis of the construction given in the previous section, we define $
D^{\scaleto{N}{3.5pt}}_{n-k}:=\prod\limits_{j=0}^{k}F^{\scaleto{N}{3.5pt}}_{n-j}$ for $k=0,1,...,n-1$, and formulate the following iterative method for $D^{\scaleto{N}{3.5pt}}$ and ${\alpha^{\scaleto{N}{3.5pt}}}$.\
\
Initial condition: $D^{\scaleto{N}{3.5pt}}_n=\alpha^{\scaleto{N}{3.5pt}}_{n}G_{n}^{-1}$.\
Iteration scheme: $D^{\scaleto{N}{3.5pt}}_{n-(k+1)}=D^{\scaleto{N}{3.5pt}}_{n-k}\alpha^{\scaleto{N}{3.5pt}}_{n-(k+1)}G_{n-(k+1)}^{-1}$ for $\,\,k=0,1,...,n-2$.\
\
Initial condition: $\alpha^{\scaleto{N}{3.5pt}}_{n}=G_{n}+\varepsilon\frac{A^{\scaleto{N}{3.5pt}}_n+(A^{\scaleto{N}{3.5pt}}_n)^T}{2}+\varepsilon^2 V^{\scaleto{N}{3.5pt}}_{n}$.\
Iteration scheme: $\alpha^{\scaleto{N}{3.5pt}}_{n-(k+1)}=G_{n-(k+1)}+G_{n-k}+\varepsilon^2 V^{\scaleto{N}{3.5pt}}_{n-(k+1)}-(\beta^{\scaleto{N}{3.5pt}}_{n-k})^T\alpha^{\scaleto{N}{3.5pt}}_{n-k}\beta^{\scaleto{N}{3.5pt}}_{n-k}$ for $k=0,1,...,n-2$.\
\
Reminding that $\beta^{\scaleto{N}{3.5pt}}_{n-k}=(\alpha^{\scaleto{N}{3.5pt}}_{n-k})^{-1}U^{\scaleto{N}{3.5pt}}_{n-k}$, and that $U^{\scaleto{N}{3.5pt}}_{n-k}=G_{n-k}+\frac{\varepsilon}{2}[(A^{\scaleto{N}{3.5pt}}_{n-k})^T-A^{\scaleto{N}{3.5pt}}_{n-(k+1)}]$, it is possible to give the explicit recurrence relation for $\alpha^{\scaleto{N}{3.5pt}}_{n-(k+1)}$, $k=0,1,...,n-2$ as $$\label{rec2}
\begin{split}
\alpha^{\scaleto{N}{3.5pt}}_{n-(k+1)}&=G_{n-(k+1)}+G_{n-k}+\varepsilon^2 V^{\scaleto{N}{3.5pt}}_{n-(k+1)}-G_{n-k}(\alpha^{\scaleto{N}{3.5pt}}_{n-k})^{-1}G_{n-k}\\
&\quad-\varepsilon\left[\frac{A^{\scaleto{N}{3.5pt}}_{n-k}-(A^{\scaleto{N}{3.5pt}}_{n-(k+1)})^T}{2}\right](\alpha^{\scaleto{N}{3.5pt}}_{n-k})^{-1}G_{n-k}-\varepsilon G_{n-k}(\alpha^{\scaleto{N}{3.5pt}}_{n-k})^{-1}\left[\frac{(A^{\scaleto{N}{3.5pt}}_{n-k})^T-A^{\scaleto{N}{3.5pt}}_{n-(k+1)}}{2}\right]\\
&\quad-\varepsilon^2\left[\frac{A^{\scaleto{N}{3.5pt}}_{n-k}-(A^{\scaleto{N}{3.5pt}}_{n-(k+1)})^T}{2}\right](\alpha^{\scaleto{N}{3.5pt}}_{n-k})^{-1}\left[\frac{(A^{\scaleto{N}{3.5pt}}_{n-k})^T-A^{\scaleto{N}{3.5pt}}_{n-(k+1)}}{2}\right].
\end{split}$$Moreover, the recurrence formula for $D^{\scaleto{N}{3.5pt}}$ provides the additional useful relations
\[Drec\] $$\begin{aligned}
\alpha^{\scaleto{N}{3.5pt}}_{n-(k+1)} &= (D^{\scaleto{N}{3.5pt}}_{n-k})^{-1}D^{\scaleto{N}{3.5pt}}_{n-(k+1)}G_{n-(k+1)} \\
(\alpha^{\scaleto{N}{3.5pt}}_{n-k})^{-1} &= G_{n-k}^{-1}(D^{\scaleto{N}{3.5pt}}_{n-k})^{-1}D^{\scaleto{N}{3.5pt}}_{n-(k-1)}.\end{aligned}$$
Finally, substituting Eqs. (\[Drec\]) in Eq. (\[rec2\]), and multiplying to the left both sides by $D^{\scaleto{N}{3.5pt}}_{n-k}$, we get the full difference equation for the matrix $D^{\scaleto{N}{3.5pt}}$, in terms of $G$, ${A^{\scaleto{N}{3.5pt}}}$ and ${V^{\scaleto{N}{3.5pt}}}$, for $k=1,2,...,n-2$:
$$\label{diff}
\begin{split}
D^{\scaleto{N}{3.5pt}}_{n-(k+1)}G_{n-(k+1)}=&D^{\scaleto{N}{3.5pt}}_{n-k}G_{n-(k+1)}+D^{\scaleto{N}{3.5pt}}_{n-k}G_{n-k}-D^{\scaleto{N}{3.5pt}}_{n-(k-1)}G_{n-k}+\varepsilon^2 D^{\scaleto{N}{3.5pt}}_{n-k}V^{\scaleto{N}{3.5pt}}_{n-(k+1)}\\
&\; -\varepsilon D^{\scaleto{N}{3.5pt}}_{n-k}\left[\frac{A^{\scaleto{N}{3.5pt}}_{n-k}-(A^{\scaleto{N}{3.5pt}}_{n-(k+1)})^T}{2}\right]G_{n-k}^{-1}(D^{\scaleto{N}{3.5pt}}_{n-k})^{-1}D^{\scaleto{N}{3.5pt}}_{n-(k-1)}G_{n-k}-\varepsilon D^{\scaleto{N}{3.5pt}}_{n-(k-1)}\left[\frac{(A^{\scaleto{N}{3.5pt}}_{n-k})^T-A^{\scaleto{N}{3.5pt}}_{n-(k+1)}}{2}\right]\\
&\; -\varepsilon^2 D^{\scaleto{N}{3.5pt}}_{n-k}\left[\frac{A^{\scaleto{N}{3.5pt}}_{n-k}-(A^{\scaleto{N}{3.5pt}}_{n-(k+1)})^T}{2}\right]G_{n-k}^{-1}(D^{\scaleto{N}{3.5pt}}_{n-k})^{-1}D^{\scaleto{N}{3.5pt}}_{n-(k-1)}\left[\frac{(A^{\scaleto{N}{3.5pt}}_{n-k})^T-A^{\scaleto{N}{3.5pt}}_{n-(k+1)}}{2}\right].
\end{split}$$
Our goal is now to take the continuous limit ($n\rightarrow\infty$, $\epsilon\rightarrow 0$) for this expression, in order to obtain a differential equation for the unknown $D^{\scaleto{N}{3.5pt}}$. To this end, remember that [*e.g.* ]{}$D^{\scaleto{N}{3.5pt}}_{n-(k+1)}$ stands for $D^{\scaleto{N}{3.5pt}}(t-s_{k+1})$ with $s_{k+1}=(k+1)\varepsilon=s_k+\varepsilon$, and that similar expressions hold for all other terms. We can therefore Taylor expand each $D^{\scaleto{N}{3.5pt}}$ around $s_k$ to second order in $\varepsilon$, and each other coefficient to first order. Then, dividing everything by $\varepsilon^2$ we obtain
$$\begin{aligned}
\frac{d}{ds}&\left[\frac{d}{ds}\left[D^{\scaleto{N}{3.5pt}}\small{(t-s)}\right]G(t-s)\right]-D^{\scaleto{N}{3.5pt}}(t-s)\frac{d}{ds}\left[(A^{\scaleto{N}{3.5pt}})^{(s)}(t-s)\right]-D^{\scaleto{N}{3.5pt}}(t-s)\left[V^{\scaleto{N}{3.5pt}}(t-s)+(A^{\scaleto{N}{3.5pt}})^{(a)}(t-s)G^{-1}(t-s)(A^{\scaleto{N}{3.5pt}})^{(a)}(t-s)\right]= \nonumber\\
&=-\frac{d}{ds}\left[D^{\scaleto{N}{3.5pt}}(t-s)\right](A^{\scaleto{N}{3.5pt}})^{(a)}(t-s)+D^{\scaleto{N}{3.5pt}}(t-s)(A^{\scaleto{N}{3.5pt}})^{(a)}(t-s)G^{-1}(t-s)(D^{\scaleto{N}{3.5pt}})^{-1}(t-s)\frac{d}{ds}\left[D^{\scaleto{N}{3.5pt}}(t-s)\right]G(t-s) \;, \label{sys.2}\end{aligned}$$
subject to the boundary conditions $D^{\scaleto{N}{3.5pt}}(t-s)\big{|}_{s=0}={\mathbb{1}}$, $\frac{d}{ds}\left(D^{\scaleto{N}{3.5pt}}\small{(t-s)}\right)\big{|}_{s=0}=(A^{\scaleto{N}{3.5pt}})^{(s)}(t)G(t)^{-1}$. These are a consequence of the recurrence relations for $D^{\scaleto{N}{3.5pt}}$ and $\alpha^{\scaleto{N}{3.5pt}}$, and they are derived as it follows. For the former we have $$\label{boun}
\alpha^{\scaleto{N}{3.5pt}}_{n}=G_{n}+\varepsilon\frac{A^{\scaleto{N}{3.5pt}}_n+(A^{\scaleto{N}{3.5pt}}_n)^T}{2}+\varepsilon^2V^{\scaleto{N}{3.5pt}}_{n}\sim G_{n}\,\;\text{as}\;\,\varepsilon\rightarrow 0 \quad\Rightarrow\quad D^{\scaleto{N}{3.5pt}}_{n}=\alpha^{\scaleto{N}{3.5pt}}_{n}G_{n}^{-1}\sim\mathbb{1}\,\;\text{as}\;\,\varepsilon\rightarrow 0.$$For the latter, note that $$\label{bounD}
\frac{D^{\scaleto{N}{3.5pt}}_{n-1}-D^{\scaleto{N}{3.5pt}}_{n}}{\varepsilon}=\frac{D^{\scaleto{N}{3.5pt}}_{n}\left(\alpha^{\scaleto{N}{3.5pt}}_{n-1}G_{n-1}^{-1}-{\mathbb{1}}\right)}{\varepsilon}\sim\frac{\left(\alpha^{\scaleto{N}{3.5pt}}_{n-1}G_{n-1}^{-1}-{\mathbb{1}}\right)}{\varepsilon}\,\,\text{as}\,\,\varepsilon\rightarrow 0 \;,$$and, because $\left[\frac{A^{\scaleto{N}{3.5pt}}_{n}-(A^{\scaleto{N}{3.5pt}}_{n-1})^T+(A^{\scaleto{N}{3.5pt}}_{n})^T-A^{\scaleto{N}{3.5pt}}_{n-1}}{2}\right]G_{n-1}^{-1}\rightarrow 0$ as $\varepsilon\rightarrow 0$, this can also be written as $$\label{Neum}
\begin{split}
\frac{\alpha^{\scaleto{N}{3.5pt}}_{n-1}G_{n-1}^{-1}-{\mathbb{1}}}{\varepsilon}&=\frac{1}{\varepsilon}\big{[}G_{n}G_{n-1}^{-1}+\varepsilon^2 V^{\scaleto{N}{3.5pt}}_{n-1}G_{n-1}^{-1}-G_{n}(\alpha^{\scaleto{N}{3.5pt}}_{n})^{-1}G_{n}G_{n-1}^{-1}\\
&\qquad-\varepsilon\left[\frac{A^{\scaleto{N}{3.5pt}}_{n}-(A^{\scaleto{N}{3.5pt}}_{n-1})^T}{2}\right](\alpha^{\scaleto{N}{3.5pt}}_{n})^{-1}G_{n}G_{n-1}^{-1}-\varepsilon G_{n}(\alpha^{\scaleto{N}{3.5pt}}_{n})^{-1}\left[\frac{(A^{\scaleto{N}{3.5pt}}_{n})^T-A^{\scaleto{N}{3.5pt}}_{n-1}}{2}\right]G_{n-1}^{-1}\\
&\qquad-\varepsilon^2\left[\frac{A^{\scaleto{N}{3.5pt}}_{n}-(A^{\scaleto{N}{3.5pt}}_{n-1})^T}{2}\right](\alpha^{\scaleto{N}{3.5pt}}_{n})^{-1}\left[\frac{(A^{\scaleto{N}{3.5pt}}_{n})^T-A^{\scaleto{N}{3.5pt}}_{n-1}}{2}\right]G_{n-1}^{-1}\big{]}\\
&\sim \frac{1}{\varepsilon}\left[\mathbb{1}-G_{n}(\alpha^{\scaleto{N}{3.5pt}}_{n})^{-1}\right]\,\,\text{as}\,\,\varepsilon\rightarrow 0 \;.
\end{split}$$Inserting now the definition for $\alpha^{\scaleto{N}{3.5pt}}_{n}$ we have $$\label{bouns}
\frac{1}{\varepsilon}\left[\mathbb{1}-G_{n}(\alpha^{\scaleto{N}{3.5pt}}_{n})^{-1}\right]=\frac{1}{\varepsilon}\left[\mathbb{1}-\left(\mathbb{1}+\varepsilon\frac{A^{\scaleto{N}{3.5pt}}_n+(A^{\scaleto{N}{3.5pt}}_n)^T}{2}G_{n}^{-1}+\varepsilon^2V^{\scaleto{N}{3.5pt}}_{n}G_{n}^{-1}\right)^{-1}\right] \;,$$which, exploiting the Neumann series $({\mathbb{1}}+\Lambda)^{-1}={\mathbb{1}}-\Lambda+\Lambda^2-\Lambda^3+...$ with $\Lambda=\varepsilon\frac{A^{\scaleto{N}{3.5pt}}_n+(A^{\scaleto{N}{3.5pt}}_n)^T}{2}G_{n}^{-1}+\varepsilon^2V^{\scaleto{N}{3.5pt}}_{n}G_{n}^{-1}$, gives $$\label{finbou}
\frac{D^{\scaleto{N}{3.5pt}}_{n-1}-D^{\scaleto{N}{3.5pt}}_{n}}{\varepsilon}\sim (A^{\scaleto{N}{3.5pt}}_n)^{(s)}G_{n}^{-1}\,\,\text{as}\,\,\varepsilon\rightarrow 0 \;.$$
To summarize, setting $\tau=t-s$ in Eq. (\[sys.2\]), this leads to the second order non-linear differential equation Eq. (\[papeq\]) subject to the boundary conditions ${D^{\scaleto{N}{3.5pt}}}(t)=\mathbb{1}$ and ${\dot{D}^{\scaleto{N}{3.5pt}}}(t)=-({A^{\scaleto{N}{3.5pt}}})^{\small{(s)}}(t)G(t)^{-1}$.
Different discretisation choices {#appB}
================================
According to Eq. (\[disc\]), the most general discretisation prescription for Eq. (\[start\]) is given as a function of $\gamma\in[0,1]$ as $$\label{startgam}
\begin{split}
\delta^2S(q^{\scaleto{N}{3.5pt}},h)\approx\varepsilon\sum\limits_{j=0}^n\delta^2S(q^{\scaleto{N}{3.5pt}},h)_{\gamma_j}&:=\frac{1}{\varepsilon}\sum\limits_{j=1}^n\left[\Delta {h}_j^TG_j\Delta {h}_j+2\varepsilon \Delta {h}_j^T\left(\gamma{A^{\scaleto{N}{3.5pt}}_j} {h}_j+(1-\gamma)A^{\scaleto{N}{3.5pt}}_{j-1}{h}_{j-1}\right)+\varepsilon^2 {h}_j^TV^{\scaleto{N}{3.5pt}}_j{h}_j \right] \;.\\
\end{split}$$From this expression, we can repeat all the steps followed in Appendix \[Ap1intN\] and \[Ap1limitN\] to see that the difference equation (\[diff\]) now becomes
$$\label{diff.gam}
\begin{split}
D^{\scaleto{N}{3.5pt}}_{n-(k+1)}G_{n-(k+1)}=&D^{\scaleto{N}{3.5pt}}_{n-k}G_{n-(k+1)}+D^{\scaleto{N}{3.5pt}}_{n-k}G_{n-k}-D^{\scaleto{N}{3.5pt}}_{n-(k-1)}G_{n-k}+\varepsilon^2 D^{\scaleto{N}{3.5pt}}_{n-k}V^{\scaleto{N}{3.5pt}}_{n-(k+1)}+2\varepsilon(2\gamma-1)D^{\scaleto{N}{3.5pt}}_{n-k}A^{\scaleto{N}{3.5pt}}_{n-(k+1)}-\\
&\;-\varepsilon D^{\scaleto{N}{3.5pt}}_{n-k}\left[\gamma{A^{\scaleto{N}{3.5pt}}_{n-k}-(1-\gamma)(A^{\scaleto{N}{3.5pt}}_{n-(k+1)})^T}\right]G_{n-k}^{-1}(D^{\scaleto{N}{3.5pt}}_{n-k})^{-1}D^{\scaleto{N}{3.5pt}}_{n-(k-1)}G_{n-k}-\\
&\;-\varepsilon D^{\scaleto{N}{3.5pt}}_{n-(k-1)}\left[\gamma{(A^{\scaleto{N}{3.5pt}}_{n-k})^T-(1-\gamma)A^{\scaleto{N}{3.5pt}}_{n-(k+1)}}\right]-\\
&\;-\varepsilon^2 D^{\scaleto{N}{3.5pt}}_{n-k}\left[\gamma{A^{\scaleto{N}{3.5pt}}_{n-k}-(1-\gamma)(A^{\scaleto{N}{3.5pt}}_{n-(k+1)})^T}\right]G_{n-k}^{-1}(D^{\scaleto{N}{3.5pt}}_{n-k})^{-1}D^{\scaleto{N}{3.5pt}}_{n-(k-1)}\left[\gamma{(A^{\scaleto{N}{3.5pt}}_{n-k})^T-(1-\gamma)A^{\scaleto{N}{3.5pt}}_{n-(k+1)}}\right].
\end{split}$$
If $A^{\scaleto{N}{3.5pt}}$ is not symmetric, it is possible to derive a differential equation from Eq. (\[diff.gam\]) through Taylor expansion only if $\gamma=\frac{1}{2}$, which gives Eq. (\[papeq\]). This is easy to check by performing the calculation. The same is also true for $\mathcal{K}$, for which the expansion is now taken around the isolated minimum $q^{\scaleto{D}{3.5pt}}$.
If $A^{\scaleto{D}{3.5pt}}$ and $A^{\scaleto{N}{3.5pt}}$ are symmetric, and if we adopt the same discretisation prescription for both $q^{\scaleto{D}{3.5pt}}$ and $q^{\scaleto{N}{3.5pt}}$, then there is a one parameter family of different equations providing the same normalized result for the transition probability density. Namely $$\label{sys.new.gam}
\frac{{\text{d}}}{{\text{d}}\tau}\left[\dot{D}{{G}}\right]+2\gamma {{D}}\dot{A}-{{D}}[{{V}}-(1-2\gamma)^2{{A}}{{G}}^{-1}A]=
(1-2\gamma)\dot{D}{{A}}+(1-2\gamma){{D}}{{A}}{{G}}^{-1}D^{-1}\dot{D}G \;,$$subject to the boundary conditions ${D^{\scaleto{D}{3.5pt}}}(t)=\mathbb{0}$, ${\dot{D}^{\scaleto{D}{3.5pt}}}(t)=-G(t)^{-1}$ and ${D^{\scaleto{N}{3.5pt}}}(t)=\mathbb{1}$, ${\dot{D}^{\scaleto{N}{3.5pt}}}(t)=-2\gamma A^{\scaleto{N}{3.5pt}}(t) G(t)^{-1}$.
The non-linear transformation Eq. {#appC}
==================================
As mentioned in the main text, the solutions $D$ of Eq. (\[papeq\]) and $L$ of Eq. (\[jactr\]) are related by the nonlinear transformation Eq. (\[tran\]) presented in [@LUD]. Here we present in detail how the boundary conditions for $D$ translate into boundary conditions for $L$, in the context of the backward integration procedure.
First, let us consider the case for $q^{\scaleto{D}{3.5pt}}$. Eq. (\[tran\]) gives us a mapping between $L^{\scaleto{D}{3.5pt}}$ and $D^{\scaleto{D}{3.5pt}}$, as far as they are invertible. If we assume $L^{\scaleto{D}{3.5pt}}$ and $D^{\scaleto{D}{3.5pt}}$ invertible for all $\tau\neq t$ (no conjugate points) the transformation is valid except for $\tau=t$, where $D^{\scaleto{D}{3.5pt}}(t)=\mathbb{0}$ because of the boundary conditions in the backward direction (see paragraph after Eq. ).
To derive the boundary conditions for $L^{\scaleto{D}{3.5pt}}$ in $\tau=t$ from the boundary conditions for $D^{\scaleto{D}{3.5pt}}$ in $\tau=t$, we consider the following reasoning. For $\tau\neq t$ we can write $\dot{L^{\scaleto{D}{3.5pt}}}=L^{\scaleto{D}{3.5pt}}(D^{\scaleto{D}{3.5pt}})^{-1}\dot{D^{\scaleto{D}{3.5pt}}}+L^{\scaleto{D}{3.5pt}}(A^{\scaleto{D}{3.5pt}})^{(a)} {G}^{-1}$, and we know that $D^{\scaleto{D}{3.5pt}}\rightarrow \mathbb{0}$, $\dot{D^{\scaleto{D}{3.5pt}}}\rightarrow -{G}(t)^{-1}$ as $\tau\rightarrow t$, for continuity of $D^{\scaleto{D}{3.5pt}}$ and $\dot{D^{\scaleto{D}{3.5pt}}}$. As a consequence, in order to obtain a finite boundary condition for $\dot{L^{\scaleto{D}{3.5pt}}}$, we necessarily want $L^{\scaleto{D}{3.5pt}}(D^{\scaleto{D}{3.5pt}})^{-1}\rightarrow {X}$ as $\tau\rightarrow t$, where ${X}$ is a finite valued matrix. This implies that $L^{\scaleto{D}{3.5pt}}(t)=\lim_{\tau\rightarrow t}{L^{\scaleto{D}{3.5pt}}}=\mathbb{0}$, for continuity of $L^{\scaleto{D}{3.5pt}}$. Furthermore, having $L^{\scaleto{D}{3.5pt}}$ invertible for $\tau\neq t$ implies $\dot{L^{\scaleto{D}{3.5pt}}}(t)$ is not singular, meaning that the matrix ${X}$ is not singular as well, since $L^{\scaleto{D}{3.5pt}}(t)=\mathbb{0}$. To summarize, we have that $$\label{sysb}
\frac{{\text{d}}}{{\text{d}}\tau}\left[\dot{L^{\scaleto{D}{3.5pt}}}{G}+L^{\scaleto{D}{3.5pt}}(A^{\scaleto{D}{3.5pt}})^T\right]-\dot{L^{\scaleto{D}{3.5pt}}}A^{\scaleto{D}{3.5pt}}-L^{\scaleto{D}{3.5pt}}V^{\scaleto{D}{3.5pt}}=\mathbb{0}$$is subject to the boundary conditions $L^{\scaleto{D}{3.5pt}}(t)=\mathbb{0}$, $\dot{L^{\scaleto{D}{3.5pt}}}(t)=-{X} {G}(t)^{-1}$.
In the same way, we now consider the case for $q^{\scaleto{N}{3.5pt}}$. Assuming $L^{\scaleto{N}{3.5pt}}$ and $D^{\scaleto{N}{3.5pt}}$ to be non-singular also for $\tau=t$ (note $D^{\scaleto{N}{3.5pt}}(t)=\mathbb{1}$), we have $$\begin{aligned}
L^{\scaleto{N}{3.5pt}}(t)^{-1}\dot{L^{\scaleto{N}{3.5pt}}}(t) &= -(A^{\scaleto{N}{3.5pt}})^{(s)}(t){G}(t)^{-1}+(A^{\scaleto{N}{3.5pt}})^{(a)}(t){G}(t)^{-1} \nonumber\\
&= -(A^{\scaleto{N}{3.5pt}})^T(t){G}(t)^{-1} \;. \label{ttr}\end{aligned}$$In addition, as we want $L^{\scaleto{N}{3.5pt}}$ to be invertible in $\tau=t$, then ${Y}:=L^{\scaleto{N}{3.5pt}}(t)$ must be a non-singular matrix. To summarize, we have that $$\label{sysbb}
\frac{{\text{d}}}{{\text{d}}\tau}\left[\dot{L^{\scaleto{N}{3.5pt}}}{G}+L^{\scaleto{N}{3.5pt}}(A^{\scaleto{N}{3.5pt}})^T\right]-\dot{L^{\scaleto{N}{3.5pt}}}A^{\scaleto{N}{3.5pt}}-L^{\scaleto{N}{3.5pt}}V^{\scaleto{N}{3.5pt}}=\mathbb{0}$$is subject to the boundary conditions $L^{\scaleto{N}{3.5pt}}(t)={Y}$, $\dot{L^{\scaleto{N}{3.5pt}}}(t)=-{Y}(A^{\scaleto{N}{3.5pt}})^T(t){G}(t)^{-1}$.
At this point, we use the observation that $\det{{L}}=c\det{{D}}$ for all $\tau$, where $c$ is a constant [@LUD]. In order to make the transformation unique, we impose $c=1$ for both $L^{\scaleto{D}{3.5pt}}$, $D^{\scaleto{D}{3.5pt}}$ and $L^{\scaleto{N}{3.5pt}}$, $D^{\scaleto{N}{3.5pt}}$, which allows us to fix the matrices $X$ and $Y$. To see this, consider the following expansions around $\tau=t$:\
\
$L^{\scaleto{D}{3.5pt}}(\tau)=-{X} {G}(t)^{-1}(\tau-t)+\frac{1}{2}\ddot{L}^{\scaleto{D}{3.5pt}}(t)(\tau-t)^2+O(\tau-t)^3$, $\quad(D^{\scaleto{D}{3.5pt}})^{-1}(\tau)=-{G}(t)\frac{1}{(\tau-t)}-\frac{1}{2}{G}(t)\ddot{D}^{\scaleto{D}{3.5pt}}(t){G}(t)+O(\tau-t)$,\
\
$L^{\scaleto{N}{3.5pt}}(\tau)={Y}-{Y}(A^{\scaleto{N}{3.5pt}})^T(t){G}(t)^{-1}(\tau-t)+\frac{1}{2}\ddot{L}^{\scaleto{N}{3.5pt}}(t)(\tau-t)^2+O(\tau-t)^3$, $\quad(D^{\scaleto{N}{3.5pt}})^{-1}(\tau)={\mathbb{1}}+(A^{\scaleto{N}{3.5pt}})^{(s)}(t){G}(t)^{-1}(\tau-t)+O(\tau-t)^2$,\
\
resulting in $\,\,\lim\limits_{\tau\rightarrow t}{\det({L^{\scaleto{D}{3.5pt}}(\tau)(D^{\scaleto{D}{3.5pt}})^{-1}(\tau)})}=\det({X})$, $\,\,\lim\limits_{\tau\rightarrow t}{\det({L^{\scaleto{N}{3.5pt}}(\tau)(D^{\scaleto{N}{3.5pt}})^{-1}(\tau)})}=\det({Y})\,\,$, so that we can set ${X}={Y}={\mathbb{1}}$.
Time evolution of the transition probability for the Van der Pol oscillator Eq. {#appD}
================================================================================
![Slices along direction $(q_1,q_2)=(q_1,5 q_1)$ of the transition probability $\rho\left((q_1,q_2),t \vert (0,0),0 \right)$ for the Van der Pol oscillator Eq. with $\xi=3$, $\lambda=0.5$. Note that, despite the semi-classical approximation, the resulting probability density is not necessairly Gaussian.[]{data-label="fig:vdp"}](plotVdP.pdf){width="12cm"}
|
---
author:
- 'Jacco Th. van Loon'
title: Observed properties of red supergiant and massive AGB star populations
---
Introduction
============
In the final stages of their evolution, stars in the birth mass range $\sim1$–7 M$_\odot$ ascend the AGB, reaching luminosities of order $10^4$ L$_\odot$ due to hydrogen and helium shell burning; Hot Bottom Burning (HBB) at the bottom of the convection zone in the most massive AGB stars can further raise the luminosity, approaching $10^5$ L$_\odot$ [@Iben83]. The nuclear evolution of AGB stars is truncated by mass loss [@vanLoon99], leaving behind a carbon–oxygen white dwarf. Stars in the birth mass range $\sim11$–30 M$_\odot$ do not develop core degeneracy and instead become core helium burning RSGs [@Maeder12]. RSGs also evolve along a (short) branch, and their mass loss may or may not change their appearance [@vanLoon99] before they explode as a core-collapse supernova [@Smartt15]. This leaves a range $\sim7$–11 M$_\odot$ unaccounted for; these are the super-AGB stars, which behave very much like massive AGB stars but ignite carbon burning in the core before leaving an oxygen–neon–magnesium white dwarf or undergoing electron-capture core-collapse [@Doherty17]. The actual mass range is much narrower, but the boundaries are uncertain. Understanding the fate of super-AGB stars, and recognising them in nature, presents one of the greatest challenges in contemporary astrophysics.
Do we know any super-AGB star?
==============================
No. Claims are sometimes made, on the basis of surface abundances (rubidium, lithium) or luminosities (above the classical AGB limit), but we lack a way to unequivocally distinguish super-AGB stars from massive AGB stars experiencing HBB. Nonetheless, we should keep a record of candidates for when we have a way to confirm, or refute, their super-AGB nature. Until then, “discoveries” of super-AGB stars are steeped in controversy.
A fascinating example is the bright Harvard Variable, HV2112 in the Small Magellanic Cloud (SMC). It is a little more luminous than the classical AGB limit, sometimes as cool as spectral type M7.5, and has a relatively long pulsation period [@Wood83] – commensurate with being such a large star. Things got more exciting when @Smith90 detected lithium, which indicated some level of HBB even if not clinching its status as a super-AGB star. In that same year, however, @Reid90 showed lithium to be absent.
Nothing much happened, until @Levesque14 proposed that HV2112 could also be a Thorne–$\dot{\rm Z}$ytkow object – a neutron star which ended up inside a stellar mantle. This conjecture was immediately put to the test by @Tout14, who note the difficulty in distinguishing between the different origins of similar peculiarities – the calcium abundance might be the discriminating feature, in favour of a Thorne–$\dot{\rm Z}$ytkow object.
But does HV2122 in fact reside in the SMC, and is a relatively massive star? @Maccarone16 concluded on the basis of proper motion measurements that HV2112 must instead belong to the Galactic Halo, and be an extrinsic S-type star upon which the anomalous abundances had been imprinted by a companion AGB star. This would explain the calcium abundance, which is generally high in the Halo. The radial velocity, whilst not excluding Halo membership, is however consistent with that of the SMC [@Gonzalez15]. Revisiting the proper motion, @Worley16 reinstate its SMC membership. The saga continues.
Converting luminosity functions into star formation histories
=============================================================
As stars of different birth masses attain different luminosities during their lives as AGB stars, super-AGB stars or RSGs, their luminosity distribution reflects the star formation history (SFH). However, the evolutionary tracks are difficult to separate especially if there is a spread in metallicity, and so the luminosity distribution will be a blend of the luminosity evolution of stars of a range of birth masses, especially on the AGB. Evolution in luminosity is curtailed, though, once stars lose mass at very high rates, so if one could identify stars in that extreme final phase the luminosity distributions would map much more cleanly onto the SFH. Luckily, these stars develop strong, long-period variability (LPV) as their atmospheres pulsate and (help) drive the mass loss [@Wood83].
A method was devised to determine the SFH from the luminosity distribution of LPVs by [@Javadi11b], and applied to a near-infrared monitoring survey of LPVs in the Local Group spiral galaxy M33 [@Javadi11a]. If $dn^\prime(t)$ is the number of LPVs associated with all stars that formed within a timespan $dt$ around lookback time $t$, then the star formation rate around that time is $$\xi(t)\ =\ \frac{dn^\prime(t)}{\delta t}\,
\frac{\int_{\rm min}^{\rm max}f_{\rm IMF}(m)m\,dm}
{\int_{m(t)}^{m(t+dt)}f_{\rm IMF}(m)\,dm},$$ where $f_{\rm IMF}$ is the initial mass ($m$) function and $\delta t$ the LPV lifetime.
The method relies on stellar evolution models; only the Padova group predicts all of the essential information [@Marigo17]. This is a problem, as different models predict a different final luminosity for a given birth mass, and different lifetimes before and during the LPV phase [@Marigo17]. It is also a blessing [*because*]{} it is model dependent, as it opens an avenue into calibrating the models and thus offers insight into the physics. This became apparent very soon, when @Javadi13 found that the inferred integrated mass loss exceeded the birth mass. If we define $$\eta\ =\ \frac{\Sigma_{i=1}^N\left(\dot{M}_i\times(\delta t)_i\right)}
{\Sigma_{i=1}^NM_i},$$ then $\eta(m)=1-m_{\rm final}/m_{\rm birth}<1$ always, but the measurements yielded $\eta>1$. At least part of the solution had to be a downward revision of the LPV lifetimes, which was later confirmed independently by the Padova group themselves [@Rosenfield16]. The revised lifetimes were used to derive a more reliable SFH for M33 [@Javadi17].
Mass loss from dusty AGB stars and red supergiants
==================================================
Mass-loss rates can be determined from dusty AGB stars and RSGs by modelling the spectral energy distribution (SED) using a radiative transfer model. The first such systematic analysis for stars with known distances was performed for populations within the Large Magellanic Cloud (LMC) by @vanLoon99, who found that mass-loss rates exceeded nuclear consumption rates for most of the AGB evolution but not for much of the RSG evolution. This means that AGB stars avoid SN but RSGs do not.
They also found evidence that the mass-loss rate increases during the evolution along the AGB as the luminosity increases and the photosphere cools, though RSGs in particular seem to exhibit more abrupt variations between mild and possibly short episodes of much enhanced mass loss. This was put on a more quantitative footing by @vanLoon05 who parameterised the mass-loss rate during the dusty phase of (massive) AGB and RSG evolution as $$\begin{aligned}
\log{\dot{M}}\ =\ & -5.65+1.05\log(L/10\,000\,{\rm L}_\odot) \nonumber \\
& -6.3\log(T_{\rm eff}/3500\,{\rm K}),\end{aligned}$$ with no evidence for a metallicity dependence. The dependence on temperature seems very strong but it must be realised that the temperatures among these kinds of stars fall within a limited range ($\sim2500$–4000 K).
Because the mass-loss rate that is derived from the SED depends on how much the dust is diluted it has a dependence on the wind speed, $v_{\rm exp}$. What is really seen directly is the optical depth of the dusty envelope: $$\tau\ \propto\ \frac{\dot{M}}{r_{\rm gd}v_{\rm exp}\sqrt{L}},$$ where $r_{\rm gd}$ is the gas:dust mass ratio. However, if the wind is driven by radiation pressure upon the dust, then $$v_{\rm exp}\ \propto\ \frac{L^{1/4}}{r_{\rm gd}^{1/2}}.$$ By measuring $v_{\rm exp}$ directly, for instance from the double-horned hydroxyl maser line profile, its value can be reconciled with the SED modelling by tuning the value for $r_{\rm gd}$. This led @Goldman17 to determine a more sophisticated formula for the mass-loss rates from AGB stars and RSGs in the LMC, Galactic Centre and Galactic Bulge with known pulsation periods $P$: $$\begin{aligned}
\log{\dot{M}}\ =\ -4.97+0.90\log(L/10\,000\,{\rm L}_\odot) \nonumber \\
+0.75\log(P/500\,{\rm d})-0.03\log(r_{\rm gd}/200).\end{aligned}$$ This time $P$ instead of $T_{\rm eff}$ measures the size of the star (in combination with $L$), but the resemblance to the formula found earlier is striking. There is no dependence on the gas:dust ratio and, by inference, on the metallicity.
The mass loss probed in the dusty phases may only be part of the story. If sustained over an extended period of time, moderate mass loss may matter too. Likewise, the metallicity of massive AGB stars and RSGs in the SMC is only $\sim0.2$ Z$_\odot$ (as opposed to $\sim0.5$ Z$_\odot$ in the LMC), and stars become dusty later in their evolution, possibly having lost more mass in other ways before. @Bonanos10 indeed found that most RSGs exhibit moderate mass-loss rates, $\sim10^{-6}$ M$_\odot$ yr$^{-1}$, which agrees with the study by @Mauron11 of Galactic RSGs among which only few exhibit mass-loss rates $>10^{-5}$ M$_\odot$ yr$^{-1}$. While this confirms the earlier findings by @vanLoon99, no clear bimodality was seen.
Larger populations of massive AGB stars and RSGs are needed to be more conclusive about the rarest, but most intense phases of mass loss in comparison to the more common, gentler phases. To that aim, following from @Javadi13 and the extended survey by @Javadi15, Javadi et al.(in prep.) have measured mass-loss rates for thousands such stars in M33. They confirm the gradual evolution in mass loss along the AGB and the bimodal mass loss on the RSG, with the rates increasing in proportion to luminosity.
These studies show no evidence for anything peculiar to be happening to super-AGB stars. If anything, they are most likely to follow the extension of the AGB sequence and attain exuberant mass-loss rates of $\sim10^{-4}$ M$_\odot$ yr$^{-1}$. An accurate assessment of the integrated mass loss might lead us to exclude – or allow – their possible fate as electron-capture SN.
An alternative route to mapping the evolution of mass loss along the AGB or RSG branch is based on the fact that populous clusters may show more than one such example. Given the fast evolution in those advanced stages, we are essentially watching a star of the same mass at different moments in its evolution, as snaphots in a movie. @Davies08 pioneered this approach in the Galactic cluster RSGC1, and more recently @Beasor16 applied the same principle to the LMC cluster NGC2100. They both confirm the increase in mass loss along the RSG branch. Such studies in clusters in which we [*know*]{} super-AGB stars should form may elucidate the properties, evolution and fate of super-AGB stars where field studies struggle to recognise them. Apart from the difficulty in finding such clusters, it may be difficult to catch them at their most extreme.
Beyond the red supergiant and asymptotic giant branches
=======================================================
Massive AGB stars and RSGs can undergo a blueward excursion as a result of core expansion before returning to the cool giant branch – a “blue loop”. This is a much slower transformation than the “jump” from the main sequence to the giant branch, and is responsible for populating the Cepheid instability strip [@Valle09]. Cepheid variables therefore could be extremely valuable probes of what may already have happened to these stars on the cool giant branch, such as any mass loss, especially as their pulsational properties depend on their current mass. It also means that the cool giant branches are populated by stars that have, and those that have not undergone a blue loop – again, these may differ as a result of the mass loss during their lives as blue (or yellow) supergiants. Typically, much fewer warm supergiants are seen than are predicted by the models [@Neugent10], which suggests that the models do not yet adequately account for certain processes that happen inside RSGs.
Pulsation periods lengthen as a star expands when it is reduced in mass, so period–luminosity diagrams of AGB and RSG LPVs have the potential to trace mass loss and possibly the effects of blue loops. Also, the mode in which the pulsation is excited depends on stellar structure. @Yang12 charted this parameterspace for RSGs in the Magellanic Clouds, M33 and the Milky Way but their combination with similar information for AGB stars left an unfortunate gap right around where super-AGB stars could be found. This must – and can – be remedied.
AGB stars and RSGs may also move irreversibly towards hotter photospheres, due to mass loss (by a wind or stripping in a binary system). The luminosity distribution over post-AGB stars and post-RSGs must be devoid of the birth masses associated with the electron-capture SN demise of the more massive among super-AGB stars, and depleted of those RSGs which encountered their end in a SN. Again, a concerted analysis is required, where in the past different communities have often concentrated on just the lower-mass (post-)AGB stars or on just the higher-mass (post-)RSGs.
The progenitors and remnants of core-collapse supernov[æ]{}
===========================================================
Having established that many RSGs most likely do not lose their envelope before the core collapses, it is comforting that all of the SN type II-P progenitors that have been discovered so far are RSGs [@Smartt15]. Their birth mass range is estimated to be $\sim9$–17 M$_\odot$ [@Smartt15], though the upper limits could allow dusty RSGs as massive as $\sim21$ M$_\odot$ to have resulted in II-P SNe. This would also be more consistent with the above findings about the mass-loss rates of RSGs, where a proportion of core-collapsing RSGs should experience high mass-loss rates. The confirmed RSG progenitors tend to be of relatively early spectral type and thus constitute those RSGs that have lost relatively little of their mantle.
What exactly determines the difference in rate of evolution of the core – setting the timing of core collapse – and of the mantle – setting the mass loss, is unclear, but larger samples of discoveries and limits on progenitors of II-P SNe should help elucidate this: their lightcurves and spectral evolution may reveal differences in the mantle mass and circumstellar density, whilst their galactic environments may reveal a dependence on metallicity. Likewise, a connection between RSG evolution and mass loss on the one hand, and SNe of types II-L and Ib on the other, may be made if the more massive RSGs ($\sim20$–30 M$_\odot$ lose most of their mantle before exploding.
A metallicity effect on the properties of the SN resulting from the explosion of a RSG (or super-AGB star) is expected even if the mass-loss rate does not change, because the winds are slower at lower metallicity [@Goldman17]. This would increase the circumburst density, with implications for the SN lightcurve [@Moriya17]. It also means that the reverse shock may be stronger.
The SN has a devastating effect on the dust produced by the RSG, even if it did not explode as a RSG as the SN ejecta move at $\sim10^3$ times the speed of the RSG wind and catch up within a century. The RSG dust, and any ISM dust in the vicinity, is sputtered in the forward shock, whilst the reverse shock sputters much of the dust produced in the SN ejecta. The net effect being that SNe are dust destroyers [@Lakicevic15; @Temim15].
Because SN remnants (SNRs) can be seen for $>10^4$ yr they might tell us something about the SN progenitors in their final $<10^4$ yr that is difficult to capture while they are still alive. The two most prolific SN factories known, NGC6946 [@Sugerman12] and M83 [@Blair15] exhibit hundreds of SNRs [@Bruursema14; @Winkler17].
The next frontier
=================
A naive estimate for a massive spiral galaxy suggests one RSG explodes every century. With a typical RSG lifetime of $\sim10^3$ centuries, we thus expect to find a population of order $10^3$ RSGs. Thus the statistics look extremely promising for studies of the luminosity distribution and the evolution of mass loss of RSGs (and super-AGB stars) in such galaxies.
The next step from the Magellanic Clouds can take us to the nearest spiral galaxies, M33 [@Drout12] and M31 [@Massey16] at $\sim0.9$ Mpc, or NGC300 and the metal-poor dwarf Sextans A, both within 2 Mpc. Indeed, the Surveying the Agents of Galaxy Evolution (SAGE) team, who have revolutionised our views of the Magellanic Clouds, are proposing an Early Release Science programme for the James Webb Space Telescope (JWST) precisely to do that. But how far could we go?
M101 is the most massive spiral galaxy within 7 Mpc, viewed face on. Bamber et al. (in prep.) have used groundbased and [*Spitzer*]{} infrared images, and [*Hubble*]{} optical images, to identify RSG candidates across the entire disc. The [*Spitzer*]{} data are heavily compromised by the limits in resolution and sensitivity, and JWST will be both necessary and sufficient to quantify the dust production by RSGs in M101. Likewise, the luminosity distribution suffers from incompleteness at the low end. Still, it reveals a tantalising first glimpse of the evolution of the most luminous RSGs (Fig. \[m101\]): a healthy number of RSGs in the $\sim17$–22 M$_\odot$ range suggests these could well be the progenitors of SNe, while the sharp drop that sets in at higher masses suggests much diminished RSG lifetimes. The latter is not unexpected if those are the RSGs that become Wolf–Rayet stars, as a result of stronger mass loss.
Will we find super-AGB stars?
=============================
Yes. We look towards our colleagues who model the structure and dynamical behaviour, and nucleosynthesis and surface enrichment of super-AGB stars to make predictions for the observable signatures that can tell super-AGB stars apart from other massive AGB stars (and RSGs). We also need to reach an agreement on the birth-mass range of super-AGB stars. Meanwhile, we look for peculiarities or more subtle hints of deviations, that indicate we may be dealing with a star of a different kind.
I would like to warmly thank Paolo Ventura, Flavia Dell’Agli and Marcella Di Criscienzo and all participants for an interesting and pleasant meeting, my collaborators and students – in particular Atefeh Javadi, Steven Goldman and James Bamber some of whose results I presented here – and two new feline friends for their affection on my walk up to the observatory.
Beasor, E. R. & Davies, B. 2016, MNRAS, 463, 1269 Blair, W. P., et al. 2015, ApJ, 800, 118 Bonanos, A. Z., et al. 2010, AJ, 140, 416 Bruursema, J., et al. 2014, AJ, 148, 41 Davies, B., et al. 2008, ApJ, 676, 1016 Doherty, C. L., et al. 2017, arXiv:1703.06895 Drout, M. R., Massey, P. & Meynet, G. 2012, ApJ, 750, 97 Goldman, S. R., et al. 2017, MNRAS, 465, 403 González-Fernández, C., et al. 2015, A&A, 578, A3 Iben, I., Jr. & Renzini, A. 1983, ARA&A, 21, 271 Javadi, A., van Loon, J. Th. & Mirtorabi, M. T. 2011a, MNRAS, 411, 263 Javadi, A., van Loon, J. Th. & Mirtorabi, M. T. 2011b, MNRAS, 414, 3394 Javadi, A., et al. 2013, MNRAS, 432, 2824 Javadi, A., et al. 2015, MNRAS, 447, 3973 Javadi, A., et al. 2017, MNRAS, 464, 2103 Lakićević, M., et al. 2015, ApJ, 799, 50 Levesque, E. M., et al. 2014, MNRAS, 443, L94 Maccarone, T. J. & de Mink, S. E. 2016, MNRAS, 458, L1 Maeder, A. & Meynet, G. 2012, Reviews of Modern Physics, 84, 25 Marigo, P., et al. 2017, ApJ, 835, 77 Massey, P. & Evans, K. A. 2016, ApJ, 826, 224 Mauron, N. & Josselin, E. 2011, A&A, 526, A156 Moriya, T. J., et al. 2017, arXiv:1703.03084 Neugent, K. F., et al. 2010, ApJ, 719, 1784 Reid, N. & Mould, J. 1990, ApJ, 360, 490 Rosenfield, P., et al. 2016, ApJ, 822, 73 Smartt, S. J. 2015, PASA, 32, 16 Smith, V. V. & Lambert, D. L. 1990, ApJ, 361, L69 Sugerman, B. E. K., et al. 2012, ApJ, 749, 170 Temim, T., et al. 2015, ApJ, 799, 158 Tout, C. A., et al. 2014, MNRAS, 445, L36 Valle, G., et al. 2009, A&A, 507, 1541 van Loon, J. Th., et al. 1999, A&A, 351, 559 van Loon, J. Th., et al. 2005, A&A, 438, 273 Winkler, P. F., Blair, W. P. & Long, K. S. 2017, ApJ, 839, 83 Wood, P. R., Bessell, M. S. & Fox, M. W. 1983, ApJ, 272, 99 Worley C. C., et al. 2016, MNRAS, 459, L31 Yang, M. & Jiang, B. W. 2012, ApJ, 754, 35
|
---
abstract: 'Energy flows in bio-molecular motors and machines are vital to their function. Yet experimental observations are often limited to a small subset of variables that participate in energy transport and dissipation. Here we show, through a solvable Langevin model, that the seemingly hidden entropy production is measurable through the violation spectrum of the fluctuation-response relation of a slow observable. For general Markov systems with timescale separation, we prove that the violation spectrum exhibits a characteristic plateau in the intermediate frequency region. Despite its vanishing height, the plateau can account for energy dissipation over a broad timescale. Our findings suggest a general possibility to probe hidden entropy production in nano systems without direct observation of fast variables.'
author:
- 'Shou-Wen Wang'
- Kyogo Kawaguchi
- 'Shin-ichi Sasa'
- 'Lei-Han Tang'
title: Entropy production of nano systems with timescale separation
---
*Introduction.—* Recent advances in technology have made it possible to investigate energetic aspects of an open nano system experimentally. Studies have been carried out to quantify, e.g., the energy conversion efficiency of molecular motors [@noji1997direct; @toyabe2010nonequilibrium], the information-energy conversion efficiency of an artificial Maxwell demon [@sagawa2010generalized; @toyabe2010experimental; @berut2012experimental; @Koski2015MaxwellDemon], the entropy production of a quantum tunneling device in a temperature gradient [@koski2013distribution], and the effective temperature of single molecules in nonequilibrium steady-states [@dieterich2015single].
In the most interesting cases, such small systems are capable of complex dynamic behavior by virtue of multiple degrees of freedom over a broad range of timescales, and furthermore by operating out of equilibrium. The functional features of these systems are usually associated with slow processes, but recent theoretical studies have provided several examples of “hidden entropy production” arising from non-equilibrium coupling between fast and slow variables [@hondou2000unattainability; @esposito2012stochastic; @celani2012anomalous; @kawaguchi2013fluctuation; @nakayama2015invariance; @Chun2015fast; @Esposito2015Stochastic; @shouwen2015adaptation; @Pablo2015Adaptation; @bo2014entropy; @puglisi2010entropy]. A better understanding of the conditions and key characteristics of this phenomenon is desirable not only from a theoretical viewpoint, but also for developing experimental protocols to uncover channels of energy dissipation without a full characterization of the possibly great many fast processes in a complex nano system.
In this Letter, we show that it is indeed possible to trace the hidden entropy production by quantifying the violation spectrum of the fluctuation-response relation (FRR) of a slow observable. In equilibrium systems, the FRR states that the spontaneous fluctuation of an observable decays in the same way as the deviation produced by an external perturbation. For a velocity observable $\dot{x}$ which is of particular interest here, the former can be measured via the autocorrelation function $C_{\dot{x}}(t)\equiv \langle [\dot{x}(t)- \langle\dot{x}\rangle_{s} ] [\dot{x}(0)-\langle\dot{x}\rangle_{s}] \rangle_{s}$ in the stationary ensemble $\langle \cdot\rangle_{s}$, while the latter is captured by the dynamic response $R_{\dot{x}}(t)\equiv \delta \langle \dot{x}(t)\rangle/\delta h$, with $h$ being the perturbing field [@kubo1966fluctuation]. In frequency, $\tilde{C}_{\dot{x}}(\omega)=2T\tilde{R}_{\dot{x}}'(\omega)$, where $T$ is the temperature of the bath and the prime on $\tilde{R}_{\dot{x}}$ denotes its real part. However, the two properties are not simply related for nonequilibirum systems [@cugliandolo1997fluctuation; @speck2006restoring; @baiesi2009fluctuations; @seifert2010fluctuation]. One significant result achieved in this direction is an equality derived by Harada and Sasa (HS) in Langevin systems. The equality connects the steady-state dissipation rate through the frictional motion of $x$, denoted as $J_x$, to the integral of the frequency-resolved FRR violation [@harada2005equality; @harada2006energy], $$J_x=\gamma \left \{ \langle \dot{x} \rangle_{s}^2+ \int_{-\infty}^\infty \frac{d\omega}{2\pi} [\tilde{C}_{\dot{x}}(\omega)-2T\tilde{R}_{\dot{x}}'(\omega)] \right \},
\label{eq:HS}$$ where $\gamma$ is the friction coefficient. This equality has been validated experimentally in a driven colloidal system [@toyabe2007experimental], and applied to F$_1$-ATPase, a biomolecular motor, to infer the dissipation rate of the rotary motion [@toyabe2010nonequilibrium; @kawaguchi2014nonequilibrium].
The known applications of the HS equality are for systems that dissipate energy on a slow timescale. Here we show that the HS equality can also be used to probe hidden entropy production that takes place on timescales faster than the relaxation time of the observed variable. We first demonstrate, in a solvable Langevin model, the use of Eq. (\[eq:HS\]) to fully account for the dissipated energy in a nonequilibrium steady-state. Interestingly, the FRR violation becomes vanishingly small for large timescale separation, while its integral with respect to frequency remains finite. We present a proof that this feature of the FRR violation spectrum, arising from the dissipative coupling between slow and fast variables, is generic for a timescale separated Markov system. Based on these findings, we suggest an experimental method to detect hidden entropy production from the fluctuation-response spectrum of a slow variable.
*Potential switching model.—* Consider the one-dimensional, over-damped motion of a bead subjected to a potential that switches stochastically between $U_0(x)$ and $U_1(x)$ at a rate $r$. Figure \[fig:violation-spectrum\](a) illustrates an experimental realization using a laser trap that produces a harmonic potential $U_0(x)=kx^2/2$ whose center position switches back and forth between $x= 0$ and $L$ \[therefore $U_1(x)=k(x-L)^2/2$\] [@toyabe2007experimental; @dieterich2015single]. This can also be viewed as a minimum Langevin model to study molecular machines which have fast binding/unbinding of currency molecules that triggers transition between different chemical states. With the potential state $\sigma_t$ $(=0,1)$ at time $t$, and under a perturbing force $h$, the Langevin equation for the bead position $x$ takes the form, $$\gamma \dot{x}=-\partial_x U_{\sigma_t}(x)+h(t)+\eta(t). \label{eq:noise}$$ Here $\gamma$ is the friction coefficient and $\eta (t)$ is the thermal noise satisfying $\langle \eta(t)\rangle=0$ and $\langle \eta(t)\eta(t')\rangle=2\gamma T\delta (t-t')$, with $T$ the temperature of the bath. The Boltzmann constant $k_B$ is set to 1.
![(a) The potential switching model with a fast switching time constant $\tau_f=1/r$ and a slow relaxation time constant $\tau_s=\gamma/k$. The effective potential $U_e(x)$ in the fast switching limit $\epsilon\ll 1$ is shown. (b) Rate of energy input against $\epsilon$. (c) Frequency spectra of the velocity correlation and response functions at various values of $\epsilon$. (d) Frequency spectra of the FRR violation whose integral yields the hidden entropy production that balances energy input in (b). Parameters: $k=\gamma=1$ and $L=5$. Open circles and stars in (c) and (d) give results from a simulated trajectory of length $T_{sp}=10^4\tau_s$ and sampling rate $2\tau_f^{-1}$. The response function is reconstructed from data using a perturbation strength $h=0.5$. (See Supplemental Material [@supp-hiddenEntropy].) []{data-label="fig:violation-spectrum"}](figure1_new2.PDF){width="8.5cm"}
The model defined above has two timescales, $\tau_s=\gamma/k$ for relaxation within a given potential and $\tau_f=1/r$ for potential switching. We introduce $\epsilon \equiv \tau_f/\tau_s$ to characterize the timescale separation between the two competing processes. In the fast switching limit $\epsilon\to 0$, it is straightforward to show that the steady-state distribution of the bead position takes the Boltzmann form $P^{s}(x)\sim \exp[-U_e(x)/T]$, where $U_e=[U_0(x)+U_1(x)]/2$ is the effective potential seen by the bead. Nevertheless, due to potential switching, energy is continuously injected into the system at an average rate $$\dot{W}=\int_{-\infty}^\infty r[P_0^{s}(x)-P_1^{s}(x)][U_1(x)-U_0(x)]dx,$$ where $P_\sigma^{s}(x)$ is the stationary distribution in the full state space $(x,\sigma)$. By analyzing the Fokker-Planck equation in the steady-state satisfied by $P_\sigma^{s}(x)$, we obtain $$\dot{W} \xrightarrow{\epsilon \to 0} \frac{1}{4\gamma} \left \langle \left \{ \partial_x \Big[ U_1(x)-U_0(x) \Big] \right \} ^2 \right \rangle_{s}, \label{eq:hidden}$$ where $\langle \cdot \rangle_{s}$ denotes the average over the reduced distribution $P^{s}(x)$. In the case of a harmonic potential, Eq. (\[eq:hidden\]) yields $\dot{W}=k^2 L^2/(2 \epsilon \gamma + 4 \gamma )$. As shown in Fig. \[fig:violation-spectrum\](b), the energy injection rate is always positive and approaches a constant in the limit $\epsilon\to 0$.
We now examine dissipation of the injected energy through viscous relaxation of the bead position $x$, which is our slow variable. To use Eq. (\[eq:HS\]) to compute the associated entropy production, we need to work out the velocity correlation spectrum $\tilde{C}_{\dot{x}}(\omega)$ and the response spectrum $\tilde{R}_{\dot{x}}'(\omega)$. It turns out that, in the harmonic case, Eq. (\[eq:noise\]) takes a linear form and can be solved analytically. Here, $\partial_x U_{\sigma_t}(x)=\partial_x U_e(x)-\xi(t)$ is decomposed into an effective force $\partial_x U_e(x)=k(x-L/2)$ and a “switching noise” $\xi(t)=kL(\sigma_t-1/2)$, with $\langle\xi(t)\rangle=0$ and $\langle\xi(t)\xi(t')\rangle=(kL/2)^2\exp(-2r|t-t'|)$. As shown in Fig. 1(c), $\tilde{R}'_{\dot{x}}(\omega)=\gamma\omega^2/(k^2+\gamma^2\omega^2)$ is independent of $\epsilon$. The correlation spectrum $\tilde{C}_{\dot{x}} (\omega)$, on the other hand, contains a term $2T\tilde{R}_{\dot{x}}'(\omega)$ from the thermal noise $\eta(t)$, and the remaining part from the switching noise $\xi(t)$. The latter is precisely the FRR violation spectrum shown in Fig. 1(d): $$\begin{aligned}
\tilde{C}_{\dot{x}}(\omega)-2T\tilde{R}_{\dot{x}}'(\omega)= \epsilon {k(L/2)^2\over 1+(\omega\tau_f/2)^2} \tilde{R}_{\dot{x}}'(\omega).
\label{eq:violation}
\end{aligned}$$ Carrying out the integral over $\omega$ in Eq. (\[eq:HS\]), we obtain $J_x=\dot{W}$. Therefore, the FRR violation spectrum of the slow variable $\dot{x}$ allows full recovery of entropy production in the present case.
A remarkable feature of the FRR violation spectrum displayed in Fig. 1(d), which is also evident from Eq. (\[eq:violation\]), is the plateau behavior in the intermediate frequency range $\tau_s^{-1} \ll \omega \ll \tau_f^{-1}$. This is the time or frequency window over which the fluctuating dynamics of $x$ deviates most from equilibrium, and also where the hidden entropy production takes place in the model. As $\epsilon$ approaches zero, the height of the plateau diminishes, leaving the apparent impression that the FRR is restored. Nevertheless, the integral in Eq. (\[eq:HS\]) remains finite so as to be consistent with the energy input shown in Fig. 1(b). Our explicit solution of the potential switching model thus exposes subtleties surrounding the limit $\epsilon\rightarrow 0$.
The above example demonstrates that, with the help of the HS equality, at least part of the hidden entropy production can be recovered through precise measurement of the FRR violation spectrum of a slow variable. To gain an impression on the feasibility of this proposal, we have explicitly reconstructed the velocity correlation and response spectrum from a simulated stochastic trajectory $x(t)$ of the potential switching model at $\epsilon=0.01$. The length of the trajectory is taken to be $T_{sp}=10^4\tau_s$, with a sampling rate of $2\tau_f^{-1}$. This is sufficient to reveal the full range of the plateau as indicated by open circles in Fig. \[fig:violation-spectrum\](d). As we show in Supplemental Material [@supp-hiddenEntropy], the relative fluctuation of $\tilde{C}(\omega)$ goes generally as $(T_{sp}/\tau_s)^{-1/2}$. Therefore, to reach a precision of order $\epsilon$, the length of the time series should be of the order of $\tau_s/\epsilon^2$. Lowering temporal resolution in the measurement will lose information on the high frequency end of the spectrum. Nevertheless, even a sampling rate of $0.1\tau_f^{-1}$ can provide good evidence of nonequilibrium fluctuations in $x$ generated by the hidden fast processes.
Below we show that the plateau behavior is a general feature of nonequilibrium Markov systems with timescale separation. Since a Langevin model can be considered as a special case of the Markov process, the aforementioned results can be extended to general potential switching models with a nonlinear force field and position-dependent switching rates, including the well-studied F$_1$-ATPase. This constitutes the main result.
![(a) Illustration of a general nonequilibrium Markov process with a dissipative cycle formed by fast and slow processes. (b) Effective system at large timescale separation $\epsilon\ll 1$, where the circulating probability flux is hidden, along with the associated entropy production. (c) Eigenvalue spectrum of the Master equation for a timescale separated system. Fast modes are well-separated in their decay rates $\lambda_j$ from the slow ones. The latter form a nearly degenerate band at the bottom that defines the slow dynamics of the effective system. []{data-label="fig:modelSystem"}](modelSystem6.PDF){width="7cm"}
*General Markov processes.—* Consider a general connected Markov system with $N$ states. Transition from state $m$ to state $n$ ($1 \leq n,m \leq N$) takes place at rate $w_{m}^{n}$. The probability $P_n(t)$ to be at state $n$ after time $t$ follows the Master equation $$\frac{d}{dt}P_n(t)=\sum_m M_{nm}P_m(t),$$ where $M_{nm}=w_m^n-\delta_{nm} \sum_{k}w_n^k $ and $\delta_{nm}$ is the Kronecker delta. The right and left eigenmodes, denoted as $x_j$ and $y_j$ respectively, satisfy $\sum_m M_{nm}x_j (m)=-\lambda_j x_j (n)$ and $\sum_n y_j (n) M_{nm}=-\lambda_j y_j (m)$, where the minus sign is introduced so that $\operatorname{Re}(\lambda_j)\ge 0$. We rank $\lambda_j$ in an ascending order by its real part, i.e., $\operatorname{Re}(\lambda_1)\le \operatorname{Re}(\lambda_2)\le \cdots$. The first eigenvalue $\lambda_1=0$, with $x_1(n)=P_n^{s}$ being the steady-state distribution and $y_1(n)=1$. Generically, the normalised eigenmodes satisfy the orthogonality relation $\sum_n x_j(n)y_{j'}(n)=\delta_{jj'}$ and the completeness relation $\sum_j x_j(n)y_j(n')=\delta_{nn'}$.
We now introduce an external perturbation of strength $h$ whose effect on the dynamics is specified by the modified transition rates $\tilde{w}_m^n=w_m^n\exp[(\mathcal{Q}_n-\mathcal{Q}_m)h/2T]$, where $\mathcal{Q}_n$ is a state variable conjugate to $h$ [@diezemann2005fluctuation; @maes2009response]. Along a stochastic trajectory $n_t$, $Q(t)\equiv \mathcal{Q}_{n_t}$. In analogy with the velocity variable for the Langevin dynamics (\[eq:noise\]), we consider the time derivative $\dot{Q}(t)$ whose correlation and dynamic response are defined as $C_{\dot{Q}}(t-\tau)\equiv \langle [\dot{Q}(t)-\langle \dot{Q}\rangle_{s}][\dot{Q}(\tau)-\langle\dot{Q}\rangle_{s}]\rangle_{s}$ and $R_{\dot{Q}}(t-\tau)\equiv \delta \langle \dot{Q}(t)\rangle/\delta h_\tau$, respectively. By following the time evolution of the state probabilities $P_n(t)$ under the eigenmode expansion, we have obtained general expressions for $C_{\dot{Q}}$ and $R_{\dot{Q}}$ which take the following form in frequency,
\[eq:CR-fre\] $$\begin{aligned}
\tilde{C}_{\dot{Q}}(\omega)&=&\sum_{j=2}^N 2\alpha_j\beta_j\lambda_j \Big[1-\frac{1}{1+(\omega/\lambda_j)^2}\Big], \\
\label{eq:Cvelo_fre}
\tilde{R}_{\dot{Q}}(\omega) &=&\sum_{j=2}^N \alpha_j\phi_j\Big[1- \frac{1-i(\omega/\lambda_j) }{1+(\omega/\lambda_j)^2}\Big] ,
\label{eq:Rvelo_fre}
\end{aligned}$$
where $i$ is the imaginary unit.
The coefficients in Eqs. (\[eq:CR-fre\]) are weighted averages of $\mathcal{Q}$, i.e., $\alpha_j\equiv \sum_{n} \mathcal{Q}_{n}x_j(n),\; \beta_j \equiv \sum_{n}\mathcal{Q}_{n} y_j(n)P_{n}^{s}$, and $\phi_j \equiv \sum_{n} B_{n} y_j(n)$, with $B_n \equiv \sum_m[ w_m^nP_m^{s}+w_n^mP_n^{s} ](\mathcal{Q}_n-\mathcal{Q}_m)/2T$. They satisfy the general sum rule [@shouwen2016PRE], $$\sum_{j=2}^N \alpha_j(\lambda_j\beta_j-T\phi_j)=0.
\label{eq:sum-rule}$$ The detailed balance condition $w_n^mP_n^{eq}=w_m^nP_m^{eq}$ implies $\lambda_j\beta_j^{eq}=T\phi_j^{eq} $ for all $j$, and hence the FRR $\tilde{C}_{\dot{Q}}(\omega)=2T\tilde{R}_{\dot{Q}}'(\omega)$. More generally, in the limit $\omega\rightarrow\infty$, $\tilde{C}_{\dot{Q}}(\infty)=2T\tilde{R}_{\dot{Q}}'(\infty)= const$ by virtue of the sum rule, confirming the behavior seen in Fig. 1(c) on the high frequency side.
We now apply the general results to a timescale separated system whose states can be partitioned into $K$ subgroups or coarse-grained states, denoted as $p$ or $q$. Each state $n$ ($m$) is alternatively labeled by $p_k$ ($q_l$), with $k$ ($l$) for a microscopic state within $p$ ($q$). Within a coarse-grained state, transitions are fast (timescale $\sim \tau_f$), whereas transitions across coarse-grained states are slow (timescale $\sim \tau_s$), as illustrated in Fig. \[fig:modelSystem\](a). Formally, this condition amounts to the statement that the transition rate matrix is split into two parts: $$M_{p_kq_l}=\epsilon^{-1}\delta_{pq}M^q_{kl}+M^{(1)}_{p_kq_l},
\label{eq:M-separated}$$ where $ \epsilon^{-1}M^{q}$ ($\epsilon \equiv \tau_f/\tau_s$) is the transition rate matrix within a coarse-grained state $q$ while $M^{(1)}$ is for slow transitions between coarse-grained states [@rahav2007fluctuation]. Below we shall analyze the eigenvalue and FRR spectra of the Markov process Eq. (\[eq:M-separated\]) using perturbation theory.
In the absence of inter-group transitions, the matrix $M$ is block diagonalized. Within each block $q$, the rate matrix $\epsilon^{-1}M^q$ has a nondegenerate eigenvalue $\lambda_1^q=0$ and the corresponding stationary distribution $P^{s}(l|q)$. All other eigenvalues of the matrix are positive and scale as $\epsilon^{-1}$, which together define the fast modes of the system.
Figure 2(c) illustrates the eigenvalue spectrum of the block diagonalized matrix and its modification by $M^{(1)}$ when inter-block transitions are introduced. Lifting of the $K$-fold degeneracy at $\lambda=0$ can be analyzed using standard perturbation theory. To the leading order in $\epsilon$, the projected transition rate matrix, $$\widehat{M}_{pq}= \sum_{k,l} M_{p_kq_l}^{(1)}P^{s}(l|q),$$ defines an emergent dynamics for the coarse-grained states. Its eigenmodes $\widehat{x}_j$ and $\widehat{y}_j$ with eigenvalue $\widehat{\lambda}_j$ ($\sim 1$) account for the leading order behavior of the slow modes $(j\le K)$ in the full state space [@shouwen2016PRE], which satisfy $ x_j(p_k)=\widehat{x}_j(p)P^{s}(k|p)+O(\epsilon)$, and $ y_j(p_k)=\widehat{y}_j(p)+O(\epsilon)$, and $\lambda_j=\widehat{\lambda}_j+O(\epsilon)$. We now focus on a slow observable $Q_{p_k}=Q_p$ that only depends on the coarse-grained state. Then, $\widehat{\alpha}_j$, $\widehat{\beta}_j$, and $\widehat{\phi}_j$ are also defined for the slow modes $(j \le K)$, which turns out to be $\alpha_j=\widehat{\alpha}_j+O(\epsilon)$, $\beta_j=\widehat{\beta}_j+O(\epsilon)$, and $\phi_j=\widehat{\phi}_j+O(\epsilon)$. In terms of the parameters from the coarse-grained dynamics, we may write $$\tilde{C}_{\dot{Q}}-2T\tilde{R}'_{\dot{Q}}=2\sum_{j=2}^K \widehat{\alpha}_j \frac{T\widehat{\phi}_j-\widehat{\beta}_j\widehat{\lambda}_j}{1+(\omega/\widehat{\lambda}_j)^2}+ \epsilon V_s(\omega).
\label{eq:velo-FRR-vio}$$ An explicit form for $V_s(\omega)$ is given in Supplemental Material [@supp-hiddenEntropy]. In the intermediate region ($\tau_s^{-1}\ll \omega\ll\tau_f^{-1}\sim \epsilon^{-1}$), $V_s(\omega) \simeq 2\epsilon^{-1}\sum_{j=2}^K \alpha_j(\beta_j\lambda_j-T\phi_j)\xrightarrow{\epsilon \to 0} const$ due to the sum rule $\sum_{j=2}^K \widehat{\alpha}_j(\widehat{\beta}_j\widehat{\lambda}_j-T\widehat{\phi}_j)=0$ under $\widehat{M}$. Besides, $V_s(\omega)$ vanishes for both $\omega\gg \tau_f^{-1}$ and $\omega\ll \tau_s^{-1}$.
Equation (\[eq:velo-FRR-vio\]) is the generalized description of the plateau behavior we found in the potential switching model \[see Eq. (\[eq:violation\])\]. The low frequency FRR violation spectrum $(\omega\sim \tau_s^{-1})$ comes from the coarse-grained dynamics, which vanishes if the detailed balance condition is fulfilled under $\widehat{M}$. In the intermediate frequency region, the non-equilibrium coupling between the fast and slow variables produces a plateau $\epsilon V_s$ whose height diminishes in the timescale separation limit $\epsilon\rightarrow 0$.
Since the HS equality holds generally for system variables that follow the Langevin dynamics, Eq. (\[eq:velo-FRR-vio\]) can be immediately used to obtain heat dissipation associated with the frictional motion of these variables. On the other hand, if the slow variable $p$ takes on discrete set of values, then the situation is more complicated. Lippiello et al. [@Lippiello2014fluctuation] considered a special class of discrete models whose dynamics follows closely that of a Langevin system. There it was shown for a one-dimensional system that, when the allowed transition rates take the symmetric form $w_m^n=\tau^{-1}e^{[S(n)-S(m)]/2}$ with $|S(n)-S(m)|\ll 1$, the HS equality holds approximately, and hence a link between the FRR violation spectrum and the heat dissipation can again be established. In Supplemental Material [@supp-hiddenEntropy], we present an example with a ladder network structure. More general discussion of the relation between the plateau behavior and the hidden entropy production is left to future work.
*Concluding Remarks.—* We have demonstrated in a fairly general setting that, for driven systems with large timescale separation, the fluctuation-response relation applied to a slow variable is close to be satisfied below its relaxation time. However, close examination should reveal a characteristic plateau behavior in the FRR violation spectrum in the intermediate frequency region. The Harada-Sasa equality can then be invoked to compute the frictional dissipation arising from nonequilibrium fluctuations using data from precise measurements of the fluctuation-response spectrum.
We believe that our findings can be applied to expose hidden entropy production in molecular motors. Examples include the F$_{1}$-ATPase mutants [@toyabe2011thermodynamic; @toyabe2015single], which are known to be less efficient in converting chemical to mechanical energy as compared to the wild-type. The fast variable in this case corresponds to the chemical states associated with ATP binding and hydrolysis, while the relatively slow variable is the rotational angle. Their chemomechanical coupling is usually modeled by a potential switching model where position-dependent ATP binding and hydrolysis shifts the potential forward, thus generating a directed rotation [@kawaguchi2014nonequilibrium]. Our work suggests that energy loss in inefficient motors may be caused by fast switching of the chemical states that produces nonequilibrium fluctuations in the rotary motion. If so, at least part of the hidden entropy production can be measured by observing the rotary motion with a high speed camera, without monitoring the chemical states. Suppose that the rotory motion has a relaxation time around $ 0.1$s, and the timescale of ATP binding/hydrolysis around $5\times 10^{-3}$s, then a sampling duration $T_{sp}=400$s and a temporal resolution $2.5\times 10^{-3}$s, which is attainable in state-of-the-art single molecule experiments [@toyabe2012recovery], would be enough to see the FRR violation spectrum. The rate of ATP binding and ADP release can be modified by varying the concentration of these molecules, yielding further information on the nature of nonequilibrium fluctuations and associated energy dissipation in the system.
The authors thank Yohei Nakayama and Takayuki Ariga for helpful suggestions on the manuscript. The work was supported in part by the NSFC under Grant No. U1430237 and by the Research Grants Council of the Hong Kong Special Administrative Region (HKSAR) under Grant No. 12301514. It was also supported by KAKENHI (Nos. 25103002 and 26610115), and by the JSPS Core-to-Core program “Non-equilibrium dynamics of soft-matter and information”.
[32]{}
H. Noji, R. Yasuda, M. Yoshida, and K. Kinosita, Nature [**386**]{}, 299 (1997).
S. Toyabe, T. Okamoto, T. Watanabe-Nakayama, H. Taketani, S. Kudo, and E. Muneyuki, Phys. Rev. Lett. [**104**]{}, 198103 (2010).
T. Sagawa and M. Ueda, Phys. Rev. Lett. [**104**]{}, 090602 (2010).
S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki, and M. Sano, Nat. Phys. [**6**]{}, 988 (2010).
A. B[é]{}rut A. Arakelyan, A. Petrosyan, S. Ciliberto, R. Dillenschneider, and E. Lutz, Nature [**483**]{}, 187 (2012).
J. V. Koski, A. Kutvonen, I. M. Khaymovich, T. Ala-Nissila, and J. P. Pekola, Phys. Rev. Lett., [**115**]{}, 260602 (2015)
J. V. Koski, T. Sagawa, O. Saira, Y. Yoon, A. Kutvonen, P. Solinas, M. M[ö]{}tt[ö]{}nen, T. Ala-Nissila, and J. Pekola, Nat. Phys. [**9**]{}, 644 (2013).
E. Dieterich, J. Camunas-Soler, M. Ribezzi-Crivellari, U. Seifert, and F. Ritort, Nat. Phys. [**11**]{}, 971 (2015).
T. Hondou and K. Sekimoto, Phys. Rev. E [**62**]{}, 6021 (2000).
M. Esposito, Phys. Rev. E [**85**]{}, 041125 (2012).
A. Celani, S. Bo, R. Eichhorn, and E. Aurell, Phys. Rev. Lett. [**109**]{}, 260603 (2012).
K. Kawaguchi and Y. Nakayama, Phys. Rev. E [**88**]{}, 022147 (2013).
Y. Nakayama and K. Kawaguchi, Phys. Rev. E [**91**]{}, 012115 (2015).
H.-M. Chun and J. D. Noh, Phys. Rev. E [**91**]{}, 052128 (2015).
M. Esposito and J. M. R. Parrondo, Phys. Rev. E [**91**]{}, 052114 (2015).
S.-W. Wang, Y. Lan, and L.-H. Tang, J. Stat. Mech. Theor. Exp. [**2015**]{}, P07025 (2015).
P. Sartori and Y. Tu, Phys. Rev. Lett. [**115**]{}, 118102 (2015).
S. Bo and A. Celani, J. Stat. Phys. [**154**]{}, 1325–1351 (2014).
A. Puglisi, S. Pigolotti, L. Rondoni and A. Vulpiani, J. Stat. Mech. Theor. Exp. [**2010**]{}, P05015 (2010).
R. Kubo, Rep. Prog. Phys. [**29**]{}, 255 (1966).
L. F. Cugliandolo, D. S. Dean, and J. Kurchan, Phys. Rev. Lett. [**79**]{}, 2168 (1997).
T. Speck and U. Seifert, Europhys. Lett. [**74**]{}, 391 (2006).
M. Baiesi, C. Maes, and B. Wynants, Phys. Rev. Lett. [**103**]{}, 010602 (2009).
U. Seifert and T. Speck, Europhys. Lett. 89, 10007 (2010).
T. Harada and S.-i. Sasa, Phys. Rev. Lett. [**95**]{}, 130602 (2005).
T. Harada and S.-i. Sasa, Phys. Rev. E [**73**]{}, 026131 (2006).
S. Toyabe, H.-R. Jiang, T. Nakamura, Y. Murayama, and M. Sano, Phys. Rev. E [**75**]{}, 011122 (2007).
K. Kawaguchi, S.-i. Sasa, and T. Sagawa, Biophys. J. [**106**]{}, 2450 (2014).
See Supplemental Material for computation of the correlation and response spectrum from data, an illustrative Markov system to which the Harada-Sasa equality can be applied, and a detailed derivation of Eq. (\[eq:velo-FRR-vio\]).
G. Diezemann, Phys. Rev. E [**72**]{}, 011104 (2005).
C. Maes and B. Wynants, Markov Processes Relat. Fields [**16**]{}, 45 (2010).
S.-W. Wang, K. Kawaguchi, S.-i. Sasa, and L.-H. Tang (in preparation).
Aspects of such a timescale separated system was studied previously by C. Jarzynski and S. Rahav, J. Stat. Mech. Theor. Exp. [**2007**]{}, P09012 (2007).
E. Lippiello, M. Baiesi and A. Sarracino, Phys. Rev. Lett., [**112**]{}, 140602 (2014).
S. Toyabe, T. Watanabe-Nakayama, T. Okamoto, S. Kudo, and E. Muneyuki, Proc. Natl. Acad. Sci. U.S.A. [**108**]{}, 17951 (2011).
S. Toyabe and E. Muneyuki, New J. Phys. [**17**]{}, 015008 (2015).
S. Toyabe, H. Ueno, and E. Muneyuki, Europhys. Lett. [**97**]{}, 40004 (2012)
addtoreset[equation]{}[section]{}
addtoreset[equation]{}[section]{}
addtoreset[figure]{}[section]{}
Supplemental Material
=====================
Computation of the correlation and response spectra from data
-------------------------------------------------------------
To determine the velocity correlation spectrum, we do the following: (a) sample the unperturbed trajectory $x(t)$ at a temporal resolution $\delta t$ for a duration $T_{sp}$ ($\gg \tau_s$); (b) compute the Discrete Fourier Transform (DFT) $\tilde{x}(\omega)$ of $x(t)$, from which the velocity correlation spectrum $\omega^2|\tilde{x}(\omega)|^2 $ is constructed; (c) following the proposal in [@berg2004power], perform local average of the spectrum to suppress noise (known as data compression).
To determine the velocity response spectrum at selected frequencies $\{\omega_k\}$ simultaneously, we do the following: (a) sample the trajectory $x(t)$ perturbed by a linear combination of periodic driving forces $h=\sum_k h_k\exp(i\omega_k t)$, at a temporal resolution $\delta t$ for a duration $T_{sp}$ ($\gg \tau_s$); (b) compute the DFT $\tilde{x}(\omega)$ of the perturbed trajectory $x(t)$; (c) at each $\omega_k$, calculate the response $\tilde{R}_x(\omega_k)=\tilde{x}(\omega_k)/h_k$, and thus the velocity response $\tilde{R}_{\dot{x}}(\omega_k)=-i\omega_k \tilde{R}_x(\omega_k)$.
Below we examine in some detail fluctuations of the spectra constructed through the above procedure when a single long trajectory is used in computation. This will allow us to devise suitable averaging procedures to suppress noise in the data without significantly hampering its information content.
### Correlation spectrum
The discussion below follows main ideas of the power spectrum analysis presented in Ref. [@berg2004power].
For a general stochastic trajectory $x(t)$ sampled at resolution $\delta t$ and for duration $T_{sp}=N_{sp}\delta t$, we introduce the following DFT, $$\tilde{x}(\omega_k)=\frac{\sqrt{\delta t}}{\sqrt{N_{sp}}}\sum_{j=1}^{N_{sp}} x(j\delta t)\exp(-i\omega_k j \delta t) ,
\label{eq:renormalization}$$ where $\omega_k=2\pi k/T_{sp}, k=0,\ldots, N_{sp}-1$. As illustrated by the harmonic potential switching model below, each Fourier amplitude $\tilde{x}(\omega_k)$ is a random variable whose value, just like the trajectory $x(t)$, changes from realization to realization. Therefore the velocity spectral function $$g(\omega)=\omega^2|\tilde{x}(\omega)|^2,$$ obtained from a single trajectory is a strongly fluctuating quantity. To estimate the ensemble averaged spectrum $$\tilde{C}_{\dot{x}}(\omega)=\langle g(\omega)\rangle,$$ one may invoke the smoothness of the function by averaging $g(\omega_k)$ over nearby frequencies in a suitable window of width $\Delta\omega$. Since the spacing between accessible frequencies is $\delta\omega =2\pi/T_{sp}$, relative error of the window-averaged $g(\omega)$ should decrease as $(\Delta\omega/\delta\omega)^{-1/2}=(T_{sp}\Delta\omega/2\pi)^{-1/2}$. On the other hand, $\Delta\omega$ should not be chosen too large to cause significant information loss. For the latter, we may consider the Taylor expansion of $\tilde{C}_{\dot{x}}(\omega)$ around a chosen $\bar\omega$, $$\tilde{C}_{\dot{x}}(\omega)=\tilde{C}_{\dot{x}}(\bar{\omega})+{d\tilde{C}_{\dot{x}}\over d\omega}\Bigr|_{\bar\omega}
(\omega-\bar\omega) +\frac{1}{2} \frac{d^2 \tilde{C}_{\dot{x}}}{d\omega^2}\Bigr|_{\bar\omega}
(\omega-\bar\omega)^2 +\ldots$$ Averaging the above expression over a frequency window of width $\Delta\omega$ centered at $\bar\omega$ yields, $$\overline{\tilde{C}_{\dot{x}}(\omega)}\simeq \tilde{C}_{\dot{x}}(\bar{\omega})+\frac{1}{24} \frac{d^2 \tilde{C}_{\dot{x}}}{d\omega^2}\Bigr|_{\bar\omega}(\Delta\omega)^2 .$$ Equating the two terms on the right-hand-side of the expression above, we obtain, $$\Delta \omega_{\rm M}= \sqrt{24} \left| \frac{\tilde{C}_{\dot{x}}(\bar{\omega})}{\partial^2_{\bar{\omega}}\tilde{C}_{\dot{x}}(\bar{\omega})} \right|^{1/2}.$$ This sets an upper bound on the window size for averaging. In the part of the spectrum where the curvature of $\tilde{C}_{\dot{x}}(\omega)$ is small, a large window is desirable so as to reduce statistical error in the original data.
We now illustrate the above general ideas with an explicit example, the harmonic potential switching model. The bead displacement $x(t)$ from the mid-point $L/2$ satisfies the Langevin equation, $$\gamma \dot{x}=-kx +\xi(t) +\eta(t),
\label{eq:Langevin}$$ where $\xi(t)$ and $\eta(t)$ are switching and thermal noise, respectively. Integrating Eq. (\[eq:Langevin\]) over the time interval $\delta t$, we obtain, $$x(t+\delta t)=\exp(-\delta t/\tau_s)x(t) +\gamma^{-1}\int_t^{t+\delta t}dt' \exp\bigl({t-t'\over\tau_s}\bigr)[\xi(t') +\eta(t')],
\label{eq:delta_t}$$ where $\tau_s=\gamma/k$ is the relaxation time constant of bead displacement.
With the help of Eq. (\[eq:delta\_t\]), and ignoring boundary terms at the beginning and end of the time series, we obtain the following equation for the DFT $\tilde{x}(\omega)$ when the sampling time $\delta t\ll \tau_s,2\pi/\omega$, $$i \gamma \omega \tilde{x}(\omega)\simeq -k\tilde{x}(\omega)+\tilde{\xi}(\omega)+\tilde{\eta}(\omega).
\label{eq:fourier-eq}$$ Here, $\tilde{\xi}(\omega)$ and $\tilde{\eta}(\omega)$ are the usual Fourier transforms of the active noise $\xi(t)$ and thermal noise $\eta(t)$, respectively, independent of $\delta t$. Therefore the equation satisfied by $\tilde{x}(\omega)$ is nearly identical to its continuous counterpart, provided the sampling time interval $\delta t$ is much shorter than both $\tau_s$ and $2\pi/\omega$ (i.e., the part of the spectrum with $\omega <2\pi/\delta t$).
Following Eq. (\[eq:fourier-eq\]), the velocity correlation spectrum from a single trajectory is given by, $$g(\omega)=\omega^2|\tilde{x}(\omega)|^2\simeq{1\over k^2}\frac{\omega^2}{1+ (\omega\tau_s)^2} \left[|\tilde{\xi}(\omega)|^2+ 2\text{Re}\left(\tilde{\xi}(\omega)\tilde{\eta}^*(\omega)\right)+|\tilde{\eta}(\omega)|^2\right],
\label{eq:fluctuating-g}$$ where $*$ denotes complex conjugate. Since the two noises are uncorrelated from each other, the cross term vanishes upon ensemble average that yields the desired correlation spectrum $\tilde{C}_{\dot{x}}(\omega)$.
The mean of $\tilde{\xi}(\omega)$ and $\tilde{\eta}(\omega)$ are zero while their variances are given by $\langle|\xi(\omega)|^2\rangle=\epsilon\gamma k(L/2)^2/[1+(\omega\tau_f/2)^2]$ and $\langle|\eta(\omega)|^2\rangle=2\gamma T$, respectively. On the high frequency end of the spectrum, i.e., $\omega\gg\tau_s^{-1}$, $\tilde{C}_{\dot{x}}(\omega)$ varies only on the scale of $\tau_f^{-1}$ which is much greater than $\tau_s^{-1}$ and even greater than $\delta\omega=2\pi/T_{sp}$. Therefore $\Delta\omega$ can be chosen to be very large to reduce the noise in the bare correlation spectrum (\[eq:fluctuating-g\]). On the other hand, for frequencies that are comparable or even smaller than $\tau_s^{-1}$, $\tilde{C}_{\dot{x}}(\omega)$ varies appreciably so that the window size is limited by $\tau_s^{-1}$. This is the main reason behind the noisy low frequency spectrum shown in Fig. 1(d) in the Main Text.
### Fluctuation of the response spectrum
Now, we consider applying a periodic driving force $h_0\exp(-i\omega_0 t)$ to the system. Similarly to Eq. (\[eq:fourier-eq\]), the equation at $\omega=\omega_0$ satisfies, $$i \gamma \omega_0 \tilde{x}(\omega_0)\simeq -k\tilde{x}(\omega_0)+\tilde{\xi}(\omega_0)+\tilde{\eta}(\omega_0)+\sqrt{T_{sp}} h_0 .$$ Here, the prefactor $\sqrt{T_{sp}}$ appears due to the rescaling scheme in Eq. (\[eq:renormalization\]). The velocity response spectrum calculated for a single perturbed trajectory is given by, $$f(\omega_0)\equiv -i\omega_0 \frac{\tilde{x}(\omega_0)}{h_0\sqrt{T_{sp}}}=-\frac{i\omega_0 }{i\gamma \omega_0+k}\left( 1+\frac{ \tilde{\xi}(\omega_0)}{h_0\sqrt{T_{sp}}}+\frac{\tilde{\eta}(\omega_0)}{h_0\sqrt{T_{sp}}}\right),
\label{eq:f-fluc}$$ which contains fluctuations. Its ensemble average converges to the correct result $$\langle f(\omega_0)\rangle= \tilde{R}_{\dot{x}}(\omega_0).$$ Its standard deviation is given by $$\sqrt{\langle |f(\omega_0)- \tilde{R}_{\dot{x}}(\omega_0)|^2\rangle}=\frac{1}{h_0\sqrt{T_{sp}}}\tilde{C}_{\dot{x}}(\omega_0).
\label{eq:f-sd}$$ Equation (\[eq:f-sd\]) suggests that fluctuations in the velocity response spectrum can be reduced by increasing $h_0$, thus the signal/noise ratio, and also by using a longer trajectory.
However, $h_0$ must be small enough to ensure that we are measuring linear response. Besides, $T_{sp}$ may also be limited in experiments. In such a situation, one may consider to perform averaging over nearby frequencies to reduce noise in the response spectrum (\[eq:f-fluc\]). For example, we may apply a perturbation with multiple frequencies inside a window of width $\Delta \omega$, i.e., $h_0\sum_{k=-n_r/2}^{n_r/2-1} \exp(i[\omega_0+k \delta \omega] t)$, where $\delta \omega\gg 2\pi/T_{sp}$ to avoid possible interference effects. Here $n_r=\Delta \omega/\delta \omega$ is the number of frequencies considered. Averaging over the response at these frequencies will further reduce the error (\[eq:f-sd\]) by a factor $1/\sqrt{n_r}$.
### Error bars on the reconstructed spectra
Fig. \[fig:error-bar\] shows error bars on the reconstructed spectra presented in Fig. 1 of the Main Text. They have been estimated from variations in the data within each window used for averaging. For the parameters chosen, our procedure yields very good results on the high frequency end, but less satisfactory results on the low frequency side close to $\omega\simeq \tau_s^{-1}$. Fluctuations in the latter case are mainly due to insufficient averaging when computing the correlation spectrum. Although we have not attempted to optimize this part of the data analysis, it is conceivable that data smoothening procedures apply to the autocorrelation function $C_x(t)$ can lead to much improved results.
Data points in Fig. S2(a) show the numerically determined FRR violation spectrum from a trajectory 10 times longer than the one used to obtain the data points in Fig. S1(b). Fig. \[fig:L100000\](b) shows the data in the same time window but with a much reduced sampling rate $0.1\tau_f^{-1}$. The low frequency part of the violation spectrum is faithfully reproduced.
![ (a) The velocity correlation and response spectrum reconstructed from a simulated trajectory, the same data as those presented in FIG.(1)(c) of the Main Text but with error bars. (b) The corresponding FRR violation spectrum, plotted with error bars. Parameters: $\gamma=k=1$, $L=5$, $\tau_f=0.01$, $T_{sp}=10^4$, $\delta t=\tau_f/2$, and $h_0=0.5$. []{data-label="fig:error-bar"}](errorBar_new.PDF){width="17cm"}
Validity of the Harada-Sasa equality in Markov systems
------------------------------------------------------
In this section, we show that the HS equality holds approximately for a discrete Markov jump system with a ladder network. Our argument is a generalization of the work in [@Lippiello2014fluctuation], where the HS equality in a one-dimensional discretized Langevin system was studied. The model we study here, which describes the chemical state dynamics of the chemotaxis-related membrane chemoreceptor in E.coli [@tu2013quantitative], is illustrated in FIG. \[fig:adaptation\](a). The chemical state of this receptor is a combination of the methylation level $m$ ($=0,1,2,3,4$), and the activity $a$ ($=0,1$). $\alpha$ is a small parameter that tunes the irreversibility of the methylation/demethylation dynamics, and we only consider the range $\alpha\le \exp(1)$. The transition from the active state $a=1$ to the inactive state $a=0$ at methylation level $m$ is denoted as $w_1(m)$, while the reverse transition rate is denoted as $w_0(m)$. The rates satisfy $$\begin{aligned}
w_0(m)&=&\frac{1}{\tau_f}\exp\left(-\frac{\Delta E(m)}{2T}\right),\\
w_1(m)&=&\frac{1}{\tau_f}\exp\left(\frac{\Delta E(m)}{2T}\right),\end{aligned}$$ where $\Delta E(m)=e_0(m_*-m)$. Here, we set $e_0=2$, $m_*=2$, $\tau_f=0.01$ and $r=1$ in the numerical study. Therefore, dynamics of $m$ is relative slow and the timescale separation index $\epsilon\approx 0.01$. For more details of this model, refer to [@shouwen2015adaptation]. Here, we study the validity of the HS equality when applied to the slow variable $m$.
![(a) The sensory adaptation network for E.coli. (b) The comparison between actual dissipation of the methylation dynamics, i.e., $J_m$, and that estimated from the HS equality, where we use $\gamma_{eff}=\lim_{\omega\to \infty}1/\tilde{R}_{\dot{m}}(\omega)$. (c) Similar as (b) except that we use $\gamma_{eff}=T/[r\sqrt{\alpha}]$ here. []{data-label="fig:adaptation"}](adaptationHS.PDF){width="16cm"}
Suppose that a transition from state $n$ to $m$ occurs with a rate $w_n^m$. Then this jump produces entropy $\Delta S_{n}^m=\ln [w_{n}^m/w_{m}^n]$ in the surrounding media, and the average dissipation rate of this transition is given by $$[P^s_nw_{n}^m-P^s_mw_{m}^n]\ln \frac{w_{n}^m}{w_{m}^n},$$ where $[P^s_nw_{n}^m-P^s_mw_{m}^n]$ is the net flux for this type of transition. The average dissipation rate due to the change of $m$, denoted as $J_m$, is obtained by summing over all dissipations that involves changes of $m$, which is given by $$J_m= \sum_m\Big( \left[ P^{s}_{1,m+1}r-P^{s}_{1,m}\alpha r \right]\ln \frac{1}{\alpha} + \left[ P^{s}_{0,m+1}\alpha r-P^{s}_{0,m} r\right]\ln \alpha\Big).
\label{eq:Jm}$$ where $P^s_{a,m}$ is the stationary distribution at state $(a,m)$.
Now, we consider how to estimate this dissipation rate from the FRR violation spectrum. From the temporal trajectory $m(t)$, we can compute its velocity correlation spectrum $\tilde{C}_{\dot{m}}(\omega)$ and response spectrum $\tilde{R}_{\dot{m}}(\omega)$. Then, we are supposed to estimate the dissipation rate by $$\gamma_{eff} \left (\langle \dot{m}\rangle+\int_{-\infty}^\infty \left[\tilde{C}_{\dot{m}}(\omega)-2T\tilde{R}_{\dot{m}}'(\omega)\right]\frac{d\omega}{2\pi} \right).
\label{eq:eff-HS}$$ The key component in this estimation is the effective friction coefficient $\gamma_{eff}$, which is in general missing for a Markov process. Below, we provide two possible definitions, both of which are rooted in the generalization of friction coefficient defined Langevin systems.
In the over-damped Langevin system, we know that the friction coefficient is the inverse of the velocity response spectrum in the high frequency limit. Therefore, we may define this effective friction coefficient $$\gamma_{eff}=\lim_{\omega\to \infty} 1/\tilde{R}_{\dot{m}}(\omega),
\label{eq:gamma-1}$$ which is directly measurable.
Alternatively, if we discretize a continuous over-damped Langevin system with spatial unit $\Delta x$, its transition rates take the form $$\frac{T}{\gamma \Delta x^2} \exp(\Delta S/2),
\label{eq:DeltaS}$$ where $\Delta S$ is to be understood as the entropy produced in the medium in this single jump. The corresponding backward jump produces medium entropy $-\Delta S$ due to time-reversal asymmetry. This is consistent with the fact that more probable transitions always tend to produce positive entropy in the medium, which is the microscopic origin for the irreversibility of the second law. In the case of methylation dynamics in FIG. \[fig:adaptation\], we may identify $\Delta S$ as the medium entropy produced by the transition with rate $r$, and rewrite the methylation/demethylation rates in the form of Eq. (\[eq:DeltaS\]), i.e., $r=\frac{T}{\gamma_{eff}} \exp(\Delta S/2)$ and $\alpha r=\frac{T}{\gamma_{eff}} \exp(-\Delta S/2)$, where the discretization unit is 1 here. Then, we can identify the effective friction coefficient $$\gamma_{eff}=\frac{T}{r\sqrt{\alpha}}.
\label{eq:gamma-2}$$
Using the estimator Eq. (\[eq:eff-HS\]) and effective friction coefficient defined in Eq. (\[eq:gamma-1\]), we obtain the approximate dissipation rate through $m$ dynamics under various $\alpha$, and compare it with the exact dissipation rate $J_m$, as shown in FIG. \[fig:adaptation\](b). It quantitatively captures the dependence of the dissipation rate on $\alpha$ in the range $\alpha\in [e^{-1} ,\;\;e^1 ]$, where $|\Delta S|=|\ln \alpha |<1$. Around $\ln \alpha\approx 0$, where the two dash lines intersect, the HS equality becomes almost exact. Alternatively, we may use the effective friction coefficient definition in Eq. (\[eq:gamma-2\]), as shown in FIG. \[fig:adaptation\](c), which is almost exact in the range $\alpha\in [e^{-1} ,\;\;e^1 ]$. Therefore, both of these two definitions are correct in the limit of small entropy production per step, i.e., $|\Delta S|=|\ln \alpha | \ll1$.
To understand why the HS estimate fails for large $\Delta S$, below we identify the higher order correction of this estimation. The HS equality is essentially to use the kinetic information to deduce the energetic information, or irreversibility of the dynamics. From the kinetic side, the biased transition rate $(w_+-w_-)$, i.e., the difference between forward and backward rate, provides the hope to extract the dissipation of this single jump, i.e., $\ln [w_+/w_-]$. This is essentially reduced to the following approximation $$\begin{aligned}
w_+-w_-&=&\frac{T}{\gamma_{eff}} \exp(\Delta S/2) -\frac{T}{\gamma_{eff}} \exp(-\Delta S/2) \nonumber\\
&=&\frac{T}{\gamma_{eff}} \Big[ \Delta S+ \frac{1}{24}(\Delta S)^3 +o(\Delta S^3)\Big]\nonumber\\
&=&\frac{T}{\gamma_{eff}} \Big[\ln [w_+/w_-]+\frac{1}{24}(\Delta S)^3 +o(\Delta S^3)\Big]\end{aligned}$$ Therefore, the HS equality is a leading order approximation, which becomes exact for Langevin dynamics. Since the correction is in the third order of $\Delta S$, the HS equality can still be reasonably good for many discrete Markov systems, as we have demonstrated in this sensory adaptation network. We will discuss this more deeply in our coming paper.
Derivation of Equation (10) in the Main Text
--------------------------------------------
Now, we study a special type of observable $Q_{p_k}=Q_p$ that only depends on the coarse-grained state, and therefore evolves on a slow timescale $\tau_s$. We ask how its violation spectrum may reveal the information of the microscopic dynamics that is hidden from our observation.
Intuitively, its dynamics seems to be equally well described in both the complete state space by matrix $M$ and the coarse-grained state space by ${\widehat{M}}$. The projection coefficients for the original system is $\alpha_j\equiv \sum_{p_k} Q_{p}x_j(p_k)$, $\beta_j\equiv \sum_{p_k} Q_{p} y_j(p_k)P^s_{p_k}$, and $\phi_j\equiv \sum_{p_k} B_{p_k} y_j(p_k)$, where $B_{p_k}\equiv \sum_{q_l} [w_{q_l}^{p_k}P^s_{q_l}+w_{p_k}^{q_l}P^s_{p_k}](Q_p-Q_q)/2T$. On the other hand, in the coarse-grained system, we may also introduce the projection coefficients ${\widehat{\alpha}}_j\equiv\sum_p Q_p {\widehat{x}}_j(p)$, ${\widehat{\beta}}_j\equiv \sum_p Q_p {\widehat{y}}_j(p) {\widehat{P}}^s_p$, and ${\widehat{\phi}}_j\equiv \sum_p {\widehat{B}}_p {\widehat{y}}_j(p) $, where ${\widehat{B}}_p\equiv \sum_q [{\widehat{w}}_q^p{\widehat{P}}^s_q+{\widehat{w}}_p^q{\widehat{P}}^s_p](Q_p-Q_q)/2T$. Here, ${\widehat{w}}_p^q$ is the transition rate of the coarse-grained dynamics, which is given by ${\widehat{w}}_p^q\equiv \sum_{k,l}w_{p_k}^{q_l}P^s(k|p)$. Note that $p\neq q$ here. Besides, we define $\lambda_j$ as $\sum_mM_{nm}x_m=-\lambda_jx_n$ and $\sum_my_m M_{mn}=-\lambda_jy_n$, which affects the sign below.
To connect the two levels of description, we use the following relations concerning eigenmodes $(j\le K)$ at the two levels,
\[eq:eigen-xy\] $$\begin{aligned}
x_j(p_k)&=&\widehat{x}_j(p)P^{s}(k|p)+O(\epsilon),
\label{eq:eigen-x}\\
y_j(p_k)&=&\widehat{y}_j(p)+O(\epsilon),
\label{eq:eigen-y}\end{aligned}$$
with $\lambda_j=\widehat{\lambda}_j+O(\epsilon)$. A special but important case of Eq. (\[eq:eigen-xy\]) is $$P^s_{p_k}={\widehat{P}}^s_pP^s(k|p)+O(\epsilon).$$ Applying these relations to the projection coefficients, we obtain the following results for the slow modes $(j\le K)$ $$\alpha_j=\widehat{\alpha}_j+O(\epsilon),\quad \beta_j=\widehat{\beta}_j+O(\epsilon),\quad \phi_j=\widehat{\phi}_j+O(\epsilon).
\label{eq:alpha-relation}$$ The sum rule of the effective system ${\widehat{M}}$ demands that $$\sum_{j=2}^K {\widehat{\alpha}}_j({\widehat{\beta}}_j{\widehat{\lambda}}_j-T{\widehat{\phi}}_j)=0.
\label{eq:sum-rule-effective}$$
For fast modes $(j>K)$, perturbative analysis shows that
$$\begin{aligned}
x_j(p_k)&=&\delta_{p,q}\bar{x}_j^q(k)+ O(\epsilon),\\
y_j(p_k)&=&\delta_{p,q}\bar{y}_j^q(k)+ O(\epsilon),\end{aligned}$$
with $\lambda_j=\epsilon^{-1} \lambda^q+O(1)$. Here, $\bar{x}_j^q(k)$ and $\bar{y}_j^q(k)$ are non-stationary eigenmodes of the matrix $\epsilon^{-1}M^q$ describing transitions within the coarse-grained state $q$, with $\epsilon^{-1}\lambda^q$ being the corresponding eigenvalue. These non-stationary eigenmodes $(\lambda^q\neq 0)$ satisfy $$\sum_k \bar{x}_j^q(k)=0,\quad \sum_k \bar{y}_j^q(k)P^s(k|q)=0,$$ which results from the orthogonal relations with the stationary eigenmodes $(\lambda^q=0)$. With these relations, we can prove that the fast modes $(j>K)$ satisfy $$\alpha_j=O(\epsilon),\quad \beta_j=O(\epsilon),\quad \phi_j=O(1).
\label{eq:alpha-relation-2}$$
With the above preparation, we now derive Eq.(10) in the Main Text. First, we note that there is a big gap between eigenvalues of the slow modes, which are of order $\tau_s^{-1}\sim 1$, and those of the fast modes, which are of order $\tau_f^{-1}\sim \epsilon^{-1}$. Therefore, in the frequency region $\omega\ll \tau_f^{-1}$, only the slow modes contribute to correlation and response spectrum according to Eq.(6) in the Main Text, i.e.,
\[eq:CR-fre\] $$\begin{aligned}
\tilde{C}_{\dot{Q}}(\omega)&=&\sum_{j=2}^K 2\alpha_j\beta_j\lambda_j \Big[1-\frac{1}{1+(\omega/\lambda_j)^2}\Big]+\epsilon^3 O([\omega \tau_s]^2), \\
\label{eq:Cvelo_fre}
\tilde{R}_{\dot{Q}}(\omega) &=&\sum_{j=2}^K \alpha_j\phi_j\Big[1- \frac{1-i(\omega/\lambda_j) }{1+(\omega/\lambda_j)^2}\Big]+\epsilon^3 O([\omega \tau_s]^2),
\label{eq:Rvelo_fre}
\end{aligned}$$
where the correction comes from the fast modes. According to Eq. (\[eq:alpha-relation-2\]), this correction is of order $ \epsilon (\omega \tau_f)^2$, or equivalently, $\epsilon^3 (\omega\tau_s)^2$. Then, the violation spectrum for $\omega\ll \tau_f^{-1}$ can be written as $$\tilde{C}_{\dot{Q}}(\omega)-2T\tilde{R}_{\dot{Q}}'(\omega)=2\sum_{j=2}^K \alpha_j\frac{T\phi_j-\beta_j\lambda_j}{1+(\omega/\lambda_j)^2}+2\sum_{j=2}^K \alpha_j(\beta_j\lambda_j-T\phi_j)+\epsilon^3 O([\omega \tau_s]^2).
\label{eq:violation-1}$$ Second, we apply the relations Eq. (\[eq:alpha-relation\]) to this violation spectrum, and obtain for $\omega\ll \tau_f^{-1}$ $$\tilde{C}_{\dot{Q}}(\omega)-2T\tilde{R}_{\dot{Q}}'(\omega)=2\sum_{j=2}^K {\widehat{\alpha}}_j\frac{T{\widehat{\phi}}_j-{\widehat{\beta}}_j{\widehat{\lambda}}_j}{1+(\omega/{\widehat{\lambda}}_j)^2}+\epsilon V_s(\omega),
\label{eq:violation-2}$$ which is exactly Eq.(10) in the Main Text, with $\epsilon V_s(\omega)$ a residual contribution defined by Eq. (\[eq:violation-2\]). $V_s(\omega)$ vanishes for $\omega\gg \tau_f^{-1}$ due to FRR in the high frequency limit. For $\omega\ll \tau_f^{-1}$, we have $$V_s(\omega)=\frac{2}{\epsilon} \sum_{j=2}^K \alpha_j(\beta_j\lambda_j-T\phi_j)+\frac{2}{\epsilon}\sum_{j=2}^K \alpha_j\frac{T\phi_j-\beta_j\lambda_j}{1+(\omega/\lambda_j)^2}-\frac{2}{\epsilon}\sum_{j=2}^K {\widehat{\alpha}}_j\frac{T{\widehat{\phi}}_j-{\widehat{\beta}}_j{\widehat{\lambda}}_j}{1+(\omega/{\widehat{\lambda}}_j)^2}+\epsilon^2 O([\omega \tau_s]^2).
\label{eq:Vs}$$ The diverging contribution from the first term vanishes due to the approximate relations Eq. (\[eq:alpha-relation\]) and the sum rule Eq. (\[eq:sum-rule-effective\]), and the other two diverging contributions from the second and the third term cancel each other due to Eq. (\[eq:alpha-relation\]). Therefore, $V_s(\omega)$ is a well-defined frequency-dependent function in the timescale separation limit $\epsilon\to 0$. Below, we analyze its frequency dependence.
In the intermediate frequency region $\tau_s^{-1}\ll\omega\ll \tau_f^{-1}$, both the second and the third term in Eq. (\[eq:Vs\]) vanish, and we obtain $$V_s(\omega) = \frac{2}{\epsilon}\sum_{j=2}^K \alpha_j(\beta_j\lambda_j-T\phi_j)\xrightarrow{\epsilon \to 0} const$$ due to Eq. (\[eq:alpha-relation\]) and the sum rule Eq. (\[eq:sum-rule-effective\]). For $\omega\ll \lambda_2\sim \tau_s^{-1}$, the third term in Eq. (\[eq:Vs\]) becomes $2\epsilon^{-1} \sum_{j=2}^K{\widehat{\alpha}}_j(T{\widehat{\phi}}_j-{\widehat{\beta}}_j{\widehat{\lambda}}_j)$, which vanishes due to the sum rule Eq. (\[eq:sum-rule-effective\]). Besides, the second term becomes $2\epsilon^{-1} \sum_{j=2}^K\alpha_j(T\phi_j-\beta_j\lambda_j)$ in this frequency region, which cancels the first term. Therefore, $V_s(\omega)=0$ for $\omega\ll \tau_s^{-1}$. To conclude, $V_s(\omega)$ vanishes both in the high ($\omega\gg\tau_f^{-1}$) and low ($\omega\ll \tau_s^{-1}$) frequency region, and it becomes a plateau in the intermediate frequency region.
[32]{} K. Berg-S[ø]{}rensen and H. Flyvbjerg, Rev. Sci. Instrum. [**75**]{}, 594–612 (2004)
E. Lippiello, M. Baiesi and A. Sarracino, Phys. Rev. Lett. [**112**]{}, 140602 (2014)
Y. Tu, Annu. Rev. Biophys. [**42**]{}, 337 (2013)
S.-W. Wang, Y. Lan, and L.-H. Tang, J. Stat. Mech. Theor. Exp. [**2015**]{}, P07025 (2015).
|
---
abstract: 'Using the continuous decomposition, we classify strongly free actions of discrete amenable groups on strongly amenable subfactors of type III$_\lambda$, $0<\lambda<1$. Winsløw’s fundamental homomorphism is a complete invariant. This removes the extra assumptions in the classification theorems of Loi and Winsløw and gives a complete classification up to cocycle conjugacy.'
author:
- |
Toshihiko Masuda[^1]\
Department of Mathematical Sciences,\
University of Tokyo, Komaba, Tokyo, 153. JAPAN
title: |
Classification of actions of discrete\
amenable groups on\
strongly amenable subfactors of type III$_\lambda$
---
amssym.def \[section\] \[th\][Definition]{} \[th\][Proposition]{} \[th\][Lemma]{} 226F 226E
Introduction
============
In the theory of operator algebras, the study of automorphisms is one of the most important topics. Especially since Connes’s work [@C2], much progress has been made on the classification of the actions of discrete amenable groups on injective factors.
In the subfactor theory, various studies of automorphisms have been done. In [@P2], Popa has introduced the notion of proper outerness of automorphisms and proved that the strongly outer actions of discrete amenable groups on strongly amenable subfactors of type II$_1$ are classified by the Loi invariant (See [@L1]). (In [@CK], Choda and Kosaki have introduced the same property independently and they call it strong outerness.)
In the case of subfactors of type III$_\lambda$, $(0<\lambda<1)$, partial results on classification of group actions have been obtained by Winsløw and Loi. ([@L2], [@W1], [@W2].) In [@W1] and [@W2], Winsløw has introduced the strong freeness and the fundamental homomorphism for actions. He has classified the strongly free actions of discrete amenable groups on subfactors of type III$_\lambda$ for groups having the character lifting property. His fundamental homomorphism is a complete invariant. In [@L2], Loi gave a classification theorem when $G$ is finite. Their idea of the proof is that they reduce the classification problem to the type II$_\infty$ case using the discrete decomposition and apply Popa’s classification result. The most difficult points of their proofs are to reduce the problem to the type II$_\infty$ case. Because of this difficulty, they made extra assumptions such as the character lifting property or finiteness for groups. But it seems difficult to generalize their method to the arbitrary discrete amenable group case.
Our idea of a proof is using the continuous decomposition instead of the discrete decomposition based on the method in [@ST1], [@ST2]. But in this case, we treat only factors of type III$_\lambda$, $0<\lambda<1$, with the trivial characteristic invariants, so the proof is less complicated than those in [@ST1] and [@ST2]. By using the continuous decomposition, we can more easily reduce the classification probelm to the type II$_\infty$ case than using the discrete decomposition and this method is valid for arbitrary discrete amenable groups.\
\
[**Acknowledgement.**]{} The author is grateful to Prof. M. Izumi for proposing this problem to him and helpful suggestions, Prof. Y. Kawahigashi for fruitful comments and constant encouragement.
Preliminaries
=============
In this section, we recall several results about group actions on subfactors, and fix notations. The facts stated in this section are found in [@CK], [@L1], [@L2], [@P2], [@W1], [@W2], [@W3].
Let $N\subset M$ be an inclusion of factors with finite index and $N\subset M \subset M_1 \subset M_2 \subset \cdots $ the Jones tower. (Throughout this paper, we always assume that conditional expectations are minimal in the sense of [@H] and inclusions of factors of type II are extremal. ) For $\alpha \in {\rm Aut}(M,N)$, we extend $\alpha$ to $M_k$ such that $\alpha(e_k)=e_k$ inductively, where $e_k$ denotes the Jones projection for $M_{k-1}\subset M_k$.
First we recall the Loi invariant and the strong outerness of group actions.
With above notations, Put $$\Phi(\alpha):=\{\alpha|_{M'\cap M_k}\}_k.$$ We call $\Phi$ the Loi invariant for $\alpha$.
An automorphism\
$\alpha\in{\rm Aut}(M,N)$ is said to be properly outer or strongly outer if we have no non-zero $a\in\bigcup_k M_k$ satisfying $\alpha(x)a=ax$ for all $x\in M$. The action $\alpha$ of $G$ on $N\subset M$ is said to be strongly outer if $\alpha_g$ is strongly outer except for $g=e$.
The most important result on classification of actions of groups on subfactors has been obtained by Popa.
Let $N\subset M$ be a strongly amenable inclusion of factor of type $II_1$ and $G$ a countable discrete amenable group.
If $\alpha$ and $\beta$ are strongly outer actions of $G$ on $N\subset M$, then $\alpha$ and $\beta$ are cocycle conjugate if and only if $\Phi(\alpha)=\Phi(\beta)$.
For type II$_\infty$ inclusions, we have the following result due to Popa and Winsløw.
\[th:infty\] Let $N\subset M$ a strongly ame- nable inclusion of factors of type II$_\infty$. If $\alpha$ and $\beta$ are actions of countable discrete amenable group $G$ on $N \subset M$, then $\alpha$ and $\beta$ are cocycle conjugate if and only if $\Phi(\alpha)=\Phi(\beta)$ and ${{{\rm mod\,}}}(\alpha)={{{\rm mod\,}}}(\beta)$.
Let $N\subset M$ be an arbitrary inclusion of factors with the common flow of weights. Fix a normal state of $N$ and take a crossed product of $N\subset M$ by the modular automorphism. Put ${\tilde N} \subset {\tilde M}:=N\rtimes_{\sigma^\phi}{\bf R} \subset
M\rtimes_{\sigma^{\phi\circ E}}{\bf R}$, where $E$ is the minimal conditional expectation from $M$ onto $N$. Let ${\tilde \alpha}$ be the canonical extension of $\alpha$ to ${\tilde N}\subset{\tilde M}$ ([@CT], [@HS]), i.e., $${\tilde \alpha}(x):=\alpha(x), \quad x\in M,$$ $${\tilde \alpha}(\lambda(t)):=(D\phi\,\alpha^{-1}:D\phi)_t\,\lambda(t),$$ where $\lambda(t)$ is the usual implementing unitary. The notions of strong freeness for automorphisms and the fundamental homomorphism are introduced by Winsløw in [@W1], [@W2].
An automorphism\
$\alpha\in{\rm Aut}(M,N)$ is said to be strongly free if we have no non-zero $a\in\bigcup_k {\tilde M_k}$ satisfying ${\tilde \alpha}(x)a=ax$ for all $x\in {\tilde M}$. For an action $\alpha$ of G on $N\subset M$ is said to be strongly free if $\alpha_g$ is strongly free except for $g=e$.
According to [@W1] and [@W2], we set $$\Upsilon(\alpha):=\{{\tilde \alpha}|_{{\tilde M}'\cap{\tilde M_k}}\}_k$$ and we call this the fundamental homomorphism.
Classification of actions
=========================
Throughout this section, we assume that inclusions of factors of type III$_\lambda$ are strongly amenable in the sence of Popa. (See [@P1] and [@P2].)
The following theorem is the main result of this paper.
\[main\] Let $N\subset M$ be a strongly amenable inclusion of factors of type III$_\lambda$, $0<\lambda<1$, with the common flow of weights. Let $G$ be a countable discrete amenable group, and $\alpha$ and $\beta$ strongly free actions of $G$ on $N\subset M$. Then $\alpha$ and $\beta$ are cocycle conjugate if and only if ${\rm \Upsilon}(\alpha)={\rm \Upsilon}(\beta)$.
Our idea of proof is that we lift actions to inclusions of type II$_\infty$ von Neumann algebras using continuous decomposition and apply Popa’s result.
The “only if” part is obvious, so we only prove the “if” part. Let $(X, F_t)$ be the flow of weights of $M$. Since $M$ is of type III$_\lambda$, $(X, F_t)$ is of the form $([0,-\log\lambda), {\rm translation})$. And we have an isomorphism $$({{{\tilde N}}}\subset {{{\tilde M}}}\subset {{{\tilde M}}}_1 \subset \cdots )\cong
( L^\infty(X)\otimes Q \subset L^\infty(X)\otimes P \subset
L^\infty(X)\otimes P_1 \subset \cdots) ,$$ where $Q\subset P\subset P_1 \subset \cdots $ is a tower of factors of type II$_\infty$ and $Q\subset P$ is strongly amenable by assumption.
Let $\theta_t$ be the usual trace scaling action of [**R**]{} on ${{{\tilde N}}}\subset{{{\tilde M}}}$. Since ${\tilde \alpha}_g, g\in G$, commutes with $\theta_t$, we can consider an action of $G\times {\bf R}$ by setting $(g,t)\to {\tilde \alpha}_g\theta_t$. If no confusion arises, we also denote this action of $G\times {\bf R}$ by ${{\tilde \alpha}}$. If we prove that two actions of $G\times {\bf R}$, ${{\tilde \alpha}}$ and ${{\tilde \beta}}$ are cocycle conjugate, the proof of [@ST2 Proposition 1.1] also works in this case and we can deduce that the canonical extensions of ${{\tilde \alpha}}$ and ${{\tilde \beta}}$ on $N\rtimes_{\sigma^\phi}{\bf R}\rtimes_\theta{\bf R} \subset
M\rtimes_{\sigma^{\phi\circ E}}{\bf R}\rtimes_\theta{\bf R}
\cong N\otimes B(L^2({\bf R}))\subset M\otimes B(L^2({\bf R}))$ are also cocycle conjugate and get the conclusion that $\alpha$ and $\beta$ are cocycle conjugate, since $N$ and $M$ are properly infinite.
So our purpose is the classification of actions of $G\times {\bf R}$ on ${{{\tilde N}}}\subset {{{\tilde M}}}$. Note that for $g\in G$, the equality ${{\rm tr}}_{{{\tilde M}}}\,{{\tilde \alpha}}_g={{\rm tr}}_{{{\tilde M}}}$ holds.
Put $H:=G\times {\bf R}$ and we consider the action ${{\tilde \alpha}}$ of the groupoid $H\ltimes X$ on $Q\subset P$ by the equality $${{\tilde \alpha}}_g(a):=\int^\oplus_X\,{{\tilde \alpha}}_{(g,g^{-1}x)}(a(g^{-1}x))\,dx.
\quad\mbox{(See \cite[Proposition 1.2]{ST2}.)}$$
Put $x_0:=0\in X$ and $H_0:=\{(g,x_0)\in H\ltimes X\, |\, gx_0=x_0\}$. Then $H_0$ is a discrete amenable group acting on $Q\subset P$. For $x\in X=[0,-\log\lambda)$, we define $h(x): X\to X$ by $h(x)y:= y+x$, where sum is taken modulo $-\log\lambda$. Especially $h(x)x_0=x$.
Here we have the following proposition.
\[prop:outer\] If the action of $G$ is strongly free, then the action of $H_0$ is strongly outer.
[**Proof.**]{} Assume that the action of $H_0$ is not strongly outer. Then there exists $g\in H\backslash \{e\}$ and a nonzero $a\in P_k$ for some $k$ such that for every $b\in Q$, we have ${{\tilde \alpha}}_g(b)a=ab$. Since ${{\tilde \alpha}}_g$ is not strongly outer and ${{{\rm mod\,}}}{{\tilde \alpha}}_g=1$, we know that $g$ is in $G$.
Set $${\tilde a}:=\int^\oplus_X {{\tilde \alpha}}_{(h(x), x_0)}(a) dx.$$ Then an easy computation shows that the equality ${{\tilde \alpha}}_g(b){\tilde a}={\tilde a}b$ holds for every $b\in Q$ and this means that action $\alpha$ is not strongly free. $\Box$\
[**Proof of Theorem \[main\]**]{} Let $\alpha$ and $\beta$ be strongly free actions of $G$ on $N\subset M$ such that $\Upsilon(\alpha)=\Upsilon(\beta)$. Then we get two actions ${{\tilde \alpha}}$ and ${{\tilde \beta}}$ of the same groupoid $H\ltimes X$. So we get two actions ${{\tilde \alpha}}$ and ${{\tilde \beta}}$ of a discrete amenable group $H_0$.
By Proposition \[prop:outer\], both actions are strongly outer and by assumption both actions have the same Loi invariant and the same module. So there exists an automorphism $\theta \in {\rm Aut}(P,Q)$ and $u_g\in Z_{{{\tilde \beta}}}(H_0, U(Q))$ such that $$\theta\,{{\tilde \alpha}}_g\,\theta^{-1}={\rm Ad}u_g\,{{\tilde \beta}}_g, \quad g\in H_0.$$
Set $$\theta_x:={{\tilde \beta}}_{(h(x),x_0)}\,\theta\,{{\tilde \alpha}}^{-1}_{(h(x),x_0)},
\quad x\in X\quad \mbox{and}$$
$$u_{(g,x)}:={{\tilde \beta}}_{(h(gx)^{-1}gh(x),x_0)}(u_{h(gx)^{-1}gh(x)}),
\quad (g,x)\in H\ltimes X.$$
Then an easy computation shows that $u_{(g,x)}\in Z_{{\tilde \beta}}(H\ltimes X, U(Q))$ and the equality $$\theta_{gx}\,{{\tilde \alpha}}_{g,x}\,\theta^{-1}_x
={\rm Ad}\,u_{(g,x)}\,{{\tilde \beta}}_{(g,x)}$$ holds.
Put ${\tilde \theta}:=\int_X^{\oplus}\theta_x\,dx$ and ${\tilde u}_g:=\int_X^{\oplus}u_{(g,x)}dx.$ Then we get $${\tilde \theta}\,{{\tilde \alpha}}_g\,{\tilde \theta}^{-1}
={\rm Ad}\,{\tilde u}_g\,{{\tilde \beta}}_g$$ and we get the conclusion. $\Box$
[99]{} Choda, M., and Kosaki, H., J. Func. Anal. [**122**]{}, 315–332, (1994). Connes, A., Ann. Sci. Ec. Norm. Sup. [**6**]{}, 133–252, (1973). Connes, A., Ann. Sci. Ec. Norm. Sup. [**8**]{}, 383–420, (1975). Connes, A., Ann. Math. [**104**]{}, 73–115, (1976). Connes, A., and Takesaki, M., Tohoku Math. J. [**29**]{}, 473–575, (1977). Jones, V. F. R., Invent. Math. [**72**]{}, 1–25, (1983) Haagerup, U. and Størmer, E., Adv. Math. [**83**]{}, 180–262, (1990). Hiai, F., Publ. RIMS. [**24**]{}, 673–678, (1990). Loi, P. H., J. Func. Anal. [**141**]{}, 275–293, (1996). Loi, P. H., to appear in Internat. J. Math. Popa, S., Acta. Math. [**172**]{}, 352–445, (1994). Popa, S., preprint (1992). Sutherland, C. E. and Takesaki, M., Publ. RIMS. [**21**]{}, 1087–1120, (1985). Sutherland, C. E. and Takesaki, M., Pac. J. Math. [**137**]{}, 405–444, (1989). Takesaki, M., Acta. Math. [**131**]{}, 249–310, (1973). Winsløw, C., Internat. J. Math. [**4**]{}, 675–688, (1993). Winsløw, C., Pac. J. Math. [**166**]{}, 385–400, (1994). Winsløw, C., Subfactors —Proceeding of the Taniguchi Symposium, Katata —, (ed, H. Araki, et al.), World Scientific, 139–152, (1994).
[^1]: 1991 [*Mathematics Subject Classification*]{} 46L37
|
---
abstract: 'Software-defined networking offers numerous benefits against the legacy networking systems through simplifying the process of network management and reducing the cost of network configuration. Currently, the management of failures in the data plane is limited to two mechanisms: *proactive* and *reactive*. Such failure recovery techniques are activated after occurrences of failures. Therefore, packet loss is highly likely to occur as a result of service disruption and unavailability. This issue is not only related to the slow speed of recovery mechanisms, but also the delay caused by the failure detection process. In this paper, we define a new approach to the management of fault tolerance in software-defined networks where the goal is to eliminate the convergence process altogether, rather than speed up failure detection and recovery. We propose a new framework, called *Smart Routing*, which works based on the forewarning signs on failures in order to compute alternative paths and isolate the risky links from the routing tables of the data plane devices. We validate our framework through a set of experiments that demonstrate how the underlying model runs.'
author:
- 'Ali Malik, Benjamin Aziz, Mo Adda and Chih-Heng Ke [^1] [^2]'
title: 'Smart Routing: Towards Proactive Fault-Handling in Software-Defined Networks'
---
[Shell : Bare Demo of IEEEtran.cls for Journals]{}
Software-Defined Networking, OpenFlow, fault management, risk management, service availability.
Introduction {#sec:Intoduction}
============
HE concern about the Internet ossification, which is a consequence of the growing number of variety networks (e.g. Internet of Things, wireless sensor, Cloud, etc.) that serve a huge number of clients (currently estimated about 9 billion) around the globe, has led to rethink about the existing rigid network infrastructure whether it can be replaced by a programmable one [@Internet_Ossification]. In this context, Software-Defined Networking (SDN) has emerged as a promising solution to tackle the inflexibility of the legacy networking systems. Unlike traditional IP networks, SDN architectures consist of two layers: A *control plane* and a *data plane*. The control plane, or sometimes called the *controller*, represents the network brain and maintain a global view on the network. While, the data plane comprises network forwarding elements, i.e. switches and routers, that constitute the network topology. All the data plane elements are dictated by the network controller and therefore the entire nodes have to disclose their status periodically toward the controller, hence the global view comes. So far, OpenFlow [@OF] is the most widely used protocol that enables the controller to govern the SDN data plane through carrying the *forwarding rules* as well as to facilitate the exchanging of signals between the two planes. Nowadays, communication networks play a vital role in human being’s life activities as it represents the backbone for most of the current modern technologies. Since networking equipment are failure prone, some aspects like availability measurements, fault management and reliability become very important. This paper is mainly focused on the availability attribute in terms of fault tolerance and forecasting of failure in SDNs. Despite SDN benefits, new challenges such as recovery from failure still require investigation in order to maximise their utility [@SDN_Challenges2015; @Challenges2016]. This paper presents a complementary approach that minimises the percentage of service unavailability through utilising an online failure prediction mechanism. This allows the network controller to perform the necessary reconfiguration prior the failure incidents. Although a number of works on SDN fault management have been proposed, none of them has exploited the feature of SDN global view in the context of failure prediction purposes.
The rest of the paper is organised as follows. Section \[sec:Previous\_work\] provides an overview of literature related to various SDN fault management techniques. We define the problem statement in Section \[sec:Problem\_statement\] and the novelty of our work. We then present our model and framework in Section \[sec:Model\]. Section \[sec:Experimental\] and \[sec:Discussion\] present the experimental procedure, observed result and comparison. Finally, a summary of this paper is provided in Section \[sec:Conclusion\] with some future research directions.
Related Work {#sec:Previous_work}
============
Link failure issues often occur as part of everyday routine network operations. Due to their negative impact on network Quality of Service (QoS), a considerable amount of research has been conducted to analyse, characterise, evaluate and recover from the frequent issues of network link failures. Such failures can either be unintentional (i.e. *unplanned*) due to various causes like human error, natural disasters, overload, software bugs or cable cuts, or intentional (i.e. *planned*) caused by the process of maintenance [@Markopoulou2008]. Failure recovery is a necessary requirement for networking systems to ensure the reliability and service availability. Generally, failure recovery mechanisms of carrier-grade networks are categorized into two types: *protection* and *restoration*. In protection, which is also know as *proactive*, alternative solutions are pre-planned and reserved in advance (i.e. before a failure occurs). By contrast, in restoration, which is also called *reactive*, possible solutions are not pre-planned and will be calculated dynamically when failures occur. Both approaches have pros and cons. For example, the authors in [@Protection2012] implemented an OpenFlow monitoring function for achieving a fast data plane recovery. In [@Protection2013], another protection method was proposed through using the OpenFlow-based Segment Protection (OSP) scheme. The main disadvantage of these approaches is that they consume the data plane storing capability since the more flow entries (i.e rules) that need to be stored, the more storage space that needs to be used. Current OpenFlow appliances in the market are able to accommodate up to 8000 flow entries only, due to known limitations of the Ternary Content-Addressable Memory (TCAM), hence making this kind of solutions costly [@RoadMap2014; @ComprehensiveSurvey2015]. The installation of many attributes in the OpenFlow forwarding elements could lead to the deterioration of the process of match-and-action for the data plane nodes. Moreover, there is no guarantee that the preserved backups are failure-free; the backup path might fail before the primary one. Following the restoration approach, the authors in [@Sharma2011] and [@CarierGrade2011] presented OpenFlow restoration methods to recover from single link failures. Experiments were conducted on small scale network topologies that did not exceed 14 nodes. In [@Sharma2013], the authors demonstrated, through extensive experiments, that OpenFlow restoration is not easily attainable within a time of 50ms, especially for large-scale networks, unless using protection techniques. In the same context, some works have utilised the concept of multiple disjoint paths to be employed as a backup. For example, CORONET [@CORONET2012] is presented as a fault-tolerance system for SDNs, in which multiple link failures can be resolved. The ADaptive Multi-Path Computation Framework (ADMPCF) [@ADMPCF2015] and HiQoS [@HiQoS2015] for large scale OpenFlow networks were produced as traffic engineering tools that are capable of holding two or more disjoint paths to be utilised when some network events (e.g. link failure) occur. Most of the existing works do not take into account the processing time of flow entries, i.e. insert, delete and modify of rules. Although the performance of OpenFlow devices is associated with their vendors, in [@OFLOPS2012] the authors stated that each single flow entry insertion ranges from 0.5ms to 10ms. However, 11ms is the minimum duration required to modify a single rule, since each modification process includes both deletion (of old rules) and insertion (of new ones) [@Dionysus2014]. Unlike existing works, the authors in [@Heydari2016] considered the problem of minimising the time of flow entries required when diverting from an affected primary path to a backup one. Although, the presented algorithms do not guarantee the shortest path from end-to-end, nonetheless, they open a new direction that is worth exploring. Within the same context, the authors in [@Malik2017] produced new algorithms for minimising the required time to update rules through reducing the solution search space from the source to the destination in the affected path. Similarly, in [@Malik2017_cliques], an approach to divide the network topology into non-overlapping cliques has been introduced to tackle the issue of failures in a localised manner, rather than taking a global view of the network. Both [@Malik2017] and [@Malik2017_cliques] took into account the time required to compute the alternative route in order to speed up the update operation. The main issue with the last three works is that they do not guarantee a shortest path from source to destination.
In summary, the previous studies demonstrated different methods to tackle the problem of data plane recovery from link failure incidents. A more recent survey [@SDN_Survey2017] outlines in detail more contributions to the area of fault management in SDNs. One can conclude that protection approaches are not ideal due to the TCAM space exhaustion problem, whereas the latency issue is the major drawback of the existing restoration approaches. As a result, we believe that more research is needed in terms of achieving efficient SDN resilience, which is the main aim of this work.
Problem Statement and Contributions {#sec:Problem_statement}
===================================
Current SDN fault tolerance mechanisms inescapably lead to a certain amount of packet loss as well as to a certain probability of service unavailability. This is due to the delay of the convergence scheme $T_C$. We define $T_C$ as the time taken by the OpenFlow controller to amend a path in response to failure scenario. Typically, the convergence time in SDNs can be defined in terms of three factors: $\bullet$ *Failure detection time* ($T_D$): This is the required time to detect a failure incident. Compared with the conventional networking systems, the centralised management and global view of an SDN eases this task by continuously monitoring network status and obtaining notifications upon failure. However, the speed of receiving a notification is sometimes associated with the nature of network design and mode of communication (i.e. in-band or out-of-band) [@inoutband2015; @coping2010]. According to [@detectiontime2014], link failure detection time ranges from tens to hundreds of milliseconds, depending on the type of commercial OpenFlow switch being used. $\bullet$ *New route computation time* ($T_{SP}$): This is the spent time when network controller runs a nominated shortest path routing algorithm (e.g. Dijkstra [@Dijkstra1959]) to compute the backup path (usually for the reactive fault tolerance strategies). The $T_{SP}$ computation time could reach 10s of milliseconds [@Malik2017] according to how big the network is.$\bullet$ *Flow entries update time* ($T_{Update}$): This is the required time to update the relevant switches (i.e. nodes who are involved in the affected path). Again, this factor depends on how many forwarding rules need to be updated after the failure scenario, where the amount of time for a single rule may exceed 10ms. Accordingly, the resulting convergence time can be calculated through the following equation: $$\label{eq:convergence} %\footnotesize
T_C = T_D + T_{SP} + \sum_{src}^{dst} T_{Update}$$
Currently, the classical SDN fault management methods aim to tackle the failure after it occurrence, therefore, the recovery mechanism is activated after the moment of failure and hence all the previous work proposals embroiled in a certain amount of delay according to (\[eq:convergence\]). The only way to completely overcome the three factors of (\[eq:convergence\]) altogether is by handling the failure before it occurs. Therefore, failure prediction is required to provide awareness about the potential future incidents as well as allowing the controller to perform the reconfiguration action in purpose of overriding failures before causing damage on some paths. Although there are a number of studies that have put efforts in the area of failure prediction, none of these (except [@Prediction_OSPF2013]) has exploited the information that can be gained from any prediction method to eliminate network incidents (e.g. link failures). To the best of our knowledge, [@Prediction_OSPF2013] is the only realistic study that discussed the advantages of failure prediction through producing a risk-aware routing method for the legacy IP networks. Our work is different from theirs in that we build a framework of proactive failure management for SDNs. Our work combines the concept of the online failure prediction with risk analysis towards maximising the network service availability.
With this context in mind, we can summarise the main contributions of this paper as follows:$\bullet$ A new network model that allows for the forecasting of link failures by predicting their characteristics in an online fashion. This model also combines the predictive capability with the decision making process using risk analysis.
$\bullet$ We provide an implementation of the new model in terms of a couple of fault tolerance algorithms. We use simulation techniques to test the efficiency of these algorithms. Our simulation results prove that the proposed model and algorithms improve the service availability of SDNs.
The Proposed Model {#sec:Model}
==================
Anticipating failures before they occur is a promising approach for further enhancement of SDN failure management techniques, i.e. the proactive and reactive, in which the controller responds to failures when they take place. The SDN proposed model for anticipating link failure events is presented in this section. We start by outlining some notations that will be used throughout this paper, as shown in Table \[tab:terms\].
Symbol Description
-------------- -------------------------------------------------------
$src$ Source router
$dst$ Destination router
$A$ Service availability
$U$ Service unavailability
$e_{ij}$ Link traversing any two arbitrary routers $i$ and $j$
$Q_{ptr}$ A pointer that points to first $e_{ij}$ in the Queue
${F}$ Failed link set
${F_R}$ Failed/affected route set
${PF_L}$ Potential failed link set
${PF_R}$ Potential failed route set
${M}$ Prediction alarm message
${CO}$ Network controller
${T_\Omega}$ Threshold of failure probability
$T_\omega$ Threshold of risk
${OF}$ OpenFlow instruction
${TP}$ True positive
${FN}$ False negative
${FP}$ False positive
$CC$ Cable cut per year
${SP_x}$ Any shortest path algorithm x in terms of hops
: List of notations[]{data-label="tab:terms"}
The network topology is modelled as an *undirected graph* $G = (V, E)$; where $V$ represents the finite set of vertices (i.e. routers) in $G$ that ranges over by $\{v_i, v_j, \dots, v_z\}$ where $\{i,j, \dots, z\} \subset \{1, \dots, n\}$ for $n\in\mathbb{N}$ , and $E$ represents the finite set of bidirectional edges (i.e. links) in $G$ that denoted as $\{e_{ij}\}$ where each $e_{ij} \in E$ is an edge that enables $v_i$ and $v_j$ to connect each other. Now, we define the following test operational function ($OP$) over a link, which reflects the link state whether it’s working or not:
$$OP(e_{ij})=
\begin{cases}
1 \ \ \ \text{the link is operational}\\
0 \ \ \ \text{otherwise}\\
\end{cases}$$Therefore, ${F}$ can be defined as follows:
${F} = \{e_{ij}~ |~ e_{ij} \in E \wedge OP(e_{ij})=0\}$ Based on $G$, we define a path $P$ as a *sequence* of vertices representing routers in the network. Each path starts from a source router, $src$, and ends with a destination router, $dst$:
${P}= (src,\ldots,dst)$ We define the set $Flow$ to represent all demand traffic flows that need to be serviced. Each $flow \in Flow$ is an instance of $P$, which associates with a particular traffic that are defined by unique $src$ and $dst$ pair. We consider $flow_{set}$ to be the set of all the possible paths between $src$ and $dst$ that can be derived from $G$, which is defined as follows:
${flow_{set}=\{{P}~ |~ (\textit{first}({P})=src) \wedge (\textit{last}({P})=dst)\}}$ and the definition of *first* and *last* is given as functions on any general sequence $(a_1,\ldots,a_n)$:
$\textit{first}((a_1,\ldots,a_n))=a_1$, $\textit{last}((a_1,\ldots,a_n))=a_n$ We also consider ${P}_{set}$ as a set that contains all the admissible paths that can be constructed from $G$, so this means that $ P \in {P}_{set}$ and therefore, $Flow \subset {P}_{set}$. When a link failure is reported in $G$, then, we identify the affected routes as follow:
${F_R} = \{flow~ |~ flow \in Flow \wedge \exists _{{v_i, v_j}} . v_i, v_j \in flow \wedge OP(v_i, v_j)=0$} In the same context, but this time we consider the case of when there is a link failure prediction message $m_i \in {M}$ such that ${M}$ set denoted by $\{m_i\}_{i=1}^{n}$ where each $m_i \in M$ is defined as $m_i= (\bar{e}_{ij}, t)$, where $t$ is the time when the system receives $m_i$. In this context, we define the following:
${PF_L} = \{\bar{e}_{ij}~ |~ \bar{e}_{ij} \in E \wedge \exists_{m_i}. m_i=(\bar{e}_{ij}, t) \wedge m_i \in {M} $} to characterise the received link, which we use $\bar{e}_{ij}$ to imply that ${e}_{ij} \in {PF_L}$ is a shorthand, with state of *potential to fail* and hence it does not belong to $F$. Now, we can define the *potential to fail route* set as follows:
${PF_R} = \{\bar{flow}~ |~ \bar{flow} \in Flow \wedge (\exists_{\bar{e}_{ij}} . {\bar{e}_{ij}} \in \bar{flow}~ \wedge {\bar{e}_{ij}} \in {PF_L})$} where $\bar{flow}$ is a $flow$ that has at least one $\bar{e}_{ij}$, in other words, $\bar{flow} \cap {PF_L} \neq \emptyset $.
SDN Predictive Model
--------------------
All the previous efforts that dealt with data plane failures have succeeded in mitigating the impact of failures (e.g. reduce the downtime) rather than attempting to obviate their effect, such as the service unavailability. Network incidents that cause routing instability, i.e. flaps, and lead to significant degrading of network service availability vary [@Origin1999; @Joint_analysis2010], however, we are merely concerned with the type of data link failure. By relying on monitoring techniques, some failures can be predicted through failure tracking, syndrome monitoring, and error reporting [@Salfner2010]. Consequently, a set of conditions can be defined as a base to trigger a failure warning when at least one of the predefined conditions is satisfied, as follows: $\textit{if} \ \Big < \textit{condition} \Big > \ \textit{then} \ \Big < \textit{warning} \ \textit{trigger} \Big >$
Online failure prediction strategies vary such as machine learning techniques (e.g. using the $\kappa$-nearest neighbor algorithm [@Prediction_machineLearning]) and statistical analysis methods (e.g. time series [@Salfner2010], Kalman and Wiener filter [@kalman_wienerbook]). Such techniques can be used to predict the incoming events through relying on the past and current state information of a system. However, in this paper, we do not intend to propose a failure prediction solution as extensive studies have been conducted in this field with remarkable achievements. Instead, employing the online failure prediction as a technique to enrich the current SDN fault management is one of the main aims of this work. A generic overview of the time relations of online failure prediction is presented in Figure \[fig:prediction\].
![Online failure prediction and time relations [@Salfner2010].[]{data-label="fig:prediction"}](Prediction)
- $\Delta {t_d}$: represents the past (historical) data upon which the predictor is forecasting the upcoming failure events.
- $\Delta {t_l}$: represents the lead time upon which a failure alarm is generated. It can also be defined as the minimum duration between the prediction and failure.
- $\Delta {t_w}$: represents the warning time in which an action may be required to find a new solution based on the predicted event. Therefore, $\Delta {t_l}$ must be greater than $\Delta {t_w}$ so that the information from prediction will be serviceable. In SDN, the $\Delta {t_w}$ should be at least adequate to the time required to set up the longest shortest path in given $G$.
- $\Delta {t_p}$: represents the time for which the prediction will be assumed to be a valid case. This should be defined carefully by the network operator so as to identify the true and false alarms after a certain time window.
The quality of the failure prediction is usually evaluated by two parameters: [${FP}$]{} and [${FN}$]{}; whereas, *Recall* and *Precision* are the two well-known metrics that are used to measure the overall performance.
$$%\footnotesize
Recall = \frac{{TP}}{{TP} + {FN}} \ , \
Precision = \frac{{TP}}{{TP} + {FP}}$$
Recall is defined as the ratio of the accurately captured failures to the total number of the certainly occurred failures. However, Precision is defined as the ratio of the correctly classified failures to the total number of the positive predictions. Correspondingly, SDN controller actions will now associate with predicted and unpredicted situations as listed in Table \[tab:Prediction\_actions\].
Prediction Action
------------ ------------------------------------
${TP}$ Select an alternative route
${FP}$ Unnecessary/needless action
${FN}$ Call the standard failure recovery
: Controller actions based on prediction[]{data-label="tab:Prediction_actions"}
On one hand, every false failure alarm will lead to an unnecessary reconfiguration for a particular set of routes in $Flow$ and this will cause unwitting network instability. On the other hand, a controller needs to deal with the undetected failures in a similar way to the classical methods. Consequently, the more precise behaviour of prediction, the higher the percentage of network stability and service availability will be gained. The relevance between the network model and the predictive model is summarised in Figure \[fig:4\].
Failure Event Model {#failure_model}
-------------------
We have implemented an approach of generating failure events as it is very difficult to find a public network dataset that includes some useful details like failures, hence, we adopted an alternative approach by developing our failure model. This work intends to enhance the SDN fault tolerance and resilience through maximising the network service availability. Two basic metrics have been exploited in this model: *Mean Time Between Failure* ($MTBF$) and *Mean Time To Recover* ($MTTR$); which are essential for calculating the availability and reliability of each network repairable component [@Reliability_modeling2017],[@Pan2003]. $MTBF$ is defined as the average time in which a particular component functions before failing, where it comes through: $\frac{\sum (start_{down\_time} - start_{up\_time})}{number \ of \ failures}$; while, $MTTR$ is the average time required to repair a failed component. Each component (i.e. link) is characterized by its own values of both $MTBF$ and $MTTR$, which are commonly independent from other components in the network. As a consequence of lacking real data, some metrics (such as cable length and $CC$) can be alternatively used for measuring the two availability metrics. According to [@Pan2003], $MTBF$ can be calculated as follows:
$$\label{eq:MTBF}
%\footnotesize
MTBF (hours) = \frac{CC \times 365 \times 24}{Cable \ Length}$$
For instance, when $CC$ is equal to 100 km, it means that per 100 km there will be on average one cut per year. Besides this, the $MTTR$ of a link is influenced by its length [@Gonzalez2012], which expresses the fact that the longer link has a higher $MTTR$ value. On this basis, we have designed the following formula for calculating the $MTTR$ value for each link in the network.
$$\label{eq:MTTR}
%\footnotesize
MTTR (hours) = \gamma \times Cable Length
%MTTR (hours) = \hat{Y} .{\frac {Cable_{len}}{All Cables_{len}}}
%MTTR(hours) = {\frac{\sum_{i=1}^{n} TR_i}{n}}$$
Where $\gamma$ is defined as a parameter indicating the time required to fix the cable, which is measured by hour/kilometer format. Due to the fact that links are physically distributed in different locations and environments, therefore, $\gamma$ differs from one link to another. In other words, even if some links have the same length, their $\gamma$ could be different as it relies on the physical location and the ambient conditions. We will discuss the use of these two values in Section \[sec:Framework\].
Risk analysis {#Risk_analysis}
=============
According to [@kaplanRisk1981], risk can be defined in terms of the following three questions: What scenario could occur? what is the likelihood that scenario would occur? and what is the consequence if the scenario does occur? We next consider these questions towards formulating failure risk in SDNs.
*What scenario could occur?* We define the *scenario* as any undesirable event, such as failure, that breaks the service down and therefore requires a solution (e.g. path change). According to [@FailureScenarios2014], there are three main types of failure scenarios, namely controller failure (including hardware and software), communication components failure (i.e. node and link) and application failure (e.g. bugs in application code), that could affect the SDN networking system. We define the set of all scenarios as $S$ ranged over by variables $s_1, s_2, \dots, s_n \in S$.
*What is the likelihood a scenario would occur?* The likelihood that a failure scenario disrupts the network services is conditional on the occurrence of the scenario. We address this question by the aid of online failure prediction that in our case works based on a scenario’s failure probability, $p\in[0,1]$. *What is the consequence if the scenario does occur?* We address this question by computing the percentage of loss or consequence, $c$, that might potentially happen when a failure scenario is predicted at an early stage. Each failure scenario might lead to some disconnections and service disruption. Therefore, the severity of adverse effects of each failure scenario varies. For instance, $c_1$ that was caused by $s_1$ might be different from $c_2$ that was caused by $s_2$, which would reflect the outage costs that would result from disrupting some of the network connections.
Over a period of time, these questions would make a list of outcomes in the form of a triplet $\langle s_i, \ p_i, \ c_i \rangle$. Utilising such information, $risk$ can then be formulated as a set of triples:$$%\footnotesize
Risk = \{\langle s_i,\ p_i,\ c_i \rangle\}, \ \ \ i= 1, 2, \dots, n
\label{eq:generalRisk}$$
Failure scenarios may have many causes and different origins. However, in this paper we focus only on one type, i.e. link failure scenarios that hit the data plane. Therefore, because we are considering the only link failure scenarios, $s_{(e_{ij})}$, we shall refine the definition of risk in (\[eq:generalRisk\]). Accordingly, we redefine risk of damage to be the combination of the probability of link failure and its consequence. $$\label{eq:risk}
%\footnotesize
Risk_{s_{(e_{ij})}} = p_{(e_{ij})} \times c_{(e_{ij})}$$ To deduce the risk value, the two factors of (\[eq:risk\]), i.e. $p$ and $c$, can be assessed independently. On one hand, the probability, $p$, depends on the efficacy of the online failure predictor at determining the likelihood of the incoming failure scenarios, which is, in this study, defined by a selective failure probability threshold value, ${T_\Omega}$. On the other hand $c$ can be measured based upon the percentage of affected routes that would result from the anticipated scenario. By utilising some global network topological characteristics, such as *Edge Betweenness Centrality (EBC)*, the consequence score can be identified. The edge betweenness centrality of a link $e_{ij}$ is the total number of shortest paths between pairs of nodes that traverse the edge $e_{ij}$ [@EdgeBetweenness], which can be formulated as follows: $$\label{eq:EBC}
%\footnotesize
EBC_{e_{ij}} = \sum_{v_i \in V} \sum_{v_j \in V}\frac{\Gamma_{vi,vj} e_{ij}}{\Gamma_{vi,vj}}
%Which is congruent to:$$ Where $\Gamma_{vi,vj}$ denotes the number of shortest paths between nodes $vi$ and $vj$, while, $\Gamma_{vi,vj}e_{ij}$ denotes the number of shortest paths between nodes $vi$ and $vj$ and go through $e_{ij} \in E$. For instance, Figure \[fig:ebc\] demonstrates an example topology with an EBC value for each link in the network, which has been calculated based on Ulrik Brandes algorithm [@Ulrik; @Brandes2008].
![Topology example with different EBC values\[fig:ebc\]](EBC_Example)
The network controller knows the demand traffic matrix between all pairs in the network, i.e. $Flow$. Therefore, equation (\[eq:EBC\]) in our case is congruent with the following: $$%\footnotesize
EBC_{e_{ij\in M}}= %\sum_{s,t \in Flow}
\frac{\Gamma_{flow} e_{ij}}{\Gamma_{flow}}$$ Where $\Gamma_{flow}$ denotes the total number of paths in $Flow$ set, while, $\Gamma_{flow} e_{ij}$ denotes the number of paths in $Flow$ set and pass through $e_{ij} \in M$. With the above context in mind, the higher the EBC value of $e_{ij}$, which is a normalised value between $0$ and $1$, the more critical the link is and therefore, the higher the score indicating the consequences. This is because the outcome of failure for a link with high EBC will definitely lead to a huge number of path failures and therefore a higher percentage of negative impacts on the availability of network services. Our goal in this analysis is to gauge the percentage of possible loss and provide such information to the concerned decision-making mechanism, i.e. the routing mechanism in our case. For more details about the existing risk analysis methods that fit SDNs, we refer the interested readers to [@SDN_Risk2018].
Framework design {#sec:Framework}
================
From a high level point of view, Figure \[fig:framework\] illustrates the main components of our proposed framework where the *Smart Routing* and *Prediction* modules are the primary contribution of our work. We discuss next in more detail the components we used to develop this framework.
\(a) *SDN Controller* Our framework currently supports the POX controller [@POX_Controller], which is an open source SDN controller written in python and it is more suitable for fast prototyping than other available controllers such as [@OpenFlow_Controllers2013]. The standard OpenFlow protocol is used for establishing the communication between the data and control planes, whereas the set of POX APIs can be used for developing various network control applications.
![Architecture of the proposed framework.[]{data-label="fig:framework"}](Framework.pdf)
\(b) *Smart Routing* Firstly, this module is responsible for maintaining and parsing the underlying network topology. Topology parameters such as the number of nodes and links, way of connection and port status can be detected via the Link Layer Discovery Protocol (LLDP) [@Topo_discovery2012], which is one of the vital features of the current OpenFlow specification. The *openflow.discovery*[^3], which is an already developed component that can be used to send crafted LLDP messages out of OpenFlow nodes so that the topological view over the data plane layer can be constructed. This module will then convert the discovered network topology into a graph $G$ representation for efficient management purposes. To do so, we utilised the Networkx tool [@Networkx2008], which is a pure python package with a set of powerful functions for manipulating network graphs. When the network starts working and after shaping the data plane topology, the shortest path for each $flow \in Flow$ is configured by the appointed ${SP_x}$ algorithm, which thereafter is stored in the *Operational Routes* table that is specified to contain all the desired working (healthy) paths. In order to perceive how the link failure incident could affect the configured paths from the perspective of service availability and convergence time, we provide a simple example in Table \[tab:service availability\] in which the service deterioration of the $flow_x$ due to link failure incident is highlighted.
Event Flow $src \rightarrow dst$ accessibility $T_C$ Serviceability Notes
-------------------- ---------- ------------------------------------- -------------- ---------------- -------------------------
– $flow_x$ Yes – Path is working
$flow_x \in {F_R}$ $flow_x$ No $T_D$ Path is not working
$flow_x \in {F_R}$ $flow_x$ No $T_{SP}$ Search for alternatives
$flow_x \in {F_R}$ $flow_x$ No $T_{Update}$ Path is restoring
– $flow_x$ Yes – Path is restored
In order to maintain the *Operational Routes* table, two algorithms have been implemented each with its own view in respect to keep the $Flow$ maintained.
Algorithm \[alg:shortestpath\] depicts the default shortest path routing strategy that is performed by the network controller. We specify Dijkstra’s algorithm [@Dijkstra1959], with complexity $O( |V| + |E| \ log \ |V|)$, as the shortest path finder approach for Algorithm \[alg:shortestpath\], which we denote by $SP_{\scriptscriptstyle{D}}$ instead of $SP_x$. So, the $SP_{\scriptscriptstyle{D}}$ is a Dijkstra function that can be applied on any $flow_{set}$ to return only one unique shortest path. When the OpenFlow controller reports a link failure event, every path suffering from that failure will be detected and then two operations will be issued by the controller. First, a *Remove*, denoted by ${OF}_{\scriptscriptstyle{Remove}}$, command is sent to all the routers that belong to each failed path in $Flow$ as a step to remove the incorrectly working entries, then an alternative route will be computed for every affected $flow$. The new flow entries of the alternative path are then forwarded to the relevant routers of each $flow$ through the *Install*, denoted by ${OF}_{\scriptscriptstyle{Install}}$, command. Each modified $flow$, i.e. assigned to alternative, will be stored in a special set that is called *the Labeled Flow* ($LF$), where: $LF \subset Flow$ and with length of $n$. This is to indicate that each $flow \in LF$ is in a sub-optimal state. The recovery from link failure procedure is demonstrated in line (1-13). However, the algorithm also includes the reversion procedure that is activated after a failure recurs (line 15-32) and it is no less important than the recovery process [@Malik2018]. This procedure is required to take into account the percentage of routing flaps that is necessary for the experimental analysis. In fact, we developed this algorithm for comparison purposes only against Algorithm \[alg:Smart\_Routing\]. Therefore, it does not reflect a contribution of this paper.
\[alg:shortestpath\]
c := 0
Algorithm \[alg:Smart\_Routing\] is one of the main contributions of this work that exploited the prediction information towards enhancing the service availability and the fault tolerance of SDNs. This algorithm depends on Bhandari’s algorithm for finding *K* edge-disjoint paths [@Bhandari1999], which has been utilised as a complementary to build the smart routing strategy. We denoted Bhandari’s algorithm as $SP_{\scriptscriptstyle{B}}$ in place of $SP_x$.
\[alg:Smart\_Routing\]
$\forall \ {flow \in Flow} :
\textit{Set Primary Path as} \
flow_{b_1} \ . \ flow_{b_1} \in SP_{\scriptscriptstyle{B}}(flow_{set})$
$EBC_{\bar{e}_{ij}} = \frac{PF_{R_{len}}}{Flow_{len}}$
$Risk_{\bar{e}_{ij}} = p(\bar{e}_{ij}) \times EBC_{\bar{e}_{ij}}$
${PF_R} = \emptyset$
Thereon, we consider $SP_{\scriptscriptstyle{B}}$ as a function specified to compute two link-disjoint paths with the least total cost for any given pair of nodes (i.e. $src$ and $dst$) or $flow_{set}$. For the purpose of distinguishing between the two returned paths of $SP_{\scriptscriptstyle{B}}$, we denote the first path as $flow_{b_1}$ and the second disjoint one as $flow_{b_2}$. The time complexity of $SP_{\scriptscriptstyle{B}}$ is different from the $SP_{\scriptscriptstyle{D}}$, which is a polynomial that is equivalent to $O((K+1).|E| + |V| \ log \ |V|)$.
The pseudo code of Smart Routing ($SR$) is demonstrated in Algorithm \[alg:Smart\_Routing\], in which the $flow_{b_1}$ is initially selected to represent the primary path for each $flow$ in the network. The network controller will then start listening to the prediction module, which will be discussed in the next section, for the potential of future incidents. When a new message ($m$) is received, the controller will firstly identify the potential failed list, which contains the information about link which is expected to fail in the near future as described in (line 2-4). Secondly, the route (or routes) which might be affected according to the predicted failure message will be computed as a preparatory step to replace them (lines 5-7). After identifying the routes that may possibly fail, the $EBC$ for the predicted link will be calculated as a step towards measuring the risk (lines 8-10). If the risk value is below the threshold, then the prediction information will be ignored and no action will be taken. Otherwise, the flow entries of the newly computed disjoint path from the second step will be installed through using the *Install* command. This is done by adjusting the disjoint path rules with lower priority than the primary path to avoid conflict of matching and action processes.
Following this step, the forwarding rules of the risky primary paths will need to be deleted in order to use TCAM resources efficiently. This needs to be done in a similar procedure to the installation but with the *Remove* command as demonstrated in (lines 11-14). After swapping the primary, $flow_{b_1}$, with the disjoint, $flow_{b_2}$, this action will be considered as the correct decision for a certain period of time (i.e. $\Delta {t_p}$) as indicated in line 15. To examine the substantiality of the changing routes decision, the link that was anticipated to get down within $\Delta {t_l}$ will be compared against the failure set $F$. On one hand, if the link exists then, the prediction will be marked as ${TP}$. In addition, each $flow \in PF_R$ will be labeled as sub-optimal and store in $LF$ (lines 16-18). On the other hand, if the link does not exist then, the prediction will be considered as ${FP}$. In such a case, it is necessary to reset the primary path to its initial state (i.e. optimal) as deliberated in (lines 19-25). However, in case when there is a failure that is not captured by the prediction module then, it is considered as ${FN}$ and such failures are tackled by calling Algorithm \[alg:shortestpath\] as outlined in (line 28-30). Finally, Algorithm \[alg:shortestpath\] will also be invoked when a failed link is repaired (lines 32-34). (c) *Prediction Module* \[subsec:prediction\_model\] In this work, this module is placed on top of the parsed network topology state that gained from the network controller as a result of lacking historical data. We consider each link in the network as an independent object of link class. The link class contains a set of attributes, which currently includes eight attributes as shown in Figure \[fig:Queue\].
![Representation of links in priority queue[]{data-label="fig:Queue"}](Queue.pdf)
The link attributes are used to control the up and down events. In the current implementation, we used the priority queue, $Q$, as a pool to hold all the non-faulty links. On one hand, equations (\[eq:MTBF\]) and (\[eq:MTTR\]) are essential for computing the two static attributes ($MTBF$ and $MTTR$) of each link. For (\[eq:MTBF\]), we rely on the topologies information in Section \[subsec:topologies\] and by assuming that $CC$ equals the minimum cable length in a network. While, for (\[eq:MTTR\]) we used the uniform distribution to generate $\gamma$ for each link independently. On the other hand, the six remaining attributes are described as follows:
$\bullet$ ID: a numerical unique value (i.e. $1,\dots, n$) assigned to the link to represent the link identification number.$\bullet$ F\_Count: registers the number of times the link has failed.$\bullet$ Length: represents the link’s length in km, which is derived from the topology specification.$\bullet$ Next\_F : refers to the *next time to failure* of link, which controls the enqueue and dequeue operations of the link. In other words, this attribute determines the link’s life span in the $Q$, where the link will be dequeued when *Next\_F*=0.$\bullet$ Probability\_F : registers the current failure probability, $p$, of the link. For instance, the *Probability\_F* of the link ($j$) is defined as:
$\frac{F\_Count(ID_j)}{\sum_{i=1}^n F\_Count(ID_i)} \times 100$
where $n$ is the $Q$ length.$\bullet$ Status : reflects the current state of the link as either operational or faulty. On this basis, we have placed our online predictor scheme, as defined by Algorithm \[alg:M\], on top of the priority queue in order to send encapsulated messages about the links which satisfy the following two conditions (as described in lines 2-9): First, the probability of failure is greater than or equal to the threshold ${T_\Omega}$ and second, the leading time (i.e. $\Delta{t_l}$) is less than or equal to the *next time to failure*.
\[alg:M\]
Experimental Setup and Design {#sec:Experimental}
=============================
Since smart routing is aimed to enhance the SDN fault tolerance in the context of network service availability, we have implemented some metrics for fair comparison between the traditional SDN and the proposed system. We also show in this section the adopted network topologies that have been utilised in our experiments.
Availability Measurements
-------------------------
Considering the convergence time that is required to shift from a failed or non-operational path to an alternative or backup one, which conforms with Equation (\[eq:convergence\]). This convergence process definitely damages the availability of some paths, as shown in Table \[tab:service availability\]. For the purpose of identifying the serviceable, which are denoted by “Yes”, and the unserviceable, which are denoted by “No”, $flows$ with respect to some failure events, we formulated this problem as follows:
$(flow \cap Q) = flow \implies Yes$
$(flow \cap Q) \subset flow \implies No$
where, “Yes” and “No” can be obtained by intersecting each $flow \in Flow$ against the $Q$. The $flow$ is subjected to “Yes” when all its forming edges reside in the $Q$, otherwise, the $flow$ will be considered as unserviceable and subjected to “No". By knowing the number of serviceable and unserviceable $flows$, the service unavailability and thus the service availability can be measured. The service unavailability of SDN ($U_{SDN}$) over a given interval time with a certain number of failure events, which are denoted by $ev$, can be arrived at as follows: $$%\footnotesize
U_{SDN}(Flow, G) = \frac{\sum\limits_{\substack i=1}^{ev}{_{flow \in Flow} No}}{ev \ \times \ Flow_{len}}$$
Whereas, for smart routing it is important to further consider the impact of $Recall$ values as well. Hence, the service unavailability of $SR$ ($U_{SR}$) can be arrived at through the following equation: $$%\footnotesize
U_{SR}(Flow, G) = (1-Recall) \times (U_{SDN}(Flow, G))$$
Consequently, the availability $A_x$, with $x = \ SDN \ or \ SR$, can be arrived at through the following: $$%\footnotesize
A_x = 1 - U_x$$
Routing Instability Measurements
--------------------------------
In traditional networks, routing protocols (e.g. IGP [@IGP2011]) perform two routing changes as a reaction to every single failure, one time when a failure occurs and another when a failure is repaired. In fact, both changes are essential for the QoS where the first change is for the purpose of service availability, while, the goal of the second one is to return back from the backup (i.e. sub-optimal) to the primary (i.e. optimal) path again. In contrast, SDN architecture brings centralisation and programmability to the scene, therefore, traditional distributed protocols are independent of the SDN architecture. Maintaining the optimal path (e.g. minimum hops in our case) of each $flow$ will require a continuously adaptive strategy that will be responsible for replacing each sub-optimal $flow$ with the optimal one after it becomes serviceable. To do so, we assume that each alternative $flow$ is additionally stored in $LF$ as mentioned in Section \[sec:Framework\]. For SDN, the routing flaps (denoted by $RF$) can be measured by the means of link *up* (denoted by $u_f$) and *down* (denoted by $d_f$) as follows:
$$\label{RF_SDN}
%\footnotesize
RF_{\scriptscriptstyle{SDN}} = \sum_{flow \in LF} u_f + \sum_{flow \in {F_R}} d_f$$
On one hand, and according to (\[RF\_SDN\]), after each link down event; a new route for each $flow \in {F_R}$ is required, which then leads to a first routing change for each $flow$. On the other hand, and after each link up announcement, the controller will need to check the state of each labeled $flow$ in $LF$ to determine if it’s still the optimal choice. If so, then no change will be made, otherwise, rerouting is required and therefore it will result in another routing change.
However, for the smart routing mechanism, it is necessary to consider the three prediction parameters also (i.e. ${FN}, {TP}$ and ${FP}$) as follows:
$$\label{RF_SR}
\footnotesize
RF_{\scriptscriptstyle{SR}} = \sum_{flow \in {F_R}} {FN}_f + \sum_{flow \in {PF_R}} {TP}_f + \sum_{flow \in {PF_R}} {FP}_f + \sum_{flow \in LF} u_f$$
According to (\[RF\_SR\]), the ${FN}_f$ is equivalent to $d_f$ in (\[RF\_SDN\]) as it reflects the actual failure events that have not been captured by the prediction module, while the remaining are as follows:$\bullet$ Each true prediction will lead to a first reroute flap that gives the advantage of avoiding an upcoming failure event. While, the second flap will be similar to the scenario of $RF_{\scriptscriptstyle{SDN}}$ through inserting the $flow$ into the $LF$ and the next flap builds upon the link restoration $u_f$.$\bullet$ Each false prediction leads into two useless flaps, one when the prediction triggers an alarm, in such a case each potential $flow$ will be added to the *temporary labeled Flow* set ($TLF$), as a transient step before it recognises the prediction was false. The second flap is performed when $\Delta {t_p}$ expires.
We provide an overview of the process of measuring the number of routing flaps in the flow chart of Figure \[fig:Routing\_Flaps\], which also shows how the $LF$ is adjusted in the scenario of the two algorithms, i.e. Algorithm \[alg:shortestpath\] and \[alg:Smart\_Routing\].
![Flow chart of routing flaps[]{data-label="fig:Routing_Flaps"}](Routing_flaps.pdf)
Since all actions are associated with the link state, in this work, we utilise the OpenFlow protocol to reflect the data plane links changing state by relying on the *Link-State Advertisement* (LSA), in addition to the proposed prediction module that will also produce additional observed information about the potential failures. Both LSA and prediction information will be delivered to the controller through the *Updater* in order to apply the appropriate action as illustrated in the flow chart.
Simulated network topologies {#subsec:topologies}
----------------------------
In order to evaluate the proposed method, we have modelled three core network topologies as illustrated in Table \[tab:top\], where both janos-us and germany50 represent a real network topology instance that was defined in [@SNDlib], while waxman synthetic topology is created by the Internet topology generator Brite [@BRITE2001] through using the well-known Waxman model [@WAXMAN1988].
[| >m[0.5in]{} || >m[0.3in]{} | | >m[0.3in]{} | | >m[0.65in]{} | | >m[0.65in]{} |]{} Topology& Nodes& Edges& Min$_{\tiny{len}(e_{ij})}$ & Max$_{\tiny{{len}}(e_{ij})}$\
janos-us& 26 &42&145 km& 1127 km\
germany50& 50 &88& 36 km& 236 km\
waxman& 70 &140& 15 km& 1099 km\
Waxman’s model is a geographical approach that connects distributed routers in a plane on the basis of the distance among them, given by the following probabilistic formula: $$%\footnotesize
\mathbb{P}(\{v_i,v_j\}) = \beta \ exp{^{\frac{-d(v_i,v_j)}{L \alpha}}}$$
where $0 < \alpha$ and $\beta \leq 1$. $d$ represents the distance between $v_i$ and $v_j$, while $L$ represents the maximum distance between any two given nodes. The number of links among the generated nodes is associated with the value of $\alpha$ in a directly proportional manner, while the edge distance increases when the value of $\beta$ is incremented. We used Brite to generate a large-scale network topology in comparison to the others (e.g. when the number of edges or nodes $\geq 100$). The characteristics of all the modelled topologies are detailed in Table \[tab:top\].
Experimental Design and Implementation
--------------------------------------
In order to validate our approach, the proposed framework is built-up on top of POX controller[^4]. We evaluated our framework prototype by using the container-based emulator, Mininet [@Mininet2010]. Mininet is a widely used emulation system, as evidenced in a recent survey [@ComprehensiveSurvey2015], for evaluating and prototyping SDN protocols and applications. It can also be used to create realistic virtual networks, running real kernel, switch and application code, on a single machine (VM, cloud or native). Our experiments were designed based on the topologies that we illustrated in the preceding section. Since one of our experimental topologies was designed via Brite, we utilised the Fast Network Simulation Setup (FNSS) [@FNSS2013]. FNSS is a python-based toolchain simulator that can be used to facilitate the process of network experiments. It provides a wide range of functions and adapters that allow network researchers to parse graphs from different topology generators, such as Brite, in order to be compatible with and/or to interface with other simulator/emulator tools, such as Mininet. Based on the *failure event model* (Section \[failure\_model\]), the general reliability theory [@reliabilityBook] has been utilised to generate failure events using the exponential distribution ($mean = MTBF$) for the next time to failure of each link, and lognormal distribution $E(\mu, \sigma)$ with:
$\mu = \log (MTTR) - ((0.5) \times \log(1 + ((0.6 \times MTTR)^2 / MTTR^2)))$
and,
$\sigma = \sqrt{\log (1+ ((0.6\times MTTR)^2/MTTR^2}$
for time to recover. Regarding failure anticipation, false and true positive have been generated during the simulated time using the uniform distribution following the specified threshold value. Figure \[fig:flow\_diagram\] summarises the simulated link queuing system that is correlated to the two metrics of reliability, i.e. MTBF and MTTR.
![Flow diagram of a link’s life cycle in the Queue[]{data-label="fig:flow_diagram"}](Updated_flowdiagram.pdf)
In order to dispatch the prediction information that is necessarily important to the smart routing module, the distributed messages framework (ZeroMQ [@ZMQ]) was exploited to carry the alarm messages, ${M}$, from the prediction module to the network controller interface. In some network $flow$ conditions it will activate the smart routing module to begin a possible reconfiguration. In the emulation environment, we employed two servers; one acts as the OpenFlow controller and the other to simulate the network topologies. For each server, we used Ubuntu version 14.04 LTS running on an Intel Core-i5 processor equipped with 8 GB RAM.
Key Advantages of Smart Routing {#sec:Discussion}
===============================
In this section, we present comparison and evaluation of the proposed method versus the default SDN technique. To do so, the study has been conducted on the three topologies that were summarised in Table \[tab:top\]. To simulate the three topologies, we ran the emulator for 144 hours, i.e. each experimental topology was simulated in the system for 48 hours. Figure \[fig:results\] shows the obtained results from the three topologies based on parameter settings of ${T_\Omega} = 0.25$, $T_\omega = 0.1$, $\Delta{t_l}=120s$ and $\Delta{t_p} = 30s$.
\[fig:flaps\_percentage\]
As discussed earlier, the ${T_\Omega}$ and $T_\omega$ values can be selected by the network operator or by using additional algorithms (i.e. machine learning) to identify the near optimal values. Since the main goal of smart routing is to enhance the network service availability, we plot for each network that which gives the default SDN and SR mechanisms for the service availability percentage (Y-axis) and the rate of routing flaps (X-axis). Furthermore, for SR, the performance of the online failure predictor represented by the values of *Recall* and *Precision* are considered and reported respectively to each topology. In fact, *Recall* value has a crucial impact on the service availability in the SR scheme, however, *Precision* value has an impact on the unnecessary routing changes. It can be clearly observed that SR outperformed the default SDN in providing network service availability for all test cases. In spite of the low *Recall* values (i.e. 0.2-0.3), there is still a gain in service availability. Similarly, *janos-us* gained the highest improvement percentage in the service availability and this is because its *Recall* value is greater than that of the other topologies.
On the other hand, the rate of the routing flaps generated by SR is always higher than the SDN. This disadvantage comes as a trade-off for improving the network service availability. Given that the routing instability by means of unnecessary flaps is correlated with the value *Precision*, we have measured the only useless flaps that were generated during the simulation time and for each topology as shown in Figure \[fig:uselese\_flaps\]. Figure \[subfig:useless\_flaps\_rate\] shows the only unnecessary routing changes that have been reported based on the $FP$ rate of each topology, where each single $FP$ is associated with two useless flaps, that is, one for the reconfiguration and the other for the reversion. However, Figure \[subfig:useless\_flaps\_percent\] shows the percentage of useless routing flaps for each topology in comparison with the total number of flaps. In the worst case scenario the routing flaps did not exceed 25%. Although *janos-us* topology has the highest *Precision* value, it yielded a relatively high percentage of useless flaps and this is because the number of links in the topology is low, hence, it is highly likely that each single link is associated with a large number of routes in contrast to the other two topologies. It is also clearly evident that the online failure prediction plays a significant role in both service availability (by $TP$) and routing flaps (by $FP$). Based upon the experiments and simulations, we have some observations, as follows:$\bullet$ Some alternative routes are considered as optimal after receiving an updater message, even though the received update is not involved in its conforming path. The reason for this is that the current system defines the optimal path based on the number of hops. Therefore, each alternative path that has the same number of hops as the optimal one will be considered to be an optimal path. It might not be the case if the obtained mechanism, i.e. using a specified cost function with different parameters such as bandwidth, congestion, energy, etc., is not relying on the number of hops.$\bullet$ In some cases the algorithm is barely able to find two-disjoint paths and therefore, sometimes if a path has faced two successive predictions on its links then, no change will be made. Hence, we used ($\approx$) instead of ($=$) in the output of Algorithm \[alg:Smart\_Routing\], to imply that an entirely empty ${PF_R}$ cannot be always guaranteed.$\bullet$ It is also possible that each $flow \in LF$ may face one or more risky links, thus in such a case the entangled $flow$ state will be the same (i.e. sub-optimal).$\bullet$ In some cases and when the $Next\_F < 2 \ min$, the controller ignores the prediction if it is generated as in such a case the $\Delta {t_l}$ is not satisfied and so the controller will not have enough time for the preparation process.
Conclusion and Future Work {#sec:Conclusion}
==========================
This paper has demonstrated how to use online failure prediction to enhance SDN service availability. We presented a new model for SDNs that tackles the problem of data plane link failures. Our work differs from the existing contributions by allowing SDN controllers to have a time window to reconfigure the network before the anticipated failure occurs and avoid the interruption in the availability of network services. The proposed model was implemented using a couple of new algorithms that extract the risky links from paths. Hence, when such risky links fail, no path will be affected. Our experiments were performed over a number of network topologies conducted with the link failure event model. The experimental findings demonstrate the effectiveness of the proposed method in enhancing the SDN service availability. A major drawback of this approach is the routing flaps rate that results from the failure prediction process, which may lead to network instability, especially when it reaches high rates. For this purpose, we measured the percentage of the unnecessary routing changes and in the worst scenario, it was 25%, which we consider requires improving in future research.
For other future work, we will position the study in the setting of machine learning algorithms in order to achieve more flexibility in the decision making process, allowing this to be gauged against optimal threshold values. We are also planning to extend this work to consider disaster situations, which involve multiple link failures.
[1]{}
Lin, P., Bi, J., Hu, H., Feng, T., & Jiang, X. (2011, November). A quick survey on selected approaches for preparing programmable networks. In *Proceedings of the 7th Asian Internet Engineering Conference* (pp. 160-163). ACM.
McKeown, N., Anderson, T., Balakrishnan, H., Parulkar, G., Peterson, L., Rexford, J., ... & Turner, J. (2008). OpenFlow: enabling innovation in campus networks. *ACM SIGCOMM Computer Communication Review, 38*(2), 69-74.
Laprie, J. C. (1992). Dependability: Basic concepts and terminology. In *Dependability: Basic Concepts and Terminology* (pp. 3-245). Springer, Vienna.
Wickboldt, J. A., De Jesus, W. P., Isolani, P. H., Both, C. B., Rochol, J., & Granville, L. Z. (2015). Software-defined networking: management requirements and challenges. *IEEE Communications Magazine, 53*(1), 278-285.
Akyildiz, I. F., Lee, A., Wang, P., Luo, M., & Chou, W. (2016). Research challenges for traffic engineering in software defined networks.*IEEE Network, 30*(3), 52-58.
Markopoulou, A., Iannaccone, G., Bhattacharyya, S., Chuah, C. N., Ganjali, Y., & Diot, C. (2008). Characterization of failures in an operational IP backbone network. *IEEE/ACM transactions on networking, 16*(4), 749-762.
Akyildiz, I. F., Lee, A., Wang, P., Luo, M., & Chou, W. (2014). A roadmap for traffic engineering in SDN-OpenFlow networks. *Computer Networks, 71,* 1-30.
Kempf, J., Bellagamba, E., Kern, A., Jocha, D., Takács, A., & Sköldström, P. (2012, June). Scalable fault management for OpenFlow. In *Communications (ICC), 2012 IEEE International Conference* on (pp. 6606-6610). IEEE.
Sgambelluri, A., Giorgetti, A., Cugini, F., Paolucci, F., & Castoldi, P. (2013). OpenFlow-based segment protection in Ethernet networks. *Journal of Optical Communications and Networking, 5*(9), 1066-1075.
Kreutz, D., Ramos, F. M., Verissimo, P. E., Rothenberg, C. E., Azodolmolky, S., & Uhlig, S. (2015). Software-defined networking: A comprehensive survey. *Proceedings of the IEEE, 103*(1), 14-76.
Sharma, S., Staessens, D., Colle, D., Pickavet, M., & Demeester, P. (2011, October). Enabling fast failure recovery in OpenFlow networks. In *Design of Reliable Communication Networks (DRCN), 2011 8th International Workshop on the* (pp. 164-171). IEEE.
Staessens, D., Sharma, S., Colle, D., Pickavet, M., & Demeester, P. (2011, October). Software defined networking: Meeting carrier grade requirements. In *Local & Metropolitan Area Networks (LANMAN), 2011 18th IEEE Workshop on* (pp. 1-6). IEEE.
Sharma, S., Staessens, D., Colle, D., Pickavet, M., & Demeester, P. (2013). OpenFlow: Meeting carrier-grade recovery requirements. *Computer Communications, 36*(6), 656-665.
Kim, H., Schlansker, M., Santos, J. R., Tourrilhes, J., Turner, Y., & Feamster, N. (2012, October). Coronet: Fault tolerance for software defined networks. In *Network Protocols (ICNP), 2012 20th IEEE International Conference on* (pp. 1-2). IEEE.
Luo, M., Zeng, Y., Li, J., & Chou, W. (2015). An adaptive multi-path computation framework for centrally controlled networks. *Computer Networks, 83,* 30-44.
Jinyao, Y., Hailong, Z., Qianjun, S., Bo, L., & Xiao, G. (2015). HiQoS: An SDN-based multipath QoS solution. *China Communications, 12*(5), 123-133.
Rotsos, C., Sarrar, N., Uhlig, S., Sherwood, R., & Moore, A. W. (2012, March). Oflops: An open framework for openflow switch evaluation. In *International Conference on Passive and Active Network Measurement* (pp. 85-95). Springer Berlin Heidelberg.
Jin, X., Liu, H. H., Gandhi, R., Kandula, S., Mahajan, R., Zhang, M., ... & Wattenhofer, R. (2014, August). Dynamic scheduling of network updates. In *ACM SIGCOMM Computer Communication Review* (Vol. 44, No. 4, pp. 539-550). ACM.
Astaneh, S. A., & Heydari, S. S. (2016). Optimization of SDN flow operations in multi-failure restoration scenarios. *IEEE Transactions on Network and Service Management, 13*(3), 421-432.
Malik, A., Aziz, B., Adda, M., & Ke, C. H. (2017). Optimisation methods for fast restoration of software-defined networks. *IEEE Access, 5*, 16111-16123.
Malik, A., Aziz, B., Ke, C. H., Liu, H., & Adda, M. Virtual Topology Partitioning Towards An Efficient Failure Recovery of Software Defined Networks. In *Machine Learning and Cybernetics (ICMLC), 2017 International Conference on* (pp. 646-651). IEEE.
Fonseca, P., & Mota, E. (2017). A Survey on Fault Management in Software-Defined Networks. *IEEE Communications Surveys & Tutorials*.
Lee, S. S., Li, K. Y., Chan, K. Y., Lai, G. H., & Chung, Y. C. (2015, October). Software-based fast failure recovery for resilient OpenFlow networks. In *Reliable Networks Design and Modeling (RNDM), 2015 7th International Workshop on* (pp. 194-200). IEEE.
Desai, M., & Nandagopal, T. (2010, January). Coping with link failures in centralized control plane architectures. In *Communication Systems and Networks (COMSNETS), 2010 Second International Conference on* (pp. 1-10). IEEE.
Lee, S. S., Li, K. Y., Chan, K. Y., Lai, G. H., & Chung, Y. C. (2014, April). Path layout planning and software based fast failure detection in survivable OpenFlow networks. In *Design of Reliable Communication Networks (DRCN), 2014 10th International Conference on the*(pp. 1-8). IEEE.
E. W. Dijkstra, E., W. (1959, December). A note on two problems in connexion with graphs. *Numerische Mathematik, 1*(1), 269-271.
Vidalenc, B., Ciavaglia, L., Noirie, L., & Renault, E. (2013, May). Dynamic risk-aware routing for OSPF networks. In *Integrated Network Management (IM 2013), 2013 IFIP/IEEE International Symposium on* (pp. 226-234). IEEE.
Labovitz, C., Malan, G. R., & Jahanian, F. (1999, March). Origins of Internet routing instability. In *INFOCOM’99. Eighteenth Annual Joint Conference of the IEEE Computer and Communications Societies. Proceedings. IEEE* (Vol. 1, pp. 218-226). IEEE.
Medem, A., Teixeira, R., Feamster, N., & Meulle, M. (2010, October). Joint analysis of network incidents and intradomain routing changes. In *Network and Service Management (CNSM), 2010 International Conference on* (pp. 198-205). IEEE.
Salfner, F., Lenk, M., & Malek, M. (2010). A survey of online failure prediction methods. *ACM Computing Surveys (CSUR), 42*(3), 10.
Medem, A., Teixeira, R., & Usunier, N. (2010, December). Predicting critical intradomain routing events. In *Global Telecommunications Conference (GLOBECOM 2010), 2010 IEEE* (pp. 1-5). IEEE.
Mangoubi, R. S. (2012). *Robust estimation and failure detection: A concise treatment*. Springer Science & Business Media.
De Maesschalck, S., Colle, D., Lievens, I., Pickavet, M., Demeester, P., Mauz, C., ... & Derkacz, J. (2003). Pan-European optical transport networks: an availability-based comparison. *Photonic Network Communications*, 5(3), 203-225.
Gonzalez, A. J., & Helvik, B. E. (2012). Characterisation of router and link failure processes in UNINETT’s IP backbone network. *International Journal of Space-Based and Situated Computing 7, 2*(1), 3-11.
Kaplan, S., & Garrick, B. J. (1981). On the quantitative definition of risk. *Risk analysis, 1*(1), 11-27.
Chandrasekaran, B., & Benson, T. (2014, October). Tolerating SDN application failures with LegoSDN. In *Proceedings of the 13th ACM workshop on hot topics in networks* (p. 22). ACM.
Lu, L., & Zhang, M. (2013). Edge betweenness centrality. In *Encyclopedia of systems biology* (pp. 647-648). Springer, New York, NY.
Brandes, U. (2008). On variants of shortest-path betweenness centrality and their generic computation. *Social Networks, 30*(2), 136-145.
Szwaczyk, S., Wrona, K., & Amanowicz, M. (2018, May). Applicability of risk analysis methods to risk-aware routing in software-defined networks. In *2018 International Conference on Military Communications and Information Systems (ICMCIS)* (pp. 1-7). IEEE.
POX Wiki. \[Online\]. Available at: https://openflow.stanford.edu/display/ONL /POX+Wiki.
Shalimov, A., Zuikov, D., Zimarina, D., Pashkov, V., & Smeliansky, R. (2013, October). Advanced study of SDN/OpenFlow controllers. In *Proceedings of the 9th central & eastern european software engineering conference in russia* (p. 1). ACM.
Huang, W. Y., Hu, J. W., Lin, S. C., Liu, T. L., Tsai, P. W., Yang, C. S., ... & Mambretti, J. J. (2012, March). Design and implementation of an automatic network topology discovery system for the future internet across different domains. In *Advanced Information Networking and Applications Workshops (WAINA), 2012 26th International Conference on* (pp. 903-908). IEEE.
Schult, D. A., & Swart, P. (2008, August). Exploring network structure, dynamics, and function using NetworkX. In *Proceedings of the 7th Python in Science Conferences (SciPy 2008)* (Vol. 2008, pp. 11-16).
Malik, A., Aziz, B., & Adda, M. (2018, November). Towards filling the gap of routing changes in software-defined networks. In *Proceedings of the Future Technologies Conference* (pp. 682-693). Springer, Cham.
Bhandari, R. (1999). *Survivable networks: algorithms for diverse routing*. Springer Science & Business Media.
Poretsky, S., Imhoff, B., & Michielsen, K. (2011). *Terminology for Benchmarking Link-State IGP Data-Plane Route Convergence* (No. RFC 6412).
SNDlib library. \[Online\]. Available at: http://sndlib.zib.de.
Medina, A., Lakhina, A., Matta, I., & Byers, J. (2001). BRITE: An approach to universal topology generation. In *Modeling, Analysis and Simulation of Computer and Telecommunication Systems, 2001. Proceedings. Ninth International Symposium on* (pp. 346-353). IEEE.
Waxman, B. M. (1988). Routing of multipoint connections. *IEEE journal on selected areas in communications, 6*(9), 1617-1622.
Lantz, B., Heller, B., & McKeown, N. (2010, October). A network in a laptop: rapid prototyping for software-defined networks. In *Proceedings of the 9th ACM SIGCOMM Workshop on Hot Topics in Networks* (p. 19). ACM.
Saino, L., Cocora, C., & Pavlou, G. (2013, March). A toolchain for simplifying network simulation setup. In *Proceedings of the 6th International ICST Conference on Simulation Tools and Techniques* (pp. 82-91). ICST (Institute for Computer Sciences, Social-Informatics and Telecommunications Engineering).
Ohring, M., & Lloyd J. R. *Reliability and failure of electronic materials and devices*. Academic Press, 2009.
ZeroMQ. \[Online\]. Available at: http://zeromq.org/.
[^1]: Ali Malik, Benjamin Aziz and Mo Adda are with the University of Portsmouth, School of Computing, Buckingham Building, Lion Terrace, Portsmouth PO1 3HE, United Kingdom, e-mail: {ali.al-bdairi; benjamin.aziz; mo.adda}@port.ac.uk
[^2]: Chih-Heng Ke is with the Department of Computer Science and Information Engineering, National Quemoy University, Taiwan, e-mail: smallko@gmail.com
[^3]: https://github.com/att/pox/blob/master/pox/openflow/discovery.py
[^4]: The implementation code of the current framework is made available on github : `https://github.com/Ali00/SDN-Prediction-Model.`
|
---
abstract: 'Introducing an isolated intermediate band (IB) into a wide band gap semiconductor can potentially improve the optical absorption of the material beyond the Shockley-Queisser limitation for solar cells. Here, we present a systematic study of the thermodynamic stability, electronic structures, and optical properties of transition metals ($M=$ Ti, V, and Fe) doped CuAlSe$_2$ for potential IB thin film solar cells, by adopting the first-principles calculation based on the hybrid functional method. We found from chemical potential analysis that for all dopants considered, the stable doped phase only exits when the Al atom is substituted. More importantly, with this substitution, the IB feature is determined by $3d$ electronic nature of $M^{3+}$ ion, and the electronic configuration of $3d^1$ can drive a optimum IB that possesses half-filled character and suitable subbandgap from valence band or conduction band. We further show that Ti-doped CuAlSe$_2$ is the more promising candidate for IB materials since the resulted IB in it is half filled and extra absorption peaks occurs in the optical spectrum accompanied with a largely enhanced light absorption intensity. The result offers a understanding for IB induced by transition metals into CuAlSe$_2$ and is significant to fabricate the related IB materials.'
address:
- 'Center for Photovoltaic Solar Energy, Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, 518055, China'
- 'Department of Physics, South University of Science and Technology of China, Shenzhen, 518055, China'
author:
- Tingting Wang
- Xiaoguang Li
- Wenjie Li
- Li Huang
- Ya Cheng
- Jun Cui
- Hailin Luo
- Guohua Zhong
- Chunlei Yang
bibliography:
- '<your-bib-database>.bib'
title: 'Transition metals doped CuAlSe$_2$ for promising intermediate band materials'
---
Intermediate band, Intermediate band solar cell, CuAlSe$_2$, first-principles calculation
Introduction
============
The conversion efficiency is one of the most important factors to estimate the performance of photovoltaic solar cells. At present, many ideas to increase the efficiency of solar cell have been developed including the third generation solar cells. As one of the third generation solar cells, intermediate band solar cell (IBSC) is believed to have bright prospects. For traditional photovoltaic solar cells, electrons are directly excited from the valence band (VB) to the conduction band (CB) by absorbing photons. For IBSC, the concept is to introduce an isolated intermediate band (IB) in the main band gap of the host semiconductor to build a three-photon absorption process such that electrons can be excited not only from the VB to the CB but also from the VB to the IB and from the IB to the CB. As a result, the upper limiting efficiency of IBSC was predicted to be as high as 63.1%, greater than the single-junction solar cell (40.7%).[@ref1; @ref2] By further increasing the number of IBs, the efficiency would reach 86.8%.[@ref3; @ref4] Theoretical and experimental reports have verified that IB materials could effectively increase the optical absorption.[@ref5; @ref6; @ref7] Numerous efforts have been made to implement IBSCs, including through quantum dots and the insertion of appropriate impurities into the bulk host semiconductor.[@ref5; @ref6; @ref7; @ref8; @ref9; @ref10; @ref11; @ref12; @ref13; @ref14; @ref15; @ref16; @ref17; @ref18]
From previous studies, the bulk IB material is easier to fabricate than quantum dots and has a stronger absorption because of the higher density of the IB states.[@ref18] Among bulk IB host materials, Cu-based chalcopyrite compounds are the important candidate host semiconductors for IBSCs with the high conversion efficiency of 46.7%.[@ref19] When the value of band gap is in the optimum region of $2.2-2.8$ eV with the IB located in the region of $0.8-1.1$ eV from VB, the photovoltaic energy conversion is becoming stronger. Hence, CuGaS$_2$ and CuAlSe$_2$ are viewed as the most promising IB host materials because their band gaps are 2.45 eV and 2.67 eV, respectively. In the past decade, many theoretical studies, based on the first-principles calculations, have been made to design and understand the bulk CuGaS$_2$ IB materials.[@ref20; @ref21] Elements of the $3d$ transition metals and group IVA have been identified as viable candidates to act as substitutes in the cation sites of the CuGaS$_2$ chalcopyrite hosts.[@ref14; @ref20; @ref21; @ref22; @ref23; @ref24; @ref25; @ref26; @ref27; @ref28] In particular, Ti[@ref29], Sn[@ref30], and Fe[@ref31] doped CuGaS$_2$ IB semiconductors have been successfully synthesized and display the enhancement of absorption coefficient. However, the photovoltaic energy conversion does not meet the expectations of higher efficiency. More efforts are needed in the future work.
With respect to CuAlSe$_2$ IB materials, the band gap of the host semiconductor is about 2.6 eV. After introducing a IB located at 1.01 eV from VB, the photovoltaic energy conversion will realize 46%.[@ref19] Comparing to CuGaS$_2$, perfect CuAlSe$_2$ has much higher adsorption coefficient in the visible region.[@ref28; @ref32] Nevertheless, few studies on CuAlSe$_2$ IB materials are reported. For transition metals, we know that introducing local $3d$ electrons into the wide band gap semiconductor often induces the impurity bands in the main band gap of the host. Considering the advantage of $3d$ electrons of transition metals and selecting three typical elements Ti, V, and Fe, in this work, we therefore employ density functional theory based on the hybrid density functional method to investigate the transition metals $M$-doped CuAlSe$_2$. The effects of doping concentrations and positions are examined. It is our aim to explore the thermodynamic stability induced by doping, understand the forming mechanism of IB, and predict the promising candidates materials for IBSC. This is significant to fabricate the CuAlSe$_2$ IB materials and develop the related IBSCs.
Computational Details
=====================
Density functional theory (DFT) calculations have been carried out on a representative structure which consists on a supercell derived from the parent body-centered tetragonal CuAlSe$_2$ structure as shown Fig. 1(a). Different concentrations of the proposed dopants inside the host semiconductor were computed. In order to obtain the expectative doping concentration, calculations for supercells containing 16, 32, 64, and 128 atoms were carried out and compared. The desired structures were achieved by substituting $M$ atom for Cu, for Al, and for Se atom respectively inside a CuAlSe$_2$ supercell. The configurations were fully optimized using a conjugate-gradient algorithm. The Monkhorst-Pack $k$-point grids are generated according to the specified $k$-point separation 0.02 [Å]{}$^{-1}$ and the convergence thresholds are set as $10^{-6}$ eV in energy and 0.005 eV/[Å]{} in force.
All calculations are based on the projector augmented wave method (PAW)[@ref33] with a cutoff energy of 400 eV as implemented in the Vienna *ab* initio simulation package (VASP)[@ref34]. In the standard DFT, the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) version[@ref35] is adopted to describe the electronic exchange-correlation (XC) interactions. However, the standard DFT usually leads to erroneous descriptions for some real systems such as underestimating the band gap. One way of overcoming this deficiency is to use Heyd-Scuseria-Ernzerhof (HSE) hybrid functional, where a part of the nonlocal Hartree-Fock (HF) type exchange is admixed with a semilocal XC functional, to give the following expression[@ref36]: $$\begin{aligned}
E^{HSE}_{xc}=&&\alpha E^{HF,SR}_{x}(\nu)+(1-\alpha)E^{PBE,SR}_{x}(\nu)\nonumber \\
&&+E^{PBE,LR}_{x}(\nu)+E^{PBE}_{c},\end{aligned}$$ where $\alpha$ is the mixing coefficient and $\nu$ is the screening parameter that controls the decomposition of the Coulomb kernel into short-range (SR) and long-range (LR) exchange contributions. In this calculation, the HSE exchange-correlation functional is set as 35% mixing of screened HF exchange to PBE functional, namely $\alpha=0.35$, which can reproduce the experimental band gap of CuAlSe$_2$. Noticeably, Spin polarization is studied due to the existence of a magnetic $M$ atom.
Results and discussion
======================
CuAlSe$_2$ crystallizes in a chalcopyrite structure with a space-group of $I\bar{4}2d$. As shown in Fig. 1(a), each atom in this structure is fourfold coordinated. Namely, each Se atom is coordinated with two Al and two Cu atoms, and each cation is coordinated with four Se ions. The optimized lattice constants of CuAlSe$_2$ are $a = 5.654$ [Å]{} and $c=11.147$ [Å]{} according with the experimental values. The calculated band gap is only 0.86 eV within the GGA method, while it reaches 2.6 eV based on HSE functional ($\nu=0.35$), which is in good agreement with the experimental value of 2.67 eV.[@ref37] From the electronic characteristics near the Fermi level shown in Fig. 1(b), the VB is mainly formed by Cu-$3d$ and Se-$4p$ electronic states, while the CB mainly results from Al-$3s$ and Se-$4s4p$ states.
To examine the probability of the formation of doped compounds, we have calculated the formation energy of doped system using the following as $$\begin{aligned}
E_{f}=&&E_{SC}(doped)-E_{SC}(host)+\sum_{i}\Delta n_i(E_i+\mu_i)\nonumber \\
=&&\Delta H_{f}+\sum_{i}\Delta n_i\mu_i,\end{aligned}$$ where $E_{SC}(doped)$ and $E_{SC}(host)$ is the total energy of the doped and host perfect supercells, respectively. $E_i$ is the total energy of the component element $i$ in its pure phase, which has been shown in Table I. $\Delta n_i$ is the number of exchanged atom $i$ between the host and the doped system ($\Delta n_i$ means that the atom $i$ moves into the host). $\mu_i$ is the chemical potential referenced to the total energy $E_i$ of the pure element phase for the $i$ species, satisfying $\mu_{i}\leq0$. $\mu_{i}=0$ represents the limit where the element is so rich that its pure phase can form. $\mu_{i}<0$ means that the formation of the compound is favorable rather than the pure elemental phase. $\Delta H_{f}$ is the formation enthalpy of doped system related to formation energy. $E_{f}$ represents the chemical reaction ongoing, while $E_{f}$ indicates that the thermodynamic balance is achieved between the host and doping compounds.
To analyze the thermodynamic stability of doped systems, we firstly start from the growth of the host CuAlSe$_2$. To obtain the CuAlSe$_2$ phase under a certain chemical environment, we have a balanced system with $E_{f}=0$ , and the chemical potentials of Cu, Al, and Se satisfy the equation $$\mu_{Cu}+\mu_{Al}+2\mu_{Se}=\Delta H_{f}(CuAlSe_2)=-4.24 ~eV,$$ where the formation enthalpy $\Delta H_{f}$(CuAlSe$_2$) can be obtained from our first-principles calculations. At the same time, to avoid the secondary phases Cu$_2$Se and Al$_2$Se$_3$, the chemical potentials satisfy the following relations $$2\mu_{Cu}+\mu_{Se} < \Delta H_{f}(Cu_2Se)=-0.78 ~eV,$$ and $$2\mu_{Al}+3\mu_{Se}<\Delta H_{f}(Al_2Se_3)=-6.79 ~eV.$$ By considering the above constraints Eqs. (3-5), we can draw the chemical potential range of stable CuAlSe$_2$ phase excluding the secondary phases of Cu$_2$Se and Al$_2$Se$_3$. Based on the analysis, as long as we know the formation enthalpy of doped systems and the chemical potential of different species $i$, the formation energy of the doped systems can be evaluated from Eq. (2). The chemical potential of different species $i$ has been listed Table I. We have also calculated the formation enthalpy of different doped systems and show in Table II. Considering the doped system with the transition metal $M$ occupying on the Cu site, namely $M_{Cu}$, we should control the chemical potentials to satisfy $$\Delta H_{f}(M_{Cu})+\mu_{Cu}-\mu_{M}<0.$$ Similarly, for the $M$ respectively occupying on the Al and Se site, $M_{Al}$ and $M_{Se}$, the chemical potentials must respectively satisfy $$\Delta H_{f}(M_{Al})+\mu_{Al}-\mu_{M}<0$$ and $$\Delta H_{f}(M_{Se})+\mu_{Se}-\mu_{M}<0.$$
By applying the above constraints in Eqs. (3-8), we can obtain the chemical potential range for the doped CuAlSe$_2$ system at $\mu_{M}=0$ (namely under $M$-rich condition). As shown in Fig. 2, the direction faced the origin satisfies the chemical potential constraints when substituting for Cu atom (plotted with solid line), while the direction deviated from the origin satisfies the chemical potential constraints when substituting for Al or Se atom (plotted with dashed line). When $\mu_{M}<0$ (namely under $M$-poor condition), the solid line will further shift toward to the origin, while the dashed lines further deviate from the origin. From the chemical potential range shown in Fig. 2(a), therefore, the doped systems of Ti$_{Cu}$ and Ti$_{Se}$ is difficult to grow. The grey region shown in Fig. 2(a) implies the chemical potential range to be able to obtain the doped system of Ti$_{Al}$. Lowering the chemical potential of Ti, namely representing the Ti-poor condition, the line determined by Ti$_{Al}$ will shift toward to the left, which results in the area of the grey region in Fig. 2(a) reducing. As a result, the Ti$_{Al}$ can be obtained when $\Delta H_{f}$(Ti$_{Al})-3.85\leq\mu_{Ti}\leq0$ eV, where $\Delta H_{f}$(Ti$_{Al})$ can be obtained from our first-principles calculations, such as the values shown in Table II. The doping concentration is just corresponding to the variation of $\mu_{Ti}$. For our considered several doping concentrations, as shown in Fig. 2(a), the phase boundary of Ti$_{Se}$ is sensitive to the doping content.
With regarded to the substitution V atom for Cu or Se atom, as shown in Fig. 2(b), the doped systems of both V$_{Cu}$ and V$_{Se}$ are difficult to grow from the chemical potential constrains. For the substitution V atom for Al atom, the line determined by V$_{Al}$ shifts toward to the left comparing with Ti$_{Al}$. The grey region shown in Fig. 2(b) marks out the range of chemical potential to grow the V$_{Al}$ system. However, the area of grey region in Fig. 2(b) slightly reduces comparing with that in Fig. 2(a). The V$_{Al}$ can be stabilized in the range of $\Delta H_{f}$(V$_{Al})-3.85\leq\mu_{V}\leq0$ eV. For the Fe doping situation, the Fe$_{Se}$ is impracticable, while the Fe$_{Cu}$ can be obtained in a very small chemical potential region. When substituting Fe atom for Al atom, the Fe$_{Al}$ can be grown in a region of chemical potential as the grey area shown in Fig. 2(c). Namely, the doping at Al site is available when the range of Fe chemical potential satisfies $\Delta H_{f}$(Fe$_{Al})-3.85\leq\mu_{Fe}\leq0$ eV. Thus far, we have cleared that the stable doped phase can be obtained only when substituting $M$ for Al atom, which is according with transition metal doped CuGaS$_2$. Additionally, when presenting the chemical potential range of achieving in growth, we assume the constraint of Cu-rich ($\mu_{Cu}=0$). However, the experimental fabrication often achieves under the Cu-poor situation ($\mu_{Cu}<0$). For $\mu_{Cu}<0$, the area of the grey region in Fig. 2 will decrease. Combining with Eqs. (3-8), to obtain the doped phase, we know that the chemical potential of Cu must satisfy $-0.85<\mu_{Cu}\leq0$ eV.
After obtaining the stable $M$-doped CuAlSe$_2$, we further investigate the electronic structures to evaluate the feasibility using as IB materials. View from the previous study[@ref19], to achieve the high conversion efficiency, the subbangap formed by IB from VB or CB should be in the range of $0.7-1.2$ eV, or near to this range. A narrow IB also leads to a high effective mass and a low carrier mobility that makes carrier transport within the band difficult. Thus IB with the finite width is perfect[@ref38]. Importantly, the IB need be half-filled with electrons since both electrons and empty states are required to supply electrons to the CB as well as accepting them from the VB[@ref18]. To examine whether the impurity bands induced by $M$ doping satisfy these requirements above, we have calculated the electronic density of states (DOS) of $M$-doped CuAlSe$_2$ for different doping situations. Figure 3(a) shows the DOSs for Ti doping concentrations of 25% and 6.25%, respectively. Clear IB localized around the Fermi level is observed in these two doped CuAlSe$_2$ systems with substituting Ti atom for Al atom. As the dopant content reaching 25%, the IB is 1.4 eV from the VB maximum (VBM) and 0.7 eV from the CB minimum (CBM). The width of IB is about 0.5 eV. Lowering the doping concentration to 6.25%, the subbandgaps correspondingly become to 1.6 eV and 1.0 eV. The IB position in the Ti-doped system is suitable for promising IB materials as proposed by Martí *et al*[@ref19]. With the increase of doping content as shown in Fig. 3(a), the electronic states to form IB increase and result in the extending of IB, which will suppress the Shockley-Read-Hall recombination[@ref28; @ref39]. Seen from the DOSs of V-doped CuAlSe$_2$ at two concentration levels of 25% and 6.25% shown in Fig. 3(b), however, no half-filled IB is observed around the Fermi level. The impurity bands are far away from the Fermi level and close to the VBM or CBM which indicates the subbandgap is very small. Furthermore, we can not observe the half-filled IB in Fe-doped CuAlSe$_2$ systems either. As the DOSs of two doping concentrations shown in Fig. 3(c), the IB is completely empty though it has the isolated feature. As a result, from the aspect of electronic characteristics of IBs, both V-doping and Fe-doping are not good candidates, though the substitutions of V or Fe atom for Al atom satisfy the conditions of thermodynamic stability.
The IB feature is related to the electronic configuration of transition metallic ion. In chalcopyrite CuAlSe$_2$, four nearest neighboring Se atoms around a Al atom form a tetrahedral crystal field. The $M$ ion substitutes the Al atom in CualSe$_2$ and therefore is located in a tetrahedral crystal field environment. Based on the crystal field theory, the fivefold degenerated $3d$ orbital of $M$ will split into two main manifolds: lower twofold degenerate $e_g$ states and upper threefold degenerate $t_{2g}$ states. Considering the electronic spin polarization on $3d$ orbitals, $e_g^{2\uparrow}t_{2g}^{3\uparrow}$, namely $d^{5\uparrow}$, represents the five $3d$ electrons with the majority-spin, while $e_g^{2\downarrow}t_{2g}^{3\downarrow}$ ($d^{5\downarrow}$) corresponding to the minority-spin situation. Substituting for Al atom in CuAlSe$_2$, Ti, V, and Fe ions exhibit the valence states of Ti$^{3+}$, V$^{3+}$, and Fe$^{3+}$, respectively. The orbital splitting in the tetrahedral crystal field combined with the DOS of $M$-$3d$ leads to the electronic configurations of Ti$^{3+}$-$3d^1$-$e_g^{1\uparrow}$, V$^{3+}$-$3d^2$-$e_g^{2\uparrow}$, and Fe$^{3+}$-$3d^5$-$e_g^{2\uparrow}t_{2g}^{3\uparrow}$. $M^{3+}$ ion is at high-spin state with all $3d$ electrons being spin up. For the substitution Ti atom for Al atom shown in Fig. 3(a), the IB nature origins from the half filling of $e_g^{\uparrow}$ band, due to the single $3d$ electron in Ti$^{3+}$. The coupling between $e_g^{\uparrow}$ band and VBs is very weak. So there is visible gap between them. For the substitution V atom for Al atom, the number of $3d$ electrons increases to two. However, as shown in Fig. 3(b), these two $3d$ electrons full fill the $e_g^{\uparrow}$ band and lower its energy, which results that the full-filled $e_g^{\uparrow}$ band strongly couples with VB. The empty $3d$ bands made up of $t_{2g}^{\uparrow}$ and $e_g^{\downarrow}t_{2g}^{\downarrow}$ are pushed up to the CB. This is reason that the favorable IB can not be observed in V-doped CuAlSe$_2$. In Fe doping case, all of five $3d$ electrons exhibit the majority spin nature, namely $e_g^{2\uparrow}t_{2g}^{3\uparrow}$. The full-filled $e_g^{\uparrow}t_{2g}^{\uparrow}$ bands strongly hybridize with the VBs of the host below the Fermi level. The $e_g^{\downarrow}t_{2g}^{\downarrow}$ bands are completely empty and form the isolated IBs in the main band gap of CuAlSe$_2$. This is just the origin that no half-filled IBs is observed in Fe-doped CuAlSe$_2$.
Along the analysis for three doping elements above and extending other $3d$ transition metals such as Cr, Mn, Co, and Ni, the $3d$ electronic configuration is respectively Cr$^{3+}$-$3d^{3}$-$e_g^{2\uparrow}t_{2g}^{1\uparrow}$, Mn$^{3+}$-$3d^{4}$-$e_g^{2\uparrow}t_{2g}^{2\uparrow}$, Co$^{3+}$-$3d^{6}$-$e_g^{2\uparrow}t_{2g}^{3\uparrow}e_g^{1\downarrow}$, and Ni$^{3+}$-$3d^{7}$-$e_g^{2\uparrow}t_{2g}^{3\uparrow}e_g^{2\downarrow}$ under substituting for Al atom in CuAlSe$_2$. For Ni$^{3+}$, we find that the orbitals are all full occupied without the half-filled band around the Fermi level. Seen from the electronic configurations, the half-filled or unfilled band is existent, $t_{2g}^{1\uparrow}$ band for Cr$^{3+}$, $t_{2g}^{2\uparrow}$ band for Mn$^{3+}$, and $e_g^{1\downarrow}$ band for Co$^{3+}$, which can be viewed as IBs. However, the full-occupied band of $e_g^{2\uparrow}$ or $t_{2g}^{3\uparrow}$ that exists will strongly couple or hybridize with the VBs of the host. As a result, the subbandgap between VB and IB will reduce or disappear. Combining this analysis with three typical doping elements of Ti, V, and Fe, therefore, we conclude that the IB characteristic is determined by the electronic nature of $M^{3+}$ ion. We further infer that $M^{3+}$ ion with the electronic configuration of $d^1$ can induce the optimum IB in wide band gap semiconductors. Just this, Ti is the most promising candidate.
Adopted the method of Gajdoš *et al.*[@ref40], the calculated optical properties further confirm the promising of Ti-doped CuAlSe$_2$ IB materials. Figure 4 shows the absorption coefficient $\alpha$ of Ti-doped CuAlSe$_2$ in visible light region, comparing with the host. After doping, we can observe a greatly enhanced light absorption intensity and additional absorption peaks in the range of $0.7-2.2$ eV, which is induced by the IB. Especially, the 25% doping concentration case (Cu$_4$Al$_3$TiSe$_8$) visibly displays the three-photon absorption process. As shown in Fig. 4, peak 1 implies the electronic transition from IB to CB, and peak 2 represents the electronic transition from VB to IB, while peak 3 results from the electronic transition from VB to CB. The optical result further supports that Ti is the promising candidate to achieving the IB material.
Conclusions
===========
To explore and design novel structural solar cell materials with higher conversion efficiency, we have investigated the feasibility of $M$ (Ti, V, and Fe) doped CuAlSe$_2$ as IBSC material from thermodynamic stability, electronic structures, and optical properties, using the first-principles calculation based on HSE hybrid functional. Based on the chemical potential analysis, we point out that the stable $M$-doped CuAlSe$_2$ only exists when substituting for Al atom, and present the chemical potential conditions to growth. Seen from electronic structures, the Ti substitution results in the optimum IB feature, such as half-filled, favorable subbandgaps of 1.4 eV from VB to IB and 0.7 eV from IB to CB, as well limit width of IB for 25% doping concentration. However, the impurity band induced by the V doping adjoins the VB or CB, while the IB formed by the Fe doping is completely empty. Stable V- and Fe-doped phase can not result in the half-filled IB. We therefore conclude that the $M^{3+}$ with the electronic configuration of $3d^1$ can drive the optimum IB in the wide bandgap semiconductor, namely the IB feature origins from the $3d$ electronic nature into the host. Combining the analysis above, Ti is considered to be the more promising candidate. The calculated absorption spectra of Ti-doped CuAlSe$_2$ illustrate three-photons absorption process. Two additional absorption peaks are formed in the band gap region of the host. And the absorption coefficient of doped systems in visible light region is obviously stronger than that of the host material. This study is helpful for the experimental research of IBSC materials from the doping stability and the forming mechanism of IB.
Acknowledgments {#acknowledgments .unnumbered}
===============
The work was supported by the National Major Science Research Program of China under Grant no. 2012CB933700, the Natural Science Foundation of China (Grant nos. 61274093, 61574157, 11274335, 11504398, 51302303, 51474132, and 11404160), and the Basic Research Program of Shenzhen (Grant nos. JCYJ20150529143500956, JCYJ20140901003939002, JCYJ20150521144320993, JCYJ20140417113430725, and JCYJ20150401145529035).
W. Shockley, H.J. Queisser, Detailed Balance Limit of Efficiency of p-n Junction Solar Cells, J. Appl. Phys. **32** (1961) 510-519. A. Luque, A. Martí, Increasing the Efficiency of Ideal Solar Cells by Photon Induced Transitions at Intermediate Levels, Phys. Rev. Lett. **78** (1997) 5014. M.A. Green, Multiple band and impurity photovoltaic solar cells: General theory and comparison to tandem cells, Prog. Photovolt.: Res. Appl. **9**(2001) 137-144. T. Nozawa, Y. Arakawa, Detailed balance limit of the efficiency of multilevel intermediate band solar cells, Appl. Phys. Lett. **98** (2011) 171108. I. Aguilera, P. Palacios, P. Wahnón, Optical properties of chalcopyrite-type intermediate transition metal band materials from first principles, Thin Solid Films, **516** (2008) 7055-7059. I. Aguilera, P. Palacios, P. Wahnón, Enhancement Of Optical Absorption In Ga-Chalcopyrite-Based Intermediate-Band Materials For High Efficiency Solar Cells, Sol. Energy Mater. Sol. Cells **94** (2010) 1903-1906. P. Chen, M. Qin, H. Chen, C. Yang, Y. Wang, F. Huang, Cr incorporation in CuGaS$_2$ chalcopyrite: A new intermediate-band photovoltaic material with wide-spectrum solar absorption, Phys. Status Solidi A **210** (2013) 1098-1102. A. Luque, A. Martí, C. Stanley, N. López, L. Cuadra, D. Zhou, A. McKee, General equivalent circuit for intermediate band devices: Potentials, currents and electroluminescence, J. Appl. Phys. **96** (2001) 903-909. A. Luque, A. Martí, N. López, E. Antolín, E. Cánovas, C. Stanley, C. Farmer, L.J. Caballero, L. Cuadra, J.L. Balenzategui, Experimental analysis of the quasi-Fermi level split in quantum dot intermediate-band solar cells, Appl. Phys. Lett. **87** (2005) 083505. A. Martí, E. Antolin, C.R. Stanley, C.D. Farmer, N. López, P. Díaz, E. Cánovas, P.G. Linares, A. Luque, Production of Photocurrent due to Intermediate-to-Conduction-Band Transitions: A Demonstration of a Key Operating Principle of the Intermediate-Band Solar Cell, Phys. Rev. Lett. **97**, (2006) 247701. A. Luque, A. Martí, N. López, E. Antolín, E. Cánovas, C. Stanley, C. Farmer, P. Díaz, Operation of the intermediate band solar cell under nonideal space charge region conditions and half filling of the intermediate band, J. Appl. Phys. **99** (2006) 094503. G. Wei, S.R. Forrest, Intermediate-Band Solar Cells Employing Quantum Dots Embedded in an Energy Fence Barrier, Nano Lett. **7** (2007) 218-222. Q. Shao, A.A. Balandin, A.I. Fedoseyev, M. Turowski, Intermediate-band solar cells based on quantum dot supracrystals, Appl. Phys. Lett. **91** (2007) 163503. N. López, A. Martí, A. Luque, C. Stanley, C. Farmer, P. Díaz, Experimental Analysis of the Operation of Quantum Dot Intermediate Band Solar Cells, J. Solar Energy Eng. **129** (2007) 319-322. A. Martí, E. Antolín, E. Cánovas, N. López, P.G. Linares, A. Luque, C.R. Stanley, C.D. Farmer, Elements of the design and analysis of quantum-dot intermediate band solar cells, Thin Solid Films **516** (2008) 6716-6722. P. Palacios, K. Sánchez, P. Wahnón, J.C. Conesa, Characterization by Ab Initio Calculations of an Intermediate Band Material Based on Chalcopyrite Semiconductors Substituted by Several Transition Metals, J. Solar Energy Eng. **129**, (2007) 314-318. Y. Seminóvski, P. Palacios, P. Wahnón, Intermediate band position modulated by Zn addition in Ti doped CuGaS$_2$, Thin Solid Films, **519** (2011) 7517-7521. A. Luque, A. Martí, The Intermediate Band Solar Cell: Progress Toward the Realization of an Attractive Concept, Adv. Mater. **22** (2010) 160-174. A. Martí, D.F. Marrón, A. Luque, Evaluation of the efficiency potential of intermediate band solar cells based on thin-film chalcopyrite materials, J. Appl. Phys. **103** (2008) 073706. P. Palacios, K. Sánche, J.C. Conesa, P. Wahnón, First principles calculation of isolated intermediate bands formation in a transition metal-doped chalcopyrite-type semiconductor, Phys. Status Solidi A **203** (2006) 1395-1401. P. Palacios, K. Sánche, J.C. Conesa, J.J. Fernández, P. Wahnón, Theoretical modelling of intermediate band solar cell materials based on metal-doped chalcopyrite compounds, Thin Solid Films **515** (2007) 6280-6284. C. Tablero, Ionization levels of doped sulfur and selenium chalcopyrites, J. Appl. Phys. **106** (2009) 073718. C. Tablero, Electronic and optical properties of the group IV doped copper gallium chalcopyrites, Thin Solid Films **519** (2010) 1435-1440. D.F. Marrón, A. Martí, A. Luque, Thin-film intermediate band photovoltaics: advanced concepts for chalcopyrite solar cells, Phys. Status Solidi A **206** (2009) 1021-1025. Y.-J. Zhao, A. Zunger, Site preference for Mn substitution in spintronic Cu$M^{III}X_{2}^{VI}$ chalcopyrite semiconductors, Phys. Rev. B **69** (2004) 075208. P. Palacios, I. Aguilera, P. Wahnón, J.C. Conesa, Thermodynamics of the Formation of Ti- and Cr-doped CuGaS$_2$ Intermediate-band Photovoltaic Materials, J. Phys. Chem. C **112** (2009) 9525-9529. C. Tablero, D.F. Marrón, Analysis of the Electronic Structure of Modified CuGaS$_2$ with Selected Substitutional Impurities: Prospects for Intermediate-Band Thin-Film Solar Cells Based on Cu-Containing Chalcopyrites, J. Phys. Chem. C **114** (2010) 2756-2763. M. Han, X. Zhang, Z. Zeng, The investigation of transition metal doped CuGaS$_2$ for promising intermediate band materials, RSC Adv. **4** (2014) 62380-62386. X. Lv, S. Yang, M. Li, H. Li, J. Yi, M. Wang, J. Zhong, Investigation of a novel intermediate band photovoltaic material with wide spectrum solar absorption based on Ti-substituted CuGaS$_2$, Sol. Energy **103** (2014) 480-487. C. Yang, M. Qin, Y. Wang, D. Wan, F. Huang, J. Lin, Observation of an Intermediate Band in Sn-doped Chalcopyrites with Wide-spectrum Solar Response, Sci. Rep. **3** (2013) 1286. M. Björn, K. Sascha, U. Thomas, S. Hans-Werner, Investigation of the Sub-Bandgap Photoresponse in CuGaS$_2$:Fe for Intermediate Band Solar Cells, Prog. Photovoltaics **20** (2012) 625-629. H.G. Zhou, H. Chen, D. Chen, Y. Li, K.N. Ding, X. Huang, Y.F. Zhang, Electronic Structures and Optical Properties of CuAl$X_2$ ($X=$ S, Se, Te) Semiconductors with a Chalcopyrite Structure, Acta Phys. Chim. Sin. **27** (2011) 2805-2813. P.E. Blöchl, Projector augmented-wave method, Phys. Rev. B **50** (1994) 17953-17979. G. Kresse, J. Furthmuller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Comput. Mater. Sci. **6** (1996) 15-50. J.P. Perdew, K. Burke, M. Ernzerhof, Generalized Gradient Approximation Made Simple, Phys. Rev. Lett. **77** (1996) 3865. J. Heyd, G.E. Scuseria, M. Ernzerhof, Erratum: ¡°Hybrid functionals based on a screened Coulomb potential¡± \[J. Chem. Phys. 118, 8207 (2003)\], J. Chem. Phys. **124** (2006) 219906. M. Bettim, Reflection measurements with polarization modulation: A method to investigate bandgaps in birefringent materials like I-III-VI$_2$ chalcopyrite compounds, Solid State Commun. **13** (1973) 599-602. M.Y. Levy, C. Honsberg, Solar cell with an intermediate band of finite width, Phys. Rev. B **78** (2008) 165122. A. Luque, A. Martí, E. Antolín, C. Tablero, Intermediate bands versus levels in non-radiative recombination, Physica B **382** (2006) 320-327. M. Gajdoš, K. Hummer, G. Kresse, J. Furthmüller, F. Bechstedt, Linear optical properties in the projector-augmented wave methodology, Phys. Rev. B **73** (2006) 045112.
Phase $E_{i}$
-------------------------------- ---------
Cu face-centered cubic bulk -3.72
Al face-centered cubic bulk -3.74
Se$_2$ molecule -2.63
Ti hexagonal close-packed bulk -7.76
V body-centered cubic bulk -8.94
Fe body-centered cubic bulk -7.83
: \[tab:table1\] Total energy per atom (eV) of the pure element phase.
Dopant ($M$) Doping concentration $\Delta H_{f}(M_{Cu})$ (eV) $\Delta H_{f}(M_{Al})$ (eV) $\Delta H_{f}(M_{Se})$ (eV)
-------------- ---------------------- ----------------------------- ----------------------------- -----------------------------
$M=$ Ti 25% 1.21 0.30 2.85
12.5% 1.25 0.32 3.27
6.25% 1.31 0.34 5.21
3.125% 1.32 0.36 5.96
$M=$ V 25% 1.72 1.24 4.13
12.5% 1.77 1.24 4.44
6.25% 1.82 1.25 5.75
3.125% 1.83 1.25 5.77
$M=$ Fe 25% 0.81 1.56 4.40
12.5% 0.82 1.56 4.48
6.25% 0.82 1.57 4.51
3.125% 0.83 1.59 4.66
: \[tab:table2\] Formation enthalpy of different doping systems at different doping concentrations.
{width="\columnwidth"}
{width="\textwidth"}
{width="\columnwidth"}
{width="\columnwidth"}
|
---
abstract: 'We investigate the influence of couplings among continuum states in collisions of weakly bound nuclei. For this purpose, we compare cross sections for complete fusion, breakup and elastic scattering evaluated by continuum discretized coupled channel (CDCC) calculations, including and not including these couplings. In our study, we discuss this influence in terms of the polarization potentials that reproduces the elastic wave function of the coupled channel method in single channel calculations. We find that the inclusion of couplings among continuum states renders the real part of the polarization potential more repulsive, whereas it leads to weaker absorption to the breakup channel. We show that the non-inclusion of continuum-continuum couplings in CDCC calculations may lead to qualitative and quantitative wrong conclusions.'
author:
- 'L.F. Canto'
- 'J. Lubian'
- 'P. R. S. Gomes'
- 'M. S. Hussein'
title: 'Continuum-continuum coupling and polarization potentials for weakly bound systesm'
---
After the elapse of almost two decades of extensive experimental and theoretical effort, full understanding of the way the coupling to the continuum influences the near-barrier fusion and other channels in collisions of weakly bound nuclei is still lacking [@CGD06; @LiS05; @KRA07]. Collisions of weakly bound projectiles can lead to different kinds of fusion. The first is the usual complete fusion (CF), when the whole projectile is absorbed by the target. The second type is incomplete fusion (ICF). This process corresponds to the situation where the projectile breaks up into fragments along the collision and some fragments are absorbed while at least one is not. In addition, the breakup process may end up as non-capture breakup (NCBU). In this case, none of the fragments is absorbed. An ideal theoretical description of the collision should take into account all these processes. In a rather detailed calculation within the Continuum Discretized Coupled Channel (CDCC) model [@SYK83; @SYK86; @AIK87], Diaz-Torres and Thompson [@DiT02], have managed to supply some very useful information about the aforementioned question. They found that the continuum coupling hinders the complete fusion cross section above and below the Coulomb barrier, with enhacement setting in only at deep sub-barrier energies. They further found that the inclusion of the continuum-continuum couplings (CCC) was of paramount importance in reaching the above conclusions about fusion. Their findings seem to concur with experimental data [@DHB99; @DGH04]. Increased fusion can arise from a lowering of the Coulomb barrier which results in a greater tunneling. On the other hand, decreased fusion can arise from an increase in the height of the barrier which results in a smaller tunneling. This latter effect would, in principle, be accompanied by an increase in the quasi-elastic scattering at backward angles.
This “common sense” argument about the effects on the non-capture, quasi-elastic, processes seem not be borne out by explicit calculation which takes into account the CCC effects [@NuT99; @LCA09]. It is difficult to discern the physics behind all of the above. Clearly the inclusion or exclusion of the CCC dictates whether one is dealing with geniune breakup or merely a collection of inelastic channels. Further, there seems to be a need to invoke concepts such as irreversibility, and doorway that funnels the flux to the continuum channel and hinders its return to the entrance channel. Attempt to develop quantum transport theory for these reactions, which incorporates these concepts at the outset, has recently been made [@DHD08]. It becomes abundantly clear that more insight into the working of the CCC and the resulting irreversibility and doorway constriction is called for. This is an important issue which goes beyond the realm of nuclear physics. In fact, as far back as 1961, Fano [@Fan61] elaborated a elastic + breakup theory for the autoionization lines in atoms.
Our aim in this paper is to further elucidate the physics of the CCC in the context of reactions involving weakly bound nuclei at near-barrier energies. For this purpose, we investigate the effects of the CCC on the cross sections for CF, NCBU and elastic scattering. We find that the CCC lowers the CF and the NCBU cross sections, but enhances the elastic cross section at large angles. The calculated polarization potential clearly indicates a repulsive real part and a reduced imaginary part.
In the CDCC method [@SYK83; @SYK86; @AIK87], the continuum states of the dissociated projectile are approximated by a discrete set of wave-packets. In this way, the coupled-channel problem in the continuum can be handled analogously to the ones containing only bound channels. The scattering wave function is expanded in components with well defined values of total angular momentum and its z-projection. The full Schrödinger equation is then projected on each intrinsic state and one gets a set of differential equations. The difference between the CDCC and a coupled-channel problem restricted to bound channels is that the configuration space of the former is much larger. The computational problem is then considerably more complex. The calculation is greatly simplified if it takes into account only the couplings among the bound channels and the couplings between one bound channel and one channel in the continuum. In this way, continuum-continuum couplings are left out. This procedure was adopted in the first CDCC calculation of the CF cross section for the $^{11}$Be + $^{208}$Pb system, performed by Hagino [*et al.*]{} [@HVD00]. The importance of CCC was investigated in a subsequent CDCC calculation for the same system, performed by Diaz-Torres and Thompson [@DiT02]. In this work, the CF cross sections evaluated with and without CCC were compared.
![(Color on line) CF cross sections in the $^{11}$Be + $^{208}$Pb collision. The solid and the dashed lines correspond respectively to CDCC calculations with and without CCC [@DiT02].[]{data-label="fusion"}](Fig1.pdf){width="8"}
The results are shown in Fig. \[fusion\]. The comparison indicates that CCC leads to a drastic reduction of the CF cross section at near- and sub-barrier energies. The reduction is of about two orders of magnitude.
![(Color on line) Effects of the real and imaginary parts of the optical and the polarization potential on the incident current.[]{data-label="schematic"}](Fig2.pdf){width="7"}
Is there a simple and intuitive explanation for this result? To answer this question we use the language of polarization potentials. The elastic wave functions obtained from a set of coupled channel equations can always be obtained from a single-channel equation with an effective potential. This potential is the sum of the optical potential and polarization potentials. The former represents the diagonal part of the interaction in channel space and an average influence of channel coupling. The latter contains the detailed influence of the strongly coupled excited channels. In this way, the coupled channel problem can be handled as a problem of potential scattering. Following the approach of Ref. [@CCD02], we use this picture and resort to the schematic representation of Fig. \[schematic\]. It shows the currents and the potentials (real and imaginary parts) involved in the collision, for some particular partial wave. The fusion barrier is the sum of the real parts of the optical and polarization potentials, plus the centrifugal term. As the incident current, $j_{\rm in}$, approaches the external turning point, it is attenuated by the long range absorptive potential $W_{\rm pol}$. The lost flux populates the channels that are responsible for this imaginary potential, that is, inelastic channels, transfer and breakup. At large distances, $W_{\rm pol}$ is dominated by Coulomb breakup. The final destination of the fragments produced by the breakup process, namely NCBU, ICF is not relevant for our discussion. The situation would be different if all the fragments were absorbed sequentially, leading to CF. Since the contribution of this process to the CF cross section is not supposed to be large, it is neglected here. When the attenuated incident current reaches the barrier, it splits into two parts. The reflected component, $j_{\rm sc}$, and the transmitted current, $j_{\rm CF}$. The reflected current is attenuated as it moves away from the barrier, until it is out of the reach of $W_{\rm pol}$. It is then responsible for the elastic scattering cross section. The transmitted current is fully absorbed by the short-range imaginary part of the optical potential inside the barrier, giving rise to CF. The probabilities of elastic scattering, $P_{\rm sc}$, and fusion $P_{\rm CF}$, at that partial wave then given by are $$P_{\rm sc}= \frac{j_{\rm sc}}{j_{\rm in}}
\qquad\qquad {\rm and} \qquad\qquad
P_{\rm CF}= \frac{j_{\rm CF}}{j_{\rm in}}. \label{Psc-PF}$$ The direct reaction probability, representing inelastic scattering + transfer + ICF + NCBU, is given by the current absorbed by $W_{\rm pol}$, $$P_{\rm DR}= 1-\frac{j_{\rm sc} + j_{\rm CF}}{j_{\rm in}}.
\label{Pqe}$$ We can now speculate on the modifications of the polarization potential arising from the inclusion of CCC in the CDCC calculations. The strong suppression observed in Fig. \[fusion\] implies that the transmitted current is drastically reduced. In principle, it could be caused by three factors:
1. the inclusion of CCC strengthens the absorptive imaginary potential $W_{\rm pol}$. In this case the quasi-elastic cross section becomes larger, due to the increase of breakup.
2. the inclusion of CCC makes the real part of the polarization potential more repulsive, so that the incident current has to cross a higher barrier to produce fusion. If this case, $j_{\rm CF}$ is reduced and $j_{\rm sc}$ increases. Therefore, the suppression CF should be followed by an enhancement of the elastic scattering cross section.
3. a combination of possibilities 1 and 2. In this case, both the breakup and the elastic cross sections could be enhanced.
![(Color on line) Angular distributions for different processes in the $^8$B + $^{58}$Ni collision, calculated with (solid lines) and without CCC (dashed lines). In panels (a) and (b) we show, respectively, results for breakup [@LuN07] and elastic scattering [@LCA09]. For details, see the text.[]{data-label="breakup"}](Fig3.pdf){width="9"}
To find out which of these possibilities is actually happening, one should check the cross section for other channels. We first consider possibility 1. In this case, the reduction of $\sigma_{\rm CF}$ would arise from some kind of [*irreversibility*]{} of the transition to the continuum. Of course, there is no irreversibility in quantum mechanics, as any transition can take place in two directions. However the elastic transition matrix is a superposition of a direct process with multistep processes of higher orders. To evaluate it, one should then perform sums over intermediate states. When CCC is taken into account, some of these sums become integrals. If these contributions have random phase, one could have destructive interference. Although this is a plausible hypothesis, there is no convincing arguments supporting it.
To settle the matter, we first look at results of CDCC calculations for the NCBU cross section, performed by Lubian and Nunes [@LuN07]. In panel (a) of Fig. \[breakup\], we show angular distributions of the center of mass of the $^8$B projectile in its breakup in the $^8$B + $^{58}$Ni collision. The solid and the dashed lines correspond respectively to results of calculations with and without CCC. These curves are similar to the ones obtained in Ref. [@NuT99] for a different collision energy. The difference is that we only show the curves involved in our discussion, leaving out other details of their calculations. Comparing calculations with and without CCC, we conclude that CCC leads to a substantial suppression of the breakup cross section. Therefore, possibilities 1 and 3 can be ruled out.
We are then inclined to believe that the reduction of the CF cross section arises from an increase of the height of the potential barrier, when CCC is taken into account. This can be checked in an investigation of the elastic cross section. Such CDCC investigation of the elastic angular distributions, which also included the elastic breakup cross scetions, was in fact performed in the past. In particular Sakuragi [*et al.*]{} [@SYK83] went at length in calculating these observables for the systems $^6$Li + $^{28}$Si and $^6$Li + $^{40}$Ca at two laboratory energies of $^6$Li: 99 MeV and 155 MeV. In the following we present our results for the elastic angular distribution of the proton halo nucleus $^8$B on $^{56}$Ni target at a much lower laboratory energy of 23.77 MeV, corresponding to $E_{\rm c.m.}$ = 20.8 MeV. The results are shown in panel (b) of Fig. 3. The calculations are the same as in Ref. [@LCA09]. In this case the excitations of the target were not included explicitly. We do not show results for other collision energies because they are qualitatively similar. The figure shows clearly that the inclusion of CCC leads to a very strong enhancement of the cross section, as compared to the results without CCC. This confirms that possibility 2 is the mechanism responsible for the suppression of complete fusion. That is, the main effect of CCC is making the polarization potential repulsive.
![(color on line) Real (left panel) and imaginary (right panel) parts of the polarization potential for the $^8$B + $^{58}$Ni system. Potentials based on CDCC calculations including and not including CCC are represented respectively by solid and dashed lines.[]{data-label="Upol"}](Fig4.pdf){width="8"}
\
We now confront the conclusions of the previous section with the results of a direct analysis of the polarization potential. The calculation of an exact polarization potential presents some serious difficulties. This potential depends strongly on the angular momentum and has poles. However, there are approximate angular momentum independent polarization potentials which are free of poles and lead to reasonable predictions for cross sections. We derive here the approximate polarization potential following the prescription of Thompson [*et al.*]{} [@TNL89]. According to this prescription, the polarization potential is written as an average over angular momentum, involving radial wave functions and S-matrix obtained from a coupled channel (in our case CDCC) calculation. We obtained polarization potentials based on CDCC calculations with and without CCC. The strengths of the real and imaginary parts of the polarization potential evaluated at the barrier radius are shown in Fig. \[Upol\], for the $^8$B + $^{58}$Ni system. First, we note that the imaginary part of the polarization potential is alway negative, both in the calculations with and without CCC. This is not surprising since it represents the flux lost to the inelastic and breakup channels. The second relevant point is that the inclusion of CCC leads to a weaker imaginary potential, reducing the absorption associated with direct reactions. This is consistent with the reduction of the NCBU cross section found directly in our CDCC calculation with CCC. We now turn to the real part of the polarization potential. Again the effect of CCC on the polarization potential confirms our previous findings. In the absence of CCC, $V_{\rm pol}$ is attractive, reducing the barrier of the optical potential. The inclusion of CCC modifies $V_{\rm pol}$ qualitatively. It becomes repulsive. In this way, the fusion barrier becomes higher and the CF cross section lower. The higher barrier increases reflection and enhances the elastic elastic scattering cross section, as discussed in a previous paragraph.
Our conclusions above are in complete qualitative agreement with those of Ref. [@SYK83], where the dynamic polarization potential was also fully investigated within CDCC taken into account the reorientation part of the the continuum-continuum coupling effects. These authors calculated the DPP for each orbital angular momentum and summed all the contribution (Eqs. (7.4) and (7.5) of Ref. [@SYK83]). They obtained for the total, $l$-summed DPP for $^6$Li + $^{28}$Si at $E_{{\rm Lab.}} = 99$ MeV a rather strong repulsive real part and a small imaginary part in the surface region. This is in agreement with our results described above.
Concluding, in this paper we investigated the role of the continuum-continuum coupling in CDCC calculation of low-energy observables in heavy-ion reactions involving weakly bound nuclei. We have found that this coupling reduces the value of the non-elastic cross sections, which includes fusion, non-capture breakup, etc., in full agreement with previous works [@DiT02; @NuT99], while it increases the purely elastic scattering cross section ratio to Rutherford at back angles. We have traced this effect to the rather peculiar behaviour of the breakup dynamic polarization potential which we found to have a repulsive real part and a weaker absorptive part, when the CCC is included. Without the CCC, the real part was found to be attractive with a stronger absorption. The latter case is a common feature of coupling to inelastic channels, which leads us to conclude that a discretized continuum can only be a loyal representative of a breakup channel if the continuum-continuum coupling is fully accounted for in a CDCC calculation.
**Acknowledgements** We thank Alexis Diaz-Torres for supplying part of figure 1. This work was supported in part by the FAPERJ, CNPq, FAPESP and the PRONEX.
[16]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, , , , , ****, ().
, ****, ().
, , , , ****, ().
Y. Sakuragi, M. Yahiro and M. Kamimura, Prog. Theo. Phys. [**70**]{}, 1047 (1983).
, , , ****, ().
, , , , , , ****, ().
, ****, ().
, , , , , , , , , , , ****, ().
, , , , , , , , , , , ****, ().
, ****, ().
, , , , , , , ****, ().
, , , , , ****, ().
, ****, ().
, , , , ****, ().
, , , , , , ****, ().
, ****, ().
, , , , ****, ().
|
---
abstract: 'We are working to develop automated intelligent agents, which can act and react as learning machines with minimal human intervention. To accomplish this, an intelligent agent is viewed as a question-asking machine, which is designed by coupling the processes of inference and inquiry to form a model-based learning unit. In order to select maximally-informative queries, the intelligent agent needs to be able to compute the relevance of a question. This is accomplished by employing the inquiry calculus, which is dual to the probability calculus, and extends information theory by explicitly requiring context. Here, we consider the interaction between two question-asking intelligent agents, and note that there is a potential information redundancy with respect to the two questions that the agents may choose to pose. We show that the information redundancy is minimized by maximizing the joint entropy of the questions, which simultaneously maximizes the relevance of each question while minimizing the mutual information between them. Maximum joint entropy is therefore an important principle of information-based collaboration, which enables intelligent agents to efficiently learn together.'
author:
- 'N. K. Malakar'
- 'K. H. Knuth'
- 'D. J. Lary'
bibliography:
- 'nkm-me11.bib'
title: 'Maximum Joint Entropy and Information-Based Collaboration of Automated Learning Machines'
---
[ address=[Department of Physics, University of Texas at Dallas]{} ]{}
[ address=[Department of Physics, University at Albany (SUNY)]{} ,altaddress = [Department of Informatics, University at Albany (SUNY)]{} ]{}
[ address=[Department of Physics, University of Texas at Dallas]{} ]{}
INTRODUCTION {#introduction .unnumbered}
============
Present day scientific explorations involve gathering data at an ever-increasing rate, thereby requiring autonomy as a vital part of exploration. For example, remote science operations require automated systems that can both act and react with minimal human intervention. Our vision is to construct an autonomous intelligent instrument system (AIIS) that collects data in an automated fashion, learns from that data, and then, based on the learning goal, decides which new measurements to take. Such a system would constitute a learning machine that could act and react with minimal human intervention. This is made possible by the comprehensive successes of Bayesian inference, the decision theoretic approach to experimental design [@fedorov1972theory; @lindley_measure_1956; @bernardo_expected_1979; @Loredo03bayesianadaptive], and the development of the inquiry calculus [@cox_inference_1979; @fry_cybernetics; @knuth_what_2003; @knuth_valuations_2006]. Our efforts to construct such autonomous systems [@knuth_arm_2007; @knuth_center2010; @Malakar2011] have considered the process of data collection and the process of learning in two distinct phases: the inquiry phase and inference phase. By coupling these processes of inference and inquiry one can form a model-based learning unit that cyclically collects data and learns from that data by updating its models. At this stage, the inference phase, which is based on Bayesian probability theory, is sufficiently well-understood so that our current focus is on inquiry. For this reason, we tend to view the AIIS as a question-asking machine.
In this paper, we build upon our previous work [@knuth_arm_2007; @knuth_center2010] and consider the problem of coordinating two question-asking intelligent agents. Without coordination, after each agent has independently solved the presented problem, there will be a redundancy in the information obtained by the two agents. In addition, during the question-asking process, there is a great potential for redundancy in terms of the questions that they pose. We show that collectively this redundancy is minimized at each step of the question-asking process by maximizing the joint entropy of the two questions that the agents plan to ask. This has the tendency to simultaneously maximize the relevance of each of the two questions posed while minimizing the mutual information between them. We illustrate the process via simulation and show that maximization of the joint entropy is an important principle of information-based collaboration, which enables intelligent agents to efficiently learn together.
THE INQUIRY CALCULUS {#the-inquiry-calculus .unnumbered}
====================
In this section, we briefly review the inquiry calculus. The development of this calculus relies on several order-theoretic notions, which are more thoroughly discussed in papers outlining the theoretical development [@knuth_what_2003; @knuth_valuations_2006]. Central to this development is the concept of a partially ordered set, which is a set of elements in conjunction with a binary ordering relation. Related, is a special case of a partially ordered set called a lattice, which is endowed with a pair of operations called the join and the meet so that the lattice can be thought of as an algebra where the join and meet are algebraic operators. Here we will consider elements that can be described in terms of sets. So that the main concepts can be described in terms of subsets ordered by subset inclusion, the set union (join) and the set intersection (meet).
We consider three spaces: the state space, the hypothesis space, and the inquiry space.
The state space describes the possible states of the system itself. In the situations we will consider, the elements of the state space are mutually exclusive so that in terms of a partially ordered set, they can be represented as an antichain.
The hypothesis space describes what can be known about a system. Its elements are sets of potential states of the system. As such, it is a *Boolean lattice* (or a Boolean algebra) constructed by taking the power set of the set of states and ordering them according to set inclusion. In this space, the logical OR operation is implemented by set union (join) and the logical AND operation by set intersection (meet). Logical deduction is straightforward in this framework, since implication is implemented by subset inclusion so that a statement in the lattice implies every statement that includes it in terms of subset inclusion. General logical induction is implemented by quantifying degree to which one statement implies another with a real-valued bi-valuation, probability, which quantifies the degree to which one statement implies another.
The inquiry space describes what can be asked about a system. Its elements are sets of statements, which are called questions, such that if a set contains a given statement, then it also contains all the statements that imply it. In this sense, a question can be thought of as a set of potential statements that can be made. It is constructed by taking down-sets of statements and ordering them by set inclusion resulting in a *free distributive lattice*. Just as some statements imply other statements, some questions answer other questions. Specifically, if question $A$ is a subset of question $B$, $A \subset B$, then by answering question $A$, we will have necessarily answered question $B$.
Questions, which include all atomic statements as potential answers, are assured to be answerable by a true statement. Cox had termed such questions as real questions [@cox_inference_1979]. If one considers the sub-space formed from the real questions, the minimal real question is defined as the central issue, $$I = \bigcup_{i=1}^{n} X_i,$$ where $X_i = \{\{x_i\}\}$ and $x_i$ is the statement ‘*The system is in state $x_i$*’. The central issue can then be expressed as the question ‘*Is the system in state $x_1$ or in state $x_2$ ... or in state $x_n$?*’ Since it is the minimal real question, answering the central issue will necessarily answer all other real questions.
In practice, however, we cannot always pose the central issue directly. A special class of real questions are the *partition questions*, which partition the set of answers. For example, given a set of atomic statements indexed by integers $1$ through $n$, we can partition this set in $p(n)$ ways, where $p(n)$ can be defined in terms of a generating function $$\sum_{n=0}^{\infty}{p(n)x^n} = \prod_{k=1}^{\infty}\Big( \frac{1}{1-x^k} \Big),$$ which blows up rapidly. For example, for three atomic statements we have the set $\{1, 2, 3\}$ that can be partitioned as $\{1\}\{2\}\{3\}$ which results in the central issue $$I = X_1 \cup X_2 \cup X_3.$$ Another possible partitioning is $\{1\}\{2,3\}$, which represents the binary question ‘*Is the system in state $x_1$ or not $x_1$?*’ denoted $$\label{Eqn:P1|23}
P_{1|23} = X_1 \cup X_2X_3,$$ where $X_1 = \{x_1\}$ and $X_2X_3 = \{x_2, x_3, x_2 \vee x_3\}$. In this way by answering $x_2$, $x_3$, or $x_2 \vee x_3 = \neg x_1$ one has provided the information that the system is not in state $x_1$. Other partition questions are written similarly.
Valuations are handled in a way that is analogous to probability in the lattice of statements comprising the hypothesis space. However, due to multiple competing constraints, a bi-valuation can only be consistently assigned to the partition sublattice of the real questions. This bi-valuation is called the relevance, and is denoted $d(Q|P)$, which is read as ‘the degree to which $P$ answers $Q$’ [@knuth_valuations_2006]. In the special case where $P \subseteq Q$, we have that the relevance is maximal, which enables one to choose a grade so that $d(Q|P) = 1$. Otherwise, the relevance takes on a value between $0$ and $1$.
The relevance of a partition question depends on the probability of its particular partition of answers. One can show that the relevance of the question $P$ with respect to the central issue is given by the entropy of that partition of probabilities [@knuth_valuations_2006]. In the case of the partition question $P_{1|23}$ described in (\[Eqn:P1|23\]) we have that $$d(I|P_{1|23}) \propto H(p_1, p_2+p_3)$$ where $H$ is the Shannon entropy, and $p_i = Pr(x_i|\top)$ which is the probability that the system is in state $x_i$. The proportionality constant is the inverse of the relevance $$d(I|I) \propto H(p_1, p_2, p_3)$$ so that $$d(I|P_{1|23}) = \frac{H(p_1, p_2+p_3)}{H(p_1, p_2, p_3)}$$ and $$d(I|I) = \frac{H(p_1, p_2, p_3)}{H(p_1, p_2, p_3)} = 1.$$
AUTOMATED EXPERIMENTAL DESIGN {#automated-experimental-design .unnumbered}
=============================
Previously, we demonstrated a robotic arm, built with the LEGO MINDSTORMS NXT system, capable of autonomously locating and characterizing a white circle on a dark background [@knuth_arm_2007; @knuth_center2010]. Here we aim to extend this problem by introducing two robots that work in a collaborative effort to solve the same problem.
Computing with Questions {#computing-with-questions .unnumbered}
------------------------
The white circle is characterized by three unknown parameters $\{x_o, y_o, r_o\}$. We are interested in asking questions about the center position of the circle $\{x_o, y_o\}$ as well as its radius $r_o$ by taking light intensity measurements centered at locations determined by the inquiry system. Model-based descriptions enable one to make predictions about the outcomes of potential experiments. Given the joint posterior probability of the circle location and radius, one can determine the probability that a given intensity measurement at a position $(x_i, y_i)$, will result in a “white" or “black" intensity reading. This is easily done with sampling by maintaining a set of sampled circles and noting how many circles contain the proposed measurement location and would result in a white intensity reading, and how many circles do not contain the measurement circle resulting in a black intensity reading. Such predictions can be made more precise by modeling the spatial sensitivity of the light sensor and computing the predicted numerical result of the sensor given the measurement location and the hypothesized characteristics of the white circle [@malakar_SSF2009].
Furthermore, the entropy associated with such a measurement can be computed as the entropy of the probability distribution of predicted measurement intensities. This can be rapidly computed by generating a set of predicted measurements from the set of circles sampled from the posterior. By generating a histogram of this set of predicted intensities, one has a model of the density function of predicted measurements. The entropy of this histogram is computed and serves as an excellent estimate of the entropy associated with the question posed by recording the intensity at a particular measurement location. By computing the entropy associated with a large set of measurement locations, one can create an entropy map based on the sampled circles and the known characteristics of the light sensor. For increased speed, we also have developed an entropy-based search algorithm to intelligently search the entropy space without computing it everywhere [@Malakar2010].
We begin by encoding the questions one might ask in terms of sets of circle parameters. The central issue considers all possible circle parameter values, and in doing so asks the question “Precisely where is the circle?” In practice, this is a finite set since one can only measure to finite precision, and in the simulations we force it to be finite by considering a discrete grid of possible circle center positions and radii. The central issue $I$ can be written as $$\label{circleQ}
I = \left\{ \left\{ x_1,y_1,r_1\right\} ,\left\{ x_2,y_2,r_2\right\} ,...\right\},$$ where each element of the set, such as $\{x_i, y_i, r_i\}$, represents a potential precise answer to the question. One way to solve this problem is to simply ask all of the binary questions ‘Is the circle in state $\{x_i, y_i, r_i\}$?’ However, this is not very efficient. Moreover, faced with measurement uncertainties, we do not know the exact answer to Eq. (\[circleQ\]), as we cannot measure the exact values of the parameters of interest.
Since we cannot directly perform a single, or even a small number, of measurements that directly answer the central issue. Instead, we must identify measurements that can be performed that are maximally relevant to the central issue. This involves finding measurement locations that have the maximum entropy as computed from the posterior probability of the circle states.
We note that any given measurement location $(x_{e1}, y_{e1})$ divides the space of circles into two regions: the set of circles that contain the measurement location, and the set of circles that do not contain the measurement location $${Q}(x,y,r) = \left\{
\bigcup_{
\substack{
(x_{e1},y_{e1}) \in circle
}}\left\{ x_i,y_i,r_i\right\} ,
\bigcup_{
\substack{
(x_{e1},y_{e1})\notin circle
}}\left\{ x_j,y_j,r_j\right\}
\right\}.$$ Similarly, a second robot choosing a different measurement location, $(x_{e2}, y_{e2})$, partitions the question space differently into two sets defining a different binary partition.
Jointly, the two distinct measurement locations partition the space of circles into four regions, say $a$, $b$, $c$ and $d$: $$a = \{white,~white\}, ~b = \{white,~black\}, ~c = \{black,~white\}, ~d=\{black,~black\} ,$$ where, for example, $\{white, black\}$ refers to the set of circles that contain $(x_{e1},y_{e1})$ so that a measurement there will be predicted to result in a white intensity, but do not contain $(x_{e2},y_{e2})$ so that a measurement there will be predicted to result in a black intensity. The circles where the first robot measures white belong to the set $a \cup b$. We can then define the elementary questions as $$\begin{aligned}
AB &= \{ a\lor b, a ,b\},
& CD = \{ c\lor d, c, d\},\nonumber \\
AC &= \{ a \lor c, a, c\},
& BD = \{ b\lor d, b, d \}.\end{aligned}$$ and write the question that the first robot poses as $$AB \cup CD = \{ a\lor b, c\lor d, a, b, c, d\} \equiv \{ \{w, \bullet \}, \{b, \bullet \} \} ,$$ and the question the second robot poses as $$AC \cup BD = \{ a\lor c, b\lor d, a, b, c, d\} \equiv \{ \{\bullet, w\}, \{ \bullet, b \} \} ,$$ where the expressions on the right illustrate what the robots are measuring with $\bullet$ signifying either black or white.
Jointly the robots partition the space into four sets,$$(AB \cup CD) \cap (AC \cup BD) = (A \cup B \cup C \cup D).$$ Therefore, the relevance of the joint question, with respect to the central issue, is given by the joint entropy of the predictions of the two measurements $E1 = (AB \cup CD)$ and $E2 = (AC \cup BD)$ [@Malakar2011] $$d(I| E1 \cap E2) = d(I | A \lor B \lor C \lor D)
= H ( Pr(A), Pr(B), Pr(C), Pr(D) ),$$ where, given that robot 1 measures at $(x_{e1}, y_{e1})$ and robot 2 measures at $(x_{e2},y_{e2})$, $Pr(A)$ denotes the probability that both the first and the second robot’s measurement locations result in a white intensity, $Pr(B)$ denotes the probability that the first measurement results in white and the second in black, $Pr (C)$ denotes the probability that the first measurement results in black and the second in white, and $Pr(D)$ denotes the probability that both the first and the second measurements result in black. Considered jointly, the predicted measurement results associated with the pair of measurement locations constitute a two-dimensional distribution at each point in the four-dimensional space of pairs of measurement locations. The relevance $d(I|E1\cap E2)$ dictates that we select measurement locations that maximize the joint entropy of the intensities predicted to be measured by the two robots.
RESULTS {#results .unnumbered}
=======
In the present case of model based exploration, given a hypothesized circle location and radius, the intensity to be measured at any point in the field can be predicted. By considering 45 posterior samples, we made predictions about the intensities which gave a distribution of 45 predicted intensities. The entropy associated each possible measurement location was computed by estimating the entropy of the histogram of predicted intensities at that position in the field. This enables us to produce an entropy map for a single proposed measurement. Joint entropy maps would require four dimensions to display. Instead, we plot the joint entropy of the two measurements for the case where the first experiment E1 is determined. This map then represents a two-dimensional slice through the four-dimensional space of pairs of measurements. The mutual information maps (not shown) can be made similarly.
![Figures illustrating two set of examples where we implement the information-based collaboration for experimental design in the problem where two robots are to characterize a circle using light sensors. Figures (a) and (b) show the cases where circles are highly correlated, whereas figures (c) and (d) show the cases where circles are less correlated. In both cases, we have drawn a set circles from the posterior samples and used these circles to make predictions about the expected measured light intensity at each point. The top figures (a and c) show the entropy map, which illustrates the optimal measurement location in the case where only one measurement is to be taken. The botom figures (b and d) illustrate the joint entropy map of measurement location $E2$ shares with measurement location $E1$ fixed. Note that the selected location of meaurement $E2$ maximizes the joint entropy, which involves finding an informative measurement location that does not provide information redundant to $E1$.[]{data-label="Figureall"}](twoCircles2b){width="100.00000%"}
Figure 1 shows two examples, where we considered different degrees of overlap of the sample circles. The figures on the left column represent the case with more correlated circles than those on the right. Each of the figures show a set of circles drawn from the posterior probability. Overlaid on this are the entropy maps (a and c), and the joint entropy maps (b and d).
The entropy maps in (Figures 1a and 1c) show the measurement location that would be selected in the event that only one measurement was being performed. The joint entropy maps (Figure 1b and 1d) show the locations of the second measurement $E2$ that maximize the relevance of the question $d(I|E1 \cap E2)$ given that the location of $E1$ has been selected. By comparing the locations of $E2$ in the joint entropy map with the corresponding values of entropy and mutual information, one can see that the selected measurement locations for $E2$ favor regions of high entropy while avoiding locations that share mutual information with $E1$. Maximizing the joint relevance naturally chooses informative measurement locations that promise to provide independent information. The two measurement locations $E1$ and $E2$ that maximize the relevance of the joint question $d(I|E1 \cap E2)$ are indicated by arrows.
CONCLUSIONS AND FUTURE APPLICATIONS {#conclusions-and-future-applications .unnumbered}
===================================
In this paper, we have presented the method of information-based collaboration for Automated Intelligent Instruments System (AIIS). We have considered the intelligent agent as a question-asking machine and have focused on the inquiry phase, where our aim has been to select maximally informative queries with respect to a given goal. We have extended the order-theoretic approach [@knuth_what_2003; @knuth_valuations_2006] to assign the relevance of questions for collaborative AIIS. We have shown that the joint entropy gives the relevance of the joint question posed by the agents. Maximum joint entropy is an important principle of information-based collaboration, which enables intelligent agents to efficiently learn together.
Currently our team in UTD is working on to develop a fleet of aircrafts to deploy in the field using the technique of collaboration developed in this paper. The aircraft fleet consists of helicopters as well as the fixed wing small aircrafts. We aim to use the fleet to characterize and help predict tornado forecasts, assist with the gas leak detection, and monitor the health of cattle. The work is in progress.
|
22.5cm-0.2 in 15.5cm
[**Global geometric properties of AdS space\
and the AdS/CFT correspondence**]{}
Qi-Keng LU
[*Institute of Mathematics\
Academy of Mathematics and Systems Sciences, Academia Sinica\
Beijing 100080, China[^1]\
*]{}
Zhe CHANG
[*Institute of High Energy Physics, Academia Sinica\
P.O.Box 918(4), Beijing 100039, China[^2]\
*]{}
Han-Ying GUO
[*Institute of Theoretical Physics, Academia Sinica\
P.O.Box 2735, Beijing 100080, China[^3]*]{}
[**Abstract**]{} $~~~~$ The Poisson kernels and relations between them for a massive scalar field in a unit ball $B^{n}$ with Hua’s metric and conformal flat metric are obtained by describing the $B^{n}$ as a submanifold of an $(n+1)$-dimensional embedding space. Global geometric properties of the AdS space are discussed. We show that the $(n+1)$-dimensional AdS space AdS$_{n+1}$ is isomorphic to $RP^1\times B^n$ and boundary of the AdS is isomorphic to $RP^1\times S^{n-1}$. Bulk-boundary propagator and the AdS/CFT like correspondence are demonstrated based on these global geometric properties of the $RP^1\times B^n$.\
\
[**Keywords: Poisson kernel, AdS/CFT correspondence, Bulk-boundary\
$~~~~~~~~~~~~~~~~~$propagator**]{}
Introduction
============
In Riemann geometry, it is known that the biggest symmetry(isometric) groups of the Minkowski space, de Sitter Space (dS) and Anti-de Sitter (AdS) space have same number of generators. They can be identified, in a unified way, as the classical manifold $D_\lambda(n+1)$ with $\lambda=0,~+1$ and $-1$, respectively. Thus, the dS and the AdS space are the simplest generalization of the Minkowski space with constant curvature. The dS, in particularly, the AdS space and quantum field theory based on it has been an interested topic of mathematicians and physicists for a long time[@01]. There has recently been a revival of interest in AdS space brought about by the conjectured duality between physics in the bulk of AdS and a conformal field theory (CFT) on the boundary[@02]–[@04]. The so-called AdS/CFT correspondence states that string theory in the AdS space is holographically dual to a CFT on boundary of the AdS[@05]. A strong support for the proposal comes from comparing spectra of Type IIB string theory on the background of AdS$_5\times S^5$ and low-order correlation functions of the $3+1$ dimensional ${\cal
N}=4$ $SU(N)$ super Yang-Mills theory. The dual super Yang-Mills theory lives on the boundary of the AdS space. This is one of the most important progresses in the superstring theory. Many new results have been obtained by making use this conjecture[@06].
However, up to now, almost all discussions on the AdS/CFT correspondence were based on the so-called Euclidean version of the AdS, or an $(n+1)$-dimensional unit ball $B^{n+1}$. To prove the AdS/CFT conjecture or to investigate its delicious implications in physics theory, one should work on the more challenging topic of duality between physics theories on the AdS space and its boundary, the compact Minkowski space.
In this paper, by describing the unit ball $B^{n}$ as a submanifold of an $(n+1)$-dimensional embedding space, we first present Poisson kernels for the Laplace operator with a general nonzero eigenvalue $m_0$ and relations between them in a unit ball $B^{n}$ with Hua’s metric and conformal flat metric. Results for Euclidean version of the AdS are recovered in a different view. Then we discuss the AdS geometry with right signature within the framework of the classical manifolds and the classical domains[@07; @08]. It is proved that the AdS$_{n+1}$ is isomorphic to $RP^1\times B^n$ and the boundary of the AdS is isomorphic to $RP^1\times
S^{n-1}$. Bulk-boundary propagator for a massive scalar field in the AdS is replaced by what in the $RP^1\times B^n$ based on the global geometric properties of AdS. The bulk/boundary correspondence in this case is also demonstrated.
Poisson kernels for a scalar field in unit ball
===============================================
A unit ball $B^{n}$ can be described as the image of a two-to-one map of the hypersurface
$$\xi^{n}\xi^{n}-\sum_{i=0}^{n-1}\xi^i\xi^i=1$$
in the space $(\xi^0, \xi^1,\cdots, \xi^{n-1})$.
An explicitly $SO(1,n)$ invariant metric of the $B^n$ can be introduced $$\label{metric}
ds^2=d\xi^n d\xi^n- \sum_{i=0}^{n-1}d\xi^id\xi^i~.$$ By denoting $$x^i=\frac{\xi^i}{\xi^{n}}~,~~~~\left(i=0,~1,~\cdots,~n-1~,~~~\xi^{n}\not= 0
\right)~,$$ $n$-dimensional vector $x\equiv (x^0,x^1,\cdots,x^{n-1})$, and $x'$ transfer of the $x$, we can rewrite the $B^n$ as usual $$xx'<1~.$$ The reduced metric from Eq.(\[metric\]) in the coordinate $\{x^i\}$ is of the form $$\label{dg01}
ds^2=-\frac{dx(I-x'x)^{-1}dx'}{1-xx'}~.$$
Another set of coordinates for the unit ball $B^n$ can be introduced $$z^i=\frac{\xi^i}{\xi^{n}+1}~,~~~~\left(i=0,~1,~\cdots,~n-1~;~~~\xi^{n}
\not=0\right)~.$$ In this coordinate, the unit ball $B^n$ is also described as usual $$zz'<1~.$$ The reduced metric from Eq.(\[metric\]) in terms of the coordinate $\{z^i\}$ is $$\label{dg02}
ds^2=\frac{-4}{(1-zz')^2}dz dz'~.$$ There is a one to one transformation between the two sets of coordinates $\{x^i\}$ and $\{z^i\}$ $$\label{tran}
x^i=\frac{2z^i}{1+zz'}~.$$ Therefore, the conformal flat metric (\[dg02\]) and the Hua’s metric (\[dg01\]) are different representations of the $SO(1,n)$ invariant metric. And we can work by using one of them and got same results include invariant differential form and Poisson kernel.
The equation of eigenvalues for the Laplace operator in the unit ball $B^n$ is $$\left[\frac{1}{\sqrt{-g}}\sum_{i,j=0}^n\frac{\partial}{\partial
x^i}\left(\sqrt{-g}g^{ij} \frac{\partial}{\partial
x^i}\right)+m_0^2\right]\Phi(x)=0~.$$ The Poisson kernel for a massless scalar field has been discussed[@09] $$G^E_{B\partial}(x,u)=\left\{\begin{array}{l}
\displaystyle\frac{(1-xx')^{n-1}}{(1-2ux'+
xx')^{n-1}}~,~~~~
{\rm for ~conformal ~flat ~metric~,}\\[1cm]
\displaystyle\frac{(1-xx')^{\frac{n-1}{2}}}
{(1-ux')^{n-1}}~,~~~~{\rm for ~Hua's ~metric}~.
\end{array}\right.$$ The bulk field $\Phi(x)$ determined by the fields living on the boundary $\phi(u)$ is of the form $$\Phi(x)=\frac{1}{\omega_{n-1}}\int_{uu'=1}\cdots\int G^E_{B\partial}
(x,u)\phi(u)\dot{u}~.$$ Now, we write down a general Poisson kernel for the Laplace operator with nonzero eigenvalue $m_0^2$ $$G^\pm_{B\partial}({x},u)=\left(G^E_{B\partial}({x},u)
\right)^{\frac{1}{2}\left(1\pm\sqrt{1+\frac{4m_0^2}
{(n-1)^2}}\right)}~.$$ The Poisson kernels $G^\pm_{B\partial}({x},u)$ satisfy the following properties:
- It is definitely positive.
- On the boundary, we have $$G^\pm_{B\partial}(v,u)=\left\{\begin{array}{c}
0~,~~~~~u\not=v~,\\
\infty~,~~~~~u=v~.
\end{array}\right.$$
- It satisfies the equation of eigenvalues for the Laplace operator with a eigenvalue $m_0^2$.
Conformal boundary and AdS
==========================
In an $(n+2)$-dimensional embedding space, the $(n+1)$-dimensional AdS space AdS$_{n+1}$ can be written as $$\label{qiu}
\xi^0\xi^0-\sum_{i=1}^n\xi^i\xi^i+\xi^{n+1}\xi^{n+1}=1~.$$ From the above definition of AdS, we know that $\xi^0$ and $\xi^{n+1}$ can not be zero simultaneously, and at least two charts of coordinates $[(n+1)$-dimensional\] ${\cal U}_1$ ($\xi^{n+1}\not= 0$) and ${\cal U}_0$ ($\xi^0\not= 0$) should be needed to describe the AdS.
In the chart AdS$_{n+1}\cap {\cal U}_1$, we introduce a coordinate $$x^i=\frac{\xi^i}{\xi^{n+1}}~,~~~~~~(i=0,~1,~2,~\cdots,~n;~~\xi^{n+1}\not=0)~.$$ The AdS$_{n+1}\cap {\cal U}_1$, in the coordinate $\{x^i\}$, is described by $$\begin{array}{l}
{\rm AdS}_{n+1}\cap {\cal U}_1:~~~~~~\sigma(x^i,x^j)>0~,\\[0.5cm]
\sigma(x^i,x^j)\equiv 1+\displaystyle\sum_{i,j=0}^n\eta_{ij}x^i x^j ~,~~~~~\eta={\rm diag}
(1,~\underbrace{-1,~-1,~\cdots,~-1}\limits_{n})~.
\end{array}$$ In the chart AdS$_{n+1}\cap {\cal U}_0$, let $$\begin{array}{l}
\displaystyle y^0=\frac{\xi^{n+1}}{\xi^0}~,\\[0.5cm]
\displaystyle y^i=\frac{\xi^{i}}{\xi^{0}}~,~~~~~~(i=1,~2,~\cdots,~n;~~
\xi^{0}\not=0)~.
\end{array}$$ And the AdS$_{n+1}\cap {\cal U}_0$ can be written in the form $${\rm AdS}_{n+1}\cap {\cal U}_0:~~~~~~\sigma(y^i,y^j)>0~.$$ At the overlap region AdS$_{n+1}\cap{\cal U}_0\cap {\cal U}_1$ of the two charts ${\cal U}_1$ and ${\cal U}_0$, one has relations $$\begin{array}{l}
\displaystyle y^0=\frac{1}{x^0}~,\\[0.5cm]
\displaystyle y^i=\frac{x^i}{x^0}~.
\end{array}$$ This shows clearly a differential structure[@10] of the AdS. The boundary $\overline{M}^n$ of the AdS$_{n+1}$ consists of infinite points not belong to AdS$_{n+1}\cap{\cal U}_1$ or AdS$_{n+1}\cap{\cal U}_0$, $$\begin{array}{r}
\overline{M}^n:~~1+\displaystyle\sum_{i,j=0}^n\eta_{jk}x^jx^k=0~,\\[0.5cm]
1+\displaystyle\sum_{i,j=0}^n\eta_{jk}y^jy^k=0~.
\end{array}$$
A new set of variables in the chart AdS$_{n+1}\cap {\cal U}_1$ can be introduced as $$\begin{array}{l}
\chi^0\equiv x^0~,\\
\chi^\mu\equiv\sqrt{1+(x^0)^2}x^\mu~,~~~(\mu=1,~2,~\cdots,~n).
\end{array}$$ Then, we have $${\rm AdS}_{n+1}\cap {\cal U}_1=\{\chi^0\in \Re,~(\chi^1,~\chi^2,~\cdots,~
\cdots,~\chi^n)\in \Re^n\vert(\chi^1)^2+(\chi^2)^2+\cdots+(\chi^n)^2<1\}~.$$ This shows that, in the chart ${\cal U}_1$, the AdS$_{n+1}$ is equivalent to $\Re\times B^n$.
In the same way, in the chart AdS$_{n+1}\cap{\cal U}_0$, let $$\begin{array}{l}
\eta^0\equiv y^0~,\\
\eta^\mu\equiv\sqrt{1+(y^0)^2}y^\mu~,~~~(\mu=1,~2,~\cdots,~n).
\end{array}$$ And subsequently, one has $${\rm AdS}_{n+1}\cap {\cal U}_0=\{\eta^0\in \Re,~(\eta^1,~\eta^2,~\cdots,~
\cdots,~\eta^n)\in \Re^n\vert(\eta^1)^2+(\eta^2)^2+\cdots+(\eta^n)^2<1\}~.$$ Therefore, both charts of the AdS$_{n+1}$, ${\cal U}_1$ and ${\cal U}_0$ are equivalent to $\Re\times B^n$.
It should be noticed that, at the overlap region of the two charts, there are relations among the two different sets of coordinate variables $$\begin{array}{l}
\displaystyle \chi^0=\frac{1}{\eta^0}~,\\
\chi^\mu=\eta^\mu~,~~~(\mu=1,~2,~\cdots,~n).
\end{array}$$ This fact presents a theorem for the AdS.\
[**Theorem**]{}: [*The AdS$_{n+1}$ is isomorphic to $RP^1\times B^n$ and its boundary is $RP^1\times S^{n-1}$.*]{}.
It is well-known that the Study-Fubini metric can be introduced on the $RP^1$ space $$ds_0^2=\frac{(d\chi^0)^2}{[1+(\chi^0)^2]^2}=(d\arctan\chi^0)^2~.$$ As presented at the previous section, on the unit ball $B^n$, we have the Hua’s metric $$ds_n^2=-\sum_{\mu,\nu=1}^nd\chi^\mu d\chi^\nu\left(\frac{\delta_{\mu\nu}}
{1-\displaystyle\sum_{\alpha=1}
^n\chi^\alpha\chi^\alpha}+\frac{\chi^\mu\chi^\nu}{(1-
\displaystyle\sum_{\alpha=1}^n
\chi^\alpha\chi^\alpha)^2}\right)~.$$ Thus, a natural metric on the $RP^1\times B^n$ is of the form $$\label{fubini}
\begin{array}{rcl}
ds^2&=&ds_0^2-ds_n^2\\
&=&\displaystyle\frac{(d\chi^0)^2}{[1+(\chi^0)^2]^2}-
\sum_{\mu,\nu=1}^n d\chi^\mu d\chi^\nu
\left(\frac{\delta_{\mu\nu}}{1-\displaystyle\sum_{\alpha=1}
^n\chi^\alpha\chi^\alpha}+\frac{\chi^\mu\chi^\nu}{(1-\displaystyle\sum_{\alpha=1}^n
\chi^\alpha\chi^\alpha)^2}\right)~.
\end{array}$$ But, with this metric the $RP^1\times B^n$ is no longer AdS group invariant. In what follows, we discuss the bulk/boundary correspondence and related topics in this case.
Eigenfunctions of Laplace operator on $RP^1\times B^n$
======================================================
Let
$$\theta={\rm arctan}\chi^0~.$$
Then ${ RP}^1\times{B}^n$ and ${RP}^1\times{S}^{n-1}$ are isomorphic to $S^1\times B^n$ and $S^1\times S^{n-1}$ respectively. Moreover,
$$\Box=\frac{\partial^2}{\partial\theta^2}-\Delta ~,$$
where
$$\Delta=\frac{1}{\sqrt g}\sum_{i,j=1}^{n}\frac{\partial}{\partial
}{\partial\chi^i}\Big(\sqrt g g^{ij}\frac{\partial}{\partial
\chi^j}\Big)$$
is the Laplace-Beltrami operator of the ball $B^n$.
Denote
$$\alpha_k(m_0)=1+\left[1+4\frac{m_0^2-k^2}{(n-1)^2}\right]^{\frac{1}{2}}$$
and
$$P_{\alpha_k(m_0^2)}(\chi,u)=\left[G_{B\partial}(\chi,u)\right]^
{\alpha_k(m_0)}~,$$
where $\chi=(\chi^1,\cdots,\chi^n)$. It can be proved that
$$\Delta P_{\alpha_k(m_0)}(\chi,u)=-(k^2-m_0^2)P_{\alpha_k(m_0)}(\chi,u)~.$$
Bulk-boundary propagator on $S^1\times B^n$
============================================
Let $\Phi_0(\varphi,u)$ be a field on the boundary $S^1\times S^{n-1}$ of $S^1\times B^n$. Develop it into Fourier series of $\varphi$ such that
$$\Phi_0(\varphi,u)=\sum_{k=0}^{\infty}\left[a_k(u)\cos k\varphi
+b_k(u)\sin k\varphi\right]~,$$
where $$a_k(u)=\frac{1}{2\pi}\int_0^\infty\Phi_0(\varphi,u) \cos k\varphi~d\varphi~,
~~~b_k(u)=\frac{1}{2\pi}\int_0^{2\pi}\Phi_0(\varphi,u)
\sin k\varphi~d\varphi~.$$ Construct a scalar field $\Phi(\theta,\chi)$ on $S^1\times B^n$ such that
$$\Phi(\theta,\chi)=\sum_{k=0}^{\infty}\left[\phi_k(\chi)\cos k\theta+
\psi_k(\chi)\sin k\theta\right]~,$$
where
$$\begin{array}{l}
\phi_k(\chi)=\displaystyle\frac{1}{\omega_{n-1}}\int_{uu'=1}a_k(u)P_{\alpha_k(m_0)}
(\chi,u)\dot{u}~,\\
\psi_k(\chi)=\displaystyle\frac{1}{2\pi}\int_{uu'=1}b_k(u)
P_{\alpha_k(m_0)}(\chi,u)\dot{u}~.
\end{array}$$
Since
$$\Delta\phi_k(\chi)=-(k^2-m_0^2)\phi_k(\chi)~~{\rm
and}~~\Delta\psi_k(\chi)=-(k^2-m_0^2)\psi_k(\chi)~,$$
then $\Phi(\theta,\chi)$ must satisfy the equation
$$\Box\Phi(\theta,\chi)=-m_0^2\Phi(\theta,\chi)~.$$
Finally, $\Phi(\theta,\chi)$ can be expressed into the form
$$\begin{array}{l}
\Phi(\theta,\chi)=\displaystyle\frac{1}{2\pi\omega_{n-1}}\sum_{k=0}^{\infty}
\int_{uu'=1}\left[a_k(u)\cos k\theta+b_k(u) \sin k\theta\right]
P_{\alpha_k(m_0)}(\chi,u)\dot{u}\\[0.5cm]
=\displaystyle\frac{1}{2\pi\omega_{n-1}}\sum_{k=0}^{\infty}\int_{uu'=1}
\int_{0}^{2\pi}\left[\Phi_0(\varphi,u)\cos k\varphi \cos k\theta
+\Phi_0(\varphi,u)\sin k\varphi\sin k\theta\right]P_{\alpha_k(m_0)}
(\chi,u)d\varphi \dot{u}~,
\end{array}$$
or
$$\Phi(\theta,\chi)=\frac{1}{V(S^1\times S^{n-1})}\int_{S^1\times S^{n-1}}
\Phi_0(\varphi,u)G_{m_0}(\theta,\chi;\varphi,u)d\varphi\dot{u}~,$$
where
$$G_{m_0}(\theta,\chi;\varphi,u)=\sum_{k=0}^{\infty}P_{\alpha_k(m_0)}
(\chi,u)\cos k(\varphi-\theta)$$
is the propagator.
**Acknowledgments**
Two of us (Z.C. and H.Y.G. ) would like to thank S. K. Wang and K. Wu for enlightening discussions. The work was supported in part by the National Science Foundation of China.
[99]{} See, for example, P. A. M. Dirac, “The electronic wave equation in de-Sitter space”, Ann. Math. [**36**]{} (1935) 657;\
C. Fronsdal, “Elementary particles in a curved space”, Rev. Mod. Phys. [**37**]{} (1965) 221;\
Q. K. Lu, Z. L. Zou and H. Y. Guo, “The kinematic effects in the classical domains and the red-shift phenomena of extra-galactic objects, Acta Physica Sinica (In Chinese) [**23**]{} (1974) 225. J. Maldacena, “The large $N$ limit of superconformal field theories and supergravity”, Adv. Theor. Math. Phys. [**2**]{} (1998) 231, hep-th/9711200. S. Gubser, I. Klebanov and A. Polyakov, “Gauge theory correlators from non-critical string theory”, Phys. Lett. B[**428**]{} (1998) 105, hep-th/9802109. E. Witten, “Anti de Sitter space and holography”, Adv. Theor. Math. Phys. [**2**]{} (1998) 253, hep-th/9802150. Z. Chang, “Holography based on noncommutative geometry and the AdS/CFT correspondence”, Phys. Rev. D[**61**]{} (2000) 044009, hep-th/9904101. For a review, see, O. Aharony, S. Gubser, J. Maldacena, H. Ooguri and Y. Oz, “Large $N$ field theories, string theory and gravity”, Physics Reports [**323**]{} (2000) 183, hep-th/9905111. L. K. Hua, “Harmonic analysis for analytic functions of several complex variables on classical domains” (in Chinese), Scientific Press (Beijing) (1958). Q. K. Lu, “Classical manifolds and classical domains” (in Chinese), Sci.& Tech. Press (Shanghai) (1963). Z. Chang and H. Y. Guo, “Symmetry realization, Poisson kernel and the AdS/CFT correspondence”, Mod. Phys. Lett. A[**15**]{} (2000) 407, hep-th/9910136. Q. K. Lu, “Differential geometry and its applications in physics” (in Chinese), Scientific Press (Beijing) (1982). G. Y. Li and H. Y. Guo, “Relativistic quantum mechanics for scalar particles and spinor particles in the Beltrami-de Sitter space” (in Chinese), Acta Physica Sinica [**31**]{} (1982) 1501.
[^1]: Email: luqik@public.bta.net.cn.
[^2]: Email: changz@alpha02.ihep.ac.cn.
[^3]: Email: hyguo@itp.ac.cn.
|
The Magellanic System (MS) provides an excellent astrophysical laboratory for studying the structure and evolution of stellar systems (Skowron 2014, Piatti 2015, Jacyszyn-Dobrzeniecka 2016). Star clusters are one of the tools for such studies. However, a complete collection of star clusters is needed to conduct such a research derived from homogeneous observational data, preferably from a single photometric survey. For more details of the scientific rationale of this research, see the Introduction in Sitek (2016, hereafter referred to as Paper I).
To date, the largest catalog of extended objects in the Magellanic System was prepared by Bica (2008) as a compilation of all the previously published catalogs. The important contribution to this sample was taken from the OGLE-II star clusters catalogs (Pietrzyñski 1998, 1999). These catalogs, however, covered only the central parts of the LMC and SMC: 5.8 and 2.5 square degrees (Udalski 1997) respectively – only 1.5–2% of the area observed toward these regions during the current OGLE-IV phase (Udalski, Szymañski and Szymañski 2015). The Magellanic Bridge has never been systematically observed at such scale before, both in terms of the area, time range and cadence. The MBR coverage in the OGLE-IV is 185 square degrees.
This paper presents the second part of the star cluster collection based on the OGLE-IV data. The central part of the SMC has already been observed or analyzed by many other projects (Piatti and Bica 2012, Piatti 2016). Thus, we decided to start our exploration from the outer parts of the galaxy. We have also analyzed the whole area of the MBR covered by the OGLE-IV survey.
The photometric data of the SMC and MBR fields analyzed in this paper are based on the images gathered during the first five years (2010–2015) of the fourth phase of the OGLE project (Udalski, Szymañski and Szymañski 2015).
We have used the “deep photometric maps” – catalogs of all objects detected on the deep images of all observed fields. For details we refer the reader to Paper I. For all 120 observed SMC fields (which were analyzed in this paper) and 132 MBR fields, the number of the stacked images used for the deep images is between 55 and 100 (86 on average), depending on the overall number of good seeing individual images in the [*I*]{}-band available for any given field. For the [*V*]{}-band, which is observed less frequently, the deep images were constructed from 2 to 100 individual images with the mean value of 13. For comparison, the reference images for 41 OGLE-III SMC fields (Udalski 2008) were constructed using 4–15 images.
{width="6cm"}{width="6cm"}
{width="6cm"}{width="6cm"}
-7mm
-.8cm
{width="9cm"}{width="9cm"}
-.9cm
The star detection limit of the deep photometric maps of the SMC and MBR reaches $I\approx23.5$ mag and $V\approx 24$ mag. The maps are complete to about 22–23 mag in the [*I*]{}-band and 22.5–23.5 mag in the [ *V*]{}-band, depending on the location and crowding of the field. These limits are determined from the histograms of the mean magnitudes of the stars by estimating the value where the numbers start to deviate from the systematic growth (Fig. 1). All the details about observations, data reductions and construction of the deep photometric maps can be found in Paper I.
The method used in this paper is well established. The first automated search of star clusters was performed by Zaritsky (1997) and has been used ever since. We used Zaritsky’s method with small modifications which are described in Paper I (Section 3). Here, we present the analysis of the fields located outside the central part of the SMC and the MBR (Fig. 2).
The examined area of 353 square degrees contains 252 OGLE-IV fields (each field has 32 subfields what gives 8064 single subfields). All analyzed fields are shown in Fig. 2 as black polygons. The 10 gray polygons mark the central SMC fields which have not been analyzed here. The list of all analyzed SMC and MBR fields and their central coordinates are available on the OGLE Web page[^1] together with other supplementary information.
Exemplary density maps are presented in Figs. 3 and 4 for both MBR and SMC subfields, respectively. As in Paper I, we constructed a false-color composition of images taken in the [*I*]{}- and [*V*]{}-bands (Fig. 5) and plotted a photometric map of the region $400\times400$ pixels ($1\zdot\arcm7\times1\zdot\arcm7$) around each star cluster candidate detected by our algorithm (see an example in Fig. 6).
Maps presented in Fig. 6 were made for the same object named OGLE-MBR-CL-0033 which is presented in Fig. 5 and shown on the density map in Fig. 3.
-1.1cm
{width="9cm"}{width="9cm"}
-1cm
{width="12.3cm"}
{width="7.5cm"}-7mm{width="7.5cm"}
For all regions found by our algorithm to be denser than the median value for a given subfield we made a visual inspection, as described in Paper I. For all objects which passed the visual inspection based on six different images, the reliability index was assigned: 34% cluster candidates received the maximum value of 1, 28% – 0.9, 17% – 0.8, 12% – 0.7 and 9% – 0.6. This index depends on the quantity of images the object was identified on. The object received the reliability index equal 1 if it was found on every image. The index was reduced by 0.1 each time the object was not found on the image from the inspected group. Object which was found only on two from six images was rejected. All pictures (images in the [*V*]{}- and [*I*]{}- bands, color images and a photometric map) of the accepted star clusters are shown on the Web page (Section 4).
All the centroids were calculated in $XY$ coordinates of the field and then converted to the equatorial coordinates. The estimation of the approximate size of star cluster candidate was made using Kernel Density Estimation (KDE) contour line at half maximum value (see an example in Fig. 6. right panel). The calculations are the same as in Paper I. Table 1 presents the cluster parameters for the new objects and Table 2 for the already known objects.
We have found 198 star clusters in the outer regions of the SMC and in the MBR. Among these, we have found 35 new star clusters in the 185 square degrees area of the MBR and 40 new star clusters in the 168 square degrees area of the outer regions of the SMC, based on observations collected by the OGLE-IV survey. Their on-sky locations are shown in Fig. 1 with red dots. The remaining 123 objects were identified in previously published catalogs – 121 objects were listed in the Bica catalog and two objects in Piatti (2017). They are marked in Fig. 1 with blue dots. As some extended objects cannot be unambiguously classified, we have performed a cross-match of our sample to both star clusters and associations from the Bica (2008) catalog. Some of their A-type objects (associations) were found by our algorithm as clusters (the classification is shown in column 6 in Table 2). Almost all previously known objects located in the area of the analyzed OGLE fields were detected by our algorithm, proving its effectiveness and the completeness of the sample. There were eight Bica objects (three star clusters and five clusters similar to associations) which were not detected by our algorithm. Those which have Bica C-type were on the edge of a subfield, and those classified as CA by Bica are not visible in our data. There was also one cluster from Piatti (2017), Field16-02, which was found on the edge of our subfield but rejected after visual inspection. All undetected objects are listed in Table 3.
All discovered cluster candidates were numbered according to the OGLE-IV naming scheme, which was presented in Paper I. The name is constructed as OGLE-MBR-CL-NNNN for the Magellanic Bridge and OGLE-SMC-CL-NNNN for the Small Magellanic Cloud, where NNNN is an object number. To make the numbering consistent with the OGLE-II catalog (Pietrzyñski 1998), we started it with 0239 for the SMC. The MBR was not observed during previous OGLE phases so for the MBR we started numbering with 0001.
Table 1 presents the OGLE collection of new candidates for star clusters. Column 1 contains the OGLE identification number, column 2 shows the field name, in columns 3 and 4 we list the equatorial coordinates (J2000) of the cluster center, column 5 contains the size of the cluster (radius) in arcseconds and the last column contains the reliability index.
Table 2 presents the OGLE collection of star clusters, which were already known. We cross-matched our detections with the Bica catalog of star clusters as well as with the catalog of associations (both are part of the Bica 2008). Column 1 contains the OGLE identification number, column 2 shows the field name. Columns 3 and 4 give our estimations of the center equatorial coordinates (J2000). Column 5 shows our estimation of the size in arcseconds (radius), in column 6 we list the cross-identification of extended objects. Column 7 contains the object type from Bica (2008): C – ordinary cluster, CN – cluster in nebula, CA – cluster similar to association, A – ordinary association, AC – association similar to cluster. Some of the objects have more than one name or type because of problems with unambiguous cross-identification. Data in Tables 1 and 2 show exactly the same parameters as in Paper I to make our collection self consistent.
OGLE-IV name OGLE-IV field RA DEC $R_{\rm KDE}$ \[\] name cluster type
------------------ --------------- --------- -------------------------------- -------------------- -------------------------- --------------
OGLE-MBR-CL-0036 MBR100.23 1480175 $-70\arcd00\arcm13\zdot\arcs1$ 20 BS196 C
OGLE-MBR-CL-0037 MBR101.16 1422905 $-71\arcd16\arcm52\zdot\arcs8$ 17 HW85 C
OGLE-MBR-CL-0038 MBR102.05 1475636 $-73\arcd07\arcm38\zdot\arcs7$ 35 BS198 CA
OGLE-MBR-CL-0039 MBR103.01 1564464 $-74\arcd13\arcm09\zdot\arcs9$ 23 NGC796,L115,WG9,ESO30SC6 C
OGLE-MBR-CL-0040 MBR103.02 1525731 $-74\arcd14\arcm56\zdot\arcs7$ 29 BS207 C
OGLE-MBR-CL-0041 MBR103.03 1502050 $-74\arcd21\arcm10\zdot\arcs3$ 28 L114,WG4,ESO30SC5 C
OGLE-MBR-CL-0042 MBR103.03 1505538 $-74\arcd10\arcm43\zdot\arcs3$ 38 WG5se CA
OGLE-MBR-CL-0043 MBR103.07 1422353 $-74\arcd10\arcm24\zdot\arcs7$ 42 HW86 C
OGLE-MBR-CL-0044 MBR103.10 1534821 $-73\arcd56\arcm09\zdot\arcs3$ 25 BS212 C
OGLE-MBR-CL-0045 MBR103.10 1533418 $-74\arcd00\arcm26\zdot\arcs7$ 38 BS210 A
OGLE-MBR-CL-0046 MBR103.10 1531257 $-73\arcd58\arcm39\zdot\arcs6$ 45 WG6 C
OGLE-MBR-CL-0047 MBR103.21 1493093 $-73\arcd43\arcm57\zdot\arcs0$ 52 L113,ESO30SC4 C
OGLE-MBR-CL-0048 MBR103.29 1480105 $-73\arcd07\arcm55\zdot\arcs7$ 23 BS198 CA
OGLE-MBR-CL-0049 MBR103.32 1425316 $-73\arcd20\arcm13\zdot\arcs6$ 21 WG1 C
OGLE-MBR-CL-0050 MBR104.17 1571653 $-74\arcd42\arcm32\zdot\arcs0$ 31 BS218 A
OGLE-MBR-CL-0051 MBR104.22 1451428 $-74\arcd41\arcm23\zdot\arcs3$ 30 WG2/BS195 CA/A
OGLE-MBR-CL-0052 MBR104.28 1494375 $-74\arcd36\arcm55\zdot\arcs3$ 26 WG3 CA
OGLE-MBR-CL-0053 MBR104.28 1492556 $-74\arcd39\arcm11\zdot\arcs5$ 26 BSBD3/BBDS2 CN/AN
OGLE-MBR-CL-0054 MBR104.28 1495227 $-74\arcd28\arcm49\zdot\arcs0$ 45 BS202 A
OGLE-MBR-CL-0055 MBR104.28 1501800 $-74\arcd21\arcm34\zdot\arcs3$ 20 L114,WG4,ESO30SC5 C
OGLE-MBR-CL-0056 MBR104.31 1435016 $-74\arcd34\arcm16\zdot\arcs7$ 41 BS192 CA
OGLE-MBR-CL-0057 MBR104.31 1435364 $-74\arcd32\arcm25\zdot\arcs2$ 36 BS193 C
OGLE-MBR-CL-0058 MBR109.03 2081940 $-74\arcd48\arcm11\zdot\arcs2$ 35 WG16 AC
OGLE-MBR-CL-0059 MBR109.03 2074430 $-74\arcd45\arcm44\zdot\arcs4$ 41 BS228 AC
OGLE-MBR-CL-0060 MBR109.04 2065082 $-74\arcd41\arcm31\zdot\arcs4$ 32 ICA11 A
OGLE-IV name OGLE-IV field RA DEC $R_{\rm KDE}$ \[\] name cluster type
------------------ --------------- --------- -------------------------------- -------------------- -------------------------------- --------------
OGLE-MBR-CL-0061 MBR109.08 2143891 $-74\arcd21\arcm30\zdot\arcs4$ 22 BSBD4 C
OGLE-MBR-CL-0062 MBR109.11 2081312 $-74\arcd31\arcm48\zdot\arcs3$ 34 WG17 A
OGLE-MBR-CL-0063 MBR109.11 2074797 $-74\arcd26\arcm31\zdot\arcs8$ 26 BS229 C
OGLE-MBR-CL-0064 MBR109.11 2074003 $-74\arcd37\arcm47\zdot\arcs1$ 34 WG15 C
OGLE-MBR-CL-0065 MBR109.12 2054086 $-74\arcd23\arcm00\zdot\arcs1$ 36 BS226 C
OGLE-MBR-CL-0066 MBR109.13 2044546 $-74\arcd30\arcm57\zdot\arcs6$ 22 WG14 C
OGLE-MBR-CL-0067 MBR109.13 2040281 $-74\arcd28\arcm46\zdot\arcs9$ 38 BS223 C
OGLE-MBR-CL-0068 MBR109.14 2003809 $-74\arcd33\arcm30\zdot\arcs8$ 20 WG11 C
OGLE-MBR-CL-0069 MBR109.15 1595909 $-74\arcd22\arcm57\zdot\arcs5$ 45 WG10 AC
OGLE-MBR-CL-0070 MBR109.18 2114950 $-74\arcd06\arcm59\zdot\arcs0$ 31 BS235 C
OGLE-MBR-CL-0071 MBR109.19 2104097 $-74\arcd09\arcm20\zdot\arcs6$ 27 BS233 CA
OGLE-MBR-CL-0072 MBR109.19 2111223 $-74\arcd16\arcm44\zdot\arcs9$ 21 BS234 AC
OGLE-MBR-CL-0073 MBR109.24 1594787 $-74\arcd16\arcm30\zdot\arcs4$ 34 BS220 A
OGLE-MBR-CL-0074 MBR109.25 1565540 $-74\arcd15\arcm20\zdot\arcs6$ 30 BS216/BS217 C/A
OGLE-MBR-CL-0075 MBR109.25 1564478 $-74\arcd13\arcm08\zdot\arcs1$ 24 NGC796,L115,WG9,ESO30SC6/BS215 CA/A
OGLE-MBR-CL-0076 MBR109.25 1563544 $-74\arcd16\arcm58\zdot\arcs3$ 25 WG8 AC
OGLE-MBR-CL-0077 MBR109.28 2092082 $-74\arcd01\arcm38\zdot\arcs3$ 24 BS232/BS231 CA/A
OGLE-MBR-CL-0078 MBR109.30 2024428 $-73\arcd56\arcm16\zdot\arcs0$ 22 WG13 C
OGLE-MBR-CL-0079 MBR110.30 2042120 $-74\arcd59\arcm01\zdot\arcs8$ 34 ICA6 A
OGLE-MBR-CL-0080 MBR113.06 2192870 $-74\arcd11\arcm45\zdot\arcs4$ 31 BS243 A
OGLE-MBR-CL-0081 MBR113.08 2311150 $-73\arcd55\arcm51\zdot\arcs0$ 28 ICA57 A
OGLE-MBR-CL-0082 MBR113.10 2271601 $-73\arcd45\arcm38\zdot\arcs6$ 40 IDK2w,ICA45 A
OGLE-MBR-CL-0083 MBR113.10 2272838 $-73\arcd58\arcm29\zdot\arcs4$ 31 BS245 CA
OGLE-MBR-CL-0084 MBR113.10 2282251 $-73\arcd48\arcm05\zdot\arcs4$ 28 ICA49/ICA48 A/A
OGLE-MBR-CL-0085 MBR113.16 2145033 $-73\arcd57\arcm10\zdot\arcs9$ 31 BS240/ICA34 C/A
OGLE-IV name OGLE-IV field RA DEC $R_{\rm KDE}$ \[\] name cluster type
------------------ --------------- --------- -------------------------------- -------------------- ----------------------------------------------------- --------------
OGLE-MBR-CL-0086 MBR113.16 2143480 $-73\arcd58\arcm56\zdot\arcs6$ 28 BS239/ICA34 CA/A
OGLE-MBR-CL-0087 MBR123.26 3013352 $-73\arcd25\arcm08\zdot\arcs5$ 24 ICA71 A
OGLE-MBR-CL-0088 MBR128.03 3102286 $-73\arcd30\arcm07\zdot\arcs5$ 18 BS247 AC
OGLE-MBR-CL-0089 MBR128.15 3013327 $-73\arcd25\arcm08\zdot\arcs2$ 24 ICA71 A
OGLE-MBR-CL-0090 MBR141.07 3442641 $-71\arcd40\arcm50\zdot\arcs2$ 47 NGC1466,SL1,LW1,ESO54SC16,KMHK1 C
OGLE-MBR-CL-0091 MBR160.11 1553602 $-77\arcd39\arcm15\zdot\arcs5$ 17 L116,ESO13SC25 C
OGLE-SMC-CL-0279 SMC738.06 1292777 $-73\arcd31\arcm56\zdot\arcs5$ 27 B164 C
OGLE-SMC-CL-0280 SMC738.06 1293482 $-73\arcd33\arcm29\zdot\arcs4$ 34 GHK24/GHK29/GKH22/GHK51/NGC602,L105,ESO29SC43,H-A68 C/C/C/C/DAN
OGLE-SMC-CL-0281 SMC738.06 1291450 $-73\arcd32\arcm02\zdot\arcs1$ 36 SGDH-cluster-A C
OGLE-SMC-CL-0282 SMC738.08 1425334 $-73\arcd20\arcm15\zdot\arcs3$ 25 WG1 C
OGLE-SMC-CL-0283 SMC738.12 1344119 $-73\arcd16\arcm27\zdot\arcs2$ 29 H86-213 C
OGLE-SMC-CL-0284 SMC738.13 1310883 $-73\arcd24\arcm51\zdot\arcs1$ 41 L107,H-A69 AC
OGLE-SMC-CL-0285 SMC738.13 1304989 $-73\arcd25\arcm45\zdot\arcs2$ 43 B165 C
OGLE-SMC-CL-0286 SMC738.13 1303340 $-73\arcd25\arcm20\zdot\arcs7$ 46 BS186 A
OGLE-SMC-CL-0287 SMC738.16 1252586 $-73\arcd22\arcm58\zdot\arcs1$ 46 BS282/L104/H-A67 C/AN/DAN
OGLE-SMC-CL-0288 SMC738.16 1243028 $-73\arcd24\arcm41\zdot\arcs9$ 46 H86-211 C
OGLE-SMC-CL-0289 SMC738.16 1240976 $-73\arcd09\arcm27\zdot\arcs2$ 62 HW81 CN
OGLE-SMC-CL-0290 SMC738.16 1242525 $-73\arcd10\arcm31\zdot\arcs2$ 46 HW82 C
OGLE-SMC-CL-0291 SMC738.16 1242537 $-73\arcd10\arcm30\zdot\arcs4$ 57 BS176/HCD99-1 C/C
OGLE-SMC-CL-0292 SMC738.21 1342567 $-72\arcd52\arcm21\zdot\arcs8$ 45 L110,ESO29SC48 C
OGLE-SMC-CL-0293 SMC738.22 1310136 $-72\arcd51\arcm03\zdot\arcs1$ 28 BS187 CA
OGLE-SMC-CL-0294 SMC739.20 1331246 $-74\arcd10\arcm02\zdot\arcs7$ 24 L109,ESO29SC46 C
OGLE-SMC-CL-0295 SMC739.29 1311193 $-73\arcd53\arcm35\zdot\arcs6$ 45 B166 C
OGLE-SMC-CL-0296 SMC740.03 1303830 $-76\arcd03\arcm15\zdot\arcs3$ 28 L106,ESO29SC44 C
OGLE-SMC-CL-0297 SMC740.18 1345599 $-75\arcd33\arcm17\zdot\arcs1$ 38 NGC643,L111,ESO29SC50 C
OGLE-IV name OGLE-IV field RA DEC $R_{\rm KDE}$ \[\] name cluster type
------------------ --------------- ---------- -------------------------------- -------------------- ------------------------- --------------
OGLE-SMC-CL-0298 SMC740.18 1355827 $-75\arcd27\arcm26\zdot\arcs0$ 23 L112 C
OGLE-SMC-CL-0299 SMC740.31 1224471 $-75\arcd00\arcm30\zdot\arcs4$ 39 HW79 C
OGLE-SMC-CL-0300 SMC737.14 1313899 $-71\arcd56\arcm49\zdot\arcs4$ 29 L108 C
OGLE-SMC-CL-0301 SMC737.17 1435243 $-71\arcd44\arcm51\zdot\arcs9$ 31 BS190 CA
OGLE-SMC-CL-0302 SMC737.21 1351161 $-71\arcd44\arcm15\zdot\arcs5$ 37 BS188 C
OGLE-SMC-CL-0303 SMC737.32 1301116 $-71\arcd20\arcm19\zdot\arcs7$ 27 BS184 CA
OGLE-SMC-CL-0304 SMC736.01 1422776 $-71\arcd16\arcm47\zdot\arcs8$ 20 HW85 C
OGLE-SMC-CL-0305 SMC736.02 1414048 $-71\arcd09\arcm53\zdot\arcs2$ 30 HW84 C
OGLE-SMC-CL-0306 SMC734.08 1224914 $-75\arcd00\arcm04\zdot\arcs9$ 41 HW79 C
OGLE-SMC-CL-0307 SMC734.12 1120482 $-75\arcd11\arcm40\zdot\arcs1$ 38 HW66,ESO29SC36 C
OGLE-SMC-CL-0308 SMC717.25 0485091 $-69\arcd52\arcm08\zdot\arcs7$ 40 L38,ESO51SC3 C
OGLE-SMC-CL-0309 SMC716.10 0585801 $-68\arcd54\arcm54\zdot\arcs0$ 23 ESO51SC9 C
OGLE-SMC-CL-0310 SMC710.26 0472456 $-68\arcd55\arcm15\zdot\arcs1$ 25 L32,ESO51SC2 C
OGLE-SMC-CL-0311 SMC706.12 0265299 $-71\arcd32\arcm56\zdot\arcs6$ 46 NGC121,K2,L10,ESO50SC12 C
OGLE-SMC-CL-0312 SMC703.01 0125734 $-73\arcd29\arcm30\zdot\arcs6$ 30 L2 C
OGLE-SMC-CL-0313 SMC703.05 0034783 $-73\arcd28\arcm43\zdot\arcs4$ 24 L1,ESO28SC8 C
OGLE-SMC-CL-0314 SMC715.28 0224273 $-75\arcd04\arcm33\zdot\arcs8$ 23 L5,ESO28SC16 C
OGLE-SMC-CL-0315 SMC761.02 23485938 $-72\arcd56\arcm43\zdot\arcs6$ 16 AM-3,ESO28SC4 C
OGLE-SMC-CL-0316 SMC707.01 0283118 $-73\arcd00\arcm40\zdot\arcs4$ 50 BS2 C
OGLE-SMC-CL-0317 SMC707.03 0245718 $-73\arcd01\arcm48\zdot\arcs4$ 40 B4 CA
OGLE-SMC-CL-0318 SMC707.09 0274416 $-72\arcd46\arcm46\zdot\arcs9$ 46 K7,L11,ESO28SC22 C
OGLE-SMC-CL-0319 SMC707.11 0244477 $-72\arcd47\arcm45\zdot\arcs0$ 50 K3,L8,ESO28SC19 C
OGLE-SMC-CL-0320 SMC707.17 0310358 $-72\arcd20\arcm21\zdot\arcs1$ 37 HW5 C
OGLE-SMC-CL-0321 SMC707.29 0213047 $-71\arcd56\arcm03\zdot\arcs5$ 35 BOLOGNA-A C
OGLE-IV name OGLE-IV field RA DEC $R_{\rm KDE}$ \[\] name cluster type
------------------ --------------- --------- -------------------------------- -------------------- -------------------------- --------------
OGLE-SMC-CL-0322 SMC708.03 0191965 $-74\arcd06\arcm23\zdot\arcs3$ 36 B1 C
OGLE-SMC-CL-0323 SMC708.04 0182579 $-74\arcd19\arcm07\zdot\arcs0$ 22 L3,ESO28SC13 C
OGLE-SMC-CL-0324 SMC714.31 0243957 $-73\arcd45\arcm11\zdot\arcs9$ 45 K5,L7,ESO28SC18 C
OGLE-SMC-CL-0325 SMC708.10 0213125 $-73\arcd45\arcm27\zdot\arcs1$ 45 K1,L4,ESO28SC15 C
OGLE-SMC-CL-0326 SMC708.18 0230383 $-73\arcd40\arcm09\zdot\arcs5$ 37 K4,L6,ESO28SC17 C
OGLE-SMC-CL-0327 SMC708.19 0212797 $-73\arcd44\arcm54\zdot\arcs1$ 41 K1,L4,ESO28SC15 C
OGLE-SMC-CL-0328 SMC708.23 0125525 $-73\arcd29\arcm27\zdot\arcs9$ 29 L2 C
OGLE-SMC-CL-0329 SMC708.29 0182344 $-73\arcd23\arcm40\zdot\arcs5$ 36 HW1 CA
OGLE-SMC-CL-0330 SMC714.12 0283966 $-74\arcd23\arcm55\zdot\arcs6$ 51 B6 C
OGLE-SMC-CL-0331 SMC714.16 0191810 $-74\arcd34\arcm26\zdot\arcs2$ 25 B2 C
OGLE-SMC-CL-0332 SMC714.22 0252681 $-74\arcd04\arcm30\zdot\arcs9$ 40 K6,L9,ESO28SC20 C
OGLE-SMC-CL-0333 SMC724.03 1104392 $-71\arcd16\arcm50\zdot\arcs2$ 51 BS144 A
OGLE-SMC-CL-0334 SMC724.07 1020110 $-71\arcd01\arcm11\zdot\arcs5$ 40 B111 C
OGLE-SMC-CL-0335 SMC724.09 1130380 $-70\arcd57\arcm46\zdot\arcs1$ 35 HW67 C
OGLE-SMC-CL-0336 SMC724.12 1074173 $-70\arcd56\arcm08\zdot\arcs4$ 26 HW56 C
OGLE-SMC-CL-0337 SMC724.31 1042497 $-70\arcd20\arcm32\zdot\arcs3$ 26 L73 C
OGLE-SMC-CL-0338 SMC731.08 1301168 $-71\arcd20\arcm17\zdot\arcs5$ 33 BS184 CA
OGLE-SMC-CL-0339 SMC731.15 1162475 $-71\arcd19\arcm36\zdot\arcs1$ 33 HW73 C
OGLE-SMC-CL-0340 SMC731.16 1145434 $-71\arcd32\arcm32\zdot\arcs6$ 48 NGC458,K69,L96,ESO51SC26 C
OGLE-SMC-CL-0341 SMC731.16 1144448 $-71\arcd20\arcm54\zdot\arcs3$ 38 L95 C
OGLE-SMC-CL-0342 SMC731.20 1245587 $-71\arcd11\arcm13\zdot\arcs3$ 26 IC1708,L102,ESO52SC2 C
OGLE-SMC-CL-0343 SMC731.27 1264270 $-70\arcd46\arcm58\zdot\arcs8$ 24 B168 C
OGLE-SMC-CL-0344 SMC739.05 1295283 $-74\arcd50\arcm48\zdot\arcs2$ 28 Field12-01 -
OGLE-SMC-CL-0345 SMC734.21 1134275 $-74\arcd45\arcm14\zdot\arcs4$ 28 Field16-01 -
The OGLE star cluster collection, the list of all analyzed SMC and MBR fields and all the graphical materials are avaliable on the OGLE web page:\
*http://ogle.astrouw.edu.pl*
We have presented a catalog of star clusters in the Magellanic Bridge and the outer regions of the Small Magellanic Cloud based on the OGLE-IV deep photometric maps. We found a total of 198 star clusters, including 75 new objects which were not listed in any of the previous catalogs, 121 clusters listed in Bica (2008) and two clusters listed in Piatti (2017). For all of them the equatorial coordinates and cross-identification with the Bica catalog are provided. The detection method presented in this paper is very effective. With our algorithm we found more than 95% of previously known clusters in this characteristic sparse region of the SMC and in the whole MBR, increasing the total number of these objects by 40%. This paper is the second of a series of publications. In the next one we will present clusters found in the central regions of the LMC and SMC, thus concluding the complete collection of star clusters in the whole Magellanic System observed by the OGLE survey.
[^1]: *http://ogle.astrouw.edu.pl*
|
---
abstract: 'The dynamics of the collective excitations of a lattice Bose gas at zero temperature is systematically investigated using the time-dependent Gutzwiller mean-field approach. The excitation modes are determined within the framework of the linear-response theory as solutions of the generalized Bogoliubov-de Gennes equations valid in the superfluid and Mott-insulator phases at arbitrary values of parameters. The expression for the sound velocity derived in this approach coincides with the hydrodynamic relation. We calculate the transition amplitudes for the excitations in the Bragg scattering process and show that the higher excitation modes give significant contributions. We simulate the dynamics of the density perturbations and show that their propagation velocity in the limit of week perturbation is satisfactorily described by the predictions of the linear-response analysis.'
author:
- 'Konstantin V. Krutitsky$^1$ and Patrick Navez$^{1,2}$'
title: |
Excitation dynamics in a lattice Bose gas\
within the time-dependent Gutzwiller mean-field approach
---
Introduction
============
Studies of excitations of ultracold atoms in optical lattices play an important role for understanding of their physical properties and dynamical behavior. For instance, celebrated quantum phase transition from the superfluid (SF) into the Mott-insulator (MI) [@Fisher] is accompanied by the opening of the gap in the excitation spectrum [@Fisher]. In the deep optical lattices the system of interacting bosons is satisfactorily described by the Bose-Hubbard model [@Fisher; @JBCGZ]. Since the model is not integrable, exact analytical results can be obtained only in few special cases like in the limit of weakly interacting gas [@Rey03] or for hard-core bosons in one dimension [@Cazalilla]. However, in general exact results can be obtained only with the aid of numerical methods.
Exact numerical results for the spectrum of low-energy excitations were obtained by means of diagonalizations of the Bose-Hubbard Hamiltonian [@RB2003R; @RB2003; @RB2004]. Although numerical diagonalization can be really done only for rather small lattices which are far from the thermodynamic limit, this allows to capture all the main characteristic features of realistic systems. Numerical results for larger systems of the same size as in real experiments with ultracold atoms [@Greiner2002] have been obtained by quantum Monte Carlo methods [@KK09] which allow to compute the spectral properties [@PEH; @CSPS08]. Simulation of the real-time dynamics of quantum systems within quantum Monte Carlo is also possible (see, e.g., [@tQMC]) but has not been yet performed for Bose systems. Ground-state properties as well as the real-time dynamics of one-dimensional systems subject to external perturbation were studied, e.g., in Refs. [@KSDZ04; @KSDZ05; @PRA09; @EFG11] by the powerful numerical density-matrix renormalization group (DMRG) method giving an access to the excitation spectrum [@Sch2005] but so far restricted to one-dimensional systems.
Mean-field theories in the dimensions higher than one allow self-consistent study of the excitations and dynamics in the lowest order with respect to quantum fluctuations. In a weakly interacting regime, the atoms are fully condensed and the system is satisfactorily described by the time-dependent discrete Gross-Pitaevskii equation (DGPE). The excitation spectrum can be calculated using Bogoliubov-de Gennes (BdG) equations [@stringari; @MKP03; @Menotti04]. It has a form of the Goldstone mode characterized at large wavelengths by the sound velocity $c$. The latter is related to the compressibility $\kappa$, the effective mass $m^*$ and the condensate density $\left|\psi\right|^2$ through the relation [@MKP03] $$c
=
\sqrt{\left|\psi\right|^2/\kappa m^*}
\;.
\label{vs}$$ The same expression for the sound velocity can be also derived from the Bogoliubov theory in the operator formalism [@Rey03] and from the hydrodynamic approach [@Fisher; @Dalibard].
In the strongly interacting regime, condensate fraction becomes suppressed and for commensurate fillings the system can undergo a transition from the SF into the MI state [@Fisher; @Dalibard; @LSADSS]. In this regime, DGPE is not valid and on the mean-field level must be replaced by more general Gutzwiller equations (GE) which are exact for a gas of infinite dimensions [@Fisher; @Rokshar; @Krauth; @Navez]. GE were successfully used to study various phenomena like creation of molecular condensate by dynamically melting a MI [@JVCWZ02], dynamical transition from the SF to Bose-glass phase due to controlled growing of the disorder [@DZSZL03], the gas dynamics in time-dependent lattice potentials [@Z], transport of cold atoms induced by the shift of the underlying harmonic potential [@SH07], dynamics of metastable states of dipolar bosons [@TML08], and the soliton propagation [@KLL10].
Excitations above the ground state described by the GE were studied using the random phase approximation (RPA) [@SKPR; @Oosten2; @Gerbier; @Menotti08; @HM], the Schwinger-boson approach [@Huber], time-dependent variational principle with subsequent quantization [@HTAB], Hubbard-Stratonovich transformation [@Oosten; @SD05], slave-boson representation of the Bose-Hubbard model [@DODS03], standard-basis operator method [@KNN06; @OKM06], and Ginzburg-Landau theory [@Pelster]. All these methods show that in the MI phase the spectrum of excitations consists of the particle- and hole-modes, both with nonvanishing gaps [@Huber; @KPS]. In the SF phase, the lowest branch is a gapless Goldstone mode [@Huber; @HTAB; @KPS; @Bissbort], while higher branches are gapped [@Huber; @Bissbort]. However, different approximations used in the calculations by different methods do not always lead to the same final results. For instance, in Ref. [@Menotti08] it was checked numerically that the RPA gives the same result for the sound velocity as Eq. (\[vs\]), whereas the analytical expressions derived in Refs. [@SD05; @KPS07] differ from that.
Excitations above the Gutzwiller ground state can be also investigated using generalization of the BdG equations directly derived from the GE within the framework of the linear response theory. This method was used for the lattice Bose gas with short-range [@KPS07; @Bissbort] as well as long-range [@KPS; @TML08] interactions. This approach, which was not so far widely used to study the lattice Bose gas allows to obtain results consistent with other mean-field approaches mentioned above and to study the ground state, stationary excitation modes as well as the dynamics of the gas on equal footing.
The excitations can be probed in experiments on inelastic light scattering (Bragg spectroscopy) which provide an information on the dynamic structure factor and one-particle spectral function. Recently such experiments were carried out with ultra-cold rubidium atoms in optical lattices of different dimensions in the SF phase [@Sengstock1; @Bissbort; @Heinze] and across the SF-MI transition [@Inguscio; @Inguscio2010]. Theoretical analysis has been developed using exact diagonalization of the Bose-Hubbard Hamiltonian [@Clark; @Batrouni], quantum Monte Carlo simulations in one dimension [@PEH; @Batrouni], perturbation theory valid deep in the MI phase [@Clark], hydrodynamic theory [@Machida], exact Bethe-ansatz solution of the Lieb-Liniger model [@CC06], extended fermionization [@Batrouni], RPA [@Menotti08]. Analogous studies were also performed within the Gutzwiller approximation [@Oosten2; @Bissbort; @Huber]. However, previous calculations are valid only for the MI [@Oosten2; @Huber] or close to it [@Huber]. It was argued that the second excitation branch in the SF cannot be detected by the Bragg spectroscopy in the linear regime [@HTAB]. In Ref. [@Bissbort], the calculations beyond the linear response theory taking into account the harmonic trapping potential and the finite duration of the probing Bragg pulse were performed which are in good quantitative agreement with the experimental data reporting the observation of the second excitation branch. In the present work, we will study in details the possibility to observe the second excitation mode in a homogeneous lattice in the linear-response regime.
Experimentally sound waves can be observed with the aid of an external potential which creates a density perturbation of the gas. Corresponding numerical simulations for the lattice gas were performed on the basis of the DGPE [@Menotti04] and for soft-core bosons in 1D making use of the DMRG method [@KSDZ05]. The sound velocity extracted from these simulations is in perfect agreement with Eq. (\[vs\]) in the case of the DGPE [@Menotti04] and has a correct asymptotic behavior in 1D in the limits of weak and strong interactions [@KSDZ05], where analytical expressions are known.
The purpose of this paper is to give a comprehensive self-consistent description of the collective excitations as well as experimental techniques for their observation within the time-dependent Gutzwiller ansatz which is gapless and satisfies the basic conservation laws, in particular, f-sum rule. Solution of GE allows not only to obtain the excitation dispersion relations but also to calculate the transition amplitudes in the Bragg scattering process. We present a derivation of Eq. (\[vs\]) from the generalized BdG equations. Furthermore, the time-dependent approach is also used to investigate the sound wave propagation in the case of a stronger perturbation generated by switching off a local potential. In this way, we can determine the speed at which this perturbation propagates and compare with the predictions of the linear-response theory. We emphasize that the Gutzwiller ansatz is the only approximation used in the present work and the results are valid in the whole range of parameters.
The paper is organized in the following manner. In Sec. \[GA\], we present the time-dependent GE. Their ground state solutions are discussed in Sec. \[GS\]. In Sec. \[E\], we determine the spectrum of collective excitations using Gutzwiller-Bogoliubov-de Gennes equations. Sec. \[T\] is devoted to the Bragg scattering. In Sec. \[SW\], we simulate the sound-wave propagation. The conclusions are presented in Sec. \[C\].
\[GA\]The time-dependent Gutzwiller ansatz
===========================================
We consider a system of ultracold interacting bosons in a $d$-dimensional isotropic lattice described by the Bose-Hubbard Hamiltonian $$\begin{aligned}
\label{BHH}
\hat H
&=&
-J
\sum_{\bf l}
\sum_{\alpha=1}^d
\left(
\hat a_{\bf l}^\dagger
\hat a_{{\bf l}+{\bf e}_\alpha}
+
{\rm h.c.}
\right)
\nonumber\\
&+&
\frac{U}{2}
\sum_{\bf l}
\hat a^\dagger_{\bf l}
\hat a^\dagger_{\bf l}
\hat a_{\bf l}
\hat a_{\bf l}
-
\mu
\sum_{\bf l}
\hat a^\dagger_{\bf l} \hat a_{\bf l}
\;,\end{aligned}$$ where the site index ${\bf l}$ is a $d$-dimensional vector, ${\bf e}_\alpha$ is a unit vector on the lattice in the direction $\alpha$, $J$ is the tunneling matrix element, $U$ is the repulsive on-site atom-atom interaction energy, and $\mu$ the chemical potential. The annihilation and creation operators at site ${\bf l}$, $\hat a_{\bf l}$ and $\hat a^\dagger_{\bf l}$, obey the bosonic commutation relations. The momentum operator $$\hat{\bf P}
=
i P_0
\sum_{\alpha=1}^d
{\bf e}_\alpha
\sum_{\bf l}
\left(
\hat a_{{\bf l}}^\dagger
\hat a_{{\bf l}+{\bf e}_\alpha} - {\rm h.c.}
\right)
\;,$$ where $P_0$ is a constant determined by the parameters of the periodic potential creating the lattice, does not commute with the Hamiltonian (\[BHH\]) due to the interaction term. Instead, the quasi-momentum operator defined as $$\hat{\bf p}
=
\sum_{\bf k}
{\bf k}
\hat a_{\bf k}^\dagger
\hat a_{\bf k}
\;,\quad
\hat a_{\bf k}
=
\sum_{\bf l}
e^{-i{\bf k}\cdot{\bf l}}\hat a_{{\bf l}}/L^{d/2}
\;,$$ where $k_\alpha=2\pi n_\alpha/L$, $n_\alpha=0,\dots,L-1$, with $L$ being the number of lattice sites in each spatial direction, commutes with the latter.
Our analysis employs the Gutzwiller ansatz. Thereby, eigenstates of the Hamiltonian (\[BHH\]) are taken as tensor products of local states $$\label{state}
|\Phi\rangle
=
\bigotimes_{\bf l}
|s_{\bf l}\rangle
\;,\quad
|s_{\bf l}\rangle
=
\sum_{n=0}^\infty
c_{{\bf l},n}
|n\rangle_{\bf l}
\;.$$ where $|n\rangle_{\bf l}$ is the Fock state with $n$ atoms at site ${\bf l}$. Normalization of the $|s_{\bf l}\rangle$ imposes $$\sum_{n=0}^\infty
\left|
c_{{\bf l},n}
\right|^2
=
1
\;.
\nonumber$$ The mean number of condensed atoms in this model is given by $|\psi_{\bf l}|^2$, where $$\label{psi}
\psi_{\bf l}
=
\langle \hat a_{\bf l}\rangle
=
\sum_{n=1}^\infty
c_{{\bf l},n-1}^* c_{{\bf l},n}
\sqrt{n}$$ is the condensate order parameter. One can easily show that $|\psi_{\bf l}|^2$ cannot be larger than the mean occupation number $$\langle\hat n_{\bf l}\rangle
=
\sum_{n=1}^\infty
n
\left|
c_{{\bf l},n}
\right|^2
\;.$$ Minimization of the functional $$i\hbar
\sum_{n=0}^\infty
(c_{{\bf l},n}^*
\partial_t c_{{\bf l}n}
-
c_{{\bf l}n}
\partial_t c^*_{{\bf l}n})
-
\langle H \rangle$$ leads to the system of GE [@Z; @BPVB07]: $$\begin{aligned}
\label{GEd}
i\hbar
\frac{d c_{{\bf l}n}}{dt}
&=&
\sum_{n'}
H_{\bf l}^{n n'}
c_{{\bf l}n'}
\;,
\\
H_{\bf l}^{n n'}
&=&
\left[
\frac{U}{2}n(n-1)
-
\mu n
\right]
\delta_{n',n}
\nonumber\\
&-&
J
\sqrt{n'}
\delta_{n',n+1}
\sum_{\alpha=1}^d
\left(
\psi^*_{{\bf l}+{\bf e}_\alpha}
+
\psi^*_{{\bf l}-{\bf e}_\alpha}
\right)
\nonumber \\
&-&
J
\sqrt{n}
\delta_{n,n'+1}
\sum_{\alpha=1}^d
\left(
\psi_{{\bf l}+{\bf e}_\alpha}
+
\psi_{{\bf l}-{\bf e}_\alpha}
\right)
\;.
\nonumber\end{aligned}$$ The Gutzwiller approximation is [*conserving*]{} since these equations do not violate conservation laws of the original Bose-Hubbard model. The expectation values of the quasi-momentum, total energy, and the total number of particles remain constant in time.
As it follows from the form of the state (\[state\]), Gutzwiller approximation neglects quantum correlations between different lattice sites but takes into account on-site quantum fluctuations. This appears to be enough for satisfactory description of the SF-MI quantum phase transition. Due to the fact that in the equations of motion (\[GEd\]) the coefficients $c_{{\bf l}n}$ for different sites are coupled to each other, Gutzwiller ansatz can be also used to study the dynamics of excitations.
>From (\[GEd\]), we deduce the following equation for the order parameter: $$\begin{aligned}
\label{GEd2}
i\hbar
\frac{d \psi_{\bf l}}{dt}
=
&-&
J
\sum_{\alpha=1}^d
\left(
\psi_{{\bf l}+{\bf e}_\alpha}
+
\psi_{{\bf l}-{\bf e}_\alpha}
\right)
-\mu \psi_{\bf l}
\nonumber \\
&+&
U
\sum_{n=0}^\infty
(n-1)\sqrt{n}
c_{{\bf l},n-1}^* c_{{\bf l},n}
\;.\end{aligned}$$ This equation becomes closed if we assume the coherent state $$\label{cs}
c_{{\bf l},n}
=
c^{coh}_{{\bf l},n}
\equiv
\exp
\left(
-|\psi_{\bf l}|^2/2
\right)
\psi_{\bf l}^n/\sqrt{n!}
\;,$$ which is an exact solution of Eq. (\[GEd\]) for $U=0$. In this case, the replacement of the last term of Eq. (\[GEd2\]) by this distribution leads to the result $$U\sum_{n=0}^\infty
(n-1)\sqrt{n}
({c^{coh}_{{\bf l},n-1}})^*
c^{coh}_{{\bf l},n}
=
U|\psi_{\bf l}|^2 \psi_{\bf l}
\;,
\nonumber$$ so that we recover the DGPE valid for small $U/J$.
\[GS\]Ground state
==================
In the ground state, the coefficients $c_{{\bf l},n}$ do not depend on the site index ${\bf l}$ so that the solution has the form $$c_{{\bf l},n}(t)
\equiv
c_n^{(0)}
\exp
\left(
-i \omega_0 t
\right)
\;,$$ so that $
\langle
\hat n_{\bf l}
\rangle
\equiv
\langle
\hat n
\rangle
$, where $\hat n= \sum_{\mathbf{l}}\hat n_{\bf l}/L^d$. The coefficients $c_n^{(0)}$ are calculated numerically solving Eq. (\[GEd\]) by means of exact diagonalization in the same manner as in Refs. [@SKPR; @Oosten]. The results are shown in Figs. \[cn\] and \[cn\_cs\]. $c_n^{(0)}$ has a broad distribution in the SF phase, where the corresponding $\psi^{(0)}=\psi_{\bf l}$ defined by Eq. (\[psi\]) does not vanish. In the MI phase, however, $$\label{gsmi}
c_n^{(0)}=\delta_{n,n_0}$$ resulting in $\psi^{(0)}=0$. $\omega_0$ is determined in both phases from: $$\hbar\omega_0
=
- 4dJ
\psi^{(0)2}
+
\sum_{n=0}^\infty
\left[
\frac{U}{2}n(n-1) - \mu n
\right]
c_n^{(0)2}
\;.$$
![Ground-state solutions for the atomic distribution $c_n^{(0)}$. The scaled chemical potential $\mu/U=1.2$ and the tunneling rates $2dJ/U$: 0.7 (i), 0.5 (ii), 0.3 (iii), 0.15 (iv), and 0.05 (v). The lines connecting the dots are to guide the eye.[]{data-label="cn"}](fig01.eps){width="10cm"}
In Figs. \[cn\_cs\] and \[deviation\], we compare the coefficients $c_n^{(0)}$ obtained by numerical solution of Eqs. (\[GEd\]) with the corresponding results for the coherent state (\[cs\]). The value of $\psi^{(0)}$ for Eq. (\[cs\]) was calculated according to Eq. (\[psi\]) using $c_n^{(0)}$. These coefficients converge towards a coherent state distribution for increasing $J/U$ according to a power law (Fig. \[deviation\]), thus, justifying the use of the DGPE in this limit.
In the numerical calculations presented in this section and later on, $n$ was restricted by some finite $N$ ($c_{n} \equiv 0$ for $n>N$). The cut-off number of atoms $N$ was chosen large enough such that its influence on the eigenstates is negligible. For example, for the plots shown in Fig. \[cn\], it was enough to use $N=10$, but for Figs. \[cn\_cs\], \[deviation\] $N=500$ was used.
![Comparison of $c_n^{(0)}$ (1,2) with $c_n^{coh}$ (a,b). The scaled chemical potential $\mu/U=1.2$ and the tunneling rates $2dJ/U$: 0.5 (1,a), 50 (2,b). []{data-label="cn_cs"}](fig02.eps){width="10cm"}
![ Deviation of the exact ground state from the coherent state for the scaled chemical potential $\mu/U=1.2$. []{data-label="deviation"}](fig03.eps){width="10cm"}
\[E\]Excitations
================
We consider small perturbation of the ground state $
c_{{\bf l}n}(t)
=
\left[
c_{n}^{(0)}
+
c_{{\bf l}n}^{(1)}(t)
+
\dots
\right]
\exp
\left(
-i \omega_0 t
\right)
$, where $$\label{sol}
c_{{\bf l}n}^{(1)}(t)
=
u_{{\bf k}n}
e^{
i
\left(
{\bf k}\cdot{\bf l}
-
\omega_{\bf k} t
\right)
}
+
v_{{\bf k}n}^*
e^{
-i
\left(
{\bf k}\cdot{\bf l}
-
\omega_{\bf k} t
\right)
}
\;.$$ Substituting this expression into GE and keeping only linear terms with respect to $u_{{\bf k}n}$ and $v_{{\bf k}n}$, we obtain the system of linear equations [@TML08] $$\label{evpexc}
\hbar\omega_{\bf k}
\left(
\begin{array}{c}
\vec{u}_{\bf k}\\
\vec{v}_{\bf k}
\end{array}
\right)
=
\left(
\begin{array}{cc}
A_{\bf k} & B_{\bf k}\\
-B_{\bf k} & -A_{\bf k}
\end{array}
\right)
\left(
\begin{array}{c}
\vec{u}_{\bf k}\\
\vec{v}_{\bf k}
\end{array}
\right)
\;,$$ where $\vec{u}_{\bf k}$ and $\vec{v}_{\bf k}$ are infinite-dimensional vectors with the components $u_{{\bf k}n}$ and $v_{{\bf k}n}$ ($n=0,1,\dots$), respectively. Matrix elements of $A_{\bf k}$ and $B_{\bf k}$ have the form $$\begin{aligned}
A_{\bf k}^{nn'}
&=&
-J_{\bf 0}
\psi^{(0)}
\left(
\sqrt{n'}\,
\delta_{n',n+1}
+
\sqrt{n}\,
\delta_{n,n'+1}
\right)
\nonumber\\
&+&
\left[
\frac{U}{2}\,n(n-1)
-
\mu n
-
\hbar\omega_0
\right]
\delta_{n',n}
\nonumber\\
&-&
J_{\bf k}
\left[
\sqrt{n+1}\,
\sqrt{n'+1}\,
c_{n+1}^{(0)}\,
c_{n'+1}^{(0)}
\right.
\nonumber\\
&+&
\left.
\sqrt{n}\,
\sqrt{n'}\,
c_{n-1}^{(0)}\,
c_{n'-1}^{(0)}
\right]
\;,
\nonumber\\
B_{\bf k}^{nn'}
&=&
-
J_{\bf k}
\left[
\sqrt{n+1}\,
\sqrt{n'}\,
c_{n+1}^{(0)}\,
c_{n'-1}^{(0)}
\right.
\nonumber\\
&+&
\left.
\sqrt{n}\,
\sqrt{n'+1}\,
c_{n-1}^{(0)}\,
c_{n'+1}^{(0)}
\right]
\;,
\nonumber\end{aligned}$$ where $J_{\bf k}=2dJ-\epsilon_{\bf k}$ with $$\epsilon_{\bf k}
=
4J
\sum_{\alpha=1}^d
\sin^2
\left(
\frac{k_\alpha}{2}
\right)$$ being the energy of a free particle. This system is valid for both phases and generalizes the BdG equations previously derived for coherent states. The dependence on vector ${\bf k}$ is determined by the variable $$\label{x}
x
=
\left(
\frac{1}{d}\sum_{\alpha=1}^d \sin^2 \frac{k_\alpha}{2}
\right)^{1/2}
\;,$$ which varies from $0$ to $1$. For small $|{\bf k}|$, $x\approx |{\bf k}|/(2\sqrt{d})$.
The energy increase due to the perturbation is given by [@stringari] $$\label{dE}
\Delta E
=
\hbar \omega_{\mathbf{k}}
\left(
|\vec{u}_{{\mathbf{k}}}|^2-|\vec{v}_{{\mathbf{k}}}|^2
\right)
\;.$$ Formally, Eqs. (\[evpexc\]) have solutions with positive and negative energies $\pm\hbar\omega_{\bf k}$, which are equivalent because Eqs. (\[sol\]), (\[dE\]) are invariant under the transformation $\omega_{\bf k}\to -\omega_{\bf k}$, ${\bf k}\to -{\bf k}$, $\vec{u}_{\bf k}\to\vec{v}^*_{\bf k}$, $\vec{v}^*_{\bf k}\to\vec{u}_{\bf k}$, so that only solutions with the positive energies will be considered in the following. The eigenvectors are chosen to follow the orthonormality relations $$\vec{u}^*_{{\mathbf{k}},\lambda'}
\cdot
\vec{u}_{{\mathbf{k}},\lambda}
-
\vec{v}^*_{{\mathbf{k}},\lambda'}
\cdot
\vec{v}_{{\mathbf{k}},\lambda}
=
\delta_{\lambda,\lambda'}
\;.
\nonumber$$
Perturbation (\[sol\]) creates plane waves of the order parameter $\psi_{\bf l}(t)=\psi^{(0)}+\psi_{\bf l}^{(1)}(t)$, where $$\begin{aligned}
\label{psiw}
\psi_{\bf l}^{(1)}(t)
&=&
{\cal U}_{\bf k}
e^{
i
\left(
{\bf k}\cdot{\bf l}
-
\omega_{\bf k} t
\right)
}
+{\cal V}^*_{\bf k}
e^{
-i
\left(
{\bf k}\cdot{\bf l}
-
\omega_{\bf k} t
\right)
}
\;,
\\
{\cal U}_{\bf k}
&=&
\sum_n
\left(
c_{n-1}^{(0)}
u_{{\bf k}n}
+
c_{n}^{(0)}
v_{{\bf k},n-1}
\right)
\sqrt{n}
\;,
\nonumber\\
{\cal V}_{\bf k}
&=&
\sum_n
\left(
c_{n}^{(0)}
u_{{\bf k},n-1}
+
c_{n-1}^{(0)}
v_{{\bf k}n}
\right)
\sqrt{n}
\;.
\nonumber\end{aligned}$$
The perturbations for the total density and the condensate density are given by $$\begin{aligned}
\label{dw}
\langle
\hat n_{\bf l}
\rangle(t)
&=&
\langle
\hat n
\rangle
+
\left[
{\cal A}_{\bf k}
e^{
i
\left(
{\bf k}\cdot{\bf l}
-
\omega_{\bf k} t
\right)
}
+
{\rm c.c.}
\right]
\;,
\\
{\cal A}_{\bf k}
&=&
\sum_n
c_{n}^{(0)}
n
\left(
u_{{\bf k}n}
+
v_{{\bf k}n}
\right)
\;,
\nonumber\end{aligned}$$ and $$\begin{aligned}
\label{cw}
\left|
\psi_{\bf l}(t)
\right|^2
&=&
{\psi^{(0)}}^2
+
\left[
{\cal B}_{\bf k}
e^{
i
\left(
{\bf k}\cdot{\bf l}
-
\omega_{\bf k} t
\right)
}
+
{\rm c.c.}
\right]
\;,
\\
{\cal B}_{\bf k}
&=&
\psi^{(0)}
\sum_n
\left[
c_{n-1}^{(0)}
\left(
u_{{\bf k}n}
+
v_{{\bf k}n}
\right)
\right.
\nonumber\\
&+&
\left.
c_{n}^{(0)}
\left(
u_{{\bf k},n-1}
+
v_{{\bf k},n-1}
\right)
\right]
\sqrt{n}
\;.
\nonumber\end{aligned}$$ In what follows we consider the properties of the excitations in the MI and SF phases. Although the results for the MI are not new, we would like to present those for completeness.
Mott insulator
--------------
For the MI phase, the coefficients $c_n^{(0)}$ have a simple analytical form (\[gsmi\]). The eigenvalue problem for the infinite-dimensional matrices (\[evpexc\]) reduces to the diagonalization of two $2\times 2$-matrices which couple $u_{{\bf k},n_0-1}$ to $v_{{\bf k},n_0+1}$ and $u_{{\bf k},n_0+1}$ to $v_{{\bf k},n_0-1}$, respectively. The lowest-energy excitation spectrum consists of two branches with the energies $$\begin{aligned}
\label{om}
\hbar\omega_{{\bf k}\pm}
&=&
\frac{1}{2}
\sqrt{
U^2
-
4J_{\bf k}
U
\left(
n_0+\frac{1}{2}
\right)
+
J_{\bf k}^2
}
\nonumber\\
&\pm&
\left[U
\left(
n_0-\frac{1}{2}
\right)
-\mu
-\frac{J_{\bf k}}{2}
\right]
\;,\end{aligned}$$ The same result was obtained using Hubbard-Stratonovich transformation [@Oosten] and within the Schwinger-boson approach [@Huber].
These two branches are shown in Fig. \[excmi\] and display a gap. According to Eqs. (\[cw\]), (\[dw\]), no density wave is created in the two modes, although the order parameter does not vanish \[see Eq. (\[psiw\])\]. Eq. (\[om\]) can be also rewritten in the form $$\begin{aligned}
\hbar\omega_{{\bf k}+}
&=&
\epsilon_{{\bf k}p}
- \mu
\;,
\nonumber\\
\hbar\omega_{{\bf k}-}
&=&
-\epsilon_{{\bf k}h}
+
\mu
\nonumber
\;.\end{aligned}$$ Therefore, the solutions with index ’$+$’ and ’$-$’ have the meaning of the particle and hole excitations, respectively [@EM99].
Other solutions of Eq. (\[evpexc\]) are independent of ${\bf k}$ with the energies $$\hbar\omega_{\lambda}
=
\frac{U}{2}
\left[
\lambda(\lambda-1)
-
n_0(n_0-1)
\right]
-
\mu
(\lambda-n_0)
\;,$$ They are denoted by $\lambda$ which are non-negative integers different from $n_0,n_0\pm 1$. If $n_0$ is the smallest integer greater than $\mu/U$, the excitation energies are always positive. The eigenvectors of these modes have the form $u_{{\bf k}n\lambda}=\delta_{n,\lambda}$, $v_{{\bf k}n\lambda}=0$, and the amplitudes of all the waves defined by Eqs. (\[psiw\]), (\[dw\]), (\[cw\]) vanish.
The boundary between the SF and MI phases is determined from the disappearance of the gap in the excitation spectrum, i.e., when $\omega_{{\mathbf{0}}-}=0$. Under this condition, we recover the critical ratio [@Sachdev]: $$\label{crit}
2d(J/U)_c
=
\frac
{(n_0-\mu/U)(\mu/U-n_0+1)}
{1+\mu/U}
\;.$$ It takes its maximal value when $$\label{Jcmax}
2d(J/U)_c^{max}
=
\left(
\sqrt{n_0+1}
-
\sqrt{n_0}
\right)^2$$ for a chemical potential given by $$(\mu/U)_c=\sqrt{n_0(n_0+1)}-1
\;.$$ For $J/U>(J/U)_c$, the lowest frequency $\omega_{{\mathbf{0}}-}$ in Eq. (\[om\]) becomes negative leading to a negative expression for Eq. (\[dE\]), so that the Mott-phase solution (\[gsmi\]) does not correspond to the ground state anymore.
![ First three branches $\hbar\omega_{{\mathbf{k}}-}$ (1), $\hbar\omega_{{\mathbf{k}}+}$ (2) and $\hbar\omega_{\lambda=0}$ (3) of the excitations spectrum of the MI for $\mu/U=1.2$ and $2dJ/U=0.05$, which corresponds to $n_0=2$. []{data-label="excmi"}](fig04.eps){width="10cm"}
The excitation spectrum has interesting features on the boundary between the MI and SF. For $(J/U)_c=(J/U)_c^{max}$, the excitation energies (\[om\]) can be rewritten as $$\hbar\omega_{{\bf k}\pm}
=
\left[
\sqrt{n_0(n_0+1)}
U
\epsilon_{\bf k}
+
\frac{\epsilon^2_{\bf k}}{4}
\right]^{1/2}
\pm
\frac{\epsilon_{\bf k}}{2}
\;.$$ For small $|{\bf k}|$, the two branches are degenerate and have linear dependence $\omega_{{\bf k}\pm}=c_s^{tip}|{\bf k}|$ with the sound velocity $$\label{cs0}
c_s^{tip}
=
\frac{U}{\hbar}
\sqrt
{
(J/U)_c^{max}
}
\left[
n_0(n_0+1)
\right]^{1/4}$$ expressed in the units of number of sites per second. For other points on the boundary, i.e., $(J/U)_c < (J/U)_c^{max}$, no degeneracy appears and the sound velocity vanishes leading to the quadratic dispersion $\omega_{{\bf k}\pm}\sim {\bf k}^2$ for small $|{\bf k}|$.
Superfluid
----------
In the SF phase, the eigenvalue problem (\[evpexc\]) is solved using the numerical values of $c_n^{(0)}$ for each $J/U$ and $\mu/U$. The energies of the lowest-energy excitations are shown in Fig. \[excsf\]. The excitation spectrum consists of several branches which form a band structure shown in Figs. \[bands\_sf\], \[bands\_sf1\]. In contrast to the MI, the lowest branch has no gap. It is a Goldstone mode which appears due to the spontaneous breaking of the phase symmetry and, therefore, can be called a phase mode [@HTAB]. As it is shown in Fig. \[AB\], the amplitude of the total-density wave is larger than the amplitude of the condensate-density wave for this mode. A value greater than unity for the ratio ${\cal A}_{\bf k}/{\cal B}_{\bf k}$ means that the condensed part and normal part oscillate in phase.
![ First three branches $\hbar\omega_{{\mathbf{k}},\lambda}$ ($\lambda=1,2,3$) of the excitations spectrum of the SF for $\mu/U=1.2$ and $2dJ/U=0.15$. The straight dashed line represents the linear approximation with the sound velocity (\[c\_s\]). []{data-label="excsf"}](fig05.eps){width="10cm"}
![ Band structure of the excitation spectrum for $\langle \hat n\rangle=0.5$ (a), $1$ (b). Shaded regions which are extremely narrow for higher bands show allowed excitation energies. Lower and upper boundaries of the bands correspond to $x=0$ and $x=1$, respectively. []{data-label="bands_sf"}](fig06.eps){width="10cm"}
![ Band structure of the excitation spectrum for $2dJ/U=0.2$ (a), $0.3$ (b) versus the density. Shaded regions show allowed excitation energies. Lower and upper boundaries of the bands correspond to $x=0$ and $x=1$, respectively. []{data-label="bands_sf1"}](fig07.eps){width="10cm"}
Higher modes ($\lambda \geq 2$) do not exist in the formalism based on the DGPE. They have gaps $\Delta_\lambda=\hbar\omega_{{\bf 0}\lambda}$ which grow with the increase of $J$ (see Fig. \[bands\_sf\]). As it is shown in Appendix \[EG\], for large $J$ they have an asymptotic form $$\label{gap2}
\Delta_\lambda
=
2dJ
\lambda
+
\frac{U}{2}
\lambda
\left(
\langle \hat n\rangle
+
\lambda - 1
\right)
\;.$$ We note that only the first two lowest-energy branches have a strong dependence on ${\bf k}$. For the second mode ($\lambda=2$), the amplitude of the total-density wave is much less than that of the condensate-density wave (see Fig. \[AB\]) which means that the oscillations of the condensate and normal components are out-of-phase. However, this does not necessarily mean that there is an exchange of particles between the condensate and normal component [@Huber; @Bissbort]. Due to the reasons explained in Refs. [@Huber; @HTAB; @Bissbort] this type of excitations is called an amplitude mode. In the rest part of this section, we will study in more details the properties of the Goldstone mode.
![ ${\cal A}_{\bf k}/{\cal B}_{\bf k}$ for $\mu/U=1.2$ and $2dJ/U=0.15$ (a), $1$ (b). []{data-label="AB"}](fig08.eps){width="10cm"}
As shown in Appendix \[Der\], the lowest-energy branch has a linear form $\omega_{{\mathbf{k}},1}=c_s^0|{\mathbf{k}}|$ for small ${\bf k}$ with the sound velocity given by $$\label{c_s}
c_s^0
=
\sqrt{\frac{2J}{\kappa}}
\left|
\psi^{(0)}
\right|
/{\hbar}
\;,$$ where $\kappa=\frac{\partial\langle\hat n\rangle}{\partial\mu}$ is the compressibility. This result proves that the Gutzwiller approximation is [*gapless*]{} and coincides with Eq. (\[vs\]).
Fig. \[sv\] shows the dependence of the sound velocity on $\mu$ and $J$ calculated numerically using Eq. (\[c\_s\]). If we approach the boundary of the MI, the sound velocity goes to zero everywhere except the tips of the lobes, where it is perfectly described by Eq. (\[cs0\]). This behavior can be understood considering the properties of $\psi^{(0)}$ and $\kappa$. If we approach the SF-MI transition from the SF part of the phase diagram, the order parameter $\psi^{(0)}$ tends always continuously to zero. The compressibility $\kappa$ reaches a finite value at every point of the boundary except the tips of the MI-lobes where it tends continuously to zero such that the ratio $\psi^{(0)}/\sqrt{\kappa}$ is finite. Therefore, the sound velocity vanishes at any point of the phase boundary except the tips of the lobes [@Fisher; @Menotti08].
![ Sound velocity calculated numerically from Eq. (\[c\_s\]). Note the discontinuities at the points $[(J/U)_c^{max},(\mu/U)_c]$ described by Eq. (\[cs0\]). []{data-label="sv"}](fig09.eps){width="10cm"}
For a weakly interacting gas ($U \ll J$), $|\psi^{(0)}|^2\approx \langle\hat n\rangle$ and $\kappa\approx 1/U$. In this limit, we recover the Bogoliubov’s dispersion relation (see Appendix \[Bdr\]) and the expression for the sound velocity [@Menotti04; @Zaremba] $$\label{csB}
c_s^B
=
\sqrt{2JU\langle\hat n\rangle}/\hbar
\;.$$ The comparison of the exact numerical values of the sound velocity calculated according to Eq. (\[c\_s\]) with the approximation (\[csB\]) is shown in Fig. \[sv2\]. As it is expected, the agreement is good at large $J$ but for small tunneling rates the behavior of $c_s^B$ is completely different.
![Comparison of the sound velocity calculated numerically from Eq. (\[c\_s\]) (solid line) with the analytical expression (\[csB\]) (dashed line) for $\langle \hat n\rangle=0.5$. []{data-label="sv2"}](fig10.eps){width="10cm"}
In the opposite limit ($J \ll U$), the superfluid regions with the atomic densities $n_0 < \langle\hat n\rangle < n_0+1$ are confined in the regions $\mu_- < \mu < \mu_+$, where $\mu_-$ is the upper boundary of the MI with $n_0$ and $\mu_+$ is the lower boundary of the MI with $n_0+1$. For $J \rightarrow 0$, $\mu$ is a linear function of the density, i.e., $$\mu
=
\mu_-
+
\left(
\mu_+ - \mu_-
\right)
\left(
\langle\hat n\rangle - n_0
\right)
\;.$$ Using Eq. (\[crit\]) up to the first order in $J$, we obtain $$\mu_\pm= U n_0 \pm 2dJ(n_0+1)$$ from which we deduce $$\label{kappa}
\kappa=1/[4dJ(n_0+1)]
\;.$$ Using $\partial E / \partial \langle\hat n\rangle = \mu$ and the condition that the energy at the phase boundaries $E_\pm =Un_0(n_0\pm 1)/2$, the total energy becomes in this limit $$\begin{aligned}
\label{En}
E
&=&
U
n_0
\left(
\langle\hat n\rangle
-
\frac{n_0+1}{2}
\right)
\\
&-&
2dJ
\left(
n_0+1
\right)
\left(
\langle\hat n\rangle-n_0
\right)
\left(
n_0+1 - \langle\hat n\rangle
\right)
\;.
\nonumber\end{aligned}$$ In order to deduce the order parameter, we use the relation $\partial E /\partial J= -2d |\psi^{(0)}|^2$. From Eq. (\[En\]) we obtain $$\label{psi2}
|\psi^{(0)}|^2
=
(n_0+1)
(\langle \hat n\rangle -n_0)
(n_0+1-\langle n \rangle)
\;.$$ Eqs. (\[En\]), (\[psi2\]) coincide with the ones obtained in Ref. [@Krauth] but using a perturbation approach. Substituting (\[kappa\]) and (\[psi2\]) into (\[c\_s\]), we obtain finally: $$\begin{aligned}
\label{csu}
c_s^0
=
2J(n_0+1)
\sqrt{2d(\langle \hat n \rangle-n_0)(n_0+1-\langle \hat n \rangle)}
/\hbar
\;.\end{aligned}$$ The sound velocity in the limit of small $J$ vanishes at $\langle\hat n\rangle=n_0,n_0+1$ and takes maximal values at $\langle\hat n\rangle=n_0+1/2$. This qualitative behavior is the same as in the case of hard-core bosons in 1D, where the sound velocity is given by $c_s^{HC}=2J\sin(\pi \langle\hat n\rangle)/\hbar$ for $0<\langle\hat n\rangle<1$ (see, e.g., [@Cazalilla]).
Close to the phase boundary, the sound velocity can be calculated analytically according to Eq. (\[c\_s\]) within the fourth-order perturbation theory [@Oosten]. The result for $c_s$ appears to be very long and cannot be displayed here. However, it describes all the qualitative features discussed above and reproduces Eq. (\[cs0\]).
\[T\]Transition processes
=========================
Bragg scattering
----------------
The Bragg scattering is a common experimental method to measure the excitation spectrum of an ultra-cold gas [@stringari]. This process is induced by the following perturbation term in the Hamiltonian: $$\begin{aligned}
\label{Hint}
\hat H'(t)
&=&
\sum_{\bf l}
V_{{\mathbf{k}},\omega}
\cos({\mathbf{k}}\cdot{\mathbf{l}}-\omega t)
\hat a^\dagger_{\bf l} \hat a_{\bf l}
\;,\end{aligned}$$ where $\omega$ and ${\mathbf{k}}$ are the frequency and the wavevector of the excitation, respectively. This perturbation changes the gas density according to: $$\begin{aligned}
\delta n_{\mathbf{l}}(t)
=
\langle \hat n_{\mathbf{l}}(t)\rangle -
\langle \hat n \rangle=\frac{1}{2}
\delta \rho_{{\mathbf{k}},\omega}
e^{
i
\left(
{\bf k}\cdot{\bf l}
-
\omega_k t
\right)
}
+
{\rm c.c.}\end{aligned}$$ Up to the first order, this change is linear in the Bragg potential, i.e., $$\begin{aligned}
\label{drho}
\delta \rho_{{\mathbf{k}},\omega}
=
\chi({\mathbf{k}},\omega)V_{{\mathbf{k}},\omega}
\;,\end{aligned}$$ where $\chi({\mathbf{k}},\omega)$ is the susceptibility. Within the Gutzwiller approximation, we obtain $$\begin{aligned}
\label{chi}
&&
\chi({\mathbf{k}},\omega)=
\\
&&
-
\left(
\begin{array}{c}
\vec{u}^{(0)}_{\bf 0}\\
\vec{u}^{(0)}_{\bf 0}
\end{array}
\right)^T
\left(
\begin{array}{cc}
A_{\bf k}-\hbar\tilde\omega & B_{\bf k}\\
B_{\bf k} & A_{\bf k}+\hbar\tilde\omega
\end{array}
\right)^{-1}
\left(
\begin{array}{c}
\vec{u}^{(0)}_{\bf 0}\\
\vec{u}^{(0)}_{\bf 0}
\end{array}
\right)
\;,
\nonumber\end{aligned}$$ where $\tilde\omega=\omega+i0$ and the vector components are defined as $u^{(0)}_{{\bf 0}n} = n c^{(0)}_{n}$. After matrix inversion, the susceptibility can be rewritten in the more simple form $$\begin{aligned}
\label{chi2}
\chi({\mathbf{k}},\omega)
=
\frac{2}{\hbar}
\sum_\lambda
\frac
{\chi_{{\mathbf{k}},\lambda} \omega_{{\mathbf{k}},\lambda}}
{
\left(
\omega + i0
\right)^2
-
\omega_{{\mathbf{k}},\lambda}^2
}
\;,\end{aligned}$$ where $\lambda$ denotes various excitation branches associated to the eigenvalues $\omega_{{\mathbf{k}},\lambda}$ and $$\chi_{{\mathbf{k}},\lambda}
=
\left|
\vec{u}^{(0)}_{\bf 0}\cdot(\vec{u}_{{\mathbf{k}},\lambda}+\vec{v}_{{\mathbf{k}},\lambda})
\right|^2$$ is the amplitude for the Bragg scattering of transition frequency $\omega_{{\mathbf{k}},\lambda}$. As we see, $\chi_{{\mathbf{k}},\lambda}$ is nothing but the square of amplitude of the density wave defined by Eq. (\[dw\]).
For long wavelength, only the lowest mode is dominant since $\chi_{{\mathbf{k}},\lambda}$ vanishes for the higher modes. This is a consequence of the orthogonality between the eigenvector components of the other mode and $(\vec{u}^{(0)}_{\bf 0}, -\vec{u}^{(0)}_{\bf 0})$ in the long-wavelength limit (see Appendix \[Der\]). Since $\lambda=1$ denotes the sound branch, we obtain the approximate expression $$\begin{aligned}
\label{chi3}
\chi({\mathbf{k}},\omega)
\stackrel{{\mathbf{k}} \rightarrow {\mathbf{0}}}{=}
\frac{2}{\hbar}
\frac{\chi_{{\mathbf{k}},1} \omega_{{\mathbf{k}},1}}{(\omega +i0)^2-\omega^2_{{\mathbf{k}},1}}
\;.\end{aligned}$$ The comparison of (\[chi3\]) with the identity $$\kappa
=
-
\chi({\mathbf{k}}={\mathbf{0}},\omega=0)
\;,$$ which follows from Eq. (\[drho\]), allows to deduce that for long wavelength $$\begin{aligned}
\label{chiapp}
\chi_{{\mathbf{k}},1}
\stackrel
{
{\mathbf{k}} \rightarrow {\mathbf{0}}
}
{=}
\frac{\kappa}{2}
c_s^0
|{\mathbf{k}}|
\;.\end{aligned}$$
The dependences of $\chi_{{\bf k},\lambda}$ on the variable $x$ defined by Eq. (\[x\]) for the excitation branches with $\lambda=1,2,3$ are shown in Fig. \[chisf\]. For the chosen values of parameters, only two lowest branches display noticeable amplitudes. Similar results have been also obtained in Ref. [@Huber]. However, the calculations in Ref. [@Huber] are valid only close to the boundaries MI-SF because the occupation numbers $n$ in Eq. (\[state\]) were restricted to $n=n_0,n_0\pm 1$.
In Fig. \[chisf1\] instead, we see that the amplitude for the third excitation branch as well as for the second one can become significant at certain densities. We would like to note that the f-sum rule is automatically fulfilled in our approach (see Sect. \[Sr\]) in contrast to Refs. [@Oosten2; @Huber].
![ Transition amplitudes $\chi_{{\mathbf{k}},\lambda}$ associated the transition frequency $\omega_{{\mathbf{k}},\lambda}$ for the lowest excitation branches ($\lambda=1,2,3$) and for $\mu/U=1.2$ and $2dJ/U=0.15$. The dashed line corresponds to the approximation (\[chiapp\]). []{data-label="chisf"}](fig11.eps){width="10cm"}
![ (color online) Transition amplitudes $\chi_{{\mathbf{k}},\lambda}$ versus the density for the first excitation branches ($\lambda=1,2,3$) and for $x=1$ and $2dJ/U=0.2$ (a), $0.3$ (b). Dashed lines show the static structure factor $S({\bf k})$. []{data-label="chisf1"}](fig12.eps){width="10cm"}
In the MI phase, the Gutzwiller approximation does not allow to observe any branches since $\chi_{{\mathbf{k}},\lambda}\equiv 0$. No Bragg response is possible, although the excitations exist in the mean-field approach. In order to allow non-vanishing response, correlations between different sites should be included which goes beyond the Gutzwiller approximation [@Navez; @Huber]. In such a description, excitations in the Bragg process are created as particle-hole pairs [@Huber; @Oosten2; @Navez]. However as pointed out in [@Navez], this last process appears to be of the second order in the inverse of coordination number $z=2d$ and, therefore, is not taken into account by the standard Gutzwiller ansatz.
One-particle Green’s function
-----------------------------
The one-particle Green’s function can be also determined in the context of the Gutzwiller approximation through the interaction term $$\begin{aligned}
\label{Hint2}
\hat H'(t)
&=&
\sum_{\bf l} \eta_{{\mathbf{k}},\omega}
e^{i({\mathbf{k}}\cdot{\mathbf{l}}-\omega t)}
\hat a^\dagger_{\bf l}
+
{\rm h.c.}
\;,\end{aligned}$$ which explicitly breaks the $U(1)$ symmetry. This interaction term induces a deviation in the order parameter $$\begin{aligned}
\left( \begin{array}{c}
\delta \psi_{\bf l}
\\
\delta \psi^*_{\bf l}
\end{array}\right)
=
\left( \begin{array}{c}
\psi_{\bf l}- \psi^{(0)}
\\
\psi^*_{\bf l}- (\psi^{(0)})^*
\end{array}\right)
\nonumber \\
=
\underline{\underline {G}}({\mathbf{k}},\omega).
\left( \begin{array}{c}
\eta_{{\mathbf{k}},\omega}
e^{i({\mathbf{k}}\cdot{\mathbf{l}}-\omega t)}
\\
\eta^*_{{\mathbf{k}},\omega}
e^{-i({\mathbf{k}}\cdot{\mathbf{l}}-\omega t)}
\end{array}\right)\end{aligned}$$ The proportionality term is the one-particle $2\times2$ matrix Green’s function with the general expression $$\begin{aligned}
\label{G}
\underline{\underline{G}}({\mathbf{k}},\omega)
=
\sum_\lambda
\frac{\underline{\underline{g}}_{{\mathbf{k}},\lambda}}
{\omega +i0 - \omega_{{\mathbf{k}},\lambda}}
\;,\end{aligned}$$ where we define the matrix transition amplitude as: $$\underline{\underline{g}}_{{\mathbf{k}},\lambda}
= \underline{b}_{{\mathbf{k}},\lambda} .\underline{b}^\dagger_{{\mathbf{k}},\lambda}$$ and $$\underline{b}_{{\mathbf{k}},\lambda}=\left(
\begin{array}{c}
\sum_{n=1}^\infty
\left(
\sqrt{n+1} u_{{{\mathbf{k}}},n+1,\lambda}+ \sqrt{n}v_{{{\mathbf{k}}},n-1, \lambda}
\right)
c^{(0)}_{n}
\\
\sum_{n=1}^\infty
\left(
\sqrt{n+1} v_{{{\mathbf{k}}},n+1,\lambda}+ \sqrt{n}u_{{{\mathbf{k}}},n-1, \lambda}
\right) c^{(0)}_{n}
\end{array}
\right)$$ Here, $\lambda$ denotes branches with both positive and negative energies. In the SF phase, the existence of the order parameter [*hybridizes*]{} the one- and the two-particle Green’s functions so that their poles are identical. On the other hand, in the MI phase, we note that the transitions forbidden in the Bragg scattering become allowed in the interaction term Eq. (\[Hint2\]). Indeed, the time-dependent Gutzwiller approach allows to recover the results previously established in the context of quantum field theory [@Oosten2]: $$\begin{aligned}
\label{G2}
\underline{\underline{G}}({\mathbf{k}},\omega)
=\underline{\underline{1}}
\sum_\pm
\frac{g_{{\mathbf{k}},\pm}}{\omega +i0 \mp \omega_{{\mathbf{k}},\pm}}
\;,\end{aligned}$$ where $$\begin{aligned}
g_{{\mathbf{k}},\pm}
=
\frac{1}{2}
\pm
\frac{(2n_0+1)U- J_{\mathbf{k}}}{\hbar (\omega_{{\mathbf{k}},+} +\omega_{{\mathbf{k}},-})}\end{aligned}$$ is the probability to create a particle (hole) excitation. Although the one-particle Green’s function is a concept of importance in the context of quantum field theory, its use in the concrete experiments is limited by the impossibility to create the $U(1)$ symmetry breaking interaction (\[Hint2\]). However, this function helps to interpret the nature of the excitation which is particle-like when one atom is added to the gas or hole-like when one atom is removed.
\[Sr\]Sum rules
---------------
Let us examine some sum rules satisfied by the susceptibility function. The compressibility sum rule is deduced from the Kramers-Kronig relation $$\begin{aligned}
\label{comp}
\int_{-\infty}^\infty \frac{d\omega}{\pi\omega}
\chi''({\mathbf{k}},\omega) = \chi({\mathbf{k}},0)
\;.\end{aligned}$$ The $f$-sum rule generalizes the one obtained for a Bose gas in continuum [@Huber] $$\begin{aligned}
\hbar^2
\int_{-\infty}^\infty
\frac{d\omega}{2\pi}
\omega \chi''({\mathbf{k}},\omega)
=
\nonumber\\
\label{fsum}
-
\sum_{\alpha=1}^d
\int
d^3{\mathbf{k}}'
\cos(k_\alpha')
\langle
\hat a^\dagger_{{\mathbf{k}}'}
\hat a_{{\mathbf{k}}'}
\rangle
4J\sin^2(k_\alpha/2)
\;.\end{aligned}$$ In order to recover the continuum limit, we have to introduce the lattice constant $a$ by means of the replacement ${\bf k}\to a{\bf k}$. In the limit of small $a$, we get [@stringari] $$\begin{aligned}
\int_{-\infty}^\infty
\frac{d\omega}{2\pi}
\omega
\chi''({\mathbf{k}},\omega)
=
-\langle \hat n \rangle \frac{{\mathbf{k}}^2}{2m}
\;,\end{aligned}$$ where $m=\hbar^2/(2Ja^2)$ corresponds to the effective mass. The application of Eq. (\[comp\]) and Eq. (\[fsum\]) in the Gutzwiller approximation leads to $$\begin{aligned}
\int_{-\infty}^\infty \frac{d\omega}{\pi\omega}
\chi''({\mathbf{k}},\omega) &\stackrel{{\mathbf{k}}\rightarrow 0}{=}& -\kappa
\;,
\\
\int_{-\infty}^\infty
\frac{d\omega}{2\pi} \omega \chi''({\mathbf{k}},\omega) &=&
-
\left|
\psi^{(0)}
\right|^2
\epsilon_{\mathbf{k}}
\;.\end{aligned}$$ Using the result (\[chi2\]), we obtain $$\begin{aligned}
\sum_\lambda
\frac{\chi_{{\mathbf{k}},\lambda}}{\omega_{{\mathbf{k}},\lambda}}
&\stackrel{{\mathbf{k}}\rightarrow 0}{=}&
\frac{\kappa}{2}
\;,
\\
\hbar^2
\sum_\lambda
\chi_{{\mathbf{k}},\lambda}
\omega_{{\mathbf{k}},\lambda}
&=&
|\psi^{(0)}|^2
\epsilon_{\mathbf{k}}
\;.\end{aligned}$$ A third sum rule concerns the static structure factor $S({\mathbf{k}})$. Using the fluctuations dissipation theorem, the dynamic structure factor is expressed as $S({\mathbf{k}},\omega)=\chi''({\mathbf{k}},\omega)/\pi$ at zero temperature so that [@stringari]: $$\begin{aligned}
\label{sfactor}
\int_{0}^\infty \frac{d\omega}{\pi}
\chi''({\mathbf{k}},\omega) = S({\mathbf{k}})
=
\langle
\delta \hat \rho_{\mathbf{k}}
\delta \hat \rho_{\mathbf{-k}}
\rangle
\;.\end{aligned}$$ where $\delta \hat \rho_{\mathbf{k}}=\sum_{\mathbf{l}}
\delta \hat n_{\mathbf{l}} e^{-i{\mathbf{k}}.{\mathbf{l}}}/L^{d/2}$. This third sum rule is not fulfilled in the Gutzwiller approximation because the correlation function is equal to the particle number fluctuations $
\langle
\delta \hat \rho_{\mathbf{k}}
\delta \hat \rho_{\mathbf{-k}}
\rangle
=
\langle
\delta^2 \hat n
\rangle
$ and thus has no ${\mathbf{k}}$-dependence. However, as in the case of a Bose gas in continuum, this sum rule allows to deduce the static structure factor. We find indeed $$\begin{aligned}
S({\mathbf{k}})
=
\sum_\lambda \chi_{{\mathbf{k}},\lambda}
\stackrel{{\mathbf{k}} \rightarrow {\mathbf{0}}}
{=}
\frac{\kappa}{2} c_s |{\mathbf{k}}|
\;.\end{aligned}$$ This last result shows an interesting feature of the sum rule approach. Starting from the lowest-order Gutzwiller approach that does not contain any correlation, the two-point correlation function is determined as a next order contribution. Similarly, starting from the time-dependent DGPE, a similar procedure has been successfully used to recover the static structure predicted from the Bogoliubov approach [@stringari; @Nozieres].
Spectra measurement
-------------------
The observation of the excitation branches in the SF phase can be made through the measurement of the total momentum ${\bf P}$. After an adequate time of flight $t$, the momentum is given by [@Dalibard] $$\begin{aligned}
{\mathbf{P}}
=
\int_V d^3 {\mathbf{x}}
\frac{M{\mathbf{x}}}{t}
n({\mathbf{x}})
=
\hbar
\int d^3 {\mathbf{k}} {\mathbf{k}} |w({\mathbf{k}})|^2
G({\mathbf{k}})
\;,\end{aligned}$$ where $w({\mathbf{p}})$ is the Fourier transform of the Wannier function and $$\begin{aligned}
G({\mathbf{k}})
=
\sum_{{\mathbf{l}},{\mathbf{l'}}}
e^{i{\mathbf{k}}\cdot({\mathbf{l}}-{\mathbf{l'}})}
\langle
\hat a^\dagger_{\bf l}
\hat a_{\bf l'}
\rangle
\;.\end{aligned}$$ For small momentum, we can assume $w({\mathbf{k}})\simeq w({\mathbf{0}})$. Calculations up to the second order in the potential allow to deduce $$\begin{aligned}
\frac{d {\mathbf{P}}}{dt}
&=&
-2{\mathbf{k}}|w({\mathbf{0}})|^2
|\frac{V_{{\mathbf{k}},\omega}}{2}|^2
{\rm Im} \chi({\mathbf{k}},\omega)
\\
&=&
2\pi
{\mathbf{k}}
|w({\mathbf{0}})\frac{V_{{\mathbf{k}},\omega}}{2}|^2
\sum_{\pm,\lambda}
\pm
\chi_{{\mathbf{k}},\lambda}
\delta(\omega \mp \omega_{{\mathbf{k}},\lambda})
\;.
\nonumber\end{aligned}$$
\[SW\]Creation of sound waves
=============================
Experimentally sound waves in a trapped Bose-Einstein condensate were created turning on and off a perturbation potential in the center of the atomic cloud [@Ketterle97; @MKS09]. The same can be done in optical lattices and numerical simulations of this kind of experiment were performed in Ref. [@Menotti04] deep in the SF phase making use of the DGPE and in Ref. [@KSDZ05] for 1D-systems using DMRG method. In this section, we do the same simulations but using the dynamical Gutzwiller ansatz. Our aim is to compare the results with that obtained by the other methods and to extract from the simulations the values of the sound velocity compatible with that calculated in Sec. \[E\].
We are interested in the solutions for $d$-dimensional lattices which have a position dependence only in one chosen spatial dimension. Then in Eq. (\[GEd\]) $\psi_{{\bf l}\pm{\bf e}_\alpha} \equiv \psi_{l\pm 1}$, if $\alpha$ is the chosen dimension, otherwise $\psi_{{\bf l}\pm{\bf e}_\alpha} \equiv \psi_{l}$. Here $l$ is the site index along the chosen dimension. Initially the atoms are prepared in the ground state of the external potential $$\label{potential}
\varepsilon_l
=
\varepsilon_0
\exp
\left[
-(l-l_0)^2/w^2
\right]
\;,\quad
l_0=(L+1)/2
\;,$$ where $\varepsilon_0$ and $w$ are the strength and the spatial width of the potential, respectively. The external potential (\[potential\]) creates a density perturbation and we calculate the corresponding ground state numerically propagating Eq. (\[GEd\]) in imaginary time [@Dalfovo] with the initial conditions $c_{ln}(0)=c_{ln}^{(0)}$, where $c_{ln}^{(0)}$ are the coefficients for the ground state of the homogeneous lattice with the local chemical potential $\mu_l=\mu-\varepsilon_l$. At $t=0$, the external potential (\[potential\]) is switched off ($\varepsilon_l\equiv 0$) and the ground state starts to evolve. We calculate numerically the evolution of the ground state in real time. Numerical calculations presented in this section are performed for $d=3$ and we used $L=200$, $N=10$, which is enough to avoid parasitic effects due to the reflection from the boundaries and due to the cut-off in the occupation numbers.
Deep in the SF phase
--------------------
First we do simulations deep in the SF phase. In this case there is no difference between $\langle\hat n_l\rangle$ and $\left|\psi_l\right|^2$. Time evolution of $\langle\hat n_l\rangle$ for two negative values of $\varepsilon_0$ is shown in Fig. \[nav-J\_6-n\_1-w\_1\]. Initially the distribution of the atoms have a maximum at the position of the potential (bright perturbation). After switching off the potential the density perturbation splits up in two wave packets propagating symmetrically outward. At longer times, the form of the propagating maxima becomes irregular which signals the creation of shock waves [@Menotti04; @KSDZ05]. The irregularities become more pronounced for larger values of $\left|\varepsilon_0\right|$. An additional dip arises at the fronts of the wave packets which might stem from the discreteness of the lattice [@KSDZ05]. For positive values of $\varepsilon_0$ (gray perturbation), the dynamics is pretty much the same except that the maxima are replaced by minima and vice versa and the distribution of the atoms is more regular (Fig. \[nav-J\_6-n\_1-a\_0.6-w\_1\]).
![ Time evolution of the mean occupation numbers $\langle\hat{n}_l\rangle$ deep in the SF phase after switching off the potential with $w=1$, $\varepsilon_0/U=-0.6$ (a), $-0.1$ (b). The parameters are $\langle \hat n \rangle=1$, $2dJ/U=6$. The curves show the spatial dependences at the dimensionless time $\tau=tU/\hbar=$ $0$ (0), $2$ (1), $4$ (2), $6$ (3), $8$ (4), $10$ (5). They are shifted by $0.4$ (a) and $0.1$ (b) in the vertical direction with respect to the previous one. []{data-label="nav-J_6-n_1-w_1"}](fig13.eps){width="10cm"}
![ Time evolution of the mean occupation numbers $\langle\hat{n}_l\rangle$ deep in the SF phase after switching off the potential with $w=1$, $\varepsilon_0/U=0.6$. The parameters are $\langle \hat n \rangle=1$, $2dJ/U=6$. The curves show the spatial dependences at the dimensionless time $\tau=tU/\hbar=$ $0$ (0), $2$ (1), $4$ (2), $6$ (3), $8$ (4), $10$ (5). They are shifted by $0.3$ in the vertical direction with respect to the previous one. []{data-label="nav-J_6-n_1-a_0.6-w_1"}](fig14.eps){width="10cm"}
If we increase the width of the external potential $w$, the atomic distribution becomes more regular (compare Figs. \[nav-J\_6-n\_1-w\_1\](a) and \[nav-J\_6-n\_1-a\_-0.6-w\_5\]). In Fig. \[nav-J\_6-n\_1-a\_-0.6-w\_5\] one can clearly see that the form of the wave packets become very asymmetric during their propagation.
Figs. \[nmax-J\_6-n\_1-w\_5\] and \[nmin-J\_6-n\_1-w\_5\] show the time dependence of the global maximum and minimum of the atomic distribution $\langle\hat n_l\rangle$ for negative and positive values of $\varepsilon_0$. When the external potential is switched off, the amplitude of the density perturbation goes down and after some finite time seen as the first plateau in Fig. \[imax-J\_6-n\_1-w\_5\] two separate wave packets are formed which propagate in opposite directions. Their amplitude decreases monotonically in time for negative $\varepsilon_0$ (Fig. \[nmax-J\_6-n\_1-w\_5\]) and for small enough positive $\varepsilon_0$ (Fig. \[nmin-J\_6-n\_1-w\_5\]). For larger positive $\varepsilon_0$, the amplitude of the minima increases before it starts to decrease. Due to the discreteness of the lattice the position of the propagating extremuma is a step-function of time (Fig. \[imax-J\_6-n\_1-w\_5\]) and the amplitude of the density perturbation shows up oscillations (Figs. \[nmax-J\_6-n\_1-w\_5\], \[nmin-J\_6-n\_1-w\_5\]) which are stronger for larger values of $\left|\varepsilon_0\right|$.
![ Time evolution of the mean occupation numbers $\langle\hat{n}_l\rangle$ deep in the SF phase after switching off the potential with $w=5$, $\varepsilon_0/U=-0.6$. The parameters are $\langle \hat n \rangle=1$, $2dJ/U=6$. The curves show the spatial dependences at the dimensionless time $\tau=tU/\hbar=$ $0$ (0), $2$ (1), $4$ (2), $6$ (3), $8$ (4), $10$ (5). They are shifted by $0.3$ in the vertical direction with respect to the previous one. []{data-label="nav-J_6-n_1-a_-0.6-w_5"}](fig15.eps){width="10cm"}
![ Time evolution of the largest mean occupation number $\langle\hat{n}_l\rangle_{max}$ after switching off the potential with $w=5$, $\varepsilon_0/U=-0.1$ (1), $-0.2$ (2), $-0.3$ (3), $-0.4$ (4), $-0.5$ (5), $-0.6$ (6). The parameters are $\langle \hat n \rangle=1$, $2dJ/U=6$. []{data-label="nmax-J_6-n_1-w_5"}](fig16.eps){width="10cm"}
![ Time evolution of the smallest mean occupation number $\langle\hat{n}_l\rangle_{min}$ after switching off the potential with $w=5$, $\varepsilon_0/U=0.1$ (1), $0.2$ (2), $0.3$ (3), $0.4$ (4), $0.5$ (5), $0.6$ (6). The parameters are $\langle \hat n \rangle=1$, $2dJ/U=6$. []{data-label="nmin-J_6-n_1-w_5"}](fig17.eps){width="10cm"}
![ Location of the largest mean occupation number $\langle\hat{n}_l\rangle_{max}$ after switching off the potential with $w=5$, $\varepsilon_0/U=-0.1$ (1), $-0.6$ (6). The parameters are $\langle \hat n \rangle=1$, $2dJ/U=6$. Here we use the same labels for the curves as in Fig. \[nmax-J\_6-n\_1-w\_5\]. Straight lines are linear fits to the numerical data. []{data-label="imax-J_6-n_1-w_5"}](fig18.eps){width="10cm"}
Before the system enters the shock wave regime, there are always pronounced global maxima (minima) in the case of negative (positive) $\varepsilon_0$ and the propagation velocity of the sound wave packets can be identified with the velocity of the global extremum. Numerical values of the propagation velocity $c_s$ in our simulations are determined with the aid of a linear fit for the location of the global extremum as a function of time (straight lines in Fig. \[imax-J\_6-n\_1-w\_5\]). Its dependence on the amplitude of the external potential is shown in Fig. \[cs-J\_6-n\_1-w\_5\]. Propagation velocity decreases monotonically with $\varepsilon_0$. This can be understood looking at the behavior of the function $c_s^0(\mu'=\mu-\varepsilon_0)$ under variation of $\varepsilon_0$ at fixed $2dJ/U$. For large values of $2dJ/U$, it is also a decreasing function of $\varepsilon_0$ (see Fig. \[sv\] and the dashed line in Fig. \[cs-J\_6-n\_1-w\_5\]), which is quite close to the data of our numerical simulations.
In order to extract the value of the sound velocity from the numerical data, we have to extrapolate to $\varepsilon_0=0$. This is done making a quadratic fit to the data points which is justified by the fact that, near $\varepsilon_0\approx 0$, $c_s^0(\mu'=\mu-\varepsilon_0)$ can be decomposed into a series in powers of $\varepsilon_0$. In the example shown in Fig. \[cs-J\_6-n\_1-w\_5\], the extrapolated value of the propagation velocity is $1.433$ which is a bit higher than $c_s^0=1.372$ predicted by the linear response theory. We have done the same calculations for different values of $w$ and found that the propagation velocity becomes more close to $c_s^0$ for larger values of $w$ (see Fig. \[cs-J\_6-n\_1\]). Therefore, the deviation from Eq. (\[c\_s\]) is due to the contribution of excitations with finite wavelengths.
![ Dependence of the propagation velocity on the strength of the external potential. The parameters are $\langle \hat n \rangle=1$, $2dJ/U=6$, $w=5$. The dots are the results of numerical calculations and the solid line is a fit by quadratic polynomial. The dashed line shows $c_s^0(\mu'=\mu-\varepsilon_0)$ as a function of $\varepsilon_0$ with $\mu$ fixed by the values of $\langle\hat n\rangle$ and $2dJ/U$. []{data-label="cs-J_6-n_1-w_5"}](fig19.eps){width="10cm"}
![ Dependence of the propagation velocity on the width of the external potential. The parameters are $\langle \hat n \rangle=1$, $2dJ/U=6$. []{data-label="cs-J_6-n_1"}](fig20.eps){width="10cm"}
Near the boundary of the SF-MI transition
-----------------------------------------
Still in the SF phase but near the boundary of the SF-MI transition, $\left|\psi_l\right|^2 < \langle\hat n_l\rangle$ (Fig \[J\_0.172-n\_1-a\_-0.3-w\_5\]). Numerical simulations in this regime show that, in order to excite only the lowest mode, much less values of $\varepsilon_0$ are required. This is consistent with the fact that the gap between the first and second excitation modes is very small near the MI lobe.
Fig. \[cs-J\_0.172-n\_1-w\_5\] shows the dependence of the propagation velocity on $\varepsilon_0$ near the tip of the MI lobe with $n_0=1$. It is quite different compared to the behavior deep in the SF regime where the propagation velocity monotonically decreases (Fig. \[cs-J\_6-n\_1-w\_5\]). Near the tip of the MI lobe, the propagation velocity has a maximum around $\varepsilon_0\approx 0$ which is qualitatively similar to the behavior of $c_s^0(\mu'=\mu-\varepsilon_0)$. However, the discrepancy between the propagation velocity and $c_s^0(\mu'=\mu-\varepsilon_0)$ is large even for small $|\varepsilon_0|$ due to the significant contribution of nonlinear effects.
![ Time evolution of the mean occupation numbers $\langle\hat{n}_l\rangle$ (a) and the mean number of condensed atoms $|\psi_l|^2$ (b) near the tip of the MI-lobe after switching off the potential with $\varepsilon_0/U=-0.3$, $w=5$. The parameters are $\langle \hat n \rangle=1$, $2dJ/U=0.172$. The curves show the spatial dependences at the dimensionless time $\tau=tU/\hbar=$ $0$ (0), $24$ (1), $48$ (2), $72$ (3), $96$ (4), $120$ (5). They are shifted by $0.1$ (a) and $0.2$ (b) in the vertical direction with respect to the previous one. []{data-label="J_0.172-n_1-a_-0.3-w_5"}](fig21.eps){width="10cm"}
![ Dependence of the propagation velocity on the strength of the external potential. The parameters are $\langle \hat n \rangle=1$, $2dJ/U=0.172$, which is close to the tip of the MI lobe \[$2d(J/U)_c^{max}=0.17157$\], $w=5$. The dots are the results of numerical calculations and the solid line is a fit by quadratic polynomial. The dashed line shows $c_s^0(\mu'=\mu-\varepsilon_0)$ as a function of $\varepsilon_0$ with $\mu$ fixed by the values of $\langle\hat n\rangle$ and $2dJ/U$. []{data-label="cs-J_0.172-n_1-w_5"}](fig22.eps){width="10cm"}
We have determined the propagation velocity in the limit $\varepsilon_0\to 0$ making again a quadratic fit to the numerical data presented in Fig. \[cs-J\_0.172-n\_1-w\_5\]. This procedure gives us the value $0.199$, while from Eq. (\[c\_s\]) we get $c_s^0=0.201$.
MI phase
--------
If the parameters of the external potential $\varepsilon_l$ are chosen such that the values of $\mu_l$ are always within the MI phase, the density is not perturbed and, therefore, there will be no time dynamics when the potential is switched off. In Fig. \[temi\], we show an example of a broader potential, where local SF regions appear near the center of the potential. When the potential is switched off, the local inhomogeneities just spread without creation of any propagating wave packets. This is consistent with the fact that the sound waves do not exist in the MI phase.
![ Time evolution of the mean occupation numbers $\langle\hat{n}_l\rangle$ (a) and the mean number of condensed atoms $|\psi_l|^2$ (b) in the MI phase after switching off the potential with $\varepsilon_0/U=1$, $w=5$. The parameters are $\mu/U=1.2$ and $2dJ/U=0.07$, which is a bit smaller than the critical value $2d(J/U)_c=0.073$. The curves show the spatial dependences at the dimensionless time $\tau=tU/\hbar=$ $0$ (0), $20$ (1), $40$ (2), $60$ (3), $80$ (4), $100$ (5). They are shifted by $0.5$ in the vertical direction with respect to the previous one. []{data-label="temi"}](fig23.eps){width="10cm"}
\[C\]Conclusion
===============
We have studied collective excitations of interacting bosons in a lattice at zero temperature within the framework of the [*gapless*]{} and [*conserving*]{} time-dependent Gutzwiller approximation. The excitation modes are calculated within the framework of the linear-response theory considering small perturbation of the many-body ground state. We demonstrated that the lowest-energy excitation of the SF has a phonon-like dispersion relation and derived an analytical expression for the sound velocity in terms of compressibility and the condensate density which coincides with the hydrodynamic relation.
We have studied the response of the lattice Bose gas in the Bragg scattering process which provides an experimental tool to observe the excitations. It is demonstrated that the susceptibility function satisfies the f-sum rule in the whole parameter region. Calculations of the transition amplitudes show that within the Gutzwiller approximation the MI does not respond to the perturbation caused by the Bragg potential. In the SF phase, we show that only three lowest excitation branches have significant transition amplitudes, the others being too small to be observed in a real experiment. The absence of response in the MI phase is a limitation of the Gutzwiller approximation and can be corrected in the next leading order in the inverse of the coordination number which would take into account the possibility of a particle-hole pair creation [@Navez; @Huber].
Finally, we have performed simulations of the sound-wave propagation solving numerically the Gutzwiller equations. The calculations show that sound waves can be created only in the SF phase and the corresponding velocity is in a good agreement with the results of the linear-response theory.
\[Der\]Derivation of Eq. (\[c\_s\])
===================================
In order to work out the sound velocity $c_s^0$, we consider the limit of small $|{\mathbf{k}}|$ and look for the lowest energy solution of Eq. (\[evpexc\]) as an expansion with respect to ${\mathbf{k}}$: $$\begin{aligned}
\label{pert}
\vec{u}_{\bf k}
&=&
\vec{u}_{\bf k}^{(0)}
+
\vec{u}_{\bf k}^{(1)}
+
\vec{u}_{\bf k}^{(2)}+ \dots
\;,
\nonumber\\
\vec{v}_{\bf k}
&=&
\vec{v}_{\bf k}^{(0)}
+
\vec{v}_{\bf k}^{(1)}
+
\vec{v}_{\bf k}^{(2)} +\dots
\;,
\nonumber\\
\omega_{\bf k}
&=&
\omega_{\bf k}^{(0)}
+
\omega_{\bf k}^{(1)}
+
\omega_{\bf k}^{(2)} +\dots
\;.\end{aligned}$$ The zeroth-order solution satisfies the equation $$\hbar\omega_{\bf k}^{(0)}
\left(
\begin{array}{c}
\vec{u}_{\bf k}^{(0)}\\
\vec{v}_{\bf k}^{(0)}
\end{array}
\right)
=
\left(
\begin{array}{cc}
A_{0} & B_{0}\\
-B_{0} & -A_{0}
\end{array}
\right)
\left(
\begin{array}{c}
\vec{u}_{\bf k}^{(0)}\\
\vec{v}_{\bf k}^{(0)}
\end{array}
\right)$$ and is non-trivial only in the SF phase with the form: $$\label{u0}
u_{{\bf k}n}^{(0)}
\equiv
\left(
n
+
\frac{\partial \hbar\omega_0}{\partial \mu}
\right)
c_n^{(0)}
\;,\quad
v_{{\bf k}n}^{(0)}
=
-u_{{\bf k}n}^{(0)}
\;,\quad
\omega_{\bf k}^{(0)}=0
\;.$$ The quantities of the first order are governed by the equation $$\hbar\omega_{\bf k}^{(1)}
\left(
\begin{array}{c}
\vec{u}_{\bf k}^{(0)}\\
\vec{v}_{\bf k}^{(0)}
\end{array}
\right)
=
\left(
\begin{array}{cc}
A_{0} & B_{0}\\
-B_{0} & -A_{0}
\end{array}
\right)
\left(
\begin{array}{c}
\vec{u}_{\bf k}^{(1)}\\
\vec{v}_{\bf k}^{(1)}
\end{array}
\right)
\;.$$ Taking into account the identity $$\sum_{n'}
\left(
A_0^{nn'}
+
B_0^{nn'}
\right)
\frac{\partial c_{n'}^{(0)}}{\partial\mu}
=
\left(n+\frac{\partial \hbar\omega_0}{\partial \mu}\right)c_n^{(0)}
\;,$$ the first-order solution can be written as $$\label{u1}
{u}_{{\bf k}n}^{(1)}
=
{v}_{{\bf k}n}^{(1)}
=
\hbar\omega_{\mathbf{k}}^{(1)}
\frac{\partial c_n^{(0)}}{\partial\mu}
\;.$$ We substitute all these results in the equation for the quantities of the second order in ${\mathbf{k}}$ $$\begin{aligned}
\label{eqexc2}
&&
\hbar\omega_{\bf k}^{(1)}
\left(
\begin{array}{c}
\vec{u}_{\bf k}^{(1)}\\
\vec{v}_{\bf k}^{(1)}
\end{array}
\right)
+
\hbar\omega_{\bf k}^{(2)}
\left(
\begin{array}{c}
\vec{u}_{\bf k}^{(0)}\\
\vec{v}_{\bf k}^{(0)}
\end{array}
\right)
\nonumber\\
&&
=
\left(
\begin{array}{cc}
A_{0} & B_{0}\\
-B_{0} & -A_{0}
\end{array}
\right)
\left(
\begin{array}{c}
\vec{u}_{\bf k}^{(2)}\\
\vec{v}_{\bf k}^{(2)}
\end{array}
\right)
\\
&&
+
\left(
\begin{array}{cc}
A_{\bf k}^{(2)} & B_{\bf k}^{(2)}\\
-B_{\bf k}^{(2)} & -A_{\bf k}^{(2)}
\end{array}
\right)
\left(
\begin{array}{c}
\vec{u}_{\bf k}^{(0)}\\
\vec{v}_{\bf k}^{(0)}
\end{array}
\right)
\;.
\nonumber\end{aligned}$$ Multiplying Eq. (\[eqexc2\]) by the vector $(\vec{u}_{\bf k}^{(0)}\;,\; -\vec{v}_{\bf k}^{(0)})$ from the left side and taking into account that, $$\vec{u}_{\bf k}^{(0)} \cdot \vec{u}_{\bf k}^{(0)}
-
\vec{v}_{\bf k}^{(0)} \cdot \vec{v}_{\bf k}^{(0)}
=
0
\;,$$ $$\sum_n
\left(
{u}_{{\bf k}n}^{(0)}
-
{v}_{{\bf k}n}^{(0)}
\right)
\frac{\partial c_n^{(0)}}{\partial\mu}
=
\frac{\partial\langle\hat n\rangle}{\partial\mu}
\equiv
\kappa
\;,$$ where $\kappa$ is the compressibility, we arrive to Eq. (\[c\_s\]) for the sound velocity.
\[Bdr\]Bogoliubov’s dispersion relation
=======================================
We look for the solution of Eq. (\[evpexc\]) for arbitrary ${\bf k}$ in the form $$\begin{aligned}
\label{ab}
\vec{u}_{{\bf k}}
&=&
\vec{u}_{{\bf k}}^{(0)}
a_{\bf k}
+
\vec{u}_{{\bf k}}^{(1)}
b_{\bf k}
\nonumber\\
\vec{v}_{{\bf k}}
&=&
\vec{v}_{{\bf k}}^{(0)}
a_{\bf k}
+
\vec{v}_{{\bf k}}^{(1)}
b_{\bf k}\end{aligned}$$ where ${u}_{{\bf k}n}^{(0,1)}$ and ${v}_{{\bf k}n}^{(0,1)}$ are given by Eqs. (\[u0\]), (\[u1\]) with $\omega_{\bf k}^{(1)}$ being replaced by $\omega_{\bf k}$.
Plugging (\[ab\]) into Eq. (\[evpexc\]) and multiplying the resulting equations by vectors $\vec{u}_{{\bf k}}^{(0)}$, $\vec{u}_{{\bf k}}^{(1)}$ we obtain linear homogeneous equations for $a_{\bf k}$ and $b_{\bf k}$: $$\begin{aligned}
\hbar\omega_{\bf k}
\left(
\vec{u}_{\bf k}^{(0)}
\cdot
\vec{u}_{\bf k}^{(1)}
\right)
a_{\bf k}
&=&
\sum_{n,n'}
{u}_{{\bf k}n}^{(1)}
\left(
A_{\bf k}^{nn'}
+
B_{\bf k}^{nn'}
\right)
{u}_{{\bf k}n'}^{(1)}
b_{\bf k}
\nonumber\\
\hbar\omega_{\bf k}
\left(
\vec{u}_{\bf k}^{(0)}
\cdot
\vec{u}_{\bf k}^{(1)}
\right)
b_{\bf k}
&=&
\sum_{n,n'}
{u}_{{\bf k}n}^{(0)}
\left(
A_{\bf k}^{nn'}
-
B_{\bf k}^{nn'}
\right)
{u}_{{\bf k}n'}^{(0)}
a_{\bf k}
\nonumber\end{aligned}$$ The relations $$\vec{u}_{\bf k}^{(0)}
\cdot
\vec{u}_{\bf k}^{(1)}
=
\hbar
\omega_{\bf k}
\kappa/2
\;,$$ $$\begin{aligned}
&&
\sum_{n,n'}
{u}_{{\bf k}n}^{(1)}
\left(
A_{\bf k}^{nn'}
+
B_{\bf k}^{nn'}
\right)
{u}_{{\bf k}n'}^{(1)}
\\
&=&
\left(
\hbar\omega_{\bf k}
\right)^2
\left[
\frac{\kappa}{2}
+
\left(
\frac
{\partial\psi^{(0)}}
{\partial\mu}
\right)^2
\epsilon_{\bf k}
\right]
\;,
\nonumber\end{aligned}$$ $$\sum_{n,n'}
{u}_{{\bf k}n}^{(0)}
\left(
A_{\bf k}^{nn'}
-
B_{\bf k}^{nn'}
\right)
{u}_{{\bf k}n'}^{(0)}
=
\epsilon_{\bf k}
{\psi^{(0)}}^2
\;,$$ lead us to the result $$\begin{aligned}
\label{dispersion}
\hbar
\omega_{\mathbf{k}}
=
\sqrt
{
\frac{{2\psi^{(0)}}^2}{\kappa}
\epsilon_{\bf k}
+
\left(\frac{\partial {\psi^{(0)}}^2}{\partial \langle \hat n \rangle}\right)^2
\epsilon_{\bf k}^2
}
\;.\end{aligned}$$ In the limit of small $U/J$, $\langle \hat n\rangle ={\psi^{(0)}}^2$ and we recover the well-known Bogoliubov’s dispersion relation. For arbitrary values of $U/J$, Eq. (\[dispersion\]) better describes the lowest excitation branch of the SF than the standard Bogoliubov’s dispersion relation but gives higher values of energies compared to the exact numerical data.
\[EG\]Energy gaps
=================
We first rewrite Eq. (\[evpexc\]) in the form $$\hbar\omega_{\bf k}
\left(
\begin{array}{c}
|{u}_{\bf k}\rangle\\
|{v}_{\bf k}\rangle
\end{array}
\right)
=
\left(
\begin{array}{cc}
\hat A_{\bf k} & \hat B_{\bf k}\\
-\hat B_{\bf k} & -\hat A_{\bf k}
\end{array}
\right)
\left(
\begin{array}{c}
|{u}_{\bf k}\rangle\\
|{v}_{\bf k}\rangle
\end{array}
\right)
\;,$$ where the operators $$\begin{aligned}
\hat A_{\bf k}
&=&
-J_{\bf 0}
\psi^{(0)}
(\hat a+{\hat a}^\dagger)
+
\frac{U}{2}
\hat n(\hat n-1)
-\mu \hat n
-\hbar\omega_0
\nonumber\\
&-&
J_{\bf k}
{\hat a}
|s^{(0)}\rangle
\langle s^{(0)}|
{\hat a}^\dagger
+
{\hat a}^\dagger
|s^{(0)}\rangle
\langle s^{(0)}|
\;,
\nonumber\\
\hat B_{\bf k}
&=&
-J_{\bf k}
{\hat a}
|s^{(0)}\rangle
\langle s^{(0)}|
{\hat a}
+
{\hat a}^\dagger
|s^{(0)}\rangle
\langle s^{(0)}|
{\hat a}^\dagger
\;,
\nonumber\end{aligned}$$ with the ground state $|s^{(0)}\rangle$ defined by Eq. (\[state\]), act on the kets $$\begin{aligned}
|u_{\bf k}\rangle
=
\sum_n
u_{{\bf k}n}|n\rangle
\;,\quad
|v_{\bf k}\rangle
=
\sum_n
v_{{\bf k}n}|n\rangle
\;.\end{aligned}$$
The zeroth order exact solution in the small parameter $U/J$ is given by $$\begin{aligned}
|u_{{\bf k}\lambda}^{(0)}\rangle
&=&
\hat D
\left(
\psi^{(0)}
\right)
|\lambda\rangle
\;,
\nonumber\\
|v_{{\bf k}\lambda}^{(0)}\rangle
&=&
0
\;,\end{aligned}$$ where $\hat D(\alpha)=\exp(\alpha \hat a^\dagger -\alpha^* \hat a)$ is the displacement operator. This can be shown using the property of the displacement operator $$\begin{aligned}
\hat D(\alpha)
\hat a
\hat D(\alpha)
=
\hat a + \alpha
\;.\end{aligned}$$ Finally, first order perturbation theory in small $U/J$ and the relation $$\begin{aligned}
\langle u_{{\bf k}\lambda}^{(0)}|
\hat B_{\bf k}
|u_{{\bf k}\lambda}^{(0)}\rangle
=0\end{aligned}$$ valid for $\lambda\geq 2$ allow to establish that $$\begin{aligned}
\hbar \omega^{(1)}_{{\bf k}\lambda}
=
\langle u_{{\bf k }\lambda}^{(0)}|
\hat A_{\bf k}
|u_{{\bf k }\lambda}^{(0)}\rangle
\;.\end{aligned}$$ Explicit calculation of the matrix element leads to the result (\[gap2\]) for the gaps in the excitation spectrum.
This work was supported by the SFB/TR 12 of the German Research Foundation (DFG). Helpful discussions with Robert Graham are gratefully acknowledged.
[999]{} M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Phys. Rev. B [**40**]{}, 546 (1989).
D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Phys. Rev. Lett. [**81**]{}, 3108 (1998).
A. M. Rey, K. Burnett, R. Roth, M. Edwards, C. J. Williams, and C. W. Clark, J. Phys. B [**36**]{}, 825 (2003).
M. A. Cazalilla, Phys. Rev. A [**70**]{}, 041604 (2004).
R. Roth and K. Burnett, Phys. Rev. A [**67**]{}, 031602(R) (2003).
R. Roth and K. Burnett, Phys. Rev. A [**68**]{}, 023604 (2003).
R. Roth and K. Burnett, J. Phys. B [**37**]{}, 3893 (2004).
M. Greiner, O. Mandel, T. Esslinger, T. Hänsch, and I. Bloch, Nature [**415**]{}, 51 (2002).
Y. Kato and N. Kawashima, Phys. Rev. E [**79**]{}, 021104 (2009).
P. Pippan, H. G. Evertz, and M. Hohenadler, Phys. Rev. A [**80**]{}, 033612 (2009).
B. Capogrosso-Sansone, Ş. G. Söyler, N. Prokof’ev, and B. Svistunov, Phys. Rev. A [**77**]{}, 015602 (2008).
L. Mühlbacher and E. Rabani, Phys. Rev. Lett. [**100**]{}, 176403 (2008); Ph. Werner, T. Oka, and A. J. Millis, Phys. Rev. B [**79**]{}, 035320 (2009).
C. Kollath, U. Schollwck, J. von Delft, and W. Zwerger, Phys. Rev. A [**69**]{}, 031601(R) (2004).
C. Kollath, U. Schollwöck, J. von Delft, and W. Zwerger, Phys. Rev. A [**71**]{}, 053606 (2005).
S. Ramanan, T. Mishra, M. S. Luthra, R. V. Pai, and B. P. Das, Phys. Rev. A [**79**]{}, 013625 (2009).
S. Ejima, H. Fehske, and F. Gebhard, Europhys. Lett. [**93**]{}, 30002 (2011).
U. Schollwöck, Rev. Mod. Phys. [**77**]{}, 259 (2005).
L. P. Pitaevskii and S. Stringari, [*Bose-Einstein Condensation*]{}, Oxford University Press, Oxford, 2003.
C. Menotti, M. Krämer, L. Pitaevskii, and S. Stringari, Phys. Rev. A [**67**]{}, 053609 (2003).
C. Menotti, M. Krämer, A. Smerzi, L. Pitaevskii, and S. Stringari, Phys. Rev. A [**70**]{}, 023609 (2004).
I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. [**80**]{}, 885 (2008).
M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen (De), and U. Sen, Adv. Phys. [**56**]{}, 243 (2007).
D. S. Rokhsar and B. G. Kotliar, Phys. Rev. B [**44**]{}, 10328 (1991).
W. Krauth, M. Caffarel, and J.-P. Bouchaud, Phys. Rev. B [**45**]{}, 3137 (1992).
P. Navez and R. Schützhold, Phys. Rev. A [**82**]{}, 063603 (2010).
D. Jaksch, V. Venturi, J. I. Cirac, C. J. Williams, and P. Zoller, Phys. Rev. Lett. [**89**]{}, 040402 (2002).
B. Damski, J. Zakrzewski, L. Santos, P. Zoller, and M. Lewenstein, Phys. Rev. Lett. [**91**]{}, 080403 (2003).
J. Zakrzewski, Phys. Rev. A [**71**]{}, 043601 (2005).
M. Snoek and W. Hofstetter, Phys. Rev. A [**76**]{}, 051603(R) (2007).
C. Trefzger, C. Menotti, and M. Lewenstein, Phys. Rev. A [**78**]{}, 043604 (2008).
K. V. Krutitsky, J. Larson, and M. Lewenstein, Phys. Rev. A [**82**]{}, 033618 (2010).
K. Sheshadri, H. R. Krishnamurthy, R. Pandit, and T. V. Ramakrishnan, Europhys. Lett. [**22**]{}, 257 (1993).
D. van Oosten, D. B. M. Dickerscheid, B. Farid, P. van der Straten, and H. T. C. Stoof, Phys. Rev. A [**71**]{}, 021601(R) (2005).
F. Gerbier, A. Widera, S. Fölling, O. Mandel, T. Gericke, and I. Bloch, Phys. Rev. A [**72**]{}, 053606 (2005).
C. Menotti and N. Trivedi, Phys. Rev. B [**77**]{}, 235120 (2008).
K. R. A. Hazzard and E. J. Mueller, Phys. Rev. A [**81**]{}, 033404 (2010).
S. D. Huber, E. Altman, H. P. Büchler, and G. Blatter, Phys. Rev. B [**75**]{}, 085106 (2007).
S. D. Huber, B. Theiler, E. Altman, and G. Blatter, Phys. Rev. Lett. [**100**]{}, 050404 (2008).
D. van Oosten, P. van der Straten, and H. T. C. Stoof, Phys. Rev. A [**63**]{}, 053601 (2001).
K. Sengupta and N. Dupuis, Phys. Rev. A [**71**]{}, 033629 (2005).
D. B. M. Dickerscheid, D. van Oosten, P. J. H. Denteneer, and H. T. C. Stoof, Phys. Rev. A [**68**]{}, 043623 (2003).
S. Konabe, T. Nikuni, and M. Nakamura, Phys. Rev. A [**73**]{}, 033621 (2006).
Y. Ohashi, M. Kitaura, and H. Matsumoto, Phys. Rev. A [**73**]{}, 033617 (2006).
T. D. Grass, F. E. A. dos Santos, and A. Pelster, arXiv:1003.4197, arXiv:1011.5639.
D. L. Kovrizhin, G. V. Pai, and S. Sinha, Europhys. Lett. [**72**]{}, 162 (2005).
U. Bissbort, Y. Li, S. Götze, J. Heinze, J. S. Krauser, M. Weinberg, C. Becker, K. Sengstock, and W. Hofstetter, arXiv:1010.2205.
D. L. Kovrizhin, G. V. Pai, and S. Sinha, arXiv:0707.2937.
P. T. Ernst, S. Götze, J. S. Krauser, K. Pyka, D.-S. Lühmann, D. Pfannkuche, and K. Sengstock, Nature Physics [**6**]{}, 56 (2009).
X. Du, S. Wan, E. Yesilada, C. Ryu, D. J. Heinzen, Z. X. Liang, and B. Wu, New J. Phys. [**12**]{}, 083025 (2010).
D. Clément, N. Fabbri, L. Fallani, C. Fort, and M. Inguscio, Phys. Rev. Lett. [**102**]{}, 155301 (2009).
D. Clément, N. Fabbri, L. Fallani, C. Fort and M. Inguscio, J. Low Temp. Phys. [**158**]{}, 5 (2010).
A. M. Rey, P. B. Blakie, G. Pupillo, C. J. Williams, and C. W. Clark, Phys. Rev. A [**72**]{}, 023407 (2005).
G. Pupillo, A. M. Rey, and G. G. Batrouni, Phys. Rev. A [**74**]{}, 013601 (2006).
T. K. Ghosh and K. Machida, J. Phys. B [**40**]{}, 2587 (2007).
J.-S. Caux and P. Calabrese, Phys. Rev. A [**74**]{}, 031605(R) (2006).
P. Buonsante, V. Penna, A. Vezzani, and P. B. Blakie, Phys. Rev. A [**76**]{}, 011602(R) (2007).
N. Elstner and H. Monien, Phys. Rev. B [**59**]{}, 12184 (1999).
S. Sachdev, [*Quantum Phase Transitions*]{} (Cambridge University Press, Cambridge, England, 2001).
E. Taylor and E. Zaremba, Phys. Rev. A [**68**]{}, 053611 (2003).
D. Pines and P. Nozières, [The Theory of Quantum Liquids]{} (Addison-Wesley, Readin, MA, 1989), Vol. I and II.
M. R. Andrews, D. M. Kurn, H.-J. Miesner, D. S. Durfee, C. G. Townsend, S. Inouye, and W. Ketterle, Phys. Rev. Lett. [**79**]{}, 553 (1997).
R. Meppelink, S. B. Koller, and P. van der Straten, Phys. Rev. A [**80**]{}, 043605 (2009).
F. Dalfovo and S. Stringari, Phys. Rev. A [**53**]{}, 2477 (1996).
|
---
abstract: |
Robust inference of a low-dimensional parameter in a large semi-parametric model relies on external estimators of infinite-dimensional features of the distribution of the data. Typically, only one of the latter is optimized for the sake of constructing a well behaved estimator of the low-dimensional parameter of interest. Optimizing more than one of them for the sake of achieving a better bias-variance trade-off in the estimation of the parameter of interest is the core idea driving the general template of the collaborative targeted minimum loss-based estimation (C-TMLE) procedure.
The original implementation/instantiation of the C-TMLE template can be presented as a greedy forward stepwise C-TMLE algorithm. It does not scale well when the number $p$ of covariates increases drastically. This motivates the introduction of a novel implementation/instantiation of the C-TMLE template where the covariates are pre-ordered. Its time complexity is $\mathcal{O}(p)$ as opposed to the original $\mathcal{O}(p^2)$, a remarkable gain. We propose two pre-ordering strategies and suggest a rule of thumb to develop other meaningful strategies. Because it is usually unclear a priori which pre-ordering strategy to choose, we also introduce another implementation/instantiation called SL-C-TMLE algorithm that enables the data-driven choice of the better pre-ordering strategy given the problem at hand. Its time complexity is $\mathcal{O}(p)$ as well.
The computational burden and relative performance of these algorithms were compared in simulation studies involving fully synthetic data or partially synthetic data based on a real world large electronic health database; and in analyses of three real, large electronic health databases. In all analyses involving electronic health databases, the greedy C-TMLE algorithm is unacceptably slow. Simulation studies indicate our scalable C-TMLE and SL-C-TMLE algorithms work well. All C-TMLEs are publicly available in a Julia software package.
author:
- |
Cheng Ju, Susan Gruber, Samuel D. Lendle, Antoine Chambaz,\
Jessica M. Franklin, Richard Wyss, Sebastian Schneeweiss,\
Mark J. van der Laan
bibliography:
- 'references.bib'
title: 'Scalable Collaborative Targeted Learning for High-Dimensional Data'
---
Introduction
============
The general template of collaborative double robust targeted minimum loss-based estimation (C-TMLE; “C-TMLE template” for short) builds upon the targeted minimum loss-based estimation (TMLE) template [@van2011targeted; @van2010collaborative]. Both the TMLE and C-TMLE templates can be viewed as meta-algorithms which map a set of user-supplied choices/hyper-parameters ([ e.g.]{}, parameter of interest, loss function, submodels) into a specific machine-learning algorithm for estimation, that we call an instantiation of the template.
Constructing a TMLE or a C-TMLE involves the estimation of a nuisance parameter, typically an infinite-dimensional feature of the distribution of the data. For a vanilla TMLE estimator, the estimation of the nuisance parameter is addressed as an independent statistical task. In the C-TMLE template, on the contrary, the estimation of the nuisance parameter is optimized to provide a better bias-variance trade-off in the inference of the targeted parameter. The C-TMLE template has been successfully applied in a variety of areas, from survival analysis [@stitelman2011targeted], to the study of gene association [@wang2011finding] and longitudinal data structures [@stitelman2010collaborative] to name just a few.
In the original instantiation of the C-TMLE template of @van2010collaborative, that we henceforth call “the greedy C-TMLE algorithm”, the estimation of the nuisance parameter aiming for a better bias-variance trade-off is conducted in two steps. First, a greedy forward stepwise selection procedure is implemented to construct a nested sequence of candidate estimators of the nuisance parameter. Second, cross-validation is used to select the candidate from this sequence which minimizes a criterion that incorporates a measure of bias and variance with respect to (wrt) the targeted parameter (the algorithm is described in Section \[sec:generalCTMLE\]). The authors show the greedy C-TMLE algorithm exhibits superior relative performance in analyses of sparse data, at the cost of an increase in time complexity. For instance, in a problem with $p$ baseline covariates, one would construct and select from $p$ candidate estimators of the nuisance parameter, yielding a time complexity of order $\mathcal{O}(p^2)$. Despite a criterion for early termination, the algorithm does not scale to large-scale and high-dimensional data. The aim of this article is to develop novel C-TMLE algorithms that overcome these serious practical limitations without compromising finite sample or asymptotic performance.\
We propose two such “scalable C-TMLE algorithms”. They replace the greedy search at each step by an easily computed data adaptive pre-ordering of the candidate estimators of the nuisance parameter. They include a data adaptive, early stopping rule that further reduces computational time without sacrificing statistical performance. In the aforementioned problem with $p$ baseline covariates where the time complexity of the greedy C-TMLE algorithm was of order $\mathcal{O}(p^2)$, those of the two novel scalable C-TMLE algorithms is of order $\mathcal{O}(p)$.
Because one may be reluctant to specify a single a priori pre-ordering of the candidate estimators of the nuisance parameter, we also introduce a SL-C-TMLE algorithm. It selects the best pre-ordering from a set of ordering strategies by super learning (SL) [@van2007super]. SL is an example of ensemble learning methodology which builds a meta-algorithm for estimation out of a collection of individual, competing algorithms of estimation, relying on oracle properties of cross-validation.
We focus on the estimation of the average (causal) treatment effect (ATE). It is not hard to generalize our scalable C-TMLE algorithms to other estimation problems.
The performance of the two scalable C-TMLE and SL-C-TMLE algorithms are compared with those of competing, well established estimation methods: G-computation [@Robins86], inverse probability of treatment weighting (IPTW) [@hernan_brumback_robins; @Robins98], augmented inverse probability of treatment weighted estimator (A-IPTW) [@Robins00a; @Robins00b; @Robins00c]. Results from unadjusted regression estimation of a point treatment effect are also provided to illustrate the level of bias due to confounding.\
The article is organized as follows. Section \[sec:background\] introduces the parameter of interest and a causal model for its causal interpretation. Section \[sec:review\] describes an instantiation of the TMLE template. Section \[sec:generalCTMLE\] presents the C-TMLE template and a greedy instantiation of it. Section \[sec:scalableCTMLE\] introduces the two proposed pre-ordered scalable C-TMLE algorithms, and SL-C-TMLE algorithm. Sections \[sec:sim\] and \[subsec:sim:five\] present the results of simulation studies (based on fully or partially synthetic data, respectively) comparing the C-TMLE and SL-C-TMLE estimators with other common estimators. Section \[sec:discussion\] is a closing discussion. The appendix presents additional material: an introduction to a Julia software that implements all the proposed C-TMLE algorithms; a brief analysis of their computational performance; the results of their application to the analysis of three large electronic health databases.
The Average Treatment Effect Example {#sec:background}
====================================
We consider the problem of estimating the ATE in an observational study where we observe on each experimental unit: a collection of $p$ baseline covariates, $W$; a binary treatment indicator, $A$; a binary or bounded continuous $(0,1)$-valued outcome of interest, $Y$. We use $O_i = (W_i, A_i, Y_i)$ to represent the $i$-th observation from the unknown observed data distribution $P_0$, and assume that $O_{1}, \ldots, O_{n}$ are independent. The parameter of interest is defined as $$\Psi(P_0) = \operatorname{\mathbb{E}}_0[\operatorname{\mathbb{E}}_0(Y \mid A = 1, W) - \operatorname{\mathbb{E}}_0(Y \mid A = 0, W)].$$
The ATE enjoys a causal interpretation under the non-parametric structural equation model (NPSEM) given by: $$\left\{ \begin{array}{l}
W=f_W(U_W),\\
A=f_A(W, U_A),\\
Y=f_Y(A, W, U_Y),
\end{array},\right.$$ where $f_{W}$, $f_{A}$ and $f_{Y}$ are deterministic functions and $U_W, U_A, U_Y$ are background (exogenous) variables. The potential outcome under exposure level $a \in \{0,1\}$ can be obtained by substituting $a$ for $A$ in the third equality: $Y_a = f_Y(a, W, U_Y)$. Note that $Y = Y_{A}$ (this is known as the “consistency” assumption). If we are willing to assume that [*(i)*]{} $A$ is conditionally independent of $(Y_1, Y_0)$ given $W$ (this is known as the “no unmeasured confounders” assumption) and [ *(ii)*]{} $0 < P(A=1 \mid W) < 1$ almost everywhere (known as the “positivity” assumption), then parameter $\Psi(P_{0})$ satisfies $\Psi(P_{0})=\operatorname{\mathbb{E}}_{0}(Y_1-Y_0)$.\
For future use, we introduce the propensity score (PS), defined as the conditional probability of receiving treatment, and define $g_0(a,W) \equiv P_{0}(A=a \mid W)$ for both $a=0,1$. We also introduce the conditional mean of the outcome: $\bar{Q}_0(A,W)= \operatorname{\mathbb{E}}_{0}(Y \mid A,W)$. In the remainder of this article, $g_n(a,W)$ and $\bar{Q}_n(A,W)$ denote estimators of $g_0(a,W)$ and $\bar{Q}_{0}(A,W)$.
A TMLE Instantiation for ATE {#sec:review}
============================
We are mainly interested in double robust (DR) estimators of $\Psi(P_{0})$. An estimator of $\Psi(P_{0})$ is said to be DR if it is consistent if either $\bar{Q}_{0}$ or $g_{0}$ is consistently estimated. In addition, an estimator of $\Psi(P_{0})$ is said to be efficient if it satisfies a central limit theorem with a limit variance which equals the second moment under $P_{0}$ of the so called efficient influence curve (EIC) at $P_{0}$. The EIC for the ATE parameter is given by $$D^*(\bar{Q}_0, g_{0})(O) = H_0(A,W) [Y - \bar{Q}_{0}(A,W)] + \bar{Q}_{0}(1, W) -
\bar{Q}_{0}(0,W) - \Psi(P_{0}),$$ where $H_{0}(a,W) = a / g_{0}(1,W) - (1-a) / g_{0}(0,W)$ ($a=0,1)$. The notation is slightly misleading because there is more to $\Psi(P_{0})$ than $(\bar{Q}_{0}, g_{0})$ (namely, the marginal distribution of $W$ under $P_{0}$). We nevertheless keep it that way for brevity. We refer the reader to [@bickel1998efficient] for details about efficient influence curves.
More generally, for every valid distribution $P$ of $O=(W,A,Y)$ such that [ *(i)*]{} the conditional expectation of $Y$ given $(A,W)$ equals $\bar{Q}(A,W)$ and the conditional probability that $A=a$ given $W$ equals $g(a,W)$, and [ *(ii)*]{} $0<g(1,W)<1$ almost surely, we denote $$D^*(\bar{Q}, g)(O) = H_{g}(A,W) [Y - \bar{Q}(A,W)] + \bar{Q}(1, W) -
\bar{Q}(0,W) - \Psi(P),$$ where $H_{g}(a,W) = a / g(1,W) - (1-a) / g(0,W)$ ($a=0,1$).
The augmented inverse probability of treatment weighted estimator (A-IPTW, or so called “double robust IPTW”; @robins1994estimation [@robins2000marginal; @van2003unified]) and TMLE [@van2006targeted; @van2011targeted] are two well studied DR estimators. Taking the estimation of ATE as an example, A-IPTW estimates $\Psi(P_0)$ by solving the EIC equation directly. For given estimators $\bar{Q}_n$, $g_n$ and with $$\label{eq:Hgn}
H_{g_n}(a,W) = a/g_{n}(1,W) - (1-a) /g_{n}(0,W) \quad (a=0,1),$$ solving (in $\psi$) $$\begin{aligned}
0
&=& \sum_{i=1}^{n} H_{g_n}(A_i,W_i) [Y_i - \bar{Q}_n(A_i,W_i)] + \bar{Q}_n(1,
W_i) - \bar{Q}_n(0,W_i) - \psi \end{aligned}$$ yields the A-IPTW estimator $$\label{eq:A-IPTW:one}
\psi_n^{A-IPTW} = \sum_{i=1}^{n}H_{g_n}(A_i,W_i) [Y_i - \bar{Q}_n(A_i,W_i)] +
\bar{Q}_n(1, W_i) - \bar{Q}_n(0,W_i).$$
A substitution (or plug-in) estimator of $\Psi(P_0)$ is obtained by plugging-in the estimator of a relevant part of the data-generating distribution $P_{0}$ into the mapping $\Psi$. Substitution estimators belong to the parameter space by definition, which is a desirable property. The A-IPTW is not a substitution estimator and can suffer from it by sometimes producing estimates outside of known bounds on the problem, such as probabilities or proportions greater than 1. On the contrary, an instantiation of the TMLE template yields a DR TMLE estimator defined by substitution. For instance, a TMLE estimator can be can be constructed by applying the TMLE algorithm below (which corresponds to the negative $\log$-likelihood loss function and logistic fluctuation submodels).
1. [**Estimating $\bar{Q}_0$.**]{} Derive an initial estimator $\bar{Q}_n^0$ of $\bar{Q}_0$. It is highly recommended to avoid making parametric assumptions, as any parametric model is likely mis-specified. Relying on SL [@van2007super] is a good option.
2. [**Estimating $g_0$.**]{} Derive an estimator $g_n$ of $g_{0}$, The same recommendation as above applies.
3. [**Building the so called “clever covariates”.**]{} For $a=0,1$ and a generic $W$, define $H_n(a,W)$ as in .
4. [**Targeting.**]{} Fit the logistic regression of $Y_{i}$ on $H_n(A_i,W_i)$ with no intercept, using $\operatorname*{logit}(\bar{Q}_n^0(A_{i}, W_{i}))$ as offset (an $i$-specific intercept). This yields a minimum loss estimator $\epsilon_{n}$. Update the initial estimator $\bar{Q}_n^0$ into $\bar{Q}_n^*$ given by $$\bar{Q}_n^*(A,W) = \operatorname*{expit}\{\operatorname*{logit}[\bar{Q}_n^0(A,W)] + \epsilon_n H_n(A,W)\}.$$
5. [**Evaluating the parameter estimate**]{}. Define $$\label{eq:TMLE}
\psi_n^{TMLE} = \frac{1}{n} \sum_{i=1}^n (\bar{Q}_n^*(1,W_i) -
\bar{Q}_n^*(0,W_i)).$$ As emphasized, TMLE is a *substitution* estimator.
The targeting step aims to reduce bias in the estimation of $\Psi(P_{0})$ by enhancing the initial estimator derived from $\bar{Q}_n^{0}$ and the marginal empirical distribution of $W$ as an estimator of its counterpart under $P_{0}$. The fluctuation is made in such a way that the EIC equation is solved: $\sum D^*(\bar{Q}^*_n, g_n)(O_i)=0$. Therefore, the TMLE estimator is double robust and (locally) efficient under regularity conditions [@van2011targeted].
Standard errors and confidence intervals (CIs) can be computed based on the variance of the influence curve. Proofs and technical details are available in the literature [@van2006targeted; @van2011targeted for instance].
In practice, bounded continuous outcomes and binary outcomes are fluctuated on the $\operatorname*{logit}$ scale to ensure that bounds on the model space are respected [@susan2010targeted].
The C-TMLE General Template and Its Greedy Instantiation for ATE {#sec:generalCTMLE}
================================================================
When implementing an instantiation of the TMLE template, one relies on a single external estimate of the nuisance parameter, $g_{0}$ in the ATE example (see Step 2 in Section \[sec:review\]). In contrast, an instantiation of the C-TMLE template involves constructing a series of nuisance parameter estimates and corresponding TMLE estimators using these estimates in the targeting step.
The C-TMLE Template {#subsec:C-TMLE}
-------------------
When the ATE is the parameter of interest, the C-TMLE template can be summarized recursively like this (see Algorithm \[algo:ctmle\] for a high-level algorithmic presentation). One first builds $(g_{n,0}, \bar{Q}_{n}^{0}=\bar{Q}_{n,0}, \bar{Q}_{n,0}^{*})$ where $g_{n,0}$ is an estimator of $g_{0}$ and $\bar{Q}_{n}^{0}=\bar{Q}_{n,0}, \bar{Q}_{n,0}^{*}$ are estimators of $\bar{Q}_{0}$, the latter being targeted toward the parameter of interest for instance as in Section \[sec:review\]. Given the previous triplets $(g_{n,0}, \bar{Q}_{n}^{0}=\bar{Q}_{n,0}, \bar{Q}_{n,0}^{*}), \ldots,
(g_{n,k-1}, \bar{Q}_{n,k-1}, \bar{Q}_{n,k-1}^{*})$ where, by construction, the empirical loss of each $\bar{Q}_{n,\ell}^{*}$ is smaller than that of $\bar{Q}_{n,\ell-1}^{*}$, one needs to generate the next triplet in the sequence. The current initial estimator of $\bar{Q}_{0}$ at the $(k+1)$-th step is set at $\bar{Q}_{n,k}=\bar{Q}_{n,k-1}$ ([*i.e.*]{}, the same as that from triplet $(g_{n,k-1}, \bar{Q}_{n,k-1}, \bar{Q}_{n,k-1}^*)$). One then has a set of moves to create candidates $g_{n,k}^j$ updating $g_{n,k-1}$ with move $j$ ([*e.g.*]{}, adding $j$-th covariate), providing better empirical fit than $g_{n,k-1}$ and yielding the corresponding $\bar{Q}_{n,k}^{j,*}$ using $\bar{Q}_{n,k}=\bar{Q}_{n,k-1}$ as initial. The candidate with the smallest empirical loss is $(g_{n,k}, \bar{Q}_{n,k}, \bar{Q}_{n,k}^{*})$. Two cases arise: if the empirical loss of the candidate $\bar{Q}_{n,k}^{*}$ is smaller than that of $\bar{Q}_{n,k-1}^{*}$, then one has derived the next triplet $(g_{n,k}, \bar{Q}_{n,k}=\bar{Q}_{n,k-1}, \bar{Q}_{n,k}^{*})$; otherwise, in our sequence, one updates the initial $\bar{Q}_{n,k}=\bar{Q}_{n,k-1}^{*}$ to the $\bar{Q}_{n,k-1}^{*}$ in the last triplet, and one repeats the above to generate $(g_{n,k}, \bar{Q}_{n,k}, \bar{Q}_{n,k}^{*})$ – since it is now guaranteed that the empirical loss of $\bar{Q}_{n,k}^{*}$ is smaller that that of $\bar{Q}_{n,k-1}^{*}$, one always gets the desired next element $(g_{n,k}, \bar{Q}_{n,k}, \bar{Q}_{n,k}^{*})$.
In the original greedy C-TMLE algorithm [@van2010collaborative], the successive nuisance parameter estimates are based on a data-adaptive forward stepwise search that optimizes a goodness-of-fit criterion at each step. Each of them then yields a specific, candidate TMLE. Finally, the C-TMLE is defined as that candidate that optimizes a cross-validated version of the criterion. The C-TMLE inherits all the properties of a vanilla TMLE estimator [@van2010collaborative]. It is double robust and asymptotically efficient under appropriate regularity conditions.
In Step 1 of Algorithm \[algo:ctmle\], we recommend using SL as described further in Section \[sec:review\]. Step 2 will be commented on in the next section. In Step 3, the best candidate is selected based on the cross-validated penalized log-likelihood and indexed by $$k_n = \operatorname*{arg\,min}_{k} \left\{{{\rm cvRSS}}+ {{\rm cvVar}}_k + n \times {{\rm cvBias}}_k^2\right\}$$ where $$\begin{aligned}
{{\rm cvRSS}}_k
&=& \sum_{v=1}^{V} \sum_{i \in {{\rm Val}}(v)} (Y_i - \bar{Q}_{n,k}^*(P^0_{nv})(W_i, A_i))^2,\\
{{\rm cvVar}}_k
&=& \sum_{v=1}^{V}\sum_{i \in {{\rm Val}}(v)}D^{*2} (\bar{Q}_{n,k}^*(P^0_{nv}),
g_{n,k}(P_n))(O_i),\\
{{\rm cvBias}}_k
&=& \frac{1}{V} \sum_{v=1}^{V}\Psi(\bar{Q}_{n,k}^*(P_{nv}^0)) -
\Psi(\bar{Q}_{n,k}^*(P_n)).\end{aligned}$$ In the above display, ${{\rm Val}}(v)$ is the set of indices of observations used for validation in the $v$-th fold, $P_{nv}^{0}$ is the empirical distribution of the observations indexed by $i \not\in {{\rm Val}}(v)$, $P_{n}$ is the empirical distribution of the whole data set, and $Z(P_{nv}^{0})$ (respectively, $Z(P_{n})$) means that $Z$ is fitted using $P_{nv}^{0}$ (respectively, $P_{n}$). The penalization terms ${{\rm cvVar}}_{k}$ and ${{\rm cvBias}}_{k}$ robustify the finite sample performance when the positivity assumption is violated [@van2010collaborative].
To achieve collaborative double robustness, the sequence of estimators $(g_{n,k} : k)$ should be arranged in such a way that the bias is monotonically decreasing while the variance is monotonically increasing such that $g_{n,k}$ converges (in $k$) to a consistent estimator of $g_0$ [@van2011targeted]. One could for instance rely on a nested sequence of models, see Section \[subsec:origCTMLE\]. By doing so, the empirical fit for $g_{0}$ improves as $k$ increases [@van2011targeted; @gruber2010application].
@porter2011relative discuss and compare TMLE and C-TMLE with other DR estimators, including A-IPTW.
The Greedy C-TMLE Algorithm {#subsec:origCTMLE}
---------------------------
We refer to the original instantiation of the C-TMLE template as the greedy C-TMLE algorithm. It uses a forward selection algorithm to build the sequence of estimators of $g_{0}$ as a nested sequence of treatment models. Let us describe it in the case that $W$ consists of $p$ covariates. For $k=0$, a one-dimensional logistic model with only an intercept is used to estimate $g_{0}$. Recursively, the $(k+1)$th model is derived by adding one more covariate from $W$ to the $k$th logistic model. The chosen covariate is selected from the set of covariates in $W$ that have not been selected so far.
More specifically, one begins with the intercept model for $g_{0}$ to construct $g_{n,0}$ then a first fluctuation covariate $H_{g_{n,0}}$ as in , which is used in turn to create the first candidate estimator $\bar{Q}_{n,0}^{*}$ based on $\bar{Q}_{n,0}$. Namely, denoting $g_{n,0}(1 \mid W) = P_{n}(A=1)$ and $g_{n,0}(0 \mid W) = P_{n}(A=0)$, we set $$\begin{aligned}
\label{eq:clever}
H_{g_{n,k}}(a,W)
&=& a/g_{n,k}(1 \mid W) - (1-a)/g_{n,k} (0 \mid W),\\
\label{eq:fluct}
\operatorname*{logit}(\bar{Q}_{n,k}^{*}(a,W))
&=& \operatorname*{logit}(\bar{Q}_{n,k}(a,W)) + \epsilon_{k}H_{g_{n,k}}(a,W) \quad (a=0,1) \end{aligned}$$ where $k=0$. Here $\epsilon_{k}$ is fitted by a logistic regression of $Y$ on $H_{g_{n,k}}(A,W)$ with offset $\bar{Q}_{n,k}(A,W)$, and $\bar{Q}_{n,1}^{*}$ is the first candidate TMLE. We denote ${{\cal L}}_{0}$ its empirical loss wrt the negative $\log$-likelihood function ${{\cal L}}$.
We proceed recursively. Assume that we have already derived $\bar{Q}_{n,0}^{*}, \ldots$, $\bar{Q}_{n,k-1}^{*}$, and denote the initial estimator used in the last TMLE $\bar{Q}_{n,k-1}^*$ with $\bar{Q}_{n,k-1}$. The $(k+1)$-th estimator $g_{n,k}$ of $g_{0}$ is based on a larger model than that we yielded $g_{n,k-1}$. It contains the intercept and the same $(k-1)$ covariates as the previous model fit $g_{n,k-1}$, with one additional covariate. Each covariate $W_j$ ($1 \leq j \leq p$ such that $W_{j}$ has not been selected yet) is considered in turn for inclusion in the model, yielding a update $g_{n,k}^{j}$ of $g_{n,k-1}$, which implies corresponding updates $H_{g_{n,k}}^{j}$ and $\bar{Q}_{n,k}^{j,*}$ as in the above display. A best update $\bar{Q}_{n,k}^{*}$ is selected among the candidate updates $\bar{Q}_{n,k}^{1,*}, \ldots, \bar{Q}_{n,k}^{p,*}$ by minimizing the empirical loss wrt ${{\cal L}}$. Its empirical loss is denoted ${{\cal L}}_{k}$. If ${{\cal L}}_{k} \leq {{\cal L}}_{k-1}$, then this $\bar{Q}_{n,k}^*$ defines the next fluctuation in our sequence, with corresponding initial estimator still $\bar{Q}_{n,k}=\bar{Q}_{n,k-1}$, the same as that used to build $\bar{Q}_{n,k-1}^*$. We can now move on to the next step. Otherwise, we reset the initial estimator $\bar{Q}_{n,k-1}$ to $\bar{Q}_{n,k-1}^*$ and repeat the above procedure: [*i.e.*]{}, we compute the candidate updates $\bar{Q}_{n,k}^{j,*}$ again for this new initial estimator, and select the best choice $\bar{Q}_{n,k}^*$. Due to the initial estimator in $\bar{Q}_{n,k}^*$ being $\bar{Q}_{n,k-1}^*$, it is now guaranteed that the new ${{\cal L}}_{k}$ is smaller than ${{\cal L}}_{k-1}$, thereby providing us with our next TMLE $\bar{Q}_{n,k}^*$ in our sequence.
This forward stepwise procedure is carried out recursively until all $p$ covariates have been incorporated into the model for $g_{0}$. In the discussed setting, choosing the first covariate requires $p$ comparisons, choosing the second covariate requires $(p-1)$ comparisons and so on, making the time complexity of this algorithm $\mathcal{O}(p^2)$.
Once all candidates $\bar{Q}_{n,0}^{*}, \ldots, \bar{Q}_{n,k}^{*}$ have been constructed, cross-validation is used to select the optimal number of covariates to include in the model for $g_{0}$. For more concrete examples, we refer to [@van2010collaborative; @van2011targeted]. @gruber2010application proposes several variations on the forward greedy stepwise C-TMLE algorithm. The variations did not improve performance in simulation studies. In this article, the greedy C-TMLE algorithm is defined by the procedure described above.
Scalable C-TMLE Algorithms {#sec:scalableCTMLE}
==========================
Now that we have introduced the background on C-TMLE, we will now introduce our scalable C-TMLE algorithm. Section \[subsec:outline\] summarizes the philosophy of the scalable C-TMLE algorithm, which hinges on a data adaptively determined pre-ordering of the baseline covariates. Sections \[subsec:logistic\] and \[subsec:corr\] present two such pre-ordering strategies. Section \[subsec:pre:order:discuss\] discusses what properties a pre-ordering strategy should satisfy. Finally, Section \[subsec:SL-CTMLE\] proposes a discrete Super Learner-based model selection procedure to select among a set of scalable C-TMLE estimators, which is itself a scalable C-TMLE algorithm.
Outline {#subsec:outline}
-------
As we have seen in the previous section, the time complexity of the greedy C-TMLE algorithm is $\mathcal{O}(p^2)$ when the number of covariates equals $p$. This is unsatisfactory for large scale and high-dimensional data, which is an increasingly common situation in health care research. For example, the high-dimensional propensity score (hdPS) algorithm is a method to extract information from electronic medical claims data that produces hundreds or even thousands of candidate covariates, increasing the dimension of the data dramatically [@schneeweiss2009high].
In order to make it possible to apply C-TMLE algorithms to such data sets, we propose to add a new pre-ordering procedure after the initial estimation of $\bar{Q}_{0}$ and before the stepwise construction of the candidate $\bar{Q}_{n,k}^{*}$, $k=0,\ldots$. We present two pre-ordering procedures in Sections \[subsec:logistic\] and \[subsec:corr\]. By imposing an ordering over the covariates only one covariate is eligible for inclusion in the PS model at each step when constructing the next candidate TMLE in the sequence, $\bar{Q}_{n,k}^{*}$. Thus, the new C-TMLE algorithm overcomes the computational issue.
Once an ordering over the covariates has been established, we add them one by one to the model used to estimate $g_{0}$, starting from the intercept model. Suppose that we are adding the $k$th covariate; we obtain a new estimate $g_{n,k}$ of $g_{0}$; we define a new clever covariate as in ; we fluctuate the current initial estimator $\bar{Q}_{n}^{k}$ as in ; we evaluate the empirical loss ${{\cal L}}_{k}$ wrt ${{\cal L}}$ of the resulting candidate $\bar{Q}_{n,k}^{*}$. If ${{\cal L}}_{k} \leq {{\cal L}}_{k-1}$, then we move on to adding the next covariate; otherwise, the current initial estimate $\bar{Q}_{n,k}$ is replaced by $\bar{Q}_{n,k-1}^{*}$ and we restart over adding the $k$th covariate. This approach guarantees that ${{\cal L}}_{k} \leq {{\cal L}}_{k-1}$.
Finally, we use cross-validation to select the best candidate among $\bar{Q}_{n,0}^{*}, \ldots$, $\bar{Q}_{n,p}^{*}$ in terms of cross-validated loss wrt ${{\cal L}}$.
Logistic Pre-Ordering Strategy {#subsec:logistic}
------------------------------
The logistic pre-ordering procedure is similar to the second round of the greedy C-TMLE algorithm. However, instead of selecting one single covariate before going on, we use the empirical losses wrt ${{\cal L}}$ to order the covariates by their ability to reduce bias. More specifically, for each covariate $W_{k}$ ($1 \leq k \leq p$), we construct an estimator $g_{n,k}$ of the conditional distribution of $A$ given $W_{k}$ only (one might also add $W_k$ to a fixed baseline model); we define a clever covariate as in using $g_{n,k}$ and fluctuate $\bar{Q}_{n}^{0}$ as in ; we compute the empirical loss of the resulting $\bar{Q}_{n,k}^{*}$ wrt ${{\cal L}}$, yielding ${{\cal L}}_{k}$. Finally, the covariates are ranked by increasing values of the empirical loss. This is summarized in Algorithm \[algo:logist\].
Partial Correlation Pre-Ordering Strategy {#subsec:corr}
-----------------------------------------
In the greedy C-TMLE algorithm described in Section \[subsec:origCTMLE\], once $k$ covariates have already been selected, the $(k+1)$th is that remaining covariate which provides the largest reduction in the empirical loss wrt ${{\cal L}}$. Intuitively, the $(k+1)$th covariate is the one that best explains the residual between $Y$ and [*the current*]{} $\bar{Q}_{n}^{0}$. Drawing on this idea, the partial correlation pre-ordering procedure ranks the $p$ covariates based on how each of them is correlated with the residual between $Y$ and [*the initial*]{} $\bar{Q}_{n}^{0}$ within strata of $A$. This second strategy is less computationally demanding than the previous one because there is no need to fit any regression models, merely to estimate $p$ partial correlation coefficients.
Let $\rho(X_{1}, X_{2})$ denote the Pearson correlation coefficient between $X_{1}$ and $X_{2}$. Recall that the partial correlation $\rho(X_{1},X_{2}|X_{3})$ between $X_{1}$ and $X_{2}$ given $X_{3}$ is defined as the correlation coefficient between the residuals $R_{X_{1}}$ and $R_{X_{2}}$ resulting from the linear regression of $X_{1}$ on $X_{3}$ and of $X_{2}$ on $X_{3}$, respectively [@hair2006multivariate]. For each $1 \leq k \leq p$, we introduce $R=Y-\bar{Q}_{n}^{0} (A,W)$, $$\rho(R,W_k | A) = \frac{\rho(R, W_k) -\rho(R, A) \times \rho(W_k,
A)}{\sqrt{(1 - \rho(R, A)^2)(1 - \rho(W_k, A)^2)}}.$$ The partial correlation pre-ordering strategy is summarized in Algorithm \[algo:corr\].
Discussion of the Design of Pre-ordering {#subsec:pre:order:discuss}
----------------------------------------
Sections \[subsec:logistic\] and \[subsec:corr\] proposed two pre-ordering strategies. In general, a rule of thumb for designing a pre-ordering strategy is to rank the covariates based on the impact of each in reducing the residual bias in the target parameter which results from the initial estimator $\bar{Q}_{n}^{0}$ of $\bar{Q}_{0}$. In this light, the logistic ordering of Section \[subsec:logistic\] uses TMLE to reflect the importance of each variable wrt its potential to reduce residual bias. The partial correlation ordering of Section \[subsec:corr\] ranks the covariates according to the partial correlation of residual of the initial fit and the covariates, conditional on treatment. Because the rule of thumb considers each covariate in turn separately, it is particularly relevant when the covariates are not too dependent. For example, consider the extreme case where two or more of the covariates are highly correlated and can greatly explain the residual bias in the target parameter. In this scenario, these dependent covariates would [*all*]{} be ranked towards the front of the ordering. However, after adjusting for [*one*]{} of them, the others would typically be much less helpful for reducing the remaining bias. This redundancy may harm the estimation. In cases where it is computationally feasible, this problem can be avoided by using the greedy search strategy, but many other intermediate strategies can be pursued as well.
Super Learner-Based C-TMLE Algorithm {#subsec:SL-CTMLE}
------------------------------------
Here, we explain how to combine several C-TMLE algorithms into one. The combination is based on a Super Learner (SL). Super learning is an ensemble machine learning approach that relies on cross-validation. It has been proven that a SL selector can perform asymptotically as well as an oracle selector under mild assumptions [@van2007super; @van2003unified; @vaart2006oracle].
As hinted at above, a SL-C-TMLE algorithm is an instantiation of an extension of the C-TMLE template. It builds upon several competing C-TMLE algorithms, each relying on different strategies to construct a sequence of estimators of the nuisance parameter. A SL-C-TMLE algorithm can be designed to select the single best strategy (discrete SL-C-TMLE algorithm), or an optimal combination thereof (ensemble SL-C-TMLE algorithm). A SL-C-TMLE algorithm can include both greedy search and pre-ordering methods. A SL-C-TMLE algorithm is scalable if all of the candidate C-TMLE algorithms in the library are scalable themselves.
We focus on a scalable discrete SL-C-TMLE algorithm that uses cross-validation to choose among candidate scalable (pre-ordered) C-TMLE algorithms. Algorithm \[algo:SL:ctmle\] describes its steps. Note that a single cross-validation procedure is used to select both the ordering procedure $m$ and the number of covariates $k$ included in the PS model. It is because computational time [*is*]{} an issue that we do not rely on a nested cross-validation procedure to select $k$ for each pre-ordering strategy $m$.
The time complexity of the SL-C-TMLE algorithm is of the same order as that of the most complex C-TMLE algorithm considered. So, if only pre-ordering strategies of order $\mathcal{O}(p)$ are considered, then the time complexity of the SL-C-TMLE algorithm is $\mathcal{O}(p)$ as well. Given a constant number of user-supplied strategies, the SL-C-TMLE algorithm remains scalable, with a processing time that is approximately equal to the sum of the times for each strategy.\
We compare the pre-ordered C-TMLE algorithms and SL-C-TMLE algorithm with greedy C-TMLE algorithm and other common methods in Sections \[sec:sim\] and Appendix \[sec:analyses\].
Simulation Studies on Fully Synthetic Data {#sec:sim}
==========================================
We carried out four Monte-Carlo simulation studies to investigate and compare the performance of G-computation (that we call MLE), IPTW, A-IPTW, greedy C-TMLE algorithm and scalable C-TMLE algorithms to estimate the ATE parameter. For each study, we generated $N = 1,000$ Monte-Carlo data sets of size $n = 1,000$. Propensity score estimates were truncated to fall within the range $[0.025, 0.975]$ for all estimators.
Denoting $\bar{Q}_{n}^{0}$ and $g_{n}$ two initial estimators of $\bar{Q}_{0}$ and $g_{0}$, the unadjusted, G-computation/MLE, and IPTW estimators of the ATE parameter are given by , and : $$\begin{aligned}
\label{eq:unadj}
\psi_n^{unadj}
&=& \displaystyle \frac{\sum_{i=1}^n I(A_i = 1)Y_i} {\sum_{i=1}^nI(A_i = 1)}
- \frac{\sum_{i=1}^n I(A_i = 0)Y_i} {\sum_{i=1}^nI(A_i = 0)},\\
\label{eq:Gcomp}
\psi_n^{MLE}
&=& \frac{1}{n} \displaystyle \sum_{i=1}^n [Q_n^0(1,W_i) - Q_n^0(0,W_i)],\\
\label{eq:IPTW}
\psi_n^{IPTW}
&=& \frac{1}{n} \displaystyle \sum_{i=1}^n \left[I(A_i=1) - I(A_i=0)\right]
\frac{Y_i}{g_n(A_i,W_i)}, \\
\notag
\psi_n^{A-IPTW}
&=&
\frac{1}{n} \displaystyle \sum_{i=1}^n \frac {\left[I(A_i=1) -
I(A_i=0)\right]}{g_n(A_i \mid W_i)} (Y_i-Q^0_n(W_i,A_i)) \\
\label{eq:A-IPTW}
&& +\frac{1}{n}\sum_{i=1}^n (Q^0_n(1,W_i)-Q^0_n(0,W_i)).
$$ The A-IPTW and TMLE estimators were presented in Section \[sec:review\]. The estimators yielded by the C-TMLE and scalable C-TMLE algorithms were presented in Section \[sec:generalCTMLE\], \[subsec:origCTMLE\] and \[sec:scalableCTMLE\].\
For all simulation studies, $g_0$ was estimated using a correctly specified main terms logistic regression model. Propensity scores incorporated into IPTW, A-IPTW, and TMLE were based on the full treatment model for $g_0$. The simulation studies of Sections \[subsec:sim:one\] and \[subsec:sim:two\] illustrate the relative performance of the estimators in scenarios with highly correlated covariates. These two scenarios are by far the most challenging settings for the greedy C-TMLE and scalable C-TMLE algorithms. The simulation studies of Section \[subsec:sim:three\] and \[subsec:sim:four\] illustrate performance in situations where instrumental variables (covariates predictive of the treatment but not of the outcome) are included in the true PS model. In these two scenarios, greedy C-TMLE and our scalable C-TMLEs are expected to perform better, if not much better, than other widely used doubly-robust methods.
Simulation Study 1: Low-dimensional, highly correlated covariates {#subsec:sim:one}
-----------------------------------------------------------------
In the first simulation study, data were simulated based on a data generating distribution published by @freedman2008weighting and further analyzed by @petersen2012diagnosing. A pair of correlated, multivariate normal baseline covariates $(W_1,W_2)$ is generated as $(W_1, W_2) \sim N(\mu, \Sigma)$ where $\mu_1 = 0.5, \mu_2 = 1$ and $\Sigma = \begin{bmatrix} 2 & 1 \\ 1 & 1
\end{bmatrix}$. The PS is given by $$P_{0}(A = 1 \mid W) = g_0(1 \mid W) = \operatorname*{expit}(0.5 + 0.25 W_{1} + 0.75 W_{2})$$ (this is a slight modification of the mechanism in the original paper, which used a probit model to generate treatment). The outcome is continuous, $Y = \bar{Q}_0(A,W) + \epsilon$, with $\epsilon \sim N(0,1)$ (independent of $A,W$) and $\bar{Q}_0(A,W) = 1 + A + W_1 + 2W_2.$ The true value of the target parameter is $\psi_0 = 1$.
Note that [*(i)*]{} the two baseline covariates are highly correlated and [ *(ii)*]{} the choice of $g_{0}$ yields practical (near) violation of the positivity assumption.
Each of the estimators involving the estimation of $\bar{Q}_{0}$ was implemented twice, using or not a correctly specified model to estimate $Q_0$ (the mis-specified model is a linear regression model of $Y$ on $A$ and $W_1$ only).
[0.4]{} ![Simulation 1: Box plot of ATE estimates with correct/mis-specified models for $\bar{Q}_{0}$. The green line indicates the true parameter value.[]{data-label="fig:Freedman2008"}](Freedman2008-correct.pdf "fig:"){width="\textwidth"}
[0.4]{} ![Simulation 1: Box plot of ATE estimates with correct/mis-specified models for $\bar{Q}_{0}$. The green line indicates the true parameter value.[]{data-label="fig:Freedman2008"}](Freedman2008-Qm.pdf "fig:"){width="\textwidth"}
Bias, variance, and mean squared error (MSE) for all estimators across 1000 simulated data sets are shown in Table \[table:Freedman2008\]. Box plots of the estimated ATE are shown in Fig. \[fig:Freedman2008\]. When $Q_{0}$ was correctly specified, all models had very small bias. As Freedman and Berk discussed, even when the correct PS model is used, near positivity violations can lead to finite sample bias for IPTW estimators [see also @petersen2012diagnosing]. Scalable C-TMLEs had smaller bias than the other DR estimators, but the distinctions were small.
When $Q_{0}$ was not correctly specified, the G-computation/MLE estimator was expected to be biased. Interestingly, A-IPTW was more biased than the other DR estimators. All C-TMLE estimators have identical performance, because each approach produced the same treatment model sequence.
Simulation Study 2: Highly correlated covariates {#subsec:sim:two}
------------------------------------------------
In the second simulation study, we study the case that multiple confounders are highly correlated with each other. We will use the notation $W_{1:k} = (W_{1}, \ldots, W_{k})$. The data-generating distribution is described as follows: $$\begin{aligned}
W_1,W_2,W_3 &\stackrel{iid}{\sim}& {{\rm Bernoulli}}(0.5),\\
W_4|W_{1:3} &\sim& {{\rm Bernoulli}}(0.2 + 0.5 \cdot W_1),\\
W_5|W_{1:4} &\sim& {{\rm Bernoulli}}(0.05 + 0.3 \cdot W_1 + 0.1 \cdot W_2 + 0.05 \cdot W_3 +
0.4 \cdot W_4),\\
W_6|W_{1:5} &\sim& {{\rm Bernoulli}}(0.2 + 0.6 \cdot W_5),\\
W_7|W_{1:6} &\sim& {{\rm Bernoulli}}(0.5 + 0.2 \cdot W_3),\\
W_8|W_{1:7} &\sim& {{\rm Bernoulli}}(0.1 + 0.2 \cdot W_2 + 0.3 \cdot W_6 + 0.1 \cdot W_7),\end{aligned}$$ $$\begin{aligned}
P_{0}(A = 1 \mid W) &=& g_0(1 \mid W) \\
&=& \operatorname*{expit}( -0.05 + 0.1 \cdot W_1 +
0.2 \cdot W_2 + 0.2 \cdot W_{3} \\
&& \qquad\qquad - 0.02\cdot W_4 - 0.6 \cdot W_5 - 0.2 \cdot W_6 - 0.1 \cdot W_7),\end{aligned}$$ and finally, for $\epsilon \sim N(0,1)$ (independent from $A$ and $W$), $$Y = 10+ A + W_1 +W_2 +W_4 +2 \cdot W_6 +W_7 + \epsilon.$$ The true ATE for this simulation study is $\psi_0 = 1$.
In this case, the true confounders are $W_1,W_2,W_4,W_6,W_7$. Covariate $W_5$ is most closely related to $W_1$ and $W_4$. Covariate $W_3$ is mainly associated with $W_7$. Neither $W_3$ nor $W_5$ is a confounder (both of them are predictive of treatment $A$, but do not influence directly outcome $Y$). Including either one of them in the PS model should inflate the variance [@brookhart2006variable].
As in Section \[subsec:sim:one\], each of the estimators involving the estimation of $\bar{Q}_{0}$ was implemented twice, a correctly specified model to estimate $Q_0$, and a mis-specified model defined by a linear regression model of $Y$ on $A$ only.
[0.4]{} ![Simulation 2: Box plot of ATE estimates with correct/mis-specified models for $\bar{Q}_{0}$. The green line indicates the true parameter value.[]{data-label="fig:Gruber2015"}](Gruber2015-correct.pdf "fig:"){width="\textwidth"}
[0.4]{} ![Simulation 2: Box plot of ATE estimates with correct/mis-specified models for $\bar{Q}_{0}$. The green line indicates the true parameter value.[]{data-label="fig:Gruber2015"}](Gruber2015-intercept.pdf "fig:"){width="\textwidth"}
Table \[table:Gruber2015\] demonstrates and compares performance across 1000 replications. Box plots of the estimated ATE are shown in Fig. \[fig:Gruber2015\]. When $\bar{Q}_{0}$ was correctly specified, all estimators except the unadjusted estimator had small bias. The DR estimators had lower MSE than the inefficient IPTW estimator. When $\bar{Q}_{0}$ was mis-specified, the A-IPTW and IPTW estimators were less biased than the C-TMLE estimators. The bias of the greedy C-TMLE was five times larger. However, all DR estimators had lower MSE than the IPTW estimator, with the TMLE outperforming the others.
Simulation Study 3: Binary outcome with instrumental variable {#subsec:sim:three}
-------------------------------------------------------------
In the third simulation, we assess the performance of C-TMLE in a data set with positivity violations. We first generate $W_1, W_2, W_3, W_4$ independently from the uniform distribution on $[0,1]$, then $A|W \sim {{\rm Bernoulli}}(g_{0}(1|W))$ with $$g_{0} (1,W) = \operatorname*{expit}(-2 + 5W_1 + 2W_2 + 1 W_3),$$ and finally $Y|(A,W) \sim {{\rm Bernoulli}}(\bar{Q}_{0}(A,W))$ with $$\bar{Q}_{0}(A,W)= \operatorname*{expit}(-3 + 2W_2+ 2 W_3 + W_4 + A).$$
As in Sections \[subsec:sim:one\] and \[subsec:sim:two\], each of the estimators involving the estimation of $\bar{Q}_{0}$ was implemented twice, once with a correctly specified model and once with a mis-specified linear regression model of $Y$ on $A$ only.
[0.4]{} ![Simulation 3: Box plot of ATE estimates with correct/mis-specified models for $\bar{Q}_{0}$. The green line indicates the true parameter value.[]{data-label="fig:Ju2015"}]({Ju2015-correct.pdf} "fig:"){width="\textwidth"}
[0.4]{} ![Simulation 3: Box plot of ATE estimates with correct/mis-specified models for $\bar{Q}_{0}$. The green line indicates the true parameter value.[]{data-label="fig:Ju2015"}](Ju2015-intercept.pdf "fig:"){width="\textwidth"}
Table \[table:Ju2015\] demonstrates the performance of the estimators across 1000 replications. Fig. \[fig:Ju2015\] shows box plots of the estimates for the different methods across 1000 simulation, with a well specified or mis-specified model for $\bar{Q}_{0}$.
When $\bar{Q}_{0}$ was correctly specified, the DR estimators had similar bias/variance trade-offs. Although IPTW is a consistent estimator when $g$ is correctly specified, truncation of the PS $g_n$ may have introduced bias. However, without truncation it would have been extremely unstable due to violations of the positivity assumption when instrumental variables are included in the propensity score model.
When the model for $\bar{Q}_{0}$ was mis-specified, the MLE was equivalent to the unadjusted estimator. The DR methods performed well with an MSE close to that observed when $\bar{Q}_{0}$ was correctly specified. All C-TMLEs had similar performance. They out-performed the other DR methods (namely, A-IPTW and TMLE) and the pre-ordering strategies improved the computational time without loss of precision or accuracy compared to the greedy C-TMLE algorithm.
### Side note. {#side-note. .unnumbered}
Because $W_1$ is an instrumental variable that is highly predictive of the PS, but not helpful for confounding control, we expect that including it in the PS model would increase the variance of the estimator. One possible way to improve the performance of the IPTW estimator would be to apply a C-TMLE algorithm to select covariates for fitting the PS model. In the mis-specified model for $\bar{Q}_{0}$ scenario, we also simulated the following procedure:
1. Use a greedy C-TMLE algorithm to select the covariates.
2. Use main terms logistic regression with selected covariates for the PS model.
3. Compute IPTW using the estimated PS.
The simulated bias for this estimator was $0.0340$, the SE was $0.0568$, and the MSE was $0.0043$. Excluding the instrumental variable from the PS model thus reduced bias, variance, and MSE of the IPTW estimator.
Simulation Study 4: Continuous outcome with instrumental covariate {#subsec:sim:four}
------------------------------------------------------------------
In the fourth simulation, we assess the performance of C-TMLEs in a simulation scheme with a continuous outcome inspired by [@gruber2011c] (we merely increased the coefficient in front of $W_1$ to introduce a stronger positivity violation). We first independently draw $W_1, W_2, W_3, W_4,W_5, W_6$ from the standard normal law, then $A$ given $W$ with $$P_0(A = 1 \mid W) = g_{0}(1,W) = \operatorname*{expit}(2W_1 + 0.2W_2 + -3 W_3),$$ and finally $Y$ given $(A,W)$ from a Gaussian law with variance 1 and mean $$\bar{Q}_{0}(A,W) = 0.5 W_1 - 8 W_2 + 9 W_3 - 2 W_5 + A.$$
The initial estimator $\bar{Q}_{n}^{0}$ was built based on a linear regression model of $Y$ on $A$, $W_1$, and $W_2$, thus partially adjusting for confounding. There was residual confounding due to $W_3$. There was also residual confounding due to $W_1$ and $W_2$ within at least one stratum of $A$, despite their inclusion in the initial outcome regression model.
![Simulation 4: Box plot of ATE estimates with mis-specified model for $\bar{Q}_{0}$.[]{data-label="fig:sim4"}]({ctmle_win.pdf}){width="60.00000%"}
Fig. \[fig:sim4\] reveals that the C-TMLEs performed much better than TMLE and A-IPTW estimators in terms of bias and standard error. This illustrates that choosing to adjust for less than the full set of covariates can improve finite sample performance when there are near positivity violations. In addition, Table \[table:sim4\] shows that the pre-ordered C-TMLEs out-performed the greedy C-TMLE. Although the greedy C-TMLE estimator had smaller bias, it had higher variance, perhaps due to its more data-adaptive ordering procedure.
Simulation Study on Partially Synthetic Data {#subsec:sim:five}
============================================
The aim of this section is to compare TMLE and all C-TMLEs using a large simulated data set that mimics a real-world data set. Section \[subsec:sim:real\] starts the description of the data-generating scheme and resulting large data set. Section \[subsec:hdPs\] presents the High-Dimensional Propensity Score (hdPS) method used to reduce the dimension of the data set. Section \[subsec:sim:real:cont\] completes the description of the data-generating scheme and specifies how $\bar{Q}_{0}$ and $g_{0}$ are estimated. Section \[subsec:results:real\] summarizes the results of the simulation study.
Data-generating scheme {#subsec:sim:real}
----------------------
The simulation scheme relies on the Nonsteroidal anti-inflammatory drugs (NSAID) data set presented and studied in [@schneeweiss2009high; @rassen2012using]. Its $n=49,653$ observations were sampled from a population of patients aged 65 years and older, and enrolled in both Medicare and the Pennsylvania Pharmaceutical Assistance Contract for the Elderly (PACE) programs between 1995 and 2002. Each observed data structure consists of a triplet $(W,A,Y)$ where $W$ is decomposed in two parts: a vector of 22 baseline covariates and a highly sparse vector of $C=9,470$ unique claim codes. In the latter, each entry is a nonnegative integer indicating how many times (mostly zero) a certain procedure (uniquely identified among $C=9,470$ by its claim code) has been undergone by the corresponding patient. The claim codes were manually clustered into eight categories: ambulatory diagnoses, ambulatory procedures, hospital diagnoses, hospital procedures, nursing home diagnoses, physician diagnoses, physician procedures and prescription drugs. The binary indicator $A$ stands for exposure to a selective COX-2 inhibitor or a comparison drug (a non-selective NSAID). Finally, the binary outcome $Y$ indicates whether or not either a hospitalization for severe gastrointestinal hemorrhage or peptic ulcer disease complications including perforation in GI patients occurred.
The simulated data set was generated as in [@gadbury2008evaluating; @franklin2014plasmode]. It took the form of $n=49,653$ data structures $(W_{i}, A_{i}, Y_{i})$ where $\{(W_{i}, A_{i}) : 1 \leq i \leq n\}$ was extracted from the above real data set and where $\{Y_{i} : 1 \leq i \leq n\}$ was simulated by us in such a way that, for each $1 \leq i \leq n$, the random sampling of $Y_{i}$ depended only on the corresponding $(W_{i}, A_{i})$. As argued in the aforementioned articles, this approach preserves the covariance structure of the covariates and complexity of the true treatment assignment mechanism, while allowing the true value of the ATE parameter to be known, and preserving control over the degree of confounding.
High-Dimensional Propensity Score Method For Dimension Reduction {#subsec:hdPs}
----------------------------------------------------------------
The simulated data set was large, both in number of observations and the number of covariates. In this framework, directly applying any version of C-TMLE algorithms would not be the best course of action First, the computational time would be unreasonably long due to the large number of covariates. Second, the resulting estimators would be plagued by high variance due to the low signal-to-noise ratio in the claim data.
This motivated us to apply the High-Dimensional Propensity Score (hdPS) method for dimension reduction prior to applying the TMLE and C-TMLE algorithms.
Introduced in [@schneeweiss2009high]), the hdPS method was proposed to reduce the dimension in large electronic healthcare databases. It is is increasingly used in studies involving such databases [@rassen2012using; @patorno2014studies; @franklin2015regularized; @toh2011confounding; @kumamaru2016comparison; @ju2016propensity]).
The hdPS method essentially consists of two main steps: [*(i)*]{} generating so called hdPS covariates from the claims data (which can increase the dimension) then [*(ii)*]{} screening the enlarged collection of covariates to select a small proportion of them (which dramatically reduces the dimension). Specifically, the method unfolds as follows [@schneeweiss2009high]:
Cluster by Resource.
: Cluster the data by resource in ${\cal C}$ clusters.
In the current example, we derived ${\cal C}=8$ clusters corresponding to the following categories: ambulatory diagnoses, ambulatory procedures, hospital diagnoses, hospital procedures, nursing home diagnoses, physician diagnoses, physician procedures and prescription drugs. See [@schneeweiss2009high; @patorno2014studies] for other examples.\
Identify Candidate Claim Codes.
: For each cluster separately, for each claim code $c$ within the cluster, compute the empirical proportion $Pr(c)$ of positive entries, then sort the claim codes by decreasing values of $\min(Pr(c), 1-Pr(c))$. Finally, select only the top $J$ claim codes. We thus go from $C$ claim codes to $J\times {\cal C}$ claim codes.
As explained below, we chose $J=50$ so the dimension of the claims data went from $9,470$ to 400.\
Assess Recurrence of Claim Codes.
: For each selected claim code $c$ and each patient $1 \leq i \leq n$, [*replace*]{} the corresponding $c_{i}$ with three binary covariates called “hdPS covariates”: $c_{i}^{(1)}$ equal to one if and only if (iff) $c_{i}$ is positive; $c_{i}^{(2)}$ equal to one iff $c_{i}$ is larger than the median of $\{c_{i} : 1\leq i \leq n\}$; $c_{i}^{(3)}$ equal to one iff $c_{i}$ is larger than the 75%-quantile of $\{c_{i} : 1\leq i \leq n\}$. This inflates the number of claim codes related covariates by a factor 3.
As explained below, the dimension of the claims data thus went from 400 to $1,200$.\
Select Among the hdPS Covariates.
: For each hdPS covariate, estimate a measure of its potential confounding impact, then sort them by decreasing values of the estimates of the measure. Finally, select only the top $K$ hdPS covariates.
For instance, one can rely on the following estimate of the measure of the potential confounding impact introduced in [@bross54]: for hdPS covariate $c^{\ell}$ $$\label{eq:Bross}
\frac{\pi_{n}^{\ell}(1) (r_{n}^{\ell} - 1) +
1}{\pi_{n}^{\ell}(0) (r_{n}^{\ell} - 1) +
1}$$ where $$\begin{aligned}
\pi_{n}^{\ell} (a)
&=& \frac{\sum_{i=1}^{n} {{\bf 1}}\{c_{i}^{\ell}=1,a_{i}=a\}}{\sum_{i=1}^{n}
{{\bf 1}}\{a_{i}=a\}} \quad (a=0,1) \quad \text{and}\\
r_{n}^{\ell}
&=& \frac{p_{n}(1)}{p_{n}(0)} \quad \text{with} \quad
p_{n}(c)
= \frac{\sum_{i=1}^{n} {{\bf 1}}\{y_{i}=1,c_{i}^{\ell}=c\}}{\sum_{i=1}^{n}
{{\bf 1}}\{c_{i}^{\ell}=c\}} \quad (c=0,1).
\end{aligned}$$ A rationale for this choice can be found in [@schneeweiss2009high], where $r_{n}^{\ell}$ in is replaced by $\max(r_{n}^{\ell}, 1/r_{n}^{\ell})$.
As explained below we chose $K=100$. As a result, the dimension of the claims data was reduced to 100 from $9,470$.
Data-generating scheme (continued) and estimating procedures {#subsec:sim:real:cont}
------------------------------------------------------------
Let us resume here the presentation of the simulation scheme initiated in Section \[subsec:sim:real\]. Recall that the simulated data set writes as $\{(W_{i}, A_{i}, Y_{i}) : 1 \leq i \leq n\}$ where $\{W_{i} : 1 \leq i \leq n\}$ is the by product of the hdPS method of Section \[subsec:hdPs\] with $J=50$ and $K=100$ and $\{A_{i} : 1 \leq i \leq n\}$ is the original vector of exposures. It only remains to present how $\{Y_{i} : 1 \leq i \leq n\}$ was generated.
First, we arbitrarily chose a subset $W^\prime$ of $W$, that consists of 10 baseline covariates ([*congestive heart failure*]{}, [*previous use of warfarin*]{}, [*number of generic drugs in last year*]{}, [*previous use of oral steroids*]{}, [*rheumatoid arthritis*]{}, [*age in years*]{}, [ *osteoarthritis*]{}, [*number of doctor visits in last year*]{}, [*calendar year*]{}) and 5 hdPS covariates. Second, we arbitrarily defined a parameter $$\begin{gathered}
\beta = (1.280, -1.727, 1.690, 0.503, 2.528, 0.549, 0.238, -1.048, 1.294,\\
0.825, -0.055, -0.784, -0.733, -0.215, -0.334)^{\top}.\end{gathered}$$ Finally, $Y_{1}, \ldots, Y_{n}$ were independently sampled given $\{(W_{i}, A_{i}) : 1 \leq i \leq n\}$ from Bernoulli distributions with parameters $q_{1}, \ldots, q_{n}$ where, for each $1 \leq i \leq n$, $$q_{i} = \operatorname*{expit}\left(\beta^{\top} W_{i}' + A_{i}\right).$$ The resulting true value of the ATE is $\psi_0 = 0.21156$.\
The estimation of the conditional expectation $\bar{Q}_{0}$ was carried out based on two logistic regression models. The first one was well specified whereas the second one was mis-specified, due to the omission of the five hdPS covariates.
For the TMLE algorithm, the estimation of the PS $g_{0}$ was carried out based on a single, main terms logistic regression model including all of the 122 covariates. For the C-TMLE algorithms, main terms logistic regression model were also fitted at each step. An early stopping rule was implemented to save computational time. Specifically, if the cross-validated loss of $\bar{Q}_{n,k}^{*}$ is smaller than the cross-validated losses of $\bar{Q}_{n,k+1}^{*}, \ldots, \bar{Q}_{n,k+10}^{*}$, then the procedure is stopped and outputs the TMLE estimator corresponding to $\bar{Q}_{n,k}^{*}$.
The scalable SL-C-TMLE library included the two scalable pre-ordered C-TMLE algorithms and excluded the greedy C-TMLE algorithm.
Results {#subsec:results:real}
-------
Table \[table:sim\_data\] reports the point estimates for $\psi_{0}$ as derived by all the considered methods. It also reports the 95% CIs of the form $[\psi_{n} \pm 1.96 \sigma_{n}/\sqrt{n}]$, where $\sigma_{n}^{2} = n^{-1} \sum_{i=1}^{n} D^{*} (\bar{Q}_{n}, g_{n})
(O_{i})^{2}$ estimates the variance of the efficient influence curve at the couple $(\bar{Q}_{n}, g_{n})$ yielding $\psi_{n}$. We refer the interested reader to [@van2011targeted Appendix A] for details on influence curve based inference. All the CIs contained the true value of $\psi_{0}$. Table \[table:sim\_data\] also reports processing times (in seconds).
The point estimates and CIs were similar across all C-TMLEs. When the model for $\bar{Q}_{0}$ was correctly specified, the SL-C-TMLE selected the partial correlation ordering. When the model for $\bar{Q}_{0}$ was mis-specified, it selected the logistic ordering. In both cases, the estimator with smaller bias was data-adaptively selected. In addition, as all the candidates in its library were scalable, the SL-C-TMLE algorithm was also scalable, and ran much faster than the greedy C-TMLE algorithm. Computational time for the scalable C-TMLE algorithms was approximately 1/10th of the computational time of the greedy C-TMLE algorithm.
Discussion {#sec:discussion}
==========
Robust inference of a low-dimensional parameter in a large semi-parametric model traditionally relies on external estimators of infinite-dimensional features of the distribution of the data. Typically, only one of the latter is optimized for the sake of constructing a well behaved estimator of the low-dimensional parameter of interest. For instance, the targeted minimum loss (TMLE) estimator of the average treatment effect (ATE) relies on an external estimator $\bar{Q}_{n}^{0}$ of the conditional mean $\bar{Q}_{0}$ of the outcome given binary treatment and baseline covariates, and on an external estimator $g_{n}$ of the propensity score $g_{0}$. Only $\bar{Q}_{n}^{0}$ is optimized/updated into $\bar{Q}_{n}^{*}$ based on $g_{n}$ in such a way that the resulting substitution estimator of the ATE can be used, under mild assumptions, to derive a narrow confidence interval with a given asymptotic level.
There is room for optimization in the estimation of $g_{0}$ for the sake of achieving a better bias-variance trade-off in the estimation of the ATE. This is the core idea driving the general C-TMLE template. It uses a targeted penalized loss function to make smart choices in determining which variables to adjust for in the estimation of $g_{0}$, only adjusting for variables that have not been fully exploited in the construction of $\bar{Q}_{n}^{0}$, as revealed in the course of a data-driven sequential procedure.
The original instantiation of the general C-TMLE template was presented as a greedy forward stepwise algorithm. It does not scale well when the number $p$ of covariates increases drastically. This motivated the introduction of novel instantiations of the C-TMLE general template where the covariates are pre-ordered. Their time complexity is $\mathcal{O}(p)$ as opposed to the original $\mathcal{O}(p^2)$, a remarkable gain. We proposed two pre-ordering strategies and suggested a rule of thumb to develop other meaningful strategies. Because it is usually unclear a priori which pre-ordering strategy to choose, we also introduced a SL-C-TMLE algorithm that enables the data-driven choice of the better pre-ordering given the problem at hand. Its time complexity is $\mathcal{O}(p)$ as well.
The C-TMLE algorithms used in our data analyses have been implemented in Julia and are publicly available at <https://lendle.github.io/TargetedLearning.jl/>. We undertook five simulation studies. Four of them involved fully synthetic data. The last one involves partially synthetic data based on a real electronic health database and the implementation of a high-dimensional propensity score (hdPS) method for dimension reduction widely used for the statistical analysis of claim codes data. In the appendix, we compare the computational times of variants of C-TMLE algorithms. We also showcase the use of C-TMLE algorithms on three real electronic health database. In all analyses involving electronic health databases, the greedy C-TMLE algorithm was unacceptably slow. Judging from the simulation studies, our scalable C-TMLE algorithms work well, and so does the SL-C-TMLE algorithm.
This article focused on ATE with a binary treatment. In future work, we will adapt the theory and practice of scalable C-TMLE algorithms for the estimation of the ATE with multi-level or continuous treatment by employing a working marginal structural model. We will also extend the analysis to address the estimation of other classical parameters of interest.
[**Appendix**]{}
We gather here some additional material. Appendix \[sec:package\] provides notes on a Julia software package that implements all the proposed C-TMLE algorithms. Appendix \[sec:time\] presents and compares the empirical processing time of C-TMLE algorithms for different sample sizes and number of candidate estimators of the nuisance parameter. Appendix \[sec:analyses\] compares the performance of the new C-TMLEs with standard TMLE on three real data sets.
C-TMLE Software {#sec:package}
===============
A flexible Julia software package implementing all C-TMLE algorithms described in this article is publicly available at <https://lendle.github.io/TargetedLearning.jl/>. The website contains detailed documentation and a tutorial for researchers who do not have experience with Julia.
In addition to the two pre-ordering methods described in Section \[sec:scalableCTMLE\], the software accepts any user-defined ranking algorithm. The software also offers several options to decrease the computational time of the scalable C-TMLE algorithms. The `Pre-Ordered` search strategy has an optional argument `k` which defaults to 1. At each step, the next $k$ available ordered covariates are added to the model used to estimate $g_0$. Large $k$ can speed up the procedure when there are many covariates. However, this approach is prone to over-fitting, and may miss the optimal solution.
An early stopping criteria that avoids computing and cross-validating the complete model containing all $p$ covariates can also save unnecessary computations. A `patience` argument accelerates the training phase by setting the number of steps to carry out after having found a local optimum. To prepare Section \[subsec:sim:real\], argument `patience` was set to 10. More details are provided in that section.
Time Complexity {#sec:time}
===============
We study here the computational time of the pre-ordered C-TMLE algorithms. The computational time of each algorithm depends on the sample size $n$ and number of covariates $p$. First, we set $n = 1,000$ and varied $p$ between 10 and 100 by steps of 10. Second, we varied $n$ from $1,000$ to $20,000$ by steps of $1,000$ and set $p = 20$. For each $(n,p)$ pair, the analysis was replicated ten times independently, and the median computational time was reported. In every data set, all the random variables are mutually independent. The results are shown in Figures \[fig:ptime\] and \[fig:ntime\].
[0.4]{} ![Computational times of the C-TMLE algorithms with greedy search and pre-ordering. []{data-label="fig:ctmle_time"}](PTime/PTime.pdf "fig:"){width="\textwidth"}
[0.4]{} ![Computational times of the C-TMLE algorithms with greedy search and pre-ordering. []{data-label="fig:ctmle_time"}](NTime/NTime.pdf "fig:"){width="\textwidth"}
Figure \[fig:ptime\] is in line with the theory: the computational time of the forward stepwise C-TMLE is $\mathcal{O}(p^2)$ whereas the computational times of the pre-ordered C-TMLE algorithms are $\mathcal{O}(p)$. Note that the pre-ordered C-TMLEs are indeed scalable. When $n=1,000$ and $p=100$, all the scalable C-TMLE algorithms ran in less than 30 seconds.
Figure \[fig:ntime\] reveals that the pre-ordered C-TMLE algorithms are much faster in practice than the greedy C-TMLE algorithm, even if all computational times are $\mathcal{O}(n)$ in that framework with fixed $p$.
Real Data Analyses {#sec:analyses}
==================
This section presents the application of variants of the TMLE and C-TMLE algorithms for the analysis of three real data sets. Our objectives are to showcase their use and to illustrate the consistency of the results provided by the scalable and greedy C-TMLE estimators. We thus do not implement the competing unadjusted, G-computation/MLE, IPTW and A-IPTW estimators (see the beginning of Section \[sec:sim\]).
In Sections \[sec:sim\] and \[subsec:sim:five\], we knew the true value of the ATE. This is not the case here.
Real data sets and estimating procedures
----------------------------------------
We compared the performance of variants of TMLE and C-TMLE algorithms across three observational data sets. Here are brief descriptions, borrowed from [@schneeweiss2009high; @ju2016propensity].
NSAID Data Set.
: Refer to Section \[subsec:sim:real\] for its description.\
Novel Oral Anticoagulant (NOAC) Data Set.
: The NOAC data were collected between October, 2009 and December, 2012 by United Healthcare. The data set tracked a cohort of new users of oral anticoagulants for use in a study of the comparative safety and effectiveness of these agents. The exposure is either “warfarin” or “dabigatran”. The binary outcome indicates whether or not a patient had a stroke during the 180 days after initiation of an anticoagulant.
The data set includes $n=18,447$ observations, $p=60$ baseline covariates and $C=23,531$ unique claim codes. The claim codes are manually clustered in four categories: inpatient diagnoses, outpatient diagnoses, inpatient procedures and outpatient procedures.\
Vytorin Data Set.
: The Vytorin data included all United Healthcare patients who initiated either treatment between January 1, 2003 and December 31, 2012, with age over 65 on day of entry into cohort. The data set tracked a cohort of new users of Vytorin and high-intensity statin therapies. The exposure is either “Vytorin” or “high-intensity statin”. The outcomes indicates whether or not any of the events “myocardial infarction”, “stroke” and “death” occurred.
The data set includes $n=148,327$ observations, $p=67$ baseline covariates and $C=15,010$ unique claim codes. The claim codes are manually clustered in five categories: ambulatory diagnoses, ambulatory procedures, hospital diagnoses, hospital procedures, and prescription drugs.
Each data set is given by $\{(W_{i}, A_{i}, Y_{i}) : 1 \leq i \leq n\}$ where $\{W_{i} : 1 \leq i \leq n\}$ is the by product of the hdPS method of Section \[subsec:hdPs\] with $J=100$ and $K=200$ and $\{(A_{i}, Y_{i}): 1 \leq i \leq n\}$ is the original collection of paired exposures and outcomes.
The estimations of the conditional expectation $\bar{Q}_{0}$ and of the PS $g_{0}$ were carried out based on logistic regression models. Both models used either the baseline covariates only or the baseline covariates [*and*]{} the additional hdPS covariates.
To save computational time, the C-TMLE algorithms relied on the same early stopping rule described in Section \[subsec:sim:real:cont\]. The scalable SL-C-TMLE library included the two scalable pre-ordered C-TMLE algorithms and excluded the greedy C-TMLE algorithm.
Results on the NSAID data set {#subsec:NSAID}
-----------------------------
Figure \[fig:nsaid-ctmle\] shows the point estimates and 95% CIs yielded by the different TMLE and C-TMLE estimators built from the NSAID data set.
The various C-TMLE estimators exhibit similar results, with slightly larger point estimates and narrower CIs compared to the TMLE estimators. All the CIs contain zero.
Results on the NOAC Data Set {#subsec:NOAC}
----------------------------
Figure \[fig:noac-ctmle\] shows the point estimates and 95% CIs yielded by the different TMLE and C-TMLE estimators built on the NOAC data set.
We observe more variability in the results than in those presented in Appendix \[subsec:NSAID\].
The various TMLE and C-TMLEs exhibit similar results, with a non-significant shift to the right for the latter. All the CIs contain zero.
Results on the Vytorin Data Set {#subsec:Vytorin}
-------------------------------
Figure \[fig:vytorin-ctmle\] shows the point estimates and 95% CIs yielded by the different TMLE and C-TMLEs built on the Vytorin data set.
The various TMLE and C-TMLEs exhibit similar results, with a non-significant shift to the right for the latter. All the CIs contain zero.
|
---
author:
- '$^{1}$, $^{1}$, $^{2}$, $^{3}$\'
title: Inverse Renormalization Group Transformation in Bayesian Image Segmentations
---
Bayesian segmentation modeling based on Markov random fields (MRF’s) is one of the interesting research topics[[@KatoZerubia2011]]{}. Image segmentations are required to classify pixels in an observed image into several regions, and such segmentations are regarded as a kind of clustering of pixels. Markov random fields are regarded as classical spin systems in statistical mechanics[[@KTanaka2002]]{}. Loopy belief propagations (LBP’s) have been applied to construct certain practical algorithms for application in Bayesian image segmentations[[@HasegawaOkadaMiyoshi2011; @TanakaKataokaYasudaWaizumiHsu2014]]{}. In the present short note, we propose a new Bayesian image segmentation algorithm based on a RSRG transformation to reduce the computational time.
We consider an image as defined on a set of pixels arranged on a square grid graph $({\cal{V}},{\cal{E}})$. Here ${\cal{V}}{\equiv}\{i|i=1,2,{\cdots},|{\cal{V}}|\}$ denotes the set of all the pixels and ${\cal{E}}$ is the set of all the nearest-neighbour pairs of pixels $\{i,j\}$. The total numbers of elements in the sets ${\cal{V}}$ and ${\cal{E}}$ are denoted by $|{\cal{V}}|$ and $|{\cal{E}}|$, respectively. The square grid graph has the periodic boundary conditions along the $x$- and $y$-directions. The label at each pixel $i$ is regarded as a state variable, and it is denoted by $a_{i}$. Each pixel $i$ takes on the values of all the possible integers in the set ${\cal{Q}}{\equiv}\{0,1,2,{\cdots},q-1\}$ as its region label. The state vector of labels is represented by ${\bm{a}}=(a_{i}|i{\in}{\cal{V}})
=(a_{1},a_{2},{\cdots},a_{|{\cal{V}}|})^{\rm{T}}$. The prior probability of a labeling configuration ${\bm{a}}$ is assumed to be specified by a constant $u$ as $$\begin{aligned}
P({\bm{a}})
\propto
{\prod_{\{i,j\}{\in}{\cal{E}}}}
{\exp}{\Big (}{\frac{1}{2}}{\alpha}
{\delta}_{a_{i},a_{j}}{\Big )},
\label{Prior}\end{aligned}$$ up to the normalization constant.
For the prior probability distribution $P({\bm{a}}|u)$, we introduce the following RSRG transformation: $$\begin{aligned}
{\exp}{\Big (}{\frac{1}{2}}{\alpha}^{(r)}
{\delta}_{a_{1},a_{3}}{\Big )}
& \varpropto &
{\sum_{a_{2}{\in}{\cal{Q}}}}
{\sum_{a_{4}{\in}{\cal{Q}}}}
{\exp}{\Big (}{\frac{1}{2}}{\alpha}^{(r-1)}
{\big (}{\delta}_{a_{1},a_{2}}
+{\delta}_{a_{2},a_{3}}
+{\delta}_{a_{1},a_{4}}
+{\delta}_{a_{4},a_{3}}{\big )}{\Big )}
\nonumber\\
& &{\hspace{3.0cm}}
(r=1,2,{\cdots},R),
\label{BlockSpinTransformation}\end{aligned}$$ where ${\alpha}^{(0)}{\equiv}{\alpha}$. The square grid graph ${\cal{G}}^{(r)}=({\cal{V}}^{(r)},{\cal{E}}^{(r)})$ is defined as shown in Fig.[\[Figure01\]]{}(b). Here ${\cal{V}}^{(r)}$ is the set of all the pixels and ${\cal{E}}^{(r)}$ represents the set of all the nearest-neighbour pairs of pixels $\{i,j\}$ in the graph ${\cal{G}}^{(r)}$. Here, we remark that ${\cal{V}}^{(r)}$ is the subset of ${\cal{V}}$. The prior probability distribution $P^{(r)}({\bm{a}}^{(r)})$ for ${\bm{a}}^{(r)}{\equiv}(a_{i}|i{\in}{\cal{V}}^{(r)})$ on ${\cal{G}}^{(r)}=({\cal{V}}^{(r)},{\cal{E}}^{(r)})$ after the $r$-iteration of the RSRG transformation is expressed as $$\begin{aligned}
P^{(r)}({\bm{a}}^{(r)})
\propto
{\prod_{\{i,j\}{\in}{\cal{E}}^{(r)}}}
{\exp}{\Big (}{\frac{1}{2}}{\alpha}^{(r)}
{\delta}_{a_{i},a_{j}}{\Big )},
\label{RGPrior}\end{aligned}$$ up to the normalization constant. The transformation in Eq.(\[BlockSpinTransformation\]) can be reduced to $$\begin{aligned}
{\alpha}^{(r)}=4
{\ln}{\Big (}
{\frac{q-1+e^{{\alpha}^{(r-1)}}}
{q-2+2e^{{\frac{1}{2}}{\alpha}^{(r-1)}}}}
{\Big )}.
\label{RGTransformationA}\end{aligned}$$
The intensities of the red, green, and blue channels at pixel $i$ in the observed image are regarded as state variables denoted by $d_{i}^{\rm{Red}}$, $d_{i}^{\rm{Green}}$ and $d_{i}^{\rm{Blue}}$, respectively. The random fields of red, green and blue intensities in the observed color image are then represented by the $3|{\cal{V}}|$-dimensional vector ${\bm{d}}{\equiv}({\bm{d}}_{1},{\bm{d}}_{2},
{\cdots},{\bm{d}}_{|{\cal{V}}|})^{\rm{T}}$, where ${\bm{d}}_{i}
\equiv (d_{i}^{\rm{Red}},d_{i}^{\rm{Green}},
d_{i}^{\rm{Blue}})^{\rm{T}}$. The state variables $d_{i}^{\rm{Red}}$, $d_{i}^{\rm{Green}}$ and $d_{i}^{\rm{Blue}}$ at each pixel $i$ can take any real numbers in the interval $(-{\infty},+{\infty})$. It is to be noted that, in the above generative process of natural color images, we assign each pixel $i$ to a labeling state $a_{i}$. For example, if the pixel $i$ is in the labeling state $a_{i}={\xi}$ (${\xi}{\in}{\cal{Q}}$), then its color intensity vector ${\bm{d}}_{i}$ is assumed to be generated from the following Gaussian distribution: $$\begin{aligned}
g({\bm{d}}_{i}|{\xi})
\equiv
{\sqrt{{\frac{1}{{\rm{det}}(2{\pi}{\bm{C}}({\xi}))}}}}
{\exp}{\Big (}
-{\frac{1}{2}}
({\bm{d}}_{i}-{\bm{m}}({\xi}))^{\rm{T}}
{\bm{C}}^{-1}({\xi})
({\bm{d}}_{i}-{\bm{m}}({\xi}))
{\Big )}.
\label{2DGaussian}\end{aligned}$$ In other words, the labeling state $a_{i}$ specifies the distribution within the set $\{g({\bm{d}}_{i}|{\xi})|{\xi}{\in}{\cal{Q}}\}$ that generates a color intensity vector ${\bm{d}}_{i}$. As mentioned in the previous section, we introduce the labeling state variable $a_{i}$ at each pixel $i$ as a Potts spin variable in statistical mechanics.
After setting $R$ as a positive integer, we construct a $3|{\cal{V}}^{(R)}|$-dimensional vector ${\bm{d}}^{(R)}=({\bm{d}}_{i}|i{\in}{\cal{V}}^{(R)})$ from our observed color image ${\bm{d}}$. We remark that $d_{i}$ in the coarse-grained image ${\bm{d}}^{(R)}$ is always the same as $d_{i}$ in the observed color image ${\bm{d}}$ in our scheme. By using the Bayes formula, we introduce the posterior probability distribution $P{\big (}{\bm{a}}^{(R)}{\big |}{\bm{d}}^{(R)}{\big )}$ for ${\bm{a}}^{(R)}=(a_{i}|i{\in}{\cal{V}}^{(R)})$ as follows: $$\begin{aligned}
P^{(R)}{\big (}{\bm{a}}^{(R)}{\big |}{\bm{d}}^{(R)}{\big )}
\propto
{\Big (}{\prod_{i{\in}{\cal{V}}^{(R)}}}
g({\bm{d}}_{i}|a_{i}){\Big )}
{\Big (}{\prod_{\{i,j\}{\in}{\cal{E}}^{(R)}}}
{\exp}{\big (}{\frac{1}{2}}{\alpha}^{(R)}
{\delta}_{a_{i},a_{j}}{\big )}{\Big )},
\label{Posterior}\end{aligned}$$ up to the normalization constant. The estimates ${\widehat{\alpha}}^{(R)}$ and $\{{\widehat{\bm{m}}}({\xi}),
{\widehat{\bm{C}}}({\xi})|{\xi}{\in}{\cal{Q}}\}$ of ${\alpha}^{(R)}$ and $\{{\bm{m}}({\xi}),{\bm{C}}({\xi})|{\xi}{\in}{\cal{Q}}\}$. are approximately computed by means of the LBP algorithm for the conditional maximum entropy framework[[@TanakaKataokaYasudaWaizumiHsu2014]]{}. After obtaining these estimates, we calculate ${\widehat{\alpha}}^{(0)}$ from ${\widehat{\alpha}}^{(R)}$ by using the following inverse RSRG transformations of Eq.(\[RGTransformationA\]): $$\begin{aligned}
{\widehat{\alpha}}^{(r-1)}
= 2{\ln}{\Big (}
e^{{\frac{1}{4}}{\alpha}^{(r)}}
+
{\sqrt{(e^{{\frac{1}{4}}{\widehat{\alpha}}^{(r)}}+Q-1)
(e^{{\frac{1}{4}}{\widehat{\alpha}}^{(r)}}-1)}}
{\Big )}
~(r=R,{\cdots},2,1),
\label{InverseRGTransformationA}\end{aligned}$$ where ${\widehat{\alpha}}={\widehat{\alpha}}^{(0)}$.
Given the estimates ${\widehat{\alpha}}$ and $\{{\widehat{\bm{m}}}({\xi}),
{\widehat{\bm{C}}}({\xi})|{\xi}{\in}{\cal{Q}}\}$, the estimate of labeling ${\widehat{\bm{a}}}({\bm{d}})
=({\widehat{a}}_{1}({\bm{d}}),
{\widehat{a}}_{2}({\bm{d}}),
{\cdots},
{\widehat{a}}_{|{\cal{V}}|}({\bm{d}}))^{\rm{T}}$ is determined by $$\begin{aligned}
{\widehat{a}}_{i}({\bm{d}})
\equiv
{\arg}
{\max_{{\zeta}{\in}{\cal{Q}}}}
{\displaystyle{{\sum_{{\bm{a}}}}}}
{\delta}_{a_{i},{\zeta}}
P^{(0)}{\big (}{\bm{a}}{\big |}{\bm{d}}{\big )}
{\hspace{2.0mm}}(i{\in}{\cal{V}}).
\label{MPM}\end{aligned}$$ This procedure to determine the estimate ${\widehat{a}}({\bm{d}})$ is approximately computed by using the LBP algorithm to $P^{(0)}{\big (}{\bm{a}}{\big |}{\bm{d}}{\big )}$.
We show numerical experiments by our proposed approach in Fig.[\[Figure02\]]{}. In our numerical experiments, the test image ${\bm{d}}$ in Fig.[\[Figure02\]]{}(a) is acquired from the Berkeley Segmentation Data Set 500 (BSDS500)[[@ArbelaezMaireFowlkesMalik2011]]{}. The size of the test image ${\bm{d}}$ is $321{\times}481$ and the sizes of ${\bm{d}}^{(R)}$ are reduced to $20{\times}30$ for $R=8$ and $10{\times}20$ for $R=10$. The labeling configurations ${\bm{\widehat{a}}}$ obtained by means of our proposed algorithm based on our inverse RSRG transformation for $R=8$ and $R=10$ are shown in Figs.[\[Figure02\]]{}(b) and (c), respectively. In our proposed algorithm, ${\widehat{a}}^{(R)}$ and $\{{\widehat{\bm{m}}}({\xi}),
{\widehat{\bm{C}}}({\xi})|{\xi}{\in}{\cal{Q}}\}$ for ${\bm{d}}^{(R)}$ are estimated by using the conditional maximum entropy framework with the LBP such that they are obtained by subjecting ${\bm{d}}^{(R)}$ to the algorithm presented in [§]{}3 of Ref.[[@TanakaKataokaYasudaWaizumiHsu2014]]{}. The estimates ${\widehat{\alpha}}^{(R)}$ for $R=8$ and $=10$ are $2.5288$ and $2.5039$, respectively. The estimates ${\widehat{\alpha}}={\widehat{\alpha}}^{(0)}$ obtained by means of the inverse RSRG group transformations given by Eq.(\[InverseRGTransformationA\]) for $R=8$ and $R=10$ are $3.6765$ and $3.6797$, respectively. The labeling configuration in Fig.[\[Figure02\]]{}(d) is computed directly by applying ${\bm{d}}$ to the algorithm in [§]{}3 of Ref.[[@TanakaKataokaYasudaWaizumiHsu2014]]{}. The estimate ${\widehat{\alpha}}$ of ${\alpha}$ in Fig.[\[Figure02\]]{}(d) is $3.0301$. The computational times corresponding to Figs.[\[Figure02\]]{}(b)-(d) are $128$(Sec), $78$(Sec) and $1331$(Sec), respectively. Our numerical experiments were performed by using a personal computer with an Intel(R) Core(TM) i7-4600U CPU with a memory of 8GB. From the results of our experiment, we observe that the computational time can be reduced to less than one-tenth of that taken by conventional methods by application of our proposed algorithm based on the inverse RSRG transformation. We obtained results similar to the above mentioned case for other test images in the database set of BSDS500.
----------------- ----------------- ----------------- -----------------
[**[(a)]{}**]{} [**[(b)]{}**]{} [**[(c)]{}**]{} [**[(d)]{}**]{}
----------------- ----------------- ----------------- -----------------
In the present short note, we have presented a novel algorithm that involve the combination of the inverse RSRG transformation with the Bayesian image segmentation method proposed in Ref.[[@TanakaKataokaYasudaWaizumiHsu2014]]{}. Our proposed method can reduce the computational time of the hyperparameter estimations significantly. We expect that our approach can also be applied to Bayesian image segmentations for three-dimensional computer vision, which remains one of significant problems.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors are grateful to Prof. Federico Ricci-Tersenghi of the Department of Physics, University of Roma La Sapienza for valuable comments. This work was partly supported by the JST-CREST and the Grants-In-Aid (No.25280089) for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan.
[9]{} Z. Kato and J. Zerubia: Foundations and Trends in Signal Processing, [**[5]{}**]{} (2012) 1. K. Tanaka: J. Phys. A [**[35]{}**]{} (2002) R81. R. Hasegawa, M. Okada and S. Miyoshi: J. Phys. Soc. Jpn [**[80]{}**]{} (2011) 093802. K. Tanaka, S. Kataoka, M. Yasuda, Y. Waizumi and C.-T. Hsu: J. Phys. Soc. Jpn [**[83]{}**]{} (2014) 124002. P. Arbelaez, M. Maire, C. Fowlkes and J. Malik: IEEE Trans. Pattern Anal. Mach. Intell. [**[33]{}**]{} (2011) 898.
|
---
abstract: 'The main physical result of this paper are exact analytical solutions of the heavenly equation, of importance in the general theory of relativity. These solutions are not invariant under any subgroup of the symmetry group of the equation. The main mathematical result is a new method of obtaining noninvariant solutions of partial differential equations with infinite dimensional symmetry groups. The method involves the compatibility of the given equations with a differential constraint, which is automorphic under a specific symmetry subgroup, the latter acting transitively on the submanifold of the common solutions. By studying the integrability of the resulting conditions, one can provide an explicit foliation of the entire solution manifold of the considered equations.'
author:
- '**L. Martina$^*$, M.B. Sheftel$^\dag$ and P. Winternitz$^\ddag$**'
title: '**Group foliation and non-invariant solutions of the heavenly equation**'
---
($^*$ Dipartimento di Fisica dell’Università di Lecce and Sezione INFN-Lecce, Lecce, Italy. [**E-mail:**]{} Luigi.Martina@le.infn.it\
$^\dag$ Feza Gürsey Institute, Istanbul, Turkey and Department of Higher Mathematics, North Western State Technical University, St. Petersburg, Russia.\
[**E-mail:**]{} sheftel@gursey.gov.tr\
$^\ddag$ Centre de Recherches Mathématiques and Département de mathématiques et statistique, Université de Montréal, Montréal, Canada.\
[**E-mail:**]{} wintern@CRM.UMontreal.ca)\
Comments: 32 pages, Latex,\
submitted to [*J. Phys. A: Math. Gen.*]{} on April 13, 2001.\
Subj-class: Mathematical Physics; Exactly Solvable and Integrable Systems\
MSC-class: 35Q75 (Primary) 57S20 (Secondary)
Introduction {#sec:intro}
============
An important problem for partial differential equations invariant with respect to an infinite Lie group is to obtain non-invariant solutions that admit no continuous symmetries of the equations. In our opinion, the old approach of S. Lie [@lie] developed by Vessiot [@vessiot] and in modern form by Ovsiannikov [@ovs], which we call [*group foliation*]{}, is an adequate tool for treating this problem in the framework of Lie theory. According to this method we foliate the solution space of the equations in question into orbits, choosing for the foliation an infinite-dimensional symmetry group. Each orbit is determined by the [*automorphic system*]{} joined to the original equations and considered as invariant differential constraints. Due to the automorphic property of this system, any of its solutions can be obtained from any other solution by a transformation of the chosen symmetry subgroup. This symmetry property makes the automorphic system completely integrable if only one of its solutions can be obtained. The collection of orbits of all solutions of the original equations is determined by the [*resolving system*]{}. Thus the problem reduces to obtaining as many particular solutions of the resolving system as possible. Each of them will fix a particular automorphic system and the corresponding orbit in the solution space of original equations.
Group theory is usually used to obtain invariant solutions. Here we show that it also provides a mechanism for obtaining non-invariant solutions. We give examples of such solutions as an application of the method.
In this paper we further develop the method of group foliation by introducing a procedure of [*invariant integration*]{}. It is used for reconstructing the solution of the original equation corresponding to the known particular solution of the resolving system. We apply the method for obtaining non-invariant solutions of the ‘heavenly" equation $$u_{xx}+u_{yy}=\kappa (e^u)_{tt}
\label{heav_xy}$$ where $\kappa =\pm 1$ and the unknown $u$ depends on the time $t$ and two space variables $x$ and $y$. Here and further subscripts of $u$ denote partial differentiation with respect to corresponding variables. This equation formally is a continuous version of the Toda lattice [@mart]. It appears in various physical theories, like the theory of area preserving diffeomorphisms [@bakas], in the theory of the so-called gravitational instantons [@eguchi] and in the general theory of relativity [@ward]. In this context it describes self-dual Einstein spaces with Euclidean signature with one rotational Killing vector. Moreover it is a completely integrable system in the sense of the existence of a Lax pair [@sav89; @sav92].
The outline of the method is the following. We determine the total group of point symmetries of the heavenly equation. For the group foliation we choose its infinite subgroup of conformal transformations. We compute differential invariants of this subgroup up to the second order inclusively and obtain $5$ functionally independent differential invariants. On account of the heavenly equation we are left with $4$ invariants. We choose three of them as new independent variables, the same number as in the heavenly equation, and one is left for the new unknown.
We obtain three first order [*operators of invariant differentiation*]{} defined by the property that acting on a differential invariant they produce again a differential invariant. These operators are determined by the condition that they should commute with an arbitrary prolongation of any element of the infinite symmetry Lie algebra chosen for the foliation.
Extensive use of operators of invariant differentiation and their commutator algebra for formulating the resolving system is a new feature of the method suggested by one of the authors (M.B.S.) in a recent article on the complex Monge-Ampère equation [@ns]. We derive the resolving system as a set of compatibility conditions for the heavenly equation and its automorphic system, using invariant cross-differentiation. Then we formulate the resolving system in terms of the [*commutator algebra of operators of invariant differentiation*]{} by discovering the fact that this algebra together with its Jacobi identities, projected on the solution manifold of the considered equation in the space of differential invariants, is equivalent to the resolving system.
We show how an Ansatz simplifying the commutator algebra of operators of invariant differentiation leads to a particular class of solutions of the resolving system. Then we use invariant integration to obtain the corresponding solution of the heavenly equation and prove that this solution is non-invariant.
Lie group of point symmetries and\
differential invariants {#sec:diffinvar}
==================================
It is convenient to work with the heavenly equation using the complex coordinates $z=(x+iy)/2, \;\bar z=(x-iy)/2$ $$u_{z\bar z}=\kappa (e^u)_{tt}.
\label{heav}$$ A standard calculation of the total symmetry group of the heavenly equation gives the following result for the symmetry generators of all one-parameter subgroups [@soliani] $$\begin{aligned}
T \!\!&=&\!\! \partial_t, \qquad G=t\partial_t+2\partial_u, \nonumber
\\ X_a \!\!&=&\!\! a(z)\partial_z +\bar a(\bar z)\partial_{\bar z}
-(a^\prime(z)+\bar a^\prime(\bar z))\partial_u ,
\label{symgen}\end{aligned}$$ where $T$ is the generator of translations in $t$, $G$ is the generator of a dilation of time accompanied by a shift of $u$: $t=\tilde t e^\tau,\;
u=\tilde u +2\tau$ and $X_a$ is a generator of the conformal transformations $$z=\phi(\tilde z),\quad \bar z=\bar \phi(\tilde{\bar
z}),\quad u(z,\bar z,t) = \tilde u(\tilde z,\tilde{\bar z},t)
-\ln\bigl(\phi^\prime(\tilde z){\bar\phi}^\prime(\tilde{\bar z})\bigr),
\label{conform}$$ where $a(z)$ and $\phi(z)$ are arbitrary holomorphic functions of $z$ (see also [@boywin]).
The Lie algebra of the symmetry generators is determined by the commutation relations $$[T,G]=T,\quad [T,X_a]=0,\quad [G,X_a]=0,\quad
[X_a,X_b]=X_{ab^\prime-ba^\prime} ,
\label{algeb}$$ which show that the generators $X_a$ of conformal transformations form an infinite-dimensional subalgebra.
We choose for the group foliation the corresponding infinite symmetry subgroup of all holomorphic transformations in $z$, [*i.e.*]{} the conformal group. Differential invariants of this group are the invariants of all its generators $X_a$ of the form (\[symgen\]) in the prolongation spaces. This means that they can depend on independent variables, the unknowns and also on the partial derivatives of the unknowns allowed by the order of the prolongation. The [*order*]{} $N$ of the differential invariant is defined as the order of the highest derivative which this invariant depends on. The determining equation for differential invariants $\Phi$ of the order $N\le 2$ has the form $$\X2(\Phi)=0 ,
\label{detinv}$$ where $\X2$ is the second prolongation of the generator $X_a$ (\[symgen\]) of the conformal group defined by the standard prolongation formulae $$\begin{aligned}
\X2\!\!&\!\!=\!\!&\!\!a\partial_z+\bar a\partial_{\bar z}
-\left(a^\prime+{\bar a}^\prime\right)\partial_u
-\left(a^{\prime\prime}+a^\prime u_z\right)\partial_{u_z}
-\left({\bar a}^{\prime\prime}+{\bar a}^\prime u_{\bar z}\right)
\partial_{u_{\bar z}}\nonumber
\\ \!\!&\!\! \!\!&\!\! \mbox{}-\left(a^{\prime\prime\prime}
+a^{\prime\prime}u_z+2a^\prime u_{zz}\right)\partial_{u_{zz}}
-\left({\bar a}^{\prime\prime\prime}
+{\bar a}^{\prime\prime}u_{\bar z}+2{\bar a}^\prime u_{\bar z\bar z}\right)
\partial_{u_{\bar z\bar z}}\nonumber
\\ \!\!&\!\! \!\!&\!\! \mbox{}-a^\prime u_{zt}\partial_{u_{zt}}
-{\bar a}^\prime u_{\bar zt}\partial_{u_{\bar zt}}
-\left(a^\prime+{\bar a}^\prime\right)u_{z\bar z}\partial_{u_{z\bar z}},
\label{X2}\end{aligned}$$ where $a=a(z)$ and $\bar a=\bar a(\bar z)$.
The integration of eq.(\[detinv\]) gives $5$ functionally independent differential invariants up to the second order inclusively $$t,\qquad u_t,\qquad u_{tt},\qquad \rho=e^{-u}u_{z\bar z},\qquad
\eta=e^{-u}u_{zt}u_{\bar zt}
\label{difinv}$$ and all of them turn out to be real. This allows us to express the heavenly equation (\[heav\]) solely in terms of the differential invariants $$u_{tt}=\kappa\rho-u_t^2 .
\label{heavinv}$$
Operators of invariant differentiation and a basis of differential invariants {#sec:invdiff}
=============================================================================
[*Operators of invariant differentiation*]{} are linear combinations of total derivative operators with respect to independent variables. Their coefficients depend on local coordinates of the prolongation space. They are defined by the special property that, acting on any (differential) invariant, they map it again into a differential invariant. Being first-order differential operators, they raise the order of a differential invariant by one. Invariance requires that these differential operators commute with any infinitely prolonged generator $X_a$ (\[symgen\]) of the conformal symmetry group. It is obvious (see [@ovs] par. 24.2 for a complete proof) that the total number of independent operators of invariant differentiation is equal to the number of total derivative operators, that is to the number of independent variables (which is three in the present case).
We look for operators of invariant differentiation in the form $$\delta=\lambda_1D_t+\lambda_2D_z+\lambda_3D_{\bar z}=\sum_{i=1}^{3}\lambda_i
D_i
\label{invdifform}$$ where $D_1=D_t,\; D_2=D_z,\; D_3=D_{\bar z}$ are operators of total derivatives with respect to the subscripts. We look for the coefficients $\lambda_i$ satisfying the condition of commutativity of $\delta$ with the infinite prolongation $\X\infty$ of the generator $X_a$ (\[symgen\]). It can be decomposed as the sum of the infinite prolongation of the symmetry generator in the evolution form $\widehat X$ [@olv] and the linear combination of the total derivative operators $$\X\infty= \widehat X +\sum_{j=1}^{3}\xi^jD_j=\widehat X +a(z)D_z +\bar a(z)
D_{\bar z},
\label{decomp}$$ where from the form of $X_a$ we take $$\xi^1=\xi^t=0,\quad \xi^2=\xi^z=a(z),\quad
\xi^3=\xi^{\bar z}=\bar a(\bar z).
\label{xi^i}$$ The generator $\widehat X$ in the evolution form commutes with all the total derivatives $D_j$: $\Bigl[D_i,\widehat X\Bigr]=0$ and hence we have the standard commutation relation $$\Bigl[D_i,\X\infty\Bigr]=\Bigl[D_i,\sum_{j=1}^{3}\xi^jD_j\Bigr]
=\sum_{j=1}^{3}D_i(\xi^j)D_j.
\label{standcom}$$ We use it in the determining relation for operators of invariant differentiation $$\begin{aligned}
&
&\Bigl[\delta,\X\infty\Bigr]=\sum_{i}^{}\Bigl[\lambda_iD_i,\X\infty\Bigr]
=\sum_{i}^{}\Bigl(\sum_{j}\lambda_iD_i[\xi^j]D_j-\X\infty(\lambda_i)D_i\Bigr)
\nonumber
\\ & &=\sum_{i}^{}\Bigl(\sum_{j}\lambda_jD_j[\xi^i]-\X\infty(\lambda_i)\Bigr)D_i=0.
\label{comdelt}\end{aligned}$$ The final equation for the coefficients $\lambda_i$ of the operators of invariant differentiation (see eq.(24.2.3) of [@ovs]) is $$\X\infty(\lambda_i)=\sum_{j=1}^{3}\lambda_jD_j[\xi^i].
\label{eqinvdif}$$ Using (\[xi\^i\]) and restricting ourselves to the second prolongation $\X2$ of the symmetry generator, the equation (\[eqinvdif\]) leads to $$\X2(\lambda_1)=0,\quad \X2(\lambda_2)=\lambda_2 a^\prime(z), \quad
\X2(\lambda_3)=\lambda_3 {\bar a}^\prime(\bar z)
\label{eqlamb}$$ where primes denote derivatives and $$\lambda_i=\lambda_i(t,z,\bar z,u,u_t,u_z,
u_{\bar z},u_{zt},u_{\bar zt},u_{z\bar z},u_{zz},u_{\bar z\bar z}).$$ Here $\X2$ is the second prolongation of the generator $X_a$ of the conformal group defined by eq.(\[X2\]).
Equations (\[eqlamb\]) are easily solved by the method of characteristics and we choose $3$ simplest linearly independent solutions for the coefficients $\lambda_i$ of the three operators of invariant differentiation $$\begin{aligned}
& \lambda^1_1=1,\; \lambda^1_2=0,\; \lambda^1_3=0,\quad
\lambda^2_1=0,\; \lambda^2_2=e^{-u}u_{\bar zt},\; \lambda^2_3=0,&\nonumber
\\ &\lambda^3_1=0,\; \lambda^3_2=0,\; \lambda^3_3=e^{-u}u_{zt}.&
\label{sollamb}\end{aligned}$$ From here we obtain a basis for the operators of invariant differentiation $$\delta=D_t,\qquad \Delta=e^{-u}u_{\bar zt}D_z,\qquad
\bar \Delta=e^{-u}u_{zt}D_{\bar z}.
\label{deltas}$$
The [*basis of differential invariants*]{} is defined as a minimal finite set of invariants of a symmetry group from which any other differential invariant of this group can be obtained by a finite number of invariant differentiations and operations of taking composite functions. The proof of the existence and finiteness of the basis was given by Tresse [@tresse] and in a more modern form by Ovsiannikov [@ovs].
In our example the basis of differential invariants is formed by the set of three invariants $t,u_t,\rho$, while two other invariants $u_{tt}$ and $\eta$ of eq.(\[difinv\]) are given by the relations $$u_{tt}=\delta(u_t),\qquad \eta\equiv e^{-u}u_{zt}u_{\bar zt}=\Delta(u_t)
=\bar\Delta(u_t).
\label{invdepend}$$ All other functionally independent higher-order invariants can be obtained by acting with operators of invariant differentiation on the basis $\{t,u_t,\rho\}$. In particular, the following third-order invariants generated from the 2nd-order invariant $\rho$ by invariant differentiations will be involved in our construction $$\sigma=\Delta(\rho),\quad \bar\sigma=\bar\Delta(\rho),\quad \tau=\delta(\rho)
\equiv \rho_t.
\label{sigtau}$$
The operators of invariant differentiation form the commutator algebra $$\begin{aligned}
& {\displaystyle
\bigl[\delta,\Delta\bigr]=\left(\kappa\,\frac{\bar\sigma}{\eta}-3u_t\right)
\Delta,\qquad
\bigl[\delta,\bar\Delta\bigr]=\left(\kappa\,\frac{\sigma}{\eta}-3u_t\right)
\bar\Delta }&\nonumber
\\ & {\displaystyle
\bigl[\Delta,\bar\Delta\bigr]=\left(\frac{\Delta(\eta)}{\eta}-(u_t\rho+\tau)
\right)\bar\Delta -\left(\frac{\bar\Delta(\eta)}{\eta}-(u_t\rho+\tau)\right)
\Delta } &
\label{comalg}\end{aligned}$$ which form a Lie algebra over the field of invariants of the conformal group, in agreement with Ovsiannikov’s lemma 24.2 [@ovs].
The commutator algebra is simplified by introducing two new operators of invariant differentiation $Y$ and $\bar Y$ instead of $\Delta$ and $\bar\Delta$ and two new variables $\lambda$ and $\bar\lambda$ instead of $\sigma$ and $\bar\sigma$, defined by $$\Delta=\eta Y,\quad \bar\Delta=\eta\bar Y,\quad \sigma=\eta\lambda,\quad
\bar\sigma=\eta\bar\lambda ,
\label{Ydef}$$ and becomes $$\begin{aligned}
& {\displaystyle \bigl[\delta,Y\bigr]=\left(\kappa\bar\lambda
-3u_t-\frac{\delta(\eta)}{\eta}\right) Y, \qquad \bigl[\delta,\bar
Y\bigr]=\left(\kappa\lambda-3u_t-\frac{\delta(\eta)}{\eta}\right)\bar Y,\quad
}& \nonumber
\\&\bigl[Y,\bar Y\bigr]={\displaystyle\frac{(u_t\rho+\tau)}{\eta}}
\left(Y - \bar Y\right).&
\label{Yalg}\end{aligned}$$ Equations (\[invdepend\]) and (\[sigtau\]) imply the following properties of the operators $Y$ and $\bar Y$ $$Y(u_t)=\bar Y(u_t)=1,\quad Y(\rho)=\lambda,\quad \bar Y(\rho)=\bar\lambda.
\label{Yact}$$
Automorphic and resolving equations {#sec:autresys}
===================================
We have four independent differential invariants $t,u_t,\rho,\eta$ on the solution manifold of the heavenly equation (\[heavinv\]). We choose three of them $t,u_t,\rho$ as new invariant independent variables, the same number as in the original equation (\[heav\]), and consider the fourth one $\eta$ as a real function $F$ of these three $$\eta=F(t,u_t,\rho)\quad\iff\quad
u_{zt}u_{\bar zt}e^{-u}=F\left(t,u_t,u_{z\bar z}e^{-u}\right),
\label{autom}$$ which gives us the general form of the automorphic equation, [*i.e.*]{} invariant differential constraint.
Our next task is to derive the [*resolving equations*]{} for the heavenly equation. This will account for all integrability conditions of the system (\[heavinv\]), (\[autom\]) in an explicitly invariant form. If we pick a particular solution of this [*resolving system*]{} for $F$ and use it in the right-hand side of (\[autom\]), then the latter equation will possess the automorphic property: each solution of it can be obtained from any other solution by an appropriate conformal symmetry transformation.
We consider the automorphic equation (\[autom\]) divided by $F$ in the form $$Y(u_t)=1 ,
\label{Yautom}$$ and the heavenly equation (\[heavinv\]) in the form $$\delta(u_t)=\kappa\rho-u_t^2.
\label{deltheav}$$ We put $\eta = F$ in the definitions (\[Ydef\]) of $Y$ and $\bar Y$ and in their commutation relations (\[Yalg\]). The integrability condition for the system (\[Yautom\]) and (\[deltheav\]) is obtained by the [*invariant cross-differentiation*]{} with $\delta$ and $Y$ with the use of their commutation relation (\[Yalg\]) $$\delta(F)=\bigl[\kappa(\lambda+\bar\lambda)-5u_t\bigr]F.
\label{1}$$ Since this equation involves $\lambda$ and $\bar\lambda$ we use their definitions in eq.(\[Yact\]) $$Y(\rho)=\lambda,\qquad \bar Y(\rho)=\bar\lambda
\label{lambdef}$$ and obtain the integrability condition for these two equations by the invariant cross-differentiation by $\bar Y$ and $Y$ using their commutation relation from eq.(\[Yalg\]) $$F\bigl(Y(\bar\lambda)-\bar
Y(\lambda)\bigr)=(u_t\rho+\tau)(\lambda-\bar\lambda).
\label{3}$$ This equation contains $\tau$, so we use its definition (\[sigtau\]) $$\delta(\rho)=\tau .
\label{taudef}$$ Using the invariant cross-differentiation with $Y$ or $\bar Y$ and $\delta$, we obtain the compatibility conditions of eq.(\[taudef\]) with each of equations (\[lambdef\]) $$\delta(\lambda)=Y(\tau)+2u_t\lambda-\kappa\lambda^2
\label{2}$$ and $$\delta(\bar\lambda)=\bar Y(\tau)+2u_t\bar\lambda-\kappa{\bar\lambda}^2 .
\label{b2}$$ These are complex conjugate to each other. There is one more differential consequence of the obtained resolving equations. This is the integrability condition of the equation (\[3\]) solved with respect to $Y(\bar\lambda)$ together with the equation (\[b2\]). It is obtained by the invariant cross-differentiation of these equations by $\delta$ and $Y$. Using the other resolving equations it can be brought to the form $$\begin{aligned}
& & F\bigl(Y(\bar\lambda)+\bar Y(\lambda)\bigr)=
-(u_t\rho+\tau)(\lambda+\bar\lambda)\nonumber
\\ & & \mbox{}+2\kappa\bigl[\delta(\tau)+2F+4u_t\tau+\kappa\rho^2+2u_t^2\rho
\bigr].
\label{4}\end{aligned}$$
The resolving equations (\[1\]), (\[3\]), (\[2\]), (\[b2\]) and (\[4\]) form a closed resolving system if we assume that not only the 2nd-order differential invariant $\eta=F$, but also the 3rd-order differential invariants $\lambda,\bar\lambda$ and $\tau$ are functions of $t,u_t,\rho$. They should be regarded as additional unknowns in these equations, so the resolving system consists of $5$ partial differential equations with $4$ unknowns $F,\lambda,\bar\lambda$ and $\tau$. The operators of invariant differentiation are projected on the solution manifold of the heavenly equation and on the space of differential invariants treated as new independent variables. We keep the same notation for the projected operators of invariant differentiation and write them in the form $$\delta=\partial_t+(\kappa\rho-u_t^2)\partial_{u_t}+\tau\partial_\rho,
\quad Y=\partial_{u_t}+\lambda\partial_\rho,\quad
\bar Y=\partial_{u_t}+\bar\lambda\partial_\rho .
\label{deltproj}$$ Here we have used the following properties of these operators $$\begin{aligned}
& \delta(t)=1,\quad \delta(u_t)=\kappa\rho-u_t^2,\quad \delta(\rho)=\tau &
\label{deltprop}
\\ & Y(t)=\bar Y(t)=0,\quad
Y(u_t)=\bar Y(u_t)=1,\quad Y(\rho)=\lambda,\quad \bar Y(\rho)=\bar\lambda, &
\nonumber\end{aligned}$$ which follow from their definitions, equations (\[invdepend\]), (\[sigtau\]), (\[Yact\]) and the heavenly equation in the form (\[deltheav\]). If we used for the operators of invariant differentiation $\delta,Y,\bar Y$ the formulae (\[deltproj\]) in the resolving equations (\[1\]), (\[3\]), (\[2\]), (\[b2\]) and (\[4\]), then we would obtain the resolving system in an explicit form as a system of $5$ first-order PDEs with $4$ unknowns $F,\lambda,\bar\lambda,\tau$ and $3$ independent variables $t,u_t,\rho$. This system is passive, [*i.e.*]{} it has no further algebraically independent first-order integrability conditions.
The commutator relations (\[Yalg\]) were satisfied identically by the operators of invariant differentiation. On the contrary, for the projected operators (\[deltproj\]) these commutation relations and even the Jacobi identity $$\bigl[\delta,[Y,\bar Y]\bigr]+\bigl[Y,[\bar Y,\delta]\bigr]
+\bigl[\bar Y,[\delta,Y]\bigr]=0
\label{jacobi}$$ are not identically satisfied, but only on account of the resolving equations. It is easy to check that even a stronger statement is valid.
The commutator algebra [(\[Yalg\])]{} of the operators of invariant differentiation $\delta,Y,\bar Y$, together with the Jacobi identity [(\[jacobi\])]{}, is equivalent to the resolving system for the heavenly equation and hence provides a commutator representation for this system.
This theorem means that the complete set of the resolving equations is encoded in the commutator algebra of the operators of invariant differentiation and provides the easiest way to derive the resolving system. In Section $6$ we shall show how the commutator representation of the resolving system can lead to a useful Ansatz for solving this system.
Invariant and non-invariant solutions {#sec:invsol}
=====================================
[*Invariant solutions*]{} are defined as solutions that are invariant with respect to a symmetry subgroup of the equation. [*Non-invariant solutions*]{} are those solutions which are not invariant with respect to any one-parameter symmetry group of the equation. We present here a simple derivation of the infinitesimal criterion of invariance of solutions.
Consider a general form of the generator of a one-parameter symmetry of the heavenly equation as a linear combination of symmetry generators (\[symgen\]) with arbitrary real constant coefficients $\alpha$ and $\beta$ $$X= \alpha\partial_t + \beta\left(t\partial_t+2\partial_u\right)
+ a(z)\partial_z + \bar a(\bar z)\partial_{\bar z}
-\left(a^\prime(z)+\bar a^\prime(\bar z)\right)\partial_u
\label{symmet}$$ where $a(z)$ is an arbitrary holomorphic function. The infinitesimal criterion for the invariance of the solution $u=f(z,\bar z,t)$ with respect to the generator $X$ has the general form (see par. 19.2.1 of [@ovs]) $$X(f-u)|_{u=f}=0 ,
\label{gencrit}$$ which for $X$ defined by the formula (\[symmet\]) becomes $$(\alpha+\beta t)f_t + a(z)f_z + \bar a(\bar z)f_{\bar z} =
2\beta - a^\prime(z) - \bar a^\prime(\bar z).
\label{critinv}$$ The invariance criterion can be summed up as follows.
If there exists a holomorphic function $a(z)$ and constants $\alpha$ and $\beta$, not all equal to zero, such that the equation (\[critinv\]) is satisfied, then the solution $u=f(z,\bar z,t)$ is invariant. Otherwise this solution is non-invariant.
From this proposition one can derive some criteria for the non-invariance of solutions. For example, we consider the case when $\alpha=0$ and $\beta=0$ so that equation (\[critinv\]) is a criterion of conformal invariance. The general solution of eq.(\[critinv\]) in this case has the form $$u = \ln{f(\xi,t)} -\ln{a(z)} -\ln{\bar a(\bar z)}
\label{invsol}$$ where $$\xi = i \left(\int\frac{dz}{a(z)} - \int\frac{d\bar z}{\bar a(\bar z)}
\right).
\label{xi}$$ The invariant $\rho$ defined by eq.(\[difinv\]) becomes $$\rho = \frac{ff_{\xi\xi}-f_\xi^2}{f^3}
\label{ro}$$ and the invariants $\sigma$ and $\bar\sigma$, defined by eqs.(\[sigtau\]) as $$\sigma = e^{-u} u_{\bar zt} D_z(\rho),\qquad
\bar\sigma = e^{-u} u_{zt} D_{\bar z}(\rho) ,$$ are equal to each other $$\bar\sigma = \sigma = \frac{(ff_{t\xi}-f_tf_\xi)}{f^3}\times
\left(\frac{ff_{\xi\xi}-f_\xi^2}{f^3}\right)_\xi ,
\label{sigm}$$ where the subscripts denote partial differentiations. Hence the necessary condition for a solution to be conformally invariant is the equality $$\bar\sigma = \sigma \quad (\iff \bar\lambda = \lambda).
\label{confinv}$$ The converse statement gives the criterion for a solution to be conformally non-invariant.
The sufficient condition for a solution of the heavenly equation to be conformally non-invariant is that the following inequality should be satisfied $$\bar\sigma \ne \sigma
\label{noninv}$$
(or equivalently $\bar\lambda \ne \lambda$).
Concerning the practical use of this statement we must remark that even if the inequality (\[noninv\]) is satisfied for a solution of the resolving system it could become the equality (\[confinv\]) on the corresponding solution of the heavenly equation. Nevertheless the above criterion is useful, meaning that we should avoid solutions of the resolving equations satisfying eq.(\[confinv\]) in order not to end up with conformally invariant solutions.
Particular solutions of the resolvingsystem {#sec:resolv}
===========================================
Here we show that the commutator representation of the resolving system can prompt Ansatzes, leading to particular solutions of the resolving equations. Attempts to solve the commutation relations by imposing relations between the operators of invariant differentiation lead to invariant solutions of the heavenly equation. This is the case with the Ansatz $\bar Y = Y$. Then the expressions (\[deltproj\]) for $Y,\bar Y$ imply $\bar\lambda =
\lambda$, so the condition (\[noninv\]) of Corollary is not satisfied. Hence we obtain a conformally invariant solution of the heavenly equation.
Another possible simplifying Ansatz is that the operators $Y$ and $\bar Y$ commute and we have $$\tau = -u_t \rho \quad \Rightarrow \quad [Y,\bar Y] = 0 ,
\label{ans}$$ but $\bar Y\ne Y$.
Before solving the resolving system with the Ansatz (\[ans\]) we keep in mind that $F\ne 0$. Indeed the case $F=0$ is singular for the derivation of the resolving equations and should be treated separately. We shall consider first the case $F=0$ and show that it leads to invariant solutions of the heavenly equation.
Putting $F=0$ in equation (\[autom\]) we obtain $$u_{zt} = 0, \qquad u_{\bar zt} =0
\label{f0}$$ and hence we have the separation $$u = \alpha(t) + \beta(z,\bar z).
\label{sepu}$$ Substituting this expression for $u$ into the heavenly equation (\[heav\]) we obtain $$e^{\alpha(t)}\left(\alpha^{\prime\prime}(t)+\left(\alpha^\prime(t)\right)^2
\right)=\kappa e^{-\beta(z,\bar z)}\beta_{z\bar z}(z,\bar z)=2l ,
\label{sepheav}$$ where $l=\bar l$ is a separation constant and the primes denote derivatives in $t$. Integrating the equation for $\alpha$ we obtain $$\alpha(t)=\ln{\left(lt^2+C_1t+C_2\right)}
\label{alp}$$ where $C_1,C_2$ are arbitrary real constants. The equation for $\beta(z,\bar z)$ $$\beta_{z\bar z}=2\kappa l e^{\beta}
\label{liouv}$$ is the Liouville equation if $\kappa=1$ and the \`$\!$pseudo-Liouville$\!$' equation for $\kappa=-1$. Its general solution is $$\beta(z,\bar z) = \ln{a^\prime(z)} + \ln{{\bar a}^\prime(\bar z)}
-2 \ln{\left(a(z)+\bar a(\bar z)\right)} -\ln{l}
\label{plus}$$ if $\kappa=1$ and $$\beta(z,\bar z) = \ln{a^\prime(z)} + \ln{{\bar a}^\prime(\bar z)}
-2 \ln{\left(a(z)\bar a(\bar z)+1\right)} -\ln{l}
\label{minus}$$ if $\kappa=-1$. Here $a(z)$ is an arbitrary holomorphic function and the primes denote derivatives. Thus, the corresponding solutions of the heavenly equation are given by the equation (\[sepu\]) with $\alpha(t)$ determined by the formula (\[alp\]) and $\beta(z,\bar z)$ determined by the formula (\[plus\]) or (\[minus\]).
To obtain the simplest representative of the orbit of solutions we apply simplifying symmetry transformations: the conformal transformation $$a(z)\mapsto z,\qquad \bar a(\bar z)\mapsto \bar z ,$$ the suitable time translation and the dilation of time accompanied by a shift of $u$ $$u\mapsto u+\ln{l},\qquad t\mapsto \frac{t}{\sqrt{l}}.$$ The resulting solutions become $$u = \ln{\left(t^2+C\right)} -2\ln{(z+\bar z)}\qquad {\rm if}\quad \kappa=1 ,
\label{f0simplus}$$ $$u = \ln{\left(t^2+C\right)} -2\ln{(z\bar z+1)}\qquad {\rm if}\quad \kappa=-1,
\label{f0siminus}$$ where $C$ is an arbitrary real constant.
To perform a check of invariance of the solutions (\[f0simplus\]) and (\[f0siminus\]), we substitute them into the criterion of invariance (\[critinv\]) and make a splitting in $t$. Then we obtain $\alpha=0$ and if $C\ne 0$, then also $\beta=0$. For $C=0$ the constant $\beta$ can be arbitrary. We also obtain a differential equation for $a(z)$ and $\bar
a(\bar z)$ $$a^\prime(z)+{\bar a}^\prime(\bar z)=2\,\frac{a(z)+\bar a(\bar z)}{z+\bar z}
\quad {\rm for}\quad \kappa=1
\label{odeplus}$$ and $$a^\prime(z)+{\bar a}^\prime(\bar z)=2\,\frac{\bar z a(z)+ z\bar a(\bar
z)}{z\bar z+1}
\quad {\rm for}\quad \kappa=-1 ,
\label{odemin}$$ with the trivial solutions $$a=i,\; \bar a=-i \quad {\rm for}\; \kappa=1;\qquad
a=iz,\; \bar a=-iz \quad {\rm for}\; \kappa=-1.
\label{a(z)}$$ Thus, we have proved that there exist $\alpha,\beta$ and $a(z),\bar a(\bar z)$ such that the criterion (\[critinv\]) of invariance of solutions is satisfied for our solutions (\[f0simplus\]) and (\[f0siminus\]). Hence the case $F=0$ corresponds to invariant solutions.
In the following we assume that $F\ne 0$ and consider the resolving equations with the Ansatz (\[ans\]). Equations (\[3\]) and (\[4\]) become respectively $$Y(\bar\lambda)=\bar Y(\lambda)\quad {\rm and}\quad
Y(\bar\lambda) + \bar Y(\lambda) = 4\kappa$$ and hence $$\bar Y(\lambda) = 2\kappa,\qquad Y(\bar\lambda) = 2\kappa.
\label{Ylamb}$$ Next we consider the compatibility condition of the system of equations (\[2\]), (\[b2\]) and the first equation in (\[Ylamb\]). Because of the formula (\[ans\]) the first equation becomes $$\delta(\lambda) = u_t\lambda -\kappa\lambda^2 -\rho .
\label{2mod}$$ Then, using cross-differentiation of the invariant operators $\delta$ and $\bar Y$, their commutator (\[Yalg\]) and eq.(\[Ylamb\]), we obtain a very simple result $$\lambda + \bar\lambda = 2\kappa u_t.
\label{labla}$$ Solving this equation with respect to $\bar\lambda$, substituting in equation (\[b2\]) and using the equation (\[2mod\]) to express $\delta(\lambda)$, we obtain a quadratic equation for $\lambda$ $$\lambda^2 -2\kappa u_t\lambda +2\kappa\rho = 0 ,$$ with the solution $$\lambda = \kappa u_t + i\sqrt{2\kappa\rho -u_t^2} ,
\label{lamb}$$ where we have chosen the $+$ sign before the square root. Equation (\[labla\]) gives the result $$\bar\lambda = \kappa u_t - i\sqrt{2\kappa\rho -u_t^2}
\label{blamb}$$ which is complex conjugate to (\[lamb\]) provided the condition $$2\kappa\rho -u_t^2 \ge 0
\label{non-neg}$$ is satisfied. The obtained expressions for $\lambda$ and $\bar\lambda$ satisfy the equations (\[Ylamb\]), (\[2mod\]) and its complex conjugate, and hence all the resolving equations apart from the equation (\[1\]). We rewrite this last equation for the new unknown $f=\ln{F}$ as $$f_t + (\kappa\rho -u_t^2) f_{u_t} -u_t\rho f_\rho = -3u_t
\label{1mod}$$ and solve it by the method of characteristics obtaining the general solution of the equation (\[1\]) $$F = \rho^3 \varphi(\xi,\theta)\quad {\rm where}\quad
\xi=\frac{2\kappa\rho-u_t^2}{\rho^2}, \quad
\theta=t-\frac{\kappa}{\rho}\,\left(u_t+\sqrt{2\kappa\rho -u_t^2}\right),
\label{sol1}$$ where $\varphi$ is an arbitrary real differentiable function.
Finally we sum up our results for the particular solution of the resolving system which follows from our Ansatz (\[ans\]) $$F = \rho^3 \varphi(\xi,\theta),\quad \tau = -u_t\rho,\quad
\lambda = \kappa u_t + i\sqrt{2\kappa\rho -u_t^2}, \quad
\bar\lambda = \kappa u_t - i\sqrt{2\kappa\rho -u_t^2}
\label{solvresolv}$$ This will be used in the next section for obtaining the corresponding solution of the heavenly equation. Since $\bar\lambda\ne\lambda$ the condition (\[noninv\]) of Corollary for non-invariance of this solution is satisfied.
Invariant integration and non-invariant solution of the heavenly equation {#sec:non-invsol}
=========================================================================
In this section we reconstruct the solution of the heavenly equation starting from the particular solution (\[solvresolv\]) of the resolving system. We demonstrate here the procedure of [*invariant integration*]{} which amounts to the transformation of equations to the form of the exact invariant derivative. Then we drop the operator of invariant differentiation adding the term playing the role of the integration constant which is an arbitrary element of the kernel of this operator.
We start from our Ansatz (\[ans\]) using the definitions $\tau=\delta(\rho)$ and $\delta=D_t$ $$D_t(\ln{\rho}) = D_t(-u).
\label{ansmod}$$ We integrate this equation in the form $$\ln{\rho} = -u +\ln{\gamma_{z\bar z}(z,\bar z)},$$ where the last term is a function to be determined. Solving this equation with respect to $\rho$ and using the definition of $\rho$ we obtain $$\rho = e^{-u} u_{z\bar z} = e^{-u}\gamma_{z\bar z}(z,\bar z)$$ and hence $u_{z\bar z} = \gamma_{z\bar z}(z,\bar z)$. This implies the following form of the solution $$u(z,\bar z,t) =
\gamma(z,\bar z) + \alpha(z,t) +\bar\alpha(\bar z,t),
\label{solform}$$ where $\gamma,\alpha$ and $\bar\alpha$ are arbitrary smooth functions of two variables. After the substitution of this expression into the heavenly equation (\[heav\]) it becomes $$e^{\alpha(z,t)+\bar\alpha(\bar z,t)}\left[\alpha_{tt}(z,t)
+\bar\alpha_{tt}(\bar z,t)+\bigl(\alpha_t(z,t)
+\bar\alpha_t(\bar z,t)\bigr)^2\right] = \kappa e^{-\gamma(z,\bar z)}
\gamma_{z\bar z}(z,\bar z).
\label{heavmod}$$
Next we rewrite the formulae (\[solvresolv\]) for $\lambda$ and $\bar\lambda$ in the form of exact invariant derivatives $$Y(\sqrt{2\kappa\rho-u_t^2}-i\kappa u_t) = 0,\quad
\bar Y(\sqrt{2\kappa\rho-u_t^2}+i\kappa u_t) = 0.
\label{YbYeq}$$ On account of the definitions (\[Ydef\]), the operators $Y$ and $\bar Y$ can be written as $$Y=\frac{1}{F}\,\Delta=\frac{e^{-u}u_{\bar zt}}{F}\,D_z,\quad
\bar Y=\frac{1}{F}\,\bar\Delta=\frac{e^{-u}u_{zt}}{F}\,D_{\bar z}$$ and the equations (\[YbYeq\]) become $$(\sqrt{2\kappa\rho-u_t^2}+i\kappa u_t)_{\bar z} = 0,\quad
(\sqrt{2\kappa\rho-u_t^2}-i\kappa u_t)_z = 0.$$ They are integrated in the form $$\sqrt{2\kappa\rho-u_t^2}+i\kappa u_t=\psi(z,t),\quad
\sqrt{2\kappa\rho-u_t^2}-i\kappa u_t=\bar\psi(\bar z,t),
\label{intYeq}$$ where $\psi,\bar\psi$ are arbitrary smooth functions. Taking the difference of two equations (\[intYeq\]) we obtain $$u_t=-\frac{i\kappa}{2}\,[\psi(z,t)-\bar\psi(\bar z,t)]
= \alpha_t(z,t) +\bar\alpha_t(\bar z,t),$$ where the last equality follows from the expression (\[solform\]) for $u$. Separation of $z,\bar z$ in the last equality leads to $$\alpha_t(z,t)+\frac{i\kappa}{2}\,\psi(z,t) =
-\left[\bar\alpha_t(\bar z,t)-\frac{i\kappa}{2}\,\bar\psi(\bar z,t)
\right] = \chi^\prime(t) = -{\bar\chi}^\prime(t) ,$$ where $\chi^\prime(t)$ is the separation ‘constant" and the prime denotes the derivative. Solving these equations with respect to $\psi,\bar\psi$ and substituting the results into the equations (\[intYeq\]) we solve them with respect to the square root with the result $$\sqrt{2\kappa\rho-u_t^2}=i\kappa\left[\alpha_t(z,t)-\bar\alpha_t(\bar z,t)
-2\chi^\prime(t)\right].$$ Solving this equation with respect to $\kappa\rho$ and multiplying the result by $e^{\alpha+\bar\alpha}$ we obtain $$\begin{aligned}
\!\!&\!\!\!\!&\!\! \kappa e^{-\gamma(z,\bar z)}\gamma_{z\bar z}(z,\bar z)
\label{roeq}
\\ \!\!&\!\!\!\!&\!\! = 2e^{\alpha(z,t)+\bar\alpha(\bar z,t)}
\left[\alpha_t(z,t) \bar\alpha_t(\bar z,t)
+\chi^\prime(t)\bigl(\alpha_t(z,t) -\bar\alpha_t(\bar z,t)\bigr)
-{\chi^\prime}^2(t) \right].
\nonumber\end{aligned}$$ Using this equation in the right-hand side of the heavenly equation in the form (\[heavmod\]) and separating $z,\bar z$ we obtain two complex conjugate equations $$\begin{aligned}
\!\!&\!\!\!\!&\!\! \alpha_{tt}(z,t) = -\alpha_t^2(z,t)
+ 2\chi^\prime(t)\alpha_t(z,t) - {\chi^\prime}^2(t) + \mu(t) ,
\label{separ}
\\ \!\!&\!\!\!\!&\!\! \bar\alpha_{tt}(\bar z,t) =
- {\bar\alpha}_t^2(\bar z,t) - 2\chi^\prime(t)\bar\alpha_t(\bar z,t)
- {\chi^\prime}^2(t) - \mu(t) ,
\label{bsepar}\end{aligned}$$ where $\mu(t)=-\bar\mu(t)$ is the separation “constant". We substitute these expressions for $\alpha_{tt}$ and $\bar\alpha_{tt}$ into the transformed heavenly equation (\[heavmod\]) to obtain $$\begin{aligned}
\!\!&\!\!\!\!&\!\!
e^{\alpha(z,t)+\bar\alpha(\bar z,t)}\left[\alpha_t(z,t)\bar\alpha_t(\bar z,t)
+\chi^\prime(t)\bigl(\alpha_t(z,t)-\bar\alpha_t(\bar z,t)
-{\chi^\prime}^2(t)\bigr)\right]
\nonumber
\\ \!\!&\!\!\!\!&\!\! = \frac{\kappa}{2}\,e^{-\gamma(z,\bar z)}
\gamma_{z\bar z}(z,\bar z).
\label{heavsubst}\end{aligned}$$ Next we take the total derivative $D_t$ of this equation and substitute again the second derivatives $\alpha_{tt}$ and $\bar\alpha_{tt}$ from the equations (\[separ\]) and (\[bsepar\]). The result is unexpectedly simple $$\left(\chi^{\prime\prime}-\mu\right)\left(\alpha_t-\bar\alpha_t
-2\chi^\prime\right) = 0.
\label{simpl}$$ This equation implies that $$\mu(t) = \chi^{\prime\prime}(t) ,
\label{mueq}$$ since the complementary assumption $$\alpha_t-\bar\alpha_t -2\chi^\prime = 0$$ leads again to the equation (\[mueq\]). Indeed, the last equation allows a separation of $z,\bar z$ and, being integrated, gives $\alpha,\bar\alpha$ $$\alpha(z,t) = \chi(t)+\nu(t)+\omega(z), \qquad
\bar\alpha(\bar z,t) = -\chi(t)+\nu(t)+\bar\omega(\bar z).$$ Substituting these expressions into the equations (\[separ\]) and (\[bsepar\]) and comparing the results we discover again the equation (\[mueq\]).
With this restriction the equations (\[separ\]) and (\[bsepar\]) are simplified and integrated to give $$\alpha(z,t) = \ln{\bigl(t+b(z)\bigr)} + \chi(t) + \omega(z), \quad
\bar\alpha(\bar z,t) = \ln{\bigl(t+\bar a(\bar z)\bigr)} - \chi(t) +
\bar\omega(\bar z),
\label{alpsol}$$ where $b(z)$ and $\omega(z)$ are arbitrary holomorphic functions and we have reserved the notation $a(z)$ only for the generators of the conformal vector field $X_a$ in eq.(\[symgen\]).
Now define a new function of $z,\bar z$ $$\Gamma(z,\bar z) = \gamma(z,\bar z) + \omega(z) + \bar\omega(\bar z),
\label{defGam}$$ so that the form (\[solform\]) of the solution becomes $$u(z,\bar z,t) = \ln{\bigl(t+b(z)\bigr)} + \ln{\bigl(t+\bar b(\bar z)\bigr)}
+ \Gamma(z,\bar z).
\label{solformnew}$$ Substituting the expressions (\[alpsol\]) for $\alpha,\bar\alpha$ into the transformed heavenly equation (\[heavmod\]) we obtain the equation for the only unknown function $\Gamma(z,\bar z)$ in the solution (\[solformnew\]) $$\Gamma_{z\bar z} = 2\kappa e^\Gamma.
\label{Liouville}$$ If $\kappa=1$ this is the Liouville equation with the general solution $$\Gamma(z,\bar z) = \ln{c^\prime(z)} + \ln{{\bar c}^\prime(\bar z)}
- 2 \ln{\bigl(c(z)+\bar c(\bar z)\bigr)}
\label{Gplus}$$ where $c(z)$ is an arbitrary holomorphic function. If $\kappa=-1$ we call the equation (\[Liouville\]) \`$\!$pseudo-Liouville$\!$' equation and its general solution is $$\Gamma(z,\bar z) = \ln{c^\prime(z)} + \ln{{\bar c}^\prime(\bar z)}
- 2 \ln{\bigl(c(z)\bar c(\bar z)+1\bigr)}.
\label{Gminus}$$ Finally, substituting these expressions for $\Gamma(z,\bar z)$ into the equation (\[solformnew\]) we obtain the solutions of the heavenly equation (\[heav\]) for the two choices of the sign $\kappa=+1$ and $\kappa=-1$.
1. The solution for $\kappa=1$: $$\begin{aligned}
\!\!&\!\!\!\!&\!\! u(z,\bar z,t) =
\ln{\bigl(t+b(z)\bigr)}+\ln{\bigl(t+\bar b(\bar z)\bigr)}
\nonumber
\\ \!\!&\!\!\!\!&\!\! \mbox{} +\ln{c^\prime(z)} +\ln{{\bar c}^\prime(\bar z)}
- 2 \ln{\bigl(c(z)+\bar c(\bar z)\bigr)}.
\label{solplus}\end{aligned}$$
2. The solution for $\kappa=-1$ (see also [@tod]): $$\begin{aligned}
\!\!&\!\!\!\!&\!\! u(z,\bar z,t) =
\ln{\bigl(t+b(z)\bigr)}+\ln{\bigl(t+\bar b(\bar z)\bigr)}
\nonumber
\\ \!\!&\!\!\!\!&\!\! \mbox{}+\ln{c^\prime(z)} + \ln{{\bar c}^\prime(\bar z)}
- 2 \ln{\bigl(c(z)\bar c(\bar z)+1\bigr)}.
\label{solminus}\end{aligned}$$
Here $b(z)$ and $c(z)$ are arbitrary holomorphic functions.
To avoid “false generality" it is sufficient to choose the simplest representative of the obtained orbits of solutions applying the conformal symmetry transformation $c(z)=z,\;\bar c(\bar z)=\bar z$ with the following results.
1. The solution for $\kappa=1$: $$u(z,\bar z,t) = \ln{\bigl(t+b(z)\bigr)}+\ln{\bigl(t+\bar b(\bar z)\bigr)}
- 2 \ln{(z+\bar z)}.
\label{simpsolpl}$$
2. The solution for $\kappa=-1$: $$u(z,\bar z,t) = \ln{\bigl(t+b(z)\bigr)}+\ln{\bigl(t+\bar b(\bar z)\bigr)}
- 2 \ln{(z\bar z+1)}.
\label{simpsolmin}$$
Here $b(z)$ is still an arbitrary holomorphic function.
Up to now we solved completely only the Ansatz (\[ans\]) defining $\tau$, but we did not check the automorphic equation (\[autom\]) and the auxiliary equations (\[lambdef\]), by using the particular solution (\[solvresolv\]) of the resolving system. Hence, though we obtained the correct solutions (\[simpsolpl\]) and (\[simpsolmin\]) of the heavenly equation (\[heav\]), we have not made a complete foliation of these solutions into separate orbits.
To do this, first we remark that due to the discrete symmetry of the heavenly equation (\[heav\]) and of our solutions with respect to the permutation $z\leftrightarrow \bar z$, we can define the holomorphic function $b(z)$ as satisfying the condition $${\rm Im}\,b(z)\ge 0\;{\rm for}\;\kappa=1 \quad {\rm and} \quad
{\rm Im}\,b(z)\le 0\;{\rm for}\;\kappa=-1.
\label{condit}$$ Then we check that the automorphic equation (\[autom\]) coincides with the auxiliary equations (\[lambdef\]) and becomes $$(z+\bar z)^2 b^\prime(z) {\bar b}^\prime(\bar z)=8\varphi(\xi,\theta)\qquad
{\rm for}\quad \kappa=1
\label{autpl}$$ and $$(z\bar z+1)^2 b^\prime(z) {\bar b}^\prime(\bar z)=-8\varphi(\xi,\theta)\qquad
{\rm for}\quad\kappa=-1 .
\label{autmin}$$ Using the solutions (\[simpsolpl\]) and (\[simpsolmin\]) in the definitions (\[sol1\]) of the characteristic variables $\xi$ and $\theta$, we discover that they depend only on $b$ and $\bar b$, [*i.e.*]{} $$\xi = -\frac{(b-\bar b)^2}{4},\qquad\theta = -\frac{(b+\bar b)}{2} -\sqrt{\xi}.
\label{xieta}$$ Hence, defining the new arbitrary function $\Phi(b,\bar b)=\varphi(\xi,\theta)$, the automorphic equations (\[autpl\]) and (\[autmin\]) become $$(z+\bar z)^2 b^\prime(z) {\bar b}^\prime(\bar z) = 8\Phi(b,\bar b)\qquad
{\rm for}\quad \kappa=1
\label{aut+}$$ and $$(z\bar z+1)^2 b^\prime(z) {\bar b}^\prime(\bar z) = -8\Phi(b,\bar b)\qquad
{\rm for}\quad \kappa=-1.
\label{aut-}$$ Sufficient conditions for solving these functional-differential equations are given by the following choices of $\Phi(b,\bar b)$ $$\Phi(b,\bar b)=
\frac{\left[f(b)+\bar f(\bar b)\right]^2}{8f^\prime(b){\bar f}^\prime(\bar b)}
\qquad {\rm for}\quad \kappa=1
\label{phi+}$$ and $$\Phi(b,\bar b)=-\frac{\left[f(b)\bar f(\bar b)
+1\right]^2}{8f^\prime(b){\bar f}^\prime(\bar b)}\qquad {\rm for}\quad
\kappa=-1,
\label{phi-}$$ where $f(b)$ is an arbitrary holomorphic function. Then the automorphic equations become $$\left\{\ln{\left[f(b)+\bar f(\bar b)\right]}\right\}_{z\bar z}
=\left[\ln{(z+\bar z)}\right]_{z\bar z}\qquad {\rm for}\quad \kappa=1
\label{autom+}$$ and $$\left\{\ln{\left[f(b)\bar f(\bar b)+1\right]}\right\}_{z\bar z}
=\left[\ln{(z\bar z+1)}\right]_{z\bar z}\qquad {\rm for}\quad \kappa=-1 .
\label{autom-}$$ Their general solutions are $$f(b)+\bar f(\bar b) = w(z)\bar w(\bar z)(z+\bar z)
\qquad {\rm for}\quad \kappa=1
\label{autsol+}$$ and $$f(b)\bar f(\bar b)+1 = w(z)\bar w(\bar z)(z\bar z+1)\qquad {\rm for}\quad
\kappa=-1 ,
\label{autsol-}$$ where $w(z)$ is an arbitrary holomorphic function. The formulae (\[phi+\]) and (\[phi-\]) for $\Phi(b,\bar b)$ become $$\Phi(b,\bar b) = \frac{w^2(z){\bar w}^2(\bar z)(z+\bar
z)^2}{8f^\prime(b){\bar f}^\prime(\bar b)} \qquad {\rm for}\quad \kappa=1$$ and $$\Phi(b,\bar b) = -\frac{w^2(z){\bar w}^2(\bar z)(z\bar
z+1)^2}{8f^\prime(b){\bar f}^\prime(\bar b)} \qquad {\rm for}\quad
\kappa=-1.$$ If we plug these formulae into the automorphic equations (\[aut+\]) and (\[aut-\]), then both automorphic equations coincide and become $$b^\prime(z){\bar b}^\prime(\bar z)=
\frac{w^2(z){\bar w}^2(\bar z)}{f^\prime(b){\bar f}^\prime(\bar b)}\,.
\label{automorph}$$ This equation admits a separation of variables, leading to the ODEs $$b^\prime(z)=\frac{w^2(z)}{f^\prime(b)},\qquad
{\bar b}^\prime(z)=\frac{{\bar w}^2(\bar z)}{{\bar f}^\prime(\bar b)}\,.
\label{autODE}$$ The obvious choice of the functions $w(z)$ and $\bar w(\bar z)$ $$w(z)=1 \quad\Longleftrightarrow \quad\bar w(\bar z)=1
\label{wsol}$$ simplifies the ODEs (\[autODE\]) to $$[f(b)]_z=1, \qquad [\bar f(\bar b)]_{\bar z}=1 ,
\label{simpODE}$$ with the solution $$f[b(z)]=z, \qquad \bar f[\bar b(\bar z)]=\bar z
\label{solODE}$$ meaning that $b(z)$ is the inverse function for $f(b)$: $b=f^{-1}$. The equations (\[autsol+\]) and (\[autsol-\]) are obviously satisfied by the solution (\[solODE\]) with our choice (\[wsol\]) of $w(z),\bar
w(\bar z)$.
Thus, any particular function $b(z)$ can be obtained for an appropriate choice of $f(b)$ as its inverse function. This fixes the function $\Phi(b,\bar b)$ according to the formulae (\[phi+\]) or (\[phi-\]), the function $\varphi(\xi,\theta)=\Phi(b,\bar b)$ and the right-hand side $F$ of the automorphic equation (\[autom\]) determined by the formulae (\[solvresolv\]). Hence any particular choice of the function $b(z)$ in our solutions (\[simpsolpl\]), (\[simpsolmin\]) means a corresponding choice of the particular orbit in the solution space of the heavenly equation.
Check of non-invariance of the solutions {#sec:check}
========================================
In this section we prove that our solutions (\[simpsolpl\]) and (\[simpsolmin\]) of the heavenly equation (\[heav\]) are non-invariant, with respect to its symmetry group generated by the vector fields in (\[symgen\]), for generic functions $a(z)$, except for some particular classes listed below in the theorems summarizing the results.
For the check of non-invariance we substitute our solutions (\[simpsolpl\]) and (\[simpsolmin\]) into the invariance criterion (\[critinv\]). The resulting equation is quadratic in $t$ and it implies the vanishing of the coefficients of $t^2,t$ and $t^0$.
We consider first the case $\kappa=1$. The term with $t^2$ gives again the equation (\[odeplus\]) of the Section 6. However, now we need the general solution of this equation.
We assume in the generic case that $a^\prime(z)+{\bar a}^\prime(\bar z)\ne
0$, otherwise the equation (\[odeplus\]) implies $a=-\bar a=constant$ and this case should be treated separately. Then we rewrite the equation (\[odeplus\]) in the form $$\frac{a(z)+\bar a(\bar z)}{a^\prime(z)+{\bar a}^\prime(\bar z)} =
\frac{z+\bar z}{2} \qquad\Longrightarrow\qquad
\left(\frac{a+\bar a}{a^\prime+{\bar a}^\prime}\right)_{z\bar z} =0 .
\label{modt^2+}$$ In order to consider the generic case we postulate $a^{\prime\prime}{\bar a}^{\prime\prime}\ne0$, then the last equation can be easily manipulated, obtaining the solution $$a(z) = C_1(z+\lambda)^2+C_2,\qquad \bar a(\bar z) =
-\left[C_1(\bar z-\lambda)^2+C_2\right] ,
\label{a,ba}$$ where $C_1\ne 0$, $C_2$ and $\lambda$ are arbitrary purely imaginary constants.
Now we consider the term without $t$ in the criterion of invariance using our result (\[a,ba\]) which gives the equation with the separated variables $z,\bar z$ $$\begin{aligned}
\!\!&\!\!\!\!&\!\!\frac{\alpha}{b(z)}+\left[C_1(z+\lambda)^2+C_2\right]
\frac{b^\prime(z)}{b(z)}-\beta =
\label{t^0+}
\\ \!\!&\!\!\!\!&\!\! -\left\{
\frac{\alpha}{\bar b(\bar z)}-\left[C_1(\bar z-\lambda)^2+C_2\right]
\frac{{\bar b}^\prime(\bar z)}{\bar b(\bar z)}-\beta\right\}
=\mu=-\bar\mu \nonumber\end{aligned}$$ where $\mu$ is a separation constant. Comparing these equations with the equation obtained from the term with $t$ in the criterion of invariance we conclude that they coincide if and only if the condition $$\mu\left(b(z)-\bar b(\bar z)\right) = 0$$ is satisfied. This implies $\mu=0$, since otherwise we have $b=\bar b=constant$ and our solution is obviously invariant depending only on two variables $t$ and $z+\bar z$. Hence the equations (\[t\^0+\]) become $$\begin{aligned}
\!\!&\!\!\!\!&\!\!\left[C_1(z+\lambda)^2+C_2\right]b^\prime(z)-\beta b(z)
=-\alpha ,
\label{t+}
\\[1mm]
\!\!&\!\!\!\!&\!\!\left[C_1(\bar z-\lambda)^2+C_2\right]
{\bar b}^\prime(\bar z) + \beta \bar b(\bar z)=\alpha .
\label{bt+}\end{aligned}$$ Consider now the case $C_2\ne 0,\;\beta\ne 0$ and introduce the notation $$\nu=\sqrt{-\frac{C_2}{C_1}},\qquad \gamma=\frac{\beta}{2\sqrt{-C_1C_2}}\,.
\label{notation+}$$ Integrating the ODEs (\[t+\]) and (\[bt+\]) we fix the function $b(z)$ in our solution of the heavenly equation which corresponds to the invariant solution in the considered case $$b(z) = C\left(\frac{z+\lambda-\nu}{z+\lambda+\nu}\right)^\gamma +
\frac{\alpha}{\beta},\quad
\bar b(\bar z) = \bar C\left(\frac{\bar z-\lambda-\nu}{\bar z-\lambda+\nu}
\right)^\gamma + \frac{\alpha}{\beta}
\label{b+}$$ where $C,\bar C$ are integration constants.
In a similar way we treat other possible cases. We sum up the results for the case of $\kappa=1$ in the following statement.
The function $$u = \ln{\bigl(t+b(z)\bigr)}+\ln{\bigl(t+\bar b(\bar z)\bigr)}
- 2 \ln{(z+\bar z)}
\label{sol+}$$ is a solution of the heavenly equation (\[heav\]) for $\kappa=+1$ for an arbitrary holomorphic function $b(z)$. This solution is a non-invariant solution of this equation iff the function $b(z)$ does not coincide with any of the following choices:
1. $$b(z) = C\left(\frac{z+\lambda-\nu}{z+\lambda+\nu}\right)^\gamma +
\frac{\alpha}{\beta}$$ where $\alpha$ and $\beta$ are arbitrary real constants, $\beta\ne 0$, $\nu$ and $\gamma$ are defined by the formulae (\[notation+\]) and $\lambda,C_1,C_2$ are complex constants which satisfy the conditions $$\bar\lambda=-\lambda,\quad \bar C_1=-C_1,\quad \bar C_2=-C_2,
\quad C_1\ne 0,\quad C_2\ne 0.$$ In this case the solution is invariant with respect to the symmetry generator $$\begin{aligned}
\!\!&\!\!\!\!&\!\!X = \alpha\partial_t +\beta\left(t\partial_t+2\partial_u
\right)+C_1\left[(z+\lambda)^2\partial_z-(\bar z-\lambda)^2\partial_{\bar z}
- 2(z-\bar z)\partial_u\right]
\\ \!\!&\!\!\!\!&\!\!\mbox{}+C_2 \left(\partial_z -\partial_{\bar z}\right).\end{aligned}$$
2. $$b(z) =
\frac{\alpha}{2\sqrt{-C_1C_2}}
\left(\frac{z+\lambda+\nu}{z+\lambda-\nu}\right) + C\qquad {\rm if}\quad
\beta=0,\;C_2\ne 0;$$ the solution is invariant with respect to the previous symmetry generator $X$ with $\beta=0$.
3. $$b(z)=C\exp{\left[-\frac{\beta}{C_1(z+\lambda)}\right]}
+\frac{\alpha}{\beta}\qquad {\rm if}\quad C_2=0,\;\beta\ne 0;$$ the solution is invariant with respect to the symmetry generator $X$ from the case $1$ with $C_2=0$.
4. $$b(z)=\frac{\alpha}{C_1(z+\lambda)} + C\qquad {\rm if}\quad \beta=0\;
{\rm and}\; C_2= 0;$$ the solution is invariant with respect to the symmetry generator $X$ from the case $1$ with $\beta=0$ and $C_2=0$.
5. $$b(z) = C(C_1z+C_2)^{\beta/C_1}+\frac{\alpha}{\beta}\qquad {\rm if}\quad
C_1\ne 0,\;\beta\ne 0;$$ the solution is invariant with respect to the symmetry generator $$X = \alpha\partial_t +\beta\left(t\partial_t+2\partial_u\right)
+C_1\left(z\partial_z+\bar z\partial_{\bar z}-2\partial_u\right)
+C_2 \left(\partial_z -\partial_{\bar z}\right).$$
6. $$b(z) = Ce^{\frac{\beta}{C_2}z}+\frac{\alpha}{\beta}
\qquad {\rm if}\quad C_1 = 0,\;\beta\ne 0;$$ the solution is invariant with respect to the symmetry generator $X$ from the case $5$ with $C_1=0$.
7. $$b(z) = -\frac{\alpha}{C_2}\,z+C\qquad {\rm if}\quad C_1 = 0,\;\beta = 0;$$ the solution is invariant with respect to the symmetry generator $X$ from the case $5$ with $C_1=0$ and $\beta=0$.
8. $$b(z) = b = constant \qquad {\rm if}\quad C_1=\alpha=\beta=0,\;C_2\ne 0;$$ the solution is invariant with respect to the symmetry generator $$X=\partial_z - \partial_{\bar z}.$$ If $b=\alpha/\beta$, then this solution is also invariant with respect to the generator $$X = \alpha\partial_t +\beta\left(t\partial_t+2\partial_u\right).$$
Now we consider the case $\kappa=-1$ and substitute the solution (\[simpsolmin\]) of the heavenly equation (\[heav\]) into the criterion of invariance (\[critinv\]). Then the resulting equation is again quadratic in $t$ and the term with $t^2$ gives us again the equation (\[odemin\]), for which we need now the general solution. First we rewrite it in the form $$a^\prime+{\bar a}^\prime + \frac{a^\prime + {\bar a}^\prime}{z\bar z}
= 2\left(\frac{a}{z}+\frac{\bar a}{\bar z}\right).$$ Differentiating this equation with respect to $z$ and $\bar z$ we obtain an equation which admits separation of $z,\bar z$ in the form $$za^{\prime\prime}(z)-a^\prime(z)=-\left[\bar z{\bar a}^{\prime\prime}(\bar z)
-{\bar a}^\prime(\bar z)\right]=\lambda=-\bar\lambda
\label{ODEs}$$ where $\lambda$ is a separation constant. Integrating these ODEs we obtain $$a(z)=C_1z^2-\lambda z+C_2,\qquad \bar a(\bar z)=\bar C_1{\bar z}^2
+\lambda\bar z+\bar C_2$$ where $C_1,C_2$ are integration constants. Substituting these solutions into the equation (\[odemin\]) we see that it is identically satisfied if and only if $\bar C_1=C_2\;\iff \;\bar C_2=C_1$, so that finally we have the solution of the equation following from the term with $t^2$ $$a(z) = C_1z^2-\lambda z+C_2,\qquad \bar a(\bar z)=C_2{\bar z}^2+\lambda\bar z
+C_1.
\label{a,ba-}$$
Next we consider the term without $t$ in the criterion of invariance using our result (\[a,ba-\]) which gives the equation with the separated variables $z,\bar z$ $$\begin{aligned}
\!\!&\!\!\!\!&\!\!\frac{\alpha}{b(z)}+\left(C_1z^2-\lambda z+C_2\right)
\frac{b^\prime(z)}{b(z)}-\beta = \nonumber
\\ \!\!&\!\!\!\!&\!\! -\left[
\frac{\alpha}{\bar b(\bar z)}+\left(C_2{\bar z}^2+\lambda\bar z+C_1\right)
\frac{{\bar b}^\prime(\bar z)}{\bar b(\bar z)}-\beta\right]=\mu=-\bar\mu
\label{t^0-}\end{aligned}$$ where $\mu$ is a separation constant. Comparing these equations with the equation obtained from the term with $t$ in the criterion of invariance we conclude that they coincide if and only if the condition $$\mu\left(b(z)-\bar b(\bar z)\right) = 0$$ is satisfied. This implies $\mu=0$ for the same reason as in the case $\kappa=1$. Hence the equations (\[t\^0-\]) become $$\begin{aligned}
\!\!&\!\!\!\!&\!\!\left(C_1z^2-\lambda z+C_2\right)b^\prime(z)-\beta b(z)
=-\alpha ,
\label{t-}
\\[1mm]
\!\!&\!\!\!\!&\!\!\left(C_2{\bar z}^2+\lambda\bar z+C_1\right)
{\bar b}^\prime(\bar z) - \beta \bar b(\bar z)=-\alpha .
\label{bt-}\end{aligned}$$ Consider now the case $C_1\ne 0$. Introduce the new constants $\tilde\lambda=-\lambda/(2C_1)$ and $\tilde C_2=C_2-\lambda^2/(4C_1)$. Then the first equation takes the form $$\left[C_1(z+\tilde\lambda)^2+\tilde C_2\right]b^\prime(z)-\beta b(z)=-\alpha
\label{tt-}$$ coinciding with the ODE (\[t+\]) in the case $\kappa=1$. Therefore we can use its solutions with an appropriate change of notation. Other possible cases are treated in a similar way. Therefore we can transfer the results of Theorem to the case $\kappa=-1$ with an appropriate change of notation and sum them up in the following statement.
The function $$u = \ln{\bigl(t+b(z)\bigr)}+\ln{\bigl(t+\bar b(\bar z)\bigr)}
- 2 \ln{(z\bar z+1)}
\label{sol-}$$ is a solution of the heavenly equation (\[heav\]) for $\kappa=-1$ for an arbitrary holomorphic function $b(z)$. This solution is a non-invariant solution of this equation iff the function $b(z)$ does not coincide with any of the $8$ forms given in Theorem with the change of notation $$\begin{aligned}
\!\!&\!\!\!\!&\!\!\lambda\mapsto \tilde\lambda=-\frac{\lambda}{2C_1},\quad
C_2\mapsto \tilde C_2=C_2-\frac{\lambda^2}{4C_1},
\\ \!\!&\!\!\!\!&\!\!\nu\mapsto \tilde\nu =
\sqrt{-\frac{\tilde C_2}{C_1}},\quad\gamma\mapsto \tilde\gamma=
\frac{\beta}{2\sqrt{-C_1\tilde C_2}}\end{aligned}$$ in the cases $1,2,3,4$ and $C_1\mapsto -\lambda$ in the case $5$. Those $8$ choices of $b(z)$ give invariant solutions with respect to the corresponding symmetry generators of Theorem with the same change of notation.
Conclusions and outlook {#sec:conclusion}
=======================
The title of this article, or rather of the research direction that it represents, could have been “Invariant methods for obtaining non-invariant solutions of partial differential equations". The main result is that we are proposing an alternative tool for obtaining particular solutions of non-linear partial differential equations with infinite dimensional symmetry algebras. As stated in the Introduction, the idea of the method is more than a hundred years old [@lie; @vessiot]. We have turned it into a usable tool by adding new elements. These are:
1. The systematic use of invariant cross-differentiation involving the operators of invariant differentiation and their commutator algebra for the derivation of the resolving equations and for obtaining their particular solutions.
2. The presentation of the resolving system as a Lie algebra of the operators of invariant differentiation (over the field of differential invariants of the symmetry group) [@ns].
3. The concept of invariant integration applied to the automorphic system.
Let us use the heavenly equation (\[heav\]) to compare different methods of obtaining exact analytical solutions of a partial differential equation, provided or at least suggested by symmetry analysis. In all of them the studied equation is embedded into a larger system of equations, to be solved simultaneously.
The most standard method is that of invariant solutions [@ovs; @olv; @wint]. One first finds the symmetry algebra realized by vector fields of the form $$X = \tau\partial_t +\xi\partial_z +\bar\xi\partial_{\bar z} +\phi\partial_u
\label{vfield}$$ where $\tau,\xi,\bar\xi$ and $\phi$ are functions of $t,z,\bar z$ and $u$. Once this algebra is found ([*i.e.*]{} the algebra (\[symgen\]) for the heavenly equation) one classifies its subalgebras into conjugacy classes and then adds one, or more, first order linear equations of the type $$\tau u_t +\xi u_z + \bar\xi u_{\bar z} -\phi = 0
\label{addeq}$$ to the studied equation. These equations are solved, their solution is substituted into the original equation. This again is solved and we obtain solutions invariant under the chosen subgroup.
Further methods are the Bluman and Cole “non-classical method" [@blucol], the Clarkson-Kruskal [@clarkrus] “direct method" and that of “conditional symmetries" [@levwin] (see [@clarwint] for a review). These methods, basically all equivalent, amount to the fact that a first order equation of the type (\[addeq\]) is added to the studied equation, without the requirement that $\tau,\xi,\bar\xi$ and $\phi$ define an element of the symmetry algebra.
Finally, we have the group foliation method [@ns] used and further developed in this article. Let us review the essential steps, performed above.
1. Find the total symmetry algebra (\[symgen\]).
2. Find all differential invariants of order up to $N$ of its infinite dimensional subalgebra which is Lie algebra of the conformal group. The number $N$ must be larger or equal to the order of the equation and must satisfy the requirement that there should be $\# N$ functionally independent invariants with $$\# N \ge p+q
\label{ineq}$$ where $p$ and $q$ are the number of independent and dependent variables, respectively. In our case we have $p=3,\; q=1,\; N=2,\; \# N=5$. The actual invariants are given in the equation (\[difinv\]).
3. Choose $p$ invariants as new independent variables and require that the remaining invariants be functions of the chosen ones. This provides us with the automorphic system that also contains the considered equation, expressed in terms of the invariants. In our case the automorphic system consists of the equation (\[heavinv\]) (the heavenly equation) and the equation (\[autom\]) (or equivalently (\[Yautom\])).
4. Find the “resolving equations". This is a set of compatibility conditions between the studied equation and those that we have added to obtain the automorphic system. In our case we require compatibility between the equations (\[heavinv\]) and (\[autom\]), [*i.e.*]{} determine the restrictions on the function $F(t,u_t,\rho)$. We have shown that this can be done in an explicitly invariant manner by using the operators of invariant differentiation, in our case $\delta,Y$ and $\bar Y$ of the equations (\[deltas\]) and (\[Ydef\]). The resolving system in our case consists of the equations (\[1\]), (\[3\]), (\[2\]), (\[b2\]) and (\[4\]). As stated by the fundamental Theorem $1$, this resolving system is best written as a system of commutator relations for the operators of invariant differentiation projected on the solution manifold of the heavenly equation in the space of differential invariants, together with the Jacobi relations for these operators.
5. Solve the resolving system and the automorphic one. This provides solutions of the original equation.
The last step, step $5$ is the most difficult one. If it can be carried out completely, we obtain “all" solutions, both invariant and non-invariant ones. In general, such a situation is too good to be true. In particular, for the heavenly equation we were not able to solve the system (\[Yalg\]), (\[jacobi\]) in general. Instead, we made various simplifying assumptions. The most obvious ones, like $Y=\bar Y$ or $F=0$, lead to invariant solutions. These we already know, or can obtain by much simpler standard methods. The assumption, or restriction, that leads to non-invariant solutions was $[Y,\bar Y]=0$. The solutions obtained are (\[solplus\]) and (\[solminus\]), for $\kappa=1$ and $\kappa=-1$, respectively. Each solution involves two arbitrary holomorphic functions. One of them, $b(z)$ is fundamental. The other is induced by a conformal transformation and can be transformed away ([*i.e.*]{} set equal to [*e.g.*]{} $c(z)=z$). In Section \[sec:check\] we show that the solutions are, in general, not invariant under any subgroup of the symmetry group. They reduce to invariant ones only for very special choices of the function $b(z)$, specified in Theorems $2$ and $3$.
It would be interesting to relate the concepts of this article to that of integrability for non-linear partial differential equations. “Integrability" means that the considered equation is viewed as an integrability condition for a Lax pair, a pair of linear operators [@lax; @ablclar]. Here we can view the equations (\[comalg\]) as a set of relations between a triplet of linear operators, subject to a non-linear constraint (\[jacobi\]) .
Acknowledgments {#acknowledgments .unnumbered}
===============
A large part of the research reported here was performed while M.B.S. and P.W. were visiting the Università di Lecce. They thank the Dipartimento di Fisica and INFN, Sezione di Lecce, for their hospitality and support. One of the authors (M.B.S.) thanks Y. Nutku for useful discussions.
The research of P.W. is partly supported by research grants from NSERC of Canada and FCAR du Québec. The research of L.M. is supported by INFN - Sezione di Lecce and by the research grant Prin - Sintesi 2000 from MURST of Italy and it is a part of the INTAS research project $99 - 1782$.
[99]{} Lie S 1884 Über Differentialinvarianten [*Math. Ann.*]{} [**24**]{} 52–89 Vessiot E 1904 Sur l’integration des sistem differentiels qui admittent des groupes continus de transformations [*Acta Math.*]{} [**28**]{} 307–349 Ovsiannikov L V 1982 [*Group Analysis of Differential Equations*]{} (New York: Academic) Martina L 2000 Lie point symmetries of discrete and $SU(\infty)$ Toda theories, International Conference SIDE III (Symmetries and Integrability of Discrete Equations), Eds. D. Levi and O. Ragnisco [*CRM Proceedings and Lecture Notes*]{} [**25**]{} 295. Bakas I 1990 Area-preserving diffeomorphisms and higher spin fields in $2+1$ dimensions [*Supermembranes and Physics in $2+1$ dimensions*]{} M. Duff, C. Pope and E. Sezgin eds. (World Scientific: Singapore) pp 352–362 Eguchi T, Gilkey P B and Hanson A J 1980 Gravitation, gauge theory and differential geometry [*Phys. Rep.*]{} [**66**]{} 213–393 Ward R S 1990 Einstein-Weil spaces and $SU(\infty)$ Toda fields [*Class. Quantum Grav.*]{} [**7**]{} L95–L98 Saveliev M V 1989 Integro-differential non-linear equations and continual Lie algebras [*Comm. Math. Phys.*]{} [**121**]{} 283–290. Saveliev M V 1992 On the integrability problem of a continuous Toda system [*Theor. Math. Phys.*]{} [**92**]{} 1024–1031 Nutku Y and Sheftel M B 2001 Differential invariants and group foliation for the complex Monge-Ampère equation [*J. Phys. A: Math. Gen.*]{} [**34**]{} 137–156 Alfinito E, Soliani G and Solombrino L 1997 The symmetry structure of the heavenly equation [*Lett. Math. Phys.*]{} [**41**]{} 379–389 Boyer C P and Winternitz P 1989 Symmetries of the self-dual Einstein equations I. The infinite dimensional symmetry group and its low-dimensional subgroups [*J. Math. Phys.*]{} [**30**]{} 1081–1094 Olver P 1986 [*Applications of Lie Groups to Differential Equations*]{} (New York: Springer-Verlag) Tresse A 1894 Sur les invariants differentiels des groupes continus de transformations [*Acta Math.*]{} [**18**]{} 1–88 Calderbank D M J and Tod P 2001 Einstein metrics, hypercomplex structures and the Toda field equation [*Differ. Geom. Appl.*]{} [**14**]{} 199–208 Winternitz P 1993 Group theory and exact solutions of nonlinear partial differential equations [*Integrable Systems, Quantum Groups and Quantum Field Theories*]{} (Dordrecht: Kluwer) pp 429–95 Bluman G W and Cole J D 1969 The general similarity solution of the heat equation [*J. Math. Mech.*]{} [**18**]{} 1025–1042 Clarkson P A and Kruskal M D 1989 New similarity solutions of the Boussinesq equation [*J. Math. Phys.*]{} [**30**]{} 2201–2213 Levi D and Winternitz P 1989 Non-classical symmetry reduction: example of the Boussinesq equation [*J. Phys. A: Math. Gen.*]{} [**22**]{} 2915–2924 Clarkson P A and Winternitz P 1999 Symmetry reduction and exact solutions of nonlinear partial differential equations [*The Painleve Property, One Century Later*]{} (Springer: New York) pp 597–668 Lax P D 1968 Integrals of nonlinear equations of evolution and solitary waves [*Comm. Pure Appl. Math.*]{} [**21**]{} 467–490 Ablowitz M J and Clarkson P A 1991 [*Solitons, Nonlinear Evolution Equations and Inverse Scattering*]{} (Cambridge University Press: Cambridge)
|
---
abstract: 'We define a notion of renormalized volume of an asymptotically hyperbolic manifold. Moreover, we prove a sharp volume comparison theorem for metrics with scalar curvature at least $-6$. Finally, we show that the inequality is strict unless the metric is isometric to one of the Anti-deSitter-Schwarzschild metrics.'
address: |
Department of Mathematics\
Stanford University\
450 Serra Mall, Bldg 380\
Stanford, CA 94305
author:
- Simon Brendle and Otis Chodosh
title: A volume comparison theorem for asymptotically hyperbolic manifolds
---
[^1]
Introduction
============
Let $(\bar{M},\bar{g})$ denote the standard three-dimensional hyperbolic space, so that $$\bar{g} = \frac{1}{1 + s^2} \, ds \otimes ds + s^2 \, g_{S^2}.$$ Let us consider a Riemannian metric $g$ on $M = \bar{M} \setminus K$, where $K$ is a bounded domain with smooth connected boundary. We assume that $g$ is asymptotically hyperbolic in the sense that $|g-\overline{g}|_{\bar{g}} = O(s^{-2-4\delta})$ for some $\delta \in (0,\frac{1}{4})$ and $|\bar{D}(g-\bar{g})|_{\bar{g}} = o(1)$. We define the renormalized volume of $(M,g)$ by $$V(M,g) := \lim_{i \to \infty} (\text{\rm vol}(\Omega_i \cap M,g) - \text{\rm vol}(\Omega_i,\bar{g})),$$ where $\Omega_i$ is an arbitrary exhaustion of $\bar{M}$ by compact sets. The condition $|g-\overline{g}|_{\bar{g}} = O(s^{-2-4\delta})$ guarantees that the quantity $V(M,g)$ does not depend on the choice of the exhaustion $\Omega_i$. Clearly, $V(\bar{M},\bar{g}) = 0$.
As an example, let us consider the Anti-deSitter-Schwarzschild manifold with mass $m > 0$. To that end, let $s_0 = s_0(m)$ denote the unique positive solution of the equation $1 + s_0^2 - m \, s_0^{-1} = 0$. We then consider the manifold $\bar{M}_m = \bar{M} \setminus \{s \leq s_0(m)\}$ equipped with the Riemannian metric $$\bar{g}_m = \frac{1}{1 + s^2 - m \, s^{-1}} \, ds \otimes ds + s^2 \, g_{S^2}.$$ The boundary $S^2 \times \{s_0(m)\}$ is an outermost minimal surface, which is referred to as the horizon. Moreover, it is easy to see that $|\bar{g}_m-\bar{g}|_{\bar{g}} = O(s^{-3})$, so $\bar{g}$ satisfies the asymptotic assumptions above. The renormalized volume of $(\bar{M}_m,\bar{g}_m)$ is given by $$V(\bar{M}_m,\bar{g}_m) = \lim_{r \to \infty} \bigg ( \int_{s_0(m)}^r \frac{4\pi s^2}{\sqrt{1+s^2-m \, s^{-1}}} \, ds - \int_0^r \frac{4\pi s^2}{\sqrt{1+s^2}} \, ds \bigg ).$$ We now state the main result of this paper.
\[main.theorem\] Let us consider a Riemannian metric $g$ on $M = \bar{M} \setminus K$, where $K$ is a compact set with smooth connected boundary. We assume that $g$ has the following properties:
- The manifold $(M,g)$ is asymptotically hyperbolic in the sense that $|g-\overline{g}|_{\bar{g}} = O(s^{-2-4\delta})$ and $|\bar{D}(g-\bar{g})|_{\bar{g}} = o(1)$.
- The scalar curvature of $g$ is at least $-6$.
- The boundary $\partial M$ is an outermost minimal surface with respect to $g$, and we have $\text{\rm area}(\partial M,g) \geq \text{\rm area}(\partial \bar{M}_m,\bar{g}_m)$ for some $m>0$.
Then $V(M,g) \geq V(\bar{M},\bar{g}_m)$. Moreover, if equality holds, then $g$ is isometric to $\bar{g}_m$.
We note that our asymptotic assumptions are quite weak: in particular, $g$ and $\bar{g}_m$ may have different mass at infinity. An immediate consequence of Theorem \[main.theorem\] is that the function $m \mapsto V(\bar{M}_m,\bar{g}_m)$ is strictly monotone increasing. This fact is not obvious, as $s_0(m)$ is an increasing function of $m$.
Theorem \[main.theorem\] is motivated in part by Bray’s volume comparison theorem [@Bray1] for three-manifolds with scalar curvature at least $6$, as well as by a rigidity result due to Llarull [@Llarull]. A survey of this and other rigidity results involving scalar curvature can be found in [@Brendle1].
The proof of Theorem \[main.theorem\] uses two main ingredients. The first is the weak inverse mean curvature flow, which was introduced in the ground-breaking work of Huisken and Ilmanen [@Huisken-Ilmanen] on the Riemannian Penrose inequality (see also [@Bray2], where an alternative proof is given). The inverse mean curvature flow has also been considered as a possible tool for proving a version of the Penrose inequality for asymptotically hyperbolic manifolds; see [@Neves], [@Wang]. More recently, the inverse mean curvature flow was used in [@Brendle-Hung-Wang] to prove a sharp Minkowski-type inequality for surfaces in the Anti-deSitter-Schwarzschild manifold.
The second ingredient in our argument is an isoperimetric principle which asserts that a coordinate sphere in the standard Anti-deSitter-Schwarzschild manifold has smallest area among all surfaces that are homologous to the horizon and enclose the same amount of volume. This inequality was established in [@Corvino-Gerek-Greenberg-Krummel] following earlier work by Bray [@Bray1]. In fact, it is known that the coordinate spheres are the only embedded hypersurfaces with constant mean curvature in the Anti-deSitter-Schwarzschild manifold (see [@Brendle2]).
Our approach also shares common features with a result of Bray and Miao [@Bray-Miao], which gives a sharp bound for the capacity of a surface in a three-manifold of nonnegative scalar curvature.
Proof of Theorem \[main.theorem\]
=================================
Let $(M,g)$ be a Riemannian manifold which satisfies the assumptions of Theorem \[main.theorem\], and let $(\bar{M}_m,\bar{g}_m)$ be an Anti-deSitter-Schwarzschild manifold satisfying $\text{\rm area}(\partial M,g) \geq \text{\rm area}(\partial \bar{M}_m,\bar{g}_m)$. For abbreviation, let $A = \text{\rm area}(\partial M,g)$ and $\bar{A} = \text{\rm area}(\partial \bar{M}_m,\bar{g}_m)$.
Let $\Sigma_t$ denote the weak solution of the inverse mean curvature flow in $(M,g)$ with the initial surface $\Sigma_0 = \partial M$. For each $t$, we denote by $\Omega_t \subset \bar{M}$ the region bounded by $\Sigma_t$.
\[barrier\] Let $\delta \in (0,\frac{1}{4})$ be as above. Then we have $\{s \leq e^{\frac{(1-\delta)t}{2}}\} \subset \Omega_t$ for $t$ sufficiently large.
**Proof.** The coordinate sphere $S^2 \times \{s\}$ has mean curvature $2 + o(1)$ for $s$ large. Hence, we can find a real number $t_0$ such that the surfaces $S_t = \{s=e^{\frac{(1-2\delta)t}{2}}\}$ move with a speed less than $\frac{1}{H}$ for $t \geq t_0$. By the Weak Existence Theorem 3.1 in [@Huisken-Ilmanen], the regions $\Omega_t$ will eventually contain every given compact set. Hence, we can find a real number $\tau$ such that $\{s \leq e^{\frac{(1-2\delta)t_0}{2}}\} \subset \Omega_\tau$. By the maximum principle (cf. Theorem 2.2 in [@Huisken-Ilmanen]), we have $\{s \leq e^{\frac{(1-2\delta)(t-\tau+t_0)}{2}}\} \subset \Omega_t$ for $t \geq \tau$. From this, the assertion follows.\
Since the boundary $\partial M$ is an outermost minimal surface, we have $\text{\rm area}(\Sigma_t,g) = e^t \, A$. Moreover, it is well known that the quantity $$m_H(\Sigma_t) = \text{\rm area}(\Sigma_t,g)^{\frac{1}{2}} \, \bigg ( 16\pi - \int_{\Sigma_t} (H_g^2-4) \, d\mu_g \bigg )$$ is monotone increasing in $t$.
\[volume.estimate\] For each $\tau \geq 0$, we have $$\text{\rm vol}(\Omega_\tau \cap M,g) \geq \int_0^\tau e^{\frac{3t}{2}} \, A^{\frac{3}{2}} \, (4 \, e^t \, A + 16\pi - e^{-\frac{t}{2}} \, A^{-\frac{1}{2}} \, m_H(\Sigma_t))^{-\frac{1}{2}} \, dt.$$
**Proof.** By the co-area formula, we have $$\int_{\Omega_\tau \cap M} \psi \, H_g \, d\text{\rm vol}_g = \int_0^\tau \bigg ( \int_{\Sigma_t} \psi \, d\mu_g \bigg ) \, dt$$ for every nonnegative measurable function $\psi$. Hence, if we put $$\psi = \begin{cases} \frac{1}{H_g} & \text{\rm if $H_g > 0$} \\ \infty & \text{\rm if $H_g=0$,} \end{cases}$$ then we obtain $$\text{\rm vol}(\Omega_\tau \cap M,g) \geq \int_0^\tau \bigg ( \int_{\Sigma_t} \psi \, d\mu_g \bigg ) \, dt.$$ Moreover, it follows from Hölder’s inequality that $$\begin{aligned}
\int_{\Sigma_t} \psi \, d\mu_g
&\geq \text{\rm area}(\Sigma_t,g)^{\frac{3}{2}} \, \bigg ( \int_{\Sigma_t} H_g^2 \, d\mu_g \bigg )^{-\frac{1}{2}} \\
&= \text{\rm area}(\Sigma_t,g)^{\frac{3}{2}} \, (4 \, \text{\rm area}(\Sigma_t,g) + 16\pi - \text{\rm area}(\Sigma_t,g)^{-\frac{1}{2}} \, m_H(\Sigma_t))^{-\frac{1}{2}} \\
&= e^{\frac{3t}{2}} \, A^{\frac{3}{2}} \, (4 \, e^t \, A + 16\pi - e^{-\frac{t}{2}} \, A^{-\frac{1}{2}} \, m_H(\Sigma_t))^{-\frac{1}{2}}. \end{aligned}$$ Putting these facts together, the assertion follows.\
\[volume.estimate.2\] We have $$2 \, \text{\rm vol}(\Omega_\tau \cap M,g) \geq \int_0^\tau e^t \, A^{\frac{3}{2}} \, ((1-e^{-\frac{3t}{2}}) \, A + 4\pi \, (e^{-t} - e^{-\frac{3t}{2}}))^{-\frac{1}{2}} \, dt.$$
**Proof.** Using the monotonicity of $m_H(\Sigma_t)$, we obtain $$m_H(\Sigma_t) \geq m_H(\Sigma_0) = 4 \, A^{\frac{1}{2}} \, (A + 4\pi).$$ This implies $$2 \, \text{\rm vol}(\Omega_\tau \cap M,g) \geq \int_0^\tau e^{\frac{3t}{2}} \, A^{\frac{3}{2}} \, ((e^t-e^{-\frac{t}{2}}) \, A + 4\pi \, (1-e^{-\frac{t}{2}}))^{-\frac{1}{2}} \, dt.$$ From this, the assertion follows.\
\[iso\] Let $\Omega$ be a domain in $\bar{M}$ such that $\{s \leq s_0(m)\} \subset \Omega$, and let $\Sigma$ denote the boundary of $\Omega$. Then $$2 \, \text{\rm vol}(\Omega \cap \bar{M}_m,\bar{g}_m) \leq \int_0^{\bar{\tau}} e^t \, \bar{A}^{\frac{3}{2}} \, ((1-e^{-\frac{3t}{2}}) \, \bar{A} + 4\pi \, (e^{-t} - e^{-\frac{3t}{2}}))^{-\frac{1}{2}} \, dt,$$ where $\bar{\tau}$ is defined by $\text{\rm area}(\Sigma,\bar{g}_m) = e^{\bar{\tau}} \, \bar{A}$.
**Proof.** If $\Sigma$ is a coordinate sphere in $(\bar{M}_m,\bar{g}_m)$, then we have $$2 \, \text{\rm vol}(\Omega \cap \bar{M}_m,\bar{g}_m) = \int_0^{\bar{\tau}} e^t \, \bar{A}^{\frac{3}{2}} \, ((1-e^{-\frac{3t}{2}}) \, \bar{A} + 4\pi \, (e^{-t} - e^{-\frac{3t}{2}}))^{-\frac{1}{2}} \, dt,$$ where $\bar{\tau}$ is defined by $\text{\rm area}(\Sigma,\bar{g}_m) = e^{\bar{\tau}} \, \bar{A}$. On the other hand, it is known (cf. [@Corvino-Gerek-Greenberg-Krummel], Theorem 4.2) that the coordinate spheres in $(\bar{M}_m,\bar{g}_m)$ enclose the largest volume for any given surface area. Putting these facts together, the assertion follows.\
Let us consider a sequence of times $\tau_i \to \infty$. Moreover, we define a sequence of times $\bar{\tau}_i \to \infty$ by $\text{\rm area}(\Sigma_{\tau_i},\bar{g}_m) = e^{\bar{\tau}_i} \, \bar{A}$. By Proposition \[barrier\], we have $s \geq e^{\frac{(1-\delta)t}{2}}$ on $\Sigma_t$ if $t$ is large enough. This implies $$|g-\bar{g}_m|_{\bar{g}_m} \leq O(s^{-2-4\delta}) \leq O(e^{-(1-\delta)(1+2\delta)t})$$ at each point on $\Sigma_t$. From this, we deduce that $$\begin{aligned}
e^{\tau_i} \, A
&= \text{\rm area}(\Sigma_{\tau_i},g) \\
&= \text{\rm area}(\Sigma_{\tau_i},\bar{g}_m) \, (1 + O(e^{-(1-\delta)(1+2\delta)\tau_i})) \\
&= e^{\bar{\tau}_i} \, \bar{A} \, (1 + O(e^{-(1-\delta)(1+2\delta)\tau_i})). \end{aligned}$$ Thus, we conclude that $$\tau_i = \bar{\tau}_i - \alpha + O(e^{-(1-\delta)(1+2\delta)\tau_i}),$$ where $\alpha = \log (A/\bar{A}) \geq 0$. Note that $(1-\delta)(1+2\delta) > 1$ since $\delta \in (0,\frac{1}{4})$.
By Corollary \[volume.estimate.2\], we have $$\begin{aligned}
2 \, \text{\rm vol}(\Omega_{\tau_i} \cap M,g)
&\geq \int_0^{\tau_i} e^t \, A^{\frac{3}{2}} \, ((1-e^{-\frac{3t}{2}}) \, A + 4\pi \, (e^{-t} - e^{-\frac{3t}{2}}))^{-\frac{1}{2}} \, dt \\
&= \int_\alpha^{\tau_i+\alpha} e^{t-\alpha} \, A^{\frac{3}{2}} \, ((1-e^{-\frac{3t-3\alpha}{2}}) \, A + 4\pi \, (e^{-t+\alpha} - e^{-\frac{3t-3\alpha}{2}}))^{-\frac{1}{2}} \, dt \\
&= \int_\alpha^{\tau_i+\alpha} e^t \, \bar{A}^{\frac{3}{2}} \, ((1-e^{-\frac{3t-3\alpha}{2}}) \, \bar{A} + 4\pi \, (e^{-t} - e^{-\frac{3t-\alpha}{2}}))^{-\frac{1}{2}} \, dt. \end{aligned}$$ On the other hand, we have $$2 \, \text{\rm vol}(\Omega_{\tau_i} \cap \bar{M}_m,\bar{g}_m) \leq \int_0^{\bar{\tau}_i} e^t \, \bar{A}^{\frac{3}{2}} \, ((1-e^{-\frac{3t}{2}}) \, \bar{A} + 4\pi \, (e^{-t} - e^{-\frac{3t}{2}}))^{-\frac{1}{2}} \, dt$$ by Proposition \[iso\]. Putting these facts together, we obtain $$\begin{aligned}
&2 \, (V(M,g) - V(\bar{M}_m,\bar{g}_m)) \\
&= \limsup_{i \to \infty} 2 \, (\text{\rm vol}(\Omega_{\tau_i} \cap M,g) - \text{\rm vol}(\Omega_{\tau_i} \cap \bar{M}_m,\bar{g}_m)) \\
&\geq \limsup_{i \to \infty} \bigg ( \int_\alpha^{\tau_i+\alpha} e^t \, \bar{A}^{\frac{3}{2}} \, ((1-e^{-\frac{3t-3\alpha}{2}}) \, \bar{A} + 4\pi \, (e^{-t} - e^{-\frac{3t-\alpha}{2}}))^{-\frac{1}{2}} \, dt \\
&\hspace{20mm} - \int_0^{\bar{\tau}_i} e^t \, \bar{A}^{\frac{3}{2}} \, ((1-e^{-\frac{3t}{2}}) \, \bar{A} + 4\pi \, (e^{-t} - e^{-\frac{3t}{2}}))^{-\frac{1}{2}} \, dt \bigg ) \\
&= \limsup_{i \to \infty} \bigg ( \int_\alpha^{\bar{\tau}_i} e^t \, \bar{A}^{\frac{3}{2}} \, ((1-e^{-\frac{3t-3\alpha}{2}}) \, \bar{A} + 4\pi \, (e^{-t} - e^{-\frac{3t-\alpha}{2}}))^{-\frac{1}{2}} \, dt \\
&\hspace{20mm} - \int_0^{\bar{\tau}_i} e^t \, \bar{A}^{\frac{3}{2}} \, ((1-e^{-\frac{3t}{2}}) \, \bar{A} + 4\pi \, (e^{-t} - e^{-\frac{3t}{2}}))^{-\frac{1}{2}} \, dt \bigg ) \\
&= \bar{A}^{\frac{3}{2}} \, I(\alpha), \end{aligned}$$ where $$\begin{aligned}
I(\alpha) &= \int_\alpha^\infty e^t \, \Big [ ((1-e^{-\frac{3t-3\alpha}{2}}) \, \bar{A} + 4\pi \, (e^{-t} - e^{-\frac{3t-\alpha}{2}}))^{-\frac{1}{2}} \\
&\hspace{20mm} - ((1-e^{-\frac{3t}{2}}) \, \bar{A} + 4\pi \, (e^{-t} - e^{-\frac{3t}{2}}))^{-\frac{1}{2}} \Big ] \, dt \\
&- \int_0^\alpha e^t \, ((1-e^{-\frac{3t}{2}}) \, \bar{A} + 4\pi \, (e^{-t} - e^{-\frac{3t}{2}}))^{-\frac{1}{2}} \, dt. \end{aligned}$$ It is shown in the appendix that the function $I(\alpha)$ is positive for all $\alpha>0$. Thus, we conclude that $V(M,g) \geq V(\bar{M}_m,\bar{g}_m)$.
Finally, we analyze the case of equality. Suppose that $V(M,g) = V(\bar{M}_m,\bar{g}_m)$. Then $I(\alpha) \leq 0$, which implies that $\alpha=0$. Moreover, the difference $$2 \, \text{\rm vol}(\Omega_{\tau_i} \cap M,g) - \int_0^{\tau_i} e^t \, A^{\frac{3}{2}} \, ((1-e^{-\frac{3t}{2}}) \, A + 4\pi \, (e^{-t} - e^{-\frac{3t}{2}}))^{-\frac{1}{2}} \, dt$$ must converge to $0$ as $i \to \infty$. Using Proposition \[volume.estimate\], we conclude that $m_H(\Sigma_t) = m_H(\Sigma_0)$ for all $t$. This implies that $g$ is the isometric to one of the standard Anti-deSitter-Schwarzschild metrics. Since $\alpha=0$, the manifolds $(M,g)$ and $(\bar{M}_m,\bar{g}_m)$ have the same boundary area. Therefore, they are isometric.
Positivity of the function $I(\alpha)$
======================================
In this section, we show that $I(\alpha)>0$ for all $\alpha>0$. We begin with a lemma:
\[aux\] Let $\varepsilon$ and $\mu$ be positive real numbers. If the ratio $\frac{\varepsilon}{\mu}$ is sufficiently small, then we have $$3\mu \int_0^\infty e^{-\frac{t}{2}} \, (\varepsilon+(1-e^{-\frac{3t}{2}}) \mu)^{-\frac{3}{2}} \, dt \geq 4 \, \varepsilon^{-\frac{1}{2}} + \mu^{-\frac{1}{2}}.$$
**Proof.** It is elementary to check that $$e^t \geq 1 + \frac{2}{3} \, (1-e^{-\frac{3t}{2}}),$$ hence $$e^{-\frac{t}{2}} \geq e^{-\frac{3t}{2}} + \frac{2}{3} \, e^{-\frac{3t}{2}} \, (1-e^{-\frac{3t}{2}})$$ for all $t \geq 0$. This implies $$\begin{aligned}
&\int_0^1 e^{-\frac{t}{2}} \, (\varepsilon+1-e^{-\frac{3t}{2}})^{-\frac{3}{2}} \, dt \\
&\geq \int_0^1 e^{-\frac{3t}{2}} \, (\varepsilon+1-e^{-\frac{3t}{2}})^{-\frac{3}{2}} \, dt + \frac{2}{3} \int_0^1 e^{-\frac{3t}{2}} \, (1-e^{-\frac{3t}{2}}) \, (\varepsilon+1-e^{-\frac{3t}{2}})^{-\frac{3}{2}} \, dt \\
&= \int_0^1 e^{-\frac{3t}{2}} \, (\varepsilon+1-e^{-\frac{3t}{2}})^{-\frac{3}{2}} \, dt +
\frac{2}{3} \int_0^1 e^{-\frac{3t}{2}} \, (1-e^{-\frac{3t}{2}})^{-\frac{1}{2}} \, dt - o(1) \\
&= \frac{4}{3} \, \varepsilon^{-\frac{1}{2}} - \frac{4}{3} \, (\varepsilon+1-e^{-\frac{3}{2}})^{-\frac{1}{2}} + \frac{8}{9} \, (1-e^{-\frac{3}{2}})^{\frac{1}{2}} - o(1) \end{aligned}$$ for $\varepsilon > 0$ sufficiently small. Hence, we obtain $$\begin{aligned}
&\int_0^\infty e^{-\frac{t}{2}} \, (\varepsilon+1-e^{-\frac{3t}{2}})^{-\frac{3}{2}} \, dt \\
&\geq \frac{4}{3} \, \varepsilon^{-\frac{1}{2}} - \frac{4}{3} \, (\varepsilon+1-e^{-\frac{3}{2}})^{-\frac{1}{2}} + \frac{8}{9} \, (1-e^{-\frac{3}{2}})^{\frac{1}{2}} + (\varepsilon+1)^{-\frac{3}{2}} \int_1^\infty e^{-\frac{t}{2}} \, dt - o(1) \\
&\geq \frac{4}{3} \, \varepsilon^{-\frac{1}{2}} - \frac{4}{3} \, (1-e^{-\frac{3}{2}})^{-\frac{1}{2}} + \frac{8}{9} \, (1-e^{-\frac{3}{2}})^{\frac{1}{2}} + 2 \, e^{-\frac{1}{2}} - o(1) \\
&\geq \frac{4}{3} \, \varepsilon^{-\frac{1}{2}} + \frac{1}{3} \end{aligned}$$ if $\varepsilon>0$ is small enough. This proves the assertion for $\mu=1$. The general case follows by scaling.\
We now consider the function $$\begin{aligned}
I_\varepsilon(\alpha) &= \int_\alpha^\infty e^t \, \Big [ (\varepsilon+(1-e^{-\frac{3t-3\alpha}{2}}) \, \bar{A} + 4\pi \, (e^{-t} - e^{-\frac{3t-\alpha}{2}}))^{-\frac{1}{2}} \\
&\hspace{20mm} - (\varepsilon+(1-e^{-\frac{3t}{2}}) \, \bar{A} + 4\pi \, (e^{-t} - e^{-\frac{3t}{2}}))^{-\frac{1}{2}} \Big ] \, dt \\
&- \int_0^\alpha e^t \, (\varepsilon+(1-e^{-\frac{3t}{2}}) \, \bar{A} + 4\pi \, (e^{-t} - e^{-\frac{3t}{2}}))^{-\frac{1}{2}} \, dt. \end{aligned}$$ Then $$\begin{aligned}
\frac{d}{d\alpha} I_\varepsilon(\alpha) &= \int_\alpha^\infty e^t \, \frac{d}{d\alpha} \Big [ (\varepsilon+(1-e^{-\frac{3t-3\alpha}{2}}) \, \bar{A} + 4\pi \, (e^{-t} - e^{-\frac{3t-\alpha}{2}}))^{-\frac{1}{2}} \Big ] \, dt - e^\alpha \, \varepsilon^{-\frac{1}{2}} \\
&= \frac{1}{4} \, (3 \, e^{\frac{3\alpha}{2}} \, \bar{A} + 4\pi \, e^{\frac{\alpha}{2}}) \\
&\hspace{5mm} \cdot \int_\alpha^\infty e^{-\frac{t}{2}} \, (\varepsilon+(1-e^{-\frac{3t-3\alpha}{2}}) \, \bar{A} + 4\pi \, (e^{-t} - e^{-\frac{3t-\alpha}{2}}))^{-\frac{3}{2}} \, dt - e^\alpha \, \varepsilon^{-\frac{1}{2}} \\
&= \frac{e^\alpha}{4} \, (3 \, \bar{A} + 4\pi \, e^{-\alpha}) \\
&\hspace{5mm} \cdot \int_0^\infty e^{-\frac{t}{2}} \, (\varepsilon+(1-e^{-\frac{3t}{2}}) \, \bar{A} + 4\pi \, e^{-\alpha} \, (e^{-t} - e^{-\frac{3t}{2}}))^{-\frac{3}{2}} \, dt - e^\alpha \, \varepsilon^{-\frac{1}{2}} \\
&\geq \frac{e^\alpha}{4} \, (3 \, \bar{A} + 4\pi \, e^{-\alpha}) \\
&\hspace{5mm} \cdot \int_0^\infty e^{-\frac{t}{2}} \, \Big ( \varepsilon+(1-e^{-\frac{3t}{2}}) \, (\bar{A} + \frac{4\pi}{3} \, e^{-\alpha}) \Big )^{-\frac{3}{2}} \, dt - e^\alpha \, \varepsilon^{-\frac{1}{2}}, \end{aligned}$$ where in the last step we have used the inequality $e^{-t} - e^{-\frac{3t}{2}} \leq \frac{1}{3} \, (1-e^{-\frac{3t}{2}})$. Hence, if the ratio $\frac{\varepsilon}{\bar{A} + \frac{4\pi}{3} \, e^{-\alpha}}$ is sufficiently small, then $$\frac{d}{d\alpha} I_\varepsilon(\alpha) \geq \frac{e^\alpha}{4} \, (\bar{A} + \frac{4\pi}{3} \, e^{-\alpha})^{-\frac{1}{2}}$$ by Lemma \[aux\]. Since $I(\alpha) = \lim_{\varepsilon \to 0} I_\varepsilon(\alpha)$ for each $\alpha \geq 0$, we conclude that the function $I(\alpha)$ is strictly monotone increasing. In particular, $I(\alpha) > 0$ for all $\alpha>0$.
[99]{} H. Bray, *The Penrose inequality in general relativity and volume comparison theorems involving scalar curvature,* PhD thesis, Stanford University (1997)
H. Bray, *Proof of the Riemannian Penrose inequality using the positive mass theorem,* J. Diff. Geom. 59, 177–267 (2001)
H. Bray and P. Miao, *On the capacity of surfaces in manifolds with nonnegative scalar curvature,* Invent. Math. 172, 459–475 (2008)
S. Brendle, *Rigidity phenomena involving scalar curvature,* Surveys in Differential Geometry, volume XVII, 179–202 (2012)
S. Brendle, *Constant mean curvature surfaces in warped product manifolds,* Publ. Math. IHÉS 117, 247–269 (2013)
S. Brendle, P.-K. Hung, and M.-T. Wang, *A Minkowski-type inequality for hypersurfaces in the Anti-deSitter-Schwarzschild manifold,* arxiv:1209.0669
J. Corvino, A. Gerek, M. Greenberg, and B. Krummel, *On isoperimetric surfaces in general relativity,* Pacific J. Math. 231, 63–84 (2007)
G. Huisken and T. Ilmanen, *The inverse mean curvature flow and the Riemannian Penrose inequality,* J. Diff. Geom. 59, 353–437 (2001)
M. Llarull, *Sharp estimates and the Dirac operator,* Math. Ann. 310, 55–71 (1998)
A. Neves, *Insufficient convergence of inverse mean curvature flow on asymptotically hyperbolic manifolds,* J. Diff. Geom. 84, 191–229 (2010)
X. Wang, *The mass of asymptotically hyperbolic manifolds,* J. Diff. Geom. 57, 273–299 (2001)
[^1]: The first author was supported in part by the National Science Foundation under grant DMS-1201924. The second author was supported in part by a National Science Foundation Graduate Research Fellowship DGE-1147470.
|
---
abstract: 'Hot subdwarf stars (sdO/Bs) are evolved core helium-burning stars with very thin hydrogen envelopes, which can be formed by common envelope ejection. Close sdB binaries with massive white dwarf (WD) companions are potential progenitors of thermonuclear supernovae type Ia (SN Ia). We discovered such a progenitor candidate as well as a candidate for a surviving companion star, which escapes from the Galaxy. More candidates for both types of objects have been found by crossmatching known sdB stars with proper motion and light curve catalogues. The Gaia mission will provide accurate astrometry and light curves of all the stars in our hot subdwarf sample and will allow us to compile a much larger all-sky catalogue of those stars. In this way we expect to find hundreds of progenitor binaries and ejected companions.'
address: |
$^1$ Department of Physics, University of Warwick, Coventry CV4 7AL, UK\
$^2$ Dr. Karl Remeis-Observatory & ECAP, Astronomical Institute, Friedrich-Alexander University Erlangen-Nuremberg, Sternwartstr. 7, D 96049 Bamberg, Germany\
$^3$Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena, CA 91125, USA
author:
- 'S. Geier$^{1,2}$, T. Kupfer$^3$, E. Ziegerer$^2$, U. Heber$^2$, P. Németh$^{2}$, A. Irrgang$^{2}$ and MUCHFUSS team'
bibliography:
- 'geier\_proc.bib'
title: Hot subdwarf stars and their connection to thermonuclear supernovae
---
Introduction
============
Hot subdwarf stars (sdO/Bs) are evolved core helium-burning stars with very thin hydrogen envelopes, which can be formed by common envelope ejection. Close sdB binaries with massive C/O-WD companions are candidates for supernova type Ia (SN Ia) progenitors, because mass-transfer can lead to the thermonuclear explosion of the WD. The project Massive Unseen Companions to Hot Faint Underluminous Stars from SDSS (MUCHFUSS) aims at finding the sdB binaries with the most massive compact companions like massive white dwarfs, neutron stars or black holes.
We selected and classified about $\sim1400$ hot subdwarf stars from the Sloan Digital Sky Survey (SDSS DR7). Stars with high velocity variations have been reobserved and analysed. In total $177$ radial velocity variable subdwarfs have been discovered and $1914$ individual radial velocities measured. We constrain the fraction of close massive companions of H-rich hot subdwarfs to be smaller than $\sim1.3\%$ [@geier15b]. Orbital parameters as well as minimum companion masses have been derived from the radial velocity curves of 30 sdB binaries [@kupfer15].
Discovery of an SNIa progenitor and an ejected companion
========================================================
We detected high RV-variability of the bright sdB CD$-$30$^\circ$11223. Photometric follow-up revealed both shallow transits and eclipses, allowing us to determine its component masses and fundamental parameters. The binary system, which is composed of a C/O-WD ($\sim0.76\,M_{\rm \odot}$) and an sdB ($\sim0.51\,M_{\rm \odot}$) has a very short orbital period of $\sim0.049\,{\rm d}$. In the future mass will be transfered from the helium star to the white dwarf. After a critical amount of helium is deposited on the surface of the white dwarf, the helium is ignited. Modelling this process shows that the detonation in the accreted helium layer should be sufficiently strong to trigger the explosion of the core. Thermonuclear supernovae have been proposed to originate from this so-called double-detonation of a WD [@fink10; @geier13]. The surviving companion star will then be ejected with its orbital velocity. The properties of such a remnant match the hypervelocity star US708, a helium-rich sdO star moving with $\sim1200\,{\rm km\,s^{-1}}$, exceeding the escape velocity of our Galaxy by far and making it the fastest unbound star known in our Galaxy [@geier15a].
Finding more progenitor and ejected companion candidates
========================================================
Since the properties of the ejected companions, especially the ejection velocity, allow us to constrain the properties of the binary progenitors such as the orbital period and the companion mass right at the moment of the explosion, we will gain an unprecedented insight into the formation of SNIa and learn about other acceleration mechanisms for hypervelocity stars, if more such objects can be found and studied. The distribution of orbital periods and WD companion masses of progenitor binaries will help us to constrain SNIa progenitor models. While binaries with periods longer than about 2hr will merge as double degenerates, closer binaries might be progenitors for the helium double-detonation channel.
To search for ejected companions we compiled a catalogue of all known sdO/B stars from the literature and our own database ($\sim4500$ stars) and crossmatch it with proper motion catalogues. Candidates with high velocities are followed-up with spectroscopy (Keck/ESI, VLT/XSHOOTER, SOAR/Goodman, CAHA/TWIN, WHT/ISIS) to measure spectroscopic distances and derive kinematics. Several good candidates for unbound hypervelocity sdO/Bs have already been found.
We found that hot subdwarf binaries with massive WDs in close orbits are quite rare. To find more of those objects, we crossmatch the hot subdwarf catalogue with light curve catalogues (e.g. CRTS, PTF, SWASP, GALEX gPhoton, Kepler K2) and search for the characteristic sinusoidal variations caused by the ellipsoidal deformation of the sdB. Several candidates have been found and will be followed-up spectroscopically and photometrically.
Eventually, the Gaia mission will provide accurate astrometry and light curves of all the stars in our hot subdwarf sample and will allow us to compile a much larger all-sky catalogue of those stars. In this way we expect to find hundreds of progenitor binaries and ejected companions.\
\
|
---
abstract: 'This paper presents a study of immiscible incompressible two-phase flow through fractured porous media. The results obtained earlier in the pioneer work by A. Bourgeat, S. Luckhaus, A. Mikelić (1996) and L. M. Yeh (2006) are revisited. The main goal is to incorporate some of the most recent improvements in the convergence of the solutions in the homogenization of such models. The microscopic model consists of the usual equations derived from the mass conservation of both fluids along with the Darcy-Muskat law. The problem is written in terms of the phase formulation, i.e. the saturation of one phase and the pressure of the second phase are primary unknowns. We will consider a domain made up of several zones with different characteristics: porosity, absolute permeability, relative permeabilities and capillary pressure curves. The fractured medium consists of periodically repeating homogeneous blocks and fractures, the permeability being highly discontinuous. Over the matrix domain, the permeability is scaled by $\ve^\theta$, where $\ve$ is the size of a typical porous block and $\theta>0$ is a parameter. The model involves highly oscillatory characteristics and internal nonlinear interface conditions. Under some realistic assumptions on the data, the convergence of the solutions, and the macroscopic models corresponding to various range of contrast are constructed using the two-scale convergence method combined with the dilation technique. The results improve upon previously derived effective models to highly heterogeneous porous media with discontinuous capillary pressures.'
author:
- 'B. Amaziane$^{1}$, M. Jurak$^{2,}\footnote{Corresponding author.}$ , L. Pankratov$^{1,3}$, A. Vrbaški$^4$'
title: 'Some remarks on the homogenization of immiscible incompressible two-phase flow in double porosity media'
---
$^1$ Laboratoire de Mathématiques et de leurs Applications, CNRS-UMR 5142 Université de Pau, Av. de l’Université, 64000 Pau, France. E-mail: [brahim.amaziane@univ-pau.fr]{}
$^2$ Faculty of Science, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia. E-mail: [jurak@math.hr]{}
$^3$ Laboratory of Fluid Dynamics and Seismic ([*RAEP 5top100*]{}), Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region, 141700, Russian Federation. E-mail: [leonid.pankratov@univ-pau.fr]{}
$^4$ Faculty of Mining, Geology and Petroleum Engineering, University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia. E-mail: [anja.vrbaski@rgn.hr]{}
[**Keywords:**]{} Homogenization, double porosity media, two-scale convergence, dilation operator.
[**2010 Mathematics Subject Classification.**]{} Primary: 35B27, 35K55, 35K65, 74Q15; Secondary: 35Q35, 76S05.
Introduction {#intrrr}
============
The modeling of displacement process involving two immiscible fluids in fractured porous media is important to many practical problems, including those in petroleum reservoir engineering, unsaturated zone hydrology, and soil science. More recently, modeling multiphase flow has received an increasing attention in connection with the disposal of radioactive waste and sequestration of $CO_2$. Furthermore, fractured rock domains corresponding to the so-called Excavation Damaged Zone (EDZ) receives increasing attention in connection with the behaviour of geological isolation of radioactive waste after the drilling of the wells or shafts, see, e.g., [@Shaw].
In this paper we use the homogenization theory to derive a double porosity model describing the flow of incompressible fluids in fractured reservoirs. The model corresponds physically to immiscible incompressible two-phase flow through fractured porous media. Naturally fractured reservoirs can be modeled by two superimposed continua, a connected fracture system and a system of topologically disconnected matrix blocks. The fracture system has low storage capacity but high conductivity, while the matrix block system has low conductivity and large storage capacity. The majority of fluid transport will occur along flow paths through the fissure system. When the system of fissures is so well developed that the matrix is broken into individual blocks or cells that are isolated from each other, there is consequently no flow directly from cell to cell, but only an exchange of fluid between each cell and the surrounding fissure system. For more details on the physical formulation of such problems see, e.g., [@bear; @panf; @van].
This paper continues the research published in [@blm] and [@yeh2], and the goal is to reformulate in a more systematic manner and in somewhat more general context the homogenization problem for an immiscible incompressible two-phase flow in double porosity media by weakening the standing assumptions. Special attention is paid to developing a general approach to incorporating highly heterogeneous porous media with discontinuous capillary pressures.
During recent decades mathematical analysis and homogenization of multiphase flows in porous media have been the subject of investigation of many researchers owing to important applications in reservoir simulation. There is an extensive literature on this subject. We will not attempt a literature review here but will merely mention a few references. A recent review of the mathematical homogenization methods developed for incompressible immiscible two-phase flow in porous media and compressible miscible flow in porous media can be viewed in [@our-siam; @HOS-2012; @hor].
Let us now turn to a brief review of the homogenization in double porosity media. Here we restrict ourself to the mathematical homogenization method as described in [@hor] for flow and transport in porous media. The interest for double porosity systems came at first from geophysics. The notion of double porosity, or double permeability is borne from studies carried out on naturally fractured porous rocks, such as oil fields. The double porosity model was first introduced in [@bar] and it is since used in a wide range of engineering specialities. The first rigorous mathematical result on the subject was obtained in [@adh], where a linear parabolic equation with asymptotically degenerating coefficients describing a single-phase flow in fractured media was considered. This result is then generalized in [@bmp; @bgpp; @mk; @sandr] for non-periodic domains and various rates of contrast. Linear double porosity models with thin fissures were considered in [@pr-AA; @pr]. A singular double porosity model was considered in [@bcp]. Notice that the works [@pr-AA; @bgpp; @mk; @pr] are done in the framework of Khruslov’s energy characteristic method which is close to the $\Gamma$-convergence method. Let us also notice that the double porosity model was obtained in [@hor] (see Chapter 3) using the two-scale convergence method. Non-linear double porosity models, elliptic and parabolic, including the homogenization in variable Sobolev spaces, were obtained in [@edinburg-app; @na-rwa-app; @choq; @edinb-catho; @clark]. A study of discrete double-porosity models in the case of elastic energies has been recently done in [@braides-valer-piat]. Finally, in order to complete this brief review, we turn to the multiphase flow double porosity models. These models were obtained e.g., in [@blm; @choq; @yeh2] (see also [@hor] and the references therein) and recently in [@latifa; @ba-lp-doubpor] for immiscible compressible two-phase flows. A fully homogenized model for incompressible two-phase flow in double porosity media was obtained in [@Jurak].
This paper is concerned with a nonlinear degenerate system of diffusion-convection equations modeling the flow and transport of immiscible incompressible fluids through highly heterogeneous porous media, capillary and gravity effects being taken into account. We will consider a domain made up of several zones with different characteristics: porosity, absolute permeability, relative permeabilities and capillary pressure curves. The model to be presented herein is formulated in terms of the wetting phase saturation and the non-wetting phase pressure, and the feature of the global pressure as introduced in [@ant-kaz-mon1990; @GC-JJ] for incompressible immiscible flows is used to establish a priori estimates. The governing equations are derived from the mass conservation laws of both fluids, along with constitutive relations relating the velocities to the pressures gradients and gravitational effects. Traditionally, the standard Muskat-Darcy law provides this relationship. Let us mention that the main difficulties related to the mathematical analysis of such equations are the coupling and the degeneracy of the diffusion term in the saturation equation. Moreover the transmission conditions are nonlinear and the saturation is discontinuous at the interface separating the two media.
We start with a microscopic model defined on a domain with periodic microstructure. We will consider a domain made up of several zones with different characteristics: porosity, absolute permeability, relative permeabilities and capillary pressure curves. The fractured medium consists of periodically repeating homogeneous blocks and fractures, the permeability being highly discontinuous. Over the matrix domain, the permeability is scaled by $\ve^\theta$, where $\ve$ is the size of a typical porous block and $\theta>0$ is a parameter. Our aim is to study the macroscopic behavior of solutions of this system of equations as $\ve$ tends to zero and give a rigorous mathematical derivation of upscaled models by means of the two-scale convergence method combined with the dilation technique. Thus, we extend the results of [@blm; @yeh2] to the case of highly heterogeneous porous media with discontinuous capillary pressures.
The rest of the paper is organized as follows. In Section \[micromodel\], we describe the physical model and formulate the corresponding mathematical problem. We also provide the assumptions on the data and a weak formulation of the problem firstly in terms of phase pressures and secondly in terms of the global pressure and the saturation. Section \[uni-est\] is then devoted to the derivation of the basic [*a priori*]{} estimates of the problem under consideration. In Section \[ex-comp-sf\] we formulate the two-scale convergence results which will be used in the derivation of the homogenized system. The key point here is the proof of the compactness result for the restriction-extension sequence of the wetting fluid saturation defined on the fracture set. It is done by using the ideas from [@yeh2]. Section \[dil-oper\] is devoted to the definition and the properties of the dilation operator and to the formulation of the convergence results for the dilated functions defined on the matrix part. The key point of the section is the proof of the compactness result for the dilated saturations which is done by using the compactness result from [@our-siam]. The formulation of the main results of the paper is given in Section \[main-res\]. The resulting homogenized problem is a dual-porosity type model that contains a term representing memory effects which could be seen as source term or as a time delay for $\theta = 2$, and it is a single porosity model with effective coefficients for $0 < \theta < 2$ or $\theta > 2$. The proof of the convergence theorem in the critical case ($\theta =2$) is done in subsection \[hig-crtic-subsec\]. The key point here is subsection \[identif-l-theta\], where we prove the uniqueness of the solution to the local problem. The proof is done by reducing the problem in the phase formulation to a boundary value problem for an imbibition equation and by using ideas from [@vazquez]. The proofs of the convergence theorems for non-critical cases ($\theta>2$ or $0<\theta<2$) are given in subsections \[very-hig-crtic-subsec\], \[moder-case-subsec\]. The effective model obtained in the case of moderate contrast ($0<\theta<2$, subsection \[moder-case-subsec\]), up to our knowledge, is for the first time proposed and rigorously justified here.
Formulation of the problem {#micromodel}
==========================
The outline of this section is as follows. First, in subsection \[micromodel-simpl\] we give a short description of the mathematical and physical model used in this study for immiscible incompressible two-phase flow in a periodic double porosity medium. The notion of the global pressure is briefly recalled in subsection \[gp-relat\]. Finally, in subsection \[def-weak-sol\], we present the main assumptions on the data and we define the weak solution to our problem, first in terms of phase pressures and then an equivalent one in terms of the global pressure and saturation.
(-0.2,-0.2) – (5.8,-0.2) – (5.8, 5.8) – (-0.2, 5.8) – cycle;
(0, 0) ; (1.5,0) ; (3, 0) ; (4.5,0) ;
(0, 1.5) ; (1.5,1.5) ; (3, 1.5) ; (4.5,1.5) ;
(0, 3.0) ; (1.5,3.0) ; (3, 3.0) ; (4.5,3.0) ;
(0, 4.5) ; (1.5,4.5) ; (3, 4.5) ; (4.5,4.5) ;
(0.5,5.2) – (0.5,6.2) node\[anchor=south\] [$\Omega_\mx^{\varepsilon}$]{}; (0.5,4.3) – (-0.5,4.3) node\[anchor=east\] [$\Omega_\fr^{\varepsilon}$]{}; (1.5,2.1) – (-0.5,2.1) node\[anchor=east\] [$\Gamma^\ve_{\fr\mx}$]{};
(1.3,5.9)–(1.3,6.2); (2.8,5.9)–(2.8,6.2); (1.3,6.05)–(2.8,6.05); (2.1,6.2) node [$\varepsilon$]{};
(5.9,5.7)–(6.2,5.7); (5.9,4.3)–(6.2,4.3); (6.05,4.3)–(6.05,5.7); (6.3,5.0) node [$\varepsilon$]{};
at (2.8,0.2) [$(a)$]{};
(8.4,-0.2) – (11.4,-0.2) – (11.4,2.8) – (8.4,2.8) – cycle; (8.7,0.1) – (11.1,0.1) – (11.1,2.5) – (8.7,2.5) – cycle;
(11.6,-0.1) node\[anchor=north\] [$1$]{}; (8.5,3.0) node\[anchor=east\] [$1$]{};
(9.5,1.85) – (9.5,3.35) node\[anchor=south\] [$Y_\mx$]{}; (10.5,2.65) – (10.5,3.35) node\[anchor=south\] [$Y_\fr$]{}; (11.15,1.1) – (12.1,1.1) node\[anchor=west\] [$\Gamma_{\fr\mx}$]{};
at (10.0,0.2) [$(b)$]{};
Microscopic model {#micromodel-simpl}
-----------------
We consider a reservoir $\Omega \subset \mathbb{R}^d$ ($d = 2, 3$) which is assumed to be a bounded, connected Lipschitz domain with a periodic microstructure. More precisely, we will scale this structure by a parameter $\ve$ which represents the ratio of the cell size to the size of the whole region $\Omega$ and we assume that $0 < \ve \ll 1$ is a small parameter tending to zero. Let $Y = (0, 1)^d$ be a basic cell of a fractured porous medium. For the sake of simplicity and without loss of generality, we assume that $Y$ is made up of two homogeneous porous media $Y_\mx$ and $Y_\fr$ corresponding to the parts of the domain occupied by the matrix block and the fracture, respectively (see Fig.\[fig:ref\] (b)). Thus $Y = Y_\mx \cup Y_\fr \cup \Gamma_{\fr\mx}$, where $\Gamma_{\fr\mx}$ denotes the interface between the two media. Let $\Omega^\ve_\ell$ with $\ell = "\fr"$ or $"\mx"$ denotes the open set corresponding to the porous medium with index $\ell$. Then $\Omega = \Omega^\ve_\mx \cup \Gamma^\ve_{\fr\mx} \cup \Omega^\ve_\fr$, where $\Gamma^\ve_{\fr\mx} \eqdef \partial \Omega^\ve_\fr \cap \partial \Omega^\ve_\mx \cap \Omega$ and the subscripts $"\mx"$, $"\fr"$ refer to the matrix and fracture, respectively (see Fig.\[fig:ref\] (a)). For the sake of simplicity, we assume that $\Omega^\ve_\mx \cap \partial \Omega =
\emptyset$. We also introduce the notation: $$\label{omeg12++}
\Omega_T \eqdef \Omega \times (0,T), \quad
\Omega^\ve_{\ell,T} \eqdef \Omega^\ve_\ell \times (0,T), \quad
\Sigma^\ve_T \eqdef \Gamma^\ve_{\fr\mx} \times (0,T), \quad {\rm where}\,\, T > 0 \,\, {\rm is \,\, fixed.}$$
Before describing the equations of the model, we give some notation: $\Phi^\ve(x) = \Phi(x, \frac{x}{\ve})$ is the porosity of the reservoir $\Omega$; $K^\ve(x) = K(x, \frac{x}{\ve})$ is the absolute permeability tensor of $\Omega$; $\varrho_w$, $\varrho_n$ are the densities of wetting and nonwetting fluids, respectively; $S^\ve_{\ell, w} = S^\ve_{\ell, w}(x, t)$, $S^\ve_{\ell, n}
= S^\ve_{\ell, n}(x, t)$ are the saturations of wetting and nonwetting fluids in $\Omega^\ve_\ell$, respectively; $k_{r,w}^{(\ell)} = k_{r,w}^{(\ell)}(S^\ve_{\ell, w})$, $k_{r,n}^{(\ell)} = k_{r,n}^{(\ell)}(S^\ve_{\ell, n})$ are the relative permeabilities of wetting and nonwetting fluids in $\Omega^\ve_\ell$, respectively; $p^\ve_{\ell, w} =
p^\ve_{\ell, w}(x,t)$, $p^\ve_{\ell, n} = p^\ve_{\ell, n}(x,t)$ are the phase pressures of wetting and nonwetting fluids in $\Omega^\ve_\ell$, respectively. Here $\ell = \fr, \mx$.
The conservation of mass in each phase can be written as (see, e.g., [@GC-JJ; @ZC-GH-YM-06; @HR]): $$\label{debut1}
\left\{
\begin{array}[c]{ll}
\displaystyle
\Phi^\ve(x) \frac{\partial}{\partial t}\left[S^\ve_{\ell, w}\,\varrho_w(p^\ve_{\ell, w})\right] +
{\rm div} \big\{\varrho_w(p^\ve_{\ell, w}) \, \vec q^{\,\ve}_{\ell, w} \big\} = F^\ve_{\ell,w}(x,t)
\quad {\rm in}\,\, \Omega^\ve_{\ell,T}; \\[3mm]
\displaystyle
\Phi(x) \frac{\partial}{\partial t} \left[S^\ve_{\ell, n}\, \varrho_n(p^\ve_{\ell, n})\right] +
{\rm div}\big\{\varrho_n(p^\ve_{\ell, n}) \, \vec q^{\,\ve}_{\ell, n} \big\} = F^\ve_{\ell,n}(x,t)
\quad {\rm in}\,\, \Omega^\ve_{\ell,T}, \\[3mm]
\end{array}
\right.$$ where the velocities of the wetting and nonwetting fluids $\vec q^{\,\ve}_{\ell, w}$, $\vec q^{\,\ve}_{\ell, n}$ are defined by Darcy-Muskat’s law: $$\vec q^{\,\ve}_{\ell, w} \eqdef -K^\ve(x) \lambda_{\ell, w}(S^\ve_{\ell, w})
\left[\nabla p^\ve_{\ell, w} - \varrho_w(p^\ve_{\ell, w})\, \vec{g}\right],
\quad \!{\rm with}\,\, \lambda_{\ell, w}(S^\ve_{\ell, w}) \eqdef
\frac{k_{r,w}^{(\ell)}}{\mu_{w}}(S^\ve_{\ell, w});
\label{eq.qw}$$ $$\vec q^{\,\ve}_{\ell, n} \eqdef - K^\ve(x) \lambda_{\ell, n}(S^\ve_{\ell, n})
\left[\nabla p^\ve_{\ell, n} - \varrho_n(p^\ve_{\ell, n})\, \vec{g}\right], \quad
{\rm with}\,\, \lambda_{\ell, n}(S_{\ell, n}) \eqdef \frac{k_{r,n}^{(\ell)}}{\mu_{n}}
(S^\ve_{\ell, n}).
\label{eq.qg}$$ Here $\vec g$, $\mu_w, \mu_n$ are the gravity vector and the viscosities of the wetting and nonwetting fluids, respectively. The source terms $F^\ve_{\ell,w}, F^\ve_{\ell,n}$ are given by: $$\label{sour1}
F^\ve_{\ell,w} \eqdef \varrho_w(p^\ve_{\ell, w}) S^I_{\ell, w} f_I(x,t) -
\varrho_w(p^\ve_{\ell, w}) S^\ve_{\ell, w} f_P(x,t);$$ $$\label{sour2}
F^\ve_{\ell,n} \eqdef \varrho_n(p^\ve_{\ell, n}) S^I_{\ell, n} f_I(x,t) -
\varrho_n(p^\ve_{\ell, n}) S^\ve_{\ell, n} f_P(x,t),$$ where $f_I, f_P \geqslant 0$ are injection and productions terms and $S^I_{\ell, w}, S^I_{\ell, n}$ are known injection saturations.
From now on we deal with two incompressible fluids, that is the densities of the wetting and nonwetting fluids are constants, which for the sake of simplicity and brevity, will be taken equal to one, i.e. $\varrho_w(p^\ve_{\ell,w}) = \varrho_n(p^\ve_{\ell,n}) = 1$. The model is completed as follows. By the definition of saturations, one has $S^\ve_{\ell, w} + S^\ve_{\ell, n} = 1$ with $S^\ve_{\ell, w}, S^\ve_{\ell, n} \geqslant 0$. We set $S^\ve_\ell \eqdef S^\ve_{\ell, w}$. Then the curvature of the contact surface between the two fluids links the difference in the pressures of the two phases to the saturation by the capillary pressure law: $$P_{\ell,c}(S^\ve_\ell) \eqdef p^\ve_{\ell, n} - p^\ve_{\ell, w} \quad {\rm with} \,\,
P^\prime_{\ell,c}(s) < 0\,\, {\rm for\,\, all}\,\,s \in (0, 1)
\,\, {\rm and} \,\, P_{\ell,c}(1) = 0,
\label{eq.pc}$$ where $P^\prime_{\ell,c}(s)$ denotes the derivative of the function $P_{\ell,c}(s)$.
Now due to the assumptions on the densities of the liquids, we rewrite the system (\[debut1\]) as follows: $$\label{debut2}
\left\{
\begin{array}[c]{ll}
0 \leqslant {S}^\ve \leqslant 1 \quad {\rm in}\,\, \Omega_T;\\[3mm]
\displaystyle
\Phi^\ve(x) \frac{\partial {S}^\ve}{\partial t} - {\rm div}\, \left\{K^\ve(x)
\lambda_{w}\left(\frac{x}{\ve}, {S}^\ve\right)
\left(\nabla {\mathsf p}^\ve_{w} - \vec{g}\right)\right\} = F^\ve_{w} \quad {\rm in}\,\, \Omega_{T}; \\[4mm]
\displaystyle
-\Phi^\ve(x) \frac{\partial {S}^\ve}{\partial t} -
{\rm div}\, \left\{K^\ve(x) \lambda_{n} \left(\frac{x}{\ve}, {S}^\ve\right)
\left(\nabla {\mathsf p}^\ve_{n} - \vec{g} \right)\right\} = F^\ve_{n} \quad {\rm in}\,\, \Omega_{T}; \\[4mm]
P_{c}\left(\frac{x}{\ve}, {S}^\ve\right) = {\mathsf p}^\ve_{n} - {\mathsf p}^\ve_{w}
\quad {\rm in}\,\, \Omega_{T},
\end{array}
\right.$$ where $\lambda_{\ell,n}(S^\ve_\ell) := \lambda_{\ell,n}(1-S^\ve_\ell)$ and each function $u^\ve := {S}^\ve, {\mathsf p}^\ve_{w}, {\mathsf p}^\ve_{n}, F^\ve_{w}, F^\ve_{n}$ is defined as: $$\label{nota-pwat-pgaz}
u^\ve \eqdef u^\ve_{\fr}(x, t)\, {\bf 1}^\ve_\fr(x) + u^\ve_{\mx}(x, t)\, {\bf 1}^\ve_\mx(x).$$ Here ${\bf 1}^\ve_\ell = {\bf 1}_\ell(\frac{x}{\ve})$ is the characteristic function of the subdomain $\Omega^\ve_\ell$ for $\ell = \fr, \mx$. The exact form of the porosity function and the absolute permeability tensor corresponding to the double porosity model will be specified in conditions [(A.1)]{}, [(A.2)]{} in subsection \[def-weak-sol\] below.
Model (\[debut2\]) have to be completed with appropriate interface, boundary and initial conditions.
Interface conditions. The continuity at the interface $\Gamma^\ve_{\fr\mx}$ of the phase fluxes and the phase pressures, gives the following transmission conditions: $$\label{inter-condit}
\left\{
\begin{array}[c]{ll}
\vec q^{\,\ve}_{\fr,w} \cdot \vec \nu = \vec q^{\,\ve}_{\mx,w}\cdot \vec \nu \quad {\rm and} \quad
\vec q^{\,\ve}_{\fr,n} \cdot \vec \nu = \vec q^{\,\ve}_{\mx,n}\cdot \vec \nu
\quad {\rm on}\,\, \Sigma^\ve_T; \\[2mm]
p^\ve_{\fr,w} = p^\ve_{\mx,w} \quad {\rm and} \quad p^\ve_{\fr,n} = p^\ve_{\mx,n} \quad {\rm on}\,\, \Sigma^\ve_T, \\
\end{array}
\right.$$ where $\Sigma^\ve_T$ is defined in (\[omeg12++\]), $\vec \nu$ is the unit outer normal on $\Gamma^\ve_{\fr\mx}$, and the fluxes $\vec q^{\,\ve}_{\ell,w}, \vec q^{\,\ve}_{\ell,n}$, under the assumption on the densities of the liquids, are equal to the velocities (\[eq.qw\]), (\[eq.qg\]).
\[interface-saturrr\] It is important to notice that in contrast to the functions ${\mathsf p}^\ve_n, {\mathsf p}^\ve_w$, the saturation ${S}^\ve$ may have a jump at the interface $\Gamma^\ve_{\fr\mx}$. Namely, it is easy to see from the transmission conditions (\[inter-condit\]) for the phase pressures that $P_{\fr,c}(S^\ve_1) = P_{\mx,c}(S^\ve_2)$ on $\Sigma^\ve_T$ which gives a discontinuity of the saturation at the interface.
Now we specify the boundary and initial conditions. We suppose that the boundary $\partial \Omega$ consists of two parts $\Gamma_{1}$ and $\Gamma_{2}$ such that $\Gamma_{1} \cap \Gamma_{2} = \emptyset$, $\partial \Omega =
\overline\Gamma_{1} \cup \overline\Gamma_{2}$.
Boundary conditions: $$\label{bc3}
\left\{
\begin{array}[c]{ll}
{\mathsf p}^\ve_{w}(x, t) = {\mathsf p}^\ve_{n}(x, t) = 0 \quad {\rm on} \,\,
\Gamma_{1} \times (0,T); \\[2mm]
\vec q^{\,\ve}_{\fr,w} \cdot \vec \nu = \vec q^{\,\ve}_{\fr,n} \cdot \vec \nu = 0
\quad {\rm on} \,\, \Gamma_{2} \times (0,T).\\
\end{array}
\right.$$ Initial conditions: $$\label{init1}
{\mathsf p}^\ve_{w}(x, 0) = {\mathsf p}_{w}^{\bf 0}(x) \quad {\rm and}
\quad
{\mathsf p}^\ve_{n}(x, 0) = {\mathsf p}_{n}^{\bf 0}(x)
\quad {\rm in} \,\, \Omega.$$
A fractional flow formulation {#gp-relat}
-----------------------------
In the sequel, we will use a formulation obtained after transformation using the concept of the global pressure introduced in [@ant-kaz-mon1990; @GC-JJ]. For each subdomain $\Omega^\ve_\ell$, the global pressure, ${\mathsf P}^\ve_\ell$, is defined by: $$\label{gp1}
p^\ve_{\ell,w} \eqdef {\mathsf P}^\ve_\ell + {\mathsf G}_{\ell,w}(S^\ve_\ell)
\quad {\rm and} \quad
p^\ve_{\ell,n} \eqdef {\mathsf P}^\ve_\ell + {\mathsf G}_{\ell,n}(S^\ve_\ell),$$ where the functions ${\mathsf G}_{\ell,w}(s)$, ${\mathsf G}_{\ell,n}(s)$ are given by: $$\label{gp3}
{\mathsf G}_{\ell,n}(S^\ve_\ell) \eqdef
{\mathsf G}_{\ell,n}(0) + \int\limits_0^{S^\ve_\ell} \frac{\lambda_{\ell,w}(s)}
{\lambda_{\ell}(s)} \,P^\prime_{\ell,c}(s)\, ds\quad {\rm and} \quad
{\mathsf G}_{\ell, w}(S^\ve_\ell) \eqdef {\mathsf G}_{\ell,n}(S^\ve_\ell)
- P_{\ell,c}\left(S^\ve_\ell\right),$$ where $\lambda_{\ell}(s) \eqdef \lambda_{\ell,w}(s) + \lambda_{\ell,n}(s)$ and ${\mathsf G}_{\ell,n}(0)$ is a constant chosen to ensure $p^\ve_{\ell,w} \leqslant {\mathsf P}^\ve_\ell \leqslant p^\ve_{\ell,n}$. Notice that from (\[gp3\]) we get: $$\label{gp+grad}
\lambda_{\ell,w}(S^\ve_\ell) \nabla {\mathsf G}_{\ell,w}(S^\ve_\ell) = \nabla\beta_\ell(S^\ve_\ell)
\quad {\rm and} \quad
\lambda_{\ell,n}(S^\ve_\ell) \nabla {\mathsf G}_{\ell,n}(S^\ve_\ell) = - \nabla\beta_\ell(S^\ve_\ell),$$ where $$\label{upsi-1}
\beta_\ell(s) \eqdef \int\limits_0^s \alpha_\ell(\xi)\, d\xi \quad {\rm with} \,\,\,
\alpha_\ell(s) \eqdef
\frac{\lambda_{\ell,n}(s)\, \lambda_{\ell,w}(s)} {\lambda_\ell(s)} \left| P^\prime_{\ell,c}(s) \right|.$$
Furthermore, we have the following important relation: $$\label{gp6-new}
\lambda_{\ell,n} (S^\ve_\ell) |\nabla p^\ve_{\ell,n}|^2 +
\lambda_{\ell,w} (S^\ve_\ell) |\nabla p^\ve_{\ell,w}|^2
=
\lambda_{\ell} (S^\ve_\ell) |\nabla {\mathsf P}^\ve_\ell |^2 +
\left|\nabla \mathfrak{b}_\ell(S^\ve_\ell) \right|^2,$$ where $$\label{bbb}
\mathfrak{b}_\ell(s) \eqdef \int\limits_0^s \mathfrak{a}_\ell(\xi)\, d\xi
\quad {\rm with} \,\, \mathfrak{a}_\ell(s) \eqdef
\sqrt{\frac{\lambda_{\ell,n}(s)\, \lambda_{\ell,w}(s)} {\lambda_\ell(s)}}\, \left| P^\prime_{\ell,c}(s) \right|.$$
Now if we use the global pressure and the saturation as new unknown functions then (\[debut2\]) reads: $$\label{gp+3-simp}
\left\{
\begin{array}[c]{ll}
0 \leqslant S^\ve_\ell \leqslant 1 \quad {\rm in}\,\, \Omega^\ve_{\ell,T};\\[3mm]
\displaystyle
\Phi^\ve(x) \frac{\partial S^\ve_\ell}{\partial t} -
{\rm div}\, \bigg\{K^\ve(x) \left[
\lambda_{\ell,w} (S^\ve_\ell) \nabla {\mathsf P}^\ve_\ell +
\nabla \beta_\ell(S^\ve_\ell) - \lambda_{\ell,w} (S^\ve_\ell) \vec{g} \right] \bigg\}
= F^\ve_{\ell,w} \quad {\rm in}\,\, \Omega^\ve_{\ell,T}; \\[4mm]
\displaystyle
-\Phi^\ve(x) \frac{\partial S^\ve_\ell}{\partial t}
-
{\rm div}\, \bigg\{K^\ve(x) \left[ \lambda_{\ell,n} (S^\ve_\ell)
\nabla {\mathsf P}^\ve_\ell - \nabla \beta_\ell(S^\ve_\ell)
-
\lambda_{\ell,n}(S^\ve_\ell) \vec{g} \right] \bigg\} =
F^\ve_{\ell,n}\quad {\rm in}\,\, \Omega^\ve_{\ell,T}.\\
\end{array}
\right.$$
The system (\[gp+3-simp\]) is completed by the following boundary, interface and initial conditions.
Boundary conditions: $$\label{bc3+gp}
\left\{
\begin{array}[c]{ll}
{S}^\ve = 1 \,\,\,{\rm and}\,\,\,
{\mathsf P}^\ve = {\mathsf P}_{\Gamma_1} \quad {\rm on} \,\,
\Gamma_{1} \times (0,T); \\[2mm]
\vec q^{\,\ve}_{\fr,w} \cdot \vec \nu = \vec q^{\,\ve}_{\fr,n} \cdot \vec \nu = 0
\quad {\rm on} \,\, \Gamma_{2} \times (0,T),\\
\end{array}
\right.$$ where ${\mathsf P}_{\Gamma_1}$ is a given constant and $\vec q^{\,\ve}_{\ell, w}, \vec q^{\,\ve}_{\ell, n}$ are defined by $$\label{fluxes+gp-w}
\vec q^{\,\ve}_{\ell, w} \eqdef -K^\ve(x) \left[
\lambda_{\ell,w} (S^\ve_\ell) \nabla {\mathsf P}^\ve_\ell +
\nabla \beta_\ell(S^\ve_\ell) - \lambda_{\ell,w} (S^\ve_\ell) \vec{g} \right];$$ $$\label{fluxes+gp-n}
\vec q^{\,\ve}_{\ell, n} \eqdef - K^\ve(x) \left[ \lambda_{\ell,n} (S^\ve_\ell)
\nabla {\mathsf P}^\ve_\ell - \nabla \beta_\ell(S^\ve_\ell)
- \lambda_{\ell,n}(S^\ve_\ell) \vec{g} \right].$$
Interface conditions: $$\label{ic-glob}
\left\{
\begin{array}[c]{ll}
\vec q^{\,\,\ve}_{\fr,w} \cdot \vec \nu =
\vec q^{\,\,\ve}_{\mx,w}\cdot \vec \nu \,\,\, {\rm and} \,\,\,
\vec q^{\,\,\ve}_{\fr,n} \cdot \vec \nu =
\vec q^{\,\,\ve}_{\mx,n}\cdot \vec \nu
\quad {\rm on}\,\, \Sigma^\ve_T; \\[3mm]
{\mathsf P}^\ve_\fr + {\mathsf G}_{\fr,j}(S^\ve_\fr) =
{\mathsf P}^\ve_\mx + {\mathsf G}_{\mx,j}(S^\ve_\mx)
\quad {\rm on}\,\, \Sigma^\ve_T \quad (j = w, n);\\[3mm]
P_{\fr,c}\left(S^\ve_\fr\right) = P_{\mx,c}\left(S^\ve_\mx\right)
\quad {\rm on}\,\, \Sigma^\ve_T.
\end{array}
\right.$$
Note that the global pressure function might be discontinuous at the interface. This makes the compactness result in Section \[ex-comp-sf\] non-trivial.
Initial conditions: $$\label{init1-gl+gp}
S^\ve_\ell(x, 0) = S^{\bf 0}_\ell(x) \,\, {\rm and} \,\,
{\mathsf P}^\ve_\ell(x, 0) = {\mathsf P}^{\bf 0}_\ell(x) \quad {\rm in} \,\, \Omega.$$
Weak formulations of the problem {#def-weak-sol}
--------------------------------
Let us begin this subsection by stating the following assumptions.
- The porosity $\Phi^\ve$ is given by $
\Phi^\ve(x) \eqdef \Phi^\ve_{\fr}(x)\, {\bf 1}^\ve_\fr(x) + \Phi^\ve_{\mx}(x)\, {\bf 1}^\ve_\mx(x) =
\Phi^\ve_{\fr}(x)\, {\bf 1}^\ve_\fr(x) + \Phi_{\mx}\left(\frac{x}{\ve}\right)\, {\bf 1}^\ve_\mx(x),
$ where $\Phi^\ve_{\fr} \in L^{\infty}(\Omega)$ and there are positive constants $0 < \phi^{\ell}_- < \phi^{\ell}_+ < 1$, $\ell = \fr, \mx$, that do not depend on $\ve$ and such that $0 < \phi^{\fr}_- \leqslant \Phi^\ve_{\fr}(x) \leqslant \phi^{\fr}_+ < 1$ a.e. in $\Omega$. Moreover, $\Phi^\ve_{\fr} \longrightarrow \Phi^{\rm H}_{\fr}$ strongly in $L^2(\Omega)$. $\Phi_\mx = \Phi_\mx(y)$ is $Y$-periodic, $\Phi_{\mx} \in L^{\infty}(Y)$ and such that $0 < \phi^{\mx}_- \leqslant \Phi_\mx(y) \leqslant \phi^{\mx}_+ < 1$ a.e. in $Y$.
- The permeability $K^\ve(x) = K^\ve(x, \frac{x}{\ve})$ is defined as $
\label{tensork-expr}
K^\ve(x, y) \eqdef K(x, y)\, {\bf 1}^\ve_\fr(x) + \varkappa(\ve)\, K(x, y)\, {\bf 1}^\ve_\mx(x),
$ where $\varkappa(\ve) \eqdef \ve^\theta$ with $\theta > 0$ and $K \in (L^{\infty}(\Omega\times Y))^{d\times d}$. Moreover, there exist constants $k_{\rm min}, k^{\rm max}$ such that $0 < k_{\rm min} < k^{\rm max}$ and $
\label{tensork}
k_{\rm min} |\xi|^2 \leq (K(x, y)\,\xi, \xi) \leq k^{\rm max} |\xi|^2 \,\, {\rm for \, all \,
\xi \in \mathbb{R}^d, \,\, a.e. \, in}\,\, \Omega\times Y.
$
- The capillary pressure function $P_{\ell,c}(s) \in C^1([0, 1]; \mathbb{R}^+)$, $\ell = \fr, \mx$. Moreover, $P_{\ell,c}^\prime(s) < 0$ in $[0, 1]$, $P_{\ell,c}(1) = 0$ and $P_{\fr,c}(0) = P_{\mx,c}(0)$.
- The functions $\lambda_{\ell,w}, \lambda_{\ell,n}$ belong to the space $C([0, 1]; \mathbb{R}^+)$ and satisfy the following properties:\
[(i)]{} $0 \leqslant \lambda_{\ell,w}, \lambda_{\ell,n} \leqslant 1$ in $[0, 1]$; [(ii)]{} $\lambda_{\ell,w}(0) = 0$ and $\lambda_{\ell,n}(1) = 0$; [(iii)]{} there is a positive constant $L_0$ such that $\lambda_{\ell}(s) = \lambda_{\ell,w}(s) + \lambda_{\ell,n}(s)
\geqslant L_0 > 0$ in $[0, 1]$.
- The functions $\alpha_\ell \in C([0, 1]; \mathbb{R}^+)$. Moreover, $\alpha_\ell (0) = \alpha_\ell (1) = 0$ and $\alpha_\ell > 0$ in $(0, 1)$.
- The functions $\beta^{-1}_\ell$, inverse of $\beta_\ell$ defined in (\[upsi-1\]) are Hölder functions of order $\gamma\in (0, 1)$ in $[0, \beta_\ell(1)]$. Namely, there exists a positive constant $C_\beta$ such that for all $s_1, s_2 \in [0, \beta(1)]$, we have: $$\left|\beta^{-1}_\ell(s_1) - \beta^{-1}_\ell(s_2) \right| \leqslant C_\beta \, |s_1 - s_2|^\gamma.$$
- The initial data for the pressures are such that ${\mathsf p}^{\bf 0}_{n},
{\mathsf p}^{\bf 0}_{w} \in L^2(\Omega)$.
- The initial data for the saturation $S^{\bf 0}$ is given by $P_{\ell,c}(S^{\bf 0}_\ell) ={\mathsf p}^{\bf 0}_{\ell, n} - {\mathsf p}^{\bf 0}_{\ell, w}$ and is such that ${S}^{\bf 0} \in L^\infty(\Omega)$ and $0 \leqslant {S}^{\bf 0} \leqslant 1$ a.e.in $\Omega$.
- The source terms $F^\ve_{w}, F^\ve_{n}$ are equal to zero on the matrix part, i.e. $
\label{sour1-A9}
F^\ve_{w} \eqdef {\bf 1}^\ve_\fr(x)\, \big[S^I_{\fr, w} f_I(x,t) - S^\ve_{\fr} f_P(x,t)\big]
$ and $
F^\ve_{n} \eqdef {\bf 1}^\ve_\fr(x)\, \big[S^I_{\fr, n} f_I(x,t) - (1 - S^\ve_{\fr}) f_P(x,t)\big],
$ where $f_I, f_P \in L^2(\Omega_T)$ and $0 \leqslant S^I_{\fr, w}, S^I_{\fr, n} \leqslant 1$.
The assumptions (A.1)–(A.9) are classical and physically meaningful for existence results and homogenization problems of two-phase flow in porous media. They are similar to the assumptions made in [@ant-kaz-mon1990; @GC-JJ] that dealt with the existence of a weak solution of the studied problem.
We next introduce the following Sobolev space: $
H^1_{\Gamma_{1}}(\Omega) \eqdef \left\{u \in H^1(\Omega) \,:\,
u = 0 \,\, {\rm on}\,\, \Gamma_{1} \right\}.
$ The space $H^1_{\Gamma_{1}}(\Omega)$ is a Hilbert space. The norm in this space is given by $\Vert u \Vert_{H^1_{\Gamma_{1}}(\Omega)} = \Vert \nabla u \Vert_{(L^2(\Omega))^d}$.
\[def-weak-simp\]
We say that the functions $\langle {\mathsf p}^\ve_{w}, {\mathsf p}^\ve_{n} , {S}^\ve\rangle$ is a weak solution of [problem (\[debut2\])]{} if
- $0 \leqslant {S}^\ve \leqslant 1$ a.e. in $\Omega_T$ and $P_{\ell,c}(S^\ve_\ell) \eqdef {\mathsf p}^\ve_{\ell, n} - {\mathsf p}^\ve_{\ell, w}$ for $\ell\in\{\fr,\mx\}$.
- The functions ${\mathsf p}^\ve_{w}, {\mathsf p}^\ve_{n}$ are such that $${\mathsf p}^\ve_{w}\,, {\mathsf p}^\ve_{n}\,,
\sqrt{\lambda_w(x, {S}^\ve)}\, \nabla {\mathsf p}^\ve_w\,,
\sqrt{\lambda_n(x, {S}^\ve)}\, \nabla {\mathsf p}^\ve_n \in L^2(\Omega_T).$$
- The boundary conditions (\[bc3\]) and the initial conditions (\[init1\]) are satisfied.
- For any $\varphi_w, \varphi_n \in C^1([0, T]; H^1_{\Gamma_{1}}(\Omega))$ satisfying $\varphi_w(T) = \varphi_n(T) = 0$, we have:
$$\label{wf-1-gl}
-\int\limits_{\Omega_{T}} \Phi^\ve(x) {S}^\ve \frac{\partial \varphi_w}{\partial t}
\, dx dt - \int\limits_{\Omega} \Phi^\ve {S}^{\bf 0} \varphi_w^{\bf 0}\, dx
+
\int\limits_{\Omega_{T}} K^\ve(x) \lambda_{w}\left(\frac{x}{\ve}, { S}^\ve\right)
\left(\nabla {\mathsf p}^\ve_w - \vec g \right) \cdot \nabla \varphi_w\, dx dt
= \int\limits_{\Omega_{T}} F^\ve_{w}\, \varphi_w\, dx dt;$$
$$\label{gf-2-gl}
\int\limits_{\Omega_{T}} \Phi^\ve(x) {S}^\ve \frac{\partial \varphi_n}{\partial t}
\, dx dt + \int\limits_{\Omega} \Phi^\ve {S}^{\bf 0} \varphi_n^{\bf 0}\, dx
+
\int\limits_{\Omega_{T}} K^\ve(x) \lambda_{n}\left(\frac{x}{\ve}, {S}^\ve\right)
\left(\nabla {\mathsf p}^\ve_n - \vec g \right) \cdot \nabla \varphi_n\, dx dt
= \int\limits_{\Omega_{T}} F^\ve_{n}\,\varphi_n\, dx dt,$$
where $\varphi_w^{\bf 0} \eqdef \varphi_w(0, x)$, $\varphi_n^{\bf 0} \eqdef \varphi_n(0, x)$, and the function ${S}^{\bf 0} = {S}^{\bf 0}(x)$ is defined by the initial condition (\[init1\]) and the capillary pressure relation (\[eq.pc\]).
Let us also give an equivalent definition of a weak solution in terms of the global pressure and the saturation.
\[def-weak-gps\]
We say that the pair of functions $\langle {S}^\ve, {\mathsf P}^\ve \rangle$ is a weak solution of [problem (\[gp+3-simp\])]{} if
- $0 \leqslant {S}^\ve \leqslant 1$ a.e. in $\Omega_T$.
- The global pressure function ${\mathsf P}^\ve_\ell \in L^2(0, T; H^1(\Omega^\ve_\ell))$ and, for any $\ve > 0$, the saturation function $S^\ve_\ell$ is such that $\beta_\ell(S^\ve_\ell) \in L^2(0, T; H^1(\Omega^\ve_\ell))$.
- The boundary conditions (\[bc3+gp\]) and the initial conditions (\[init1-gl+gp\]) are satisfied.
- For any $\varphi_w, \varphi_n \in C^1([0, T]; H^1_{\Gamma_{1}}(\Omega))$ satisfying $\varphi_w(T) = \varphi_n(T) = 0$, we have:
$$-\int\limits_{\Omega_{T}} \Phi^\ve(x) {S}^\ve \frac{\partial \varphi_w}{\partial t}
\, dx dt - \int\limits_{\Omega} \Phi^\ve(x) {S}^{\bf 0} \varphi_w^{\bf 0}\, dx +
\int\limits_{\Omega^\ve_{\fr,T}} K^\ve(x)\bigg\{ \lambda_{\,\fr,w} (S^\ve_\fr)
\left(\nabla {\mathsf P}^\ve_\fr - \vec g \right) + \nabla \beta_\fr(S^\ve_\fr)\bigg\}
\cdot \nabla \varphi_w\, dx dt +$$ $$\label{wf-1-gl-GP}
+ \varkappa(\ve)\,
\int\limits_{\Omega^\ve_{\mx,T}} K^\ve(x)\bigg\{ \lambda_{\,\mx,w} (S^\ve_\mx)
\left(\nabla {\mathsf P}^\ve_\mx - \vec g \right) + \nabla \beta_\mx(S^\ve_\mx)\bigg\}
\cdot \nabla \varphi_w\, dx dt
= \int\limits_{\Omega^\ve_{\fr,T}} F^\ve_{w}\, \varphi_w\, dx dt;$$ $$\int\limits_{\Omega_{T}} \Phi^\ve(x) {S}^\ve \frac{\partial \varphi_n}{\partial t}
\, dx dt + \int\limits_{\Omega} \Phi^\ve(x) {S}^{\bf 0} \varphi_n^{\bf 0}\, dx +
\int\limits_{\Omega^\ve_{\fr,T}} K^\ve(x)\bigg\{ \lambda_{\,\fr,n} (S^\ve_\fr)
\left(\nabla {\mathsf P}^\ve_\fr - \vec g \right) - \nabla \beta_\fr(S^\ve_\fr)\bigg\}
\cdot \nabla \varphi_n\, dx dt +$$ $$\label{gf-2-gl-GP}
+ \varkappa(\ve)\,
\int\limits_{\Omega^\ve_{\mx,T}} K^\ve(x)\bigg\{ \lambda_{\,\mx,n} (S^\ve_\mx)
\left(\nabla {\mathsf P}^\ve_\mx - \vec g \right) - \nabla \beta_\mx(S^\ve_\mx)\bigg\}
\cdot \nabla \varphi_n\, dx dt = \int\limits_{\Omega^\ve_{\fr,T}} F^\ve_{n}\, \varphi_n\, dx dt.$$
Existence theorem for the weak solutions defined in Definition \[def-weak-simp\] and Definition \[def-weak-gps\] is given in [@APP-ex2013] in more general case of compressible fluids.
[**Notational convention.**]{} In what follows $C, C_1,..$ denote generic constants that do not depend on $\ve$.
A priori uniform estimates {#uni-est}
==========================
The uniform estimates for the initial system (\[debut2\]) or the equivalent one (\[gp+3-simp\]) are given by the following lemma:
\[lem-uniform\] Let $\langle {\mathsf p}^\ve_w, {\mathsf p}^\ve_n , S^\ve\rangle$ be a solution to [problem (\[debut2\])]{}. Then under assumptions [(A.1)-(A.9)]{} the following uniform in $\ve$ estimates hold true: $$\big\Vert \sqrt{\lambda_{\fr,w} \left(S^\ve_\fr\right)}\, \nabla p^\ve_{\fr,w}\big\Vert_{L^2(\Omega^\ve_{\fr,T})}
+
\big\Vert \sqrt{\lambda_{\fr,n} \left(S^\ve_\fr\right)}\, \nabla p^\ve_{\fr,n} \big\Vert_{L^2(\Omega^\ve_{\fr,T})}
+$$ $$\label{hhh8-cor1}
+
\varkappa^{\frac12}(\ve)\,\big\Vert \sqrt{\lambda_{\mx,w} \left(S^\ve_\mx\right)}\, \nabla p^\ve_{\mx,w}\big\Vert_{L^2(\Omega^\ve_{\mx,T})}
+
\varkappa^{\frac12}(\ve)\,\big\Vert \sqrt{\lambda_{\mx,n} \left(S^\ve_\mx\right)}\, \nabla p^\ve_{\mx,n} \big\Vert_{L^2(\Omega^\ve_{\mx,T})} \leqslant C;$$ $$\label{beta-un}
\left\Vert \nabla \beta_\fr(S^\ve_\fr)\right\Vert_{L^2(\Omega^\ve_{\fr,T})}
+
\left\Vert \nabla {\mathsf P}^\ve_\fr \right\Vert_{L^2(\Omega^\ve_{\fr,T})}
+
\varkappa^{\frac12}(\ve)\,\left\Vert \nabla \beta_\mx(S^\ve_\mx)\right\Vert_{L^2(\Omega^\ve_{\mx,T})}
+
\varkappa^{\frac12}(\ve)\,\left\Vert \nabla {\mathsf P}^\ve_\mx \right\Vert_{L^2(\Omega^\ve_{\mx,T})}
\leqslant C,$$ where $\varkappa(\ve) \eqdef \ve^\theta$ with $\theta > 0$.
[**Proof of Lemma \[lem-uniform\].**]{} Notice that the uniform boundedness results (\[hhh8-cor1\]), (\[beta-un\]) were already proved by many authors (see, e.g., [@yeh2] and the references therein) in the case when the source terms in (\[debut2\]) were assumed to be zero. We also refer here to [@our-siam] and the references therein, where the uniform boundedness results were obtained in the case of compressible two-phase flows in porous media. Here, for reader’s convenience, we recall the proof of the bounds (\[hhh8-cor1\]), (\[beta-un\]) focusing on the terms involving the source functions $F^\ve_{w}, F^\ve_{n}$.
We start our analysis by obtaining the uniform bound (\[hhh8-cor1\]). To this end we multiply the first equation in (\[debut2\]) by ${\mathsf p}^\ve_{w}$, the second equation in (\[debut2\]) by ${\mathsf p}^\ve_{n}$ and then integrate over the domain $\Omega$. Taking into account the boundary conditions (\[bc3\]) after integration by parts, we get the following energy equality: $$- \frac{d}{d t}\int\limits_\Omega \Phi^\ve(x)\,\digamma({S}^\ve) \, dx +
\int\limits_\Omega
\left\{K^\ve(x) \lambda_{w}\left(\frac{x}{\ve}, {S}^\ve\right)
\left(\nabla {\mathsf p}^\ve_{w} - \vec{g}\right)\right\} \cdot \nabla {\mathsf p}^\ve_{w} \, dx +$$ $$\label{uff-2}
+
\int\limits_\Omega
\left\{K^\ve(x) \lambda_{n}\left(\frac{x}{\ve}, {S}^\ve\right)
\left(\nabla {\mathsf p}^\ve_{n} - \vec{g}\right)\right\}
\cdot \nabla {\mathsf p}^\ve_{n} \, dx
=
\int\limits_\Omega \left[F^\ve_{w}(x, t)\,{\mathsf p}^\ve_{w} + F^\ve_{n}(x, t)\,{\mathsf p}^\ve_{n}\right],$$ where $$\label{uff-2-digam}
\digamma({S}^\ve) \eqdef \digamma_\fr(S^\ve_\fr)\, {\bf 1}^\ve_\fr(x) +
\digamma_\mx(S^\ve_\mx)\, {\bf 1}^\ve_\mx(x)
\eqdef {\bf 1}^\ve_\fr(x)\, \int\limits_1^{S^\ve_\fr} P_{\fr,c}(u)\, du +
{\bf 1}^\ve_\mx(x)\, \int\limits_1^{S^\ve_\mx} P_{\mx,c}(u)\, du.$$
The equality (\[uff-2\]) is the desired energy equality which will be used below to obtain the necessary bounds that are uniform in $\ve$. To this end we integrate (\[uff-2\]) over the interval $(0, T)$ to get: $$- \int\limits_\Omega \Phi^\ve(x)\,\digamma({S}^\ve) \, dx +
\int\limits_{\Omega_T}
\left\{K^\ve(x) \lambda_{w}\left(\frac{x}{\ve}, {S}^\ve\right)
\left(\nabla {\mathsf p}^\ve_{w} - \vec{g}\right)\right\}
\cdot \nabla {\mathsf p}^\ve_{w} \, dx dt +$$ $$\label{uff-3}
+
\int\limits_{\Omega_T}
\left\{K^\ve(x) \lambda_{n}\left(\frac{x}{\ve}, {S}^\ve\right)
\left(\nabla {\mathsf p}^\ve_{n} - \vec{g}\right)\right\}
\cdot \nabla {\mathsf p}^\ve_{n} \, dx dt = \mathbb{J}^\ve_{w,n} -
\int\limits_\Omega \Phi^\ve(x)\,\digamma\left({S}^\ve(x, 0)\right) \, dx,$$ where $$\label{uff-4}
\mathbb{J}^\ve_{w,n} \eqdef \int\limits_{\Omega_T} \left[F^\ve_{w}(x, t)\,{\mathsf p}^\ve_{w} +
F^\ve_{n}(x, t)\,{\mathsf p}^\ve_{n}\right]\, dx dt.$$
First, we notice that due to the positiveness of the porosity function $\Phi^\ve$ and the definition of the function $\digamma\left({S}^\ve\right)$ we have that the first term on the left-hand side of (\[uff-3\]) is bounded from below by a constant which does not depend on $\ve$. It is also easy to see from conditions [(A.1)]{}, [(A.3)]{} that the second term on the right-hand side of (\[uff-3\]) is uniformly bounded in $\ve$. Then from (\[uff-3\]) we get the following inequality: $$\int\limits_{\Omega_T}
K^\ve(x) \lambda_{w}\left(\frac{x}{\ve}, {S}^\ve\right)
\nabla {\mathsf p}^\ve_{w} \cdot \nabla {\mathsf p}^\ve_{w} \, dx dt +
\int\limits_{\Omega_T}
K^\ve(x) \lambda_{n}\left(\frac{x}{\ve}, {S}^\ve\right)
\nabla {\mathsf p}^\ve_{n} \cdot \nabla {\mathsf p}^\ve_{n} \, dx dt \leqslant$$ $$\label{uff-7}
\leqslant C+
\int\limits_{\Omega_T}
K^\ve(x) \lambda_{w}\left(\frac{x}{\ve}, {S}^\ve\right) \vec{g} \cdot
\nabla {\mathsf p}^\ve_{w} \, dx dt + \int\limits_{\Omega_T}
K^\ve(x) \lambda_{n}\left(\frac{x}{\ve}, {S}^\ve\right) \vec{g} \cdot
\nabla {\mathsf p}^\ve_{n} \, dx dt + \mathbb{J}^\ve_{w,n}.$$ With the help of Young’s inequality the second and the third terms in the right-hand side of (\[uff-7\]) can be absorbed by the first and second term in the left-hand side of (\[uff-7\]). Namely, we get: $$\label{uff-8}
\int\limits_{\Omega_T}
K^\ve(x) \lambda_{w}\left(\frac{x}{\ve}, {S}^\ve\right)
\nabla {\mathsf p}^\ve_{w} \cdot \nabla {\mathsf p}^\ve_{w} \, dx dt +
\int\limits_{\Omega_T} K^\ve(x) \lambda_{n}\left(\frac{x}{\ve}, {S}^\ve\right)
\nabla {\mathsf p}^\ve_{n} \cdot \nabla {\mathsf p}^\ve_{n} \, dx dt
\leqslant C \big[1 + \mathbb{J}^\ve_{w,n}\big].$$
Now it remains to estimate $\mathbb{J}^\ve_{w,n}$. Due to condition [(A.9)]{}, it can be written as: $$\mathbb{J}^\ve_{w,n} =
\int\limits_{\Omega^\ve_{\fr,T}} \big[S^I_{\fr, w} f_I(x,t) - S^\ve_{\fr} f_P(x,t)\big]\,p^\ve_{\fr,w}
\, dx dt
+
\int\limits_{\Omega^\ve_{\fr,T}}
\big[S^I_{\fr, n} f_I(x,t) - (1 - S^\ve_{\fr}) f_P(x,t)\big] \,p^\ve_{\fr,n}\, dx dt \eqdef$$ $$\label{uff-9}
\eqdef \mathbb{J}^\ve_{w} + \mathbb{J}^\ve_{n}.$$ Consider, first, the term $\mathbb{J}^\ve_{w}$. From the boundedness of the saturation functions, Cauchy’s inequality and condition [(A.9)]{}, we get: $$\label{uff-11}
\big|\mathbb{J}^\ve_{w} \big|
\leqslant \left[\Vert f_I \Vert_{L^2(\Omega_T)}
+
\Vert f_P \Vert_{L^2(\Omega_T)}\right] \Vert p^\ve_{\fr,w} \Vert_{L^2(\Omega^\ve_{\fr,T})}
\leqslant C_1\, \Vert p^\ve_{\fr,w} \Vert_{L^2(\Omega^\ve_{\fr,T})}.$$ In a similar way, $$\label{uff-12}
\big|\mathbb{J}^\ve_{n} \big| \leqslant C_2\, \Vert p^\ve_{\fr,n} \Vert_{L^2(\Omega^\ve_{\fr,T})}.$$
Now using condition [(A.2)]{}, (\[uff-9\]), (\[uff-11\]), and (\[uff-12\]), from the inequality (\[uff-8\]), we get: $$\mathbb{L}^\ve \eqdef
k_{\rm min}\, \int\limits_{\Omega^\ve_{\fr,T}} \lambda_{\fr,w}(S^\ve_\fr) \big|\nabla p^\ve_{\fr,w} \big|^2\, dx dt +
k_{\rm min}\, \int\limits_{\Omega^\ve_{\fr,T}} \lambda_{\fr,n}(S^\ve_\fr) \big|\nabla p^\ve_{\fr,n} \big|^2\, dx dt +$$ $$+
\varkappa(\ve)\, k_{\rm min}\, \int\limits_{\Omega^\ve_{\mx,T}} \lambda_{\mx,w}(S^\ve_\mx) \big|\nabla p^\ve_{\mx,w} \big|^2\, dx dt +
\varkappa(\ve)\, k_{\rm min}\, \int\limits_{\Omega^\ve_{\mx,T}} \lambda_{\mx,n}(S^\ve_\mx) \big|\nabla p^\ve_{\mx,n} \big|^2\, dx dt
\leqslant$$ $$\label{uff-13}
\leqslant C_3 \left[1 + \Vert p^\ve_{\fr,w} \Vert_{L^2(\Omega^\ve_{\fr,T})} +
\Vert p^\ve_{\fr,n} \Vert_{L^2(\Omega^\ve_{\fr,T})}\right].$$ Consider the right-hand side of (\[uff-13\]). From (\[gp1\]) we have: $$\Vert p^\ve_{\fr,w} \Vert_{L^2(\Omega^\ve_{\fr,T})} + \Vert p^\ve_{\fr,n} \Vert_{L^2(\Omega^\ve_{\fr,T})}
\leqslant$$ $$\label{uff-14}
\leqslant
\left[\Vert {\mathsf P}^\ve_\fr \Vert_{L^2(\Omega^\ve_{\fr,T})} +
\Vert {\mathsf G}_{\fr,w}(S^\ve_\fr) \Vert_{L^2(\Omega^\ve_{\fr,T})} +
\Vert {\mathsf P}^\ve_\fr \Vert_{L^2(\Omega^\ve_{\fr,T})} +
\Vert {\mathsf G}_{\fr,n}(S^\ve_\fr) \Vert_{L^2(\Omega^\ve_{\fr,T})}
\right].$$ Then, taking into account that the functions ${\mathsf G}_{\fr,w}(S^\ve_\fr), {\mathsf G}_{\fr,n}(S^\ve_\fr)$ are uniformly bounded in $\ve$, the inequality (\[uff-13\]) takes the form: $$\label{uff-15}
\mathbb{L}^\ve
\leqslant C_4 \left[1 + \Vert {\mathsf P}^\ve_\fr \Vert_{L^2(\Omega^\ve_{\fr,T})} \right].$$ Taking into account the boundary condition ${\mathsf P}^\ve = {\mathsf P}_{\Gamma_1} = {\rm Const}$ on $\Gamma_{1} \times (0,T)$ and applying Friedrich’s inequality we obtain that $$\label{uff-16}
\Vert {\mathsf P}^\ve_\fr \Vert_{L^2(\Omega^\ve_{\fr,T})} \leqslant C_5\, \left[1 +
\Vert \nabla {\mathsf P}^\ve_\fr \Vert_{L^2(\Omega^\ve_{\fr,T})} \right].$$ Finally, in view of (\[uff-16\]), the inequality (\[uff-15\]) takes the form: $$\int\limits_{\Omega^\ve_{\fr,T}} \lambda_{\fr,w}(S^\ve_\fr) \big|\nabla p^\ve_{\fr,w} \big|^2\, dx dt +
\int\limits_{\Omega^\ve_{\fr,T}} \lambda_{\fr,n}(S^\ve_\fr) \big|\nabla p^\ve_{\fr,n} \big|^2\, dx dt +
\varkappa(\ve)\, \int\limits_{\Omega^\ve_{\mx,T}} \lambda_{\mx,w}(S^\ve_\mx) \big|\nabla p^\ve_{\mx,w} \big|^2\, dx dt +$$ $$\label{uff-17}
+
\varkappa(\ve)\, \int\limits_{\Omega^\ve_{\mx,T}} \lambda_{\mx,n}(S^\ve_\mx) \big|\nabla p^\ve_{\mx,n} \big|^2\, dx dt
\leqslant C_6 \left[1 + \Vert \nabla {\mathsf P}^\ve_\fr \Vert_{L^2(\Omega^\ve_{\fr,T})} \right].$$ In order to complete the derivation of the uniform estimate, we make use of the equality (\[gp6-new\]). We estimate the norm of $\nabla {\mathsf P}^\ve_\fr$ using the Cauchy inequality as follows: $$\label{uff-19}
C_6\, \Vert \nabla {\mathsf P}^\ve_\fr \Vert_{L^2(\Omega^\ve_{\fr,T})} \leqslant
C_6\, \frac{\eta}{2}\, \Vert \nabla {\mathsf P}^\ve_\fr \Vert^2_{L^2(\Omega^\ve_{\fr,T})} +
C_6\, \frac{1}{2\eta},$$ where $\eta > 0$ is an arbitrary number. Moreover, it follows from (\[gp6-new\]) that $$\label{uff-20}
\lambda_{\fr} (S^\ve_\fr) |\nabla {\mathsf P}^\ve_\fr |^2 \leqslant
\lambda_{\fr,n} (S^\ve_\fr) |\nabla p^\ve_{\fr,n}|^2 +
\lambda_{\fr,w} (S^\ve_\fr) |\nabla p^\ve_{\fr,w}|^2.$$ Now (\[uff-19\]) allows us to rewrite (\[uff-17\]) in the form: $$\int\limits_{\Omega^\ve_{\fr,T}} \lambda_{\fr,w}(S^\ve_\fr) \big|\nabla p^\ve_{\fr,w} \big|^2\, dx dt +
\int\limits_{\Omega^\ve_{\fr,T}} \lambda_{\fr,n}(S^\ve_\fr) \big|\nabla p^\ve_{\fr,n} \big|^2\, dx dt +
\varkappa(\ve)\, \int\limits_{\Omega^\ve_{\mx,T}} \lambda_{\mx,w}(S^\ve_\mx) \big|\nabla p^\ve_{\mx,w} \big|^2\, dx dt +$$ $$\label{uff-21}
+
\varkappa(\ve)\, \int\limits_{\Omega^\ve_{\mx,T}} \lambda_{\mx,n}(S^\ve_\mx) \big|\nabla p^\ve_{\mx,n} \big|^2\, dx dt
\leqslant C_6 + C_6\, \frac{\eta}{2}\, \Vert \nabla {\mathsf P}^\ve_\fr \Vert^2_{L^2(\Omega^\ve_{\fr,T})} +
C_6\, \frac{1}{2\eta}.$$ Let us estimate the second term on the right-hand side of (\[uff-21\]). From condition [(A.4)]{} and (\[uff-20\]), we have: $$\begin{split}
C_6\, \frac{\eta}{2}\, \Vert \nabla {\mathsf P}^\ve_\fr \Vert^2_{L^2(\Omega^\ve_{\fr,T})}
&\leqslant
\frac{C_6\eta}{2 L_0}\, \int\limits_{\Omega^\ve_{\fr,T}} \lambda_{\fr} (S^\ve_\fr)\,
\big|\nabla {\mathsf P}^\ve_\fr \big|^2\, dx dt\\
&\leqslant
\frac{C_6\eta}{2 L_0}\, \int\limits_{\Omega^\ve_{\fr,T}}
\left[\lambda_{\fr,w}(S^\ve_\fr) \big|\nabla p^\ve_{\fr,w} \big|^2 +
\lambda_{\fr,n}(S^\ve_\fr) \big|\nabla p^\ve_{\fr,n} \big|^2 \right]\, dx dt.
\end{split}
\label{uff-22}$$ We set $\eta = \frac{L_0}{C_6}$ and, finally, obtain from (\[uff-21\]) the desired inequality (\[hhh8-cor1\]).
Now we turn to the uniform bound (\[beta-un\]). It immediately follows from (\[hhh8-cor1\]) equality (\[gp6-new\]) and the following inequality: $\left|\nabla \beta_\ell(S^\ve_\ell) \right| \leqslant
C\,\left|\nabla \mathfrak{b}_\ell(S^\ve_\ell) \right|$. This completes the proof of Lemma \[lem-uniform\].
\[lem-uniform-ppp\] Let $\langle {\mathsf p}^\ve_w, {\mathsf p}^\ve_n, S^\ve \rangle$ be a solution to [problem (\[debut2\])]{} and $\varkappa(\ve) \eqdef \ve^\theta$ with $\theta \leqslant 2$. Then under assumptions [(A.1)-(A.9)]{} the following uniform in $\ve$ estimate holds true: $$\label{pglob-un}
\left\Vert {\mathsf P}^\ve_\mx \right\Vert_{L^2(\Omega^\ve_{\mx,T})} \leqslant C.$$
[**Proof of Lemma \[lem-uniform-ppp\].**]{} In contrast to the papers [@blm; @yeh2], where the standing assumptions allow to prove the continuity of the global pressure on the interface $\Sigma^\ve_T$, in our case the global pressure is discontinuous on $\Sigma^\ve_T$. So the method which allowed to prove (\[pglob-un\]) by use of the extension operator from the subdomain $\Omega^\ve_\fr$ to the whole $\Omega$ cannot be applied here. To avoid this difficulty we make use of the ideas from [@ene] (see also [@app-2]). Since ${\mathsf P}^\ve_\mx \in L^2(0, T; H^1(\Omega^\ve_\mx))$ and ${\mathsf P}^\ve_\fr -{\mathsf P}_{\Gamma_1} \in L^2(0, T; H^1_{\Gamma_{1}}(\Omega^\ve_\fr))$, then we have: $$\label{suka-1}
\Vert {\mathsf P}^\ve_\mx \Vert_{L^2(\Omega^\ve_{m,T})} \leqslant C\, \left[\ve\, \Vert \nabla {\mathsf P}^\ve_\mx \Vert_{L^2(\Omega^\ve_{\mx,T})}
+ \sqrt{\ve}\, \Vert {\mathsf P}^\ve_\mx \Vert_{L^2(\Sigma^\ve_T)} \right].$$ Then due to the definition of the global pressure ${\mathsf P}^\ve_\mx$, (\[gp1\]), and the interface condition (\[inter-condit\]) written in terms of the global pressure, one obtains the following estimate: $$\Vert {\mathsf P}^\ve_\mx \Vert_{L^2(\Sigma^\ve_T)} \leqslant \Vert {\mathsf P}^\ve_\mx +
{\mathsf G}_{\mx, w}(S^\ve_\mx) \Vert_{L^2(\Sigma^\ve_T)}
+
\Vert {\mathsf G}_{\mx, w}(S^\ve_\mx) \Vert_{L^2(\Sigma^\ve_T)} =
\Vert {\mathsf P}^\ve_\fr + {\mathsf G}_{\fr, w}(S^\ve_\fr) \Vert_{L^2(\Sigma^\ve_T)}
+$$ $$\label{suka-2}
+ \Vert {\mathsf G}_{\mx, w}(S^\ve_\mx) \Vert_{L^2(\Sigma^\ve_T)}
\leqslant
\Vert {\mathsf P}^\ve_\fr \Vert_{L^2(\Sigma^\ve_T)}
+
\Vert {\mathsf G}_{\fr, w}(S^\ve_\fr) \Vert_{L^2(\Sigma^\ve_T)}+
\Vert {\mathsf G}_{\mx, w}(S^\ve_\mx) \Vert_{L^2(\Sigma^\ve_T)}.$$ Now, taking into account the boundedness of ${\mathsf G}_{\ell, w}(S^\ve_\ell)$, the geometry of $\Omega^\ve_{\mx,T}$, (\[suka-2\]), and the estimate: $$\label{suka-3}
\sqrt{\ve}\, \Vert {\mathsf P}^\ve_\fr \Vert_{L^2(\Sigma^\ve_T)} \leqslant
C\, \left[\ve\, \Vert \nabla {\mathsf P}^\ve_\fr \Vert_{L^2(\Omega^\ve_{\fr,T})}
+ \Vert {\mathsf P}^\ve_\fr \Vert_{L^2(\Omega^\ve_{\fr,T})} \right]$$ we obtain $$\label{suka-3-1}
\Vert {\mathsf P}^\ve_\mx \Vert_{L^2(\Omega^\ve_{\mx,T})} \leqslant
C \left( \ve \Vert \nabla {\mathsf P}^\ve_\mx \Vert_{L^2(\Omega^\ve_{\mx,T})} + 1\right)
= C \left( \frac{\ve}{\varkappa^{\frac12}(\ve)} \, \varkappa^{\frac12}(\ve) \Vert \nabla {\mathsf P}^\ve_\mx \Vert_{L^2(\Omega^\ve_{\mx,T})} + 1\right).$$ By using , from we get $$\label{suka-3-2}
\Vert {\mathsf P}^\ve_\mx \Vert_{L^2(\Omega^\ve_{\mx,T})} \leqslant
C \left( {\ve}{\varkappa^{-\frac12}(\ve)} + 1\right),$$ which means that for $\varkappa(\ve) \eqdef \ve^\theta$ with $\theta \leqslant 2$ the desired inequality (\[pglob-un\]) is obtained. Lemma \[lem-uniform-ppp\] is proved.
Let us pass to the uniform bounds for the time derivatives of ${S}^\ve$. In a standard way (see, e.g., [@ba-lp-doubpor]) we get:
\[cor-ps-l5\] Let $\langle {\mathsf p}^\ve_w, {\mathsf p}^\ve_n, S^\ve \rangle$ be a solution to [problem (\[debut2\])]{}. Then under assumptions [(A.1)-(A.9)]{} the following uniform in $\ve$ estimate holds true: $$\label{theta-tt-2-cor}
\left\{\partial_t(\Phi^\ve_\ell S^\ve_\ell) \right\}_{\ve>0} \quad {\rm is \,\, uniformly \,\,
bounded \,\, in} \,\, L^2(0,T;H^{-1}(\Omega^\ve_\ell)),$$ where the functions $\Phi^\ve_\fr, \Phi^\ve_\mx$ are defined in condition [(A.1)]{}.
Compactness and convergence results {#ex-comp-sf}
===================================
The outline of this section is as follows. First, in subection \[ss-sf-1\] we extend the function $S^\ve_\fr$ from the subdomain $\Omega^\ve_\fr$ to the whole $\Omega$ and obtain uniform estimates for the extended function $\widetilde S^\ve_\fr$. Then in subsection \[ss-sf-2\], using the uniform estimates for the function $\widetilde {\mathsf P}^\ve_\fr$ and the corresponding bounds for $\widetilde S^\ve_\fr$, we prove the compactness result for the family $\{\widetilde S^\ve_\fr\}_{\ve>0}$. Finally, in subsection \[con-results\] we formulate the two-scale convergence which will be used in the derivation of the homogenized system.
Extensions of the functions ${\mathsf P}^\ve_\fr$, $S^\ve_\fr$ {#ss-sf-1}
--------------------------------------------------------------
The goal of this subsection is to extend the functions ${\mathsf P}^\ve_\fr$, $S^\ve_\fr$ defined in the subdomain $\Omega^\ve_\fr$ to the whole $\Omega$ and derive the uniform in $\ve$ estimates for the extended functions.
Extension of the function ${\mathsf P}^\ve_\fr$. First, we introduce the extension operator from the subdomain $\Omega^\ve_{\fr}$ to the whole $\Omega$. Taking into account the results of [@acdp] we conclude that there exists a linear continuous extension operator $\Pi^\ve : H^1(\Omega^\ve_{\fr}) \longrightarrow H^1(\Omega)$ such that: [(i)]{} $\Pi^\ve u = u$ in $\Omega^\ve_{\fr}$ and [(ii)]{} for any $u \in H^1(\Omega^\ve_{\fr})$, $$\label{ue-15}
\Vert \Pi^\ve u \Vert_{L^2(\Omega)} \leqslant C\, \Vert u \Vert_{L^2(\Omega^\ve_{\fr})}
\quad {\rm and} \quad
\Vert \nabla (\Pi^\ve u) \Vert_{L^2(\Omega)} \leqslant C\, \Vert \nabla u \Vert_{L^2(\Omega^\ve_{\fr})},$$ where $C$ is a constant that does not depend on $u$ and $\ve$. Now it follows from (\[beta-un\]) and the Dirichlet boundary condition on $\Gamma_{1}$, that $$\label{hhh9-new2}
\left\Vert \nabla (\Pi^\ve P^\ve_\fr) \right\Vert_{L^2(\Omega_{T})} +
\left\Vert \Pi^\ve P^\ve_\fr \right\Vert_{L^2(\Omega_{T})} \leqslant C.$$
[Notational convention.]{} In what follows the extension of any function $f$ will be denoted by $\widetilde f$ instead of $\Pi^\ve f$.
Extension of the function $S^\ve_\fr$. In order to extend $S^\ve_\fr$, following the ideas of [@blm], we make use of the function $\beta_\fr$ defined in (\[upsi-1\]). It is evident that $\beta_\fr$ is a monotone function of $s$. Let us introduce the function: $$\label{ue-23}
\beta^\ve_\fr(x, t) \eqdef \beta_\fr(S^\ve_\fr) = \int\limits_0^{S^\ve_\fr} \alpha_\fr(u)\, du.$$ Then it follows from condition [(A.5)]{} that $$\label{ue-24}
0 \leqslant \beta^\ve_\fr \leqslant \max_{s\in[0, 1]}
\alpha_\fr(s) \quad {\rm a.e.\,\, in} \,\, \Omega^\ve_{\fr,T}.$$ It is also clear from (\[beta-un\]) that $$\label{ue-25}
\Vert \nabla \beta^\ve_\fr \Vert_{L^2(\Omega^\ve_{\fr,T})} \leqslant C.$$ Hence, $$\label{ue-26}
0 \leqslant \widetilde\beta^\ve_\fr \eqdef \Pi^\ve \beta^\ve_\fr \leqslant \max_{s\in[0, 1]}
\alpha_{\fr}(s) \,\, {\rm a.e.\,\, in} \,\, \Omega_T
\quad {\rm and} \quad
\Vert \nabla \widetilde \beta^\ve_\fr \Vert_{L^2(\Omega_{T})} \leqslant C.$$
Now we can extend $S^\ve_\fr$ from $\Omega^\ve_\fr$ to the whole $\Omega$. We denote this extension by $\widetilde S^\ve_\fr$ and define it as follows: $$\label{ue-29}
\widetilde S^\ve_\fr \eqdef\, (\beta_\fr)^{-1}(\widetilde \beta^\ve_\fr).$$ This implies that $$\label{ue-31}
\int\limits_{\Omega_T} \big|\nabla \beta_\fr\big(\,\widetilde S^\ve_\fr\,\big)\big|^2 \,\,dx\, dt =
\int\limits_{\Omega_T}
\big|\nabla \widetilde \beta^\ve_\fr \big|^2 \,\,dx\, dt \leqslant C
\quad {\rm and} \quad
0 \leqslant \widetilde S^\ve_\fr \leqslant 1 \,\, {\rm a.e.\,\, in}
\,\, \Omega_T.$$
Compactness results for the sequence $\{\widetilde S^\ve_\fr\}_{\ve>0}$ {#ss-sf-2}
-----------------------------------------------------------------------
In this subsection we establish the compactness and corresponding convergence results for the sequence $\{\widetilde S^\ve_\fr\}_{\ve>0}$ constructed in the previous section.
\[prop-s\] Under our standing assumptions there is a function $S$ such that $0 \leqslant S \leqslant 1$ in $\Omega_T$ and (up to a subsequence) $$\label{comp-beta-4}
\widetilde S^\ve_\fr \longrightarrow S \,\, {\rm strongly\,\, in}\,\, L^q(\Omega_T)\,\,
{\rm for\, all} \,\, 1 \leqslant q < +\infty.$$
[**Proof of Proposition \[prop-s\].**]{} In the proof of Proposition \[prop-s\] we follow the lines of [@blm; @yeh2]. Namely, first, we establish the modulus of continuity in time for $\widetilde \beta^\ve_\fr$ and then apply the compactness result from [@sim1987]. The derivation of the modulus of continuity in time is based on the lemma obtained earlier in [@yeh2], (see also [@ba-lp-doubpor]).
\[lem-yeh-06\] For $h$ sufficiently small, we have: $$\label{cor-1}
\int\limits_h^T \int\limits_{\Omega^\ve_\fr} \big[S^\ve_\fr(t) - S^\ve_\fr(t - h)\big]\,
\big[\beta^\ve_\fr(t) - \beta^\ve_\fr(t - h) \big]\, dx\, dt \leqslant C\, h
\quad {\rm with} \,\, \beta^\ve_\fr \eqdef \beta_\fr(S^\ve_\fr),$$ where $C$ is a constant that does not depend on $\ve, h$.
\[cor-yeh-06\] For $h$ sufficiently small, we have: $$\label{cor-f-2}
\int\limits_{\Omega^h_T}
\big|\widetilde\beta^\ve_\fr(t) - \widetilde\beta^\ve_\fr(t - h) \big|^2\, dx\, dt
\leqslant C h
\quad {\rm with} \,\, \Omega^h_T \eqdef \Omega \times (h, T).$$
[**Proof of Corollary \[cor-yeh-06\].**]{} First, let us show that the bound (\[cor-1\]) implies: $$\label{cor-f}
\int\limits_h^T \int\limits_{\Omega^\ve_\fr}
\big|\beta^\ve_\fr(t) - \beta^\ve_\fr(t - h) \big|^2\, dx\, dt \leqslant C\, h.$$ In fact, it is clear that due to the definition of the function $\beta_\fr$ and condition [(A.6)]{} we have: $$\left|\beta_\fr(S^\ve_\fr(t)) - \beta_\fr(S^\ve_\fr(t-h)) \right| =
\left|\int\limits^{S^\ve_\fr(t)}_{S^\ve_\fr(t-h)} \alpha_\fr(\xi)\,d\xi \right|
\leqslant \max_{s\in[0,1]} \alpha_\fr(s)\, |S^\ve_\fr(t) - S^\ve_\fr(t - h)|.$$ Then from (\[cor-1\]) we get: $$\int\limits_h^T \int\limits_{\Omega^\ve_\fr}
\big|\beta^\ve_\fr(t) - \beta^\ve_\fr(t - h) \big|^2\, dx\, dt \leqslant C\,
\int\limits_h^T \int\limits_{\Omega^\ve_\fr} \big[S^\ve_\fr(t) - S^\ve_\fr(t - h)\big]\,
\big[\beta^\ve_\fr(t) - \beta^\ve_\fr(t - h) \big]\, dx\, dt \leqslant C\, h$$ and the desired bound (\[cor-f\]) is obtained.
Now using the property (\[ue-15\]) of the extension operator, from (\[cor-f\]) we get (\[cor-f-2\]). This completes the proof of Corollary \[cor-yeh-06\].
Now we are in position to complete the proof of Proposition \[prop-s\]. First, we observe that the sequence $\{\widetilde\beta^\ve_\fr\}_{\ve>0}$ is uniformly bounded in the space $L^2(0, T; H^1_{\Gamma_1}(\Omega))$ and this sequence satisfies (\[cor-f-2\]). Then it follows from [@sim1987] that $\{\widetilde\beta^\ve_\fr\}_{\ve>0}$ is a compact set in the space $L^2(\Omega_T)$ and we have that $\widetilde\beta^\ve_\fr \to \beta^\star$ strongly in $L^2(\Omega_T)$ and due to the uniform boundedness of the function $\widetilde\beta^\ve_\fr$ in the space $L^\infty(\Omega_T)$, $$\label{comp-beta-1+}
\widetilde\beta^\ve_\fr \to \beta^\star \,\, {\rm strongly\,\, in}\,\, L^q(\Omega_T)\,\,
{\rm for\, all} \,\, 1 \leqslant q < +\infty.$$ Now we recall that the extended saturation function $\widetilde S^\ve_\fr$ is defined by $\widetilde S^\ve_\fr \eqdef (\beta_\fr)^{-1}(\widetilde \beta^\ve_\fr)$. We set $$\label{comp-beta-3}
S \eqdef (\beta_\fr)^{-1}(\beta^\star).$$ Then from condition [(A.6)]{} we have: $$\Vert \widetilde S^\ve_\fr - S \Vert_{L^q(\Omega_T)} =
\Vert (\beta_\fr)^{-1}(\widetilde \beta^\ve_\fr) - (\beta_\fr)^{-1}(\beta^\star) \Vert_{L^q(\Omega_T)}
\leqslant C_\beta\, \Vert \widetilde \beta^\ve_\fr - \beta^\star \Vert^{\gamma}_{L^{q\gamma}(\Omega_T)}.$$ This inequality along with (\[comp-beta-1+\]) implies (\[comp-beta-4\]) and Proposition \[prop-s\] is proved.
Two-scale convergence results {#con-results}
-----------------------------
In this subsection, taking into account the compactness results from the previous section, we formulate the convergence results for the sequences $\{\widetilde P^\ve_\fr\}_{\ve>0}$, $\{\widetilde S^\ve_\fr\}_{\ve>0}$. In this paper the homogenization process for the problem is rigorously obtained by using the two-scale approach, see, e.g., [@al]. For the reader’s convenience, let us recall the definition of the two-scale convergence.
\[two\] A sequence of functions $\{v^\ve\}_{\ve>0} \subset L^2(\Omega_T)$ two-scale converges to $v \in L^2(\Omega_T\times Y)$ if $\Vert v^\ve\Vert_{L^2(\Omega_T)} \leqslant C$, and for any test function $\varphi \in C^\infty(\overline{\Omega_T}; C_\#(Y))$ the following relation holds: $$\lim_{\ve\to 0} \int\limits_{\Omega_T} v^\ve(x, t)\, \varphi \left(x, \frac{x}{\ve}, t\right)\, dx\, dt =
\int\limits_{\Omega_T \times Y} v(x, y, t)\,
\varphi(x, y, t) \, dy\, dx\, dt.$$
This convergence is denoted by $v^\ve(x, t) \stackrel {2s}
\rightharpoonup v(x, y, t)$.
Following [@al-95] we also introduce [*the two-scale convergence on periodic surfaces:*]{}
\[two-surf\] A sequence of functions $\{v^\ve\}_{\ve>0} \subset L^2(\Sigma^{\ve}_T)$ two-scale converges to $v \in L^2(\Omega_T; L^2(\Gamma_{\fr\mx}))$ on $\Gamma_{\fr\mx}$ if for any test function $\varphi \in C^\infty(\overline{\Omega_T}; C_\#(Y))$ the following relation holds: $$\lim_{\ve\to 0} \ve \int\limits_{\Sigma^{\ve}_T} v^\ve(x, t)\,
\varphi \left(x, \frac{x}{\ve}, t\right)\, dH^{d-1}(x)\, dt =
\int\limits_{\Omega_T}\int\limits_{\Gamma_{\fr\mx}} v(x, y, t)\,
\varphi(x, y, t) \, dH^{d-1}(y)\, dx\, dt,$$ where, as before $\Sigma^{\ve}_T \eqdef \Gamma_{\fr\mx}^{\ve}\times (0,T)$, and $dH^{d-1}$ is the $(d-1)$-dimensional Hausdorff measure.
This convergence is denoted by $v^\ve(x, t) \stackrel {2s-\Gamma_{\mx\fr}}
\rightharpoonup v(x, y, t)$.
Now we summarize the convergence results for the sequences $\{\widetilde P^\ve_\fr\}_{\ve>0}$ and $\{\widetilde S^\ve_\fr\}_{\ve>0}$. We have:
\[2scale\] For any rate of contrast there exist a function $S$ such that $0 \leqslant S \leqslant 1$ a.e. in $\Omega_T$, $\beta_\fr(S)-\beta_\fr(1) \in L^2(0, T; H^1_{\Gamma_1}(\Omega))$, and functions ${\mathsf P}-{\mathsf P}_{\Gamma_1} \in L^2(0, T; H^1_{\Gamma_1}(\Omega))$, ${\mathsf w}_p, {\mathsf w}_s \in L^2(\Omega_T; H^1_{per}(Y))$ such that up to a subsequence: $$\label{2s-1}
\widetilde S^\ve_\fr(x, t) \longrightarrow S(x, t) \,\,
{\rm strongly\,\, in}\,\, L^q(\Omega_T)\,\, \forall \ 1 \leqslant q < +\infty;$$ $$\label{2s-2}
\widetilde {\mathsf P}^\ve_\fr(x, t) \rightharpoonup {\mathsf P}(x, t) \,\,
{\rm weakly\,\, in}\,\, L^2(0, T; H^1(\Omega));$$ $$\label{2s-3}
\nabla \widetilde {\mathsf P}^\ve_\fr(x, t)\stackrel {2s}
\rightharpoonup \nabla {\mathsf P}(x, t) + \nabla_y {\mathsf w}_p(x, t, y);$$ $$\label{2s-40}
\beta_\fr(\widetilde S^\ve_\fr) \longrightarrow \beta_\fr(S) \,\,
{\rm strongly\,\, in}\,\, L^q(\Omega_T)\,\, \forall \ 1 \leqslant q < +\infty;$$ $$\label{2s-4}
\nabla \beta_\fr(\widetilde S^\ve_\fr) (x, t) \stackrel {2s} \rightharpoonup
\nabla \beta_\fr(S)(x, t) + \nabla_y {\mathsf w}_s(x, t, y);$$ $$\label{2s-5}
\widetilde {\mathsf P}^\ve_\fr(x, t) \stackrel {2s-\Gamma_{\mx\fr}}\rightharpoonup {\mathsf P}(x, t);$$ $$\label{2s-6}
\beta_\fr(\widetilde S^\ve_\fr(x, t)) \stackrel {2s-\Gamma_{\mx\fr}}\rightharpoonup \beta_\fr(S(x, t)) .$$
The [**Proof of Lemma \[2scale\]**]{} is based on the [*a priori estimates*]{} for the functions $\beta_\fr(S^\ve_\fr)$ and ${\mathsf P}^\ve_\fr$ obtained in Section \[uni-est\], the extension results from Subsection \[ss-sf-1\], and Proposition \[prop-s\]. The two-scale convergence results (\[2s-3\]) and (\[2s-4\]) are obtained by arguments similar to those in [@al]. The two-scale convergence (\[2s-5\]) and (\[2s-6\]) can be proved by applying Proposition 2.6 in [@al-95]. Lemma \[2scale\] is proved.
Note also that the notion of strong two-scale convergence on periodic surfaces can be introduced in analogy with the ordinary strong two-scale convergence.
\[str-conv-per-surf\] A sequence $\{v^\ve\}_{\ve>0} \subset L^2(\Sigma^{\ve}_T)$ converges the two-scale strongly to $v \in L^2(\Omega_T; L^2(\Gamma_{\fr\mx}))$ on $\Gamma_{\fr\mx}$ if $$\lim_{\ve\to 0} \ve \int\limits_{\Sigma^{\ve}_T} | v^\ve(x, t)- v\left(x, \frac{x}{\ve}, t\right)|^2\,
dH^{d-1}(x)\, dt = 0.$$
It is easy to verify that the strong two-scale convergence on periodic surfaces implies the two-scale convergence on periodic surfaces with the same limit.
Using the strong convergence (\[2s-40\]) and the boundedness of $\nabla\beta_\fr(\widetilde S^\ve_\fr)$ given in Lemma \[lem-uniform\] we get: $$\begin{aligned}
\ve \| \beta_\fr(\widetilde S^\ve_\fr) - \beta_\fr(S)\|_{L^2(\Sigma^{\ve}_T)}^2 \leqslant C \left[
\ve^2 \| \nabla \beta_\fr(\widetilde S^\ve_\fr) -\nabla \beta_\fr(S)\|_{L^2(\Omega_{\fr,T}^\ve)}^2 +
\|\beta_\fr(\widetilde S^\ve_\fr) - \beta_\fr(S) \|_{L^2(\Omega_{\fr,T}^\ve)}^2
\right],\end{aligned}$$ which tends to zero on a given subsequence as $\ve\to 0$. Therefore, we conclude that the sequence $\{\beta_\fr(\widetilde S^\ve_\fr)\}_{\ve>0}$ converges strongly two-scale on the surface $\Gamma_{\fr\mx}$ to $\beta_\fr(S)$. Furthermore, we have:
Let $\{\beta_\fr(\widetilde S^\ve_\fr)\}$ be a subsequence from Lemma \[2scale\]. Then for any Lipschitz function ${\cal M}\colon [0, \beta_\fr(1)]\to \mathbb{R}$ the sequence $\{{\cal M}(\beta_\fr(\widetilde S^\ve_\fr))\}_{\ve>0}$ converges strongly two-scale on the surface $\Gamma_{\fr\mx}$ to ${\cal M}(\beta_\fr(S))$. \[lema:2sc-surf\]
Lemma \[lema:2sc-surf\] follows immediately from the estimate $$\begin{aligned}
\| {\cal M}(\beta_\fr(\widetilde S^\ve_\fr)) - {\cal M}(\beta_\fr(S))\|_{L^2(\Sigma^{\ve}_T)}^2 \leq
L_{\cal M}^2 \| \beta_\fr(\widetilde S^\ve_\fr) - \beta_\fr(S)\|_{L^2(\Sigma^{\ve}_T)}^2,\end{aligned}$$ where $L_{\cal M}$ is the Lipschitz constant which does not depend on $\ve$.
Dilation operator and convergence results {#dil-oper}
=========================================
It is known that due to the nonlinearities and the strong coupling of the problem, the two-scale convergence does not provide an explicit form for the source terms appearing in the homogenized model, see for instance [@blm; @choq; @yeh2]. To overcome this difficulty the authors make use of the dilation operator. Here we refer to [@adh; @blm; @choq; @yeh2] for the definition and main properties of the dilation operator. Let us also notice that the notion of the dilation operator is closely related to the notion of the unfolding operator. We refer here, e.g., to [@ddg] for the definition and the properties of this operator.
The outline of this section is as follows. First, in subsection \[def-base-dil\] we introduce the definition of the dilation operator and describe its main properties. Then in subsection \[dil-func-prop\] we obtain the equations for the dilated saturation and the global pressure functions and the corresponding uniform estimates. Finally, in subsection \[dil-func-conv\] we consider the convergence results for the dilated functions.
Definition and preliminary results {#def-base-dil}
----------------------------------
\[def-dilop\] For a given $\ve > 0$, we define a dilation operator $\mathfrak{D}^\ve$ mapping measurable functions defined in $\Omega^\ve_{\mx,T}$ to measurable functions defined in $\Omega_T \times Y_\mx$ by $$\label{do-1}
\left(\mathfrak{D}^\ve \varphi \right)(x, y, t) \eqdef
\left\{
\begin{array}[c]{ll}
\varphi\left( c^\ve(x) + \ve\, y, t\right),
\quad {\rm if}\,\, c^\ve(x) + \ve\, y \in \Omega^\ve_\mx; \\[2mm]
0,
\quad {\rm elsewhere}, \\
\end{array}
\right.$$ where $c^\ve(x) \eqdef \ve\, k$ if $x \in \ve\, (Y + k)$ with $k \in \mathbb{Z}^d$ denotes the lattice translation point of the $\ve$-cell domain containing $x$.
The basic properties of the dilation operator are given by the following lemma (see [@adh; @yeh2]).
\[lemma-dilop1\]
Let $\varphi, \psi \in L^2(0, T; H^1(\Omega^\ve_\mx))$. Then we have: $$\label{dilll}
\nabla_y \mathfrak{D}^\ve \varphi = \ve\, \mathfrak{D}^\ve (\nabla_x \varphi)
\quad {\rm a.e.\,\,in}\,\,\Omega_T \times Y_\mx;$$ $$\Vert \mathfrak{D}^\ve \varphi \Vert_{L^2(\Omega_T \times Y_\mx)} =
\Vert \varphi \Vert_{L^2(\Omega^\ve_{\mx,T})}; \,\,
\Vert \nabla_y \mathfrak{D}^\ve \varphi \Vert_{L^2(\Omega_T \times Y_\mx)}
= \ve\,
\Vert \mathfrak{D}^\ve \nabla_x\, \varphi \Vert_{L^2(\Omega_T \times Y_\mx)}
= \ve\,
\Vert \nabla_x\, \varphi \Vert_{L^2(\Omega^\ve_{\mx,T})};$$ $$\left(\mathfrak{D}^\ve \varphi, \mathfrak{D}^\ve \psi
\right)_{L^2(\Omega_T \times Y_\mx)} =
\left(\varphi, \psi \right)_{L^2(\Omega^\ve_{\mx,T})}.$$
The following lemma gives the link between the two-scale and the weak convergence (see, e.g., [@blm]).
\[lemma-dilop\] Let $\{\varphi^\ve\}_{\ve>0}$ be a uniformly bounded sequence in $L^2(\Omega^\ve_{\mx,T})$ satisfying: [(i)]{} $\mathfrak{D}^\ve \varphi^\ve \rightharpoonup \varphi^0$ weakly in $L^2(\Omega_T; L^2_{per}(Y_\mx))$; [(ii)]{} ${\bf 1}^\ve_\mx(x)
\varphi^\ve \stackrel {2s} \rightharpoonup \varphi^* \in L^2(\Omega_T; L^2_{per}(Y_\mx))$. Then $\varphi^0 = \varphi^*$ a.e. in $\Omega_T \times Y_\mx$.
Finally, we also have the following result (see, e.g., [@choq; @yeh2]).
\[lemma-dilop2\] If $\varphi^\ve \in L^2(\Omega^\ve_{\mx,T})$ and ${\bf 1}^\ve_\mx(x) \varphi^\ve \stackrel {2s} \to \varphi \in L^2(\Omega_T; L^2_{per}(Y_\mx))$ then $\mathfrak{D}^\ve \varphi^\ve$ converges to $\varphi$ strongly in $L^2(\Omega_T \times Y_\mx)$. Here $\stackrel {2s} \to$ denotes the strong two-scale convergence. If $\varphi \in L^2(\Omega_T)$ is considered as an element of $L^2(\Omega_T \times Y_\mx)$ constant in $y$, then $\mathfrak{D}^\ve \varphi$ converges strongly to $\varphi$ in $L^2(\Omega_T \times Y_\mx)$.
The dilation operator shows the same properties with respect to the two-scale convergence on periodic surfaces. For a given function $v\in L^2(\Sigma^{\ve}_T)$ and from definition of the dilation operator we have $\mathfrak{D}^\ve(v)\in L^2(\Omega_T; L^2(\Gamma_{\fr\mx}))$ and $$\sqrt{\ve}\| v\|_{L^2(\Sigma^{\ve}_T)} = \| \mathfrak{D}^\ve(v)\|_{L^2(\Omega_T; L^2(\Gamma_{\fr\mx}))}.$$ We have also the following lemma:
If $\{v^\ve\}_{\ve>0} \subset L^2(\Sigma^{\ve}_T)$ is a sequence that converges to $v \in L^2(\Omega_T; L^2(\Gamma_{\fr\mx}))$ in the two-scale sense on $\Gamma_{\fr\mx}$, then the sequence $\{\mathfrak{D}^\ve(v^\ve)\}_{\ve>0}$ converges weakly to the same limit, that is $\mathfrak{D}^\ve(v^\ve) \rightharpoonup v$ in $L^2(\Omega_T; L^2(\Gamma_{\fr\mx}))$. If $\{v^\ve\}_{\ve>0} \subset L^2(\Sigma^{\ve}_T)$ converges strongly to $v \in L^2(\Omega_T; L^2(\Gamma_{\fr\mx}))$ in the two-scale sense on $\Gamma_{\fr\mx}$, then the sequence $\{\mathfrak{D}^\ve(v^\ve)\}_{\ve>0}$ converges strongly to the same limit in $L^2(\Omega_T; L^2(\Gamma_{\fr\mx}))$. \[lemma-dilop-3\]
Due to Lemma \[lema:2sc-surf\], one can apply Lemma \[lemma-dilop-3\] to the sequence $\{ {\cal M}(\beta_\fr(\widetilde S^\ve_\fr))\}_{\ve>0}$ and find a subsequence, such that $$\begin{aligned}
\int\limits_{\Omega_T} \int\limits_{\Gamma_{\fr\mx}} \left|
{\cal M}(\beta_\fr(\mathfrak{D}^\ve(\widetilde S^\ve_\fr))) - {\cal M}(\beta_\fr(S))
\right|^2 \, dH^{d-1}(y)\, dx\, dt \to 0\end{aligned}$$ when $\ve\to 0$, for any Lipschitz function ${\cal M}$. As a consequence we have.
\[corol-dilop\] Let ${\cal M}\colon [0, \beta_\fr(1)]\to \mathbb{R}$ be a Lipschitz function. Then there is a subsequence $\ve = \ve_k$ of the sequence $\{{\cal M}(\beta_\fr(\widetilde S^\ve_\fr))\}_{\ve>0}$, still denoted by $\ve$, such that for a.e. $x\in\Omega$ $$\begin{aligned}
\int\limits_0^T \int\limits_{\Gamma_{\fr\mx}} \left|
{\cal M}(\beta_\fr(\mathfrak{D}^\ve(\widetilde S^\ve_\fr(x,y,t)))) - {\cal M}(\beta_\fr(S(x,y,t)))
\right|^2 \, dH^{d-1}(y) dt \to 0 \quad\text{as }\; \ve\to 0.\end{aligned}$$
The dilated functions $\mathfrak{D}^\ve S^\ve_\mx, \mathfrak{D}^\ve P^\ve_\mx$ and their properties {#dil-func-prop}
---------------------------------------------------------------------------------------------------
In this section we derive the equations for the dilated functions $\mathfrak{D}^\ve S^\ve_\mx,
\mathfrak{D}^\ve P^\ve_\mx$ and obtain the corresponding uniform estimates. In what follows we also make use of the notation: $$\mathfrak{D}^\ve S^\ve_\mx \eqdef s^\ve_\mx \quad {\rm and} \quad
\mathfrak{D}^\ve {\mathsf P}^\ve_\mx \eqdef p^\ve_\mx.$$ The equations for the dilated functions $s^\ve_\mx, p^\ve_\mx$ are given by the following lemma.
\[lem-dil\] For $x \in \Omega$, the functions $s^\ve_\mx, p^\ve_\mx$ satisfy the following system of equations: $$\label{dilo-1}
\Phi_\mx(y) \frac{\partial s^\ve_\mx}{\partial t} - \frac{\varkappa(\ve)}{\ve^2}\,
{\rm div}_y\, \bigg\{K(x, y) \left[
\lambda_{\mx,w} (s^\ve_\mx) \nabla_y p^\ve_\mx +
\nabla_y \beta_\mx(s^\ve_\mx) - \ve\,\lambda_{\mx,w} (s^\ve_\mx) \vec{g}\, \right] \bigg\}
= 0;$$ $$\label{dilo-2}
- \Phi_\mx(y) \frac{\partial s^\ve_\mx}{\partial t}
- \frac{\varkappa(\ve)}{\ve^2}\, {\rm div}_y\, \bigg\{K(x, y)\, \left[ \lambda_{\mx,n} (s^\ve_\mx)
\nabla_y p^\ve_\mx - \nabla_y \beta_\mx(s^\ve_\mx)
- \ve\,\lambda_{\mx,n}(s^\ve_\mx)\,\vec{g}\,
\right] \bigg\} = 0,$$ in the space $L^2(0, T; H^{-1}(Y_\mx))$.
The [**Proof of Lemma \[lem-dil\]**]{} is given in [@blm; @yeh2].
The system of equations (\[dilo-1\])-(\[dilo-2\]) is provided with the following boundary conditions: $$\label{dilo-6}
\beta_\mx(s^\ve_\mx) = {\cal M}(\beta_\fr(\mathfrak{D}^\ve \tilde{S^\ve_\fr}))
\quad {\rm on}\; \Gamma_{\fr\mx}$$ for $(x, t) \in \Omega^\ve_\mx \times (0,T)$, where $${\cal M} \eqdef \beta_{\mx} \circ ( P_{\mx,c})^{-1}\circ P_{\fr,c} \circ (\beta_{\fr})^{-1}.
\label{function:M}$$ Note that under our hypothesis function $ {\cal M}$ is Lipschitz continuous. We also have $$\label{dilo-6a}
{\mathsf p}^\ve_\mx + {\mathsf G}_{\mx, w}(s^\ve_\mx) =
\mathfrak{D}^\ve {\mathsf P}^\ve_\fr + {\mathsf G}_{\fr, w}(\mathfrak{D}^\ve \tilde{S^\ve_\fr})
\,\,
{\rm and} \,\,
{\mathsf p}^\ve_\mx + {\mathsf G}_{\mx,n}(s^\ve_\mx)
= \mathfrak{D}^\ve{\mathsf P}^\ve_\fr + {\mathsf G}_{\fr,n}(\mathfrak{D}^\ve\tilde{S^\ve_\fr})$$ on $\Gamma_{\fr\mx}$ for $(x, t) \in \Omega^\ve_\mx \times (0,T)$.
The initial conditions are $$\label{dilo-7}
s^\ve_\mx(x, y, 0) = (\mathfrak{D}^\ve S^{\bf 0}_\mx)(x, y) \quad {\rm and} \quad
p^\ve_\mx(x, y, 0) = (\mathfrak{D}^\ve {\mathsf P}^{\bf 0}_\mx)(x, y) \quad {\rm in} \,\,
\Omega^\ve_\mx \times Y_\mx,$$ where $S^{\bf 0}_\mx, {\mathsf P}^{\bf 0}_\mx$ are the restrictions to the domain $\Omega^\ve_\mx$ of the functions $S^{\bf 0}, {\mathsf P}^{\bf 0}$ defined in (\[init1-gl+gp\]) and the dilations of the functions defined on the fracture system can be defined in a way similar to one already used for the functions defined on the matrix part.
Now we establish [*a priori*]{} estimates for the functions $s^\ve_\mx, p^\ve_\mx$. They are given by the following lemma.
\[lem-dil-est\] Let $\langle s^\ve_\mx, p^\ve_\mx \rangle$ be a solution to [problem (\[dilo-1\])-(\[dilo-2\])]{}. Then:
- [For any rate of contrast ($\theta > 0$)]{}, $$\label{dilo-8}
0 \leqslant s^\ve_\mx \leqslant 1 \quad {\rm a.e.\,\, in\,\,} \Omega_T \times Y_\mx;$$ $$\label{dilo-12}
\Vert \partial_t(\Phi_\mx\, s^\ve_\mx) \Vert_{L^2(\Omega_T; H^{-1}_{per}(Y_\mx))}
\leqslant C.$$
- [For the high contrast in the critical case ($\theta = 2$)]{}, $$\label{dilo-9}
\Vert \nabla_y \beta_\mx(s^\ve_\mx) \Vert_{L^2(\Omega_T; L^2_{per}(Y_\mx))} \leqslant C;$$ $$\label{dilo-10}
\Vert p^\ve_\mx \Vert_{L^2(\Omega_T; H^1_{per}(Y_\mx))} \leqslant C.$$
- [For the moderate contrast ($0 < \theta < 2$)]{}, $$\label{dilo-10<2}
\ve^{\frac{\theta}{2}-1}\,
\Vert \nabla_y \beta_\mx(s^\ve_\mx) \Vert_{L^2(\Omega_T; L^2_{per}(Y_\mx))}
+
\Vert \beta_\mx(s^\ve_\mx) \Vert_{L^2(\Omega_T; L^2_{per}(Y_\mx))}
\leqslant C;$$ $$\label{dilo-10<2p}
\ve^{\frac{\theta}{2}-1}\,
\Vert \nabla_y p^\ve_\mx \Vert_{L^2(\Omega_T; L^2_{per}(Y_\mx))}
+
\Vert p^\ve_\mx \Vert_{L^2(\Omega_T; L^2_{per}(Y_\mx))}
\leqslant C.$$
[**Proof of Lemma \[lem-dil-est\].**]{} Statement (\[dilo-8\]) is evident. The bound (\[dilo-12\]) with $\Phi_\mx = \Phi_\mx(y)$ follow from Lemma \[cor-ps-l5\] and Lemma \[lemma-dilop1\]. The uniform estimates for $p^\ve_\mx$ in (\[dilo-10\]) and (\[dilo-10<2p\]) follow from the uniform bound (\[pglob-un\]) and Lemma \[lemma-dilop1\]. The uniform estimates for the gradients of the functions $\beta_\mx(s^\ve_\mx)$ and $p^\ve_\mx$ easy follow from the uniform bounds (\[beta-un\]) and Lemma \[lemma-dilop1\]. Lemma \[lem-dil-est\] is proved.
\[rem-supercrit-1\] Notice that in what follows we do not need the uniform estimates for the dilated functions in the case of the very high contrast.
Convergence results for the dilated functions {#dil-func-conv}
---------------------------------------------
In this subsection we establish convergence results which will be used below to obtain the homogenized system. From Lemmas \[lemma-dilop\], \[lem-dil-est\] we get the following convergence results.
\[conv-lemma-dil\] Let $\langle s^\ve_\mx, p^\ve_\mx \rangle$ be a solution to [problem (\[dilo-1\])-(\[dilo-2\]), (\[dilo-6\])-(\[dilo-7\])]{}. Then (up to a subsequence),
- [For the high contrast in the critical case ($\theta = 2$)]{}, $$\label{weak-dil-10}
{\bf 1}^\ve_\mx(x) S^\ve_\mx \stackrel {2s} \rightharpoonup s \in L^2(\Omega_T; L^2_{per}(Y_\mx))
\quad {\rm and} \quad s^\ve_\mx \rightharpoonup s \,\, {\rm weakly\,\, in}\,\, L^2(\Omega_T \times Y_\mx);$$ $$\label{weak-dil-2}
{\bf 1}^\ve_\mx(x)\, {\mathsf P}^\ve_\mx \stackrel {2s} \rightharpoonup p \in L^2(\Omega_T; L^2_{per}(Y_\mx))
\quad {\rm and} \quad
p^\ve_\mx \rightharpoonup p \,\, {\rm weakly\,\, in}\,\, L^2(\Omega_T; H^1(Y_\mx));$$ $$\label{weak-dil-201}
{\bf 1}^\ve_\mx(x)\, \nabla_x {\mathsf P}^\ve_\mx \stackrel {2s} \rightharpoonup \nabla_y p \in L^2(\Omega_T; L^2_{per}(Y_\mx));$$ $$\label{weak-dil-30}
{\bf 1}^\ve_\mx(x) \beta_\mx(S^\ve_\mx) \stackrel {2s} \rightharpoonup \beta^*
\quad {\rm and} \quad
\beta_\mx(s^\ve_\mx) \rightharpoonup \beta^* \,\, {\rm weakly\,\, in}\,\, L^2(\Omega_T; H^1(Y_\mx));$$ $$\label{weak-dil-301}
{\bf 1}^\ve_\mx(x)\, \nabla_x \beta_\mx(S^\ve_\mx) \stackrel {2s} \rightharpoonup
\nabla_y \beta^*.$$
- [For the very high contrast ($\theta > 2$)]{}, $$\label{weak-dil-10very}
{\bf 1}^\ve_\mx(x) S^\ve_\mx \stackrel {2s} \rightharpoonup s \in L^2(\Omega_T; L^2_{per}(Y_\mx)).$$
- [For the moderate contrast ($0 < \theta < 2$)]{}, $$\label{weak-dil-10mod}
{\bf 1}^\ve_\mx(x) S^\ve_\mx \stackrel {2s} \rightharpoonup s \in L^2(\Omega_T; L^2_{per}(Y_\mx))
\quad {\rm and} \quad s^\ve_\mx \rightharpoonup s \,\, {\rm weakly\,\, in}\,\, L^2(\Omega_T \times Y_\mx);$$ $$\label{weak-dil-30mod}
{\bf 1}^\ve_\mx(x) \beta_\mx(S^\ve_\mx) \stackrel {2s} \rightharpoonup \beta^*_1
\quad {\rm and} \quad
{\bf 1}^\ve_\mx(x)\, \ve^{\theta} \nabla \beta_\mx(S^\ve_\mx) \stackrel {2s}
\rightharpoonup \beta_1;$$ $$\label{weak-dil-301mod}
\beta_\mx(s^\ve_\mx) \rightharpoonup \beta^*_1 \,\, {\rm weakly\,\, in}\,\, L^2(\Omega_T; H^1(Y_\mx)).$$
It is important to notice that the convergence results of Lemma \[conv-lemma-dil\] are not sufficient for derivation of the equations for the limit functions $\langle s, p \rangle$ which involve only these functions and not the undefined limits appearing in (\[weak-dil-30\]), (\[weak-dil-301\]), (\[weak-dil-30mod\]) and (\[weak-dil-301mod\]). In order to overcome this difficulty, we introduce the restrictions of the functions $s^\ve_\mx$, $p^\ve_\mx$ which are defined below. For these functions we obtain more estimates which allow us to obtain the desired equations. For this, we make use of the density arguments. Namely, following [@choq] (see also [@ba-lp-doubpor]), we fix $x \in \Omega$ and define the restrictions of $s^\ve_\mx$, $p^\ve_\mx$ to the $\ve$-cell containing the point $x$. These functions are defined in the domain $Y_\mx \times (0, T)$ and are constants in the slow variable $x$. In order to obtain the uniform estimates for the restricted functions (they are similar to the corresponding estimates for ${\mathsf P}^\ve_\fr$, $S^\ve_\fr$ from Section \[uni-est\]) we make use of the estimates (\[dilo-8\])-(\[dilo-10<2p\]).
The scheme is as follows. First, for any natural ${\bf n}$, we introduce the set of points $x \in \Omega$ such that the corresponding norms for the restricted functions are not uniformly bounded in $\ve$. It turns out that the measure of this set is asymptotically small as ${\bf n} \to +\infty$ (see Propositions \[proppi-1\], \[proppi-2\] below). Then taking into account this fact and using the estimates (\[dilo-8\])-(\[dilo-10<2p\]), we, finally, obtain the desired uniform estimates for the restricted functions (see Lemma \[lem-hitro-vyeb\] below).
Let us first denote a periodicity cell $ \ve \big(Y + k\big)$ which contains point $x_0$ by $K^\ve_{x_0}$. For given $x_0$ and $\ve$ the index $k\in \mathbb{Z}^d$ which defines the cell $K^\ve_{x_0}$ can be uniquely defined and therefore we have a well defined function $k(x_0, \ve) \in \mathbb{Z}^d$ such that $K^\ve_{x_0} \eqdef \ve \big(Y + k(x_0, \ve)\big)$. Due to the definition of the dilation operator dilated functions are constant in $x$ on $K^\ve_{x_0}$. The restricted functions are given by: $$\label{new-smx}
s^\ve_{\mx,x_0}(y, t) \eqdef
\left\{
\begin{array}[c]{ll}
s^\ve_\mx,
\quad {\rm for}\,\, x \in K^\ve_{x_0}; \\[2mm]
0, \quad {\rm if \,\, not}; \\
\end{array}
\right.
\quad
p^\ve_{\mx,x_0}(y, t) \eqdef
\left\{
\begin{array}[c]{ll}
p^\ve_\mx,
\quad {\rm for}\,\, x \in K^\ve_{x_0}; \\[2mm]
0, \quad {\rm if \,\, not}. \\
\end{array}
\right.$$ For any $\ve > 0$, the pair $\langle s^\ve_{\mx,x_0}, p^\ve_{\mx,x_0} \rangle$ is a solution to problem (\[dilo-1\])-(\[dilo-2\]), (\[dilo-6\])-(\[dilo-7\]) in $Y_\mx \times (0,T)$.
Now we estimate the measure of the set of points $x \in \Omega$ such that the corresponding norms for the restricted functions are not uniformly bounded in $\ve$. The following result holds true.
\[proppi-1\] Let $f^\ve_\mx = f^\ve_\mx(x, y, t)$ be a dilated function such that $$\label{rev-fff}
\Vert f^\ve_\mx \Vert_{L^2(\Omega_T; L^2_{per}(Y_\mx))} \leqslant C$$ and let $A_{\bf n}$ be a set of points defined by $$\label{rev-fff-5}
A_{\bf n} \eqdef \left\{x \in \Omega\,:\, \liminf_{\ve\to0}\Vert \widehat f^\ve_{\mx,k(x,\ve)}
\Vert_{L^2(0,T; L^2_{per}(Y_\mx))} \geqslant {\bf n} \right\},$$ where for fixed $k\in \mathbb{Z}^d$ $$\label{rev-fff-2}
\widehat f^\ve_{\mx,k}(y, t) \eqdef
\left\{
\begin{array}[c]{ll}
f^\ve_\mx(\ve k, y, t),
\quad {\rm if}\,\, k \,\, {\rm is \,\, such\,\, that}\,\,\,
\ve(Y_\mx + k) \cap \Omega \not=\emptyset; \\[2mm]
0, \quad {\rm if \,\, not}. \\
\end{array}
\right.$$ Then $\sqrt{|A_{\bf n}|} \leqslant {C}/{\bf n}$.
[**Proof of Proposition \[proppi-1\].**]{} Let $f^\ve_\mx = f^\ve_\mx(x, y, t)$ be a function that satisfies (\[rev-fff\]). Then we can write $$\label{rev-fff-1}
\Vert f^\ve_\mx \Vert^2_{L^2(\Omega_T; L^2_{per}(Y_\mx))} =
\sum_{k=1}^{N_\ve} \big|\ve Y_\mx\big|\, \Vert \widehat f^\ve_{\mx,k}
\Vert^2_{L^2(0,T; L^2_{per}(Y_\mx))},$$ where, due to (\[rev-fff\]), we have that $$\label{rev-fff-3}
\sum_{k=1}^{N_\ve} \big|\ve Y_\mx\big|\, \Vert \widehat f^\ve_{\mx,k}
\Vert^2_{L^2(0,T; L^2_{per}(Y_\mx))} \leqslant C^2.$$ Now, for any ${\bf n} \in \mathbb{N}$ and $\ve >0$, let us introduce the set of “bad points” $A^\ve_{\bf n}$ defined by: $$\label{rev-fff-4}
A^\ve_{\bf n} \eqdef \left\{x \in \Omega\,:\, \Vert \widehat f^\ve_{\mx,k(x,\ve)}
\Vert_{L^2(0,T; L^2_{per}(Y_\mx))} > {\bf n} \right\}.$$ Let us estimate the measure of the set $A^\ve_{\bf n}$. It follows from (\[rev-fff-3\]) and (\[rev-fff-4\]) that $$C^2 \geqslant \sum_{k=1}^{N_\ve} \big|\ve Y_\mx\big|\, \Vert \widehat f^\ve_{\mx,k(x,\ve)}
\Vert^2_{L^2(0,T; L^2_{per}(Y_\mx))} \geqslant
\sum_{x \in A^\ve_{\bf n}} \big|\ve Y_\mx\big|\, {\bf n}^2 = {\bf n}^2 \, | A^\ve_{\bf n}|.$$ Therefore, $| A^\ve_{\bf n}|\leqslant C^2/{\bf n}^2$. By definition of limit inferior, for any $\eta >0$ we have $A_{\bf n}\subseteq \liminf_{\ve\to 0} A^\ve_{{\bf n}-\eta}$, (where $\ve$ denotes a [*sequence*]{} of real numbers). Due to the continuity of the measure we get $|A_{\bf n}|\leqslant \liminf_{\ve\to 0} |A^\ve_{{\bf n}-\eta}|\leqslant C^2/({\bf n}-\eta)^2$. Proposition \[proppi-1\] is proved.
We note that previously defined restricted functions are linked to ones appearing in Proposition \[proppi-1\] by the following relation: $$f^\ve_{\mx,x_0}(y,t) = \widehat f^\ve_{\mx,k(x_0,\ve)}(y,t).$$
In a similar way, taking into account the uniform estimate (\[dilo-12\]), we prove the following proposition.
\[proppi-2\] Let $f^\ve_\mx = f^\ve_\mx(x, y, t)$ be a dilated function such that $\Vert f^\ve_\mx \Vert_{L^2(\Omega_T; H^{-1}_{per}(Y_\mx))} \leqslant C$ and let $B_{\bf n}$ be a set of points defined by $$B_{\bf n} \eqdef \left\{x \in \Omega\,:\, \liminf_{\ve\to0}\Vert \widehat f^\ve_{\mx,k(x,\ve)}
\Vert_{L^2(0,T; H^{-1}_{per}(Y_\mx))} \geqslant {\bf n} \right\},$$ where the function $\widehat f^\ve_{\mx,k}$ is defined in (\[rev-fff-2\]). Then $\sqrt{|B_{\bf n}|} \leqslant C/{\bf n}$.
Now let us introduce ${\EuScript A}_{\bf n}$, the set of “bad points” for the functions appearing in (\[dilo-8\])-(\[dilo-10<2p\]). We set: $${\EuScript A}_{1,{\bf n}} \eqdef \left\{x \in \Omega\,:\,
\liminf_{\ve\to0} \ve^{\theta/2 -1}\, \Vert \nabla_y \beta_\mx(s^\ve_{\mx,x})
\Vert_{L^2(0,T; L^2_{per}(Y_\mx))} \geqslant {\bf n} \right\};$$ $${\EuScript A}_{2,{\bf n}} \eqdef \left\{x \in \Omega\,:\,
\liminf_{\ve\to0}\Vert p^\ve_{\mx,x}
\Vert_{L^2(0,T; L^2_{per}(Y_\mx))} \geqslant {\bf n} \right\};$$ $${\EuScript A}_{3,{\bf n}} \eqdef \left\{x \in \Omega\,:\,
\liminf_{\ve\to0} \ve^{\theta/2 -1}\, \Vert \nabla_y p^\ve_{\mx,x}
\Vert_{L^2(0,T; L^2_{per}(Y_\mx))} \geqslant {\bf n} \right\};$$ $${\EuScript A}_{4,{\bf n}} \eqdef \left\{x \in \Omega\,:\,
\liminf_{\ve\to0}\Vert \partial_t(\Phi_\mx\, s^\ve_{\mx,x})
\Vert_{L^2(0, T; H^{-1}_{per}(Y_\mx))} \geqslant {\bf n} \right\}.$$ Here $s^\ve_{\mx,x}, p^\ve_{\mx,x}$ are defined in (\[new-smx\]). Then $$\label{A_n}
{\EuScript A}_{\bf n} \eqdef \bigcup_{\ell=1}^4 {\EuScript A}_{\ell,{\bf n}}$$ and, due to Propositions \[proppi-1\], \[proppi-2\], the measure of this set satisfies the estimate $\sqrt{|{\EuScript A}_{\bf n}|} \leqslant {C}/{\bf n}$.
The following result holds.
\[lem-hitro-vyeb\] Let $s^\ve_{\mx,x_0}, p^\ve_{\mx,x_0}$ be the functions defined in (\[new-smx\]) and $0 <\theta \leqslant 2$. Then for any $x_0 \in \Omega \setminus
{\EuScript A}_{\bf n}$, there is a subsequence $\ve = \ve_k$ still denoted by $\ve$ such that: $$\label{dilo-8-vye}
0 \leqslant s^\ve_{\mx,x_0} \leqslant 1 \quad {\rm a.e.\,\, in\,\,} Y_\mx \times (0,T);$$ $$\label{dilo-9-vye}
\Vert \nabla_y \beta_\mx(s^\ve_{\mx,x_0}) \Vert_{L^2(0, T; L^2_{per}(Y_\mx))} \leqslant C \ve^{1-\theta/2};$$ $$\label{dilo-10-vye}
\Vert p^\ve_{\mx,x_0} \Vert_{L^2(0, T; L^2_{per}(Y_\mx))} \leqslant C;\quad
\Vert \nabla p^\ve_{\mx,x_0} \Vert_{L^2(0, T; L^2_{per}(Y_\mx))} \leqslant C\ve^{1-\theta/2};$$ $$\label{dilo-12-vye}
\Vert \partial_t(\Phi_\mx\, s^\ve_{\mx,x_0}) \Vert_{L^2(0, T; H^{-1}_{per}(Y_\mx))}
\leqslant C,$$ where $C = C({\bf n})$ is constant that does not depend on $x_0$ and $\ve$, and ${\bf n}$ is an arbitrary natural number.
[**Proof of Lemma \[lem-hitro-vyeb\].**]{} First, we notice that the estimate (\[dilo-8-vye\]) follows immediately from (\[dilo-8\]). Let us prove, for example, (\[dilo-9-vye\]). Taking into account that $x_0 \in \Omega\setminus {\EuScript A}_{\bf n}$, from the definition of the set ${\EuScript A}_{1,{\bf n}}$, we obtain immediately the existence of a subsequence on which (\[dilo-9-vye\]) holds with constant $C$ depending only on ${\bf n}$. The estimates (\[dilo-10-vye\])-(\[dilo-12-vye\]) are obtained in a similar way. Lemma \[lem-hitro-vyeb\] is proved.
Using these estimates and applying Lemma 4.2 from [@our-siam], we obtain the following compactness result.
\[prop-vyeb\] Assume $0 < \theta \leqslant 2$. For any $x_0 \in \Omega\setminus {\EuScript A}_{\bf n}$, on a subsequence extracted in Lemma \[lem-hitro-vyeb\], the family $\{s^\ve_{\mx,x_0}\}_{\ve>0}$ is a compact set in the space $L^q(Y_\mx \times (0,T))$ for all $q \in [1, \infty)$. In the case $\theta < 2$ every limit point of the sequence $\{s^\ve_{\mx,x_0}\}_{\ve>0}$ is independent of the fast variable $y$.
Homogenization results {#main-res}
======================
In this section we formulate and prove the main results of the paper corresponding to the homogenized models for various rates of contrast. First, we introduce the notation.
- $S$, $P_w$, $P_n$ denote the homogenized wetting liquid saturation, wetting liquid pressure, and nonwetting liquid pressure, respectively.
- $\Phi^\star = \Phi^\star(x)$ denotes the effective porosity and is given by: $$\label{H-1}
\Phi^\star(x) \eqdef \Phi^{\rm H}_{\fr}(x)\, \frac{|Y_\fr|}{|Y_\mx|},$$ where $\Phi^{\rm H}_{\fr}$ is defined in condition [(A.1)]{} and $|Y_\ell|$ is the measure of the set $Y_\ell$ ($\ell = \fr, \mx$).
- $F^\star_w, F^\star_n$ denote the effective source terms and are given by: $$\label{H-1FFF}
F^\star_w(x, t) \eqdef F^{\rm H}_{w}(x, t)\, \frac{|Y_\fr|}{|Y_\mx|} \quad {\rm and} \quad
F^\star_n(x, t) \eqdef F^{\rm H}_{n}(x, t)\, \frac{|Y_\fr|}{|Y_\mx|},$$ where $$\label{H-1FFF-source}
F^{\rm H}_w(x, t) \eqdef S^I_{\fr, w}\, f_I(x,t) - S\, f_P(x,t) \quad {\rm and}
\quad F^{\rm H}_n(x, t) \eqdef S^I_{\fr, n}\, f_I(x,t) - (1 - S)\, f_P(x,t)$$ and where the functions $S^I_{\fr, w}, S^I_{\fr, n}, f_I, f_P$ are defined in (\[sour1\]), (\[sour2\]), respectively (see also (A.9)).
- $\mathbb{K}^\star = \mathbb{K}^\star(x)$ is the homogenized tensor with the entries $\mathbb{K}^\star_{ij}$ defined by: $$\label{H-2}
\mathbb{K}^{\star}_{ij}(x) \eqdef \frac{1}{|Y_\mx|}\, \int\limits_{Y_\fr}\, K(x, y)\,
\left[\nabla_y \xi_i + \vec e_i \right]\cdot\left[\nabla_y \xi_j + \vec e_j \right]\, dy,$$ where $\xi_j = \xi_j(x, y)$ ($j = 1,\ldots,d$) is a $Y$-periodic solution to the auxiliary cell problem: $$\label{H-20}
\left\{
\begin{array}[c]{ll}
- {\rm div}_y\, \big\{K(x, y) \nabla_y \xi_j\big\} = 0 \quad {\rm in} \,\, Y_{\fr}; \\[2mm]
\nabla_y \xi_j \cdot \vec \nu_y = - \vec e_j \cdot \vec \nu_y
\quad {\rm on} \,\, \Gamma_{\fr\mx};\\[2mm]
y \mapsto \xi_j(y)\quad Y-{\rm periodic}. \\
\end{array}
\right.$$
High contrast media: critical case, ${\boldsymbol \theta}{\bf=2}$ {#hig-crtic-subsec}
-----------------------------------------------------------------
We study the asymptotic behavior of the solution to problem (\[debut2\]), (\[bc3\])-(\[init1\]) in the case $\varkappa(\ve) = \ve^2$ as $\ve \to 0$. In particular, we are going to show that the effective model, expressed in terms of the homogenized phase pressures, reads: $$\label{H-0}
\left\{
\begin{array}[c]{ll}
0 \leqslant S \leqslant 1 \quad {\rm in} \,\, \Omega_T; \\[2mm]
\displaystyle
\Phi^\star(x)\, \frac{\partial S}{\partial t}
- {\rm div}_x\, \bigg\{\mathbb{K}^\star(x)\, \lambda_{\,\fr,w}(S) \big(\nabla P_w - \vec g \big) \bigg\}
=
{\EuScript Q}_w + F^\star_w \quad {\rm in} \,\, \Omega_T;
\\[5mm]
\displaystyle
- \Phi^\star(x)\, \frac{\partial S}{\partial t}
- {\rm div}_x\, \bigg\{\mathbb{K}^\star(x)\, \lambda_{\,\fr,n}(S)
\big(\nabla P_n - \vec g \big) \bigg\}
=
{\EuScript Q}_n + F^\star_n \quad {\rm in} \,\, \Omega_T;\\[5mm]
P_{\fr,c}(S) = P_n - P_w \quad {\rm in} \,\, \Omega_T.
\end{array}
\right.$$
For almost every point $x \in \Omega$ a matrix block $Y_\mx \subset \mathbb{R}^d$ is suspended topologically. The system for flow in a matrix block is given by the so-called imbibition equation: $$\label{H-4}
\left\{
\begin{array}[c]{ll}
\displaystyle
\Phi_m(y) \frac{\partial s}{\partial t}\, - {\rm div}_y \big\{ K(x, y) \nabla_y \beta_m(s)\big\} = 0
\quad {\rm in}\,\, Y_\mx \times \Omega_T; \\[3mm]
s(x, y, t) = {\EuScript P}(S(x,t)) \quad {\rm on}\,\, \Gamma_{\fr\mx} \times\Omega_T; \\[2mm]
s(x, y, 0) = S_m^{\,\bf 0}(x) \quad {\rm in}\,\, Y_\mx \times \Omega.\\
\end{array}
\right.$$ Here $s$ denotes the wetting liquid saturation in the block $Y_\mx$ and the function ${\EuScript P}(S)$ is defined by $$\label{p-callig-moder}
{\EuScript P}(S)\, \eqdef\, (P_{c,m}^{-1} \circ P_{c,f})(S).$$
For any $x \in \Omega$ and $t > 0$, the matrix-fracture sources are given by: $$\label{H-6}
{\EuScript Q}_w \eqdef - \frac{1}{|Y_\mx|}\, \int\limits_{Y_\mx} \Phi_\mx(y)
\frac{\partial s}{\partial t}(x, y, t) \,dy = - {\EuScript Q}_n.$$ The boundary conditions for the effective system (\[H-0\]) are given by: $$\label{H-7}
\left\{
\begin{array}[c]{ll}
P_{w} = P_{n} = 0 \quad {\rm on} \,\, \Gamma_{1} \times (0,T); \\[3mm]
\mathbb{K}^\star\,\lambda_{n}(S) \left(\nabla P_w - \vec g \right) \cdot \vec \nu =
\mathbb{K}^\star\, \lambda_{w}(S) (\nabla P_n - \vec g)\cdot \vec \nu = 0 \quad {\rm on} \,\,
\Gamma_{2} \times (0,T).\\
\end{array}
\right.$$ Finally, the initial conditions read: $$\label{H-9}
P_{w}(x, 0) = {\mathsf p}_{w}^{\bf 0}(x) \quad {\rm and}
\quad
P_{n}(x, 0) = {\mathsf p}_{n}^{\bf 0}(x)
\quad {\rm in} \,\, \Omega.$$
The first main result of the paper is given by the following theorem.
\[t-hom-main\] Let $\varkappa(\ve) = \ve^2$ and let assumptions [(A.1)-(A.9)]{} be fulfilled. Then the solution of the [initial problem (\[debut2\]), (\[inter-condit\])-(\[init1\])]{} converges (up to a subsequence) in the two-scale sense to a weak solution of the [homogenized problem (\[H-0\]), (\[H-4\]), (\[H-6\])-(\[H-9\])]{}.
[**Proof of Theorem \[t-hom-main\].**]{} It is done in several steps. We start our analysis by considering the system (\[debut2\]). The main difficulty with the initial unknown functions ${\mathsf p}^\ve_{w}, {\mathsf p}^\ve_{n}$ in this system is that they do not possess the uniform $H^1$-estimates (see Lemma \[lem-uniform\]). It is important to notice that in the case of two-phase incompressible flow it is possible to find appropriate but rather strong conditions which allow us to deal directly with the phase pressures in a space wider than $H^1$ (see [@yeh2]). To overcome the difficulties appearing due to the absence of the uniform $H^1$-estimates, the authors usually pass to the equivalent formulation of the problem in terms of the global pressure and saturation. In our case it is done in subsection \[gp-relat\] and the corresponding weak formulation of the problem is then given in subsection \[def-weak-sol\]. Then using the convergence and compactness results from subsection \[ex-comp-sf\] we pass to the limit in equations (\[wf-1-gl-GP\]), (\[gf-2-gl-GP\]). This is done in subsections \[passage1\] and \[passage2\]. In order, to pass to the [homogenized phase pressures]{} we make use of the change of the unknown functions. Namely, we set, by the definition of the global pressure: $P_w \eqdef {\mathsf P} + {\mathsf G}_{\fr,w}(S)$ and $P_n \eqdef {\mathsf P} + {\mathsf G}_{\fr,n}(S)$. Then we rewrite the limit system obtained in terms of the global pressure and saturation in terms of the homogenized phase pressures. The passage to the limit in the matrix blocks makes use of the dilation operator (see Section \[dil-oper\] above). Then we pass to the equivalent problem for the imbibition equation and, finally, obtain system (\[H-4\]).
### Passage to the limit in equation (\[wf-1-gl-GP\]) {#passage1}
We set: $$\label{paslim-1}
\varphi_w\left(x, \frac{x}{\ve}, t \right) \eqdef
\varphi(x, t) + \ve\, \zeta\left(x, \frac{x}{\ve}, t \right)
=
\varphi(x, t) + \ve\, \zeta_1(x, t)\, \zeta_2\left( \frac{x}{\ve}\right)
\eqdef \varphi(x, t) + \ve\, \zeta^\ve(x, t),$$ where $\varphi \in {\EuScript D}(\Omega_T), \zeta_1 \in {\EuScript D}(\Omega_T),
\zeta_2 \in C^\infty_{per}(Y)$, and plug the function $\varphi_w$ in (\[wf-1-gl-GP\]). This yields: $$-\int\limits_{\Omega_{T}} {\bf 1}^\ve_\fr(x)\, \Phi^\ve_\fr(x)\, \widetilde S^\ve_\fr
\left[ \frac{\partial \varphi}{\partial t} +
\ve \frac{\partial \zeta^\ve}{\partial t} \right]\,\, dx\, dt
+$$ $$+
\int\limits_{\Omega_{T}} {\bf 1}^\ve_\fr(x)\, K\left(x, \frac{x}{\ve}\right)
\bigg\{\lambda_{\fr,w} (\widetilde S^\ve_\fr)
\left(\nabla \widetilde {\mathsf P}^\ve_\fr - \vec g\right)
+ \nabla \beta_\fr(\widetilde S^\ve_\fr) \bigg\} \cdot \left[\nabla \varphi +
\ve \nabla_x \zeta^\ve + \nabla_y \zeta^\ve \right]\,\, dx\, dt
-$$ $$-\int\limits_{\Omega^\ve_{\mx,T}} \Phi_\mx\left(\frac{x}{\ve}\right) S^\ve_\mx
\left[ \frac{\partial \varphi}{\partial t} +
\ve \frac{\partial \zeta^\ve}{\partial t} \right]\,\, dx\, dt
+$$ $$+
\ve^2\, \int\limits_{\Omega^\ve_{\mx,T}} K\left(x, \frac{x}{\ve}\right) \bigg\{\lambda_{\mx,w} (S^\ve_\mx)
\left(\nabla {\mathsf P}^\ve_\mx - \vec g\right)
+ \nabla \beta_\mx(S^\ve_\mx) \bigg\} \cdot \left[\nabla \varphi +
\ve \nabla_x \zeta^\ve + \nabla_y \zeta^\ve \right]\,\, dx\, dt =$$ $$\label{cr-10+1}
= \int\limits_{\Omega^\ve_{\fr,T}} \big(S^I_{\fr, w} f_I(x,t) - S^\ve_{\fr} f_P(x,t)\big)\,
\left[\varphi + \ve\, \zeta^\ve \right]\, dx dt.$$
Taking into account Lemma \[lem-uniform\] and the convergence results of Lemma \[2scale\] and Lemma \[conv-lemma-dil\], we pass to the limit in (\[cr-10+1\]) as $\ve \to 0$ and obtain the following homogenized equation: $$-\, |Y_\fr|\, \int\limits_{\Omega_T} \Phi^{\rm H}_\fr(x) S(x, t) \frac{\partial \varphi}{\partial t} \,dx\,dt +$$ $$+
\int\limits_{\Omega_T \times Y_\fr} K(x, y) \bigg\{ \lambda_{\,\fr,w}(S) \left[\nabla {\mathsf P}
+ \nabla_y {\mathsf w}_p - \vec g\right] + \nabla \beta_\fr(S) + \nabla_y {\mathsf w}_s \bigg\} \cdot \left[\nabla \varphi + \zeta_1 \nabla_y \zeta_2 \right] \,dy\, dx\, dt
=$$ $$\label{fin-cut-1}
=
\int\limits_{\Omega_T \times Y_\mx} \Phi_\mx(y)\, s(x, y, t) \, \frac{\partial \varphi}{\partial t}
\,\,dy\, dx\, dt + |Y_\fr|\, \int\limits_{\Omega_T} F^{\rm H}_w\, \varphi\, dx\, dt,$$ where $F^{\rm H}_w$ is given by (\[H-1FFF-source\]).
### Passage to the limit in equation (\[gf-2-gl-GP\]) {#passage2}
Equation (\[gf-2-gl-GP\]) is treated in the same way as equation (\[wf-1-gl-GP\]). Taking the test function of the form (\[paslim-1\]) and using the same arguments we can pass to a limit $\ve \to 0$ and obtain the following homogenized equation: $$|Y_\fr|\, \int\limits_{\Omega_T} \Phi^{\rm H}_\fr(x) S(x, t)
\frac{\partial \varphi}{\partial t} \,dx\,dt +$$ $$+
\int\limits_{\Omega_T \times Y_\fr} K(x, y) \bigg\{ \lambda_{\fr,n}(S) \left[\nabla {\mathsf P}
+ \nabla_y {\mathsf w}_p - \vec g\right] - \nabla \beta_\fr(S) - \nabla_y {\mathsf w}_s \bigg\}
\cdot \left[\nabla \varphi + \zeta_1 \nabla_y \zeta_2 \right] \,dy\, dx\, dt
=$$ $$\label{fin-cut-2}
= -
\int\limits_{\Omega_T \times Y_\mx} \Phi_\mx(y)\, s(x, y, t) \, \frac{\partial \varphi}{\partial t}
\,\,dy\, dx\, dt + |Y_\fr|\, \int\limits_{\Omega_T} F^{\rm H}_n\, \varphi\, dx\, dt.$$
### Identification of the corrector functions ${\mathsf w}_p$, ${\mathsf w}_s$ and homogenized equations {#ident-p-beta}
In this section we identify the corrector functions ${\mathsf w}_p$, ${\mathsf w}_s$ appearing in the equations (\[fin-cut-1\]), (\[fin-cut-2\]) and obtain the desired homogenized system (\[H-0\]).
Consider the equations (\[fin-cut-1\]), (\[fin-cut-2\]). Setting $\varphi \equiv 0$, we get: $$\label{hernya-3}
\int\limits_{Y_{\fr}} K(x, y)\,\bigg\{ \lambda_{\,\fr,w}(S)
\big[\nabla P + \nabla_{y} {\mathsf w}_p - \vec{g} \big]
+ \big[\nabla\beta_\fr + \nabla_{y} {\mathsf w}_s\big] \bigg\} \cdot \nabla_y \zeta_2(y)\,dy = 0$$ and $$\label{hernya-4}
\int\limits_{Y_{\fr}} K(x, y)\,\bigg\{ \lambda_{\,\fr,n}(S)
\big[\nabla P + \nabla_{y} {\mathsf w}_p - \vec{g} \big]
- \big[\nabla\beta_\fr + \nabla_{y} {\mathsf w}_s\big] \bigg\} \cdot \nabla_y \zeta_2(y)\,dy = 0.$$ Now adding (\[hernya-3\]) and (\[hernya-4\]) and taking into account condition [(A.4)]{} and the fact that the saturation $S$ does not depend on the fast variable $y$, we obtain: $$\label{hernya-5}
\int\limits_{Y_{\fr}} K(x, y)\,\bigg\{\nabla P + \nabla_{y} {\mathsf w}_p -
\vec{g} \bigg\} \cdot \nabla_y \zeta_2(y)\,dy = 0.$$ Then we proceed in a standard way (see, e.g., [@hor]). Let $\xi_j = \xi_j(x, y)$ ($j = 1,..,d$) be the $Y$-periodic solution of the auxiliary cell problem (\[H-20\]). Then the function ${\mathsf w}_p$ can be represented as: $$\label{1bvp-2}
{\mathsf w}_p(x, y, t) = \sum^d_{j=1} \xi_j(x, y) \left[\frac{\partial\, P}{\partial x_j}(x, t) - g_j\right].$$
Now we turn to the identification of the function ${\mathsf w}_s$. From (\[hernya-3\]) and (\[hernya-5\]), we get: $$\label{wbet-1}
\int\limits_{Y_{\fr}} K(x, y)\, \bigg\{ \nabla\beta_{\fr} + \nabla_{y} {\mathsf w}_s \bigg\} \cdot \nabla_y \zeta_2(y)\,dy = 0.$$ Then as in the previous case, we obtain that $$\label{wbet-2}
{\mathsf w}_s(x, y, t) = \sum^d_{j=1} \xi_j(x, y) \frac{\partial\, \beta_\fr(S)}{\partial x_j}(x, t).$$
### Effective equations in terms of the global pressure and saturation
We start by obtaining the corresponding homogenized equation for the wetting phase. Choosing $\zeta_2 = 0$ in (\[fin-cut-1\]), we get: $$\Phi^\star(x) \, \dfrac{\partial S}{\partial t} -
{\rm div}_x \bigg\{\mathbb{K}^\star(x)\,
\big[\lambda_{\,\fr,w}(S)\,\nabla P + \nabla\beta_{\fr}(S) - \lambda_{\,\fr,w}(S)\, \vec{g}\big]
\bigg\} =$$ $$=
- \frac{1}{|Y_{m}|}\, \int\limits_{Y_{\mx}} \Phi_{m}(y)
\dfrac{\partial s}{\partial t}(x, y, t)\,dy + F^\star_w(x, t),
\label{1newpoint-7:2ok}$$ where the effective porosity $\Phi^\star$, the effective source term $F^\star_w$, and the homogenized permeability tensor $\mathbb{K}^\star$ are defined in , and , respectively.
In a similar way, choosing $\zeta_2 = 0$ in equation (\[fin-cut-2\]), we derive the second homogenized equation: $$- \Phi^\star(x) \, \dfrac{\partial S}{\partial t} -
{\rm div}_x \bigg\{\mathbb{K}^\star(x)\,
\big[\lambda_{\,\fr,n}(S)\,\nabla P + \nabla\beta_{\fr}(S) - \lambda_{\,\fr,n}(S)\, \vec{g}\big]
\bigg\} =$$ $$=
\frac{1}{|Y_{m}|}\, \int\limits_{Y_{\mx}} \Phi_{m}(y)
\dfrac{\partial s}{\partial t}(x, y, t)\,dy + F^\star_n(x, t),
\label{hernya-18ok}$$ where $F^\star_n$ denotes the effective source term defined in .
### Effective equations in terms of the phase pressures {#hom-eq-in-ph-pres}
Let us introduce now the functions that is naturally to call the [homogenized phase pressures]{}. Namely, we set, by the definition: $$\label{phase-pres}
P_{w} \eqdef {\mathsf P} + {\mathsf G}_{\fr,w}(S) \quad {\rm and}
\quad P_n \eqdef {\mathsf P} + {\mathsf G}_{\fr,n}(S),$$ where the functions ${\mathsf G}_{\fr,w}, {\mathsf G}_{\fr,n}$ are defined in Section \[gp-relat\]. Then it easy to see that the homogenized equations can be rewritten as follows: $$\label{final-h-press}
\left\{
\begin{array}[c]{ll}
\displaystyle
\Phi^\star(x)\, \frac{\partial S}{\partial t}
- {\rm div}_x\, \bigg\{\mathbb{K}^\star(x)\, \lambda_{\,\fr,w}(S) \big(\nabla P_w - \vec g \big) \bigg\}
=
{\EuScript Q}_w + F^\star_w \quad {\rm in} \,\, \Omega_T;
\\[5mm]
\displaystyle
- \Phi^\star(x)\, \frac{\partial S}{\partial t}
- {\rm div}_x\, \bigg\{\mathbb{K}^\star(x)\, \lambda_{\,\fr,n}(S)
\big(\nabla P_n - \vec g \big) \bigg\}
=
{\EuScript Q}_n + F^\star_n \quad {\rm in} \,\, \Omega_T;\\[5mm]
P_c(S) = P_n - P_w \quad {\rm in} \,\, \Omega_T.
\end{array}
\right.$$
### Flow equations in the matrix block {#identif-l-theta}
In this section, following the ideas of the papers [@ba-lp-doubpor; @blm; @choq], we obtain the system (\[H-4\]) describing the behavior of the function $s$ which is involved in the definition of the matrix-fracture source term. Briefly, we pass to the limit in the equations for the dilated functions for fixed $k$ and then by density arguments the limit equations will be obtained. We recall that the equations for the dilated functions are already obtained in Lemma \[lem-dil\] from subsection \[dil-func-prop\]. Namely, for almost all $x \in \Omega$, the functions $s^\ve_\mx, p^\ve_\mx$ satisfy the following variational problem:
[for all $\phi_n,\phi_w \in L^2(0,T;H^1_0(Y_{\mx}))\cap H^1(0,T; L^2(Y_{\mx}))$, $\phi_n(T)=\phi_w(T) =0$,]{} $$- \int\limits_0^T\int\limits_{Y_{\mx}} \Phi_\mx(y) s^\ve_\mx \frac{\partial \phi_w}{\partial t}\, dy
- \int\limits_0^T\int\limits_{Y_{\mx}} \Phi_\mx(y) (\mathfrak{D}^\ve S^{\bf 0}_\mx)
\frac{\partial \phi_w}{\partial t}(0)\, dy$$ $$\label{ebio-ident-1v-w}
+ \int\limits_0^T\int\limits_{Y_{\mx}} \bigg\{K(x, y) \left[
\lambda_{\mx,w} (s^\ve_\mx) \nabla_y p^\ve_\mx +
\nabla_y \beta_\mx(s^\ve_\mx) - \ve\,\lambda_{\mx,w} (s^\ve_\mx) \vec{g}\, \right] \bigg\}
\cdot \nabla_y \phi_w \, dy = 0;$$ $$\int\limits_0^T\int\limits_{Y_{\mx}} \Phi_\mx(y) s^\ve_\mx \frac{\partial \phi_n }{\partial t}\, dy
+ \int\limits_0^T\int\limits_{Y_{\mx}} \Phi_\mx(y) (\mathfrak{D}^\ve S^{\bf 0}_\mx)
\frac{\partial \phi_n }{\partial t}\, dy$$ $$\label{ebio-ident-1v-n}
+\int\limits_0^T\int\limits_{Y_{\mx}} \bigg\{K(x, y) \left[ \lambda_{\mx,n} (s^\ve_\mx) \nabla_y p^\ve_\mx -
\nabla_y \beta_\mx(s^\ve_\mx) - \ve\, \lambda_{\mx,n}(s^\ve_\mx)\,\vec{g}\,
\right] \bigg\} \cdot \nabla_y \phi_n \, dy= 0$$ [with the boundary conditions (\[dilo-6\])]{}.
The uniform estimates for the functions $s^\ve_\mx, p^\ve_\mx$ imply the convergence results of $\langle s^\ve_\mx, p^\ve_\mx \rangle$ to $\langle s, p \rangle$ in a weak sense (see Lemma \[conv-lemma-dil\]). Thus, the limit behavior of the dilated functions $s^\ve_\mx$, $p^\ve_\mx$ is determined. However, the convergence results of Lemma \[conv-lemma-dil\] are not sufficient for derivation of the equations for the limit functions $\langle s, p \rangle$. To overcome this difficulty, in Section \[dil-func-conv\] we pass to the restrictions of the functions $s^\ve_\mx, p^\ve_\mx$ to $K^\ve_{x_0}$ defined in (\[new-smx\]). Evidently, they are constants in the slow variable $x$. Introducing the set ${\EuScript A}_{\bf n}$ of “bad points” (\[A\_n\]), by Lemma \[lem-hitro-vyeb\] we have the uniform estimates (\[dilo-8-vye\])-(\[dilo-12-vye\]) for the functions $s^\ve_{\mx,x_0}, p^\ve_{\mx,x_0}$. For any $\ve > 0$, the pair of functions $\langle s^\ve_{\mx,x_0}, p^\ve_{\mx,x_0} \rangle$ is a solution to problem (\[ebio-ident-1v-w\]), (\[ebio-ident-1v-n\]) in $Y_\mx \times (0,T)$. Moreover, the compactness result, i.e., Proposition \[prop-vyeb\] is established for the family $\{s^\ve_{\mx,x_0}\}_{\ve>0}$. Having established these results, we are in position to complete the proof of Theorem \[t-hom-main\]. The uniform estimates for the functions $s^\ve_{\mx,x_0}, p^\ve_{\mx,x_0}$ from Lemma \[lem-hitro-vyeb\] and the compactness result formulated in Proposition \[prop-vyeb\] allow us to obtain the following convergence results.
\[conv-k-dil\] Let $x_0\in \Omega\setminus {\EuScript A}_{\bf n}$. There exist functions $s_{x_0}, p_{x_0}$, and $\beta_\mx(s_{x_0})$ such that up to a subsequence: $$\label{2s-k-1}
s^\ve_{\mx,x_0} \to s_{x_0} \,\, {\rm strongly\,\, in}\,\, L^q(Y_\mx \times (0,T))\,\,
\forall \ 1 \leqslant q < +\infty;$$ $$\label{2s-k-2}
p^\ve_{\mx,x_0} \rightharpoonup p_{x_0} \,\, {\rm weakly\,\, in}\,\, L^2(0, T; H^1_{per}(Y_\mx));$$ $$\label{2s-k-40}
\beta_\mx(s^\ve_{\mx,x_0}) \rightharpoonup \beta_\mx(s_{x_0}) \,\, {\rm weakly\,\, in}\,\,
L^2(0, T; H^1_{per}(Y_\mx));$$ $$\label{2s-k-4}
\beta_\mx(s^\ve_{\mx,x_0}) \to \beta_\mx(s_{x_0}) \,\, {\rm strongly\,\, in}\,\,
L^q(Y_\mx \times (0,T))\,\, \forall \ 1 \leqslant q < +\infty;$$ $$\label{2s-k-4-front}
\beta_\mx(s^\ve_{\mx,x_0})\big|_{\Gamma_{\mx\fr}} \to \beta_\mx(s_{x_0})\big|_{\Gamma_{\mx\fr}}
\,\, {\rm weakly\,\, in}\,\, L^2(0, T; L^2(\Gamma_{\mx\fr}));$$ $$\label{2s-k-5-front}
p^\ve_{\mx,x_0}\big|_{\Gamma_{\mx\fr}} \to p_{x_0}\big|_{\Gamma_{\mx\fr}} \,\,
{\rm weakly\,\, in}\,\, L^2(0, T; L^2(\Gamma_{\mx\fr})).$$
As in subsections \[passage1\], \[passage2\] we can easily pass to the limit in (\[ebio-ident-1v-w\]) and (\[ebio-ident-1v-n\]). We get the following system of equations: $$\begin{aligned}
\label{ebio-ident-3v-w}
- \int\limits_0^T\int\limits_{Y_{\mx}} \Phi_\mx(y) &s_{x_0} \frac{\partial \phi_w}{\partial t}\, dy
- \int\limits_0^T\int\limits_{Y_{\mx}} \Phi_\mx(y) S^{\bf 0}_{x_0}
\frac{\partial \phi_w}{\partial t}(0)\, dy \\
&+ \int\limits_0^T\int\limits_{Y_{\mx}} \bigg\{K(x, y) \big[
\lambda_{\mx,w} (s_{x_0}) \nabla_y p_{x_0} +
\nabla_y \beta_\mx(s_{x_0})\big] \bigg\}\cdot \nabla_y \phi_w \, dy
= 0; \nonumber \\
\int\limits_0^T\int\limits_{Y_{\mx}} \Phi_\mx(y) & s_{x_0} \frac{\partial \phi_n }{\partial t}\, dy
+ \int\limits_0^T\int\limits_{Y_{\mx}} \Phi_\mx(y) S^{\bf 0}_{x_0}
\frac{\partial \phi_n }{\partial t}\, dy
\label{ebio-ident-3v-n} \\
&+\int\limits_0^T\int\limits_{Y_{\mx}} \bigg\{K(x, y)
\big[ \lambda_{\mx,n} (s_{x_0}) \nabla_y p_{x_0} -
\nabla_y \beta_\mx(s_{x_0}) \big] \bigg\} \cdot \nabla_y \phi_n \, dy= 0,\nonumber\end{aligned}$$ where we have used the fact that $\mathfrak{D}^\ve S^{\bf 0}_{x_0}\to S^{\bf 0}_{x_0}$ strongly in $L^2(Y_m)$ for almost all $x_0\in \Omega$.
Now we turn to the boundary condition for $s_{x_0}$ on $\Gamma_{\mx\fr}$. From Corollary \[corol-dilop\] we know that for a.e. $x_0$, $$\begin{aligned}
{\cal M}(\beta_\fr(\mathfrak{D}^\ve(\widetilde S^\ve_\fr(x_0,\cdot,\cdot)))) \to
{\cal M}(\beta_\fr(S(x_0,\cdot,\cdot)))\quad\text{strongly in }\; L^2(0,T; L^2(\Gamma_{\mx\fr})),\end{aligned}$$ where ${\cal M}$ is the function given in (\[function:M\]). Therefore, for a.e. $x_0 \in \Omega\setminus {\EuScript A}_{\bf n}$, from (\[dilo-6\]) and (\[2s-k-4-front\]) we have: $$\begin{aligned}
\beta_\mx(s_{x_0})\big|_{\Gamma_{\mx\fr}} =
{\cal M}(\beta_\fr(S(x_0,\cdot,\cdot)))\big|_{\Gamma_{\mx\fr}},\end{aligned}$$ or, equivalently $$\begin{aligned}
s_{x_0} = {\EuScript P}(S(x_0,\cdot)) \quad \text{on }\; \Gamma_{\mx\fr}\times (0,T).
\label{BC:matrix:eff}\end{aligned}$$ Note also that it follows from (\[dilo-6\]) that the convergence in (\[2s-k-4-front\]) is strong in $L^2(0, T; L^2(\Gamma_{\mx\fr}))$. This, together with convergence (\[2s-k-5-front\]) and Lipschitz continuity of the functions ${\mathsf G}_{\ell,g} $, ${\mathsf G}_{\ell,w} $, enables us to pass to the limit in the boundary condition for dilated global pressure (\[dilo-6a\]) using the two-scale convergence on $\Gamma_{\fr\mx}$, and get $$\begin{aligned}
p_{x_0} + {\mathsf G}_{\mx, w}(s_{x_0})={\mathsf P}(x_0, \cdot)+ {\mathsf G}_{\fr, w}(S(x_0,\cdot))
\quad \text{on }\; \Gamma_{\mx\fr}\times (0,T).
\label{BC:matrix:ess:P}\end{aligned}$$ In the same way we also get $$\begin{aligned}
p_{x_0} + {\mathsf G}_{\mx, g}(s_{x_0})={\mathsf P}(x_0, \cdot)+ {\mathsf G}_{\fr, g}(S(x_0,\cdot))
\quad \text{on }\; \Gamma_{\mx\fr}\times (0,T).
\label{BC:matrix:ess:P-g}\end{aligned}$$
Thus the system which is satisfied by the limit $\langle s_{x_0}, p_{x_0}\rangle$ is obtained for any $x_0 \in \Omega\setminus {\EuScript A}_{\bf n}$. Now it remains to make the link between the functions $s_{x_0}, p_{x_0}$ and the limits $s, p$ of the sequences $\{s^\ve_\mx\}_{\ve>0}$, $\{p^\ve_\mx\}_{\ve>0}$. First, we observe that the convergent subsequence in Lemma \[conv-k-dil\] depends on point $x_0 \not\in {\EuScript A}_{\bf n}$. To avoid this difficulty we will prove (see subsection \[identif-l-theta\] below) that the problem (\[ebio-ident-3v-w\]), (\[ebio-ident-3v-n\]) with the corresponding boundary conditions (\[BC:matrix:eff\]), (\[BC:matrix:ess:P\]), and (\[BC:matrix:ess:P-g\]) has a unique weak solution. Then the convergence results from Lemma \[conv-k-dil\] hold for the whole sequences, as $\ve\to0$. Since the functions $s_{x_0} = s(x_{x_0}, y, t), p_{x_0} = p(x_{x_0}, y, t)$ satisfy (\[ebio-ident-3v-w\])-(\[BC:matrix:ess:P-g\]) for almost all $x_0 \in \Omega \setminus {\EuScript A}_{\bf n}$, we conclude that $s$ and $p$ are weak solution of the following system of equations: $$\label{ebio-ident-4}
\left\{
\begin{array}[c]{ll}
0 \leqslant s \leqslant 1 \quad {\rm in} \,\, Y_\mx\times\Omega_T; \\[2mm]
\displaystyle
\Phi_\mx(y) \frac{\partial s}{\partial t} -
{\rm div}_y\, \bigg\{K(x, y) \left[
\lambda_{\mx,w} (s) \nabla_y p + \nabla_y \beta_\mx(s) \right] \bigg\}
= 0 \quad {\rm in} \,\, Y_\mx\times\Omega_T; \\[5mm]
\displaystyle
- \Phi_\mx(y) \frac{\partial s}{\partial t}
-
{\rm div}_y\, \bigg\{K(x, y) \left[ \lambda_{\mx,n} (s)
\nabla_y p - \nabla_y \beta_\mx(s) \right] \bigg\} = 0
\quad {\rm in} \,\, Y_\mx\times\Omega_T.\\[2mm]
\end{array}
\right.$$
The system is completed by the corresponding boundary and initial conditions: $$\label{last-bc-s}
\left\{
\begin{array}[c]{ll}
{\mathsf P} + {\mathsf G}_{\fr,w}(S) =
p + {\mathsf G}_{\mx,w}(s) \quad {\rm on}\,\, \Gamma_{\fr\mx} \times\Omega_T;\\[4mm]
{\mathsf P} + {\mathsf G}_{\fr,n}(S) =
p + {\mathsf G}_{\mx,n}(s) \quad {\rm on}\,\, \Gamma_{\fr\mx} \times\Omega_T; \\[4mm]
s(x, y, t) = {\EuScript P}(S(x,t)) \quad {\rm on}\,\, \Gamma_{\fr\mx} \times\Omega_T ,\\[2mm]
s(x, y, 0) = S^{\,\bf 0}(x) \quad {\rm in}\,\, Y_\mx \times \Omega.
\end{array}
\right.$$
Thus, we have identified $s$ and $p$ for $x\in \Omega \setminus {\EuScript A}_{\bf n}$. Since by Propositions \[proppi-1\], \[proppi-2\], the measure of the set ${\EuScript A}_{\bf n}$ goes to zero as ${\bf n}\to \infty$ we conclude that our conclusion holds a.e. in $\Omega$.
The proof of the uniqueness of the solution to problem (\[ebio-ident-4\]) will be done as follows. First, we reduce the system (\[ebio-ident-4\]) to a boundary value problem for the so-called [imbibition equation]{} and then make use of the uniqueness result from [@vazquez]. Equation (\[H-4\])$_1$ is the well known [generalized porous medium equation]{} (see, e.g., [@vazquez]).
\[lemma-der-imb-eqn\] Let $s = s(x, y, t)$ be the solution of the cell problem -. Then $s$ satisfies the boundary value problem (\[H-4\]).
[**Proof of Lemma \[lemma-der-imb-eqn\].**]{} First we observe that it follows from the boundary conditions (\[last-bc-s\]) that the function $s$ does not depend on $y$ on $\Gamma_{\fr\mx} \times\Omega_T$. Then the global pressure $p$ does not depend on $y$ on $\Gamma_{\fr\mx} \times\Omega_T$. Namely, we can write that $$\label{H-9-then-2}
p(x, y, t) = p_{\Gamma}(x, t) \quad {\rm on} \,\, \Gamma_{\fr\mx} \times\Omega_T.$$ By summing the two equations in we get: $$\label{H-18+}
- {\rm div}\, \big\{K(x, y)\, \lambda_{\mx}(s) \nabla_y p \big\} = 0 \quad {\rm in}\,\,
Y_\mx \times \Omega.$$ Then multiplying the equation by $(p - p_{\Gamma})$ and integrating over $Y_\mx \times \Omega_T$, using and conditions [(A.2)]{}, [(A.4)]{} we obtain: $$0 = \int\limits_{Y_\mx \times \Omega_T} K(x, y)\,
\lambda_{\mx}(s) \nabla_y p \cdot \nabla_y p \,\, dx\,dy\,dt \geqslant
k_{\rm min}\,\, L_0\, \int\limits_{Y_\mx \times \Omega_T} |\nabla_y p|^2 \,\, dx\,dy\,dt,$$ which gives $\nabla_y p = 0 \quad {\rm a.e.\,\, in} \,\, Y_\mx \times \Omega_T$. This result allows us to reduce the two equations in the problem to only one, as announced in (\[H-4\]). This completes the proof of Lemma \[lemma-der-imb-eqn\].
Now we turn to the proof of the uniqueness of the solution to (\[H-4\]). This proof is given in Theorem 5.3 from [@vazquez]. For reader’s convenience we discuss it briefly in the following lemma.
\[uniqueness-imbib\] Under our standing assumptions, there is a unique weak solution to problem (\[H-4\]).
[**Proof of Lemma \[uniqueness-imbib\].**]{} First, we introduce the weak formulation of problem (\[H-4\]). Omitting, for the sake of simplicity, the dependence on the slow variable $x$, we have:
[*for any function $\eta \in C^1(\overline{Y}_{\mx,T})$, where $Y_{\mx,T} \eqdef Y_\mx \times (0,T)$, vanishing on $\Gamma_{\fr\mx}$ and such that $\eta(x, T) = 0$*]{}, $$\label{uni-im-1}
\int\limits_{Y_{\mx,T}} \left\{K(y)\, \nabla_y \beta_\mx(s) \cdot \nabla_y \eta -
s\, \frac{\partial \eta}{\partial t} \right\}\, dy\, dt =
\int\limits_{Y_\mx} S_m^{\bf 0}(x)\, \eta(y, 0)\, dy,$$ .
Suppose now that we have two solutions $s_1$ and $s_2$ satisfying (\[uni-im-1\]). Then denoting $W_i \eqdef \beta_\mx(s_i)$, from (\[uni-im-1\]), we have: $$\label{uni-im-2}
\int\limits_{Y_{\mx,T}} \left\{K(y)\, \nabla_y (W_1 - W_2) \cdot \nabla_y \eta -
(s_1 - s_2)\, \frac{\partial \eta}{\partial t} \right\}\, dy\, dt = 0$$ for all $\eta$. Then we use as a special test function $\eta = \widehat\eta$, see e.g. [@vazquez]: $$\label{uni-im-3}
\widehat\eta \eqdef \left\{
\begin{array}[c]{ll}
\displaystyle
\int_t^T \big[W_1(x, \varsigma) - W_2(x, \varsigma)\big]\, d\varsigma \quad
{\rm if}\,\, 0 < t < T;\\[5mm]
0, \quad {\rm if}\,\, t \geqslant T.
\end{array}
\right.$$ Then, plugging (\[uni-im-3\]) in (\[uni-im-2\]), we get: $$\label{uni-im-4}
\int\limits_{Y_{\mx,T}} (s_1 - s_2)\, (W_1 - W_2) \, dy\, dt +
\int\limits_{Y_{\mx,T}} K(y)\, \nabla_y (W_1 - W_2) \cdot
\left\{ \int\limits_t^T \nabla_y (W_1 - W_2)\, d\varsigma \right\}\, dy\, dt = 0.$$ Integration of the last term leads to the following relation: $$\label{uni-im-5}
\int\limits_{Y_{\mx,T}} (s_1 - s_2)\, \big(\beta_\mx(s_1) - \beta_\mx(s_2)\big) \, dy\, dt +
\frac12\, \int\limits_{Y_\mx} K(y)\,
\left[\int\limits_0^T \nabla_y \big(\beta_\mx(s_1) - \beta_\mx(s_2)\big)\, d\varsigma \right]^2\, dy = 0.$$ Due to the monotonicity of the function $\beta_\mx$, the first term in (\[uni-im-2\]) is non-negative. Therefore we can conclude that $s_1 = s_2$ a.e. in $Y_{\mx,T}$. Lemma \[uniqueness-imbib\] is proved.
This completes the proof of Theorem \[t-hom-main\].
Very high contrast media: ${\boldsymbol \theta}{\bf>2}$ {#very-hig-crtic-subsec}
-------------------------------------------------------
We study the asymptotic behavior of the solution to problem (\[debut2\]), (\[bc3\])-(\[init1\]) as $\ve \to 0$ in the case $\varkappa(\ve) = \ve^\theta$ with $\theta > 2$. In particular, we are going to show that the effective model reads: $$\label{H-0-veryhigh}
\left\{
\begin{array}[c]{ll}
0 \leqslant S \leqslant 1 \quad {\rm in} \,\, \Omega_T; \\[2mm]
\displaystyle
\Phi^\star(x)\, \frac{\partial S}{\partial t}
- {\rm div}_x\, \bigg\{\mathbb{K}^\star(x)\, \lambda_{\,\fr,w}(S) \big(\nabla P_w - \vec g \big) \bigg\}
=
F^\star_w \quad {\rm in} \,\, \Omega_T;
\\[5mm]
\displaystyle
- \Phi^\star(x)\, \frac{\partial S}{\partial t}
- {\rm div}_x\, \bigg\{\mathbb{K}^\star(x)\, \lambda_{\,\fr,n}(S)
\big(\nabla P_n - \vec g \big) \bigg\}
=
F^\star_n \quad {\rm in} \,\, \Omega_T;\\[5mm]
P_{\fr,c}(S) = P_n - P_w \quad {\rm in} \,\, \Omega_T,
\end{array}
\right.$$ where the effective porosity $\Phi^\star$, the effective source terms $F^\star_w, F^\star_n$, and the homogenized permeability tensor $\mathbb{K}^\star$ in are defined in , and , respectively.
The boundary and the initial conditions for the system (\[H-0-veryhigh\]) are given by (\[H-7\]), (\[H-9\]).
We see that in this case the matrix blocks have a vanishing, as $\ve \to 0$, influence on the effective flow. This means that in the case of very high contrast, the medium behaves as a perforated one.
The second main result of the paper is as follows.
\[t-hom-main>2\] Let $\varkappa(\ve) = \ve^\theta$ with $\theta > 2$ and let assumptions [(A.1)-(A.9)]{} be fulfilled. Then the solution of the [initial problem (\[debut2\]), (\[inter-condit\])-(\[init1\])]{} converges (up to a subsequence) in the two-scale sense to a weak solution of the [homogenized problem , (\[H-7\]), (\[H-9\])]{}.
[**Proof of Theorem \[t-hom-main>2\].**]{} Let $\theta>2$. In the proof of Theorem \[t-hom-main>2\] we follow the lines of the proof of Theorem \[t-hom-main\]. Namely, arguing as in Sections \[passage1\], \[passage2\], \[ident-p-beta\], we obtain the homogenized equations (\[1newpoint-7:2ok\]), (\[hernya-18ok\]).
Now we want to show that in the case of the very high contrast, the model behaves as in the perforated media, i.e., the matrix blocks are totally impermeable and the additional matrix-source term equals zero. As in the paper [@yeh2], we prove the following result.
\[nosource\] The following equation holds true: $$\Phi_\mx(y)\, \frac{\partial s}{\partial t}(x, y, t) = 0 \quad
{\rm in} \,\,\, Y_\mx \times \Omega_T.$$
[**Proof of Lemma \[nosource\].**]{} Let us define the function: $${\EuScript F}^{\,\ve}(x,t) \eqdef \ve^{\frac{\theta}{2}}\,K^\ve(x)\, \bigg\{\lambda_{\mx,w} (S^\ve_\mx)
\left(\nabla {\mathsf P}^\ve_\mx - \vec g\right)
+ \nabla \beta_\mx(S^\ve_\mx) \bigg\}.$$ By using the estimate and the assumptions [(A.2)]{}, [(A.4)]{}, we get the uniform bound: $$\label{Fve}
\left\Vert {\EuScript F}^{\,\ve} \right\Vert_{L^2(\Omega^\ve_{\mx,T})} \leqslant C.$$
Let define a function: $$\varphi_w\left(x, \frac{x}{\ve}, t \right) \in {\EuScript D}(\Omega_T; C^\infty_{per}(Y))
\quad {\rm such\,\, that\,\,} \varphi_w = 0 \,\,\, {\rm for}\,\,\, y\in Y_\fr.$$ Plugging $\varphi_w$ in (\[wf-1-gl-GP\]), and taking into account condition [(A.9)]{}, we get: $$\label{eqn}
-\int\limits_{\Omega_{T}} {\bf 1}^\ve_\mx(x)\, \Phi^\ve_\mx(x)\, S^\ve_\mx\,
\frac{\partial \varphi_w}{\partial t} \, dx\, dt
+
\ve^{\frac{\theta}{2}}\, \int\limits_{\Omega^\ve_{\mx,T}} {\EuScript F}^{\,\ve} \, \nabla_x \varphi_w \, dx\, dt
+
\ve^{\frac{\theta}{2}-1}\, \int\limits_{\Omega^\ve_{\mx,T}} {\EuScript F}^{\,\ve} \, \nabla_y \varphi_w \, dx\, dt = 0,$$ We pass to the two-scale limit in using . We obtain: $$\label{eqn-1}
\int\limits_{\Omega_{T}\times Y_\mx} \Phi_\mx (y)\, s(x, y, t)\,
\frac{\partial \varphi_w}{\partial t} \, dx\, dt\, dy = 0.$$ This completes the proof of Lemma \[nosource\].
Finally, from the equations (\[1newpoint-7:2ok\]), (\[hernya-18ok\]) in view of Lemma \[nosource\], arguing as in subsection \[hom-eq-in-ph-pres\], we arrive to the desired system (\[H-0-veryhigh\]). This completes the proof of Theorem \[t-hom-main>2\].
Moderate contrast media: ${\bf 0<}{\boldsymbol \theta}{\bf<2}$ {#moder-case-subsec}
--------------------------------------------------------------
We study the asymptotic behavior of the solution to problem (\[debut2\]) as $\ve \to 0$ in the case $\varkappa(\ve) = \ve^\theta$ with $0 < \theta < 2$. In particular, we are going to show that the effective model reads: $$\label{H-0-moderate}
\left\{
\begin{array}[c]{ll}
0 \leqslant S \leqslant 1 \quad {\rm in} \,\, \Omega_T; \\[2mm]
\displaystyle
\frac{\partial }{\partial t}
\left[\Phi^\star(x)\, S + \widehat \Phi_\mx\, {\EuScript P}(S) \right]
- {\rm div}_x\, \bigg\{\mathbb{K}^\star(x)\, \lambda_{\,\fr,w}(S) \big(\nabla P_w - \vec g \big) \bigg\}
= F^\star_w \quad {\rm in} \,\, \Omega_T;
\\[5mm]
\displaystyle
- \frac{\partial }{\partial t}
\left[\Phi^\star(x)\, S + \widehat \Phi_\mx\, {\EuScript P}(S) \right]
- {\rm div}_x\, \bigg\{\mathbb{K}^\star(x)\, \lambda_{\,\fr,n}(S)
\big(\nabla P_n - \vec g \big) \bigg\}
= F^\star_n \quad {\rm in} \,\, \Omega_T;\\[5mm]
P_{\fr,c}(S) = P_n - P_w \quad {\rm in} \,\, \Omega_T,
\end{array}
\right.$$ where the effective porosity $\Phi^\star$, the effective source terms $F^\star_w, F^\star_n$, and the homogenized permeability tensor $\mathbb{K}^\star$ in are defined in , and , respectively.
The boundary conditions and the initial conditions for the system (\[H-0-moderate\]) are given by (\[H-7\]), (\[H-9\]).
In this case we observe a complete decoupling between microscale and macroscale, which is not the case for the critical scaling $\theta = 2$.
The third main result of the paper is as follows.
\[t-hom-main<2\] Let $\varkappa(\ve) = \ve^\theta$ with $0 < \theta < 2$ and let assumptions [(A.1)-(A.9)]{} be fulfilled. Then the solution of the [initial problem (\[debut2\]), (\[inter-condit\])-(\[init1\])]{} converges (up to a subsequence) in the two-scale sense to a weak solution of the [homogenized problem , (\[H-7\]), (\[H-9\])]{}.
Let $0 < \theta < 2$. In the proof of Theorem \[t-hom-main<2\] we follow the lines of the proof of Theorem \[t-hom-main\]. Namely, arguing as in Sections \[passage1\], \[passage2\], \[ident-p-beta\], we obtain the homogenized equations (\[1newpoint-7:2ok\]), (\[hernya-18ok\]). Namely, in the case of the moderate contrast we have: $$\label{H-0-moderate-proof}
\left\{
\begin{array}[c]{ll}
0 \leqslant S \leqslant 1 \quad {\rm in} \,\, \Omega_T; \\[2mm]
\displaystyle
\Phi^\star(x)\, \frac{\partial S}{\partial t}
- {\rm div}_x\, \bigg\{\mathbb{K}^\star(x)\, \lambda_{\,\fr,w}(S) \big(\nabla P_w - \vec g \big) \bigg\}
=
\widehat{{\EuScript Q}}_w + F^\star_w \quad {\rm in} \,\, \Omega_T;
\\[5mm]
\displaystyle
- \Phi^\star(x)\, \frac{\partial S}{\partial t}
- {\rm div}_x\, \bigg\{\mathbb{K}^\star(x)\, \lambda_{\,\fr,n}(S)
\big(\nabla P_n - \vec g \big) \bigg\}
=
\widehat{{\EuScript Q}}_n + F^\star_n \quad {\rm in} \,\, \Omega_T;\\[5mm]
P_{\fr,c}(S) = P_n - P_w \quad {\rm in} \,\, \Omega_T,
\end{array}
\right.$$ where the effective porosity $\Phi^\star$, the effective source terms $F^\star_w, F^\star_n$, and the homogenized permeability tensor $\mathbb{K}^\star$ in are defined in , and , respectively. For any $x \in \Omega$ and $t > 0$, the matrix-fracture source terms $\widehat{{\EuScript Q}}_w$, $\widehat{{\EuScript Q}}_n$ in have the form: $$\widehat{{\EuScript Q}}_w \eqdef - \widehat \Phi_\mx\,\frac{\partial s}{\partial t}(x, t)
= - \widehat{{\EuScript Q}}_n \quad {\rm with} \,\,\,
\widehat \Phi_\mx \eqdef \frac{1}{|Y_\mx|}\, \int\limits_{Y_\mx} \Phi_\mx(y)\,dy.$$
In order to complete the proof of Theorem \[t-hom-main<2\], we have to identify the saturation function $s$ appearing on the right-hand side of equations in (\[H-0-moderate-proof\]). The following result holds true:
\[our-m3as-version\] Let $s$ be the weak limit of $\{\mathfrak{D}^\ve S_{\mx}^\ve\}_{\ve>0}$ and $S$ is the saturation function defined in (\[2s-1\]). Then $$\label{+0-3}
s = {\EuScript P}(S) \quad {\rm a.e.\,\, in}\,\, \Omega_T
\quad {\rm with} \,\, {\EuScript P}(S) = (P_{c,m}^{-1} \circ P_{c,f})(S).$$
[**Proof of Lemma \[our-m3as-version\]**]{}. Applying Lemma \[lem-hitro-vyeb\] and Proposition \[prop-vyeb\] we conclude that, for any $x_0 \in \Omega\setminus {\EuScript A}_{\bf n}$, $$\begin{aligned}
\beta_\mx(s^\ve_{\mx,x_0}) &\to \beta_\mx(s_{x_0})\quad \text{weakly in }\;
L^2(0,T; H^1(Y_{\mx})),\\
s^\ve_{\mx,x_0} &\to s_{x_0} \quad \text{a.e. in }\; Y_{\mx}\times (0,T),\end{aligned}$$ and the limit $s_{x_0}$ does not depend of the fast variable $y$. Due to continuity of the trace operator we also have: $$\begin{aligned}
\beta_\mx(s^\ve_{\mx,x_0})\big|_{\Gamma_{\mx\fr}} \to
\beta_\mx(s_{x_0})\big|_{\Gamma_{\mx\fr}}
\quad \text{weakly in }\; L^2(0,T; L^2(\Gamma_{\mx\fr})).\end{aligned}$$ On the other hand we know that, for a.e. $x_0\in \Omega$, $$\begin{aligned}
{\cal M}(\beta_\fr(\mathfrak{D}^\ve(\widetilde S^\ve_\fr(x_0,\cdot,\cdot))))
\big|_{\Gamma_{\mx\fr}} = \beta_{\mx}(s^\ve_{\mx,x_0})
\big|_{\Gamma_{\mx\fr}} \quad {\rm with} \,\,\,
{\cal M} \eqdef \beta_{\mx} \circ ( P_{\mx,c})^{-1}\circ P_{\fr,c} \circ (\beta_{\fr})^{-1}\end{aligned}$$ a.e. on $\Gamma_{\mx\fr}\times (0,T)$. For a.e. $x_0\in \Omega$, from Corollary \[corol-dilop\] we have that $$\begin{aligned}
{\cal M}(\beta_\fr(\mathfrak{D}^\ve(\widetilde S^\ve_\fr(x_0,\cdot,\cdot)))) \to
{\cal M}(\beta_\fr(S(x_0,\cdot,\cdot)))\quad\text{strongly in }\;
L^2(0,T; L^2(\Gamma_{\mx\fr}))\end{aligned}$$ and therefore, for a.e. $x_0 \in \Omega\setminus {\EuScript A}_{\bf n}$, $$\begin{aligned}
\beta_\mx(s_{x_0})\big|_{\Gamma_{\mx\fr}} =
{\cal M}(\beta_\fr(S(x_0,\cdot,\cdot)))\big|_{\Gamma_{\mx\fr}}.\end{aligned}$$ Since these functions are independent of $y$ we have that $\beta_\mx(s_{x_0}) =
{\cal M}(\beta_\fr(S(x_0,\cdot))$ in $L^2(0,T)$, or, equivalently, $s_{x_0} = {\EuScript P}(S(x_0,\cdot))$. Now, for a chosen $x_0 \in \Omega\setminus {\EuScript A}_{\bf n}$, we can find a subsequence such that $$\begin{aligned}
s^\ve_{\mx,x_0} &\to {\EuScript P}(S(x_0,\cdot)) \quad \text{a.e. in }\;
Y_{\mx}\times (0,T).\end{aligned}$$ Since the limit is uniquely defined by the limit $S$ of the sequence $\mathfrak{D}^\ve(\widetilde S^\ve_\fr)$ we conclude that the whole sequence converge to the same limit (that is the whole subsequence for which $\mathfrak{D}^\ve(\widetilde S^\ve_\fr)$ converges). Now we can repeat our procedure for almost any $x_0 \in \Omega\setminus {\EuScript A}_{\bf n}$ and conclude that $s = {\EuScript P}(S)$ a.e. in $(\Omega\setminus {\EuScript A}_{\bf n})\times (0,T)$. Thanks to Propositions \[proppi-1\], \[proppi-2\], the measure of the set ${\EuScript A}_{\bf n}$ goes to zero as ${\bf n}\to\infty$ and the desired equality (\[+0-3\]) is proved.
Now we complete easily the proof of Theorem \[t-hom-main<2\]. Taking into account (\[+0-3\]) we can rewrite (\[H-0-moderate-proof\]) and thereby obtain (\[H-0-moderate\]). Theorem \[t-hom-main<2\] is proved.
Acknowledgments {#acknowledgments .unnumbered}
===============
The work of M. Jurak and A. Vrbaški was funded by Croatian science foundation project no 3955. The work of L. Pankratov was funded by the RScF research project N 15-11-00015. This work was partially supported by ISIFoR (http://www.carnot-isifor.eu/) France. The supports are gratefully acknowledged. Most of the work on this paper was done when M. Jurak and L. Pankratov were visiting the Applied Mathematics Laboratory of the University of Pau & CNRS.
[99]{}
E. Acerbi, V. Chiadò Piat, G. Dal Maso, D. Percival, An extension theorem from connected sets, and homogenization in general periodic domains, [*J. Nonlinear Analysis*]{}, [**18**]{}, (1992), 481–496.
L. Ait Mahiout, B. Amaziane, A. Mokrane, L. Pankratov, Homogenization of immiscible compressible two-phase flow in double porosity media, [*Electronic Journal of Differential Equations*]{}, [**2016**]{}:52 (2016), 1–28.
G. Allaire, Homogenization and two-scale convergence, [*SIAM J. Math. Anal.*]{}, [**28**]{}, (1992), 1482–1518.
G. Allaire, A. Damlamian, U. Hornung, Two-scale convergence on periodic surfaces and applications, In Proceedings of the International Conference on Mathematical Modelling of Flow through Porous Media, A. Bourgeat et al. eds., pp. 15–25, World Scientific Pub., Singapore (1996).
B. Amaziane, S. Antontsev, L. Pankratov, A. Piatnitski, Homogenization of immiscible compressible two-phase flow in porous media : application to gas migration in a nuclear waste repository, [*SIAM MMS*]{}, [**8**]{}, (2010), 2023–2047.
B. Amaziane, L. Pankratov, Homogenization of a model for water-gas flow through double-porosity media, [*Math. Meth. Appl. Sci.*]{}, [**39**]{}, (2016), 425–451.
B. Amaziane, L. Pankratov, A. Piatnitski, Homogenization of a class of quasilinear elliptic equations in high-contrast fissured media, [*Proc. of the Royal Society of Edinburgh*]{}, [**136A**]{} (2006), 1131–1155.
B. Amaziane, L. Pankratov, A. Piatnitski, Nonlinear flow through double porosity media in variable exponent Sobolev spaces, [*Nonlinear Analysis: Real World Applications*]{}, [**10**]{} (2009), 2521–2530.
B. Amaziane, L. Pankratov, A. Piatnitski, Homogenization of immiscible compressible two-phase flow in highly heterogeneous porous media with discontinuous capillary pressures, [*Math. Models and Methods in Applied Sci.*]{}, [**24**]{}, (2014), 1421–1451.
B. Amaziane, L. Pankratov, A. Piatnitski, The existence of weak solutions to immiscible compressible two-phase flow in porous media: the case of fields with different rock-types, [*DCDS B*]{} [**18**]{}, 5 (2013), 1217–1251.
B. Amaziane, L. Pankratov, V. Rybalko, On the homogenization of some double porosity models with periodic thin structures, [*Appl. Anal.*]{}, [**88**]{} (2009), 1469–1492.
S. N. Antontsev, A. V. Kazhikhov, V. N. Monakhov, [*Boundary Value Problems in Mechanics of Nonhomogeneous Fluids*]{}, North-Holland, Amsterdam, 1990.
T. Arbogast, J. Douglas, U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, [*SIAM J. Math. Anal.*]{}, [**21**]{}, (1990), 823–826.
G. I. Barenblatt, Yu. P. Zheltov, I. N. Kochina, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, [*J. Appl. Math. Mech.*]{}, [**24**]{} (1960), 1286–1303.
J. Bear, C.F. Tsang, G. de Marsily, [*Flow and Contaminant Transport in Fractured Rock*]{}, Academic Press Inc, London, 1993.
A. Bourgeat, S. Luckhaus, A. Mikelić, Convergence of the homogenization process for a double-porosity model of immicible two-phase flow, [*SIAM J. Math. Anal.*]{}, [**27**]{}, (1996), 1520–1543.
A. Bourgeat, A. Mikelic, A. Piatnitski, Modèle de double porosité aléatoire, [*C. R. Acad. Sci. Paris, Sér. 1*]{}, [**327**]{} (1998), 99–104.
A. Bourgeat, M. Goncharenko, M. Panfilov, L. Pankratov, A general double porosity model, [*C. R. Acad. Sci. Paris, Série IIb*]{}, [**327**]{} (1999), 1245–1250.
A. Bourgeat, G. Chechkin, A. Piatnitski, Singular double porosity model, [*Appl. Anal.*]{}, [**82**]{} (2003), 103–116.
A. Braides, V. Chiadò Piat, A. Piatnitski, Homogenization of discrete high-contrast energies, [*SIAM J. Math. Anal.*]{}, [**47**]{} (2015), 3064–3091.
G. Chavent, J. Jaffré, [*Mathematical Models and Finite Elements for Reservoir Simulation*]{}, North-Holland, Amsterdam, 1986.
Z. Chen, G. Huan, Y. Ma, [*Computational Methods for Multiphase Flows in Porous Media*]{}, SIAM, Philadelphia, 2006.
C. Choquet, Derivation of the double porosity model of a compressible miscible displacement in naturally fractured reservoirs, [*Appl. Anal.*]{}, [**83**]{}, (2004), 477–500.
C. Choquet, L. Pankratov, Homogenization of a class of quasilinear elliptic equations with non-standard growth in high-contrast media, [*Proc. of the Royal Society of Edinburgh*]{}, [**140**]{} (2010), 495–539.
D. Cioranescu, A. Damlamian, G. Griso, Periodic unfolding and homogenization, [*C. R. Acad. Sci. Paris, Ser. I*]{}, [**335**]{}, (2002), 99–104.
G. W. Clark, R.E. Showalter, Two-scale convergence of a model for flow in a partially fissured medium, [*Electronic Journal of Differential Equations*]{}, [**1999**]{} (1999), 1–20.
H. I. Ene, D. Polisevski, Model of diffusion in partially fissured media, [*Z. angew. Math. Phys.*]{}, [**53**]{}, (2002), 1052–1059.
R. Helmig, [*Multiphase Flow and Transport Processes in the Subsurface*]{}, Springer, Berlin, 1997.
P. Henning, M. Ohlberger, B. Schweizer, Homogenization of the degenerate two-phase flow equations, [*Math. Models Methods Appl. Sci.*]{}, [**23**]{}, (2013), 2323–2352.
U. Hornung, [*Homogenization and porous media*]{}, Springer-Verlag, New York, 1997.
M. Jurak, L. Pankratov, A. Vrbaški, A fully homogenized model for incompressible two-phase flow in double porosity media, [*Applicable Analysis*]{} (2015), DOI: 10.1080/00036811.2015.1031221.
V. A. Marchenko, E. Ya. Khruslov, [*Homogenization of Partial Differential Equations*]{}, Boston, Birkhäuser, 2006.
M. Panfilov, [*Macroscale Models of Flow Through Highly Heterogeneous Porous Media*]{}, Dordrecht-Boston-London, Kluwer Academic Publishers, 2000.
L. Pankratov, V. Rybalko, Asymptotic analysis of a double porosity model with thin fissures, [*Mat. Sbornik*]{}, [**194**]{}, (2003), 121–146.
G. Sandrakov, Homogenization of parabolic equations with contrast coefficients, [*Izvestiya: Mathematics*]{}, [**63**]{}, (1999), 1015–1061.
R. P. Shaw, Gas Generation and Migration in Deep Geological Radioactive Waste Repositories. Geological Society, 2015.
J. Simon, Compact sets in the space $L^p(0,t; B)$, [*Ann. Mat. Pura Appl., IV. Ser.*]{}, [**146**]{}, (1987), 65–96.
J. L. Vázquez, [*The porous medium equation*]{}, Oxford University Press Inc., New York, 2007.
T. D. Van Golf-Racht, [*Fundamentals of Fractured Reservoir Engineering*]{}, Elsevier Scientific Pulishing Company, Amsterdam, 1982.
L. M. Yeh, Homogenization of two-phase flow in fractured media, [*Math. Models and Methods in Applied Sci.*]{}, [**16**]{}, (2006), 1627–1651.
|
---
abstract: 'We show that if a coloring of the plane has the properties that any two points at distance one are colored differently and the plane is partitioned into uniformly colored triangles (under certain conditions), then it requires at least seven colors. This is also true for a coloring using uniformly colored convex polygons if it has a point bordering at least four polygons.'
author:
- 'Michael N. Manta'
title: Triangle colorings require at least seven colors
---
Introduction
============
The chromatic number of the plane is the minimum number of colors needed to color all points of ${\mathbb R}^2$ in such a way that any two points at distance one are colored differently. See Soifer [@Soi] for a detailed history of this problem. Although the problem was proposed by Nelson in the 1950s, the best lower bound has only recently been proven by De Grey [@Gre], who showed that at least five colors are needed to color the plane. On the other hand, the plane can be colored with seven colors by tiling the plane with hexagons or with squares (see Figure \[fig:tilings\]), so the chromatic number of the plane is 5, 6, or 7.
It is natural to ask what the chromatic number is for colorings with certain restrictions. For instance, before De Grey’s work, Falconer [@Fal] proved that five colors are needed for colorings that only use measurable sets. Woodall [@Woo] and Townsend [@Tow] proved that certain map-type colorings require at least six colors. See Soifer [@Soi Chapters 8, 9, and 24] for more details about the chromatic number of restricted colorings.
Because the upper bound for the general problem follows from restricted colorings that only use polygons, it is natural to ask for the chromatic number of such colorings. Coulson [@Cou] proved that all polygon colorings require at least six colors, under the condition that all polygons are convex and have an area greater than some positive constant. Furthermore, Moustakis [@Mou] showed that at least seven colors are needed if the plane is tiled with congruent squares under certain conditions, which matches the square coloring in Figure \[fig:sqtile\].
[0.5]{}
(-1.25,2.165063509461097) – (-2.5,0.) – (-1.25,-2.1650635094610973) – (1.25,-2.165063509461098) – (2.5,0.) – (1.25,2.1650635094610955) – cycle; (1.25,2.1650635094610955) – (2.5,0.) – (5.,0.) – (6.25,2.1650635094610924) – (5.,4.330127018922189) – (2.5,4.330127018922193) – cycle; (-1.25,2.165063509461097) – (1.25,2.1650635094610955) – (2.5,4.330127018922195) – (1.25,6.495190528383295) – (-1.25,6.495190528383297) – (-2.5,4.330127018922199) – cycle; (-1.25,2.165063509461097) – (-2.5,4.330127018922199) – (-5.,4.330127018922201) – (-6.25,2.1650635094611) – (-5.,0.) – (-2.5,0.) – cycle; (-2.5,0.) – (-5.,0.) – (-6.25,-2.1650635094611097) – (-5.,-4.330127018922215) – (-2.5,-4.330127018922212) – (-1.25,-2.1650635094611066) – cycle; (-1.25,-2.1650635094610973) – (-2.5,-4.330127018922212) – (-1.25,-6.495190528383315) – (1.25,-6.4951905283833025) – (2.5,-4.330127018922188) – (1.25,-2.1650635094610853) – cycle; (1.25,-2.165063509461098) – (2.5,-4.330127018922188) – (5.,-4.330127018922157) – (6.25,-2.1650635094610364) – (5.,0.) – (2.5,0.) – cycle;
(5,4.330127018922212) – (6.25,6.495190528383295); (6.25,2.165063509461098) – (7.5,2.165063509461098); (6.25,-2.165063509461098) – (7.5,-2.165063509461098); (5,-4.330127018922212) – (6.25,-6.495190528383295);
(-5,4.330127018922212) – (-6.25,6.495190528383295); (-6.25,2.165063509461098) – (-7.5,2.165063509461098); (-6.25,-2.165063509461098) – (-7.5,-2.165063509461098); (-5,-4.330127018922212) – (-6.25,-6.495190528383295);
(-1.15,1.4) node\[anchor=north west\] [1]{}; (2.55,3.625) node\[anchor=north west\] [2]{}; (2.55,-0.65) node\[anchor=north west\] [3]{}; (-1.25,-2.8) node\[anchor=north west\] [4]{}; (-4.95,-0.65) node\[anchor=north west\] [5]{}; (-4.95,3.625) node\[anchor=north west\] [6]{}; (-1.2,5.75) node\[anchor=north west\] [7]{};
(6.25,5.75) node\[anchor=north west\] [6]{}; (6.25,1.4) node\[anchor=north west\] [5]{}; (6.25,-2.8) node\[anchor=north west\] [7]{};
(-8.65,5.75) node\[anchor=north west\] [4]{}; (-8.65,1.4) node\[anchor=north west\] [2]{}; (-8.65,-2.8) node\[anchor=north west\] [3]{};
(-4.95,-4.35) node\[anchor=north west\] [7]{};
(2.45,-4.35) node\[anchor=north west\] [6]{};
(-4.95,7.025) node\[anchor=north west\] [3]{};
(2.45,7.025) node\[anchor=north west\] [4]{};
[0.5]{}
(-5.,0.) – (0.,0.) – (0.,5.) – (-5.,5.) – cycle; (0.,0.) – (5.,0.) – (5.,5.) – (0.,5.) – cycle; (-10.,0.) – (-5.,0.) – (-5.,5.) – (-10.,5.) – cycle; (-15.,0.) – (-10.,0.) – (-10.,5.) – (-15.,5.) – cycle; (5.,0.) – (10.,0.) – (10.,5.) – (5.,5.) – cycle; (-20.,0.) – (-15.,0.) – (-15.,5.) – (-20.,5.) – cycle; (-25.,0.) – (-20.,0.) – (-20.,5.) – (-25.,5.) – cycle; (-22.5,0.) – (-22.5,-5.) – (-17.5,-5.) – (-17.5,0.) – cycle; (-17.5,0.) – (-17.5,-5.) – (-12.5,-5.) – (-12.5,0.) – cycle; (-12.5,0.) – (-12.5,-5.) – (-7.5,-5.) – (-7.5,0.) – cycle; (-7.5,0.) – (-7.5,-5.) – (-2.5,-5.) – (-2.5,0.) – cycle; (-2.5,0.) – (-2.5,-5.) – (2.5,-5.) – (2.5,0.) – cycle; (2.5,0.) – (2.5,-5.) – (7.5,-5.) – (7.5,0.) – cycle; (7.5,0.) – (7.5,-5.) – (12.5,-5.) – (12.5,0.) – cycle; (-20.,-5.) – (-25.,-5.) – (-25.,-10.) – (-20.,-10.) – cycle; (-15.,-5.) – (-20.,-5.) – (-20.,-10.) – (-15.,-10.) – cycle; (-10.,-5.) – (-15.,-5.) – (-15.,-10.) – (-10.,-10.) – cycle; (-5.,-5.) – (-10.,-5.) – (-10.,-10.) – (-5.,-10.) – cycle; (0.,-5.) – (-5.,-5.) – (-5.,-10.) – (0.,-10.) – cycle; (5.,-5.) – (0.,-5.) – (0.,-10.) – (5.,-10.) – cycle; (10.,-5.) – (5.,-5.) – (5.,-10.) – (10.,-10.) – cycle;
(-23.9,4.25) node\[anchor=north west\] [2]{}; (-18.9,4.25) node\[anchor=north west\] [3]{}; (-13.9,4.25) node\[anchor=north west\] [4]{}; (-8.9,4.25) node\[anchor=north west\] [5]{}; (-3.9,4.25) node\[anchor=north west\] [6]{}; (1.1,4.25) node\[anchor=north west\] [7]{}; (6.1,4.25) node\[anchor=north west\] [1]{};
(-21.4,-0.75) node\[anchor=north west\] [5]{}; (-16.4,-0.75) node\[anchor=north west\] [6]{}; (-11.4,-0.75) node\[anchor=north west\] [7]{}; (-6.4,-0.75) node\[anchor=north west\] [1]{}; (-1.4,-0.75) node\[anchor=north west\] [2]{}; (3.6,-0.75) node\[anchor=north west\] [3]{}; (8.6,-0.75) node\[anchor=north west\] [4]{};
(-23.9,-5.75) node\[anchor=north west\] [7]{}; (-18.9,-5.75) node\[anchor=north west\] [1]{}; (-13.9,-5.75) node\[anchor=north west\] [2]{}; (-8.9,-5.75) node\[anchor=north west\] [3]{}; (-3.9,-5.75) node\[anchor=north west\] [4]{}; (1.1,-5.75) node\[anchor=north west\] [5]{}; (6.1,-5.75) node\[anchor=north west\] [6]{};
In this paper, we continue the study of restricted colorings that use polygons. In particular, we look at colorings that use triangles (see Figure \[fig:besttri\] for an example), and we prove that at least seven colors are needed for any triangle coloring of the plane. We do this by showing that if any convex polygon coloring has a point that borders at least four polygons, then at least seven colors are needed. This, together with the observation that any triangle coloring has at least one point that borders at least four triangles, implies that triangle colorings require at least seven colors.
Definitions and theorem
=======================
We begin by defining the colorings using polygons that we study.
A *convex polygon coloring* is a mapping from ${\mathbb R}^2$ to a finite set of colors, such that any two points a unit distance apart are colored differently, and the plane is partitioned into *regions*, *borders*, and *vertices* with the following properties. Vertices are points, borders are line segments that connect vertices, and regions are uniformly colored convex polygons that are enclosed by borders and vertices. Regions that share a border or a vertex are colored differently, and the coloring is locally finite in the sense that any disk intersects finitely many regions, borders, and vertices.
Note that all borders and vertices may be colored arbitrarily, meaning that they may be colored independently of the regions that they border. The arguments in this paper are not affected by how the border points and vertices are colored.
The *degree* of a vertex is the number of borders that are connected to it. Observe that for any convex polygon coloring, the degree of any vertex is at least 3. We say that a color *occurs* at a border point or vertex $P$ if a region bordering $P$ contains that color. If $n$ colors occur at a vertex, then we call it an *$n$-colored vertex*. Since in our definition regions that share a vertex must be colored differently, if a vertex has degree $n$ then it is $n$-colored.
We now state the first of our two main theorems, which applies to all convex polygon colorings.
\[thm:genthm\] If a convex polygon coloring of the plane contains a vertex of degree at least $4$, then at least $7$ colors are required.
We define a *triangle coloring* of the plane as a convex polygon coloring in which all the regions are triangles. We now state our second main theorem that applies to all triangle colorings, which is a consequence of Theorem \[thm:genthm\].
\[thm:trithm\] Any triangle coloring of the plane requires at least $7$ colors.
Definitions and lemmas
======================
In this section, we introduce some definitions and lemmas that we use in our proof. These all refer to a convex polygon coloring with a fixed vertex $O$ and the circle $C_O$ of unit radius centered at $O$.
Crossings
---------
The proof relies on considering certain border points and vertices on $C_O$ which we call crossings.
A *crossing* is either a point on $C_O$ that lies on a border not tangent to $C_O$ or a vertex on $C_O$ connected to at least one border that has points inside $C_O$.
[0.5]{}
(0.,0.) circle (5.cm); (-1.1255025201300173,0.8193951095066375)– (0.,0.); (0.,0.)– (0.8530955439712657,0.9136140649400318); (0.,0.)– (-0.9056582907854304,-0.9707650437278539); (0.,0.)– (1.0729397733158528,-1.0963903176390464);
(2.5,4.330127018922193) circle (3pt); (1.6226830776696684,3.4397254141432962)– (3.377316922330328,5.220528623701084);
(0.,0.) circle (2.0pt); (-1.1255025201300173,0.8193951095066375) circle (2.0pt); (0.8530955439712657,0.9136140649400318) circle (2pt); (-0.9056582907854304,-0.9707650437278539) circle (2pt); (1.0729397733158528,-1.0963903176390464) circle (2pt);
(-2.5,4.330127018922193) circle (3pt); (-1.3946013658158076,3.7465228212473294)– (-2.5,4.330127018922193); (-2.5,4.330127018922193)– (-2.5333825456094092,3.0805728561710795); (-2.5,4.330127018922193)– (-3.0572006030941794,5.449067360455862);
(-2.3,4.930127018922193) node [$P_1$]{};
(2.4,4.930127018922193) node [$P_2$]{};
(0, 0.5) node [$O$]{};
[0.5]{}
(0.,0.) circle (5.cm); (-1.1255025201300173,0.8193951095066375)– (0.,0.); (0.,0.)– (0.8530955439712657,0.9136140649400318); (0.,0.)– (-0.9056582907854304,-0.9707650437278539); (0.,0.)– (1.0729397733158528,-1.0963903176390464);
(0.,0.) circle (2.0pt); (-1.1255025201300173,0.8193951095066375) circle (2.0pt); (0.8530955439712657,0.9136140649400318) circle (2pt); (-0.9056582907854304,-0.9707650437278539) circle (2pt); (1.0729397733158528,-1.0963903176390464) circle (2pt);
(-2.5,4.330127018922193) circle (3pt); (-2.5,4.330127018922193)– (-0.7679491924311237,5.330127018922193); (-2.5,4.330127018922193)– (-4.232050807568877,3.330127018922193);
(2.5,4.330127018922193) circle (3pt); (2.5,4.330127018922193)– (0.7679491924311237,5.330127018922193); (2.5,4.330127018922193)– (4.232050807568877,3.330127018922193); (2.5,4.330127018922193)– (3.6963144919952007,4.532884529041009);
(0, 0.5) node [$O$]{};
(-2.6,4.830127018922193) node [$P_3$]{};
(2.5,4.830127018922193) node [$P_4$]{};
Note that a point on $C_O$ which lies on a border that is tangent to $C_O$ is not a crossing. Also, a vertex on $C_O$ connected to borders that do not have any points inside $C_O$ is not a crossing (see Figure \[fig:noncrossings\]). We call these points *non-crossings*.
We now record the obvious fact that $C_O$ must contain at least one crossing. It would not be difficult to prove that there are at least six crossings on $C_O$ for any convex polygon coloring of the plane, but this is not needed in our proof.
\[lem:arclem\] The circle $C_O$ must contain at least one crossing.
Assume that there are no crossings on $C_O$. This implies that all but finitely many points on $C_O$ lie within the same region, with non-crossings being the only possible exceptions. Since the convex polygon coloring is locally finite, there must be two points on $C_O$ at distance one that are not exceptions. Thus, these two points are in the same region and must be colored the same. However, this is a contradiction since points at distance one from each other must be colored differently.
Now we state the following lemma about points that are not crossings on $C_O$.
\[lem:neighlem\] If a point $P$ on $C_O$ is not a crossing, then there is a region that $P$ lies in or borders that cannot be colored any of the colors that occur at $O$.
If $P$ lies within a region, consider a small circular neighborhood of $P$. Within each of the regions bordering $O$, there is a point a unit distance away from a point in the neighborhood of $P$. Each of these points must be colored differently, so the region $P$ lies in cannot be colored any of the colors that occur at $O$. On the other hand, if $P$ does not lie within a region and is a non-crossing, then we can apply the previous argument to a point on $C_O$ in a region that $P$ borders.
Point types and special arcs
----------------------------
The following definitions and lemmas all refer to a convex polygon coloring with a fixed 4-colored vertex $O$ and the circle $C_O$ of unit radius centered at $O$.
We define an *inside neighborhood* of a point $P$ on $C_O$ as an open semi-circular neighborhood that lies on the side closer to $O$ of the tangent line to $C_O$ at $P$. Similarly, we define an *outside neighborhood* of a point $P$ on $C_O$ as an open semi-circular neighborhood that lies on the side away from $O$ of the tangent line to $C_O$ at $P$. We say that if any points within a neighborhood of a point cannot be colored any of $n$ colors, then those $n$ colors are *excluded* in that neighborhood.
An *inward point* is a point $P$ on $C_O$ that has an inside neighborhood in which all $4$ colors occurring at $O$ are excluded (see Figure \[fig:inpoint\]).
An *outward point* is a point $P$ on $C_O$ that has an outside neighborhood in which all $4$ colors occurring at $O$ are excluded (see Figure \[fig:outpoint\]).
An *alternative point* is a point $P$ on $C_O$ that has an outside neighborhood in which $3$ of the $4$ colors occurring at $O$ are excluded (see Figure \[fig:altpoint\]).
We can determine whether a point $P$ is inward, outward, or alternative as follows. Take the tangent line at $O$ to the unit circle $C_P$ centered at $P$ and count the borders on each side of the tangent. Adding one to this number usually gives the number of excluded colors on the corresponding side of $P$. The only exception to this heuristic is when two borders lie on one side of the tangent line and the other two borders lie on the side closer to $P$ (see Figure \[fig:collinearpair\] for an example).
[0.33]{}
(0.,0.) circle (5.cm); (2.403815059142447,4.635067688461732)– (4.618286161413787,2.4850613702588413); plot\[domain=-0.7706289045528925:2.3709637490368998,variable=\]([-0.5\*cos(r)]{},[-0.5\*sin(r)]{});
(-3.5355339059327373,3.5355339059327378)– (3.5355339059327373,-3.5355339059327378);
(0.,0.)– (0.7907474566524066,0.6769231762356426); (0.,0.)– (0.41556302464404626,-0.906696601244491); (0.,0.)– (-1.288589877504911,-0.05069266962571086); (0.,0.)– (-0.7027798059051065,-1.0080819404959127);
(0.,0.) circle (2.0pt); (3.482932991091283,3.5873636252222783) circle (2.0pt); (0.7907474566524066,0.6769231762356426) circle (2pt); (0.41556302464404626,-0.906696601244491) circle (2pt); (-1.288589877504911,-0.05069266962571086) circle (2pt); (-0.7027798059051065,-1.0080819404959127) circle (2pt);
(3.4,4.2) node\[anchor=north west\] [$P$]{}; (-0.4,0.85) node\[anchor=north west\] [$O$]{};
[0.33]{}
(0.,0.) circle (5.cm); (2.403815059142447,4.635067688461732)– (4.618286161413787,2.4850613702588413); plot\[domain=-0.7706289045528925:2.3709637490368998,variable=\]([0.5\*cos(r)]{},[0.5\*sin(r)]{});
(-3.5355339059327373,3.5355339059327378)– (3.5355339059327373,-3.5355339059327378);
(0.,0.)– (0.7907474566524066,0.6769231762356426); (0.,0.)– (-0.41556302464404626,0.906696601244491); (0.,0.)– (1.288589877504911,-0.05069266962571086); (0.,0.)– (-0.7027798059051065,-1.0080819404959127);
(0.,0.) circle (2.0pt); (3.482932991091283,3.5873636252222783) circle (2.0pt); (0.7907474566524066,0.6769231762356426) circle (2pt); (-0.41556302464404626,0.906696601244491) circle (2pt); (1.288589877504911,-0.05069266962571086) circle (2pt); (-0.7027798059051065,-1.0080819404959127) circle (2pt);
(2.8,3.6) node\[anchor=north west\] [$P$]{}; (-0.44,-0.05) node\[anchor=north west\] [$O$]{};
[0.33]{}
(0.,0.) circle (5.cm); (2.403815059142447,4.635067688461732)– (4.618286161413787,2.4850613702588413); plot\[domain=-0.7706289045528925:2.3709637490368998,variable=\]([0.5\*cos(r)]{},[0.5\*sin(r)]{});
(-3.5355339059327373,3.5355339059327378)– (3.5355339059327373,-3.5355339059327378);
(0.,0.)– (0.5,1.2); (0.,0.)– (-1.21556302464404626,-0.5); (0.,0.)– (1.3,0.5); (0.,0.)– (-0.7027798059051065,-1.0080819404959127);
(0.,0.) circle (2.0pt); (3.482932991091283,3.5873636252222783) circle (2.0pt); (-1.21556302464404626,-0.5) circle (2pt); (0.5,1.2) circle (2pt); (1.3,0.5) circle (2pt); (-0.7027798059051065,-1.0080819404959127) circle (2pt);
(2.8,3.6) node\[anchor=north west\] [$P$]{}; (-0.44,-0.05) node\[anchor=north west\] [$O$]{};
With these definitions, we can show that these are the only kinds of points on $C_O$.
\[lem:threetypes\] Any point on $C_O$ is an inward, outward, or alternative point.
Let $P$ be a point on $C_O$ and $C_P$ be the unit circle centered at $P$. The circle $C_P$ passes through $O$. We examine whether the borders connected to $O$ within a neighborhood of $O$ lie inside or outside of $C_P$. There are four possibilities for how many of these borders lie inside and outside of $C_P$: all four outside, three outside and one inside, two outside and two inside, and one outside and three inside. It is not possible for all four borders to lie inside of $C_P$ because then two of the borders form an angle that is greater than $\pi$, which contradicts the coloring being convex.
In the first case of four borders outside of $C_P$ and the second case of three borders outside of $C_P$ and one border inside of $C_P$, $4$ regions are intersected by any unit circle centered at a point within a sufficiently small inside neighborhood of $P$. Thus, the $4$ colors occurring at $O$ are excluded in an inside neighborhood of $P$, which makes $P$ an inward point.
In the third case of two borders outside of $C_P$ and two borders inside of $C_P$, $3$ regions are intersected by any unit circle centered at a point within a sufficiently small outside neighborhood of $P$. Thus, $3$ of the $4$ colors occurring at $O$ are excluded in an outside neighborhood of $P$, which makes $P$ an alternative point.
Lastly, in the case of three borders inside of $C_P$ and one border outside of $C_P$, $4$ regions are intersected by any unit circle centered at a point within a sufficiently small outside neighborhood of $P$. Thus, the $4$ colors occurring at $O$ are excluded in an outside neighborhood of $P$, which makes $P$ an outward point.
Since these are the only possible cases, all points on $C_O$ must be inward, outward, or alternative points.
With the understanding that all points on $C_O$ come in three different types, we can define arcs that consist of points of the same type.
A maximal arc of $C_O$ consisting of only inward points is an *inward arc*. Similarly, a maximal arc consisting of only outward points is an *outward arc*. Lastly, a maximal arc consisting of only alternative points in which the same three colors are excluded in outside neighborhoods of each point is called an *alternative arc*.
Note that an arc could be a single point of $C_O$ and endpoints may or may not be included in an arc.
With this definition, we can introduce the following lemma, which determines the maximum number of alternative arcs on $C_O$.
\[lem:fouralt\] The circle $C_O$ has at most four alternative arcs on it.
A point $P$ is an alternative point if and only if a unit circle $C_P$ centered at $P$ intersects two opposite regions bordering $O$. This is because within a neighborhood of $O$ two borders connected to $O$ must lie inside $C_P$ and two borders must lie outside $C_P$. There are two cases that are considered for the borders connected to $O$. One is a special case where only two borders are collinear and all other borders lie on one side of the pair, and the other case is where the borders are oriented in any other way.
We first consider the case of one pair of collinear borders. In this case, there are two alternative arcs and one alternative arc consisting of one point such that the unit circle centered at any point on each arc intersects two opposite regions bordering $O$.
We next consider the general case. In this case, there are four alternative arcs because each pair of opposite regions forces two alternative arcs.
Therefore, there are at most four alternative arcs on $C_O$.
[0.5]{}
(0.,0.) circle (5.cm); (-2.,0.)– (2.,0.); (-1.4142135623730951,1.414213562373095)– (0.,0.); (0.,0.)– (1.4142135623730951,1.414213562373095); plot\[domain=3.141592653589793:6.283185307179586,variable=\]([1.\*5.\*cos(r)+0.\*5.\*sin(r)]{},[0.\*5.\*cos(r)+1.\*5.\*sin(r)]{}); plot\[domain=-0.7853981633974483:0.7853981633974484,variable=\]([1.\*5.\*cos(r)+0.\*5.\*sin(r)]{},[0.\*5.\*cos(r)+1.\*5.\*sin(r)]{}); plot\[domain=2.356194490192345:3.9269908169872414,variable=\]([1.\*5.\*cos(r)+0.\*5.\*sin(r)]{},[0.\*5.\*cos(r)+1.\*5.\*sin(r)]{});
(0.,0.) circle (2.0pt); (-2.,0.) circle (2pt); (2.,0.) circle (2pt); (1.4142135623730951,1.414213562373095) circle (2pt); (-1.4142135623730951,1.414213562373095) circle (2pt);
(0.,5.) circle (3.0pt); (3.5355339059327373,3.5355339059327378) circle (3pt); (3.5355339059327373,-3.5355339059327378) circle (3pt);
(-3.5355339059327373,-3.5355339059327378) circle (3pt); (-3.5355339059327373,3.5355339059327378) circle (3pt);
(-.35,-0.05) node\[anchor=north west\] [$O$]{};
[0.5]{}
(0.,0.) circle (5.cm); (-1.0970414663416408,1.672273907327081)– (0.,0.); (0.,0.)– (1.4643382564060556,1.362245745387253); (0.,0.)– (1.7953230740095407,-0.8813711249688936); (0.,0.)– (-1.1070833348303741,-1.665642965866592); plot\[domain=3.141592653589793:6.283185307179586,variable=\]([1.\*5.\*cos(r)+0.\*5.\*sin(r)]{},[0.\*5.\*cos(r)+1.\*5.\*sin(r)]{}); plot\[domain=2.5549817831941235:3.722186694582359,variable=\]([1.\*5.\*cos(r)+0.\*5.\*sin(r)]{},[0.\*5.\*cos(r)+1.\*5.\*sin(r)]{}); plot\[domain=1.1144340759794773:2.3200914943461477,variable=\]([1.\*5.\*cos(r)+0.\*5.\*sin(r)]{},[0.\*5.\*cos(r)+1.\*5.\*sin(r)]{}); plot\[domain=-0.5866108703956696:0.5805940409925657,variable=\]([1.\*5.\*cos(r)+0.\*5.\*sin(r)]{},[0.\*5.\*cos(r)+1.\*5.\*sin(r)]{}); plot\[domain=4.25602672956927:5.461684147935941,variable=\]([1.\*5.\*cos(r)+0.\*5.\*sin(r)]{},[0.\*5.\*cos(r)+1.\*5.\*sin(r)]{});
(0.,0.) circle (2.0pt); (-1.0970414663416408,1.672273907327081) circle (2pt); (1.4643382564060556,1.362245745387253) circle (2pt); (1.7953230740095407,-0.8813711249688936) circle (2pt); (-1.1070833348303741,-1.665642965866592) circle (2pt);
(-4.164107414666481,2.767708337075936) circle (3.0pt); (-4.180684768317702,-2.742603665854102) circle (3.0pt); (-3.4056143634681324,3.660845641015139) circle (3.0pt); (2.203427812422234,4.488307685023852) circle (3pt); (4.16410741466648,-2.7677083370759354) circle (3pt); (4.180684768317702,2.742603665854102) circle (3pt); (-2.203427812422234,-4.488307685023852) circle (3pt); (3.405614363468133,-3.6608456410151398) circle (3pt);
(-.35,-0.05) node\[anchor=north west\] [$O$]{};
Note that the only case in which the number of alternative arcs on $C_O$ is fewer than four occurs when only two borders are collinear with all other borders on one side of the pair as shown in Figure \[fig:collinearpair\].
Lemmas about crossings
----------------------
The following lemmas all refer to a convex polygon coloring with a fixed 4-colored vertex $O$ and the circle $C_O$ of unit radius centered at $O$.
\[lem:crossinglem\] The regions that border a crossing $P$ on $C_O$ must be colored a fifth and sixth color that do not occur at $O$.
Because the coloring is locally finite, there must be two regions that border $P$. By Lemma \[lem:neighlem\], these regions cannot be colored any of the colors which occur at $O$. Since both regions cannot be colored the same fifth color by our definition, one region must be colored a fifth color and the other must be colored a sixth color.
Using Lemma \[lem:crossinglem\], we can now introduce lemmas that prove that certain crossings necessitate seven colors for the coloring.
\[lem:inoutlem\] If a crossing $P$ on $C_O$ is an inward or outward point, then at least seven colors are needed for the coloring.
Let $Q$ be a point at distance one from $P$ on $C_O$. Since a fifth and sixth color occur at $P$ by Lemma \[lem:crossinglem\], if $Q$ is not a crossing, then the region $Q$ lies in or borders must be colored a seventh color by Lemma \[lem:neighlem\].
Hence, we consider the case where $Q$ is a crossing (see Figure \[fig:inout\]). Since a fifth and sixth color occur at $Q$ by Lemma \[lem:crossinglem\], we consider the points on $C_Q$ within a neighborhood of $P$. Regardless of whether $P$ is an inward or outward point, there are points on $C_Q$ that lie inside a region and also in an inside or outside neighborhood of $P$. Since these points cannot be colored any of the six colors that occur at $O$ and $Q$ by Lemma \[lem:neighlem\], at least seven colors are needed.
In addition to considering the case where crossings are inward or outward points, we also introduce a lemma that considers the case of crossings that are alternative points.
\[lem:altlem\] If two crossings lie at distance one on the same alternative arc, then at least seven colors are needed for the coloring.
Let $P$ and $Q$ be the crossings and let $C_P$ and $C_Q$ be the unit circles centered at $P$ and $Q$ respectively. Without loss of generality, if the points on $C_Q$ within an outside neighborhood of $P$ lie within a region that $C_O$ passes through, then at least seven colors are needed by Lemma \[lem:neighlem\] since a fifth and sixth color occur at $Q$ by Lemma \[lem:crossinglem\].
Therefore, we consider the case in which points on $C_P$ and $C_Q$ within outside neighborhoods at $P$ and $Q$ lie within regions that $C_O$ does not pass through. Without loss of generality, consider a region bordering $Q$ that only $C_P$ passes through. If this region is colored a seventh color, then we are done.
Suppose that this region is not colored a seventh color. Since $Q$ is an alternative point, this region can only be colored one color not excluded from the colors occurring at $O$ (see Figure \[fig:altalt\]). By Lemma \[lem:neighlem\], a region which borders $P$ and contains points on $C_Q$ cannot be colored any of the six colors occurring at $Q$ and $O$. Thus, at least seven colors are needed.
[0.5]{}
(0.,0.) circle (5.cm); (-0.6816229033466099,1.0478025661513422)– (0.,0.); (0.,0.)– (-1.2387888894394545,-0.16703917924056844); (0.,0.)– (0.4500586699083075,1.1661677382093731); (0.,0.)– (0.5506276500054399,-1.122189463080761); (-3.254745078836226,5.3265505563662305)– (-1.7452549211637716,3.3337034814781568); (1.6504184119090541,3.4132240114143837)– (2.5,4.330127018922193); (2.5,4.330127018922193)– (2.890644357935366,3.1427361878925497); (2.5,4.330127018922193)– (3.0412302605588284,5.456878901572239); plot\[domain=2.7862605813657595:5.4534430523534745,variable=\]([1.\*5.\*cos(r)+0.\*5.\*sin(r)]{},[0.\*5.\*cos(r)+1.\*5.\*sin(r)]{});
(0.,0.) circle (2.0pt);
(-1.0699336262195465,0.5821198678064264) node\[anchor=north west\] [1]{}; (-0.42442144355598228,1.1097017386609105) node\[anchor=north west\] [2]{}; (0.4276430052507715,0.4934979134723176) node\[anchor=north west\] [3]{}; (-0.60476464512641364,-0.1144271589525982) node\[anchor=north west\] [4]{}; (1.9503886178142224,5.324844522833219) node\[anchor=north west\] [5]{}; (2.6248659563823267,4.982088431501437) node\[anchor=north west\] [6]{}; (-2.5633708813080254,5.175879366928679) node\[anchor=north west\] [7]{};
(0.3752301154894207,0.18953537725985969) node [$O$]{}; (-2.9675102576898646,4.34950394095487) node [$P$]{}; (1.9455628382915209,4.222951674811681) node [$Q$]{};
(2.5,4.330127018922193) circle (3pt); (-2.5,4.330127018922193) circle (3.0pt); (-0.6816229033466099,1.0478025661513422) circle (2pt); (-1.2387888894394545,-0.16703917924056844) circle (2pt); (0.4500586699083075,1.1661677382093731) circle (2pt); (0.5506276500054399,-1.122189463080761) circle (2pt);
[0.5]{}
(0.,0.) circle (5.cm); (-0.6816229033466099,1.0478025661513422)– (0.,0.); (0.,0.)– (-0.8590635758968608,-0.908025204808379); (0.,0.)– (0.4500586699083075,1.1661677382093731); (0.,0.)– (0.5506276500054399,-1.122189463080761);
(-3.254745078836226,5.3265505563662305)– (-1.7452549211637716,3.3337034814781568); (1.6504184119090541,3.4132240114143837)– (2.5,4.330127018922193); (2.5,4.330127018922193)– (2.890644357935366,3.1427361878925497); (2.5,4.330127018922193)– (3.421857703399558,5.174329823159951);
(-2.5,4.330127018922193)– (-2.124468540622823,5.522383754285029); (2.5,4.330127018922193)– (2.5081458592324717,5.580100476631341);
plot\[domain=2.7862605813657595:5.4534430523534745,variable=\]([1.\*5.\*cos(r)+0.\*5.\*sin(r)]{},[0.\*5.\*cos(r)+1.\*5.\*sin(r)]{});
(0.,0.) circle (2.0pt);
(-0.9699336262195465,0.4221198678064264) node\[anchor=north west\] [1]{}; (-0.42442144355598228,1.1097017386609105) node\[anchor=north west\] [2]{}; (0.4276430052507715,0.4934979134723176) node\[anchor=north west\] [3]{}; (-0.45476464512641364,-0.1144271589525982) node\[anchor=north west\] [4]{};
(1.8003886178142224,5.324844522833219) node\[anchor=north west\] [5]{}; (2.4003886178142224,5.424844522833219) node\[anchor=north west\] [4]{}; (2.7248659563823267,4.882088431501437) node\[anchor=north west\] [6]{};
(-3.3633708813080254,4.875879366928679) node\[anchor=north west\] [6]{}; (-3.0633708813080254,5.475879366928679) node\[anchor=north west\] [7]{}; (-2.4033708813080254,5.275879366928679) node\[anchor=north west\] [5]{};
(0.3752301154894207,0.18953537725985969) node [$O$]{}; (-1.9875102576898646,4.23950394095487) node [$P$]{}; (1.9455628382915209,4.222951674811681) node [$Q$]{};
(2.5,4.330127018922193) circle (3pt); (-2.5,4.330127018922193) circle (3.0pt); (-0.6816229033466099,1.0478025661513422) circle (2pt); (-0.8590635758968608,-0.908025204808379) circle (2pt); (0.4500586699083075,1.1661677382093731) circle (2pt); (0.5506276500054399,-1.122189463080761) circle (2pt);
Proof of Theorem \[thm:genthm\]
===============================
Let $\alpha$ be a convex polygon coloring of the plane and suppose it contains a vertex $O$ of degree 5 or more. By Lemma \[lem:arclem\] there is at least one crossing $P$ on the unit circle $C_O$ centered at $O$. The regions that $C_O$ passes through which border $P$ cannot be colored any five colors that occur at $O$ by Lemma \[lem:neighlem\]. Therefore, by our definition, the bordering regions must be colored using a sixth and seventh color.
Hence, we consider the case where all vertices in $\alpha$ have degree at most $4$, and we let vertex $O$ be a vertex of degree 4. Let $P$ be a crossing on $C_O$ and $Q$ be a point at distance one clockwise from $P$ on $C_O$. If $Q$ is not a crossing, then by Lemma \[lem:neighlem\] the region $Q$ lies in or borders must be colored a seventh color. Therefore, we can assume that $Q$ is a crossing. Similarly, if we repeat the same reasoning clockwise starting from point $Q$, we can assume that all points on the inscribed hexagon on $C_O$ that contains $P$ and $Q$ are crossings.
If any of the points on the hexagon are inward or outward points, then by Lemma \[lem:inoutlem\] at least seven colors are needed for $\alpha$. Hence, we can assume all points on the hexagon are alternative points. By Lemma \[lem:fouralt\], there are at most four alternative arcs on $C_O$. Therefore, by the pigeonhole principle, there must be two adjacent points on the inscribed hexagon that lie on the same alternative arc, so by Lemma \[lem:altlem\] more than six colors are needed for $\alpha$.
Hence, if a polygon coloring has at least one vertex of degree at least 4, then at least seven colors are needed.
Triangle colorings
==================
In this section, we introduce an observation about triangle colorings that proves that Theorem \[thm:trithm\] is a consequence of Theorem \[thm:genthm\]. Though this is a basic fact, we provide a proof to keep this paper self-contained.
Any triangle coloring has a vertex of degree 4 or more.
Consider a triangle coloring of the plane. Assume that all vertices on the plane have degree 3. Let $\triangle ABC$ be the triangle with greatest angle of any triangle on the plane. Without loss of generality, let $\angle ABC$ be the angle with the greatest angle measure.
(-5.,-1.)– (0.,4.); (0.,4.)– (6.,-1.); (6.,-1.)– (-5.,-1.);
(6.,-1.)– (10.34559791123893,-4.621331592699107); (6.,-1.)– (11.656700779645746,-1.);
(6.,-1.)– (10.477337524054995,-2.672821398861612);
(-5.,-1.) circle (2pt); (0.,4.) circle (2pt); (6.,-1.) circle (2pt); (10.477337524054995,-2.672821398861612) circle (2pt);
(-5.6,-1.) node\[anchor=north west\] [$A$]{}; (-0.25,4.75) node\[anchor=north west\] [$B$]{}; (5.6,-1.1) node\[anchor=north west\] [$C$]{}; (10.4,-2.67) node\[anchor=north west\] [$D$]{};
Consider the border that is connected to $C$, not connected to $A$ or $B$, and let $D$ be the vertex closest to $C$ on it. Since no angle between any adjacent borders can be greater than $\pi$ because the coloring is convex and all vertices have degree 3, $D$ must lie on or between the lines $\overleftrightarrow{AC}$ and $\overleftrightarrow{BC}$ (see Figure \[fig:triabc\]). The angle $\angle BCD$ must be at least the sum of $\angle BAC$ and $\angle ABC$ by the exterior angle theorem. However, since $\angle BCD$ must be in a triangle on the plane, this is a contradiction because $\angle BCD$ is larger than $\angle ABC$, which is assumed to be the biggest angle of any triangle on the plane. Therefore, there must be at least one vertex of degree 4 on the plane.
Hence, by Theorem \[thm:genthm\], at least $7$ colors are needed to color the plane for any triangle coloring, which proves Theorem \[thm:trithm\].
Discussion and open problems
============================
Since Theorem \[thm:genthm\] requires the polygon coloring to have a vertex with degree at least 4, the only possible colorings that are left to be examined are those with only vertices of degree 3. This type of coloring can no€™t be analyzed using the same method.
For instance, let $O$ be a vertex of degree 3 and $C_O$ be the unit circle centered at $O$. By Lemma \[lem:arclem\] there is at least one crossing on $C_O$. Let $P$ be the crossing and $Q$ be a point one unit away on $C_O$. In the proof of Theorem \[thm:genthm\], we can assume $Q$ is a crossing because otherwise seven colors are needed immediately.
However, in a polygon coloring with vertices that only have degree 3, $Q$ does no€™t have to be a crossing because if $Q$ lies in a region, then it does not immediately imply that seven colors needed. Therefore, each crossing on $C_O$ can be a unit apart from non-crossings and points within regions.
This leads us to ask for the fewest number of colors needed for polygon colorings in which all vertices have degree 3 (see Figure \[fig:tilings\] for examples). If it is shown that at least seven colors are needed for this case, then the chromatic number of convex polygon colorings is 7 as a consequence of Theorem \[thm:genthm\].
What is the minimum number of colors needed for convex polygon colorings with only vertices of degree 3?
Though Theorem \[thm:trithm\] shows that triangle colorings require at least seven colors, this does not imply that the chromatic number of these colorings is $7$. The best coloring that we have found for triangle colorings (see Figure \[fig:besttri\]) sets the upper bound for the chromatic number of triangle colorings to $8$.
(-1.25,2.165063509461097) – (1.25,2.165063509461097) – (0,4.330127018922194) – cycle; (1.25,2.165063509461097) – (2.5,0) – (3.75,2.1650635094610955) – cycle; (2.5,0) – (1.25,-2.165063509461097) – (3.75,-2.165063509461098) – cycle; (1.25,-2.165063509461097) – (-1.25,-2.165063509461097) – (0,-4.330127018922194) – cycle; (-1.25,-2.165063509461097) – (-2.5,0) – (-3.75,-2.1650635094610955) – cycle; (-2.5,0) – (-1.25,2.165063509461097) – (-3.75,2.165063509461098) – cycle; (-3.75,2.165063509461098) – (-1.25,2.165063509461097) – (-2.5,4.3301270189221945) – cycle; (1.25,2.165063509461097) – (3.75,2.1650635094610955) – (2.5,4.330127018922194) – cycle; (3.75,-2.165063509461098) – (1.25,-2.165063509461097) – (2.5,-4.3301270189221945) – cycle; (-1.25,-2.165063509461097) – (-3.75,-2.1650635094610955) – (-2.5,-4.330127018922194) – cycle; (0,4.330127018922194) – (-2.5,4.3301270189221945) – (-1.25,2.165063509461099) – cycle; (2.5,4.330127018922194) – (0,4.330127018922194) – (1.25,2.1650635094610955) – cycle; (0,-4.330127018922194) – (2.5,-4.3301270189221945) – (1.25,-2.165063509461099) – cycle; (-2.5,-4.330127018922194) – (0,-4.330127018922194) – (-1.25,-2.1650635094610955) – cycle; (2.5,0) – (3.75,-2.165063509461098) – (5,0) – cycle; (2.5,0) – (5,0) – (3.75,2.1650635094610973) – cycle; (-2.5,0) – (-3.75,2.165063509461098) – (-5,0) – cycle; (-2.5,0) – (-5,0) – (-3.75,-2.1650635094610973) – cycle; (-1.25,2.165063509461097) – (0,0) – (1.25,2.165063509461097) – cycle; (-2.5,0) – (0,0) – (-1.25,2.165063509461097) – cycle; (-2.5,0) – (-1.25,-2.165063509461097) – (0,0) – cycle; (0,0) – (2.5,0) – (1.25,2.165063509461097) – cycle; (0,0) – (1.25,-2.165063509461097) – (2.5,0) – cycle; (-1.25,-2.165063509461097) – (1.25,-2.165063509461097) – (0,0) – cycle;
(-1.7575, 1.4500635095) node\[anchor=north west\] [1]{}; (-.515, 1.9950635095) node\[anchor=north west\] [2]{}; (0.7275, 1.4500635095) node\[anchor=north west\] [3]{};
(-1.7875, -.1500635095) node\[anchor=north west\] [6]{}; (-.515, -.7200635095) node\[anchor=north west\] [5]{}; (0.7275, -.1500635095) node\[anchor=north west\] [4]{};
(1.9805, -.7200635095) node\[anchor=north west\] [7]{}; (1.9805, 1.9950635095) node\[anchor=north west\] [8]{};
(-2.97525, -.7200635095) node\[anchor=north west\] [7]{}; (-2.97525, 1.9950635095) node\[anchor=north west\] [8]{};
(-3.025, 3.5950635095) node\[anchor=north west\] [5]{}; (-.515, 3.5950635095) node\[anchor=north west\] [7]{}; (1.9805, 3.5950635095) node\[anchor=north west\] [5]{};
(-3.025, -2.2950635095) node\[anchor=north west\] [2]{}; (-.515, -2.2950635095) node\[anchor=north west\] [8]{}; (1.9805, -2.2950635095) node\[anchor=north west\] [2]{};
(-1.7575, -2.850635095) node\[anchor=north west\] [3]{}; (0.7775, -2.850635095) node\[anchor=north west\] [1]{};
(-1.7875, 4.150635095) node\[anchor=north west\] [4]{}; (0.71475, 4.150635095) node\[anchor=north west\] [6]{};
(-4.28775, 1.4500635095) node\[anchor=north west\] [3]{}; (-4.28775, -.1500635095) node\[anchor=north west\] [4]{};
(3.252775, 1.4500635095) node\[anchor=north west\] [1]{}; (3.252775, -.1500635095) node\[anchor=north west\] [6]{};
Since this coloring uses more than seven colors, naturally one asks what the chromatic number is for triangle colorings. We conjecture that the chromatic number for such colorings is $8$.
What is the chromatic number of triangle colorings of the plane?
Finally, we note that it may be possible to extend our approach to polygon-like colorings that use differentiable curves instead of line segments. In such a coloring, we can approximate the curved borders connected at vertices by the tangent lines of those curves, which allows us to use most of the arguments for colorings with straight line borders. It would thus follow that a coloring with regions that are curved triangles also requires seven colors.
Acknowledgements {#acknowledgements .unnumbered}
================
I am incredibly grateful for the guidance Frank de Zeeuw has given me throughout the process of developing this paper. Also, I would like to thank Adam Sheffer and Dan Stefanica for giving me the opportunity to work on this project. All the diagrams in this paper have been made with GeoGebra.
[99]{}
D. Coulson, *On the chromatic number of plane tilings*, Journal of the Australian Mathematical Society [**77**]{}, 191–196, 2004.
A. de Grey, *The chromatic number of the plane is at least 5*, Geombinatorics [**28**]{}, 18–31, 2018.
K.J. Falconer, *The realization of distances in measurable subsets covering ${\mathbb R}^n$*, Journal of Combinatorial Theory, Series A [**31**]{}, 184–189, 1981.
J. Moustakis, *On the chromatic number of the plane and square colorings*, Master thesis at EPFL, Lausanne, Switzerland, 2016.
A. Soifer, *The mathematical coloring book*, Springer, New York, 2009.
S.P. Townsend, *Every 5-coloured map in the plane contains a monochrome unit*, Journal of Combinatorial Theory, Series A [**30**]{}, 114–115, 1981.
D.R. Woodall, *Distances realized by sets covering the plane*, Journal of Combinatorial Theory, Series A [**14**]{}, 187–200, 1973.
|
---
---
**[Christian P. H. Salas]{}**
Faculty of Mathematics & Statistics, The Open University, Milton Keynes
and Department of Mathematics, Waltham Forest College, London
e-mail: c.p.h.salas@open.ac.uk
\[section\]
\[Theorem\][Definition]{}
\[Theorem\][Corollary]{}
\[Theorem\][Lemma]{}
\[Theorem\][Example]{}
**[Abstract]{}**
*An apparent paradox in Einstein’s Special Theory of Relativity, known as a Thomas precession rotation in atomic physics, has been verified experimentally in a number of ways. However, somewhat surprisingly, it has not yet been demonstrated algebraically in a straightforward manner using Lorentz-matrix-algebra. Authors in the past have resorted instead to computer verifications, or to overly-complicated derivations, leaving undergraduate students in particular with the impression that this is a mysterious and mathematically inaccessible phenomenon. This is surprising because, as shown in the present note, it is possible to use a basic property of orthogonal Lorentz matrices and a judicious choice for the configuration of the relevant inertial frames to give a very transparent algebraic proof. It is pedagogically useful for physics students particularly at undergraduate level to explore this. It not only clarifies the nature of the paradox at an accessible mathematical level and sheds additional light on some mathematical properties of Lorentz matrices and relatively-moving frames. It also illustrates the satisfaction that a clear mathematical understanding of a physics problem can bring, compared to uninspired computations or tortured derivations.*
[**Keywords:**]{} *Special relativity, Rotation paradox, Thomas precession*
[**Mathematics Subject Classification:**]{} *83A02*
Introduction
============
Multiplying two Lorentz boosts whose velocity vectors are collinear gives a third Lorentz boost whose velocity can be calculated from the first two using the Einstein velocity addition law. If the two original velocities are not collinear, however, we do not get a pure Lorentz boost as the product, but rather a Lorentz boost multiplied by a certain $4 \times 4$ matrix whose columns and rows are orthonormal. This orthogonal matrix has the effect of rotating the spatial components of vectors in spacetime, while leaving their temporal component unaffected.
An interesting review paper written by the eminent British mathematician I. J. Good [@Good] discusses this relativistic rotation paradox, but Good clearly struggles to provide a straightforward algebraic proof in the 3+1 case. Mathematically, it is necessary to show in a four-dimensional Minkowski spacetime that a certain matrix product involving two Lorentz boosts with linearly independent velocity vectors is generally equivalent to an orthogonal Lorentz matrix. In section 8 of [@Good], which seeks to prove the paradox ‘beyond any doubt’, Good admits to being unable to do this algebraically and instead resorts to providing numerical confirmations. Such computational confirmations are easy to carry out, so he suggests that ‘only a short and elegant algebraic proof would be worthwhile’. One of the motivations for the present note is that a proof like this still seems to be lacking in the literature.
What we have at present in the way of mathematical demonstrations are either elaborate approximations involving power series and extraneous assumptions such as infinitesimally small relative velocities (see, e.g., section 7.3 in [@Goldstein]), or otherwise lengthy and overly-sophisticated expositions usually not easily accessible to, say, undergraduate physics students. Some expositions are intended to be more accessible but still seem rather involved and/or do not make use of Lorentz-matrix-algebra, e.g., [@Ferraro] and [@ODonnell].
Due to the lack of a short and transparent algebraic treatment, this interesting relativistic phenomenon is simply left unmentioned and unexplored in almost all undergraduate texts, which seems a pity. The following argument could be used shortly after introducing Lorentz transformation matrices and their properties to budding relativists.
Proof of the rotation paradox
=============================
Let $G = \text{diag}(1, -1, -1, -1)$ be the metric tensor in a four-dimensional Minkowski manifold with events specified by a time coordinate $x^0 = ct$ and rectangular spatial coordinates $x^1, x^2, x^3$. A $4 \times 4$ Lorentz matrix $\Lambda$ preserves the quadratic form $x^T G x$ in the sense that if $y = \Lambda x$ then $y^T G y = x^T G x$, so $$\Lambda^T G \Lambda = G$$ The set of all Lorentz matrices thus defined constitutes a group under matrix multiplication, so inverses and products of Lorentz matrices are also Lorentz.
Let $O$, $\overline{O}$ and $\overline{\overline{O}}$ be three inertial frames with collinear axes and with their origins initially coinciding. Let $\beta = (\beta_i) = \big(\frac{v_i}{c}\big)$, $i = 1, 2, 3$, be the $3 \times 1$ velocity vector of $\overline{O}$ relative to $O$ with corresponding Lorentz factor $\gamma = \frac{1}{\sqrt{1 - \beta^2}}$, where $\beta^2 \equiv \beta^T \beta$. Similarly, let a vector $\overline{\beta} = (\overline{\beta}_i)$, which is not collinear with $\beta$, be the velocity vector of $\overline{\overline{O}}$ relative to $\overline{O}$ with corresponding Lorentz factor $\overline{\gamma}$. Using a standard formula, e.g., formula (24) in [@Good] or formula (2.59) in [@Moller], the velocity vector of $\overline{\overline{O}}$ relative to $O$ is given by $$\overline{\overline{\beta}} = \frac{\overline{\beta} + \beta[\gamma + (\gamma - 1)(\overline{\beta}^T\beta)/\beta^2]}{(1 + \overline{\beta}^T\beta)\gamma}$$ with corresponding Lorentz factor $\overline{\overline{\gamma}}$. Using a simplification similar to one described in section 7.3 of [@Goldstein], we can let the plane defined by the vectors $\beta$ and $\overline{\beta}$ be the $\overline{x}^1\overline{x}^2$-plane of $\overline{O}$ so that $\overline{\beta}_3 =0$, and we can arrange the frames $O$ and $\overline{O}$ so that the vector $\beta$ is along the $x^1$ axis of $O$, implying $\beta_2 = \beta_3 = 0$. We can do this for any given pair of velocity vectors which are not collinear, so there is no loss of generality here. Then (2) gives $$\overline{\overline{\beta}}_1 = \frac{\overline{\beta}_1 + \beta_1}{1 + \overline{\beta}_1 \beta_1}$$ $$\overline{\overline{\beta}}_2 = \frac{\overline{\beta}_2}{(1 + \overline{\beta}_1 \beta_1) \gamma}$$ $$\overline{\overline{\beta}}_3 = 0$$ and a standard formula, e.g., formula (7.11) in [@Goldstein], allows us to write Lorentz transformations $L$ and $\overline{L}$ from $O$ to $\overline{O}$ and from $\overline{O}$ to $\overline{\overline{O}}$ respectively as $$L =
\begin{pmatrix}
\gamma & -\gamma \beta_1 & 0 \ & 0 \ \\
-\gamma \beta_1 & \gamma & 0 \ & 0 \ \\
0 & 0 & 1 \ & 0 \\
0 & 0 & 0 \ & 1
\end{pmatrix}$$ and $$\overline{L} =
\begin{pmatrix}
\overline{\gamma} \ \ & -\overline{\gamma} \overline{\beta}_1 \ \ & -\overline{\gamma} \overline{\beta}_2 \ \ & 0 \ \\
-\overline{\gamma} \overline{\beta}_1 \ \ & 1+(\overline{\gamma}-1) \frac{\overline{\beta}_1^2}{\overline{\beta}^2} \ \ & (\overline{\gamma}-1) \frac{\overline{\beta}_1\overline{\beta}_2}{\overline{\beta}^2} \ \ & 0 \ \\
-\overline{\gamma} \overline{\beta}_2 \ \ & (\overline{\gamma}-1) \frac{\overline{\beta}_1\overline{\beta}_2}{\overline{\beta}^2} \ \ & 1+(\overline{\gamma}-1) \frac{\overline{\beta}_2^2}{\overline{\beta}^2} \ \ & 0 \\
0 \ \ & 0 \ \ & 0 \ \ & 1
\end{pmatrix}$$ A Lorentz boost $\overline{\overline{L}}$ from $O$ to $\overline{\overline{O}}$ with velocity vector $\overline{\overline{\beta}}$ would be a matrix like (7), but with $\overline{\overline{\gamma}}$, $\overline{\overline{\beta}}_1$, $\overline{\overline{\beta}}_2$ and $\overline{\overline{\beta}}$ replacing $\overline{\gamma}$, $\overline{\beta}_1$, $\overline{\beta}_2$ and $\overline{\beta}$ respectively.
The relativistic rotation paradox is that, in general, $\overline{\overline{L}} \neq \overline{L} \times L$, but rather $$\overline{\overline{L}} = R \times \overline{L} \times L$$ or equivalently $$R = \overline{\overline{L}} \times L^{-1} \times \overline{L}^{-1}$$ where (9) is equation (28) in [@Good]. Numerical evidence in [@Good] suggests that $$R =
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & r_1 & s_1 & t_1 \\
0 & r_2 & s_2 & t_2 \\
0 & r_3 & s_3 & t_3
\end{pmatrix}$$ where the $3 \times 3$ submatrix in (10) is orthogonal. An approximation to (10) is also provided in equation (7.21) of [@Goldstein] under the assumptions that the components of $\overline{\beta}$ are small and only need to be retained to first order, that $\overline{\gamma} \approx 1$, and that the distinction among $\gamma$, $\overline{\gamma}$ and $\overline{\overline{\gamma}}$ can be ignored to first order.
However, it is straightforward to obtain an exact algebraic proof that $R$ in (9) is indeed an orthogonal matrix of the type given in (10) by observing that $R$ must be Lorentz, since it is a product of Lorentz matrices. Therefore all that is required to prove the rotation paradox is to show that the 00-element of $\overline{\overline{L}} \times L^{-1} \times \overline{L}^{-1}$ is equal to $1$, and that all the remaining elements in the first row are equal to zero, since any Lorentz matrix with a first row of this form must necessarily be an orthogonal matrix of the type given in (10). This assertion can easily be verified by substituting a generic $4 \times 4$ matrix with first row of the form $(1 \ 0 \ 0 \ 0)$ into the left-hand side of (1), setting the result equal to $G$ on the right-hand side, and then comparing corresponding elements.
Note that $L^{-1}$ and $\overline{L}^{-1}$ are immediately obtained from (6) and (7) simply by removing the negative signs in the first row and first column. To prove that the 00-element of $\overline{\overline{L}} \times L^{-1} \times \overline{L}^{-1}$ equals 1, multiply the first row of $\overline{\overline{L}}$ by each of the columns of $L^{-1}$ to get the $1 \times 4$ row vector $$\begin{pmatrix}
\overline{\overline{\gamma}} \gamma (1 - \overline{\overline{\beta}}_1 \beta_1) \ \ & \overline{\overline{\gamma}} \gamma (\beta_1 - \overline{\overline{\beta}}_1) \ \ & -\overline{\overline{\gamma}} \ \overline{\overline{\beta}}_2 \ \ & 0
\end{pmatrix}$$ and then multiply this row vector by the first column of the matrix $\overline{L}^{-1}$ to get $$\gamma \ \overline{\gamma} \ \overline{\overline{\gamma}}(1 - \overline{\overline{\beta}}_1 \beta_1 + \overline{\beta}_1 \beta_1 - \overline{\beta}_1 \overline{\overline{\beta}}_1) - \overline{\gamma} \ \overline{\overline{\gamma}} \ \overline{\beta}_2 \overline{\overline{\beta}}_2$$ $$= \frac{(1 + \beta_1 \overline{\beta}_1)^2 - (\beta_1 + \overline{\beta}_1)^2 - \overline{\beta}_2^2(1 - \beta_1^2)}{\sqrt{1 - \beta_1^2}\sqrt{1 - \overline{\beta}_1^2 - \overline{\beta}_2^2}\sqrt{(1 + \beta_1 \overline{\beta}_1)^2 - (\beta_1 + \overline{\beta}_1)^2 - \overline{\beta}_2^2(1 - \beta_1^2)}} = 1$$ as required. To prove that the 01-element of $\overline{\overline{L}} \times L^{-1} \times \overline{L}^{-1}$ equals 0, multiply the row vector in (11) by the second column of $\overline{L}^{-1}$ to get $$\gamma \ \overline{\gamma} \ \overline{\overline{\gamma}}(\overline{\beta}_1 - \beta_1 \overline{\beta}_1 \overline{\overline{\beta}}_1) +
\bigg[1+(\overline{\gamma}-1) \frac{\overline{\beta}_1^2}{\overline{\beta}^2}\bigg]
\gamma \ \overline{\overline{\gamma}} \ (\beta_1 - \overline{\overline{\beta}}_1)
- \overline{\overline{\gamma}} \ \overline{\overline{\beta}}_2 \bigg[(\overline{\gamma}-1) \frac{\overline{\beta}_1\overline{\beta}_2}{\overline{\beta}^2}\bigg]$$ $$= \frac{\overline{\beta}_1(1 - \beta_1^2)}{(1 - \beta_1^2)(1 - \overline{\beta}_1^2 - \overline{\beta}_2^2)} -
\frac{\overline{\beta}_1(\overline{\beta}_1^2 + \overline{\beta}_2^2)}{(\overline{\beta}_1^2 + \overline{\beta}_2^2)(1 - \overline{\beta}_1^2 - \overline{\beta}_2^2)} = 0$$ as required. To prove that the 02-element of $\overline{\overline{L}} \times L^{-1} \times \overline{L}^{-1}$ equals 0, multiply the row vector in (11) by the third column of $\overline{L}^{-1}$ to get $$\gamma \ \overline{\gamma} \ \overline{\overline{\gamma}}(\overline{\beta}_2 - \beta_1 \overline{\beta}_2 \overline{\overline{\beta}}_1) + \bigg[(\overline{\gamma}-1) \frac{\overline{\beta}_1\overline{\beta}_2}{\overline{\beta}^2}\bigg]\gamma \ \overline{\overline{\gamma}} \ (\beta_1 - \overline{\overline{\beta}}_1) -
\overline{\overline{\gamma}} \ \overline{\overline{\beta}}_2\bigg[1+(\overline{\gamma}-1) \frac{\overline{\beta}_2^2}{\overline{\beta}^2}\bigg]$$ $$= \frac{\overline{\beta}_2(1 - \beta_1^2)}{(1 - \beta_1^2)(1 - \overline{\beta}_1^2 - \overline{\beta}_2^2)} -
\frac{\overline{\beta}_2(\overline{\beta}_1^2 + \overline{\beta}_2^2)}{(\overline{\beta}_1^2 + \overline{\beta}_2^2)(1 - \overline{\beta}_1^2 - \overline{\beta}_2^2)} = 0$$ as required. Finally, to prove that the 03-element of $\overline{\overline{L}} \times L^{-1} \times \overline{L}^{-1}$ equals 0, multiply the row vector in (11) by the fourth column of $\overline{L}^{-1}$. This equals 0 by inspection, so the relativistic rotation paradox is proved.
[6]{} Good, I. J., Lorentz Matrices: A Review, Int. J. Theor. Phys., Vol. 34, pp. 779-799 (1995). Møller, C., The Theory of Relativity, 2nd ed., Clarendon Press, Oxford (1972). Goldstein, H., Poole, C., and Safko, J., Classical Mechanics, 3rd ed., Addison-Wesley, London (2002). Ferraro, R., Thibeault, M., Generic composition of boosts: an elementary derivation of the Wigner rotation, Eur. J. Phys., Vol. 20, pp. 143-151 (1999). O’Donnell, K., Visser, M., Elementary analysis of the special relativistic combination of velocities, Wigner rotation and Thomas precession, Eur. J. Phys., Vol. 32, pp. 1033-1047 (2011).
|
---
abstract: 'We experimentally explore the state space of three qubits on an NMR quantum information processor. We construct a scheme to experimentally realize a canonical form for general three-qubit states up to single-qubit unitaries. This form involves a non-trivial combination of GHZ and W-type maximally entangled states of three qubits. The general circuit that we have constructed for the generic state reduces to those for GHZ and W states as special cases. The experimental construction of a generic state is carried out for a nontrivial set of parameters and the good fidelity of preparation is confirmed by complete state tomography. The GHZ and W-states are constructed as special cases of the general experimental scheme. Further, we experimentally demonstrate a curious fact about three-qubit states, where for almost all pure states, the two-qubit reduced states can be used to reconstruct the full three-qubit state. For the case of a generic state and for the W-state, we demonstrate this method of reconstruction by comparing it with the directly tomographed three-qubit state.'
author:
- Shruti Dogra
- Kavita Dorai
- Arvind
title: 'Experimental construction of generic three-qubit states and their reconstruction from two-party reduced states on an NMR quantum information processor'
---
Introduction {#intro}
============
While a qubit is considered to be a building block for quantum information processing, the actual quantum computer invariably involves complex states of multiple qubits [@nielsen-book-02]. The transition from one to two qubits is of fundamental importance because it is the two-qubit system for which we can have entangled states and hence a nontrivial quantum advantage for information processing [@horodecki-rmp-2009; @ladd-nature-2010]. The manipulation of two-qubit states is qualitatively more difficult than that for a single qubit. As a matter of fact, the dynamics of a single qubit finds a classical analog in polarization optics [@arvind-josab-2007], and it is only when we create entangled states of two qubits, do the nontrivial quantum aspects emerge [@wootters-prl-98]. It may appear that moving from two qubits to several qubits is merely a matter of detail. However, this is not the case and new quantum aspects emerge for a three-qubit system, which is the simplest system for which the concept of multi-partite entanglement can be introduced. Unlike the two-qubit case, the maximally entangled states of three qubits are not equivalent up to local unitary transformations and instead fall into two inequivalent classes, namely the GHZ and W classes of states [@chen-pra-06]. In contradistinction to the two-qubit case, a canonical form for three qubits turns out to be nontrivial and involves a combination of GHZ and W states. It has been shown that all pure states of a system of three qubits are equivalent under local unitary transformations to a canonical state with five independent non-zero real parameters [@acin-prl-00; @carteret-jmp-00; @acin-prl-01; @vicente-prl-12; @vertesi-pra-14]. While one-qubit reduced states have information about the amount of entanglement in a two-qubit pure state, they do not uniquely determine the state. On the other hand, it turns out that almost every three-qubit pure state is completely determined by its two-qubit reduced density matrices and there is no more information in the full quantum state than what is already contained in the three possible two-qubit reduced states [@linden-prl-1-02; @diosi-pra-04; @cavalcanti-pra-05]. It is indeed somewhat surprising that even when nontrivial multi-partite entanglement is present, the “parts” can determine the “whole”.
There have been several experimental implementations of tripartite-entangled W and GHZ states using different physical resources [@laflamme-proc-98; @nelson-pra-00; @mikami-prl-05; @resch-prl-05; @roos-science-04]. GHZ and W states have been used as a resource in a quantum prisoner’s dilemma game [@han-pla-02], to simulate the violation of Bell-type inequalities [@ren-pla-09], in quantum erasers [@teklemariam-chaos-03; @teklemariam-pra-02] and complementarity measurements [@peng-pra-08], quantum key distribution [@kempe-pra-99], quantum secret sharing [@hillery-pra-99] and quantum teleportation [@yeo-prl-06]. In the context of NMR quantum computing, GHZ and W states have been generated on a one-dimensional Ising chain [@rao-ijqi-12; @gao-pra-13], their decoherence properties studied [@kawamura-ijqc-06], and their ground state phase transitions investigated in a system with competing many-body interactions [@peng-prl-09; @peng-pra-10].
This work has two main results: (a) We prescribe a scheme to create generic states of three qubits and implement it on an NMR quantum computer. The complete class of separable, biseparable and maximally entangled three-qubit states can be generated using our scheme; (b) We experimentally demonstrate the reconstruction of generic three-qubit states from their two-qubit reduced marginals. The material in this paper is organized as follows: Section \[implement\] describes the NMR implementation of a generic state with a nontrivial five-parameter set, and the implementations of the GHZ and W-states as special cases of the general scheme. The density matrices of all the states are reconstructed by using an optimal set of NMR state tomography experiments. Section \[2tomo\] describes the three-qubit state reconstruction from their two-party reduced states for a generic state and for the W-state. By comparing the state tomographs obtained from the two-qubit marginals and by a full tomography of the three-qubit state we demonstrate that, reduced two-qubit density matrices are indeed able to capture all information about the full three-qubit state. Section \[concl\] contains some concluding remarks.
NMR implementation {#implement}
==================
![ Molecular structure, NMR parameters and ${}^{19}$F thermal equilibrium spectrum of trifluoroiodoethylene. The three fluorine spins in the molecule are marked as the corresponding qubits. The table summarizes the relevant NMR parameters i.e. resonance frequencies $\nu_i$ and J-coupling constants. The ${}^{19}$F spectrum is obtained after a $\pi/2$ readout pulse on the thermal equilibrium state. The resonance lines of each qubit are labeled by the corresponding logical states of the other two qubits in the computational basis. \[system\] ](shruti_fig1.eps)
The three-qubit system that we use for NMR quantum information processing is the molecule trifluoroiodoethylene dissolved in deuterated acetone. The three qubits were encoded using the ${}^{19}$F nuclei. The Hamiltonian of the three-qubit system in the rotating frame is given by $$H = \sum_{i=1}^{3} \nu_i I_{iz} + \sum_{i <
j, i=1}^{3} J_{ij} I_{iz}
I_{jz}$$ where $I_{iz}$ is the single-spin Pauli angular momentum operator, $\nu_i$ are the Larmor frequencies of the spins and $J_{ij}$ are the spin-spin coupling constants. The coupling constants recorded are $J_{12}=69.8 $ Hz, $J_{23}= -129.0$ Hz, and $J_{13}=47.5$ Hz. Decoherence is not a major issue in this system, with average fluorine longitudinal $T_1$ relaxation times of $5.0$ seconds and $T_2$ relaxation times of $1.0$ seconds respectively. The structure of the three-qubit molecule as well as the equilibrium NMR spectrum obtained after a $\pi/2$ readout pulse are shown in Fig. \[system\]. The resonance lines of each qubit are labeled by the corresponding states of the other two coupled qubits. All experiments were performed at room temperature on a Bruker Avance III 400 MHz NMR spectrometer equipped with a z-gradient BBO probe. The three fluorine nuclei cover a very large bandwidth of 68 ppm. Standard shaped pulses (of duration $400 \mu$s) were hence modulated to achieve uniform excitation of all the three qubits by exciting smaller bandwidths simultaneously at different offsets. Individual qubits were addressed using low power ’Gaussian’ shaped selective pulses of $265 \mu$s duration. Before implementing the entangling circuits, the system was first initialized into the $\vert 000
\rangle$ pseudopure state by the spatial averaging technique [@cory-physicad], with the density operator given by $$\rho_{000} = \frac{1-\epsilon}{8} I_8
+ \epsilon \vert 000 \rangle \langle 000 \vert$$ with a thermal polarization $\epsilon \approx
10^{-5}$ and $I_8$ being an $8 \times 8$ identity matrix. The experimentally created pseudopure state $\vert 000 \rangle$ was tomographed with a fidelity of $0.99$. All experimentally generated states were completely characterized by performing NMR state tomography [@chuang-proc-98]. A modified tomographic protocol has been proposed [@leskowitz-pra-04], wherein a set of 7 operations defined by $\{ {\rm III, XXX, IIY, XYX, YII, XXY, IYY} \}$ is performed on the system before recording the signal. Here $X (Y)$ denotes a single spin operator and $I$ is the identity operator. These operators can be implemented by applying the corresponding spin selective $\pi/2$ pulses. Motivated by this modified tomographic protocol, we used an expanded set of 11 operations defined by $\{ {\rm III, IIX, IXI, XII, IIY,
IYI, YII, YYI, IXX, XXX, YYY} \}$ to determine all the 63 variables for our system of three qubits. We needed a slightly expanded set in order to perform experimentally accessible measurements that were sufficient to completely characterize the experimental density matrix with good fidelity. As a measure of the fidelity of the experimentally reconstructed density matrices, we use [@weinstein-prl-01]: $$F =
\frac{Tr(\rho_{\rm theory}^{\dag}\rho_{\rm expt})}
{\sqrt(Tr(\rho_{\rm theory}^{\dag}\rho_{\rm theory}))
\sqrt(Tr(\rho_{\rm expt}^{\dag}\rho_{\rm expt}))}
\label{fidelity}$$ where $\rho_{\rm theory}$ and $\rho_{\rm expt}$ denote the theoretical and experimental density matrices respectively.
Generic state implementation
----------------------------
![(Color online) (a) Quantum circuit showing the specific sequence of implementation of the controlled-rotation, controlled-NOT, controlled-controlled-NOT and controlled-controlled-phase gates required to construct a generic state and (b) NMR pulse sequence to implement a general three-qubit generic state; $\tau_{ij}$ is the evolution period under the $J_{ij}$ coupling. The $180^{0}$ pulses are represented by unfilled rectangles. The other pulses are labeled with their specific flip angles and phases. The last pulse (gray shaded) on the third qubit is a transition-selective $180^0$ pulse on the $\vert
110\rangle$ to $\vert 111\rangle$ transition about an arbitrary axis ${\oldhat{\mathbf{n}}}$ which is inclined at angle ($\phi+90$) with the $x$-axis. The last two rectangular pulses on the first and second qubits are $90^{0}$ $z$-rotations, to compensate the extra phases acquired (as described in the text). \[gen-ckt\] ](shruti_fig2_new.eps)
The canonical (generic) state for three qubits proposed in [@acin-prl-00] is given by: $$\begin{aligned}
& \vert \psi \rangle = a_1 \vert 000 \rangle
+ a_2 \vert 001 \rangle + a_3 \vert 010 \rangle
+ a_4 \vert 100 \rangle + a_5 e^{i \phi} \vert 111
\rangle &
\nonumber \\
& a_i \ge 0; \quad \sum_i a_i^2=1 &
\label{generic_state}\end{aligned}$$ The normalization condition leads to reduction of one parameter and hence the state has five independent non-zero, real parameters (four modulii and one phase). The state is symmetric under permutations of the qubits and the five components which are invariant under local unitaries (single-qubit operations) are the minimal number of non-local parameters required to completely specify the state. Any three-qubit state up to local unitaries, can hence be written in the form given in Eqn. (\[generic\_state\]). We base our experimental construction on this canonical form and will henceforth refer to it as the generic three-qubit state. The generic three-qubit state can be constructed by a sequence of gates, starting from the system in a pseudopure state. These gates are one-parameter unitary transformations and as will be shown, have elegant decompositions in terms of NMR pulses. The normalization condition is automatically satisfied as the normalization will be preserved under these unitary operations.
The sequence of gates with four real parameters $\alpha, \beta, \gamma,
\delta$ representing the amplitude parameters $a_1
\cdots a_5$ and the phase $\phi$ leading to the construction of a generic three-qubit state is detailed below: $$\begin{aligned}
&\vert 0 0 0 \rangle& \stackrel{U^{1}_{2
\alpha}}{\longrightarrow}
\cos{\alpha} \vert 0 0 0 \rangle +
\sin{\alpha}
\vert 1 0 0 \rangle \nonumber \\
&\stackrel{{\rm CROT}_{12}^{2 \beta}}{\longrightarrow} &
\cos{\alpha} \vert 0 0 0 \rangle +
\sin{\alpha}
\cos{\beta}
\vert 1 0 0 \rangle
+ \sin{\alpha} \sin{\beta} \vert 1 1 0 \rangle \nonumber \\
&\stackrel{{\rm CNOT}_{21}}{\longrightarrow} &
\cos{\alpha} \vert 0 0 0 \rangle +
\sin{\alpha}
\cos{\beta}
\vert 1 0 0 \rangle
+ \sin{\alpha} \sin{\beta} \vert 0 1 0 \rangle \nonumber \\
&\stackrel{{\rm CROT}_{13}^{2 \gamma}}{\longrightarrow} &
\cos{\alpha} \vert 0 0 0 \rangle +
\sin{\alpha}
\cos{\beta} \cos{\gamma}
\vert 1 0 0 \rangle \nonumber \\
&& + \sin{\alpha} \cos{\beta} \sin{\gamma}
\vert 1 0 1 \rangle
+ \sin{\alpha} \sin{\beta}
\vert 0 1 0 \rangle \nonumber \\
&\stackrel{{\rm CNOT}_{31}}{\longrightarrow} &
\cos{\alpha} \vert 0 0 0 \rangle +
\sin{\alpha}
\cos{\beta} \cos{\gamma}
\vert 1 0 0 \rangle \nonumber \\
&& + \sin{\alpha} \cos{\beta} \sin{\gamma}
\vert 0 0 1 \rangle
+ \sin{\alpha} \sin{\beta}
\vert 0 1 0 \rangle \nonumber \\
&\stackrel{{\rm CROT}_{12}^{2 \delta}}{\longrightarrow} &
\cos{\alpha} \vert 0 0 0 \rangle +
\sin{\alpha}
\cos{\beta}
\cos{\gamma} \cos{\delta} \vert 1 0 0 \rangle \nonumber \\
&& + \sin{\alpha} \cos{\beta} \cos{\gamma} \sin{\delta}
\vert 1 1 0 \rangle \nonumber \\
&& + \sin{\alpha} \cos{\beta}
\sin{\gamma} \vert 0 0 1 \rangle
+ \sin{\alpha} \sin{\beta}
\vert 0 1 0 \rangle \nonumber \\
&\stackrel{{\rm CCN}_{12,3}}{\longrightarrow} &
\cos{\alpha} \vert 0 0 0 \rangle +
\sin{\alpha}
\cos{\beta}
\cos{\gamma} \cos{\delta} \vert 1 0 0 \rangle \nonumber \\
&& + \sin{\alpha} \cos{\beta} \cos{\gamma} \sin{\delta}
\vert 1 1 1 \rangle \nonumber \\
&& + \sin{\alpha} \cos{\beta}
\sin{\gamma} \vert 0 0 1 \rangle
+ \sin{\alpha} \sin{\beta}
\vert 0 1 0 \rangle \nonumber \\
&\stackrel{{\rm Ph}_{12,3}^{\phi}}{\longrightarrow} &
\cos{\alpha} \vert 0 0 0 \rangle +
\sin{\alpha} \cos{\beta}
\sin{\gamma} \vert 0 0 1 \rangle \nonumber \\
&& + \sin{\alpha} \sin{\beta}
\vert 0 1 0 \rangle
+ \sin{\alpha}
\cos{\beta}
\cos{\gamma} \cos{\delta} \vert 1 0 0 \rangle \nonumber \\
&& + e^{\iota \phi} \sin{\alpha} \cos{\beta} \cos{\gamma}
\sin{\delta}
\vert 1 1 1 \rangle \nonumber \\
\label{genstate-nmr}\end{aligned}$$ The operator $U_{2 \alpha}^{1}$ is a separable, non-entangling transformation belonging to the $SU(2)$ group which implements a rotation by an arbitrary angle $\alpha$ on the first qubit, leading to a generalized superposition state of the qubit. The global phase is not detectable in NMR experiments and is thus ignored throughout in gate implementation; CROT$_{ij}^{2 \theta}$ implements a controlled rotation by an arbitrary angle $\theta$, with the $i^{th}$ qubit as control and $j^{th}$ as target; CNOT$_{ij}$ implements a controlled-NOT gate, with the $i^{th}$ qubit as control and $j^{th}$ as target; CCN$_{12,3}$ implements a controlled-controlled-NOT (Toffoli) gate on the $3$rd qubit i.e. it flips the state of qubit $3$, if and only if both qubits $1$ and $2$ are in the $\vert 1 \rangle$ state; Ph$_{12,3}^{\phi}$ is a controlled-controlled-phase shift gate with $1,2$ as control qubits and $3$ being the target qubit. The state thus obtained has five variables: $\alpha \in [0, \pi/2], \beta \in [0, \pi/2],
\gamma \in [0, \pi/2], \delta \in [0, \pi/2]$ and $\phi \in [0, 2\pi]$.
The quantum circuit for generic state construction is given in Fig. \[gen-ckt\](a). The circuit consists of a single-qubit rotation gate, followed by several two-qubit controlled-rotation and controlled-NOT gates, a three-qubit controlled-controlled NOT (Toffoli) gate, and finally a controlled-controlled phase gate that introduces a relative phase in the $\vert 1 1 1
\rangle$ state.
The NMR pulse sequence to construct the generic three-qubit state starting from the pseudopure state $\vert 0 0 0 \rangle$ is given in Fig. \[gen-ckt\](b). Refocusing pulses are used in the middle of all J-evolution periods to compensate for chemical shift evolution. Pairs of $\pi$ pulses have been inserted at 1/4 and 3/4 of the J-evolution intervals to eliminate undesirable evolution due to other J-couplings. The $180^{0}$ pulses are represented by unfilled rectangles, while the other pulses are labeled with their specific flip angles and phases. An ideal controlled rotation gate CROT$_{ij}$, where ‘$i$’ is control and ‘$j$’ is the target qubit ($i < j$) is implemented by the sequence : $(\theta)_{-y}^j
\, (\frac{\pi}{2})_{z}^{i,j} \, \frac{1}{4J_{ij}}
\, (\pi)_{y}^{i,j} \frac{1}{4J_{ij}} \,
(\pi)_{y}^{i,j} \, (\theta)_{-y}^j \,
(\pi)_{z}^{i,j}$ [@jones-jmr-1998]; here $(\theta)_{\alpha}^{i}$ denotes an rf pulse of flip angle $\theta$ and phase $\alpha$ applied on the $i$th qubit, $(\beta)_{\alpha}^{i,j}$ denotes an rf pulse of flip angle $\beta$ and phase $\alpha$ applied simultaneously on both the $i$th and $j$th qubits, and $\frac{1}{4 J_{ij}}$ denotes an evolution period under the coupling Hamiltonian (using standard NMR notation). The above sequence for the ideal CROT$_{ij}$ gate contains two $z$-rotations on each of the control and target qubits, which are of long duration and give rise to experimental imperfections. In order to shorten the gate duration and hence reduce experimental artifacts, we implemented a shorter pulse sequence corresponding to $(\theta)_{-y}^j \,
\frac{1}{4J_{ij}} \, (\pi)_{y}^{i,j}
\frac{1}{4J_{i,j}} \, (\pi)_{y}^{i,j} \,
(\theta)_{-x}^j $, which creates the desired state alongwith a relative phase. We keep track of the relative phase gained at the end of each controlled operation and implement $z$-rotations on the spins at the end of the sequence to compensate for the relative phases acquired. The last two gates in the circuit, namely the controlled-controlled NOT (Toffoli) gate and the controlled-controlled phase gate were simultaneously implemented using a single transition-selective $\pi$ pulse, applied about an arbitrary axis of rotation ${\oldhat{\mathbf{n}}}$ (gray-shaded in Fig. \[gen-ckt\](b)) [@dorai-jmr-1995; @peng-cpl-2001]. A three-qubit controlled-controlled NOT (Toffoli) gate can be experimentally realized by a transition-selective $(\pi)_y$ pulse between energy levels $\vert 110
\rangle$ and $\vert 111 \rangle$. A transition-selective pulse $(\pi)_{{\oldhat{\mathbf{n}}}}$ about an arbitrary axis of rotation ${\oldhat{\mathbf{n}}}=\cos
\phi^{'} {\oldhat{\mathbf{x}}} + \sin \phi^{'} {\oldhat{\mathbf{y}}}$, on the other hand, introduces an extra phase of $e^{\iota
\phi}$ ($\phi^{'}=\phi+\pi/2$). Hence, $(\pi)_{{\oldhat{\mathbf{n}}}}^{\vert 110 \rangle \rightarrow
\vert 111 \rangle}$ when applied on the basis vector $\vert 110 \rangle$, results in the state $e^{\iota \phi} \vert 111 \rangle$. This is an ingenious method to reduce the experimental time, and comes in handy in completing the circuit implementation before the decoherence begins to introduce significant distortions.
To demonstrate our general method to create generic three-qubit states, we implement our scheme to create a state with a nontrivial structure. We chose a state in which all the terms in the generic state expression given in Eqn. \[genstate-nmr\] are involved in a nontrivial way. We have chosen $\alpha=45^0,
\beta= 55^0,
\gamma= 60^0,
\delta= 58^0$ and $\phi=125^0$. This set of parameters leads to the creation of the generic state: $$\begin{aligned}
&& 0.707 \vert 000 \rangle +
0.351 \vert 001 \rangle +
0.579 \vert 010 \rangle +
0.107 \vert 100 \rangle + \nonumber \\
&& 0.172 e^{i (125^0)} \vert 111 \rangle\end{aligned}$$ The tomograph corresponding to this state is shown in Fig. \[gentomo\], wherein the experimentally tomographed state (Fig. \[gentomo\](b)) is compared with the theoretically expected state (Fig. \[gentomo\](a)). The fidelity of the experimentally tomographed state (by the definition given in Eqn. \[fidelity\]) in this case is $0.92$.
![The real (Re) and imaginary (Im) parts of the (a) theoretical and (b) experimental density matrices for the three-qubit generic state, reconstructed using full state tomography. The values of the parameters are $\alpha=45^{0},
\beta=55^{0}, \gamma=60^{0}, \delta=58^{0}, \phi=125^{0}$. The rows and columns encode the computational basis in binary order, from $\vert 000 \rangle$ to $\vert 111 \rangle$. The experimentally tomographed state has a fidelity of $0.92$. \[gentomo\] ](shruti_fig3_new.eps)
Our method is quite general and can be used to construct any generic state of the three-qubit system. Given that the relaxation times for our system are quite long and the qubits are well separated in frequency space, it is also possible to perform single-qubit operations to transform the state further.
GHZ state implementation
------------------------
Generalized GHZ states are a special case of the generic state given in Eqn. \[generic\_state\], corresponding to the parameter values $\alpha= \alpha, \beta= \gamma=0, \delta= \pi/2, \phi=0$, and the circuit given in Fig. \[gen-ckt\](a) reduces to the circuit given in Fig. \[ghz-ckt\](a). The two controlled-rotation gates CROT$_{12}^{2\beta}$ and CROT$_{13}^{2\gamma}$ are hence redundant for the state implementation and the simplified experimental circuit is given in Fig. \[ghz-ckt\](b), with a single-qubit rotation followed by two controlled-NOT gates. An arbitrarily weighted GHZ kind of entangled state can be prepared from the initial pseudopure state $\vert
0 0 0 \rangle$ by the sequence of operations $$\begin{aligned}
\vert 0 0 0 \rangle &\stackrel{U_{2\alpha}^1}{\longrightarrow}&
\cos{\alpha} \vert 0 0 0 \rangle +
\sin{\alpha} \vert 1 0 0 \rangle \nonumber \\
&\stackrel{\rm CNOT_{12}}{\longrightarrow}&
\cos{\alpha} \vert 0 0 0 \rangle +
\sin{\alpha} \vert 1 1 0 \rangle \nonumber \\
&\stackrel{\rm CNOT_{13}}{\longrightarrow}&
\cos{\alpha} \vert 0 0 0 \rangle +
\sin{\alpha} \vert 1 1 1 \rangle \end{aligned}$$ For $\alpha = \pi/4$, the above sequence leads to a pure GHZ state [@laflamme-proc-98; @nelson-pra-00; @teklemariam-pra-02]: $$\vert \psi_{\rm GHZ} \rangle = \frac{1}{\sqrt{2}}(\vert 000 \rangle
+ \vert 111 \rangle)$$
. (b) Simplified circuit for experimental implementation of the GHZ state. (c) NMR pulse sequence corresponding to the circuit in (b). The $\tau_d = \frac{\tau_{13} - \tau_{12}}{2}$ period is tailored such that the system evolves solely under the $J_{13}$ coupling term. \[ghz-ckt\] ](shruti_fig4_new.eps)
![The real (Re) and imaginary (Im) parts of the (a) theoretical and (b) experimental density matrices for the GHZ state, reconstructed using full state tomography. The rows and columns encode the computational basis in binary order, from $\vert 000 \rangle$ to $\vert 111 \rangle$. The experimentally tomographed state has a fidelity of 0.97. \[ghztomo\] ](shruti_fig5.eps)
The quantum circuit and the NMR pulse sequence used to create an arbitrary GHZ-like entangled state beginning from the pseudopure state $\vert 0
0 0 \rangle$ and ignoring overall phase factors are given in Fig. \[ghz-ckt\](b) and (c) respectively. The CNOT$_{12}$ and CNOT$_{13}$ in the circuit are controlled-NOT gates with qubit 1 as the control and qubit 2 (3) as the target. Since the target qubits are different in both these cases, these gates commute and can be applied in parallel, leading to a reduction in experimental time. For our system $\tau_{13} > \tau_{12}$, where $\tau_{ij}$ denotes the evolution period under the $\frac{1}{2J_{ij}}$ coupling term. Hence, during the period $\tau_{12}$, both qubits 2 and 3 evolve under the the J-couplings $J_{12}$ and $J_{13}$ (Fig. \[ghz-ckt\](c)). The evolution in the intervals $\displaystyle \tau_d = \frac{\tau_{13}-\tau_{12}}{2}$ is solely governed by the $J_{13}$ coupling term, and by the end of the evolution period, the system evolves under $J_{12}$ and $J_{13}$ couplings for durations $\frac{1}{2J_{12}}$ and $\frac{1}{2J_{13}}$ respectively. The state generated experimentally (Fig. \[ghztomo\](b)) was tomographed and lies very close to the theoretically expected state (Fig. \[ghztomo\](a)) with a computed fidelity of 0.97.
W-state implementation
----------------------
Generalized W-states are another special case of the generic state given in Eqn. \[generic\_state\], corresponding to the parameter values $ \alpha=\pi/2, \beta, \gamma \in
[0, \pi/2], \delta=0, \phi=0$, leading to the state $\vert \psi \rangle = \cos{\gamma}
\cos{\beta} \vert 1 0 0 \rangle + \sin{\gamma}
\cos{\beta} \vert 0 0 1 \rangle + \sin{\beta}
\vert 0 1 0 \rangle$. The circuit for generalized W-states derived from the circuit in Fig. \[gen-ckt\](a) is given in Fig. \[wstate-ckt\](a) and can be constructed by the sequential operation of the gates:
$$\begin{aligned}
\vert 000 \rangle & \stackrel{U^{1}_{\pi}}{\longrightarrow} &
\vert 100 \rangle \nonumber \\
& \stackrel{\rm CROT_{12}^{2\beta}} {\longrightarrow} & \cos \beta
\vert 100 \rangle + \sin \beta \vert 110 \rangle
\nonumber \\
& \stackrel{\rm CNOT_{21}} {\longrightarrow} & \cos \beta \vert 100
\rangle +
\sin \beta \vert 010 \rangle
\nonumber \\
& \stackrel{\rm CROT_{13}^{2\gamma}}{\longrightarrow} & \cos \gamma
\cos \beta \vert 100 \rangle + \sin \gamma \cos \beta
\vert 101 \rangle + \sin \beta \vert 010 \rangle
\nonumber \\
& \stackrel{\rm CNOT_{31}} {\longrightarrow} & \cos \gamma \cos \beta
\vert 100 \rangle + \sin \gamma \cos \beta \vert
001 \rangle +
\sin \beta \vert 010 \rangle \nonumber \\
\label{weqn}\end{aligned}$$
The first gate in the circuit, namely a rotation by $\pi$ on the first qubit, can be avoided by starting the implementation on a different initial state. We hence begin with the pseudopure state $\vert 1 0 0 \rangle$ as the initial state in our experiments. We also avoid implementing the second gate in the circuit in Eqn.( \[weqn\]), namely the controlled-rotation ${\rm
CROT}_{12}^{2\beta}$ gate, and instead implement the much simpler $U^{2}_{2\beta}$ gate on the second qubit, which in this case yields the same result. For $2 \beta=
2 \sin^{-1}{(1/\sqrt{3})}$ and $\gamma=45^{0}$, the circuit leads to implementation of the standard W-state upto a phase factor $$\vert \psi_{\rm W} \rangle = \frac{1}{\sqrt{3}}(i \vert 001 \rangle
+ \vert 010 \rangle + \vert 100 \rangle)$$ One can get rid of the extra phase factor by a single-qubit unitary gate. The simplified experimental circuit and the NMR pulse sequence for the creation of an arbitrary W-like entangled state beginning from the pseudopure state $\vert 1 0 0 \rangle$ and ignoring overall phase factors, are given in Figs. \[wstate-ckt\](b) and (c) respectively. The experimentally reconstructed density matrix (Fig. \[wtomo\](b)) matches well with the theoretically expected values (Fig. \[wtomo\](a)), with a computed state fidelity of 0.96.
. (b) Simplified circuit for experimental implementation of the W-state. (c) NMR pulse sequence to experimentally implement the W-state, starting from the initial pseudopure state $\vert 1 0 0 \rangle$. The first pulse on the second qubit implements a $U^{2}_{2 \beta}$ rotation, with $2 \beta = 2 \sin^{-1}{(1/\sqrt{3})}
\equiv 70.53^{0}$. \[wstate-ckt\] ](shruti_fig6.eps)
![The real (Re) and imaginary (Im) parts of the (a) theoretical and (b) experimental density matrices for the W state, reconstructed using full state tomography. The rows and columns encode the computational basis in binary order, from $\vert 000 \rangle$ to $\vert 111 \rangle$. The experimentally tomographed state has a fidelity of 0.96. \[wtomo\] ](shruti_fig7.eps)
Three-qubit state reconstruction from two-party reduced states {#2tomo}
==============================================================
Linden et al. discovered a surprising fact about multiparty correlations, namely, that “the parts determine the whole for a generic pure state” [@linden-prl-1-02; @linden-prl-2-02]. For three qubits, this implies that all the information in a generic three-party state is contained in its three two-party reduced states, which then uniquely determine the full three-party state. The only exceptions to the above hypothesis are the generalized GHZ states, and no set of their reduced states can uniquely determine such entangled states. This is an important result which sheds some light on how information is stored in multipartite entangled states. In a related work, Diosi et al. [@diosi-pra-04] presented a tomographic protocol to completely characterize almost all generic three-qubit pure states, based only on pairwise two-qubit detectors.
![The real (Re) and imaginary (Im) parts of the density matrix for the W-state: (a) The two-qubit reduced density matrix $\rho_{AB}$. (b) The two-qubit reduced density matrix $\rho_{BC}$. (c) The entire three-qubit density matrix $\rho_{ABC}$, reconstructed from the corresponding two-qubit reduced density matrices. The rows and columns encode the computational basis in binary order, from $\vert 00 \rangle$ to $\vert 11 \rangle$ for two qubits and from $\vert 000 \rangle$ to $\vert 111 \rangle$ for three qubits. The tomographed state has a fidelity of 0.97. \[w-2bit\] ](shruti_fig8.eps)
In this paper we describe the first experimental demonstration of this interesting quantum mechanical feature of three-qubit states. We use the same algorithm delineated by Diosi et al. [@diosi-pra-04], to reconstruct three-qubit states from their two-party reduced states. Let us consider a three-qubit pure state $\rho_{ABC}={\left\vert{\psi_{ABC}}\right\rangle}{\left\langle{\psi_{ABC}}\right\vert}$, with $\rho_{AB}$, $\rho_{BC}$, $\rho_{AC}$ being its two-party reduced states. The single-qubit reduced states $\rho_{A}$, $\rho_{B}$ and $\rho_{C}$ can be further obtained from the two-party reduced states. Since $\rho_{ABC}$ is pure, $\rho_{A}$ and $\rho_{BC}$ share the same set of eigen values, and can be written as $$\begin{aligned}
\rho_A &=&\sum_i
p_A^i {\left\vert{i}\right\rangle} {\left\langle{i}\right\vert} \nonumber \\
\rho_{BC} &=& \sum_i p_A^i {\left\vert{i;BC}\right\rangle} {\left\langle{i;BC}\right\vert}\end{aligned}$$ where $\{{\left\vert{i}\right\rangle}\}$ are the eigenvectors of $\rho_{A}$ with eigenvalues $\{p_A^i\}$, and $\{ {\left\vert{i;BC}\right\rangle}\}$ are the eigenvectors of $\rho_{BC}$ with eigenvalues $\{ p_A^i\}$. The three-qubit states compatible with $\rho_{A}$ and $\rho_{BC}$ are $${\left\vert{\psi_{ABC};\alpha}\right\rangle}=\sum_i
e^{\iota \alpha_i}
\sqrt{p_A^i}{\left\vert{i}\right\rangle}\otimes{\left\vert{i;BC}\right\rangle}$$ Using a similar argument, the set of three-qubit pure states obtained from $\rho_{AB}$ and $\rho_C$ is given by $${\left\vert{\psi_{ABC};\gamma}\right\rangle}=\sum_k
e^{\iota \gamma_k}
\sqrt{p_c^k}{\left\vert{k;AB}\right\rangle}\otimes{\left\vert{k}\right\rangle}$$ where $\{{\left\vert{k}\right\rangle}\}$ are the eigenvectors of $\rho_C$ with eigenvalues $\{p_c^k\}$ and $\{{\left\vert{k;AB}\right\rangle}\}$ are the corresponding eigenvectors of $\rho_{AB}$. Since the pure state ${\left\vert{\psi_{ABC}}\right\rangle}$ is compatible with both $\rho_{AB}$ and $\rho_{BC}$, we can determine the values of $\alpha_i$ and $\gamma_k$ such that ${\left\vert{\psi_{ABC};\alpha}\right\rangle}={\left\vert{\psi_{ABC};\gamma}\right\rangle}$. We thus obtain almost all three-qubit pure states from any two of their corresponding two-party reduced states. The set ($\rho_{AB}$, $\rho_{AC}$) or the equivalent set ($\rho_{AB}$, $\rho_{BC}$) can be used to reconstruct $\rho_{ABC}$.
![The real (Re) and imaginary (Im) parts of the density matrix for the generic state: (a) The two-qubit reduced density matrix $\rho_{AB}$. (b) The two-qubit reduced density matrix $\rho_{BC}$. (c) The entire three-qubit density matrix $\rho_{ABC}$, reconstructed from the corresponding two-qubit reduced density matrices. The parameter set includes $\alpha=45^{0}, \beta=55^{0}, \gamma=60^{0},
\delta=58^{0}, \phi=125^{0}$. The rows and columns encode the computational basis in binary order, from $\vert 00 \rangle$ to $\vert 11\rangle$ for two qubits and from $\vert 000 \rangle$ to $\vert 111 \rangle$ for three qubits. The tomographed state has a fidelity of 0.90. \[gen-2bit\] ](shruti_fig9_new.eps)
The two-party reduced states $\rho_{AB}$, $\rho_{BC}$ and $\rho_{AC}$ were computed by performing partial state tomography. The set of tomography operations performed to experimentally reconstruct all three two-party reduced states include: {III, IXI, IYI, XXI} to reconstruct $\rho_{AB}$; {III, IIX, IIY, IXX} to reconstruct $\rho_{BC}$ and {III, IIX, IIY, XIX} to reconstruct $\rho_{AC}$. Almost any three-qubit pure state $\rho_{ABC}$ (except those belonging to the generalized GHZ class) can be determined by choosing any two sets from the above. The three-party state $\rho_{ABC}$ reconstructed using the $(\rho_{AB}, \rho_{BC})$ set of two-party reduced states was compared with the same state reconstructed using complete tomography, and the results match well. For the W state we tomographed $\rho_{AB}$ and $\rho_{BC}$ to give us $$\begin{aligned}
\rho_{AB}&=&
\left(
\begin{array}{cccc}
0.36 & 0. & 0. & 0.\, -0.01 i \\
0. & 0.21 & 0.2\, +0.05 i & -0.01 \\
0. & 0.2\, -0.05 i & 0.21 & 0.01 \\
0.\, +0.01 i & -0.01 & 0.01 & 0.22 \\
\end{array}
\right)
\nonumber \\
\rho_{BC}&=&
\left(
\begin{array}{cccc}
0.34 & -0.01 & 0.\, +0.01 i & 0 \\
-0.01 & 0.3 & 0.\, +0.24 i & 0.02 \\
-0.01 & 0.\, -0.24 i & 0.2 & 0. \\
0 & 0.02 & 0 & 0.16 \\
\end{array}
\right)\end{aligned}$$ These experimental tomographed density matrices were then used to reconstruct the three-qubit W-state density matrix $\rho_{ABC}$. The thus reconstructed $\rho_{ABC}$ is given by
$$\rho_{ABC}=
\left(
\begin{array}{cccccccc}
0. & -0.01-0.02 i & 0.02\, -0.01 i & 0. & 0.02 & 0. & 0. & 0. \\
-0.01+0.02 i & 0.36 & 0.\, +0.29 i & 0.02 & -0.1+0.37 i & -0.01+0.01 i & -0.02-0.02 i & 0. \\
0.02\, +0.01 i & 0.\, -0.29 i & 0.23 & 0.\, -0.02 i & 0.3\, +0.08 i & 0.01 & -0.01+0.01 i & 0. \\
0. & 0.02 & 0.\, +0.02 i & 0. & -0.01+0.02 i & 0. & 0. & 0. \\
0.02 & -0.1-0.37 i & 0.3\, -0.08 i & -0.01-0.02 i & 0.4 & 0.02 & -0.01+0.02 i & 0. \\
0. & -0.01-0.01 i & 0.01 & 0. & 0.02 & 0. & 0. & 0. \\
0. & -0.02+0.02 i & -0.01-0.01 i & 0. & -0.01-0.02 i & 0. & 0. & 0. \\
0. & 0. & 0. & 0. & 0. & 0. & 0. & 0. \\
\end{array}
\right)$$
The reconstructed density matrix for the W-state is shown in Fig. \[w-2bit\], computed from two sets of the corresponding two-qubit reduced density matrices. The tomographed state has a fidelity of 0.97, which matches well with the fidelity of the original three-qubit density matrix of the W-state (Fig. \[wtomo\](b)). As another illustration of reconstructing the whole state from its parts, the reconstructed density matrix of the experimentally generated generic state with a parameter set: $\alpha=45^{0}, \beta=55^{0},
\gamma=60^{0}, \delta=58^{0},
\phi=125^{0}$, is shown in Fig. \[gen-2bit\]. The two-party reduced states were able to reconstruct this three-qubit state with a fidelity of 0.90, which compares well with the full reconstruction of the entire three-qubit state given in Fig. \[gentomo\](b).
Concluding Remarks {#concl}
==================
We have proposed and implemented an NMR-based scheme to construct a generic three-qubit state from which any general pure state of three-qubits (including separable, biseparable and maximally entangled states) can be constructed, up to local unitaries. Full tomographic reconstruction of the experimentally generated states showed good fidelity of preparation and we have achieved a high degree of control over the state space of three-qubit quantum systems. Generating generic three-qubit states with a nontrivial phase parameter was an experimental challenge and we archived it by crafting a special pulse scheme. It has been previously shown that in a system of three qubits, no irreducible three-party correlations exist and that all information about the full quantum state is completely contained in the three two-party correlations. We have demonstrated this important result experimentally in a system of three qubits. The three-qubit density operator $\rho_{ABC}$ is obtained by complete quantum state tomography and compared with the same three-qubit state reconstructed from tomographs of the two-party reduced density operators given by $\rho_{AB}$, $\rho_{BC}$ and $\rho_{AC}$. It is expected that our experiments will pave the way for an understanding of how information is stored in multi-partite entangled systems.
All experiments were performed on a Bruker Avance-III 400 MHz FT-NMR spectrometer at the NMR Research Facility at IISER Mohali. SD acknowledges UGC India for financial support.
[40]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\
12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty @noop [**]{} (, , ) @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{}
|
[Matthias Beck and Thomas Zaslavsky[^1]]{}
[State University of New York at Binghamton\
Binghamton, NY, U.S.A. 13902-6000]{}
[matthias@math.binghamton.edu\
zaslav@math.binghamton.edu ]{}
[*Abstract:*]{} [Meshalkin’s theorem states that a class of ordered $p$-partitions of an $n$-set has at most $\max \binom{n}{a_1,\hdots, a_p}$ members if for each $k$ the $k^{\textrm{th}}$ parts form an antichain. We give a new proof of this and the corresponding LYM inequality due to Hochberg and Hirsch, which is simpler and more general than previous proofs. It extends to a common generalization of Meshalkin’s theorem and Erdős’s theorem about $r$-chain-free set families.]{}
*Keywords*: Sperner’s theorem, Meshalkin’s theorem, LYM inequality, antichain, $r$-family, $r$-chain-free, composition of a set.
*2000 Mathematics Subject Classification.* [*Primary*]{} 05D05; [*Secondary*]{} 06A07. *Running head*: Meshalkin–Hochberg–Hirsch Bounds
*Address for editorial correspondence*:\
Thomas Zaslavsky\
Department of Mathematical Sciences\
State University of New York\
Binghamton, NY 13902-6000\
U.S.A.
An *antichain of sets* is a class of sets of which none contains another. Sperner [@sperner] bounded the size of an antichain ${{\mathcal A}}$ of subsets of an $n$-element set $S$: $$|{{\mathcal A}}| \leq {n\choose {\lfloor n/2 \rfloor }} \ ,$$ with equality if ${{\mathcal A}}= {{\mathcal P}}_{\lfloor n/2 \rfloor} (S)$ or ${{\mathcal P}}_{\lceil n/2 \rceil}(S)$, where by ${{\mathcal P}}_k(S)$ we mean the class of $k$-element subsets of $S$. Subsequently, Lubell [@lubell], Yamamoto [@yamamoto], and Meshalkin [@meshalkin] independently obtained a stronger result (of which Bollobás independently proved a generalization [@bollobas]): any antichain ${{\mathcal A}}$ satisfies $$\sum_{A\in {{\mathcal A}}}\ \frac {1}{\binom n {|A|}} \leq 1 \ ,$$ and equality holds if ${{\mathcal A}}= {{\mathcal P}}_{\lfloor n/2 \rfloor} (S) $ or ${{\mathcal P}}_{\lceil n/2 \rceil}(S)$.
We give a very short proof of a considerable generalization of these results. A class ${{\mathcal A}}$ of subsets of $S$ is *$r$-chain-free* if ${{\mathcal A}}$ contains no chain of length $r$.[^2] (A *chain* is a class of mutually comparable sets, that is, $T\subset T' \subset \cdots \subset T^{(l)}$. Its *length* is $l$.) A *weak composition of $S$ into $p$ parts* is an ordered $p$-tuple $A = (A_1,\ldots,A_p)$ such that the $A_k$ are pairwise disjoint subsets of $S$ and their union is $S$. We call $A_k$ the *$k^{\textrm{th}}$ part* of $A$. A part $A_k$ may be void (hence the word “weak”). If ${{\mathcal M}}= \{ A^1, \hdots, A^m\}$ is a class of weak compositions of $S$ into $p$ parts, we write ${{\mathcal M}}_k = \{ A^i_k\}^m_{i=1}$ for the class of distinct $k^{\textrm{th}}$ parts of members of ${{\mathcal M}}$. A multinomial coefficient of the form $\binom{n}{a_1, \hdots, a_p}$ is called a *$p$-multinomial coefficient for $n$*. Our result is:
[**Theorem.**]{}
*Let $n\geq 0$, $r\geq 1$, $p\geq 2$, and let $S$ be an $n$-element set. Suppose ${{\mathcal M}}$ is a class of weak compositions of $S$ into $p$ parts such that, for each $k<p$, ${{\mathcal M}}_k$ is $r$-chain-free. Then*
[(a)]{} $ \displaystyle \sum_{A\in {{\mathcal M}}} \dfrac{1}{\binom{n}{|A_1|,\hdots,|A_p|}} \leq r^{p-1} $, and
[(b)]{} $|{{\mathcal M}}|$ is bounded by the sum of the $r^{p-1}$ largest $p$-multinomial coefficients for $n$.
The number of $p$-multinomial coefficients for $n$ is $\binom{n+p-1}{p-1}$; if $r^{p-1}$ exceeds this we extend the sequence of coefficients with zeros.
Our theorem is a common generalization of results of Meshalkin and Erdős. The case $r=1$ of part (b) (with the added assumption that every ${{\mathcal M}}_k$ is an antichain) is the relatively neglected theorem of Meshalkin [@meshalkin]; later Hochberg and Hirsch [@hh] found (a) for this case, which implies (b). Our extension to $r>1$ is inspired by the case $p=2$, which is equivalent to Erdős’s theorem [@erdos] that for an $r$-chain-free family ${{\mathcal A}}$ of subsets of $S$, $|{{\mathcal A}}|$ is bounded by the sum of the $r$ largest binomial coefficients ${n\choose k}$, $0\leq k \leq n$, and its LYM companion due to Rota and Harper [@rota]. We need the latter for our theorem; we sketch its proof for completeness’ sake.
[**Lemma**]{} [@rota p. 198, ($^*$)]. [ *For an $r$-chain-free family ${{\mathcal A}}$ of subsets of $S$, $$\sum_{A\in {{\mathcal A}}} \frac {1}{\binom n {|A|}} \leq r \ .$$* ]{}
Each of the $n!$ maximal chains in ${{\mathcal P}}(S)$ contains at most $r$ members of ${{\mathcal A}}$. On the other hand, there are $|A|! (n-|A|)!$ maximal chains containing $A \in {{\mathcal P}}(S)$. Now count: $$\sum_{A\in {{\mathcal A}}} |A|! (n-|A|)! \leq r n! \ .$$ The lemma follows.
Our proof of our whole theorem is simpler than the original proofs of the case $r=1$. The proof of (a) here (which is different from the more complicated although equally short proof by Hochberg and Hirsch) is inspired by the beginning of Meshalkin’s proof of (b) for $r=1$. We proceed, as did Meshalkin, by induction on $p$. The case $p=2$ is equivalent to the lemma because if $(A_1, A_2)$ is a weak composition of $S$, then $A_2 = S\setminus A_1$. Suppose then that $p>2$ and (a) is true for $p-1$. Let ${{\mathcal M}}(F) = \{ (A_2, \hdots, A_p): (F, A_2, \hdots, A_p) \in {{\mathcal M}}\}$. Since ${{\mathcal M}}(F)_k \subseteq {{\mathcal M}}_{k+1}$, ${{\mathcal M}}(F)_k$ is $r$-chain-free for $k<p-1$. Thus, $$\begin{aligned}
\sum_{A\in {{\mathcal M}}} \frac {1}{\binom{n}{|A_1|, \hdots, |A_p|}}
&= \sum_{A\in {{\mathcal M}}} \frac{1}{\binom{n}{|A_1|}}
\frac{1}{\binom{n-|A_1|}{|A_2|, \hdots, |A_p|}} \\
&= \sum_{F\in {{\mathcal M}}_1} \frac{1}{\binom{n}{|F|}} \sum_{A' \in {{\mathcal M}}(F)}
\frac{1}{\binom{n-|F|}{|A_2|, \hdots, |A_p|}}
\intertext{where $A' = (A_2,\hdots,A_p)$,}
&\leq \sum_{F\in {{\mathcal M}}_1} \frac{1}{\binom{n}{|F|}} r^{p-2}
\intertext{by the induction hypothesis,}
&\leq r \cdot r^{p-2}\end{aligned}$$ by the lemma. This proves (a).
To deduce (b), write the $p$-multinomial coefficients for $n$ in any weakly decreasing order as $M_1, M_2, \hdots$, extended by $0$’s as necessary to a sequence of length $r^{p-1}$. In the left-hand side of (a), replace each of the $M_1$ terms with largest denominators by $1/M_1$. Their sum is now $1$. Amongst the remaining terms all denominators are at most $M_2$; replace the $M_2$ of them with the largest denominators by $1/M_2$. Now their sum is $1$. Continue in this fashion. The number of terms could be less than $T = M_1+ \cdots + M_{r^{p-1}}$; in that case, $|{{\mathcal M}}|<T$. Otherwise, after $r^{p-1}$ steps we have replaced $T$ terms and have on the left side of (a) a sum equal to $r^{p-1}$ plus any further terms. As the total is no more than $r^{p-1}$, there cannot be more than $T$ terms. Thus we have proved (b).
Our proof is naturally general: that of (a) would be no shorter even if restricted to $r=1$ (but the deduction of (b) would become trivial). What is more, it is applicable to projective geometries [@bz]. Furthermore, our proof, even restricted to $r=1$, is simpler than the original proofs by Meshalkin and Hochberg–Hirsch.
The upper bounds in the theorem can be attained only in limited circumstances. When $r=1$, the maxima are attained if for each $k$, ${{\mathcal M}}_k = {{\mathcal P}}_{\lfloor n/p \rfloor } (S)$ or ${{\mathcal P}}_{\lceil n/p \rceil}(S)$ [@meshalkin]. When $p=2$, the upper bounds are attained if ${{\mathcal M}}_1$ is the union of the $r$ largest classes ${{\mathcal P}}_m(S)$ [@erdos]. When $r>1$ and $p>2$, the upper bounds are only sometimes attainable, but proving this is complicated.
[9]{}
, A Meshalkin theorem for projective geometries. Submitted.
, On generalized graphs. [*Acta Math. Acad. Sci. Hung.*]{} [**16**]{} (1965), 447–452.
, On a lemma of Littlewood and Offord. *Bull. Amer. Math. Soc.* [**51**]{} (1945), 898–902.
, Sperner families, s-systems, and a theorem of Meshalkin. *Ann. New York Acad. Sci.* [**175**]{} (1970), 224–237.
, A short proof of Sperner’s theorem. *J. Combinatorial Theory* [**1**]{} (1966), 209–214.
, Generalization of Sperner’s theorem on the number of subsets of a finite set. (In Russian.) *Teor. Verojatnost. i Primenen* [**8**]{} (1963), 219–220. English trans.: *Theor. Probability Appl.* [**8**]{} (1963), 203–204.
, Matching theory, an introduction. In P. Ney, ed., *Advances in Probability and Related Topics*, Vol. 1, pp. 169–215. Marcel Dekker, New York, 1971.
, Ein Satz über Untermengen einer endlichen Menge. *Math. Z.* [**27**]{} (1928), 544–548.
, Logarithmic order of free distributive lattices. *J. Math. Soc. Japan* [**6**]{} (1954), 343–353.
[^1]: Research supported by National Science Foundation grant DMS-70729.
[^2]: An $r$-chain-free family has been called an “$r$-family” or “$k$-family”, depending on the name of the forbidden length, but we think it is time for a distinctive name.
|
---
abstract: 'The MSSM candidates arising from the heterotic MiniLandscape feature a very constrained supersymmetry breaking pattern. This includes a fully predictable gaugino mass pattern, which is compressed compared to the CMSSM, and an inverted sfermion hierarchy due to distinct geometric localisation, featuring stops as light as $ 1{\:\text{Te{\kern-0.06em}V}}$. The observed Higgs mass sets a lower bound $m_{\widetilde{g}}>1.2 {\:\text{Te{\kern-0.06em}V}}$ on the gluino mass. The electroweak fine-tuning is reduced by a UV relation between the scalar mass of the two heavy families and the gluino mass. While large parts of the favoured parameter space escape detection at the LHC, the prospects to test the MiniLandscape models with future dark matter searches are very promising.'
bibliography:
- 'heterotichiggsbib.bib'
---
DESY 12-230
[**The heterotic MiniLandscape and the\
126 GeV Higgs boson** ]{}
**Marcin Badziak$^a$, Sven Krippendorf$^b$, Hans Peter Nilles$^b$, Martin Wolfgang Winkler$^c$**\
*$^a$ Institute of Theoretical Physics, Faculty of Physics, University of Warsaw,\
ul. Hoża 69, PL–00–681 Warsaw, Poland.*\
*$^b$ Bethe Center for Theoretical Physics [and]{} Physikalisches Institut der Universität Bonn,\
Nussallee 12, 53115 Bonn, Germany.*\
*$^c$ Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, 22607 Hamburg, Germany.*
Introduction
============
In the context of heterotic orbifold compactifications a large class of MSSM models has been constructed, known as the MiniLandscape [@Lebedev:2006kn; @Lebedev:2008un; @Lebedev:2007hv]. In this MiniLandscape supersymmetry is broken by a mixture of moduli mediation and anomaly mediation, referred to as Mirage scheme [@0411066; @0503216; @0504036; @Lowen:2008fm]. As discussed in [@Krippendorf:2012ir], the moduli mediated contribution to scalar masses depends on the localisation of matter fields in the extra dimensions. There are two distinct classes of matter fields, namely matter fields arising in the untwisted sector and fields in twisted sectors. The Higgs fields and the top quark arise in the untwisted sector whereas the other fields generally arise in the twisted sector. The top quark is located in the untwisted sector to generate a large top quark Yukawa coupling. This results in soft scalar masses for untwisted fields that are suppressed compared to scalar masses for twisted fields, a UV realisation of the scheme known as [*natural SUSY.*]{}
Furthermore, the breaking to the Standard Model gauge group by turning on appropriate Wilson lines fixes the rank of the hidden sector gaugino condensate. In a significant fraction (more than 70 percent) of MiniLandscape models it leads to a gravitino mass in the multi-TeV range, i.e. low-energy SUSY is realised without additional fine-tuning of the gravitino mass [@Lebedev:2006tr]. This breaking also fixes the ratio between anomaly mediated and moduli mediated supersymmetry breaking, leading to a clear phenomenological relation between gravitino and gaugino masses, allowing a prediction for the ratio among the gaugino masses.
The resulting pattern of soft-masses has only very few parameters, the gravitino mass $m_{3/2}$ (within the multi-TeV range), the twisted and untwisted sector scalar masses $m_1$ and $m_3$ (which should satisfy $m_1\gg m_3$) as well as the $\mu$ and $B\mu$ terms. A special field theoretical engineering of the MSSM twisted sector fields could in principle decouple their mass from the gravitino mass, but we restrict ourselves to the case $m_1\leq m_{3/2}$, i.e. no enhancement compared to the scale of supersymmetry breaking.
Given the recent experimental results on supersymmetry [@:2012mfa; @CMS-PAS-SUS-12-028; @ATLAS-CONF-2012-109; @ATLAS-CONF-2012-145] and in particular the Higgs results [@atlashiggs; @cmshiggs], the goal of this paper is to identify the distinct phenomenological features of this constrained model which can be tested by current LHC and dark matter experiments. In particular we identify the following properties:
- Depending on the rank of the condensing hidden sector gauge group, the ratio among the gaugino masses is fixed to be $$(m_{\widetilde{B}}:m_{\widetilde{W}}:m_{\widetilde{g}})~=~ (1:1.3:2.6) \;\text{ or }\; (1:1.4:2.9)\;.$$
- A Higgs mass $m_h=125-126{\:\text{Ge{\kern-0.06em}V}}$ limits the possible gravitino mass to $m_{3/2}>15{\:\text{Te{\kern-0.06em}V}}$. This translates into a lower bound on the gluino mass which is given by $1.2{\:\text{Te{\kern-0.06em}V}}$.[^1]
- Parts of the parameter space lead to a realisation of suppressed third generation squark masses, in particular the stops can be as light as $\sim 1{\:\text{Te{\kern-0.06em}V}},$ while $m_1$ is of order $m_{3/2}.$ In this range a sufficiently large Higgs mass can be realised through large stop mixing which is generated radiatively similar as in [@Badziak:2012rf].
- Dark matter can be a bino-like or a higgsino-like neutralino where the correct relic density can be obtained by stop coannihilations (bino case) or non-thermal production (higgsino-case). The composition of the lightest neutralino is influenced by the geometric location of $\tau_R$ in the twisted or untwisted sector. In particular we find the lightest neutralino to be higgsino-like mostly in models with $\tau_R$ in the untwisted sector and bino-like for models with $\tau_R$ in the twisted sector.
A light higgsino for models with $\tau_R$ in the untwisted sector, arises due to the UV relation between the masses of the gauginos and the scalars of the twisted sector. This relation results in a significant cancellation among radiative contributions to the soft mass of the up-type Higgs $m_{H_u}$. This, in terms, leads to a suppressed $|\mu|$ (higgsino mass) which is related to $m_{H_u}$ through electroweak symmetry breaking conditions. Intriguingly, a direct consequence of this effect is a reduced electroweak fine-tuning.
The rest of this article is structured as follows. We first review the UV structure of SUSY breaking in the heterotic MiniLandscape, highlighting the relation between the constraints on hidden sector gaugino condensation and the appearing suppression of soft masses (section \[sec:topdownnatural\]), i.e. fix the relation between gravitino and gaugino masses. In section \[sec:phenoanalysis\] we then analyse the phenomenological properties of this class in detail before concluding in section \[sec:conclusions\].
UV SUSY breaking in the MiniLandscape {#sec:topdownnatural}
=====================================
In models of the heterotic MiniLandscape SUSY is broken in the process of moduli stabilisation, leading to a scenario with mixed moduli and anomaly mediated SUSY breaking. In [@Lowen:2008fm] it was shown that given a supersymmetric stabilisation of Kähler and complex structure moduli, a mirage pattern of suppressed gaugino masses and A-parameters compared to the gravitino mass (originally found in the context of type IIB KKLT compactifications [@0411066; @0503216; @0504036]) also arises in models of the heterotic MiniLandscape. This mirage pattern for the gaugino masses is augmented by distinct scalar masses for MSSM matter fields arising from the twisted and untwisted sector [@Krippendorf:2012ir]. In particular the geometric localization of the third generation can lead to a UV realization of the bottom-up scheme known as [*natural SUSY.*]{} Here we would like to discuss briefly how the suppression of the gauginos is fixed in this UV scheme and then shortly summarize the soft-mass pattern with its free parameters.
The SUSY breaking scales in Mirage mediation
--------------------------------------------
The suppression of gaugino masses (and A-parameters) is determined in the UV scheme as follows (see [@0411066; @0503216; @0504036] for the corresponding discussion in the type IIB KKLT setup). After stabilising the Kähler and complex structure moduli supersymmetrically [@Kappl:2010yu; @Anderson:2011cza], we are left with an effective four-dimensional theory for the yet unfixed dilaton $S,$ which should be of the following form to achieve a de Sitter stabilisation of the dilaton $$\begin{aligned}
K&=& -\log{(S+\bar{S})}+K_{\rm up}(X,\bar{X})\;,\\
W&=& C+ P e^{-bS}+ W_{\rm up}(X)\;,\end{aligned}$$ where $P$ and $C$ denote constants, $b$ is linked to the rank of the condensing hidden sector gauge group (for SU(N): $b=8\pi^2/N$), and $K_{\rm up}(X,\bar{X})$ and $W_{\rm up}(X)$ specify the Kähler and superpotential of the hidden sector matter uplifting sector. For an explicit example of such an uplifting sector along with more details on the appearance/construction of the dS minimum we refer the reader to appendix \[sec:detrho\]. The gravitino mass is set by the gaugino condensate $$m_{3/2}=\frac{|W|}{\sqrt{2s}}=\sqrt{2} P b e^{- b s_0} s_0=\frac{16\pi^2\, P}{N}\ e^{-\frac{16\pi^2}{N}}\, ,$$ where we used in the last step that the dilaton is stabilised at $s_0=2$ which is required to correctly reproduce the value of the unified gauge coupling at the high scale. Assuming $P$ to be ${\mathcal O}(1),$ the rank of the gaugino condensate sets the overall scale of the soft parameters. As discussed in [@Lebedev:2006tr], the models in the MiniLandscape (more than 70 percent) feature a hidden sector gaugino condensate with rank $N=4,5$ and hence lead to realisation of low-energy supersymmetry without any additional fine-tuning.
In this approach to moduli stabilisation, supersymmetry is predominantly broken by the hidden uplifting sector $F_X\neq 0$ and the dilaton F-term $F_S$ is only non-vanishing at sub-leading order. Explicitly one finds $$\begin{aligned}
F_X&\simeq&\sqrt{3}m_{3/2}=2 \sqrt{3} P b e^{-2b}\, ,\\
F_S &=& e^{K/2}K^{SS}D_SW\simeq 6P e^{-2b}=\frac{3 m_{3/2}}{b}\, .\end{aligned}$$
The scales of soft-masses in the MiniLandscape {#sec:scalesinminilandscape}
----------------------------------------------
[**Gaugino masses:**]{} This difference in the F-terms becomes important when looking at the soft scalar masses, in particular at the gaugino masses and A-terms which are not generated by $F_X$ but by $F_S.$ For instance, the moduli mediated gaugino masses are found to be $$M_a=\frac{F_S}{2 s_0}=\frac{3m_{3/2}}{2 b s_0}\, .$$ The suppression of the gaugino masses with respect to the gravitino mass is usually parametrised by $\varrho,$ which is defined by $$\varrho:=\frac{16 \pi^2}{m_{3/2}}\frac{F_S}{2 s_0}=\frac{12 \pi^2 }{b}=\frac{3N}{2}\, ,$$ where we have used in the last step that the hidden gauge group is $SU(N).$ For $N=4,5$ one then immediately fixes $\varrho=6,7.5.$\
[**Scalar Masses:**]{} The moduli mediated contributions to the scalar masses are determined by the coupling of the matter fields $Q_i$ to the hidden sector uplifting-field $X$ in the Kähler potential, which can be parametrised as follows $$\label{eq:Kaehler1}
K_{\rm matter}~=~ Q_\alpha \overline{Q}_\alpha\, \Bigl[
1+ \xi_\alpha\, X\, \overline{X} + \mathcal{O} (|X|^4)
\Bigr] \;,$$ where $\xi_\alpha$ denotes the effective modular weight of the matter field. As in the type IIB case, the resulting moduli mediated scalar masses are typically not loop-suppressed [@Lebedev:2006qq] and are given by $$m_{\alpha}^2=m_{3/2}^2 (1-3\xi_\alpha)\, .$$ As discussed in [@Krippendorf:2012ir] the modular weights $\xi_\alpha$ need to be distinguished for twisted and un-twisted matter fields. Typically we find the Higgses, $t_{R}$, $Q_3$ in the untwisted sector and in part of the models also $\tau_R$. So we distinguish two scenarios depending on whether $\tau_R$ is in the twisted or untwisted sector. The modular weight of fields in the untwisted sector is denoted by $\xi_3$ and their moduli mediated mass by $m_3.$ The other matter fields are in the twisted sector with modular weight $\xi_1$ and moduli mediated mass $m_1$.
Untwisted sector
------------ --------------------------------
Scenario 1 $H_{u,d},\ Q_3,\ t_R$
Scenario 2 $H_{u,d},\ Q_3,\ t_R,\ \tau_R$
We expect $\xi_3$ to be close to the no-scale value of $1/3$ resulting in almost vanishing soft scalar masses, whereas $\xi_1$ is arbitrary. Note, however, that large negative values of $\xi_1$ would lead to soft-scalar masses for the twisted fields that are not related to the gravitino mass. Although theoretically not excluded, we shall exclude this possibility at this stage as this would require a non-minimal UV construction to achieve such values for $\xi_1$ which is beyond the scope of this article.
Due to the suppression of tree-level moduli mediated contributions, anomaly mediated contributions become important, leading to a mixture of moduli and anomaly mediated soft-masses, referred to as the mirage scheme. As presented for example in [@Krippendorf:2012ir], the soft-masses including the anomaly-mediated contributions can be summarised as follows: $$\begin{aligned}
\label{eq:softmassscheme}
M_a & = & \frac{m_{3/2}}{16 \pi^2}\,\left[\varrho+b_a\,g_a^2\right]\;,\\
A_{\alpha\beta\delta} & = &
\frac{m_{3/2}}{16 \pi^2}\,\left[-\varrho+\left(\gamma_\alpha+\gamma_\beta+\gamma_\delta\right)\right]
\;,\\
m_\alpha^2 & = & \frac{m_{3/2}^2}{(16\pi^2)^2}\,\left[\varrho^2\xi_\alpha-\Dot{\gamma}_\alpha
+2\varrho\,\left(S+\overline{S}\right)\,\partial_S\gamma_\alpha
+(1-3\xi_\alpha)(16\pi^2)^2\right]\; ,\end{aligned}$$ where $\gamma_i$ denote the standard anomalous dimensions as present in anomaly mediated supersymmetry breaking (for a detailed explanation of the notation see [@Falkowski:2005ck] and in particular appendix A therein).
Summary of UV parameters {#sec:summaryuv}
------------------------
As discussed previously, the value of $\varrho$ is fixed to $\varrho=6,7.5$ depending on whether the hidden sector gauge group is SU(4) or SU(5). The modular weights for the twisted fields $\xi_1$ are a priori undetermined due to the missing UV understanding of their Kähler potential. However, the largest mass available for supersymmetry breaking masses is typically the gravitino mass and this equips us in turn with a lower limit for $\xi_1,$ i.e. $\xi_1>0.$ In contrast, $\xi_3$ for the untwisted fields is expected to be close to the no-scale value $\xi_3=1/3$ as argued in [@Krippendorf:2012ir]. In the following, we discuss different values of $\xi_{1,3}$ in terms of their respective moduli mediated scalar masses $m_{1,3}.$ As the value of $\xi_1$ determines the exact ratio between the gaugino mass and soft scalar masses, the value can become important for reducing the amount of fine-tuning in a given model. Of the remaining free parameters, $B\mu$ can be traded against $\tan\beta$, the ratio of the Higgs vacuum expectation values, while $|\mu|$ can be fixed at the low scale by requiring proper electroweak symmetry breaking. Note that within the MiniLandscape models, a weak scale $\mu$-term can emerge from an underlying approximate R-symmetry [@Kappl:2008ie].
Low-energy phenomenology from the heterotic MiniLandscape {#sec:phenoanalysis}
=========================================================
In this section, we study the phenomenological implications of the MiniLandscape models with the soft terms and parameters introduced in the previous section. As already mentioned, we distinguish two phenomenological scenarios, depending on whether $\tau_R$ is located in the twisted (scenario 1) or the untwisted sector (scenario 2).
For both scenarios we have calculated the low energy mass spectra with the modified version of Softsusy 3.2.4 [@Allanach:2001kg] described in [@Badziak:2012rf] which avoids a code problem occurring for light stops. Electroweak precision observables as well as the thermal relic density and the direct detection cross section of the lightest supersymmetric particle (LSP) were determined with MicrOMEGAs 2.4.5 [@Belanger:2010gh]. In order to illustrate the main properties of the models, we provide parameter scans in figure \[fig:scans\], where we apply various phenomenological constraints. The resulting superpartner mass spectra and other relevant observables for three benchmark points as indicated in figure \[fig:scans\] are shown in table \[tab:benchmark\]. Based on the scans we will describe the most important properties of the MiniLandscape models in the remainder of this section.
{height="8.95cm"} {height="7cm"}\
{height="8.95cm"} \[fig:scans\]
[|l|c|c|c|]{} & **BP1a** & **BP1b** & **BP2**\
\
\
$m_{3/2}$ \[TeV\] & 34 & 50 & 33\
\
\
$m_h$ \[GeV\]& 125.1 & 125.4 & 125.0\
$m_H$ \[TeV\]& 9.55 & 6.60 & 2.55\
$m_a$ \[TeV\]& 9.55 & 6.60 & 2.55\
\
\
$m_{\widetilde{\chi}_1}$ \[TeV\]& 0.941 & 1.586 & 0.493\
$m_{\widetilde{\chi}_2}$ \[TeV\]& 0.959 & 2.17 & 0.503\
$m_{\widetilde{\chi}_3}$ \[TeV\]& 1.08 & 3.78 & 1.02\
$m_{\widetilde{\chi}_4}$ \[TeV\]& 1.51 & 3.78 & 1.43\
gaugino fraction $\chi_1$& 7.7% & 99.97% & 0.9%\
higgsino fraction $\chi_1$& 92.3% & 0.03% & 99.1%\
\
\
$m_{\widetilde{\chi}_1^+}$ \[TeV\]& 0.95 & 2.17 & 0.498\
$m_{\widetilde{\chi}_2^+}$ \[TeV\]& 1.51 & 3.78 & 1.43\
\
\
$m_{\widetilde{g}}$ \[TeV\]& 3.12 & 4.42 & 2.99\
\
\
$m_{\widetilde{t}_1}$ \[TeV\] & 6.22 & 1.591 & 0.95\
$m_{\widetilde{t}_2}$ \[TeV\] & 7.53 & 4.80 & 1.51\
$m_{\widetilde{b}_1}$ \[TeV\] & 7.53 & 1.591 & 1.47\
$m_{\widetilde{b}_2}$ \[TeV\] & 16.9 & 24.7 & 16.5\
$m_{\widetilde{\tau}_1}$ \[TeV\] & 16.6 & 24.3 & 2.90\
$m_{\widetilde{\tau}_2}$ \[TeV\] & 17.1 & 25.2 & 16.5\
\
\
$m$ \[TeV\]& $\sim 17$ & $\sim 25$ & $\sim 17$\
\
\
$\Omega_{\text{LSP,thermal}} \:h^2$ & 0.1 & 0.1 & 0.03\
$\sigma_n$ \[cm$^2$\]& $1.4\cdot 10^{-44}$ & $< 10^{-50}$ & $2.6\cdot 10^{-45}$\
\
\
$\text{Br}(b \rightarrow s \gamma)$& $3.3\cdot 10^{-4}$ & $3.2\cdot 10^{-4}$ & $2.9\cdot 10^{-4}$\
$\text{Br}(B_s \rightarrow \mu\mu)$ & $3.1\cdot 10^{-9}$ & $3.1\cdot 10^{-9}$ & $3.1\cdot 10^{-9}$\
\
\
$h \rightarrow \gamma \gamma$& 1.0 & 1.0 & 1.01\
$h \rightarrow gg$& 1.0 & 1.0 & 0.98\
\[tab:benchmark\]
The superpartner spectrum {#sec:inverted}
-------------------------
In both scenarios, the sfermions of the twisted sector become very heavy from an LHC perspective. Their natural mass scale is the gravitino mass which typically takes values $m_{3/2}>10{\:\text{Te{\kern-0.06em}V}}$ in the models under consideration (see section \[sec:massbounds\]). The sfermions of the untwisted sector are expected to be significantly lighter due to their localization properties. But note that this is also a phenomenological requirement as electroweak symmetry breaking is absent for too large values of $m_3$ (yellow regions in figure \[fig:scans\]).
On the other hand, the hierarchy between the untwisted and twisted sector sfermions cannot be too large. This is because the heavy scalars of the twisted sector enter the renormalisation group equations (RGEs) of $m_{\widetilde{Q}_3}$ and $m_{\widetilde{t}_R}$ at the two-loop level. More specifically, heavy twisted sector sfermions tend to reduce the stop masses through the RGE running. If the hierarchy gets too large, the lighter stop becomes the LSP (orange regions in figure \[fig:scans\]) or even tachyonic (blue regions).
Since the soft third-generation sfermion masses are non-universal in the heterotic MiniLandscape, the heavy twisted sector sfermions may enter the RGEs of $\widetilde{Q}_3$ and $\widetilde{t}_R$ also at the one-loop level through the combination: $$S_Y = m_{H_u}^2-m_{H_d}^2+\sum_{i=1}^3[m_{\widetilde{Q}_i}^2-2m_{\widetilde{U}_i}^2+m_{\widetilde{D}_i}^2-m_{\widetilde{L}_i}^2+m_{\widetilde{E}_i}^2] \,.$$ The value of $S_Y$ is the main source of phenomenological differences between the two scenarios. For $\tau_R$ in the untwisted sector (scenario 2) $S_Y$ vanishes and does not affect the low-energy spectrum. As a result, for heavy enough twisted sector sfermions both stops can be relatively light and strongly mixed (see section \[sec:Higgs\] and [@Badziak:2012rf]). On the other hand, for $\tau_R$ in the twisted sector (scenario 1) $S_Y=m_1^2-m_3^2$ at the high scale; so it is typically positive and very large since generically $m_1\sim{\mathcal{O}}(m_{3/2})\gg m_3$. The contribution from $S_Y$ to the electroweak scale soft scalar masses is determined by the hypercharge assignment and is approximately given by $$m^2_i = -0.05 Y_i S_Y \,,$$ where $Y_i$ is the hypercharge of the sfermion $i$. In particular, $S_Y$ gives a positive contribution to $m_{\widetilde{U}}^2\simeq0.035 S_Y$ and compensates the negative two-loop effect, $(m_{\widetilde{U}}^2)^{\text{2-loop}}\simeq-0.02 m_1^2$. The contribution to $m_{\widetilde{Q}}^2\simeq-0.008 S_Y$ is negative and relatively small. In consequence, in scenario 1 only one stop can be light and the left-right stop mixing is smaller than in scenario 2. This can be seen in the benchmark point BP1b in table \[tab:benchmark\] where $\widetilde{t}_1$ becomes relatively light through the RGE effects, while $\widetilde{t}_2$ stays heavy. Moreover, since $m_{\widetilde{Q}}$ is significantly smaller than $m_{\widetilde{U}}$ at the electroweak scale, the lighter stop is mostly left-handed, in contrast to conventional models such as the CMSSM, in which $m_{\widetilde{U}}< m_{\widetilde{Q}}$ typically holds. For scenario 2, both stops may become light as in the benchmark scenario BP2.
Another consequence of a large and positive $S_Y$ in scenario 1 typically is a heavy higgsino which follows from the fact that $S_Y$ gives a large negative contribution to the electroweak scale value of $m_{H_u}^2\simeq-0.025 S_Y$ and $\mu^2\simeq -m_{H_u}^2$ from the condition of proper electroweak symmetry breaking. Therefore the LSP is typically a bino-like neutralino in scenario 1 (cf. benchmark point BP1b). However, there is some parameter space close to the “No EWSB” region where the higgsino gets light due to accidental cancellations in the RGE of $m_{H_u},$ which can be seen in the benchmark point BP1a. In scenario 2, the $\mu$ parameter is suppressed due to the vanishing $S_Y$ and due to generic cancellations in the RGE of $m_{H_u}$ which we shall discuss in more detail in the context of reduced fine-tuning in section \[sec:redfine-tuning\]. As a result, the LSP is typically an almost pure higgsino in scenario 2 (cf. benchmark point BP2).
The Higgs sector {#sec:Higgs}
----------------
In MSSM models, a Higgs mass of $m_h\simeq 126{\:\text{Ge{\kern-0.06em}V}}$ as measured by ATLAS and CMS [@atlashiggs; @cmshiggs], can only be accommodated in the presence of large loop corrections to $m_h$. Applying the decoupling limit on the MSSM Higgs bosons and assuming $\tan\beta \gg 1$, one finds [@Haber:1996fp] $$m_h\simeq M_Z^2 + \frac{3\,g_2^2\,m_t^4}{8\,\pi^2\,m_W^2}\left[ \log\left(\frac{m_{\widetilde{t}}^2}{m_t^2}\right)+\frac{A_t^2}{m_{\widetilde{t}}^2}\left(1-\frac{A_t^2}{12\,m_{\widetilde{t}}^2}\right)\right]\;,$$ where we included the dominant one-loop contributions from the top/stop sector and introduced $m_{\widetilde{t}}=\sqrt{m_{\widetilde{t}_1} m_{\widetilde{t}_2}}$. A sufficiently large $m_h$ requires either heavy stops ($m_{\widetilde{t}}\gg 1{\:\text{Te{\kern-0.06em}V}}$) or sizeable stop mixing. For a given stop mass, $m_h$ is maximized for $|A_t|=\sqrt{6}\,m_{\widetilde{t}}$.
In the MiniLandscape models, there exists the interesting possibility to generate large stop mixing through RGE effects (see also [@Badziak:2012rf] for more details on stop-mixing). As discussed previously, the RGE running of stops is affected by the heavy twisted sector sfermions. With increasing hierarchy in the scalar sector, $m_{\widetilde{t}}$ is reduced, while the trilinear coupling $A_t$ is insensitive to the choice of $m_1$ (see left panel of figure \[fig:stopmixing\]). Therefore, the stop mixing grows with increasing $m_1$. The corresponding effect on the Higgs mass is shown on the right panel of figure \[fig:stopmixing\]. The Higgs mass gets larger until a maximum is reached at $|A_t|\simeq\sqrt{6}\,m_{\widetilde{t}}$. Beyond this so-called “maximum-mixing” case, the Higgs mass decreases again.
{height="4.8cm"} {height="4.8cm"}
\[fig:stopmixing\]
In the scans in figure \[fig:scans\], we have marked the parameter space consistent with the observed Higgs mass ($m_h=125-126{\:\text{Ge{\kern-0.06em}V}}$) in green. The light green region becomes viable if we include an additional theoretical uncertainty of $3{\:\text{Ge{\kern-0.06em}V}}$ on the calculation of the Higgs mass. The RGE effects on the stop masses, which we just discussed, show up for light stop masses. This can be seen by the fact that the Higgs mass grows towards the region where the stop becomes the LSP. The effects are more pronounced in scenario 2 where $\tau_R$ resides in the untwisted sector. The reason is, that in scenario 1 only the left-handed stop gets light through the RGE effects due to the non-zero $S_Y$-parameter, i.e. maximal stop mixing cannot be reached (see section \[sec:inverted\]). In the regime with relatively large $m_3$, which is only accessible in scenario 1, the effects of stop mixing become negligible as $m_{\widetilde{t}}\gg|A_t|$ and the Higgs mass simply increases with growing $m_{\widetilde{t}}$.
Turning to the decay properties of the light Higgs $h$ in models of the MiniLandscape, we find them to be very similar to the Standard Model. This is because the decoupling limit on the Higgs sector generically applies where the couplings of $h$ are Standard Model-like. However, in the presence of light stops, the radiative decays $h\rightarrow \gamma\gamma,gg\,$ may get affected by the interference of the stop loop with the relevant Standard Model processes [@Djouadi:1998az]. For large left-right stop-mixing, the stop loop enters the amplitude with the same sign as the $W$ loop (which dominates the decay to photons), but with opposite sign as the top loop (which dominates the decay to gluons). Therefore, light stops tend to enhance $\text{Br}(h\rightarrow \gamma\gamma)$, but reduce $\text{Br}(h\rightarrow gg).$ The latter also suppresses the Higgs production through gluon fusion. The Higgs production is more affected by the stop loops than $\text{Br}(h\rightarrow \gamma\gamma)$ so the $\gamma\gamma$ production rate, $\sigma(pp\rightarrow h)\times\text{Br}(h\rightarrow \gamma\gamma)$, is reduced as compared to the SM prediction.
We have determined the branching fractions $\text{Br}(h\rightarrow \gamma\gamma,gg)$ including the SUSY contributions from the stop sector as given e.g. in [@Djouadi:2005gj]. We find them to be very close to the corresponding SM branching fractions in the parameter regions which satisfy the phenomenological constraints and are consistent with the Higgs mass bounds. In the benchmark scenarios of table \[tab:benchmark\], the deviation is at the percent level for BP2, while it is totally negligible for BP1a and BP1b. Only in the region which is already excluded either by a stop LSP or a too light $m_h,$ the branching fractions $\text{Br}(h\rightarrow \gamma\gamma,gg)$ may deviate by more than ten percent from the Standard Model values.
Mirage pattern in the gaugino sector
------------------------------------
The gaugino pattern in the MiniLandscape models is fully predictable and markedly different from standard schemes like the CMSSM. As discussed in section \[sec:scalesinminilandscape\], the gaugino masses $M_a$ receive comparable contributions from modulus and anomaly mediation, the latter being non-universal among the $M_a$. As the splitting of the gaugino masses at the high scale is fixed by the same $\beta$-function coefficients which determine their RGE running, the gaugino masses unify at an intermediate so-called “mirage scale”. At the low scale, the gaugino hierarchy in the MiniLandscape models is the same as in the CMSSM (bino lightest, gluino heaviest), but their spectrum is considerably compressed. Up to two-loop corrections, the pattern is completely fixed by the parameter $\varrho$ which determines the relative size of modulus and anomaly mediated contributions to the gaugino masses. As discussed in section \[sec:topdownnatural\], $\varrho$ can take the discrete values $\varrho=6$ and $\varrho=7.5$ depending on whether the hidden sector gauge group is a SU(4) or a SU(5). Therefore, we obtain a prediction for the physical gaugino mass pattern in the MiniLandscape models which is compared to the CMSSM in table \[tab:gauginopattern\]. Note that the compressed gaugino spectrum has important implications for SUSY searches at the LHC as discussed e.g. in [@Dreiner:2012gx; @Dreiner:2012sh].
$\boldsymbol{m_{\widetilde{B}}:m_{\widetilde{W}}:m_{\widetilde{g}}}$
------------------------ ---------------------------------------------------------------------- ---------------------
\[-4mm\] MiniLandscape ($\varrho=6$) $1\,:\,1.3\,:\,2.6$
($\varrho=7.5$) $1\,:\,1.4\,:\,2.9$
\[-4mm\] $1\,:\,1.9\,:\,5.3$
\[tab:gauginopattern\]
Lower limit on the gravitino and gluino mass {#sec:massbounds}
--------------------------------------------
As can be seen for example in figure \[fig:scans\], the measured Higgs mass sets a lower bound on the gravitino mass. This can easily be understood as the gravitino sets the overall scale of the soft terms which enter the loop contribution to $m_h$. In order to determine the limit on the gravitino mass we have chosen the remaining free parameters ($m_3$, $m_1$, $\tan\beta$, $\varrho$) within the theoretical limits as described in section \[sec:summaryuv\] such that for a given Higgs mass the gravitino mass is minimised.
For example, an increase in the twisted sector scalar masses $m_1$ compared to the gravitino mass generally speaking leads to lower $m_{3/2}$ consistent with the observed Higgs mass. If we now require $m_h>125{\:\text{Ge{\kern-0.06em}V}}$, we obtain the limit $$m_{3/2} > 15{\:\text{Te{\kern-0.06em}V}}$$ which is saturated for scenario 1 with $\tau_R$ in the twisted sector (in scenario 2 the constraint is even stronger). Note that such large gravitino masses are desirable from a cosmological perspective as the gravitino would decay before primordial nucleosynthesis, thus considerably ameliorating the gravitino problem [@Weinberg:1982zq].
This bound on the gravitino mass can be directly translated into a lower limit on the gluino mass which reads $$\label{gluinobound}
m_{\widetilde{g}} > 1.2{\:\text{Te{\kern-0.06em}V}}\;.$$ The above bound incidentally coincides with the recent lower bound on the gluino mass from the ATLAS search for gluino pair production in final states with multiple b-jets [@ATLAS-CONF-2012-145] which was found at $ m_{\widetilde{g}} \simeq 1.2{\:\text{Te{\kern-0.06em}V}}$ for our ratio of gluino to LSP mass $m_{\widetilde{\chi}_1}\lesssim 0.4 \,m_{\widetilde{g}}$ (cf. table \[tab:gauginopattern\]). Even though this limit was set under the assumption of a simple gluino decay chain, $\widetilde{g}\rightarrow t\bar{t}\widetilde{\chi}_1$ or $\widetilde{g}\rightarrow b\bar{b}\widetilde{\chi}_1$, we expect a rather similar bound[^2] in the heterotic MiniLandscape scenario since typically the gluino decays via off-shell stops to $t\bar{t}$ pairs associated by (not necessarily the lightest) neutralino or to $b\bar{t}$ (or $\bar{b}t$) and a chargino so that the final states are rich in b-jets. Although the lower bound is very close to the current experimental lower gluino mass limit, large fractions of the parameter space favour a gluino mass in the multi TeV range, i.e. in particular above the energy accessible at the LHC (see e.g. [@Baer:2012vr] for the gluino mass reach of the LHC). Note, however, that the constraints on $m_{3/2}$ and $m_{\widetilde{g}}$ get weaker if we take into account the theoretical uncertainty of the Higgs mass calculation by the spectrum calculator. For instance, if we allow for a Higgs mass $m_h=122{\:\text{Ge{\kern-0.06em}V}}$ (corresponding to a theoretical uncertainty of 3 GeV), we find the limit $m_{3/2}>7.5{\:\text{Te{\kern-0.06em}V}}$ which implies $m_{\widetilde{g}}>600{\:\text{Ge{\kern-0.06em}V}}$. However this would be below the current LHC search limits for gluinos.
Flavour constraints
-------------------
Flavour constraints on the MiniLandscape models mainly arise from the decay $b\rightarrow s \, \gamma$. The present experimental value of the branching fraction [@Asner:2010qj] $$\text{Br}(b\rightarrow s \, \gamma)=\left(3.55 \pm 0.24 \pm 0.09\right)\cdot 10^{-4}$$ is consistent with the Standard Model expectation $\text{Br}^\text{SM}(b\rightarrow s \, \gamma)=(3.2\pm 0.2)\cdot 10^{-4}$ [@Misiak:2006zs]. Large SUSY contributions to $b\rightarrow s \, \gamma$ may be generated through charged Higgs or chargino/squark loops. In the MiniLandscape models, mainly the chargino/stop contribution is relevant as these fields are relatively light in some part of the parameter space (while the charged Higgs bosons are typically heavy). This is very similar as in the inverted hierarchy models discussed in [@Badziak:2012rf]. We have used MicrOMEGAs 2.4.5 [@Belanger:2010gh] to calculate $\text{Br}(b\rightarrow s \, \gamma)$ for the MiniLandscape models, the corresponding exclusions on the parameter space are shown in figure \[fig:scans\]. As it is difficult to estimate the theoretical uncertainty of the calculation by MicrOMEGAs, we have chosen a rather conservative interval of $3\sigma$ around the experimental central value to obtain the bound. As can be seen in figure \[fig:scans\], a small region of parameter space which otherwise would be viable is indeed excluded by the $b\rightarrow s \, \gamma$ constraint. The higgsino-like chargino is typically lighter in scenario 2 ($\tau_R$ in the untwisted sector) and, correspondingly, the exclusion is stronger for this case. Note that the SUSY contribution to $b\rightarrow s \, \gamma$ grows with increasing $\tan\beta$, and that it would flip sign for a negative choice of $\mu$.
We have verified that within the MiniLandscape models, the SUSY effects on $B_s\rightarrow \mu\mu$ are negligible unless for very large $\tan\beta$. In addition, the anomalous magnetic moment of the muon is not affected due to the sleptons being heavy.
Reduced fine-tuning {#sec:redfine-tuning}
-------------------
The absence of SUSY signals at the LHC as well as the rather large mass of the Higgs boson $m_h\simeq 126{\:\text{Ge{\kern-0.06em}V}}$ seem to prefer the superpartner mass scale to be considerably above the weak scale. On the other hand, heavy superpartners threaten the naturalness of supersymmetric theories as the mass of the $Z$ boson is connected to the scale of the soft terms. Any splitting between the electroweak scale and the scale of the soft terms requires fine-tuning; this is the so-called little hierarchy problem of the MSSM.
The little hierarchy problem can considerably be ameliorated within the models from the heterotic MiniLandscape, which we would like to specify in the following. As discussed in section \[sec:Higgs\], the hierarchy in the scalar sector may induce large stop mixing through the RGE running. The effect is especially strong in scenario 2 ($\tau_R$ in the untwisted sector), where a Higgs mass $m_h\sim 126{\:\text{Ge{\kern-0.06em}V}}$ can still be realised with both stops at around $1{\:\text{Te{\kern-0.06em}V}}$. In this scenario there exists a mechanism to reduce the fine-tuning even further. To illustrate this, it is instructive to express $M_Z^2$ in terms of the high scale parameters. For scenario 2, we find (for $\tan{\beta}=10$) $$\label{eq:zmass}
M_Z^2\simeq 2.8\, M_3^2 - 0.4\, M_2^2 - 0.7\,A_t\,M_3 + 0.2\, A_t^2 - 0.1\,m_3^2 - 0.01\,m_1^2 - 2 \mu^2\;.$$ The fine-tuning can then be defined as the sensitivity of $M_Z$ with respect to a certain high scale parameter.
One might think that the fine-tuning is always dominated by the gluino due to the large coefficient in front of $M_3$. Note, however, that the twisted sector sfermions are much heavier than the other superpartners in the considered scheme. Their contribution to $M_Z$ is therefore non-negligible despite the small coefficient in front of $m_1$. In addition, $M_3$ and $m_1$ are related as they both depend on the gravitino mass (cf. equation ) $$M_3 = \frac{m_{3/2}}{16\,\pi^2}\,(\varrho-1.5)\;, \qquad m_1 = \sqrt{1-3\,\xi_1}\:m_{3/2}\;.$$ As the parameter $\varrho$ may only take the discrete values $6$ or $7.5$ (depending on the hidden sector gauge group), it can easily be verified that, especially for $\varrho=7.5,$ there arise strong cancellations between the gluino and sfermion contributions to $M_Z.$ In this case the fine-tuning is considerably reduced. The most favourable situation is achieved for a small, but non-zero $\xi_1$ which constitutes the best compromise between a low fine-tuning with respect to $m_{3/2}$ and a low fine-tuning with respect to $\xi_1$. Note that such cancellations do not arise if $\tau_R$ resides in the twisted sector (scenario 1) as in this case $M_Z$ receives an additional contribution $\sim 0.05 m_1^2$ due to the non-zero $S_Y$ parameter (see section \[sec:inverted\]). Therefore, in the MiniLandscape models, the fine-tuning is generically lower if $\tau_R$ is in the untwisted sector.
Dark matter
-----------
#### Thermal production {#thermal-production .unnumbered}
In scenario 1 with $\tau_R$ in the twisted sector, the higgsino is typically quite heavy and, consequently, we obtain a mostly bino LSP in wide regions of parameter space. As there exists no bino-bino-gauge boson vertex at tree-level, its annihilation cross section is suppressed. Therefore, we encounter thermal overproduction of dark matter in most of the parameter space shown in the upper panel of figure \[fig:scans\]. However there are two exceptions:
- If the stop becomes light through RGE running, stop coannihilations may suppress the bino abundance. Indeed, there is a very thin stripe at the border of the stop LSP region where the thermal LSP density is equal or less than the dark matter density. The benchmark point BP1b in table \[tab:benchmark\] is chosen from this region.
- On the other hand, the individual contributions in the RGE of $m_{H_u}$ may cancel accidentally in which case the higgsino gets lighter. In the upper panel of figure \[fig:scans\], a light higgsino occurs close to the region where electroweak symmetry breaking is absent. For a sufficient higgsino fraction, the LSP abundance again (under)matches the dark matter abundance (cf. benchmark point BP1a).
In scenario 2 with $\tau_R$ in the untwisted sector, the LSP is typically a higgsino with a small bino and wino admixture. In the early universe the higgsinos undergo very efficient annihilations into gauge bosons and third generation quarks. Their effective cross section is further enhanced by coannihilation processes including the charged higgsinos. Therefore, the thermal higgsino density $\Omega_{\text{LSP,thermal}}$ is generically suppressed. In the lower panel of figure \[fig:scans\] we thus find $\Omega_\text{LSP,thermal}<\Omega_\text{DM}$, where $\Omega_\text{DM}\,h^2\simeq 0.1$ is the dark matter density (cf. benchmark point BP2). This holds except for a tiny region at low $m_{3/2}$ where the LSP contains a considerable bino fraction.
#### Non-thermal production and dilution {#non-thermal-production-and-dilution .unnumbered}
As in any locally supersymmetric theory, the energy content of the universe may be affected by the presence of moduli fields. We assume that the matter field which dominantly breaks supersymmetry decouples from the low-energy theory (in the toy model presented in appendix \[sec:detrho\], $X$ receives a large mass through the effective $(X\bar{X})^2$ term in the Kähler potential, see also [@Dine:1983ys; @Coughlan:1984yk; @Greene:2002ku]). Therefore, we merely have to deal with the dynamics of the dilaton. In the early universe, the latter gets displaced from its zero temperature minimum by inflation [@Coughlan:1983ci] and by finite temperature effects during the reheating phase [@Buchmuller:2004xr]. The dilaton amplitude at reheating $\delta s_\text{RH}$ due to thermal effects can be approximated as $$\label{eq:thermaldisplacement}
\delta s_\text{RH}\sim \frac{T_R^4}{m_s^2\,M_P}\,,$$ where $T_R$ denotes the reheating temperature, $m_s$ the dilaton mass and $M_P$ the Planck mass. The amplitude induced by inflation depends on the details of the inflationary model.[^3] Subsequent to reheating, the dilaton undergoes coherent oscillations. The corresponding energy density decreases as $a^{-3}$ where $a$ is the scale factor of the universe. Especially, it redshifts slower than radiation and may contribute significantly to, or even dominate the energy content of the universe at late times. By its decay, the dilaton produces Standard Model fields and superpartners, the latter cascading to neutralino LSPs. Therefore, we are left with a non-thermal contribution to the dark matter density. Depending on the dilaton mass and energy density (prior to decay), the so-produced neutralino LSPs may considerably reduce their abundance through annihilations if the corresponding cross section is large enough.
If we assume that the dilaton density never dominates the energy content of the universe, and that neutralino annihilation after dilaton decay is negligible, the non-thermal neutralino relic density can be estimated as $$\Omega_\text{LSP,non-thermal} \sim \frac{m_\chi\,\mathcal{S}_0}{3\,H_0^2\,M_P^2}\,\frac{m_s\,{(\delta s_\text{RH})}^2}{T_R^3}\,,$$ where $m_\chi$ denotes the mass of the neutralino LSP, $\mathcal{S}_0\simeq 2900{\:\text{cm}}^{-3}$ the present entropy density and $H_0=71{\:\text{km}}{\:\text{s}}^{-1}/\text{Mpc}$ the Hubble parameter. The total dark matter density then simply reads $\Omega_\text{LSP}=\Omega_\text{LSP,thermal}+\Omega_\text{LSP,non-thermal}$. Given a dilaton mass of $\mathcal{O}(1000{\:\text{Te{\kern-0.06em}V}})$ and considering only the thermal effects (cf. equation ) $\Omega_\text{LSP,non-thermal}$ is in the range of the dark matter density for $T_R\sim10^8-10^9{\:\text{Ge{\kern-0.06em}V}}$.
If the dilaton dominates the energy content at its decay, it dilutes the thermal dark matter abundance. However, this scenario typically suffers from the overproduction of non-thermal dark matter and/or from a moduli induced gravitino problem [@Endo:2006zj].
The settings with $\tau_R$ in the untwisted sector typically come with a higgsino LSP, i.e. $\Omega_\text{LSP,thermal}<\Omega_\text{DM}$. Non-thermal higgsinos may easily account for the remaining fraction of the dark matter. The class of models with $\tau_R$ in the twisted sector, which typically have a bino LSP, suffer from thermal overproduction of dark matter unless in the small parameter regions with stop coannihilations or bino/higgsino mixing. In the remaining parameter space, a consistent picture with bino dark matter arises only if the thermal abundance is sufficiently diluted by modulus decay. In turn, this causes problems associated with the regeneration of binos and gravitinos (see above). Nevertheless, we should not completely exclude this possibility as these follow-up problems may be solved through minimal extensions of the hidden sector [@Dine:2006ii].
#### Direct detection {#direct-detection .unnumbered}
In the following, we will consider the constraints which arise from direct dark matter detection. In the considered models, the scattering of the lightest neutralino off nucleons is typically dominated by the exchange of the light Higgs. The corresponding cross section can be written as[^4] $$\sigma_n \simeq \frac{4 \, m_n^4}{\pi} \,f_q^2\,\left(f^n_u + f^n_d + f^n_s + \frac{6}{27} \,f^n_G \right)^2\;.$$ Here $f^n_u$, $f^n_d$, $f^n_s$ and $f^n_G$ specify the up-, down-, strange-quark and gluon contribution to the nucleon mass $m_n$.[^5] The effective neutralino quark coupling divided by the quark mass reads $$f_q = \frac{g_{h\chi_1\chi_1}}{\sqrt{2} \, v_\mathrm{EW}}\,\frac{1}{m_h^2}\,,$$ with $v_\mathrm{EW}$ being the Higgs vacuum expectation value and the neutralino-Higgs coupling $g_{h\chi_1\chi_1}$ can be taken from [@Rosiek:1995kg] for instance. Important for us is that $g_{h\chi_1\chi_1}$ vanishes in the limit of a pure gaugino or higgsino LSP, while it becomes large for a strongly mixed state. Therefore, we expect strong direct detection signals, whenever the mass splitting between gauginos and higgsinos is small.
We have systematically calculated $\sigma_n$ with MicrOMEGAs which automatically takes into account further sub-dominant contributions to $\sigma_n$ arising e.g. from squark exchange in the s-channel. On these results for the cross sections, we applied the constraints from the XENON100 direct dark matter search [@Aprile:2012nq]. The corresponding exclusions on the parameter space of the considered models are indicated in figure \[fig:scans\]. Note that in order to apply the constraints, we have assumed that the neutralino LSPs make up the total dark matter density ($\Omega_\text{LSP}=\Omega_\text{DM}$), irrespective whether they have to be produced thermally or non-thermally. In case the neutralinos only account for a sub-dominant fraction of the dark matter, the constraints would become correspondingly weaker.
In scenario 2 with $\tau_R$ in the untwisted sector, $\sigma_n$ is generically sizeable. The LSP is dominantly higgsino, but through the non-negligible bino and wino admixture the cross section with nucleons is enhanced as described above. The region with $m_{3/2}\leq 20{\:\text{Te{\kern-0.06em}V}}$ where the gauginos are rather light, is already excluded by the latest XENON100 search. The entire parameter space shown in the lower panel of figure \[fig:scans\] is in reach of the next generation direct detection experiments like XENON1T.
If $\tau_R$ resides in the twisted sector (scenario 1), the cross section $\sigma_n$ is typically suppressed. Only close to the region were electroweak symmetry breaking is absent, the higgsino becomes sufficiently light to induce a sizeable gaugino-higgsino mixing, increasing $\sigma_n$. In the upper panel of figure \[fig:scans\], a large part of the region where the thermal neutralino abundance matches the dark matter abundance is therefore excluded by XENON100. In the remaining parameter space, the higgsino becomes rather heavy and decouples. Especially, in the region with bino stop coannihilations, the cross section $\sigma_n$ is so small that it is not even in reach for the next generation direct detection experiments.
#### Indirect detection {#indirect-detection .unnumbered}
Dark matter annihilations in the galactic halo or within dense substructures give rise to cosmic rays which can potentially be observed in the vicinity of the earth. As the measured fluxes of antiprotons and gamma rays are consistent with astrophysical backgrounds, limits on the dark matter annihilation cross section can be set (see e.g. [@Donato:2008jk; @Kappl:2011jw; @Ackermann:2011wa]). Most relevant for the MiniLandscape models are the constraints from the search for gamma rays from dwarf spheroidal galaxies performed by the Fermi-LAT collaboration [@Ackermann:2011wa]. These are applied in figure \[fig:scans\] and determine the Fermi-LAT excluded region (cyan). Scenarios with light higgsinos are excluded as they yield strong annihilations into $W$ and $Z$ boson pairs which induce a too large flux of photons. However, only the models with $\tau_R$ in the untwisted sector (scenario 2) have a light higgsino and are constrained by Fermi.[^6] Note also that for the exclusion to hold, we have assumed that the entire dark matter is in the form of the lightest neutralino which requires non-thermal production.
Conclusions {#sec:conclusions}
===========
The MiniLandcape models of heterotic orbifold compactifications give rise to a very predictive MSSM soft mass pattern. In particular, the interplay between supersymmetry breaking and dilaton stabilisation fixes the ratio of the gaugino masses and their relation to the gravitino mass. In the scalar sector, the distinction between bulk fields (untwisted sector) and localised fields (twisted sector) induces an inverted hierarchy, a scheme also known as [*natural SUSY.*]{} The sfermions of the first two generations as well as $\widetilde{L}_3$ and $\widetilde{b}_R$ receive masses in the multi-TeV range, all of which are well beyond the LHC reach. The stops and Higgs fields as well as – depending on the construction – $\widetilde{\tau}_R$ remain lighter.
Experimental constraints considerably restrict the heterotic MSSM models. Especially we find that, in order to accommodate the observed Higgs mass, a gluino with $m_{\widetilde{g}}>1.2{\:\text{Te{\kern-0.06em}V}}$ is required. In most of the viable parameter space the gluino is substantially heavier than the lower bound, implying that the gluino may not be accessible at the LHC. Despite the heavy gluino, the electroweak fine-tuning can be ameliorated as there tend to be cancellations in the RGE of $m_{H_u}$ between the contributions of the gluino and the sfermions of the two heavy families.
In some part of the parameter space the stops are as light as $\sim 1{\:\text{Te{\kern-0.06em}V}}$, where a sufficiently large Higgs mass follows from strong stop mixing. This mixing is generated by a two-loop RGE effect which is driven by the heavy twisted sector sfermions and enhances $A_t$ relative to the stop soft masses. Upcoming dedicated stop and sbottom searches by ATLAS and CMS will potentially allow to test this particular corner of the parameter space.
The composition of the neutralino LSP depends strongly on the localisation properties of $\tau_R$. We typically find a higgsino-like LSP in the models with $\tau_R$ in the untwisted sector and a bino-like LSP in the models with $\tau_R$ in the twisted sector. We have shown that for a higgsino LSP the correct dark matter density can be obtained non-thermally by dilaton decay, while in the bino case it can be produced thermally through stop coannihilations. Still, a large fraction of the parameter space with a bino LSP is disfavoured by thermal overproduction unless one allows for very special cosmological histories.
If the neutralino LSP accounts for the observed dark matter, the direct detection cross section is generically large enough to be probed by the next generation direct detection experiments for the models with $\tau_R$ in the untwisted sector, while those with $\tau_R$ in the twisted sector would partially escape detection.
Acknowledgments {#acknowledgments .unnumbered}
===============
In particular we would like to thank Michael Ratz for extended conversations at early stages of this project. Furthermore it is a pleasure to thank Ben Allanach and Jamie Tattersall for useful conversations. This work was partially supported by the SFB–Transregio TR33 “The Dark Universe” (Deutsche Forschungsgemeinschaft) and the European Union 7th network program “Unification in the LHC era” (PITN–GA–2009–237920). MB also acknowledges partial support by the National Science Centre in Poland under research grant DEC-2011/01/M/ST2/02466.
Determining $\varrho$ in heterotic orbifold compactifications {#sec:detrho}
=============================================================
Let us review in this appendix the scheme of moduli stabilisation for the heterotic orbifold compactifications in some detail, discussing an explicit example for a hidden matter sector that can be used for uplifting. Here we are interested in a setup that stabilises complex structure and Kähler moduli supersymmetrically, leaving only the dilaton $S$ unstabilised. The superpotential and leading order Kähler potential dependence of the dilaton are given by $$\begin{aligned}
K&=&-\log{(S+\bar{S})}\, , \\
W&=& P e^{-b S}\, ,\end{aligned}$$ where the coefficient $b$ in the exponent is determined by the rank of the gauge group inducing the hidden sector gaugino condensate; for $SU(N)$ $b$ is given by $b=8\pi^2/N.$ To obtain the correct gauge coupling at the unification scale, we need to stabilise the dilaton at $s_0 \simeq 2\, .$ The potential around this desired minimum is given by the typical run-away potential for the dilaton (also shown in figure \[fig:runaway\] on the left panel) $$V=2 P^2 b^2 s e^{-2bs}\, ,
\label{eq:runawaypot}$$ where $s$ is the real part of the dilaton. Note that we are in the limit $bs\gg 1$ for any reasonable value of the hidden sector gauge group.
![Run-away potential using the parameters $N=4$ and $P=1$ in equation (left panel). The AdS potential for $N=4$ and $P=1$ with C adjusted using the relation from equation (right panel).[]{data-label="fig:runaway"}](dilatonrunaway "fig:"){height="4.4cm"} ![Run-away potential using the parameters $N=4$ and $P=1$ in equation (left panel). The AdS potential for $N=4$ and $P=1$ with C adjusted using the relation from equation (right panel).[]{data-label="fig:runaway"}](dilatonads "fig:"){height="4.4cm"}
To stabilise the dilaton, it is sufficient to include a constant contribution to the superpotential as the flux contribution in the type IIB context $$W=C+P e^{-b S}\, .$$ Such a hierarchically small constant can appear in the presence of approximate R-symmetries as discussed in [@Kappl:2008ie]. This allows to construct a supersymmetric minimum as in the type IIB framework of KKLT [@Kachru:2003aw] by demanding a supersymmetric stabilisation of the dilaton $D_S W=0,$ leading to the following condition $$-P e^{-b s} (1+2 b s)= C\, .
\label{eq:crel}$$ Numerically this requires a hierarchically small constant $C\simeq 5.72\times 10^{-16}$ for $A=1$ and $N=4$ to obtain a minimum at $s=2.$ Applying this relation for $C$ for any value of $s$ leads to an AdS minimum (as shown in figure \[fig:runaway\] in the right panel) with a vacuum energy $$V_0=-\frac{3|W|^2}{2 s}= - \frac{3|C+P e^{-2b}|^2}{4}=-3|Pbs|^2 e^{-2bs}\, .$$ The gravitino mass is given by $$m_{3/2}=\frac{|W|}{\sqrt{2s}}=\sqrt{2} P b e^{- b s_0} s_0=2P b e^{-2b}\, .
\label{eq:m32}$$ To uplift the vacuum energy a positive contribution to the potential of the following type needs to be added $$V_{\rm up}=\frac{r}{(2s)^m}\, ,$$ where $m$ depends on the choice of the uplifting mechanism and $$r=(2s_0)^m |V_0|\, .$$ Depending on $m$ the minimum is slightly shifted through uplifting. This shift in $s,$ denoted by $\Delta s,$ can be determined by looking at the vanishing of the first derivative of the potential to be given by $$\Delta s=\frac{3 m s_0}{-1+b s_0+2 b^2 s_0^2-3 m (1+2 b s_0)}\, .
\label{eq:ds}$$ We are interested in the case $s_0=2$ and $m=1$ for which the shift in $s$ becomes[^7] $$\Delta s=-\frac{3}{2+5 b-4 b^2}\simeq \frac{3}{4 b^2}\, .$$ Given this potential for the dilaton, one can in principle use various mechanisms to uplift the vacuum energy with F-terms or D-terms as discussed in the type IIB context. In realistic models from heterotic orbifold compactifications such a hidden sector for uplifting needs to be generated, but this is beyond the scope of this article and we restrict ourselves to a field theoretical toy-model at this stage. Here we discuss the example of a quantum corrected O’Raifeartaigh model as in [@Kallosh:2006dv], which allows for a minimum near the origin of the additional matter field $X.$ Such a matter construction with a minimum near the origin is desired as it guarantees the absence (respectively suppresses hierarchically) off-diagonal entries in the Kähler metric between the uplifting sector and the dilaton. If such an off-diagonal coupling is not sufficiently suppressed, additional contributions to the potential are generated and the original stabilisation procedure is changed. We take the following potential for the additional field $X$ $$W_X=-\mu^2 X\, , \qquad K_X= X \bar{X}-\frac{(X\bar{X})^2}{\Lambda^2}\, .$$ We are interested in the limit (more justification on this limit can be found in [@Kallosh:2006dv]) $$\mu^2,\Lambda^2\ll 1\, .$$ Near $x\simeq 0$ and using the above relations for the coefficients, the potential is approximated by $$V_X=\mu^4\left(1+\frac{x^2}{\Lambda^2}\right)\, .$$ The potential clearly has a positive minimum around $x\simeq 0$ and hence can be used as an uplifting potential for the dilaton field.
Let us now discuss this combined potential of $X$ and $S$ $$\begin{aligned}
K&=& -\log{(S+\bar{S})}+X \bar{X}-\frac{(X\bar{X})^2}{\Lambda^2}\, ,\\
W&=& C + P e^{-b S}-\mu^2 X\, .\end{aligned}$$ The imaginary parts are stabilised as before at zero, given that $C$ is negative which we assume from now on, otherwise the phases adjust such that this situation applies (see [@Kallosh:2006dv] for more details). Focusing on the real parts, the potential can be written as $$V=|F_S|^2+|F_X|^2-3 m_{3/2}^2\, ,$$ where the gravitino mass is the one from equation . To cancel the vacuum energy $\mu$ is adjusted such that $$F_X=\frac{\mu^2}{2}\simeq\sqrt{3}m_{3/2}=2 \sqrt{3} P b e^{-2b}\, ,$$ where the approximation is due to the slight shift in $X$ and $S$ and the resulting shift in their F-terms. Due to the small shift in $s,$ its F-term is no longer vanishing and we find to leading order $$F_S=e^{K/2}K^{SS}D_SW\simeq 6P e^{-2b}=\frac{3 m_{3/2}}{b}\, .$$ Given this structure in the F-terms on finds, as discussed in section \[sec:scalesinminilandscape\], various phenomenologically interesting properties of the soft-masses, such as for example the suppression of the gaugino masses compared to the scalar masses.
[^1]: This bound can in principle be lowered depending on the theoretical uncertainty for the Higgs mass.
[^2]: The exact limit may be slightly weaker than in the simplified model because the gluino usually does not decay directly to the LSP.
[^3]: Note that, assuming the mechanism of dilaton stabilization we employ here, strong constraints on the model of inflation arise from the requirement that the dilaton does not get destabilised during the inflationary phase (see e.g. [@Kallosh:2004yh]).
[^4]: We neglect the small differences between the masses of the neutron and proton.
[^5]: Note that the quantity $f^n_s$ is subject to large experimental uncertainties. In order not to overestimate the direct detection constraints, we made the rather conservative choice $f^n_s=0.13$ consistent with [@Gasser:1990ce]. Larger values of $f^n_s$ were suggested by [@Pavan:2001wz].
[^6]: In scenario 1 there is an extremely thin band of parameter space with a light higgsino which is excluded by Fermi at the border of the “No EWSB” region (yellow). We do not show this in figure \[fig:scans\] to avoid clutter.
[^7]: In KKLT constructions the uplifting usually scales as $m=3$ and hence leads to $$\Delta \tau\simeq\frac{9}{2 b^2 \tau}\, ,$$ where $\tau$ represents the real part of the Kähler modulus $T$ that is stabilised by non-perturbative effects.
|
---
abstract: 'When is a topological branched self-cover of the sphere equivalent to a rational map on $\CC\PP^1$? William Thurston gave one answer in 1982, giving a negative criterion (an obstruction to a map being rational). We give a complementary, positive criterion: the branched self-cover is equivalent to a rational map if and only if there is an elastic spine that gets “looser” under backwards iteration.'
address: 'Department of Mathematics, Indiana University, Bloomington, Indiana 47405, USA'
author:
- 'Dylan P. Thurston'
bibliography:
- 'dylan.bib'
- 'conformal.bib'
- 'geom.bib'
- 'topo.bib'
- 'misc.bib'
- 'curves.bib'
date: 'December 13, 2016'
title: A positive characterization of rational maps
---
|
phyzzx =0.2truein =0.1truein =6truein \#1[to ]{} \#1
.5in
\#1
\#1
.2in
\#1
\#1
.2in
**Abstract**
\#1
=10000
å[a\_]{}
Introduction
============
It is now well-known that anyons—particles with arbitrary statistics can exist in 2+1 dimensions. Owing to the topological gauge potential, even non-interacting two-anyon states are not (symmetrized) tensor products of single anyon states. Thus the quantum mechanics of systems of many anyons presents a challenge to field theorists. In general, the center of mass motion of the system is not sensitive to the statistics and can be factored out. For two anyons, the relative coordinates present us with a one-body problem which can be solved in many cases. For $ N$ anyons, there are $N-1$ relative coordinates, whereas there are $N(N-1)/2$ pairs of particles. These two numbers match only when $N=2$. For $N>2$, the various pair-separation coordinates (in terms of which the statistical gauge potential is easy to write down) are not independent of each other. For this reason, not a single three-anyon system has been completely solved. The many-anyon system has been studied in the mean field approach. Other methods that have been employed include the semi-classical approximation, numerical studies, perturbative analysis from the bosonic or fermionic ends, and the most interesting of all, the ladder operator approach: In the last couple of years, a substantial subset of the exact multi-anyon wavefunctions in a magnetic field has been found with this systematic analysis. This soon generalized to free anyons and anyons of multi-species. While a lot of states are still missing, all the states in the lowest Landau level are obtained by this method.
In this paper, we apply the ladder operator approach to non-Abelian Chern-Simons particles which may be regarded as a generalisation of anyons. In the gauge that the Hamiltonian is a free Hamiltonian, the wavefunctions (which have more than one component) are multivalued with non-trivial monodromy properties given by a monodromy matrix. By introducing statistical gauge potentials, one has the liberty to work with single-valued wavefunctions. However, since we find multivalued wavefunctions convenient to work with, we will stick to them in the rest of this paper.
In section 2, we review the ladder operator formalism as applied to anyons. This helps to highlight the differences between the cases of anyons and non-Abelian C-S particles, the object under study in section 3. In particular, of the operators used for anyons, only a subclass of operators, which preserve the monodromy properties of the wavefunctions, are allowed to act on the C-S particles. Nonetheless, our wavefunctions do cover the lowest Landau level. As an application of our formalism, we compute the second virial coefficient of NACS particles. The same set of ladder operators apply to free NACS particles with minor modifications. We also consider systems of multi-species NACS particles. Finally, the relevance of our work to systems of vortices of finite gauge groups is also discussed.
Anyons
======
The Hamiltonian for $N$ anyons with charge $e$ and mass $m$ moving on a plane with a constant magnetic field $B$ (perpendicular to the plane) is given by $$H=\suma -{1 \over 2m} (\nabla_{\alpha} -i {\bf a}_{\alpha}-
ie{\bf A})^2,\eqno(1)$$ where the external gauge field $A^i=-{1\over 2}B \epsilon^{ij} x^j $ in the symmetric gauge and the statistical gauge potential $$a^i_\alpha ({\bf x}_1, \dots,{\bf x}_N)=\nu \sumba
\epsilon^{ij}{x^j_\alpha -x^j_\beta \over |{\bf x}_\alpha -{\bf x}_\beta|^2}.
\eqno(2)$$
By a singular gauge transformation, we can remove ${\bf a}_\alpha$ from the Hamiltonian at the expense of using multi-valued wavefunction $$\psi_{new} ({\bf x}_1,\dots,{\bf x}_N)=\exp \biggl(i \nu \sumalb
\theta_{\alpha \beta} \biggr) \psi_{old} ({\bf x}_1,\dots,
{\bf x}_N).
\eqno(3)$$ Using the complex notation $z=x^1+ix^2, \zbar=x^1-ix^2, \partial=
{\partial \over \partial z},\parbar={\partial \over \partial \bar z} ,$ the gauge transformed Hamiltonian becomes $$H=\suma \biggl(-{2 \over m} \parbara \para + {e^2 B^2 \over 8m}
|\za|^2 \biggr) -{eB \over 2m}J, \eqno(4)$$ where $J$ is the angular momentum operator in the singular gauge $$J=\suma (\za \para -\zbara \parbara). \eqno(5)$$ Its eigenvalues are shifted from those in the symmetric gauge by a constant ${1 \over
2} \nu N ( N-1).$ (See (10.a).) It is convenient to extract a factor $\exp(-{1 \over 4}eB \suma
|\za|^2)$ from the wavefunction. Then the eigenvalue problem becomes $$\hat H \hat \psi =(E-{1 \over 2}N \omega) \hat \psi ,\eqno(6.a)$$ $$J \hat \psi =j \hat \psi ,\eqno(6.b)$$ (with $\omega \equiv {eB \over m}$) where the new Hamiltonian $\hat H$ and wavefunctions $\hat \psi$ are defined by $$\hat H= \suma \biggl(- {2 \over m} \parbara \para +
{eB \over m} \zbara \parbara \biggr) , \eqno(7.a)$$ $$\hat \psi =exp \biggl({eB \over 4} \suma |\za|^2 \biggr) \psi.
\eqno(7.b)$$ Note that the ground state energy is shifted by ${1\over 2}N\omega.$ We impose two physical requirements for the wavefunctions. Firstly, they must vanish at points of coincidences if $\nu \neq 0$ due to the centrifugal potentials (hard-core requirement). Secondly, they form Abelian representations of the braid group.
Now we introduce the operators $$\adaga =\zbara-{2 \over eB} \para, \aa=\parbara , \eqno(8.a)$$ $$\bdaga =\za -{2 \over eB} \parbara, \ba=\para ,\eqno(8.b)$$ which satisfy $[\aa , \adagb]=[\ba , \bdagb]=\delta_{\alpha \beta}$, all other commutators being zero. With respect to these operators, the Hamiltonian $\hat H$ in (7.a) and the angular momentum $J$ in (7.b) can be rewritten as $$\hat H=\omega \suma \adaga \aa, \eqno(9.a)$$ $$J=\suma ( \bdaga \ba -\adaga \aa). \eqno(9.b)$$
It is trivial to construct two distinct base states (for $0 \leq \nu <2 $) with energy and angular momentum eigenvalues: $$\eqalign{\psia &= \prodalb (\za-\zb)^{\nu},\cr
E_I^0 &= {1 \over 2} N \omega,\cr
j_I^0 &= {1 \over 2} \nu N(N-1),\cr} \eqno(10.a.)$$ $$\eqalign{\psib &= \prodalb (\zbara- \zbarb)^{2-\nu},\cr
E_{II}^0 &={1 \over 2} N \omega+{2-\nu \over 2}N(N-1)\omega,\cr
j_{II}^0 &=-{2-\nu \over 2} N(N-1).\cr} \eqno(10.b)$$ The general strategy of the ladder operator approach is to construct multi-anyon wavefunctions by acting with step operators on the base states in (10.a) and (10.b). We must, however, respect the statistics and hard-core requirements for the resulting wavefunctions. In order to respect the statistics, we use only symmetric combinations of the step operators. Consider the symmetric operators $$C_{ln}= \suma {\adaga}^l {\bdaga}^n , \eqno(11)$$ where $l,n$ are non-negative integers such that $l+n \le N$. (These operators form a basis in the ring of symmetric polynomials in $2N$ variables.) They are step operators in energy and angular momentum which respect the statistics properties of the base states $$\eqalignno{[\hat H,C_{ln}]&=\omega l C_{ln}, &(12.a)\cr
[J, C_{ln}]&=(n-l)C_{ln}. &(12.b)\cr}$$ They may, however, produce singular states (states with non-vanishing wavefunctions at points of coincidences) which have to be excluded by hand. We must identify which particular $C_{ln}$ produce regular states. These operators can be safely applied to the base states. Consider $C_{0n}$ first. With (8.b), we have $$C_{0n}\psia =\bigl( \suma \za^n \bigr) \psia. \eqno(13)$$ Thus they can be safely applied. For $C_{1m}$, we have $$\eqalign{C_{1m}\psia &=\suma (\zbara -\para) (\za -\parbara)^m \psia \cr
&=\suma (\zbara -\para) \za^m \psia \cr
&=-\suma \za^m \para \psia +
\suma (\zbara \za^m -m \za^{m-1} ) \psia, \cr} \eqno(14)$$ where we set $eB=2$ for simplicity. The seemingly singular first term is in fact regular because $$\suma \za^m \para \psia=\suma \za^m\sumba {\nu \over \za - \zb} \psia
=\nu \sumalb {\za^m -\zb^m \over \za -\zb} \psia .\eqno(15)$$ By a similar proof, one can apply a sequence of operators of the form $C_{0n}$ followed by a sequence of operators of the form $C_{1m}$ to $\psia$ without generating any singularities. Moreover, states of the form $$\psi_I^{(l)}=\prodalb (\zab)^{\nu +2l} , l=1,2,\dots \eqno(16)$$ are obtained from the action of $C_{on}$ on $\psia$. We can apply a string of operators $C_{n_1 m_1} C_{n_2 m_2} \dots C_{n_i m_i}$ with $\sum_{j=1}^i n_j \leq 2l$ to $\psi_I^{(1)}$ without generating singularities, because such a string contains at most derivatives of order $\sum_{j=1}^i n_j$ with respect to $\za$.
Thus we see that under suitable conditions, the step operators $C_{ln}$ can be safely applied to the base state $\psia$ to generate regular new wavefunctions. A similar analysis holds for the other base state $\psib$. Furthermore, closed-form eigenfunctions generated by the action of combinations of operators $C_{11},C_{10}$ and $C_{0m}$ only have been found, and they can be expressed in terms of the Laguerre functions. In particular, they do not involve the operators $C_{1m}$ with $m>1$, which however are allowed to act on $\psia$ to produce regular wavefunctions.
One should also note that the step operator approach only generates a subset of the whole spectrum of wavefunctions. If we naively set $\nu $ to be zero or one, we obtain only a subset of the bosonic and fermionic wavefunctions. Unlike the states generated by the step operators, the energies of the missing states show non-linear dependence on the statistical parameter $\nu$ in recent numerical studies. However, this is unimportant for what follows.
Non-Abelian Chern-Simons Particles
==================================
Recently, there has been much interest in the non-Abelian generalization of anyons. Non-Abelian Chern-Simons (NACS) particles carry non-Abelian charges and interact with each other through the non-Abelian Chern-Simons term. It has been argued that they may have applications in the fractional quantum Hall effect. Consider a system of N particles each of which carries a statistical charge corresponding to a representation $R_{l_\alpha} , \alpha=1, \dots ,N$ of a non-Abelian gauge group, which for definite we take to be $G=SU(2)$. In the holomorphic gauge, the dynamics of N free $SU(2)$ NACS particles is governed by the Hamiltonian $$\eqalign{\hat H&=-\suma {1 \over m_{\alpha}} ( \nabla_{\zbara} \nabla_{\za}+
\nabla_{\za} \nabla_{\zbara}),\cr
\nabla_{\za}&={\partial \over \partial \za}+{2 \over k} \sumba
{T_{\alpha}^a T_{\beta}^a \over \za -\zb},\cr
\nabla_{\zbara}&={\partial \over \partial \zbara},\cr} \eqno(17)$$ where k, a positive integer, is a parameter of the theory and $T_{\alpha}^a $ are the $SU(2)$-generators in the representation $R_{l_\alpha}$.
The wavefunctions take values in the tensor product of these representations. $$\Psi \in R_{l_1} \otimes \dots \otimes R_{l_N}. \eqno(18)$$ We expand the single-valued wavefunction ${\bf \psi}$ in terms of the conformal blocks ${\it F}_i \in R_{l_1} \otimes \dots \otimes R_{l_N}$ (which satisfy $\nabla_{\alpha} {\it F}_i=0$): $$\Psi=\sum_i \psi_i {\it F}_i. \eqno(19)$$ The Hamiltonian acting on the new wavefunction is just the free Hamiltonian. However, the complexity of the problem is hidden in the multivaluedness of the wavefunctions $\psi_i$. (In fact, it is more “natural” to work with the multi-valued wavefunctions $\psi_i$ than the original single-valued wavefunctions ${\bf \psi},$ partly because in the holomorphic gauge the Hamiltonian is not hermitian with respect to the usual inner product. Instead, the inner product is defined in the singular gauge and transformed back to the holomorphic gauge by a non-unitary transformation function which has to be taken into account in the definition of the inner product. )
From now on, we stick to the singular gauge. Consider $N$ NACS particles in the same irreducible representation $R_l$ of $SU(2)$ moving in a uniform external magnetic field $B$. We introduce operators $\aa,
\adaga,\ba,\bdaga$ as in eqn.(8) of section 2 and find that the Hamiltonian is again given by eqn.(9). The only difference lies in the constraints of the monodromy properties of the wavefunctions. In the case of anyons, the wavefunctions have only one component and monodromy leads to acquisition of phases, whereas NACS particles have multi-component wavefunctions whose monodromy properties are given by matrices.
We define $$\Oab \equiv {2 \over k} \sum_a T_{\alpha}^a T_{\beta}^a .\eqno(20)$$ Note that $\sumalb \Oab$, $J$ and $\hat H$ commute with each other and are thus good quantum numbers. ($J- \sumalb \Oab$ is the angular momentum operator in the holomorphic gauge.) We will discuss the diagonalization of $\sumalb \Oab$ later. For the time being, let us assume this has been done and let $\psi_{I} \in R_{l_1}\otimes
R_{l_2}\otimes \dots \otimes R_{l_N}$ be a (position-independent) eigenvector of $\sumalb \Oab$ with eigenvalue $\Omega$. In analogy with the anyon case, we propose applying the same ladder operator approach with the following base states which are expressed as path-ordered line integrals: $$\eqalignno{\psia(z_1, \dots, z_N)&=
P \exp \biggl( \int_{\Gamma} \sumalb (\Oab -2 m_{\alpha
\beta}I)
d \log(\za- \zb) \biggr) \psi_I , &(21)\cr
\psib(z_1, \dots, z_N)&=
P \exp \biggl( \int_{\Gamma} \sumalb (2n_{\alpha \beta} I-
\Omega_{\alpha \beta}) d \log(\zbara -\zbarb) \biggr)
\psi_I ,
&(22)\cr}$$ where $\Gamma$ is a path in the $N$-dimensional complex space with one end point fixed and the other being $\zeta =(z_1, \dots, z_N).$ The $\mab$ ($\nab$) depend on $\psi_I$ and are the maximal (minimal) integers which make the wavefunctions non-singular at the points of coincidences. This is analogous to the requirement $0 \leq \nu <2$ in the anyon case. Modulo the terms involving the identity matrix, the first integrand is just the flat Knizhnik-Zamolodchikov connection whereas the second is related to its antiholomorphic analogue. One can easily check that these base states have the desirable monodromy properties. From (5),(6) and (7.a), we have $$\eqalign{\hat H \psia&=0,\cr
E_I^0 &={1 \over 2} N \omega ,\cr
J\psia&=\suma \za \sumba {(\Oab-2 \mab I)
\over \za -\zb}\psia\cr
&=\sumalb (\Oab-2 \mab) \psia\cr
&=(\Omega-2 \sumalb \mab )\psia, \cr
\sumalb \Oab \psia&=\Omega \psia,\cr
} \eqno(23)$$ and $$\eqalign{\hat H \psib&=\suma {eB \over m} \zbara \sumba
{(2 \nab I-\Oab) \over \zbara - \zbarb }\psib \cr
&= \sumalb {eB \over m}(2 \nab I-\Oab)\psib\cr
&={eB \over m} [2 \sumalb \nab-\Omega]\psib\cr}$$ $$\eqalign{J\psib &=- \suma \zbara \sumba {(2 \nab I- \Oab)
\over \zbara-\zbarb}
\psib \cr
&= \sumalb (-2\nab+ \Oab )\psib\cr
&=[-2 \sumalb \nab +\Omega] \psib,\cr
\sumalb \Oab \psib &=\Omega \psib, \cr} \eqno(24)$$ where we have used the relation$$[\sumalb \Oab, \Omega_{\gamma \delta}]=0 ,\eqno(25)$$ and the fact that $\psi_I$ is an eigenstate of the operator $\sumalb \Oab$. Mathematically, these commutator relations are just consequences of the integrability condition (infinitesimal pure braid relations) satisfied by the connection. Physically, they follow from the fact that $\Omega$ is related to the angular momentum $J$ which is invariant upon monodromy.
Now that we have found the analogous base states, we will apply the ladder operators to them to generate new states. As before, the new states have to respect the statistics. (The NACS particles in the same irreducible representation are regarded as indistinguishable.) Thus, we may only use symmetric combinations of step operators. Also, we have to check that the wavefunctions produced are regular at the point of coincidence. There is, however, one crucial difference between the cases of anyons and NACS particles. Even with symmetric step operators, there is no guarantee that the monodromy properties of the wavefunctions are preserved. Any combination of step operators which do not preserve the monodromy properties of the wavefunctions are to be rejected.
First of all, let us consider $C_{0n}$. As before, we get $$C_{0n}\psia= \biggl( \suma \za^n \biggr) \psia . \eqno(26)$$ This shows that $C_{0n}$ can be safely applied to $\psia$ without changing its monodromy properties or producing singularities. Next we consider $C_{1m}$. $$\eqalign{C_{1m} \psia&=\suma (\zbara - \para) \za^m \psia \cr
&=-\suma \za^m \para \psia +
\suma (\zbara \za^m -m \za^{m-1} ) \psia \cr
&=-\sumalb (\Oab-2\mab I) {\za^m -\zb^m \over \za - \zb}
\psia+ \suma ( \zbara \za^m -m \za^{m-1})\psia. \cr}
\eqno(27)$$ For $m=0$, $$C_{10} \psia = \suma \zbara \psia ,\eqno(28)$$ which clearly preserves the monodromy property of $\psia$. When $m=1$, we have $$\eqalign{C_{11} \psia&=-\sumalb (\Oab-2\mab I) \psia + \suma (\zbara \za
-1) \psia \cr
&=[-\Omega+ 2\sumalb \mab +\suma(\zbara \za-1)] \psia.\cr}
\eqno(29)$$ This shows that $C_{11}$ can be safely applied to the base state. We can say more: strings made up of combinations of the operators $C_{0n}$ (n=1,2,...), $C_{10}$ and $C_{11}$ act on the base state to generate physical states. On the other hand, the operators $C_{1m}$ with $m>1$ and $C_{nm}$ with $n>1$ generally change the monodromy property of base state. There is no obvious way of constructing an admissible combination of operators involving them which would preserve the monodromy property of the base state. We therefore reject them as being unphysical and restrict the admissible set of operators to those generated by $C_{0n}$,$C_{10}$ and $C_{11}$.
The crucial reason why the argument for $C_{1m}$ (with $m>1$) and $C_{nm}$ (with $n>1$) as physical operators for anyons do not carry over to the case of non-Abelian C-S particles is that the the various monodromy matrices do not commute. In other words, $$[\Oab, \Omega_{\gamma \delta}]\neq 0. \eqno(30)$$
As in the anyon case, there are again missing states in the spectrum. However, our wavefunctions do cover the entire lowest Landau level as they involve the operators $C_{0n}$ only.
We now consider the construction of closed-form eigenfunctions. In the case of anyons, for $P(z_1, \dots,z_N,\bar z_1,\dots,\bar z_N)
\prodalb (\za -\zb)^{\nu}$ to be an eigenfunction of $\hat H$, it follows that the function $P$ has to satisfy a modified differential equation. $$\suma \biggl(-{2 \over m} \para +{eB \over m} \zbara \biggr) \parbara
P -{2 \over m} \sumalb \biggl( {\parbara -\parbarb \over \za -\zb}
\biggr) \nu P=\biggl(E-\omega {N \over 2} \biggr)P \eqno(31)$$ An ansatz has been made to construct closed-form eigenfunctions. All the solutions constructed can be expressed in terms of the Laguerre Polynomials. They are generated by strings of operators $C_{0n}$,$C_{10}$ and$C_{11}$ only. For NACS particles, we get a similar equation for $ P$, but with $\nu$ replaced by $\Oab-2\mab$. Nevertheless, since these operators are chosen to preserve the monodromy properties of the states. There is every reason to believe that the construction of closed-form solutions will go through with ${1 \over 2} N(N-1) \nu$ replaced by $\Omega-2 \sumalb \mab$.
Finally, we come to the diagonalization of $\sumalb \Oab$. Consider the identity $$\sum_{a} ( T_1^a + T_2^a + \dots + T_N^a )( T_1^a +T_2^a+ \dots +T_N^a)
=\suma T^a_{\alpha}T^a_{\alpha} +2 \sumalb T^a_{\alpha} T^a_{\beta}.
\eqno(32)$$ For $SU(2)$, the left-hand side gives the Casimir operator $J(J+1)$ of the “spin” of the composite made up of the $N$ particles, and the first term on the right-hand side gives the sum of the Casimir operators $\suma J_{\alpha} (J_{\alpha}+1)$ of the “spins” for the individual particles. (Here we abuse the word “spin” for the internal $SU(2)$ symmetry group. The physical spin (which is a scalar in 2+1 d) of a NACS particle in the $j$ representation is given by ${2 \over k} J(J+1)$. Thus, for $SU(2)$, we have $$\Omega \equiv \sumalb \Oab={1 \over k}[J(J+1)- \suma J_{\alpha}
(J_{\alpha}+1)].\eqno(33)$$
We just decompose the composite state into irreducible representations and $\sumalb \Oab$ would be diagonal in that basis. Actually, we can do better than that. It is easy to check that the operator $T^z_1 +T^z_2+ \dots +T^z_N$ commutes with $\hat H$, $J$ and $\sumalb \Oab$. Thus they can be simultaneously diagonalised.
Second virial coefficient and the large k limit
===============================================
In this section, we compute the second virial coefficients for some simple systems of NACS particles. To do so, we need to know all the two-particle states only.
First of all, consider two identical NACS particles in the $j={1 \over 2}$ representation of $SU(2)$. From the addition rule for angular momenta, we find that the resulting states consist of a triplet with $\Omega={1 \over
2k}$ and a singlet with $\Omega=-{3 \over 2k}.$ For $N=2$, $\Omega$ plays the role of the anyon phase, $\nu$. Let us recall the formula derived by Arovas [*et al.*]{} for the second virial coefficient of anyons, $$B(\nu=2j+\delta,T)=\lambda^2_T (-{1 \over 4}
+ | \delta | -{1 \over 2} \delta^2),\eqno(34)$$ where $|\delta| <2$. Note that it has a cusp at Bose values $\nu
=2j.$ By taking the average over the four two-body states, the second virial coefficient of the NACS particles is given by $$B(j={1 \over 2},T)=\lambda^2_T [-{1 \over 4}+{3 \over 4k} -{3 \over
8 k^2}]. \eqno(35)$$
For two particles with $j=1$, the resulting states have “spins” $2$, $1$, and $0$ (with $\Omega = {2\over k}$, $-{2 \over k}$, and $-{4 \over k}$ and degeneracies $5$, $3$ and $1$ respectively). We remark that all these states are bosonic if $k=1$. When $k=2$, the singlet is a bosonic state whereas others are fermionic. For $k>4$, all the states are anyonic with $|\nu| <1 $. For $k>1$, we have $$B(j=1,T)=\lambda^2_T [-{1 \over 4} +{20 \over 9 k}-{8 \over 3 k^2}].
\eqno(36)$$
Now we come to the large $k$ limit. For two particles belonging to a representation $j$ with $\lim_{{j \to \infty}\atop {k \to \infty} }
{j^2 \over k}=a <1$, we approximate the sum over all the resulting “spins” $r \le 2j$ by an integral. For example, the $|\delta|$ term is given by $$\eqalign{&{1
\over (2j+1)^2} \sum_{r=0}^{2j} (2r+1) {1 \over k}\bigl|[r(r+1)-
2j(j+1)] \bigr|\cr
\sim &{1 \over kj^2} \int_{r=0}^{\sqrt2 j} -r (r^2- 2j^2)\cr
= &{j^2 \over k}, \cr} \eqno(37)$$ where in the second line we approximate the sum by an integral and divide it into two parts (which happen to be equal) according to the sign of $r^2 -2 j^2$. The $\delta^2$ term can be evaluated in a similar manner. Hence, we get $$B=\lambda^2_T [-{1 \over 4}+{j^2 \over k}- { j^4 \over 3 k^2}]
=\lambda^2_T [-{1 \over 4}+a-{a^2 \over 3}].
\eqno(38)$$
If $a \to 0$, the last term may be discarded and the second virial coefficient of the NACS particle in the $j$ representation (with physical spin ${2 \over k} j(j+1)$) is the same as that of an anyon with [*half*]{} of the physical spin as its statistical parameter.
Concluding Remarks
==================
\(1) The $N$-free-NACS-particle problem can be solved by a similar method. The free Hamiltonian in the “anyon” gauge is given by $$H=\suma -{2 \over m} \parbara \para . \eqno(39)$$ The subtlety is that our base states become unnormalizable. Let us define $r =(\suma |\za|^2)^{1 \over 2}$ and consider $$H M(r)\psia = \psia \times -{1\over 2m} [ d^2/dr^2 +(1/r)(2N-1 +2 \Omega
-4 \sumalb \mab)]
M(r). \eqno(40)$$ We have eigenfunctions of the form $$M_{\mu}(r)=r^{-\mu} J_{\mu} (kr),\eqno(41)$$ (where $\mu=2(N-1+\Omega-2 \sumalb \mab)$) with eigenvalues $\hbar k^2/ 2m$ for $H$.
\(2) Let us now consider the construction of $C_{lm}$ for multi-species non-Abelian C-S particles (particles in various irreducible representations). In this case, when we construct $C_{0n}$, we do so for each irreducible representation $R$ and symmetrize over particles in this irreducible representation only. Let us call the resulting operator $C^R_{0n}$. If we construct $C^R_{10}$ and $C^R_{11}$ in a similar manner, we find that these operators have to be rejected: They do not preserve the monodromy properties of the base states because $[\Oab,\Omega_{\gamma \delta}] \ne 0,$ and we are no longer summing over all the particles. Therefore, for $C_{10}$ and $C_{11}$ we do sum over all the particles in the various irreducible representations.
\(3) Note that $C_{01}$ and $C_{10}$ represent center of mass excitations. The operator $C_{10}$ was also analyzed by Johnson and Canright, while $C_{11}$ is directly related to the Lie group generator of $SU(1,1)$.
\(4) We remark that the operators of $C_{1m}$ and $C_{nm}$ ($m,n>2$) do preserve the monodromy property of the base state, if $\psi_I$ in eqns.(20) and (21) is chosen to be a simultaneous eigenstate of all $\Oab$. The statistics is “Abelianized” in this case. This situation occurs, for example, for some models of non-Abelian vortices of finite gauge groups such as the quarternion group.
\(5) The same ladder operator approach may well apply to non-abelian vortices of finite gauge groups. Unfortunately, we generally do not know how to construct connections which would produce the desirable monodromy in this case.
\(6) In a recent paper, Dasnieres de Veigy and Ourvy derived the equation of state of an anyon gas in a strong magnetic field at low temperatures. The idea is that at sufficiently low temperatures, excitations to higher Landau levels can be neglected. Thus one may consider only the lowest Landau states of the anyons, which are covered by the step operators. In fact, apart from the statistical phase factor, the multi-anyon states in the lowest Landau level are tensor-product states of the individual anyon states. By regularizing the grand partition function with a harmonic potential, the equation of state can be obtained. The same decoupling principle should apply to NACS particles. For a fixed base state, modulo the statistical term involving $\Oab$, the multi-particle wavefunctions in the lowest Landau level are again tensor products of individual particle states. Therefore, in principle, one should be able to derive the equation of state of NACS particles in a strong magnetic field at low temperatures. We thank John Preskill and Piljin Yi for useful discussions.
|
---
author:
- 'S. Recchi [^1]'
- 'F. Calura[^2]'
- 'P. Kroupa[^3]'
date: 'Received; accepted'
title: 'The chemical evolution of galaxies within the IGIMF theory: the \[$\alpha$/Fe\] ratios and downsizing.'
---
Introduction
============
It is nowadays widely accepted that most stars in galaxies form in star clusters (Tutukov [@tutu78]; Lada & Lada [@ll03]). This has been observed in a number of different galaxies; from the Milky Way to the dwarf galaxies of the Local Group (Wyse et al. [@wyse]; Massey [@massey]; Piskunov et al. [@piskunov]). Within each star cluster, the initial mass function (IMF) can be well approximated by the canonical two-part power-law form $\xi(m) \propto m^{-\alpha}$ (e.g. Pflamm-Altenburg, Weidner & Kroupa [@pwk07], hereafter PWK07). Massey & Hunter ([@mh98]) have shown that for stellar masses $m>$ a few M$_\odot$ a slope similar to the Salpeter ([@salpeter]) index (i.e. $\alpha=2.35$) can approximate well the IMF in clusters and OB associations for a wide range of metallicities, whereas many studies have shown that the IMF flattens out below $m$ $\sim$ 1 M$_\odot$ (Kroupa, Tout & Gilmore [@ktg93]; Chabrier [@chab01]).
On the other hand, star clusters are also apparently distributed according to a single-slope power law, $\xi_{\rm ecl} \propto M_{\rm
ecl}^{-\beta}$, where $M_{\rm ecl}$ is the stellar mass of the embedded star cluster. There is a general consensus that this slope $\beta$ should be of the order of $\sim$ 2 (Zhang & Fall [@zf99]; Lada & Lada [@ll03]; Hunter et al. [@hunter]), although a $\beta$ as high as 2.4 can also be realistic (Weidner, Kroupa & Larsen [@wkl04]). According to this correlation, small embedded clusters are more numerous in galaxies. They provide therefore most of the stars but not most of the massive ones, since they are preferentially formed in massive clusters (Weidner & Kroupa [@wk06]). As a consequence of this mass distribution of embedded clusters, the integrated IMF in galaxies, the IGIMF, can be steeper than the stellar IMF within each single star cluster (Kroupa & Weidner [@kw03]; Weidner & Kroupa [@wk05]).
The Salpeter IMF slope has been used in a very wide range of modelling, providing good fits with observations concerning the cosmic star formation history (Calura, Matteucci & Menci [@cmm04]), the X-ray properties of elliptical galaxies (Pipino et al. [@pipino05]), the chemical evolution of dwarf galaxies (Larsen, Sommer-Larsen & Pagel [@lslp]) and of the Milky Way (Pilyugin & Edmunds [@pe06], but see also Romano et al. [@romano05]). Broadly speaking, a flatter than Salpeter IMF produces a larger fraction of massive stars. The large production of oxygen (and of $\alpha$-elements in general) leads to lower \[Z/O\] metallicity ratios. A steep IMF slope would instead be biased towards low- and intermediate-mass stars, underproducing oxygen and therefore resulting in larger \[N/O\] and \[C/O\] abundance ratios. On the other hand, iron will also be overproduced compared to $\alpha$-elements, since it comes mainly from Type Ia SNe which originate from C-O deflagration of binary systems of intermediate mass. Therefore, galaxies characterized by a steep IMF will tend to have \[$\alpha$/Fe\] ratios lower than models in which the IMF is flat.
The scenario of a variable integrated galactic initial mass function (IGIMF) has been applied in models of chemical evolution (Köppen, Weidner & Kroupa [@kwk07]), producing an excellent agreement with the mass-metallicity relation found by Tremonti et al. ([@t04]). However, these authors consider only the effect of the IGIMF on the global metallicity and the evolution of abundance ratios has not yet been explored in the literature. In a series of papers we plan to study the impact of the IGIMF on the abundance ratios in different classes of galaxies, using different methodologies. In this paper we study, by means of simple analytical and semi-analytical models, the evolution of \[$\alpha$/Fe\] ratios in galaxies, in particular in early-type ones. It is, in fact, now well established that the \[$\alpha$/Fe\] ratios in the cores of elliptical galaxies increase with galactic mass (Weiss, Peletier & Matteucci [@wpm95]; Kuntschner et al. [@kunt01]) and this poses serious problems to the current paradigm of hierarchical build-up of galaxies (see e.g. Thomas et al. [@thom05], hereafter THOM05; Nagashima et al. [@naga05]; Pipino, Silk & Matteucci [@psm09]; Calura & Menci, in preparation). In fact, in the classical hierarchical models the most massive ellipticals take a longer time to assemble and therefore form stars for a longer time than less massive galaxies, thus producing a a trend of \[$\alpha$/Fe\] vs. mass which is opposite of what is observed (see Thomas, Maraston & Bender [@thom02]; Matteucci [@matt07]).
We will show in this paper that the trend of increasing \[$\alpha$/Fe\] vs. galaxy mass is naturally accounted for in models of elliptical galaxies in which the IGIMF is implemented. The second paper of this series will be devoted to the study of the chemical evolution of the Solar Neighborhood and of the local dwarf galaxies and in this case we will make use of detailed chemical evolution models. Another paper of this series will study the evolution of galaxies by means of chemodynamical models, in order to analyze how the IGIMF changes the feedback of the ongoing star formation in galaxies and how this affects the chemical evolution.
The plan of the present paper is as follows. In Sect. 2 we summarize the IGIMF theory and the formulations we adopt. In Sect. 3 we describe how we calculate the Type Ia and Type II SN rates in galaxies in which the SFR is given. Once we know the Type Ia and Type II SN rates, it is possible to calculate the \[$\alpha$/Fe\] ratios. This has been done in Sect. 4 for ellipticals and early-type galaxies in general. A discussion and the main conclusions are presented in Sect. 5.
The determination of the integrated galactic initial mass function {#sec:igimf}
==================================================================
The determination of the IGIMF has been described previously (Kroupa & Weidner [@kw03]; Weidner & Kroupa [@wk05]; PWK07). The IGIMF theory is based on the assumption that all the stars in a galaxy form in star clusters. Surveys of star-formation in the local Milky Way disk have shown that 70 to 90 % of all stars appear to form in embedded clusters (Lada & Lada [@ll03]; Evans et al. [@evans08]). The remaining 10-30 % of the apparently distributed population may stem from a large number of short-lived small clusters that evolve rapidly by dissolving through energy equipartition and residual gas expulsion. It is therefore reasonable to assume that star formation occurs in embedded clusters with masses ranging from a few M$_\odot$ upwards. The IGIMF, integrated over the whole population of embedded clusters forming in a galaxy, becomes
$$\xi_{\rm IGIMF}(m;{\rm SFR} (t)) =
\int_{M_{\rm ecl, min}}^{M_{\rm ecl, max} ({\rm SFR} (t))}
\hspace{-0.6cm}\xi (m \leq m_{\rm max}) \xi_{\rm ecl} (M_{\rm ecl})
d M_{\rm ecl},$$
where $M_{\rm ecl, min}$ and $M_{\rm ecl, max} ({\rm SFR} (t))$ are the minimum and maximum possible masses of the embedded clusters in a population of clusters and $m_{\rm max} = m_{\rm max} (M_{\rm ecl})$ (eqs. \[eq:normmecl\] and \[eq:normxi\]). For $M_{\rm ecl, min}$ we take 5 M$_\odot$ (the mass of a Taurus-Auriga aggregate, which is arguably the smallest star-forming “cluster” known), whereas the upper mass of the embedded cluster population depends on the SFR. The correlation between $M_{\rm ecl, max}$ and SFR has been determined observationally (Larsen & Richtler [@lr00]; Weidner et al. [@wkl04]) and can be expressed with the correlation
$$\log M_{\rm ecl, max} = \log k_{\rm ML} + 0.75 \log \psi + 6.77,
\label{eq:mecl}$$
where $\psi$ is the SFR in M$_\odot$ yr$^{-1}$ and $k_{\rm ML}$ is the mass-to-light ratio, typically 0.0114 for young stellar populations (Smith & Gallagher [@sg01]). This empirical finding can be understood to result from the sampling of clusters from the embedded cluster mass function given the amount of gas mass being turned into stars per unit time (Weidner et al. [@wkl04]).
The stellar IMF (i.e. the IMF within each embedded cluster) has the canonical form $\xi(m) = k m^{-\alpha}$, with $\alpha = 1.3$ for 0.08 M$_\odot \le$ $m$ $<$ 0.5 M$_\odot$ and $\alpha = 2.35$ (i.e. the Salpeter slope) for 0.5 M$_\odot \le$ $m$ $< m_{\rm max}$, where $m_{\rm
max}$ depends on the mass of the embedded cluster. In order to determine $m_{\rm max}$ and the proportionality constant $k$ we have to solve the following two equations (Kroupa & Weidner [@kw03]):
$$M_{\rm ecl} = \int_{m_{\rm low}}^{m_{\rm max}} m \xi (m) dm,
\label{eq:normmecl}$$
$$\int_{m_{\rm max}}^{m_{\rm max *}} \xi (m) dm = 1,
\label{eq:normxi}$$
where $m_{\rm low}$ is the smallest considered stellar mass (0.08 M$_\odot$ in our case) and $m_{\rm max *}$ is the upper physical stellar mass and its value is assumed to be 150 M$_\odot$ (Weidner & Kroupa [@wk04]). Eq. \[eq:normxi\] indicates that, by definition of $m_{\rm max}$, there is only one and exactly one star in the embedded cluster with mass $M_{\rm ecl}$ whose mass is larger than or equal to $m_{\rm max}$.
The last ingredient we need is the distribution function of embedded clusters, $\xi_{\rm ecl} (M_{\rm ecl})$, which, as we have mentioned in the Introduction, we can assume proportional to $M_{\rm
ecl}^{-\beta}$. In this work we have assumed 3 possible values of $\beta$: 1.00 (model BETA100), 2.00 (model BETA200) and 2.35 (model BETA235). In Fig. \[igimf\_ext\] we have plotted the resulting IGIMFs for different values of SFR. In particular, we have tested 20 SFRs, ranging from 10$^{-4}$ to 100 M$_\odot$ yr$^{-1}$, equally spaced in logarithm. To appreciate better the differences between various models, we have plotted in Fig. \[igimf\_comp\] IGIMFs for 3 different values of the SFR: SFR $\simeq$ 10$^{-2}$ M$_\odot$ yr$^{-1}$ (heavy lines), SFR $\simeq$ 1 M$_\odot$ yr$^{-1}$ (middle lines), SFR $\simeq$ 10$^{2}$ M$_\odot$ yr$^{-1}$ (light lines). We have considered all the possible values of $\beta$: model BETA100 (dashed lines), BETA200 (dotted lines), BETA235 (solid lines). For clarity, we have plotted the IGIMFs only for masses larger than $\sim$ 2 M$_\odot$, since in the range of low mass stars the IGIMFs do not vary. As expected, the model with the steepest distribution of embedded clusters (model BETA235) produces also the steepest IGIMFs. This is due to the fact that model BETA235 is biased towards embedded clusters of low mass, therefore the probability of finding high mass stars in this cluster population is lower.
We can also notice from Fig. \[igimf\_comp\] that the differences between IGIMFs with SFR $\simeq$ 1 M$_\odot$ yr$^{-1}$ (middle lines) and SFR $\simeq$ 10$^{2}$ M$_\odot$ yr$^{-1}$ (light lines) are not very pronounced. This is due to the fact that for both these SFRs, the maximum possible mass of the embedded cluster is very high (see eq. \[eq:mecl\]), therefore in both cases the upper possible stellar mass of the whole galaxy is very close to the theoretical limit of 150 M$_\odot$. This can be seen in Fig. \[slope\] (lower panel) in which we plot the variation of $m_{\rm max}$ as a function of SFR as deduced from eqs. \[eq:normxi\] and \[eq:normmecl\]. This correlation is valid for all the possible values of $\beta$ because it is determined by $\xi (m)$ and not by $\xi_{\rm ecl} (M_{\rm ecl})$. As we can see from Figs. \[igimf\_ext\] and \[igimf\_comp\], the IGIMFs are characterized by a nearly uniform decline, which follows approximately a power law, and a sharp cutoff when $m$ gets close to $m_{\rm max}$. In Fig. \[slope\] (upper panel) we plot therefore also the slope that better approximates the IGIMF in the range 3 - 16 M$_\odot$. This is the range of masses where most of the progenitors of SNeII and SNeIa originate (see Sect. 3). Of course, the steeper the distribution of embedded clusters is, the steeper the corresponding IGIMFs are. Fig. \[slope\] shows also what we have noticed before, namely that the various IGIMFs saturate for SFR $>$ 1 M$_\odot$ yr$^{-1}$. Finally, in Fig. \[slope\] (middle panel) $k_{\alpha}$ is shown as a function of the SFR for the various models. $k_{\alpha}$ is the number of stars per unit mass in one stellar generation (see e.g. Greggio [@greggio05]) and its value is given by
$$k_{\alpha} = {{\int_{m_{\rm low}}^{m_{\rm max}} \xi_{\rm IGIMF} (m) dm}
\over
{\int_{m_{\rm low}}^{m_{\rm max}} m \xi_{\rm IGIMF} (m) dm}}.$$
This parameter is useful to calculate the SNII rates (see Sect. 3).
The determination of Type Ia and Type II SN rates
=================================================
Type II SN rates {#subs:SNII}
----------------
Stars in the range $m_{\rm up} < m < m_{\rm max}$ (where $m_{\rm
up}$ is the mass limit for the formation of a degenerate C–O core) are generally supposed to end their lives as [*core-collapse*]{} SNe. These SNe divide into SNeII, SNeIb and SNeIc according to their spectra. For our purposes, this distinction is not useful and we will suppose that all the core-collapse supernovae are indeed SNeII. These SNe produce the bulk of $\alpha$-elements and some iron (one third approximately). The standard value of $m_{\rm up}$ is 8 M$_\odot$ but stellar models with overshooting predict lower values (e.g. Marigo [@marigo01]). However, stars more massive than $m_{\rm up}$ can still develop a degenerate O–Ne core and end their lives as electron-capture SNe (Siess [@siess07]). We will assume for simplicity that all the stars with masses larger than 8 M$_\odot$ end their lives as SNeII, therefore the SNII rate is simply given by the rate at which massive stars die, namely:
$$R_{SNII} (t) = \int_8^{m_{\rm max}} \psi (t - \tau_m) \xi_{\rm IGIMF} [m, \psi
(t - \tau_m)] dm,
\label{eq:snii}$$
where $\psi$ is the SFR and $\tau_m$ is the lifetime of a star of mass $m$. The lifetime function is adapted from the work of Padovani & Matteucci ([@pm93]),
$$\tau_m = \cases{1.2 m^{-1.85}+0.003\;{\rm Gyr} &if $m\geq 6.6$ M$_{\odot}$\cr
10^{f(m)}
\;{\rm Gyr} &if $m<6.6$ M$_{\odot}$,\cr}$$
where
$$f(m)={{\bigl\lbrack 0.334-\sqrt{1.79-0.2232\times(7.764-\log(m))}
\bigr\rbrack\over 0.1116}}.$$
In eq. \[eq:snii\] the IGIMF is calculated by considering the SFR at the time $t - \tau_m$, therefore it depends both on time and on mass.
It is instructive to analyze models in which SFR is constant during the whole evolution of the galaxy. In this way, eq. \[eq:snii\] simplifies into
$$R_{SNII} = \psi \int_8^{m_{\rm max}} \xi_{\rm IGIMF} (m, \psi) dm =
\psi k_\alpha n_{> 8},
\label{eq:r2}$$
where, as we have seen in Sect. 2, $k_{\alpha}$ is the number of stars per unit mass in one stellar generation (therefore $\psi k_\alpha$ is the number of stars formed per unit time) and $n_{> 8}$ is the number fraction of stars with masses larger than 8 M$_\odot$. In Fig. \[sn2\_cen\] (lower panel) we plot $R_{SNII}$ (in cen$^{-1}$) as a function of SFR for the 3 adopted values of $\beta$. It is worth pointing out that, even if SFR is constant, $R_{SNII}$ starts increasing only after the star with mass $m_{\rm max}$ ends its life and reaches a constant value only after the lifetime of a 8 M$_\odot$ star. This lifetime ($\sim$ 28 Myr with the adopted lifetime function) is however negligible compared to the Hubble time, therefore it is reasonable to consider $R_{SNII}$ constant with time. For SFRs larger than $\sim$ 10$^{-2}$ M$_\odot$ yr$^{-1}$ $R_{SNII}$ increases almost monotonically with SFR, whereas it drops dramatically for SFR $<$ 10$^{-2}$ M$_\odot$ yr$^{-1}$. This is mostly due to the drop of $n_{>8}$ at low SFRs (Fig. \[sn2\_cen\], upper panel) which in turn depends on the fact that the upper mass $m_{\rm max}$ for these values of SFR starts reducing significantly and it gets very close to 8 M$_\odot$ for a SFR of 10$^{-4}$ M$_\odot$ yr$^{-1}$ (Fig. \[slope\]), therefore only a very narrow interval of stellar masses gives rise to SNII explosions. From Fig. \[slope\] we can see instead that the variation of $k_{\alpha}$ with SFR is not very significant, therefore $k_{\alpha}$ affects only mildly the Type II SN rates.
It is nowadays getting popular to consider SN rates normalized to the stellar mass of the considered galaxy. The usually chosen unit of measure is the SNuM (1 SNuM = 1 SN cen$^{-1}$ 10$^{-10}$ M$_*$$^{-1}$, where M$_*$ is the current stellar mass of the galaxy). In this case, models in which the SFR is constant cannot attain a constant Type II SN rate in SNuM since the stellar mass of the galaxy increases with time. We therefore calculated $R_{SNII}$ in SNuM as a function of time for the various models. The stellar mass of the galaxy at each time $t$ is given by $\int_0^t \psi \cdot f_{< m (t)} dt$, where $f_{< m (t)}$ is the mass fraction of stars, born until the time $t$ that have not yet died.
Fig. \[sn2\_snum\] shows the evolution with time of the Type II SN rate for different models and different SFRs, assuming a constant SFR for 14 Gyr. These results are compared with the average SNeII rates (in SNuM), observationally derived by Mannucci et al. ([@mann05]) in S0a/b galaxies (solid boxes), Sbc/d galaxies (dotted boxes) and irregular ones (dashed boxes). We can notice that, for the models BETA100 and BETA200 only the mildest SFRs can reproduce the final SNII rates in S0a/b galaxies, whereas model BETA235 can fit the final SNII rate of S0a/b galaxies for a wide range of SFRs. On the other hand, all the models predict final SNII rates significantly below the observations of irregular galaxies and the final values for model BETA235 fail also to fit the observed rates in Sbc/d galaxies. It is important to note, however, that the stellar mass in galaxies is usually calculated assuming some (constant) IMF. Under the assumption that the IMF changes with the SFR, the determinations of the stellar masses must be revisited. PWK07 showed that the IGIMF effect (i.e. the suppression of the number of massive stars with respect of low-mass stars) can be very significant in dwarf galaxies, whereas in large galaxies it tends to be very small. Moreover, a constant SFR for 14 Gyr is not a reasonable description of the star formation history of irregular (and Sbc/d) galaxies which often experience an increase of the SFR in the last Gyrs of their evolution (see e.g. Calura & Matteucci [@cm06]). For this reason, the calculated SNII rates of late type galaxies tend to fit the observations at smaller ages.
Type Ia SN rates {#subs:SNIa}
----------------
In order to calculate the SNIa rates, we assume the so-called Single Degenerate Scenario of SNIa formation. It is commonly assumed that a SNIa explodes when a C-O white dwarf in a binary system reaches the Chandrasekhar mass after mass accretion from a companion star. According to the Single Degenerate channel of SNIa explosion, the accretion of matter occurs via mass transfer from a non-degenerate companion (a red giant or a main sequence star) filling its Roche lobe (Whelan & Iben [@wi73]). In this way, the SNIa rate depends on the number distribution of C-O white dwarfs, but also on the mass ratio between primary and secondary stars in a binary system. The SNIa rate in the framework of the Single Degenerate Scenario has been analytically calculated by a number of authors assuming a universal IMF (see Valiante [@vali09] and references therein). Here we follow the formulation of Greggio & Renzini ([@gr83]) and Matteucci & Recchi ([@mr01]) but we modify it to take into account that, in the framework of the IGIMF, the IMF changes according to the SFR. The SNIa rate in this case turns out to be:
$$R_{\rm Ia} (t)=A \hspace{-0.1cm}\int_{m_{\rm B, inf}}^{m_{\rm B, sup}}
\hspace{-0.25cm}\int_{\mu_{\rm min}}^{0.5}\hspace{-0.25cm}f(\mu)
\psi(t-\tau_{m_2})\xi_{\rm IGIMF} [m_{\rm B}, \psi
(t - \tau_{m_2})]d\mu\, dm_{\rm B},
\label{eq:r1a}$$
where $A$ is a normalization constant (assumed to be 0.09 in the following). Although theoretical arguments demonstrate that $A$ should be small (e.g. Maoz [@maoz08]) its value is usually calibrated with the Milky Way. Unfortunately, our analytical approach does not allow us to simulate the Milky Way within the IGIMF theory, therefore we take 0.09 as a reference value and postpone a more careful discussion about it to the follow-up numerical paper (but see also Sec. \[sec:results\] for a study of the variation of A for early-type galaxies). $m_{\rm B}$ is the total mass of the binary system, $m_2$ is the mass of the secondary star, $\mu = m_2 / m_{\rm
B}$ and $f(\mu)$ is the distribution function of mass ratios (see below). It is commonly assumed that the maximum stellar mass able to produce a degenerate C-O white dwarf is a 8 M$_\odot$ star, therefore the maximum possible binary mass is 16 M$_\odot$. The minimum possible binary mass is assumed to be 3 M$_\odot$ in order to ensure that the smallest possible white dwarf can accrete enough mass from the secondary star to reach the Chandrasekhar mass. With these assumptions, the limits of integration in eq. (\[eq:r1a\]) are:
$$m_{\rm B, inf} = max [2 m_2 (t), 3 {\rm M}_\odot]$$
$$m_{\rm B, sup} = 8 {\rm M}_\odot + m_2 (t),$$
$$\mu_{\rm min} = max \biggl[{m_2 (t) \over m_B},
{{m_B - 8 {\rm M}_\odot}\over m_B}\biggr].$$
The distribution function of mass ratios is generally described as a power law ($f(\mu) \propto \mu^{\gamma}$), but the value of $\gamma$ is still much debated in the literature (Duquennoy & Mayor [@dm91]; Shatsky & Tokovinin [@st02]; Kouwenhoven et al. [@kouw05]) and therefore we will take it as a free parameter. [^4]
Fig. \[sn1a\_snum\] shows the evolution with time of the Type Ia SN rate for different models and different SFRs, analogously to Fig. \[sn2\_snum\] for SNeII rates. Also shown (dashed lines) for comparison are SNIa rates obtained for a model with fixed (i.e. not SFR-dependent) IMF. We assume the canonical stellar IMF (i.e. the IMF within each embedded cluster) which, as mentioned in Sect. \[sec:igimf\], has the form $\xi(m) = k m^{-\alpha}$, with $\alpha = 1.3$ for 0.08 M$_\odot <$ $m$ $<$ 0.5 M$_\odot$ and $\alpha
= 2.35$ above 0.5 M$_\odot$. As we can see, at large SFRs model BETA100 produces rates almost indistinguishable from the ones obtained with the fixed canonical IMF (see also Kroupa & Weidner [@kw03]). In this figure $\gamma$ is assumed to be 2 (Tutukov & Yungelson [@ty80]). This large value of $\gamma$ favors the occurrence of SNeIa in binary systems with similar masses. Such a steep mass ratio distribution that favors equal-mass binaries may result from dynamical evolution of stellar populations in long-lived star clusters (Shara & Hurley [@sh02]). We can notice again that only model BETA235 at very low SFRs seems able to reproduce the SNIa rates in S0a/b and E/S0 galaxies. However, we point out that the comparison with the observed SNIa rates in elliptical galaxies is meaningless because they have stopped forming stars several Gyr ago, therefore they have evolved passively since then. For them we cannot therefore assume a constant SFR for 14 Gyr (see Sect. \[sec:results\]). On the other hand, model BETA235 produces SNIa rates that only match the observed rates in dwarf irregular galaxies at their peak. Therefore, assuming $\gamma = 2$, the best value for $\beta$ seems to be 2 (but see the comment in Sect. \[subs:SNII\] about the possible inconsistency of the published determination of stellar masses, at least for irregular galaxies). To show the dependence of the results on $\gamma$ we show in Fig. \[sn1a\_snum\_0.3\] the SNeIa rates obtained assuming $\gamma
= 0.3$. This flatter distribution function implies that a larger fraction of binary systems with small mass ratios end up as SNeIa. We can notice from this figure that the observed SNIa rates in spiral galaxies are reproduced by a larger range of SFRs, whereas the disagreement with the observed rates in irregular galaxies worsens.
A test of the IGIMF: \[$\alpha$/Fe\] ratios in early-type galaxies {#sec:results}
==================================================================
The study of the average stellar \[$\alpha$/Fe\] ratio in galaxies represents an important constraint for our models, since this quantity depends both on the adopted galactic star formation history and on the stellar IMF (Matteucci [@mat01]). In local ellipticals, the observed correlation between the central velocity dispersion $\sigma$, which reflects the total stellar mass, and the stellar \[$\alpha$/Fe\] is interpreted as due to the shorter star formation timescales in the most massive galaxies (Pipino & Mattuecci [@pm04]; THOM05) which in turn implies also that the most massive galaxies experience the most intense episodes of star formation. For this reason, the average stellar \[$\alpha$/Fe\] vs $\sigma$ relation represents a valuable test for the IGIMF, since the IGIMF is a function of the galactic star formation rate. The issue of a variable IMF among elliptical galaxies to explain the \[$\alpha$/Fe\] vs. $\sigma$ relation has already been explored with success by Matteucci ([@matt94]) but assuming ad hoc variations of the IMF slope. In this section we test, using well-established and observationally constrained star formation histories of early-type galaxies of various masses, if the physically motivated IGIMF can equally well reproduce this correlation.
To simplify the calculations, the SFR is assumed to be constant over a period of time $\Delta t$. We have numerically tested that this crude approximation about the star formation history does not affect significantly the results. The value of $\Delta t$ as a function of galaxy luminous mass is adopted from the work of THOM05, who, on the basis of the observational relation between \[$\alpha$/Fe\] and $\sigma$, showed the existence of a [*downsizing*]{} pattern for elliptical galaxies, according to which the smaller ellipticals form over longer timescales (see also Matteucci [@matt94]; Cowie et al. [@cowie96]; Kodama et al. [@koda04]). Since the present-day stellar mass is given in this case by $\psi \cdot
\int_o^{\Delta t}$ $f_{\rm low} (t) dt$ (where $f_{\rm low} (t)$ is the fraction of long-living stars, namely the stars, born at the time $t$, that live until the present day), it is possible to derive a relation between the SFR and the duration of the star formation activity $\Delta t$, which we show in Fig. \[deltat\]. This relation saturates at 14 Gyr since this is assumed to be the age of the Universe. A similar relation can be recovered from the work of Pipino & Matteucci ([@pm04]) assuming that the star formation occurs only until the onset of the galactic wind, however the two SFR-$\Delta t$ relations do not significantly differ.
For each galaxy (characterized by a specific SFR over a period $\Delta
t$) we calculate the average yield from SNeII of a chemical element $i$,
$$\overline{y^{\rm II}_{\rm i}} =
{{\int_8^{m_{\rm max}} y_{\rm i} (m) \xi_{\rm IGIMF} (m, \psi) dm}
\over {\int_8^{m_{\rm max}} \xi_{\rm IGIMF} (m, \psi) dm}},$$
where $y_{\rm i} (m)$ is the yield of chemical element $i$ produced by a single star of mass $m$. The nucleosynthetic prescriptions are taken from Woosley & Weaver ([@ww95]). We have however halved the iron yields, in accordance with Timmes, Woosley & Weaver ([@tww95]) and Chiappini, Matteucci & Gratton ([@cmg97]), because it is known that only in this way it is possible to reproduce the \[$\alpha$/Fe\] in Galactic stars. Unfortunately, Woosley & Weaver ([@ww95]) calculated yields only for stellar masses up to 40 M$_\odot$. We assume that the yields of stars more massive than 40 M$_\odot$ are equal to the 40 M$_\odot$ star yields. Given the very limited amount of stars in the range 40 M$_\odot$ $<$ $m$ $<$ $m_{\rm
max}$ the results are not sensitive to this assumption. In Fig. \[yields\] we show the IGIMF-averaged SNII yields of oxygen (solid lines), iron (dotted lines) and magnesium (dashed lines) as a function of the SFR for different values of $\beta$. For what concerns SNeIa, we assume the yields reported by Gibson, Loewenstein & Mushotzky ([@glm97]), based on the work of Thielemann, Nomoto & Hashimoto ([@tnh93]).
Once we know the SNIa yields and the IGIMF-averaged SNII yields for each galaxy, we can calculate the mass fraction ${\alpha \over Fe}
(t)$ (where $\alpha$ is O or Mg) produced until the time $t$ by using the formula:
$${\alpha \over Fe} (t) = {{\int_0^t \bigl(R_{\rm Ia} (t) y_\alpha^{\rm
Ia} + R_{\rm SNII} \overline{y^{\rm II}_\alpha}\bigr) dt} \over
{\int_0^t \bigl(R_{\rm Ia} (t) y_{Fe}^{\rm Ia} + R_{\rm SNII}
\overline{y^{\rm II}_{Fe}}\bigr) dt} },
\label{eq:alphafe_gas}$$
where $R_{\rm Ia}$ and $R_{\rm SNII}$ are the SNIa and SNII rates, given by eqs. \[eq:r1a\] and \[eq:r2\], respectively, and $y^{\rm
Ia}$ are the SNIa yields.
At this point, we can compute the theoretical average stellar abundances by means of:
$$[\alpha/Fe] = \log_{10} \frac{\psi \cdot \int_{0}^{\Delta t}
{\alpha \over Fe} (t)
\cdot f_{\rm low} (t) dt}{M_{tot}} - \log_{10}{\alpha_\odot
\over Fe_\odot}.$$
where $M_{tot}$ is the total present-day stellar mass and $\alpha_\odot$ and $Fe_\odot$ are the solar abundances of $\alpha$-elements and Fe, respectively, taken from Anders & Grevesse ([@ag89]). This means that our theoretical abundance ratios represent values mass-averaged over the stars which survive until the present time (see also Thomas, Greggio & Bender [@thom99]).
The observable in elliptical galaxies is the velocity dispersion instead of the mass, so in order to properly compare our results with observations we need to assume a correlation between the stellar mass and the velocity dispersion of galaxies (Faber-Jackson relation). We assume:
$$\sigma = 0.86 M_{tot}^{0.22},$$
(Burstein et al. [@burs97]), where $\sigma$ is the velocity dispersion in km/s. The resulting relation between $\sigma$ and $\Delta t$ can be seen from Fig. \[deltat\] where we have indicated in the upper scale the $\sigma$ corresponding to each SFR.
In Fig. \[alphafe\_gamma0.3\] we show our results for $\gamma = 0.3$ comparing our models with observations taken from THOM05 and references therein (filled squares). We can first notice that, as expected, the model BETA100 (heavy dashed lines), giving rise to flatter IGIMFs (see Fig. \[igimf\_ext\]), produces larger \[$\alpha$/Fe\] ratios. In fact, flatter IGIMFs result in a larger fraction of massive stars and, therefore, to a larger production of $\alpha$-elements. We can also appreciate that the models reproduce quite well the \[$\alpha$/Fe\] (both \[O/Fe\] and \[Mg/Fe\]) ratios in elliptical galaxies, at least for the models BETA200 and BETA235. To appreciate the effect of the IGIMF approach, we plot also (long-dashed line) a model with the fixed canonical IMF which, as mentioned in Sect. \[sec:igimf\] and reminded in Sect. \[subs:SNIa\], has the form $\xi(m) = k m^{-\alpha}$, with $\alpha = 1.3$ for 0.08 M$_\odot
<$ $m$ $<$ 0.5 M$_\odot$ and $\alpha = 2.35$ above 0.5 M$_\odot$. The curves obtained with the IGIMF tend to flatten out at large $\sigma$, whereas the curve obtained with the constant IMF shows a constant slope. This demonstrates once more that the adoption of the IGIMF is particularly remarkable in the low-mass (and low-$\sigma$) galaxies. The curve with a constant IMF approaches asymptotically the model BETA100 since this model at large SFRs produces the flattest IMFs (see Fig. \[igimf\_ext\]). Besides a small shift of a few tenths of dex (which can be fixed increasing the parameter $A$ in Eq. \[eq:r1a\]), the curve with a constant IMF reproduces well the trend of \[$\alpha$/Fe\] vs. $\sigma$ of the THOM05 sample, demonstrating that the downsizing (or inverse-wind) models (Matteucci [@matt94]; Pipino & Matteucci [@pm04]) are also perfectly capable of explaining this trend in large elliptical galaxies. However, evidence is mounting that \[$\alpha$/Fe\] ratios in early-type dwarf galaxies are solar or sub-solar. For instance, van Zee, Barton & Skillman ([@vbs04]) showed that \[$\alpha$/Fe\] ratios (derived from Lick indices) of a sample of Virgo dwarf irregular galaxies range between -0.3 and solar. Also in the cluster Abell 496 the smallest galaxies show \[Mg/Fe\] to be solar or sub-solar (Chilingarian et al. [@chil08]). To show that we have also plotted in Fig. \[alphafe\_gamma0.3\] (open triangles) the data of a sample of low-mass early-type galaxies by Sansom & Northeast ([@sn08]). These data confirm that the \[$\alpha$/Fe\] vs. $\sigma$ relation is probably steeper in the low-mass regime and that our IGIMF results can naturally predict this behavior. However, in order to properly test our results in the low-mass regime more data are needed.
It is worth pointing out that in this figure (and in the following ones) we have considered only model galaxies for which the SFR is smaller than 100 M$_\odot$ yr$^{-1}$. This is the reason why the data points reach larger $\sigma$ than the results of our model. In extreme starbursts the IMF might become top-heavy as evident by the mass-to-light ratios in ultra-compact dwarf galaxies which are ultra-massive “star clusters” that form when the SFR is very high (Dabringhausen, Kroupa & Baumgardt [@dkb09]) and this will need to be incorporated in the IGIMF calculations (work in preparation).
In general, in local early-type galaxies the stellar abundances are measured by means of various absorption-line Lick indices, such as Mg$\, b$ and $<Fe> = 0.5 (Fe52720 + Fe5335)$ (THOM05). To properly compare predictions to observational abundance data obtained for local ellipticals, in general one should derive the luminosity-weighted average abundances. The real abundances averaged by mass are larger than the luminosity-averaged ones, owing to the fact that, at constant age, metal-poor stars are brighter (Greggio [@greg97]). To calculate the luminosities we have made use of the Starburst99 package (Leitherer et al. [@leit99]; V[á]{}zquez & Leitherer [@vl05]), producing $L (t)$ for each value of SFR and $\beta$. The results are shown in Fig. \[lum\] for the first 100 Myr (the luminosities remain almost constant after 100 Myr). As expected, since model BETA100 is characterized by the flattest IGIMFs, it produces also the largest luminosities. We have then calculated luminosity-weighted mass ratios by using the formula: $$[\alpha/Fe] = \log_{10} \frac{\int_{0}^{\Delta t} {\alpha \over Fe} (t)
L (t) dt}{\int_{0}^{\Delta t} L (t) dt} - \log_{10}{\alpha_\odot
\over Fe_\odot}.
\label{eq:alphafe_lw}$$ The results are shown in Fig. \[alphafe\_lumw\]. As we can see, the results differ very little (by a few hundredths of dex at most) compared to the mass-averaged abundance ratios. We have checked these results also using the spectro-photometric code of Jimenez et al. ([@jime98]) (see also Calura & Matteucci [@cm03]) but the results do not differ appreciably compared with the ones obtained with the Starburst99 package. Indeed, it has been already shown in the literature (but for constant IMFs) that the discrepancy for the \[Mg/Fe\] ratio computed by averaging by mass and by luminosity is very small, with typical values of 0.01 dex (Matteucci, Ponzone & Gibson [@mpg98]; Thomas et al. [@thom99]). We have confirmed this finding also in the case of the IGIMF.
To check how much our results depend on the assumption of a variable $\Delta t$ with stellar mass, we plot in Fig. \[alphafe\_dtc\] the \[$\alpha$/Fe\] obtained assuming a constant value of $\Delta t$ = 1 Gyr. The agreement with the observations is still quite good; in particular the models maintain an increasing trend of \[$\alpha$/Fe\] with $\sigma$. However, the curves tend to flatten out too much at larger $\sigma$, at variance with the trend shown by the observations. This is due to the fact that, as pointed out in Sect. \[sec:igimf\], the various IGIMFs for rates of star formation larger than 1 M$_\odot$ yr$^{-1}$ do not show very large differences. Therefore, the assumption of a star formation duration inversely proportional to the stellar mass of the galaxy (or in other words the downsizing) is a key ingredient to understand the chemical properties of large elliptical galaxies.
To appreciate the dependence on the distribution function of mass ratios in binary stars (the parameter $\gamma$ introduced in Sect. \[subs:SNIa\]) we plot in Figs. \[alphafe\_gamma2.0\] and \[alphafe\_gamma-0.3\] the results of models with $\gamma$ = 2.0 and $\gamma$ = -0.3, respectively. The curves obtained with $\gamma$ = 2.0 tend to be slightly steeper than the ones shown in Fig. \[alphafe\_gamma0.3\] (and slightly steeper than the observations) but the agreement remains still good, in particular for the models BETA100 and BETA200. If we assume $\gamma$ = -0.3 an excellent match with the observations is instead provided by the model BETA235. Models BETA100 and BETA200 show the same slope of the observational data but shifted by a few tenths of dex. A slight increase of the parameter $A$ in eq. \[eq:r1a\] would make these models perfectly compatible with the observations.
It is particularly remarkable that the trend of \[$\alpha$/Fe\] ratios vs. $\sigma$ (namely an increase of \[$\alpha$/Fe\] with $\sigma$) is naturally reproduced using the IGIMF approach, without any further assumption or fine-tuning of parameters. This is for instance at variance with what hierarchical clustering models of structure formation would tend to produce, since in this case larger elliptical galaxies are formed later, out of building blocks in which the \[$\alpha$/Fe\] ratio has already dropped (e.g. Thomas et al. [@thom02]). It is worth mentioning that De Lucia et al. ([@delu06]), by means of a semi-analytical model adopting the concordance $\Lambda$ CDM cosmology, suggested that more massive ellipticals should have shorter star formation timescales, but lower assembly (by dry mergers) redshift than less luminous systems. This is one of the first works based on the hierarchical paradigm for galaxy formation producing downsizing in the star formation histories of early-type galaxies through the inclusion of AGN feedback (see also Bower et al. [@bower06]; Cattaneo et al. [@catta06]), although they did not compute the \[$\alpha$/Fe\]–$\sigma$ relation for ellipticals. However, the lower assembly redshift for the most massive system is still in contrast to what is concluded by Cimatti, Daddi & Renzini ([@cdr06]), who show that the downsizing trend should be extended also to the mass assembly, in the sense that the most massive ellipticals should have assembled before the less massive ones. Very recently, Pipino et al. ([@pipi08]) showed that even in semi-analytical models able to account for the downsizing, the \[$\alpha$/Fe\] vs. $\sigma$ relation is not reproduced.
Although the agreement between our results and the observations is good, none of the models presented so far fits perfectly the data at low and high $\sigma$ simultaneously. In order to work out an overall best model, for each value of $\gamma$ and $\beta$ we have checked, by means of a minimization of the normalized chi square, which normalization constant $A$ fits better the data. The results are shown in Fig. \[testA\]. As we can see, model BETA235 seems to be preferable and the best agreement between data and models is obtained for the model BETA235 with $\gamma$ = 2.5 and $A$ = 0.036. In general, the best fits are obtained with large values of $\gamma$, although that requires low values of $A$. A large value of $\gamma$, favoring equal-mass binary systems, is consistent with the results of Shara & Hurley ([@sh02]), although observational surveys cited in Sect. \[subs:SNIa\] seem to indicate lower values of $\gamma$.
We should however not forget that the $\Delta t$–luminous mass relation we have used in this work has been obtained by THOM05 [*assuming a constant IMF.*]{} We have therefore checked, starting from our best model, namely a model with $\beta = 2.35$, $\gamma = 2.5$ and $A = 0.036$, how this relation should change in order to best fit the data. It turns out that, within the IGIMF theory, the best $\Delta
t$–luminous mass relation is given by:
$$\log \Delta t = 2.38 - 0.24 \log M_{tot},
\label{eq:newdtl}$$
where $\Delta t$ is in Myr and $M_{tot}$ in M$_\odot$, and this relation produces the results shown in Fig. \[bestfit\]. We have also plotted in this figure a model with canonical stellar IMF (heavy long-dashed line) in which we have increased the value of $A$ up to 0.13 in order to better reproduce the data. We can notice again that the canonical IMF can perfectly fit the \[$\alpha$/Fe\] ratios in large ellipticals but it shows a constant slope and therefore it cannot equally well reproduce the \[$\alpha$/Fe\] ratios in dwarf galaxies, at variance with the IGIMF model. The comparison between relation \[eq:newdtl\] and eq. 5 of THOM05 is displayed in Fig. \[deltat\_comp\]. We can notice here that the [*downsizing effect*]{} (namely the shorter duration of the star formation in larger galaxies) is milder, in the sense that the $\Delta t$ for large galaxies is (slightly) larger than the timescale calculated by THOM05, whereas at low $\sigma$ the star formation durations are significantly lower than the ones predicted by THOM05.
Discussion and conclusions
==========================
In this paper we have studied, by means of analytical and semi-analytical calculations, the evolution of \[$\alpha$/Fe\] ratios in early-type galaxies and in particular their dependence on the luminous mass (or equivalently on the velocity dispersion $\sigma$). We have applied the so-called integrated galactic initial mass function (IGIMF; Kroupa & Weidner [@kw03]; Weidner & Kroupa [@wk05]) theory, namely we have assumed that the IMF depends on the star formation rate (SFR) of the galaxy, in the sense that the larger the SFR is, the flatter is the resulting slope of the IGIMF. This kind of behavior would naturally tend to form more massive stars (and therefore more SNeII) in large galaxies, which are characterized by more intense star formation episodes. Therefore, it is expected that, since $\alpha$-elements are mostly formed by SNeII, the most massive galaxies are also the ones which attain the largest \[$\alpha$/Fe\] ratios, in agreement with the observations. One of the main aims of this paper was to quantitatively check whether the chemical evolution of galaxies within the IGIMF theory is able to accurately fit the observed \[$\alpha$/Fe\] vs. $\sigma$ relation.
We have analytically calculated the SNII and SNIa rates with the IGIMF assuming 3 possible slopes of the distribution function of embedded clusters, $\xi_{\rm ecl} \propto M_{\rm ecl}^{-\beta}$, where $M_{\rm
ecl}$ is the stellar mass of the embedded star cluster; in particular we have considered $\beta = 1.00$ (model BETA100); $\beta = 2.00$ (model BETA200); $\beta = 2.35$ (model BETA235). We have seen that, if we consider constant SFRs over the whole Hubble time, the final SNIa and SNII rates agree quite well with the observations of spiral galaxies (in particular the S0a/b ones). The agreement with the observed rates in irregular galaxies is not good, but a constant SFR over the whole Hubble time is not likely in irregular galaxies, which probably have experienced an increase of the SFR in the last Gyrs of their evolution.
To calculate the \[$\alpha$/Fe\] ratios with the IGIMF we assumed that early-type galaxies form stars at a constant rate over a period of time $\Delta t$ which depends on the total luminous mass of the considered galaxy. This hypothesis is based on the work of THOM05 who, on the basis of observational grounds, showed the existence of a downsizing pattern for elliptical galaxies, i.e. that the most massive galaxies are the ones with the shortest $\Delta t$. We have then calculated the production of $\alpha$-elements and Fe by SNeII (in particular we have calculated IGIMF-averaged SNII yields) and by SNeIa and we have calculated mass-weighted and luminosity-weighted \[$\alpha$/Fe\] ratios for each model galaxy, characterized by different SFRs and $\beta$.
The resulting mass-averaged \[$\alpha$/Fe\] vs. $\sigma$ relations show the same slope as the observations in massive galaxies as reported by THOM05, irrespective of the value of $\beta$ and of the distribution function of mass ratios in binaries $f(\mu) \propto \mu^{\gamma}$ (which affects the SNIa rates), although models with $\beta = 2.35$ and large values of $\gamma$ seem to be preferable. Some models show a shift (of a few tenths of dex) compared with the observations but this can be fixed increasing (or decreasing) the fraction, $A$, of binary systems giving rise to SNeIa, which is an almost unconstrained parameter. It is however remarkable that all the models we have calculated show the same trend of the observations because if, as commonly argued, large elliptical galaxies form out of mergers of smaller sub-structures (hierarchical clustering), it would be natural to expect that they are the ones with the lowest \[$\alpha$/Fe\] ratios because they form later, out of building blocks where \[$\alpha$/Fe\] has already dropped.
It is worth pointing out that the \[$\alpha$/Fe\] ratios do not depend on the gas flows (infall and outflow) experienced by the galaxy (Recchi et al. [@recc08]) therefore our results do not depend on specific infall and outflow parameters, which make them particularly robust. However, these parameters affect the overall metallicity of the galaxy, therefore they need to be taken into account in order to check whether our models can correctly reproduce the mass-metallicity relation. As mentioned in the Introduction, Köppen et al. ([@kwk07]) have already shown that the IGIMF theory is able to reproduce the mass-metallicity relation found by Tremonti et al. ([@t04]) in star-forming galaxies. We are checking, by means of detailed numerical models, that the IGIMF theory is able to reproduce at the same time the mass-metallicity relation and the \[$\alpha$/Fe\]-$\sigma$ relation in early-type galaxies. This study will be presented in a forthcoming paper.
We have also considered models in which the IMF does not vary with the SFR and, because of the variations of $\Delta t$ with SFR, these models are compatible with the observations of large elliptical galaxies as well. However, these models produce a \[$\alpha$/Fe\] vs. $\sigma$ relation that can be described as a single-slope power-law, whereas the IGIMF models bend down significantly at low masses (and low $\sigma$). This is because the IGIMF becomes particularly steep in the galaxies with the mildest SFRs and this adds to the downsizing effect (namely the decreasing duration of the SFR with increasing mass). From our study therefore, an important conclusion is that a very reliable observable to test the validity of the IGIMF theory is the observation of the \[$\alpha$/Fe\] ratios in dwarf galaxies. The available data on \[$\alpha$/Fe\] ratios in low-mass early-type galaxies show indeed some steepening of the \[$\alpha$/Fe\] vs. $\sigma$ relation, in agreement with the IGIMF predictions.
We have also tested how much our results depend on the assumption of a variable $\Delta t$ with stellar mass by computing models with $\Delta
t$ = 1 Gyr irrespective of the stellar mass. The agreement between models and observational data is still reasonably good but the curves tend to flatten out too much at large stellar masses compared with the observations (and with the IGIMF models). This indicates that the downsizing remains a fundamental ingredient to understand the chemical properties of early-type galaxies. However, if we check for which $\Delta t$–luminous mass relation we obtain the best fit between data and models, it turns out that the downsizing effect must be milder than predicted by THOM05, in the sense that large galaxies form stars for a slightly longer timescale than calculated by THOM05, whereas low-mass galaxies have star formation durations significantly shorter. Although the exact form of the best-fit $\Delta t$–luminous mass relation is subject to a number of parameters (IGIMF parameters; parameters regulating the SNIa rate etc.) and might change once larger and more detailed abundance measurements are available, the result of a milder downsizing effect compared to the findings of THOM05 is robust.
Eventually, we have seen that luminosity-weighted \[$\alpha$/Fe\] ratios agree very well with the mass-weighted ones (with relative differences of a few hundredths of dex at most), in accordance with the results of Matteucci et al. ([@mpg98]).
We remind the reader that, with our analytical approach to chemical evolution, we are making some important simplifying assumptions. For instance, our computation of the interstellar ${\alpha \over Fe} (t)$ given by eq. \[eq:alphafe\_gas\] does not take into account in detail the lifetimes of massive stars. Furthermore, our present calculations do not take into account the variation with time of the metallicity in galaxies, which should also influence the stellar yields. From the various tests performed so far, and from the comparison of our results with numerical results (Thomas et al. [@thom99]; Pipino & Matteucci [@pm04]), we have verified that these assumptions may play only some minor role in determining the zero-point, but not the slope of the predicted \[$\alpha$/Fe\] vs $\sigma$ relation. All of these simplifying assumptions will be relaxed in our forthcoming paper, where we will present a numerical approach to the role of the IGIMF in galactic chemical evolution.
The main results of our paper can be summarized as follows:
- Models in which the IGIMF theory is implemented naturally reproduce an increasing trend of \[$\alpha$/Fe\] with luminous mass (or $\sigma$), as observed in early-type galaxies.
- However, models with constant duration of the star formation produce a \[$\alpha$/Fe\] vs. $\sigma$ relation which flattens out too much at large $\sigma$. Only models in which the star formation duration inversely correlates with the galactic luminous mass (downsizing) can quantitatively reproduce the observations.
- Models in which the IGIMF is implemented show (at variance with the constant IMF models) a steepening of the \[$\alpha$/Fe\] vs. $\sigma$ relation for small galaxies, therefore the IGIMF theory can be tested by observing the \[$\alpha$/Fe\] in dwarf galaxies. The observations available so far are in agreement with our predictions.
- Luminosity-weighted abundance ratios differ from the mass-weighted ones by a few hundredths of dex at most. This result, already known for constant IMF models, has been confirmed in the IGIMF framework.
- In order to obtain the best fit between our results and the observed \[$\alpha$/Fe\] ratios in early-type galaxies, the downsizing effect (namely the shorter duration of the star formation in larger galaxies) has to be milder than previously thought.
- The best results are obtained for a cluster mass function $\xi_{\rm ecl} \propto M_{\rm ecl}^{-2.35}$, indicating that the embedded cluster mass function should have a Salpeter slope.
We deeply thank Francesca Matteucci for help and support. Discussions with Antonio Pipino are also acknowledged. S.R. acknowledges generous financial support from the FWF through the Lise Meitner grant M1079-N16. F.C. and S.R. acknowledge financial support from PRIN2007 (Italian Ministry of Research) Prot. N. 2007JJC53X. We thank the referee, Daniel Thomas, for very useful comments.
Abt, H.A., & Levy, S.G. 1976, ApJS, 30, 273
Anders, E., & Grevesse, N. 1989, Geochim. Cosmochim. Acta 53, 197
Bower, R.G., Benson, A.J., Malbon, R., et al. 2006, MNRAS, 370, 645
Burstein, D., Bender, R., Faber, S., & Nolthenius, R. 1997, AJ, 114, 1365
Calura, F., & Matteucci, F. 2003, ApJ, 596, 734
Calura, F., & Matteucci, F. 2006, ApJ, 652, 889
Calura, F., Matteucci, F., & Menci, N. 2004, MNRAS, 353, 500
Cattaneo, A., Dekel, A., Devriendt, J., Guiderdoni, B., & Blaizot, J. 2006, MNRAS, 370, 1651
Chabrier, G. 2001, ApJ, 554, 1274
Chiappini, C., Matteucci, F., & Gratton, R. 1997, ApJ, 477, 765
Chilingarian, I.V., Cayatte, V., Durret, F., et al. 2008, A&A, 486, 85
Cimatti, A., Daddi, E., & Renzini, A. 2006, A&A 453,29
Cowie, L.L., Songalia, A., Hu, E.M., & Cohen, J.G. 1996, AJ, 112, 839
Dabringhausen, J., Kroupa, P., & Baumgardt, H. 2009, MNRAS, accepted ([arXiv:0901.0915]{})
De Lucia, G., Springel, V., White, S.D.M., Croton, D., & Kauffmann, G. 2006, MNRAS, 366, 499
Duquennoy, A., & Mayor, M. 1991, A&A, 248, 485
Evans, N.J., Dunham, M.M., J[ø]{}rgensen, J.K., et al. 2008, [arXiv:0811.1059]{}
Gibson, B.K., Loewenstein, M., & Mushotzky, R.F. 1997, MNRAS, 290, 623
Greggio, L. 1997, MNRAS, 285, 151
Greggio, L. 2005, A&A, 441, 1055
Greggio, L., & Renzini, A. 1983, A&A, 118, 217
Hunter, D.A., Elmegreen, B.G., Dupuy, T.J., & Mortonson, M. 2003, AJ, 126, 1836
Jimenez, R., Padoan, P., Matteucci, F., & Heavens, A.F. 1998, MNRAS, 299, 123
Kodama, T., Yamada, T., Akiyama, M., et al. 2004, MNRAS, 350, 1005
Köppen, J., Weidner, C., & Kroupa, P. 2007, MNRAS, 375, 673
Kouwenhoven, M.B.N., Brown, A.G.A., Zinnecker, H., Kaper, L., & Portegies Zwart, S.F. 2005, A&A, 430, 137
Kroupa, P., Tout, C.A., & Gilmore, G. 1993, MNRAS, 262, 545
Kroupa, P., & Weidner, C. 2003, ApJ, 598, 1076
Kuntschner, H., Lucey, J.R., Smith, R.J., Hudson, R.J., & Davies, R.L. 2001, MNRAS, 323, 615
Lada, C.J., & Lada, E.A. 2003, ARA&A, 41, 57
Larsen, S.S., & Richtler, T. 2000, A&A, 354, 836
Larsen, T.I., Sommer-Larsen, J., & Pagel, B.E.J. 2001, MNRAS, 323, 555
Leitherer, C., Schaerer, D., Goldader, J.D., et al. 1999, ApJS, 123, 3
Mannucci, F., Della Valle, M., Panagia, N., et al. 2005, A&A, 433, 807
Maoz, D. 2008, MNRAS, 384, 267
Marigo, P. 2001, A&A, 370, 194
Massey, P. 2003, ARA&A, 41, 15
Massey, P., & Hunter, D.A. 1998, ApJ, 493, 180
Matteucci, F. 1994, A&A, 288, 57
Matteucci, F. 2001, The Chemical Evolution of the Galaxy, ASSL, Kluwer Academic Publisher
Matteucci, F. 2007, [arXiv:0704.0770]{}
Matteucci, F., Ponzone, R., & Gibson, B.K. 1998, A&A, 335, 855
Matteucci, F., & Recchi, S. 2001, ApJ, 558, 351
Nagashima, M., Lacey, C.G., Okamoto, T., et al. 2005, MNRAS, 363, L31
Padovani, P., & Matteucci F. 1993, ApJ, 416, 26
Pflamm-Altenburg, J., Weidner, C., & Kroupa, P. 2007, ApJ, 671, 1550 (PWK07)
Pilyugin, L.S., & Edmunds, M.G. 1996, A&A, 313, 783
Pipino, A., Kawata, D., Gibson, B.K., & Matteucci, F. 2005, A&A, 434, 553
Pipino, A., & Matteucci, F. 2004, MNRAS, 347, 968
Pipino, A., Silk, J., & Matteucci, F. 2009, MNRAS, 392, 475
Pipino, A., Devriendt, J.E.G., Thomas, D., Silk, J., & Kaviraj, S. 2008, [astro-ph/0810.5753]{}
Piskunov, A.E., Belikov, A.N., Kharchenko, N.V., Sagar, R., & Subramaniam, A. 2004, MNRAS, 349, 1449
Recchi, S., Spitoni, E., Matteucci, F., & Lanfranchi, G.A. 2008, A&A, 489, 555
Romano, D., Chiappini, C., Matteucci, F., & Tosi, M. 2005, A&A, 430, 491
Salpeter, E.E. 1955, ApJ, 121, 161
Sansom, A.E., & Northeast, M.S. 2008, MNRAS, 387, 331
Shara, M.M., & Hurley, J.R. 2002, ApJ, 571, 830
Shatsky, N., & Tokovinin, A. 2002, A&A, 382, 92
Siess, L. 2007, A&A, 476, 893
Smith, L.J., & Gallagher, J.S. III 2001, MNRAS, 326, 1027
Thielemann, F.-K., Nomoto, K., & Hashimoto, M. 1993, in Origin and Evolution of the Elements, eds. N. Prantzos, E. Vangioni-Flam and M. Cassé, Cambridge Univ. Press, Cambridge, p. 297
Thomas, D., Greggio, L., & Bender, R. 1999, MNRAS, 302, 537
Thomas, D., Maraston, C., & Bender, R. 2002, Ap&SS, 281, 371
Thomas, D., Maraston, C., Bender, R., & Mendes de Oliveira, C. 2005, ApJ, 621, 673 (THOM05)
Timmes, F.X., Woosley, S.E., & Weaver, T.A. 1995, ApJS, 98, 617
Tremonti, C.A., Heckman, T.M., Kauffmann, G., et al. 2004, ApJ, 613, 898
Tutukov, A.V. 1978, A&A, 70, 57
Tutukov, A.V., & Yungelson, L.R. 1980, in Close Binary Stars, eds. M.J. PLavec, D.M. Popper and R.K. Ulrich, Reidel, Dordrecht, p. 15
Valiante, R., Matteucci, F., Recchi, S., & Calura, F. 2009, NewA, accepted ([arXiv:0807.2354]{})
van Zee, L., Barton, E.J., & Skillman, E.D. 2004, AJ, 128, 2797
V[á]{}zquez, G.A., & Leitherer, C. 2005, ApJ, 621, 695
Weidner, C., & Kroupa, P. 2004, MNRAS, 348, 187
Weidner, C., & Kroupa, P. 2005, ApJ, 625, 754
Weidner, C., & Kroupa, P. 2006, MNRAS, 365, 1333
Weidner, C., Kroupa, P., & Larsen, S.S. 2004, MNRAS, 350, 1503
Weiss, A. Peletier, R.F., & Matteucci, F. 1995, A&A, 296, 73
Whelan, J., & Iben, I.Jr. 1973, ApJ, 186, 1007
Woosley, S.E., & Weaver, T.A. 1995, ApJS, 101, 181
Wyse, R.F.G., Gilmore, G., Houdashelt, M.L., et al. 2002, New Astron., 7, 395
Zhang, Q., & Fall, S.M. 1999, ApJ, 527, L81
[^1]: simone.recchi@univie.ac.at
[^2]: fcalura@oats.inaf.it
[^3]: pavel@astro.uni-bonn.de
[^4]: Note that usually observative papers adopt mass ratios $q = m_2/m_1$ instead of $\mu$ (Abt & Levy [@al76]). The mass ratio distribution function $f (q)$ can be obtained from $f
(\mu)$ by means of a simple change of variable.
|
---
abstract: |
The propagation of a streamer near an insulating surface under the influence of a transverse magnetic field is theoretically investigated. In the weak magnetic field limit it is shown that the trajectory of the streamer has a circular form with a radius that is much larger than the cyclotron radius of an electron. The charge distribution within the streamer head is strongly polarized by the Lorentz force exerted perpendicualr to the streamer velocity. A critical magnetic field for the branching of a streamer is estimated. Our results are in good agreement with available experimental data.
PACS numbers: 52.80.Mg, 52.35.Lv, 51.60.+a
address:
- 'Department of Chemistry. Technion, Haifa 32000, ISRAEL'
- |
Grenoble High Magnetic Laboratory\
Max-Planck-Institute für Festkorperforschung and\
Center National de la Recherche Scientific,\
25 Avenue des Martyres, F-38042, Cedex 9, FRANCE\
.
author:
- 'V.N.Zhuravlev, T.Maniv'
- 'I.D.Vagner and P.Wyder'
date: 'August 15, 1997'
title: STREAMER PROPAGATION IN MAGNETIC FIELD
---
\#1
epsf.sty
Recent experiments [@uhlig] on gas breakdown near an insulating surface in a high magnetic field $\overrightarrow{B}$ have shown new remarkable properties of such discharges. The channel of discharge in a magnetic field appears to have a circular form with radius $R_s$ several orders of magnitude larger than the electronic cyclotron radius. It decreases with increasing $B$ and reaches $R_s$ $\sim 1cm$ at $B$ $\sim 7T$. At higher magnetic field the discharge has a branched structure. These experiments have shown that the streamer propagation cannot be treated as the motion of a charge particle in crossed external electric ${\cal {E}}$ and magnetic $B$ fields.
Interest in the theory of streamers is usually associated with investigation of gas breakdown phenomena. On an insulating surface near a point electrode, in the region with strong electric field, the discharge has a filamentary structure. The tip of the filament moves with high velocity $v_0$ ( $v_0$ $\sim 10^8cm/s$ ), that exceeds the drift velocity $v_d$ of electrons in the streamer head field $%
\overrightarrow{E}_s$. The increase of external electric field $%
\overrightarrow{{\cal {E}}}$ results in penetration of some filaments ( so called ”leaders” ) deep into the surrounding gas. Although the plasma parameters in streamers are different from those in leaders, their propagation is associated with the same physical processes and is defined mainly by the parameters of the streamer head. Since we are interested in the behavior of the streamer front only, we will not distinguish here between a streamer and a leader.
The streamer propagation mechanism was suggested by Raether, Loeb, and Meek [@raether]-[@loeb], and was further developed by other authors [@turc]-[@tidman]. According to this theory the charged head induces in its vicinity a strong electric field. This field leads to the increase of the electron density ahead of the streamer front due to impact ionization . The charge is displaced from this region via Maxwell relaxation. It is assumed that the free electron density ahed of the streamer front is not zero due, e.g. to absorbtion of the streamer head radiation produced, for example, by the streamer head radiation. The simple model which takes into account only these main processes was considered by M.I. D’yakonov and V.Y. Kashrovskii [@mid1][@mid2]. They have estimated theoretically the streamer parameters and have shown that the streamer velocity $v_0$ and radius $r_s$ change smoothly with the external electric field $\overrightarrow{{\cal {E}}}$, such that the propagation of the streamer head can be treated as a quasistationary process.
In the present paper we generalize the streamer model [@mid1][@mid2] to include an external magnetic field. It is assumed that the plasma filaments propagate in a plane perpendicular to the external magnetic field and the streamer parameters do not change in the direction parallel to the magnetic field. It is shown that , in the weak magnetic field limit, a quasistationary streamer in the frame of reference rotating with a constant angular velocity $\omega _s=v_0/R_s$ , proportional to the head charge density, can be considered as a streamer in the absence of the magnetic field . We estimate the main parameters of a streamer head, and show that the obtained value for the radius of curvature is in close agreement with experimental data [@uhlig]. The influence of the magnetic field on the charge distribution within the streamer head is discussed, and a critical magnetic field for the onset of branching is estimated and compared to the experimental data.
Since the energy relaxation time of electrons is much larger than the electron-ion relaxation time we ignore the gas heating processes. Thus the concentration of atoms changes smoothly on the distance of the order of the streamer head size and is assumed to be constant inside the head. We neglect also the ion drift velocity in comparison with electron drift velocity $v_d$ and streamer velocity $v_0$.
The system of equations for the electron density $n$, the ion density $N$, and the electric field $\overrightarrow{E}$ is $$\frac{\partial n}{\partial t}+div\left( n\overrightarrow{v}_d\right) =\beta
\left( E\right) n$$ $$\frac{\partial N}{\partial t}=\beta \left( E\right) n$$ $$div\overrightarrow{E}=4\pi \rho \left( x\right) ,\text{ }rot\overrightarrow{E%
}=0 \label{e1}$$ where $\rho \left( x\right) =e\left( N-n\right) $ is the charge distribution, $\beta \left( E\right) =v_d\alpha \left( E\right) $. The impact-ionization coefficient $\alpha \left( E\right) $ increases very sharply with the field and saturates at some field value $E_0$ [@engel] $$\alpha \left( E\right) =\alpha _0e^{-E_0/E} \label{e2}$$
We assume for simplicity that the electron drift velocity is proportional to electric field $\overrightarrow{E}$, $v_{di}=\sum_{k}\mu _{ik}E_k$. Without external magnetic field the mobility $\mu _{ik}$ is a diagonal tensor, $\mu _{ik}=\mu
_0\delta _{ik}$. In a weak magnetic field the mobility $\mu _{ik}$ is a function of $\overrightarrow{B}$, which can be written as $$\mu _{ik}=\frac{\mu _0}{1+\gamma ^2}\left( \delta _{ik}+\gamma \varepsilon
_{ik}\right) , \label{e3}$$ where $\varepsilon _{ik}$ is the antisymmetric tensor in a plane perpendicular to $\overrightarrow{B}$. The parameter $\gamma =\omega _B\tau
_{ea}$ , where $\omega _B=\frac{eB}{m_ec}$ is the cyclotron frequency and $%
\tau _{ea}$ is the time of electron-atom collisions, is assumed to be small, $\gamma \ll 1$. In what follows we will be interested only in linear corrections in $\gamma $ to the solution of Eq. (\[e1\]). Since the parameters $\alpha _0,$ $E_0$, and $v_d$ depend on $\gamma ^2$ [@brown] they are magnetic field independent, in our approximation.
Let us consider the streamer propagation equation (\[e1\]) in the frame of reference with the origin at $\overrightarrow{r}_h=\frac{\int
\overrightarrow{r}\rho \left( \overrightarrow{r},t\right) d\overrightarrow{r}%
}{\int \rho \left( \overrightarrow{r},t\right) d\overrightarrow{r}}$ and rotating with an angular velocity $\omega \left( t\right) $ . The resulting new coordinates are: $$\eta _i=\sum_k\Omega _{ik}\left( t\right) \left( x_k-r_{hk}\left( t\right)
\right) \label{e4}$$ where $\Omega _{ik}\left( t\right) $ is the rotation matrix for the angle $%
\varphi =\int \omega \left( t\right) dt$ in the plane perpendicular to $%
\overrightarrow{B}$: $$\Omega _{ik}\left( t\right) =\left(
\begin{array}{cc}
\cos \varphi & \sin \varphi \\
-\sin \varphi & \cos \varphi
\end{array}
\right) \label{e4a}$$
The transformation of derivatives are $$\frac \partial {\partial x_i}\rightarrow \sum_k\frac \partial {\partial \eta
_k}\Omega _{ki},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac \partial {%
\partial t}\rightarrow \frac \partial {\partial t}+\sum_{ikl}\frac \partial {%
\partial \eta _k}\left[ \stackrel{.}{\Omega }_{ik}\Omega _{kl}^{-1}\eta
_l-\Omega _{ki}\stackrel{.}{r}_{hi}\right] \label{e5}$$ Here the dot denotes time derivative. It follows from Eq. (\[e5\]) that the streamer charge propagation is quasistationary if $\sum\limits_k\stackrel{.}{\Omega }_{ik}\Omega
_{kl}^{-1}=\varepsilon _{il}\omega =const$ and $\sum\limits_i\Omega _{ki}%
\stackrel{.}{r}_{hi}=v_{0k}=const$. Thus the head of the quasistationary propagating streamer moves with constant velocity $v_0$ along a circle with radius $R_s=v_0/\omega $.
Rewriting Eq.(\[e1\]) for quasistationary propagation in the rotating frame we obtain $$\sum_{il}\frac \partial {\partial \eta _i}n\left[ -v_{0i}+\omega \varepsilon
_{il}\eta _l+\mu _0\left( E_i+\gamma \varepsilon _{il}E_{0l}\right) \right]
=\beta \left( E\right) n$$ $$-\sum_iv_{0i}\frac{\partial N}{\partial \eta _i}=\beta \left( E\right) n$$ $$\sum_i\frac{\partial E_i}{\partial \eta _i}=4\pi \rho \left( \eta \right) ,%
\text{ }\sum_{ik}\varepsilon _{ik}\frac{\partial E_i}{\partial \eta _k}=0
\label{e6}$$ Eq. (\[e6\]) should be solved with the following boundary condition $$\rho _sv_0=en_s\mu _0E_{s\text{,}} \label{e7}$$ which follows from the charge conservation on the surface of the streamer front. Here $n_s$ and $\rho _s=e\left( N-n_s\right) $ are the electron and the charge densities on the front.
Appearing in the first order term with $\gamma ,$ the electric field $E_{0k}$ is the field of a streamer propagating in the absence of external magnetic field. It consists of two parts: the external field $\overrightarrow{{\cal {E%
}}}$ and field $\overrightarrow{E}_\rho $ created by the head charge. Field $%
\overrightarrow{E}_\rho $ is a symmetrical function with respect to the streamer axis. Usually $\left| \overrightarrow{{\cal {E}}}\right| $ is negligible in comparison with $\left| \overrightarrow{E}_\rho \right| $, but near the electrode it can strongly influence the streamer propagation.
Let us expand $E_{0k}\left( \eta \right) $ near the central point $\eta _i=0$$$E_{0k}\left( \eta \right) =E_{0k}\left( 0\right) +a\sum_{l}\delta_{kl}
\eta_l+\sum_{l}b_{kl}\eta_l +D_k \label{e8}$$ where $a=2\pi \rho \left( 0\right) $, $b_{kl}$ - symmetrical matrix with $%
Sp\left( b\right) =0$. The term $E_{0k}\left( 0\right)+\sum_{l}b_{kl}
\eta_{l} $ corresponds to the potential field, which satisfies the Laplace equation and can be absorbed into $E_k$ as a correction. The field $D_k$, defined by Eq.(\[e8\]), is proportional to the deviation of the charge distribution from the uniform one. It is small in the central region and becomes large near the surface of the streamer head. Assuming that the streamer propagation is determined mainly by the central region, one can discard the terms with $\overrightarrow{D}$ while determining the streamer trajectory.
The second term in (\[e8\]) leads to the streamer curving. At $\omega
=-2\pi \rho \left( 0\right) \gamma \mu _0$ the equations (\[e6\]) turn to a system of equations describing the quasistationary streamer propagation without magnetic field. Thus streamers moving from cathode or anode will curve in opposite directions with frequency $\omega _s=\left| \omega \right|
=$ $\left| 2\pi \rho \left( 0\right) \gamma \mu _0\right| $.
Introducing Maxwell relaxation time $\tau _m^{-1}=4\pi \mu _0en_s$, one obtains $\omega _s\tau _m=\frac{\gamma \rho \left( 0\right) }{2en_s}$. If the streamer radius $r_s$ is of the same order of magnitude as the characteristic distance of the increase of electric field from internal region to the front, one can estimate $r_s$ as $r_s\simeq \tau _mv_0$. So we have $$\frac{r_s}{R_s}=\frac{\gamma \rho \left( 0\right) }{2en_s} \label{e9}$$ This value is very small, since $\gamma \ll 1$ and $\frac{\rho \left(
0\right) }{en_s}\sim \frac{\mu E_s}{v_0}\ll 1$. The last inequality can be obtained from Eq. (\[e7\]) at $\rho \left( 0\right) \simeq \rho _s$. Parameter $\gamma $ is proportional to magnetic field $B$, so the streamer radius decreases as $1/B$. This form of $R_s\left( B\right) $ is somewhat different from the experimentally observed field dependence reported in [@uhlig]$, i.e. R_s\left( B\right)\sim 1/B^\alpha $, where $\alpha \sim 1.3-1.5$. Such a disagreement is connected probably with the approximate description of the ionization coefficient and mobility by formulas (\[e2\]), (\[e3\]). It must be especially noticeable at high magnetic field.
To evaluate the streamer head charge we will consider the one dimensional streamer equations (\[e1\]). Such approximation holds if the width $\delta
$ of the streamer front is much smaller than the head size $r_s$: $\delta
\ll r_s$. This is the case if the electron density in front of the streamer is much smaller than inside [@mid2]. Equations (\[e1\]) at $B=0$ have a simple analytical solution. Assuming that the streamer moves along the $x$-axis and choosing the boundary conditions as $n\left( -\infty \right)
=n_\infty $, $E\left( -\infty \right) =0,$ and $n\left( E=E_s\right) =0$ we can easily obtain the relation between the equilibrium electron density $%
n_\infty $ and the electric field $E_s$ on the front $$\frac{n_\infty }{n_0}=\frac 1{1-\frac{\mu _0E_s}{v_0}}\int%
\limits_0^{E_s/E_0}e^{-1/x}dx \label{e10}$$ Here $n_0=\frac{\alpha _0}{4\pi e}$. This solution describes a plane wave with narrow front if $\mu _0E_s\ll v_0$. In the opposite case $\mu
_0E>>v_0$,one can neglect time derivatives in (\[e1\]) and obtain the stationary solution.
The equilibrium electron density $n_\infty $ and the propagation velocity $%
v_0$ are defined by the conditions of the streamer formation. This stage of the discharge development should be described by essentially nonstationary equations. Their solution depends on external electric field and parameters of initial ”seed”. On the quasistationary stage of the streamer propagation the electron density $n_\infty $ can be estimated with the help of the relation $$r_s\simeq \frac{v_0}{4\pi e\mu _0n_s} \label{e11}$$ Here it is supposed that $n_\infty \simeq n_s$ in agreement with $\rho
\left( 0\right) \sim \rho _s\ll en_s$. Equation (\[e7\]) allows to relate the charge density $\rho _s$ with $E_s$. Note that $E_s$ and $\rho _s/en_s$ have logarithmic dependence on the density $n_s$ and radius $r_s$. Thus, the experimental error in $r_s$ gives rise to small logarithmic correction to the relative charge $\rho _s/en_s$.
Let us now compare our results with the experiment. For the streamer and plasma parameters in the absence of a magnetic field we have used the data from the paper of Dhali and Williams [@dhali] for the streamer in $N_2$ at atmosphere pressure: $v_0=2\cdot 10^8cm/s$, $r_s\simeq 10^{-2}cm$. Substituting these values to (\[e10\]) and (\[e11\]) one obtains $%
n_s\simeq 3\cdot 10^{13}cm^{-3}$, $E_s/E_0\simeq 0.6$, $\rho _s/en_s\simeq
\mu _0E_s/v_0\simeq 0.2$. Estimating $\gamma =0.04$ at $B=1T$ we have from Eq. (\[e9\]) for the trajectory curvature radius $R_s\simeq 0.5cm$ at $%
B=5T $. The experimental value of $R_s$ [@uhlig] for the same conditions is slightly larger, i.e. $R_s^{ex}\simeq 1.2cm$. This discrepancy can be explained e.g. by the growth of the charge density from external region towards the streamer front.
Let us consider the streamer in the limit of an infinitely narrow front in the system of reference where the streamer head is at rest . The electric field $E$ inside the head is sufficiently small so that the ionization process can be safely neglected. The ions are assumed not to be affected by the electromagnetic field, i.e. $N=const$. The corresponding equations are $$\sum_{kl}\frac \partial {\partial \eta _k}\left[ n\left( -v_{0k}+\mu
_{kl}E_l+\omega \varepsilon _{kl}\eta _l\right) \right] =0$$ $$\sum_k\frac{\partial E_k}{\partial \eta _k}=4\pi \left( N-n\right)
\label{e12}$$ For simplicity the streamer body will be represented as a cylinder with radius $r_s$ and axis directed along the $\eta _x$-axis. Writing the content of the square brackets in Eq.(\[e12\]) by: $$\sum_ln\left( -v_{0k}+\mu _{kl}E_l+\omega \varepsilon _{kl}\eta _l\right)
=v_0N\frac{\partial \Phi }{\partial \eta _k} \label{e13}$$ the function $\Phi $ satisfies the Laplace equation $\Delta \Phi =0$ inside the streamer body with boundary conditions $$\frac{\partial \Phi }{\partial \eta _x}=1,\frac{\partial \Phi }{\partial
\eta _y}=0\text{ at }\eta _x=0\text{; }\frac{\partial \Phi }{\partial \eta _k%
}=0\text{ at }\eta _y=r_s\text{; }\Phi \rightarrow 0\text{ at }\eta
_x\rightarrow \infty \label{e14}$$
Using $E_l$ from (\[e13\]) and substituting it into Poisson equation we obtain the following equation for the normalized electron density $\overline{%
n}\equiv n/N$$$\left( \nu _{ik}\frac{\partial \Phi }{\partial \eta _k}\right) \frac{%
\partial \overline{n}}{\partial \eta _i}=-\frac{\overline{n}^2\left(
\overline{n}-1\right) }{L_0}+\frac{2\gamma \overline{n}}{R_s} \label{e15}$$ with the boundary condition $$\overline{n}\left( \eta _x=0,\eta _y=0\right) =1+\Delta \overline{n}
\label{e16}$$ where $\nu _{ik}=\delta _{ik}-\gamma \varepsilon _{ik}$, $L_0=\frac{v_0}{%
4\pi e\mu _0N}$ is the characteristic length of the charge relaxation and $%
\Delta \overline{n}=\rho _s/eN$. Since $\gamma \ll 1$ and $L_0\simeq r_s\ll
R_s$ the second term on the RHS of (\[e15\]) may be neglected. In the case of small charge density $e\Delta \overline{n}\ll e\overline{n}$ the equation (\[e15\]) has a simple analytical solution $$\overline{n}\left( \eta _x,\eta _y\right) =1+\Delta \overline{n}\exp \left( -%
\frac{s\left( \eta _x,\eta _y\right) }{L_0}\right) \label{e17}$$ where the effective path $s\left( \eta _x,\eta _y\right) $ is defined by an integral over a path from the point $\left( 0,0\right) $ to the point $\eta
\equiv \left( \eta _x,\eta _y\right) $, i.e: $$s\left( \eta _x,\eta _y\right) =\int\limits_{\left( 0,0\right) }^{\left(
\eta _x,\eta _y\right) }\frac{\sum\limits_{ik}\nu _{ik}\frac{\partial \Phi }{%
\partial \eta _k}d\eta _i}{\sum\limits_i\left| \sum\limits_k\nu _{ik}\frac{%
\partial \Phi }{\partial \eta _k}\right| ^2} \label{e18}$$
According to (\[e14\]) near the front surface $\frac{\partial \Phi }{%
\partial \eta _i}=\delta _{ix}$, so that to the first order with $\gamma $, $s\left( \eta \right) =\eta _x-\gamma \eta _y$ and $$\overline{n}\left( \eta _x,\eta _y\right) =1+\Delta \overline{n}\exp \left( -%
\frac{\eta _x}{L_0}+\gamma \frac{\eta _y}{L_0}\right) \label{e19}$$ The corresponding electric field is $$E_{\eta _x}\left( \eta _x,\eta _y\right) =E_s\exp \left( -\frac{\eta _x}{L_0}%
+\gamma \frac{\eta _y}{L_0}\right)$$ $$E_{\eta y}\left( \eta _x,\eta _y\right) =\gamma E_s\exp \left( -\frac{\eta _x%
}{L_0}+\gamma \frac{\eta _y}{L_0}\right) \label{e20}$$ where $E_s=e\Delta \overline{n}\frac{v_0}{\mu _0}$.
Thus, an electric field $E$ of the order of $E_s$,in the $\eta _x$-direction at the streamer front stimulates its propagation in this direction. It is therefore reasonable to suggest that a sufficiently strong electric field component $E_{\eta _y}$ $\sim E_s$, perpendicular the streamer propagation, will lead to the breakdown of the streamer head and the formation of a new streamer deflected along the $\eta _y$-direction with respect to the original one. The new streamers will arise only from one side in the plane transverse to $%
\overrightarrow{B}$ . Substituting $\eta _x=0$, $\eta _y=r_s\simeq L_0$ in (\[e20\]) we conclude that the condition $E_{\eta _y}$ $\sim E_s$ is fulfilled at $\gamma e^\gamma \sim 1$, i.e. $\gamma \simeq 0.6$. This value for the atmosphere discharge in $N_2$ corresponds to $B=12T$, which closely agree with the experimental result $B^{ex}\simeq 7T$ [@uhlig].
In conclusion we have shown that the simple model taking into account only the main processes provides a reasonably good description of the streamer discharge in a magnetic field. The streamer head propagation is very similar to the movement of a free charged ”particle”. Without magnetic field this ”particle” moves with a constant velocity $v_0=const$. In the presence of magnetic field the trajectory has a circular form. Such a simple picture occurs when the external electric field ${\cal {E}}$ is negligible in comparison with the field $E_s$ of the charged streamer head. Nevertheless, the role of the external field ${\cal {E}}$ is very important not only for maintenance of the discharge, but also for the definition of the streamer parameters on the initial stage of the development. Strong electric field $\cal {E}\sim E_{s}\sim E_0$ distorts the circular trajectory making it similar to the trajectory of a charged particle in crossed electric and magnetic fields. This phenomenon was observed in [@uhlig].
To estimate the ”particle” mass density $\rho _m$ we compare the expression for the radius $R_s$ in the form $R_s=\frac{\rho _mv_0c}{\rho _sB}
$ with Eq.(\[e9\]). We obtain the following expression for the ratio of the mass and the charge densities $$\frac{\rho _m}{\rho _s}=\frac 2{\Delta \overline{n}}\frac{\tau _m}{\tau _{ea}%
}\frac{m_e}e \label{e21}$$ Because of the large parameter $\frac 2{\Delta \overline{n}}\frac{\tau _m}{%
\tau _{ea}}>>1$ the streamer head turns in magnetic field more slowly than the particle with charge density $\rho _s$ and mass density $\rho _m\simeq
m_e\rho _s/e$, whose radius does not depend on the plasma parameters of the streamer head.
ACKNOWLEDGMENT: We are grateful to P.Uhlig for useful discussions. This research was supported by a grant from the US-Israeli Binational Science Foundation grant no. 94-00243, by the fund for the promotion of research at the Technion, and by the center for Absorption in Science, Ministry of Immigrant Absorption State of Israel.
P.Uhlig, J.C.Maan, and P.Wyder, Phys. Rev. Lett. [ **[63]{}**]{},1968 (1989)
M.Raether, [*[Electron Analanches and Breakdown in Gases]{}. (Butterworths,1964)* ]{}
J.M.Meek, J.D.Craggs, [*[Electron Breakdown of Gases]{}*]{}, (Oxford, 1953)
L.B.Loeb, Science [**[148]{}**]{}, 1417 (1965)
D.L.Turcotte, R.S.B.Ong, J. Plasma Physics [**[2]{}**]{}, part 2, 145 (1968)
G.A.Dawson, W.P.Winn, Zeittschrift für Physik [**[183]{}**]{}, 159 (1965)
R.Kligbell, D.A.Tidman, R.F.Fernsler, Physics of fluids [**[15]{}**]{}, 1969 (1972)
M.I.D’yakonov, V.Yu.Kachorovskii, Zh. Eksp. Teor. Fiz. [**[94]{}**]{}, 321 (1988)
M.I.D’yakonov, V.Yu.Kachorovskii, Zh. Eksp. Teor. Fiz. [**[95]{}**]{}, 1850 (1989)
A.von. Engel, [*[Ionized Gases]{}*]{}, (Oxford, 1965)
Sanborn C.Brown, [*[Introduction to Electrical Discharge in Gases]{}*]{}, ( New York, 1966 )
S.K.Dhali, P.F.Willians, J. Appl. Phys. [**[62]{}**]{}, 4696 (1987)
|
---
abstract: 'The occurrence of the modulational instability (MI) in transverse dust lattice waves propagating in a one-dimensional dusty plasma crystal is investigated. The amplitude modulation mechanism, which is related to the intrinsic nonlinearity of the sheath electric field, is shown to destabilize the carrier wave under certain conditions, possibly leading to the formation of localized envelope excitations. Explicit expressions for the instability growth rate and threshold are presented and discussed.'
author:
- 'I. Kourakis[^1] and P. K. Shukla[^2]'
date: 'Submitted 16 December 2003; revised version 30 January 2003'
title: |
Nonlinear modulation of transverse dust lattice waves\
in complex plasma crystals[^3]
---
#### Introduction.
Studies of numerous collective processes [@psbook] in dust contaminated plasmas (DP) have been of significant interest in connection with linear and nonlinear waves that are observed in laboratory and space plasmas. An issue of particular importance is the formation of strongly coupled DP crystals by highly charged dust grains, for instance in the sheath region above a horizontal negatively biased electrode in experiments [@Chu; @Thomas; @Melzer; @Hayashi]. Low-frequency oscillations occurring in these mesoscopic dust grain quasi-lattices, in both longitudinal and transverse directions, have been theoretically predicted [@Melandso; @farokhi; @vladimirov1; @tskhakaya; @Wang; @Morfill], and later experimentally observed [@psbook; @Morfill; @Homann; @Ivlev2000; @Misawa; @Piel]. We note that the observation of the characteristics of transverse vibrations around a levitated equilibrium position, where the electric and gravity forces are in balance, has been suggested as a diagnostic tool, enabling the determination of the grain charge [@Melzer; @Melzer2; @PS1996]. Recent generalizations taking into account dust charge variations [@Morfill2], layer coupling [@Vladimirov2] and two-dimensional crystal anisotropy [@anisotropic] are also worth mentioning.
It is known from solid state physics [@solid] that lattice vibrations are inevitably subject to amplitude modulation due to intrinsic nonlinearities of the medium. Furthermore, the wave propagation in crystals are often characterized by the Benjamin-Feir-type modulational instability (MI), a well-known mechanism for the energy localization related to the wave propagation in nonlinear dispersive media. The MI mechanism has been thoroughly studied in the past, mostly in one-dimensional (1d) solid state systems, where nonlinearities of the substrate potential and/or particle coupling may be seen to destabilize waves and possibly lead to localized excitations (solitary waves) [@Remo; @Scott]. In the context of plasma wave theory, this nonlinear mechanism has been investigated in a variety of contexts since long ago [@redpert; @Hasegawa]. In a weakly coupled dusty (or complex) plasma (DP), in particular, new electrostatic wave modes arise [@Verheest; @psbook], whose modulation has been studied quite recently [@AMS; @IKPSDIAW]; instability conditions were shown to depend strongly on modulation obliqueness, dust concentration and the ion temperature [@IKPSDIAW].
In principle, nonlinearity is always present in dusty plasma vibrations, due to the form of the inter-grain electrostatic interaction potential, which may be of the Debye-Hückel–type [@Melandso; @tskhakaya] or else [@Ignatov; @IKPSPLA]. Furthermore, the electric potential dominating oscillations in the transverse direction, yet often thought to be practically parabolic near the levitated equilibrium position [@Tomme], is intrinsically anharmonic [@tskhakaya2], as suggested by experimental results [@Zafiu; @Ivlev2000]. Despite this evidence, knowledge of nonlinear mechanisms related to low-frequency DP lattice modes is still in a preliminary stage. Small amplitude localized longitudinal excitations (described by a Korteweg-de Vries equation) were considered in Refs. [@Melandso; @PS2003] based on which longitudinal dust lattice wave (LDLW) amplitude modulation was considered in Ref. [@AMS2]. However, to the best of our knowledge, no study has been carried out, from first principles, of the amplitude modulation of transverse dust lattice waves (TDLWs) because of the sheath electric field nonlinearity. This Letter aims in making a first analytical step towards the study of DP crystals in this framework.
About three years ago, Misawa [*et al.*]{} [@Misawa] reported the observations of vertical nonlinear oscillations of a dust grain in a plasma sheath, and interpreted their results in terms of a position-dependent delayed grain charging effect. Zafiu *et al.* [@Zafiu] studied vertical dust-grain oscillations in a rf-discharge plasma, and succeeded in pointing out their strongly nonlinear behavior due to the sheath potential anharmonicity. Recently, Ivlev [*et al.*]{} [@Ivlev2003] investigated the nonlinear coupling between high-frequency transverse (vertical) dust lattice oscillations (TDLOs) and slow longitudinal dust lattice vibrations (LDLVs). Within the framework of a slowly varying envelope approximation, they derived a pair of equations for the modulated TDLOs and driven (by the ponderomotive force of the latter) slow LDLVs. The coupled system of equations admit envelope soliton solutions. Finally, compressional pulses were studied, both theoretically and experimentally, in Ref. [@Nosenko]; however, these excitations are related to longitudinal grain motion.
In this Brief Communication, we consider the amplitude modulation of transverse dust lattice waves (TDLWs), taking into account self-interaction nonlinearities associated with harmonic generation, involving the intrinsic nonlinearity of the sheath electric potential. Thus, the underlying physics of our nonlinear process is entirely different from those considered in Refs. [@Misawa; @Zafiu; @Ivlev2003; @Nosenko]. By adopting the reductive perturbation technique, we derive a cubic nonlinear Schrödinger equation for the modulated TDLWs. It is shown that the latter may be modulationally unstable depending on the plasma parameters, and that they can propagate in the form of envelope localized excitations due to a balance between nonlinearity and dispersion. Explicit forms of localized excitations are presented.
#### Equation of motion.
Let us consider TDLWs (vertical, off–plane) propagating in the one-dimensional (1d) DP crystal. The dust grain charge $q$ and the mass $M$ are assumed to be constant. DP crystals have been shown to support low-frequency optical-mode-like oscillations in both transverse and longitudinal directions [@psbook; @Melandso; @farokhi; @vladimirov1; @tskhakaya; @Wang]. Focusing on the former and summarizing previous results, let us recall that transverse motion of a charged dust grain (mass $M$, charge $q$, both assumed constant for simplicity) in a DP crystal (lattice constant $r_0$) obeys an equation of the form $$M \, \frac{d^2 \delta z_n}{dt^2} = M \, \omega_0^2 \, (2 \,\delta
z_n - \,\delta z_{n-1} - \,\delta z_{n+1}) + F_e - Mg \, ,
\label{eqmotion}$$ where $\delta z_n = z_n - z_0$ denotes the small displacement of the $n-$th grain around the equilibrium position $z_0$, in the transverse ($z-$) direction, propagating in the longitudinal ($x-$) direction. Assuming that the neighboring dust grains (situated at a distance $x = |x_i - x_j|$) interact via an electrostatic potential $\Phi(x)$, we obtain the DP oscillation ‘eigenfrequency’ $\omega_0^2\, = - \frac{q}{M r_0} \biggl.
\frac{\partial \Phi(x)}{\partial x}\biggr|_{x=r_0}$, e.g. in the case of a Debye-Hückel potential: $ \Phi(x) = ({q}/{x}) \,e^{-
{x/\lambda_D}}$, $$\omega_0^2\, = \frac{q^2}{M r_0^3} \, \biggl(1 +
\frac{r_0}{\lambda_D}\biggr) \,e^{-r_0/\lambda_D} \, ,
\label{Debye-frequency}$$ where $\lambda_D$ denotes the effective DP Debye radius [@psbook]. The force $F_e = q\, E(z)$ is due to the electric field $E(z) = - \partial V(z)/\partial z$; the potential $V(z)$ is obtained by solving Poisson’s equation, taking into account the sheath potential and also (in a more sophisticated description) the wake potential generated by supersonic ion flows towards the electrode [@PS1996]. The potential $V(z)$ thus obtained, actually a [*nonlinear*]{} function of $z$, can be developed around the equilibrium position $z_0$ as $$\begin{aligned}
V(z) \approx V(z_0) + V_{(1)} \, \delta z + \frac{1}{2}\, V_{(2)}
\, (\delta z)^2 + \frac{1}{6} \,V_{(3)} \, (\delta z)^3 \nonumber
\\ + {\cal O}[(\delta z)^4] \,; \label{Vseries}\end{aligned}$$ obviously $V_{(j)} \equiv \biggl. \frac{\partial^j V(z)}{\partial
z^j}\biggr|_{z=z_0}$; the electric force therefore reads $$F_e(z) \approx F_e(z_0) + \gamma_{(1)} \, \delta z + \gamma_{(2)}
\, (\delta z)^2 + \gamma_{(3)} \, (\delta z)^3 + {\cal O}[(\delta
z)^4] \, ,$$ where all coefficients are defined via the derivatives of $V(z)$, i.e. $\gamma_{(j)} \equiv - q\, \frac{1}{j!} V_{(j+1)}$. The zeroth-order term balances gravity at (and actually defines the value of) $z_0$, viz. $ F_e(z_0) - M g = 0 $, while $-
\gamma_{(1)} = q \, V_{(2)}= \gamma \equiv M \, \omega_g^2 $ is the effective width of the potential well; the value of the gap frequency $\omega_g = \omega(k=0)$ may either be evaluated from first theoretical principles [@vladimirov1] or determined experimentally [@Ivlev2000], and is typically of the order of $\omega_g/2 \pi \approx 20\, Hz$ in laboratory experiments. Collisions with neutrals and dust charge dynamics are omitted, at a first step, in this simplified model.
Retaining only the linear contribution and considering phonons of the type, $x_n = A_n \,exp[i \,(k n r_0 - \omega t)] + c.c.$, we obtain an optical-mode-like dispersion relation $$\omega^2\, = \omega_g^2\, - 4 \omega_0^2\, \sin^2 \biggl( \frac{k r_0}{2}
\biggr)
\, , \label{dispersion-discrete}$$ where $\omega$ and $k = 2 \pi/\lambda$ denote, respectively, the wave frequency and the wavenumber. We will not go into further details concerning the linear regime, since it is sufficiently covered in Refs. [@Melandso; @farokhi; @vladimirov1; @tskhakaya; @Wang]. Let us now see what happens if the [*nonlinear*]{} terms are retained.
#### Derivation of a Nonlinear Schrödinger Equation.
For analytical tractability, we shall limit ourselves to a quasi–continuum limit, by considering an amplitude which varies over a scale $L$ which is significantly larger than the inter-grain distance $r_0$ (i.e. $L/r_0 \ll 1$). Equation (\[eqmotion\]) takes the form $$\frac{d^2 u_{n}}{dt^2} + \, c_0^2 \,(2 \,\delta u_{n} - \,u_{n-1}
- \,u_{n+1}) + \omega_g^2 \, u_{n} + \alpha \, u_{n}^2 + \beta \,
u_{n}^3 = 0 \, , \label{NL-eq}$$ where we set $\delta z \equiv u(x, t)$ for simplicity; $c_0 =
\omega_0 \, r_0$ is a characteristic propagation speed related to the interaction \[e.g. Debye–type, see (\[Debye-frequency\])\] potential; the nonlinearity coefficients $\alpha$, $\beta$ are related to the anharmonicity of the electric potential, viz. $$\alpha = -
\frac{\gamma_{(2)}}{M} \equiv \frac{q V_{(3)}}{2 M}\, , \qquad
\beta = - \frac{\gamma_{(3)}}{M} \equiv \frac{q V_{(4)}}{6 M} \, .$$ Remember that inter-grain interactions are [*repulsive*]{}, hence the difference in structure from the nonlinear Klein-Gordon equation used to describe one-dimensional oscillator chains. ‘Phonons’ in this chain are stable only in the presence of the electric field (i.e. for $\gamma \ne 0 $).
We now proceed by considering small-amplitude oscillations of the form $$u =
\epsilon \, u_{1} + \epsilon^2 \, {u_{2}}^2 + \,...$$ at each lattice site. Introducing multiple scales in time and space, i.e. $X_n = \epsilon^n \, x$, $T_n = \epsilon^n \, t$ (n = 0, 1, 2, ...), we develop the derivatives in Eq. (\[NL-eq\]) in powers of the smallness parameter $\epsilon$ and then collect terms arising in successive orders. The equation thus obtained in each order can be solved and substituted to the subsequent order, and so forth. This reductive perturbation technique is a standard procedure for the study of the nonlinear wave propagation (e.g. in hydrodynamics, in nonlinear optics, etc.) often used in the description of localized pulse propagation, prediction of instabilities, etc. [@Remo; @Scott]. This procedure leads to a solution of the form $$\begin{aligned}
u(x, t) = \epsilon \,\biggl[A \,e^{i \,(k x - \omega t)} +
c.c.\biggr] \qquad \qquad \qquad \qquad \nonumber \\ + \,
\epsilon^2 \,\alpha\, \biggl[ - \frac{2 \, |A|^2}{\omega_g^2} \, +
\frac{A^2}{3 \omega_g^2} \,e^{2 i \,(k x - \omega t)} + c.c.
\biggr] \, + {\cal O}(\epsilon^3) \, ,\end{aligned}$$ where *c.c.* denotes the complex conjugate; recall that $\omega$ obeys the dispersion relation (\[dispersion-discrete\]).
The slowly-varying amplitude $A = A(X_1 - v_g \, T_1)$ moves at the (negative) group velocity $v_g = d\omega/dk = - \omega_0^2 r_0
\,\sin (k r_0/\omega)$ in the direction opposite to the phase velocity; this [*[backward]{}*]{} wave has been observed experimentally: see the discussion in Ref. [@Misawa]. The amplitude $A$ obeys a [*Nonlinear Schrödinger Equation*]{} (NLSE) of the form $$i\, \frac{\partial A}{\partial T} + P\, \frac{\partial^2
A}{\partial X^2} + Q \, |A|^2\,A = 0 \, , \label{NLSE}$$ where the ‘slow’ variables $\{ X, T \}$ are $\{ X_1 - v_g \, T_1,
T_2 \}$, respectively. The [*dispersion coefficient*]{} $P$, which is related to the curvature of the phonon dispersion relation (\[dispersion-discrete\]) as $P = \,({d^2 \omega}/{d k^2})/2$, reads $$P = - \frac{c_0^2 \omega_0^2}{4 \omega^3}\biggl[ 2 \,
\biggl(\frac{\omega_g^2}{\omega_0^2} - 2 \biggr) \, \cos (k r_0)\, +
\cos (2 k r_0) \, + 3 \biggr] \, , \label{Pcoeff}$$ and the [*nonlinearity coefficient*]{} $$Q = \frac{1}{2 \omega} \biggl( \frac{10 \alpha^2}{3 \omega_g^2} -
3\, \beta \biggl) \, = \,
\frac{1}{2 M \omega} \biggl[ - \frac{10 \gamma_{(2)}^2}{3 \gamma_{(1)}} +
3\, \gamma_{(3)} \biggl]
\label{Qcoeff}$$ is related to the electric field nonlinearity considered above. Notice that the sign of $P$ depends on the ratio $\omega_g/\omega_0 \equiv \lambda \, > 0$, which appears naturally as an order parameter; see for instance in (\[dispersion-discrete\]) that $\lambda \ge 2$ is a stability criterion (necessary for $\omega$ to be real in the whole range of the first Brillouin zone). For long wavelength values, $P \approx
- ({c_0^2 \omega_g^2})/({2 \omega^3}) \, < 0$, given the parabolic form of $\omega(k)$ close to $k=0$ (continuum case). In the general (discrete) case, we see that the coefficient $P$ becomes positive at some critical value of $k$, say $k_{cr}$, inside the first Brillouin zone. Some simple algebra shows that the [*zero-dispersion point*]{} $k_{cr}$ satisfies the relation: $\cos (k_{cr} r_0) = (2 - \lambda^2 + \lambda \sqrt{\lambda^2 -
4})/2$, for $\lambda > 2$ ($0 < k_{cr} r_0 < \pi$); otherwise, for $\lambda < 2$, $P$ remains negative everywhere.
#### Modulational instability.
In a generic manner, a modulated wave whose amplitude obeys the NLS equation (\[NLSE\]), is stable (unstable) to perturbations if the product $ P Q $ is negative (positive). To see this, one may first check that the NLSE accepts the monochromatic solution (Stokes’ wave) $ A(X, T) = A_0\, e^{i Q
|A_0|^2 T} \, + \, c.c. $ The standard (linear) stability analysis then shows that a linear modulation with the frequency $\Omega$ and the wavenumber $\kappa$ obeys the dispersion relation $$\Omega^2(\kappa) = P^2\, \kappa^2\, \biggl( \kappa^2\, - 2
\frac{Q}{P}\,|A_0|^2 \biggr) \, ,
\label{pert-disp}$$ which exhibits a purely growing mode for $\kappa \geq \kappa_{cr} =
({Q}/{P})^{1/2}\,|A_0|$. The growth rate attains a maximum value of $ \gamma_{max} =
{Q}\,|A_0|^2 $. This mechanism is known as the [*Benjamin-Feir instability*]{} [@Remo]. For $P Q < 0$, the wave is modulationally stable, as evident from (\[pert-disp\]).
One now needs to deduce the sign of Q, given by (\[Qcoeff\]), in order to determine the stability profile of the TDL oscillations. In fact, given the above definitions of the parameters $\omega$, $\alpha$, $\beta$, one easily finds that $Q$ is related to the (derivatives of the) electric potential $V(z)$ via $$Q = \frac{q}{4 M \omega} \biggl[ \frac{5 V_{(3)}^2}{3 V_{(2)}} -
V_{(4)} \biggl] \, = \, \frac{\omega_g^2}{4 \omega} \biggl[
\frac{5 V_{(3)}^2}{3 V_{(2)}^2} - \frac{V_{(4)}}{V_{(2)}}
\biggl] \, .
\label{Qcoeff-reduced}$$ The exact form of the potential $V(z)$ may be obtained from *ab initio* calculations or by experimental data fitting. For instance, in Ref. [@Ivlev2000], the dust grain potential energy ${\mathcal{U}}(z) = q\, V(z)$ was reconstructed from experimental data as $$\begin{aligned}
{\mathcal{U}}(z) \simeq M \omega_0^2\, \biggl[ - \, 0.9 \,\delta z
\,+\, \frac{1}{2} \,(\delta z)^2 \,- \,\frac{1}{3} \, 0.5
\,(\delta z)^3 \, \nonumber \\ +\,\frac{1}{4} \, 0.07 \,(\delta
z)^4 \,+\,... \, \biggr] \, ,\end{aligned}$$ which is Eq. (9) in Ref. [@Ivlev2000]; upon simple inspection from (\[Vseries\]), we obtain $ V_{(3)}/V_{(2)} = -1\, , \quad
V_{(4)}/V_{(2)} = 0.42 $, so the value of $Q$ is positive, as may be checked from (\[Qcoeff-reduced\]). Therefore, the transverse oscillation considered in Ref. [@Ivlev2000] would propagate as a *stable* wave, for large wavelength values $\lambda$. However, for shorter wavelengths, the coefficient $P = \,
\omega''(k)/2$ - as defined in (\[dispersion-discrete\]) - may become positive (and so will the product $P Q$, in this case); the TDL wave may thus be potentially unstable. These results may *a priori* be checked experimentally.
#### Localized excitations.
A final comment concerns the possibility of the existence of localized excitations related to transverse dust-lattice waves. It is known that the NLSE (\[NLSE\]) supports pulse-shaped localized solutions (envelope solitons) of the [*bright*]{} ($P Q > 0$) or [*dark/grey*]{} ($P Q < 0$) type [@Fedele; @Fedele2]. The former (*continuum breathers*) are $$\begin{aligned}
A = (2 D/P Q)^{1/2} \, sech \bigl[ (2 D/P Q)^{1/2}\, (X - v_e \,
T)\bigr]\, \nonumber \\
\times \, \exp\bigl[ i \, v_e \, (X - v_c \, T)/ 2 P\bigr] \, + \,
c.c. \, ,\label{breather}\end{aligned}$$ where $v_e$ ($v_c$) is the envelope (carrier) velocity and $D =
(v_e^2 - 2 \,v_e \,v_c)/(4 P^2)$; they may occur and propagate in the lattice if a sufficiently short wavelength is chosen, so that the product $P Q$ is positive. We note that the pulse width $L$ and the amplitude $\rho$ satisfy $L \rho \sim (|P/Q|)^{1/2} = const.$
For $P Q < 0$, we have the [*grey*]{} envelope soliton [@Fedele] $$A \, = \, \rho_1 \, \{ 1 - a^2 \, sech^2\{[X - (v_e \, + 2 \alpha)
T]/L_1\}\}^{1/2} \, \exp ( i\, \sigma ) \, , \label{greysoliton}$$ where $\sigma = \sigma(X, T)$ is a nonlinear phase correction to be determined. This excitation represents a localized region of negative wave density (a *void*), with finite amplitude $(1 -
a) \rho_1$ at $X = 0$; $0 \le a \le 1$). Again, the pulse width $L_1 = (|P/Q|)^{1/2}/(a \rho_1)$ is inversely proportional to the amplitude $\rho_1$. Notice the (dimensionless) parameter $a$, which regulates the depth of the excitation. For $a = 1$, one obtains a [*dark*]{} envelope soliton, which describes a localized density *hole*, characterized by a vanishing amplitude at $\zeta = 0$. The latter excitations (of grey/dark type), yet apparently privileged in the continuum limit (where $P Q < 0$), are rather physically irrelevant in our (infinite chain) model, since they correspond to an infinite energy stored in the lattice. Nevertheless, their existence locally in a finite–sized chain may be considered (and possibly confirmed) either numerically or experimentally.
*In conclusion*, we have shown that the modulational instability is, in principle, possible for transverse DP lattice waves. Long wavelength modes seem to ensure wave stability, while shorter wavelength modes may be modulationally unstable. The existence of localized excitations and the occurrence of the modulational instability rely on the same criterion, which needs to be thoroughly examined for a given exact form of the sheath electric potential. These results may be investigated and will hopefully be confirmed by appropriate experiments. Of course, for a more complete description, one has to take into account certain factors that are ignored in this simple model, viz. collisions with neutral particles, dust charge variations and, eventually, transverse to longitudinal mode coupling. The effect of inter-layer coupling may also be investigated. Work in this direction is in progress and the results will be reported soon.
This work was supported by the European Commission (Brussels) through the Human Potential Research and Training Network via the project entitled: “Complex Plasmas: The Science of Laboratory Colloidal Plasmas and Mesospheric Charged Aerosols” (Contract No. HPRN-CT-2000-00140).
[10]{} P. K. Shukla and A. A. Mamun, *Introduction to Dusty Plasma Physics* (Institute of Physics Publishing Ltd., Bristol, 2002).
J. H. Chu and Lin I, Phys. Rev. Lett. **72**, 4009 (1994).
H. Thomas, G. E. Morfill, V. Demmel and J. Goree, Phys. Rev. Lett. **73**, 652 (1994).
A. Melzer, T. Trottenberg and A. Piel, Phys. Lett. A **191**, 301 (1994).
Y. Hayashi and K. Tachibana, Jpn. J. Appl. Phys. **33**, L84 (1994).
F. Melandsø, Phys. Plasmas **3**, 3890 (1996).
B. Farokhi, P. K. Shukla, N. L. Tsintsadze and D. D. Tskhakaya, Phys. Lett. A **264**, 318 (1999); *idem*, Phys. Plasmas [**7**]{}, 814 (2000).
S. V. Vladimirov, P. V. Shevchenko, and N. F. Cramer, Phys. Rev. E **56**, R74 (1997).
D. Tskhakaya and P. K. Shukla, Phys. Lett. A **286**, 277 (2001).
X. Wang, A. Bhattacharjee and S. Hu, Phys. Rev. Lett. **86**, 2569 (2001).
G. E. Morfill, H. M. Thomas and M. Zuzic, in *Advances in Dusty Plasma Physics*, Eds. P. K. Shukla, D. A. Mendis and T. Desai (Singapore, World Scientific) p. 99; G. E. Morfill, H. M. Thomas, U. Konopka, H. Rothermel, M. Zuzic, A. Ivlev and J. Goree, Phys. Rev. Lett. **83**, 1598 (1999).
A. Homann, A. Melzer, S. Peters and A. Piel, Phys. Rev. E **56**, 7138 (1997).
A. V. Ivlev, R. Sütterlin, V. Steinberg, M. Zuzic and G. Morfill, Phys. Rev. Lett. **85**, 4060 (2000).
T. Misawa, N. Ohno, K. Asano, M. Sawai, S. Takamura and P. K. Kaw, Phys. Rev. Lett. [**86**]{}, 1219 (2001).
A. Piel, A. Homann, M Klindworth, A Melzer, C Zafiu, V Nosenko and J Goree, J. Phys. B [**36**]{}, 533 (2003).
A. Melzer, A. Homann and A. Piel, Phys. Rev. E **53**, 2757 (1996).
P. K. Shukla and N. N. Rao, Phys. Plasmas [**3**]{}, 1770 (1996).
G. E. Morfill, A. V. Ivlev and J. R. Jokipii, Phys. Rev. Lett. **83**, 971 (1999); A. V. Ivlev, U. Konopka and G. Morfill, Phys. Rev. E **62**, 2739 (2000).
S. V. Vladimirov, P. V. Shevchenko and N. F. Cramer, Phys. Plasmas **5**, 4 (1998).
A. V. Ivlev and G. Morfill, Phys. Rev. E **63**, 016409 (2000); S. K. Zhdanov, D. Samsonov and G. E. Morfill, Phys. Rev. E **66**, 026411 (2002).
Y. S. Kivshar and M. Peyrard, Phys. Rev. A [**46**]{}, 3198 (1992).
M. Remoissenet, *Waves Called Solitons* (Springer-Verlag, Berlin, 1994).
A. C. Scott and D. W. McLaughlin, Proc. IEEE **61** (10), 1443 (1973).
T. Taniuti and N. Yajima, J. Math. Phys. [**10**]{}, 1369 (1969); N. Asano, T. Taniuti and N. Yajima, *ibid.* [**10**]{}, 2020 (1969).
A. Hasegawa, *Plasma Instabilities and Nonlinear Effects* (Springer-Verlag, Berlin, 1975).
F. Verheest, *Waves in Dusty Space Plasmas* (Kluwer Academic Publishers, Dordrecht, 2001).
M. R. Amin, G. E. Morfill and P. K. Shukla, Phys. Rev. E [**58**]{}, 6517 (1998).
I. Kourakis and P. K. Shukla, Phys. Plasmas **10** (9), 3459 (2003).
A. M. Ignatov, Plasma Physics Reports [**29**]{} (4), 296 (2003).
I. Kourakis and P. K. Shukla, Phys. Lett. A **317**, 156 (2003).
E.B. Tomme, D. A. Law, B. M. Annaratone and J. E. Allen, Phys. Rev. Lett. **85**, 2518 (2000).
D. Tskhakaya and P. K. Shukla, Phys. Lett. A **279**, 243 (2001).
C. Zafiu, A. Melzer and A. Piel, Phys. Rev. E **63**, 066403 (2001).
P. K. Shukla, Phys. Plasmas [**10**]{}, 1619 (2003); P. K. Shukla and A. A. Mamun, New J. Phys. [**5**]{}, 17 (2003).
M. R. Amin, G. E. Morfill and P. K. Shukla, Phys. Plasmas [**5**]{}, 2578 (1998); *idem*, Phys. Scripta [**58**]{}, 628 (1998).
A. V. Ivlev, S. K. Zhdanov and G. E. Morfill, Phys. Rev. E **68**, 066402 (2003).
V. Nosenko, S. Nunomura and J. Goree, Phys. Rev. Lett. **88**, 215002 (2002); K. Avinash, P. Zhu, V. Nosenko and J. Goree, Phys. Rev. E **68**, 046402 (2003).
R. Fedele and H. Schamel, [*Eur. Phys. J. B*]{} [**27**]{} 313 (2002); R. Fedele, H. Schamel and P. K. Shukla, [*Phys. Scripta*]{} T [**98**]{} 18 (2002).
R. Fedele, [*Phys. Scripta*]{} [**65**]{} 502 (2002).
[^1]: On leave from: U.L.B. - Université Libre de Bruxelles, Physique Statistique et Plasmas C. P. 231, Boulevard du Triomphe, B-1050 Brussels, Belgium; also: Faculté des Sciences Apliquées - C.P. 165/81 Physique Générale, Avenue F. D. Roosevelt 49, B-1050 Brussels, Belgium;\
Electronic address: `ioannis@tp4.rub.de`
[^2]: Electronic address: `ps@tp4.rub.de`
[^3]: Preprint, submitted to *Physics of Plasmas.*
|
---
abstract: |
In this paper we study the strong convergence for the Euler-Maruyama approximation of a class of stochastic differential equations whose both drift and diffusion coefficients are possibly discontinuous.
**2010 Mathematics Subject Classification**: 60H35; 41A25; 60C30;\
\
**Keywords**: Euler-Maruyama approximation $\cdot$ Strong rate of convergence $\cdot$ Stochastic differential equation $\cdot$ Discontinuous coefficients
author:
- 'Hoang-Long Ngo[^1] $\quad $ and $\quad$ Dai Taguchi[^2]'
title: 'Strong convergence for the Euler-Maruyama approximation of stochastic differential equations with discontinuous coefficients'
---
Introduction {#Sec_1}
============
Let us consider the one-dimensional stochastic differential equation (SDE) $$\begin{aligned}
\label{SDE_1}
X_t=
x_0
+\int_0^t b(X_s)ds
+\int_0^t \sigma(X_s)dW_s,
~x_0 \in {\mathbb{R}},
~t \in [0,T],\end{aligned}$$ where $W:=(W_t)_{0\leq t \leq T}$ is a standard Brownian motion defined on a probability space $(\Omega, \mathcal{F},{\mathbb{P}})$ with a filtration $(\mathcal{F}_t)_{0\leq t \leq T}$ satisfying the usual conditions. Since the solution of (\[SDE\_1\]) is rarely analytically tractable, one often approximates $X=(X_t)_{0 \leq t \leq T}$ by using the Euler-Maruyama (EM) scheme given by $$\begin{aligned}
X_t^{(n)}
&= x_0 +\int_0^tb\left(X_{\eta _n(s)}^{(n)}\right)ds +\int_0^t \sigma\left(X_{\eta _n(s)}^{(n)}\right) dW_s,~t \in [0,T],\end{aligned}$$ where $\eta _n(s) = kT/n=:t_k^{(n)}$ if $ s \in \left[kT/n, (k+1)T/n \right)$.
It is well-known that if $b$ and $\sigma$ are Lipschitz continuous, the EM approximation for converges at the strong rate of order $1/2$ (see [@KP]). On the other hand, when $b$ and $\sigma$ are not Lipschitz continuous, the strong rate is less known and it has been a subject of extensive study. In the recent articles [@JMY] and [@HHJ], it has been shown that for every arbitrarily slow convergence speed there exist SDEs with infinitely often differentiable and globally bounded coefficients such that neither the EM approximation nor any approximation method based on finitely many observations of the driving Brownian motion can converge in absolute mean to the solution faster than the given speed of convergence. The approximation for SDEs with possibly discontinuous drift coefficients was first studied in [@G98]. It is shown that if the drift satisfies the monotonicity condition and the diffusion coefficient is Lipschitz continuous, then the EM scheme converges at the rate of $1/4$ in pathwise senses. In [@HK], the strong convergence of EM scheme is shown for SDEs with discontinuous monotone drift coefficients. If $\sigma$ is uniformly elliptic and $(\alpha + 1/2)$-Höder continuous, and $b$ is of locally bounded variation, it has been shown that the strong rate of the EM in $L^1$-norm is $n^{-\alpha}$ for $\alpha \in (0,1/2]$ and $(\log n)^{-1}$ for $\alpha=0$ (see [@NT_MCOM; @NT_2015_2]). The strong rate of convergence for SDEs whose drift coefficient $b$ is Hölder continuous is studied in [@Gyongy; @MeTa; @NT_2015_2]. The above mentioned papers contain just a few selected results and a number of further and partially significantly improved approximation results for SDEs with irregular coefficients are available in the literature; see, e.g., [@AKU; @CS; @HaTsu; @HJK; @KLY; @LeSz; @MT; @NT_2015_1; @Y] and the references there in.
In this paper we are interested in strong approximation of SDEs with discontinuous diffusion coefficients. These SDEs appears in many applied domains such as stochastic control and quantitative finance (see [@CE; @AI]). For such SDEs, the existence and uniqueness of solution was studied in [@Nakao; @LeGall; @CE]; the weak convergence of EM approximation was shown in [@Y]. To the best of our knowledge, the strong convergence of the EM approximation of SDEs with discontinuous diffusion coefficient has not been considered before in the literature. It is worth noting that the key ingredients to establish the strong rate of convergence of EM approximation for SDEs with discontinuous drift are either the Krylov estimate (see [@KLY; @Gyongy]) or the Gaussian bound estimate for the density of the numerical solution ([@Lemaire; @NT_MCOM; @NT_2015_2]). However, these estimates seem no longer available for SDEs with discontinuous diffusion coefficients. Therefore in this paper we develop another method, which is based on an argument with local time, to overcome this obstacle.
The remainder of the paper is structured as follows. In the next section we introduce some notations and assumptions for our framework together with the main results. All proofs are deferred to Section 3.
Main results
============
Notations
---------
Throughout this paper the following notations are used. For any continuous semimartingale $Y$, we denote $L^x_t(Y)$ the symmetric local time of $Y$ up to time $t$ at the level $x \in {\mathbb{R}}$ (see [@LeGall]). For bounded measurable function $f$ on ${\mathbb{R}}$, we define $\|f\|_{\infty}:=\sup_{x \in {\mathbb{R}}} |f(x)|$. We denote by $L^1({\mathbb{R}})$ the space of all integrable functions with respect to Lebesgue measure on ${\mathbb{R}}$ with semi-norm $\|f\|_{L^1({\mathbb{R}})}:=\int_{{\mathbb{R}}}|f(x)|dx$. For each $\beta \in (0,1]$ and $\kappa >0$, we denote by $H^{\beta, \kappa}$ the set of all functions $f:{\mathbb{R}}\to {\mathbb{R}}$ such that there exists a measurable subset $S(f)$ of ${\mathbb{R}}$ satisfying
- $\|f\|_{\beta} := \|f\|_\infty+ \sup_{x<y; [x,y]\cap S(f) = \emptyset} \dfrac{|f(x)-f(y)|}{|x-y|^{\beta}} < \infty$; and
- $C_{\beta,\kappa}:= \sup_{K\geq 1}\sup_{\varepsilon>0} \dfrac{\lambda(S(f)^\varepsilon \cap [-K,K])}{K \varepsilon^\kappa} < +\infty$ where $\lambda$ denotes the Lebesgue measure on ${\mathbb{R}}$ and $S(f)^\varepsilon$ is the $\varepsilon$-neighbourhood of $S(f)$, i.e., $S(f)^\varepsilon = \{y \in {\mathbb{R}}: \text{ there exists } x \in S(f) \text{ such that } |x-y|\leq \varepsilon\}.$
Here are some remarks on the class $H^{\beta, \kappa}$.
1. $H^{\beta, \kappa}$ is a vector space on $\mathbb{R}$, i.e., if $a, b \in {\mathbb{R}}$ and $f, g \in H^{\beta, \kappa}$ then $af + bg \in H^{\beta, \kappa}$.
2. A bounded function $f$ is called piecewise $\beta$-Hölder if there exist a positive constant $L$ and a sequence $-\infty = s_0 < s_1 < s_2 < \ldots < s_m < s_{m+1}= \infty$ such that $|f(u) - f(v)| \leq L|u-v|^\beta$ for any $u,v$ satisfying $s_k < u < v < s_{k+1}$. It is easy to verify that such function $f \in H^{\beta,1}, \ S(f) = \{s_1, \ldots, s_m\}$ and $C_{\beta, 1} \leq 2m$.
3. The following $\zeta$ is a non-trivial example of function of $H^{\beta, \kappa}$ with $\kappa < 1$. For each ${\hat{\beta}}, \kappa \in (0,1)$, we denote $$\label{exp:zeta} \zeta(x) =
\begin{cases} \frac{x-1}{2x-1} & \text{ if } x \leq 0,\\
1+\frac{\log 2}{\log(n+1)}x^{{\hat{\beta}}} & \text{ if } (n+1)^{-1/(1-\kappa)} \leq x < n^{-1/(1-\kappa)} \text{ and } \ n \in {\mathbb{N}}, \\
\frac{3x+1}{x+1} &\text{ if } x \geq 1.\end{cases}$$ It can be shown that $\zeta$ is a strictly increasing function with an infinite number of discontinuous points which are cumulative at $0$, $\frac{1}{2} < \zeta < 3$, and $\zeta \in H^{\beta,\kappa}$ with $\beta = \frac{1+{\hat{\beta}}-\kappa}{2-\kappa}$, $S(\zeta) = \{ n^{-1/(1-\kappa)}, n = 1, 2,\ldots \}$ and $C_{\beta,\kappa} \leq 3$.
Main results
------------
We need the following assumptions on the diffusion coefficient $\sigma$.
\[Ass\_1\]
- There exists a bounded and strictly increasing function $f_{\sigma}$ such that for any $x,y \in {\mathbb{R}}$, $$\begin{aligned}
|\sigma(x)-\sigma(y)|^2 \leq |f_{\sigma}(x)-f_{\sigma}(y)|.
\end{aligned}$$
- $\sigma$ is bounded and uniformly positive, i.e. there exist positive constants $\overline{\sigma}$ and $\underline{\sigma}$ such that for any $x \in {\mathbb{R}}$, $$\begin{aligned}
\underline{\sigma} \leq \sigma(x) \leq \overline{\sigma}.
\end{aligned}$$
Le Gall [@LeGall] has shown that if $b$ is bounded measurable, and $\sigma$ satisfies Assumption \[Ass\_1\], then there exists a unique strong solution to SDE (see also [@Nakao]). We now give some remarks on the Assumption \[Ass\_1\].
1. The function $\sigma(x) = 1 + {{\bf 1}}_{x \geq 0}$ satisfies Assumption \[Ass\_1\] and belongs to $H^{1,1}$.
2. The function $\zeta$ defined in also satisfies Assumption \[Ass\_1\].
3. If $a, b >0$ and $\sigma_1, \sigma_2$ satisfies Assumption \[Ass\_1\], then $a\sigma_1 + b\sigma_2$ also satisfies Assumption \[Ass\_1\].
4. Let $f_1, f_2$ be two strictly increasing, piecewise $1$-Hölder functions. Let $\rho$ be a $1/2$-Hölder continuous function satisfying $0 < \inf_{x \in {\mathbb{R}}} \rho(x) \leq \sup_{x \in {\mathbb{R}}} \rho(x) < \infty$. Then $\sigma := \rho\circ (f_1-f_2)$ is piecewise $1/2$-Hölder and it satisfies Assumption \[Ass\_1\] with $f_\sigma = C(f_1 + f_2)$ for some positive constant $C$.
We are now in the position to state the main result of this paper.
\[Main\_1\] Let Assumption \[Ass\_1\] hold, and $b, \sigma \in H^{\beta, \kappa}$ for some $\beta \in (0,1]$ and $\kappa >0$.
1. There exists a constant $C$ such that for all $n \geq 3$, $$\label{logn2}
\sup_{0\leq t \leq T} {\mathbb{E}}[|X_t - X^{(n)}_t|] \leq \frac{Ce^{C\sqrt{\log \log n}}}{\log n}.$$
2. Moreover, if $ b \in L^1({\mathbb{R}})$, then there exists a constant $C$ such that for all $n \geq 3$, $$\label{logn}
\sup_{0\leq t \leq T} {\mathbb{E}}[|X_t - X^{(n)}_t|] \leq \frac{C}{\log n}.$$
The estimates and were obtained in [@Gyongy; @NT_MCOM; @NT_2015_2] under a stronger assumption that $\sigma$ is $1/2$-Hölder continuous on $\mathbb{R}$.
Proof of main results
=====================
Some auxiliary estimates
------------------------
In this section, we derive a key estimation (Lemma \[key\_sigma\_0\]) for proving the main theorem. We first introduce the following standard estimation (see Remark 1.2 in [@Gyongy]).
\[Lem\_1\] Suppose that $b$ and $\sigma$ are bounded, measurable. Then for any $q>0$, there exists $C_q \equiv C(q,\|b\|_{\infty}, \|\sigma\|_{\infty}, T) $ such that for all $n \in {\mathbb{N}}$, $$\begin{aligned}
\sup_{t \in [0,T]} {\mathbb{E}}[|X_t^{(n)}-X_{\eta_n(t)}^{(n)}|^q]\leq \frac{C_q}{n^{q/2}}.
\end{aligned}$$
The next estimation is a uniform $L^2$-bounded of the local time of solution of SDE and its EM approximation.
\[local\_time\] Suppose that $b$ is bounded, measurable and $\sigma$ is measurable and satisfies Assumption \[Ass\_1\]-(ii). For each $\theta \in [0,1]$, define $$\begin{aligned}
&V_t^{(n)}(\theta):=(1-\theta)X_t+\theta X_t^{(n)}.\\
&=x_0
+\int_{0}^{t} \left\{ (1-\theta)b(X_s) + \theta b(X_{\eta_n(s)}^{(n)}) \right\} ds
+\int_{0}^{t} \left\{ (1-\theta)\sigma(X_s) + \theta \sigma(X_{\eta_n(s)}^{(n)}) \right\} dW_s.
\end{aligned}$$ Then it holds that $$\begin{aligned}
\label{esti_local_time_0}
\sup_{\theta \in [0,1], x \in {\mathbb{R}}}
{\mathbb{E}}[|L_T^x(V^{(n)}(\theta))|^2]
&\leq 12\|b\|_{\infty}^2T^2+ 6 \overline{\sigma}^2 T.
\end{aligned}$$
By using the symmetric Itô-Tanaka formula, we have $$\begin{aligned}
L_T^x(V^{(n)}(\theta))
&=|V_T^{(n)}(\theta)-x|-|x_0-x|-\int_0^T \left( {{\bf 1}}(V_s^{(n)}(\theta)>x)-{{\bf 1}}(V_s^{(n)}(\theta)<x) \right) dV_s^{(n)}(\theta)\\
&\leq |V_T^{(n)}(\theta)-x_0|+\left| \int_0^T \left( {{\bf 1}}(V_s^{(n)}(\theta)>x)-{{\bf 1}}(V_s^{(n)}(\theta)<x) \right) dV_s^{(n)}(\theta) \right|\\
&\leq 2\int_{0}^{T} \left| (1-\theta)b(X_s) + \theta b(X_{\eta_n(s)}^{(n)}) \right| ds
+\left| \int_{0}^{T} \left\{ (1-\theta)\sigma(X_s) + \theta \sigma(X_{\eta_n(s)}^{(n)}) \right\} dW_s\right|\\
&+\left| \int_{0}^{T} \left( {{\bf 1}}(V_s^{(n)}(\theta)>x)-{{\bf 1}}(V_s^{(n)}(\theta)<x) \right) \left\{ (1-\theta)\sigma(X_s) + \theta \sigma(X_{\eta_n(s)}^{(n)}) \right\} dW_s\right|.
\end{aligned}$$ Since $b$ and $\sigma$ are bounded, it follows from inequality $(a+b+c)^2\leq 3(a^2+b^2+c^2)$ and the $L^2$-isometry that, $$\begin{aligned}
\sup_{\theta \in [0,1], x \in {\mathbb{R}}}
{\mathbb{E}}[|L_T^x(V^{(n)}(\theta))|^2] \notag
&\leq
12\|b\|_{\infty}^2T^2
+ 6 \sup_{\theta \in [0,1], x \in {\mathbb{R}}}\int_0^T {\mathbb{E}}\Big[ \big| (1 - \theta) \sigma(X_s) + \theta \sigma(X^{(n)}_{\eta(s)})\big|^2 \Big] ds\\
&\leq 12\|b\|_{\infty}^2T^2 + 6 \overline{\sigma}^2T.
\end{aligned}$$ This concludes the statement.
The following lemma, which is similar to Lemma 2.2 in [@Y], plays a crucial role in our argument.
\[tight\_1\] Assume that $b$ and $\sigma$ are bounded measurable. For any $\varepsilon, \chi>0$ such that $\delta:=\frac{\chi \varepsilon^4}{8(T\|b\|_{\infty}^4+2^7\overline{\sigma}^4)}\leq T$, it holds that for any $t \geq 0$ and $n \in {\mathbb{N}}$, $ {\mathbb{P}}(\sup_{t \leq r \leq t+\delta}|X_r^{(n)}-X_t^{(n)}| \geq \varepsilon) \leq \delta \chi.$
Let $t\in [0,T]$ be fixed. We define $Z_s^{(n)}:=X_{t+s}^{(n)}-X_{t}^{(n)}$. Then using Burkholder-Davis-Gundy’s inequality, it holds that for any $\delta \in [0,T]$, $$\begin{aligned}
{\mathbb{E}}\left[\sup_{0\leq s \leq \delta}|Z_s^{(n)}|^4\right]
&\leq 8 {\mathbb{E}}\left[\sup_{0 \leq s \leq \delta} \left|\int_{t}^{t+s} b(X_{\eta_n(r)}^{(n)})dr\right|^4\right]
+8 {\mathbb{E}}\left[\sup_{0 \leq s \leq \delta} \left|\int_{t}^{t+s} \sigma(X_{\eta_n(r)}^{(n)})dW_r\right|^4\right]\\
&\leq 8 \delta^3 {\mathbb{E}}\left[ \int_{t}^{t+\delta} \left| b(X_{\eta_n(r)}^{(n)})\right|^4 dr\right]
+2^{10} \delta {\mathbb{E}}\left[ \int_{t}^{t+\delta} \left|\sigma(X_{\eta_n(r)}^{(n)})\right|^4dr\right]\\
&\leq 8 \|b\|^4_{\infty} \delta^4
+2^{10} \overline{\sigma}^4 \delta^2
\leq 8\left( \|b\|^4_{\infty}T^2+ 2^7 \overline{\sigma}^4 \right) \delta^2.
\end{aligned}$$ Hence, for any $\varepsilon, \chi>0$ such that $\delta:=\frac{\chi \varepsilon^4}{8(T^2\|b\|_{\infty}^4+2^7\overline{\sigma}^4)} \leq T$, from Markov’s inequality, we have $$\begin{aligned}
{\mathbb{P}}\left(\sup_{t \leq s \leq t+\delta}|X_s^{(n)}-X_t^{(n)}| \geq \varepsilon \right)
&\leq \frac{1}{\varepsilon^4} {\mathbb{E}}\left[\sup_{t \leq s \leq t+\delta}|X_s^{(n)}-X_t^{(n)}|^4\right]
= \frac{1}{\varepsilon^4} {\mathbb{E}}\left[\sup_{0 \leq s \leq \delta} |Z_s^{(n)}|^4 \right]\\
&\leq \frac{8\left( \|b\|^4_{\infty}T^2+ 2^7 \overline{\sigma}^4 \right) \delta^2}{\varepsilon^4}
=\delta \chi,
\end{aligned}$$ which concludes the statement.
Lemma \[tight\_1\] directly implies the following result.
\[tight\_3\] Assume that $b$ and $\sigma$ are bounded measurable. Let $(\gamma_n)_{n\in{\mathbb{N}}}$ be a decreasing sequence such that $\gamma_n \in (0,1]$ and $\gamma_n \downarrow 0 $ and $\gamma_n n^2 \to \infty$ as $n \to \infty$. Denote $ \varepsilon_n
:=\frac{\widetilde{c}}{\gamma_n^{1/4} n^{1/2}},~
\widetilde{c}
:=2^{3/4}T^{1/2}\{T^2\|b\|_{\infty}^4+ 2^7 \overline{\sigma}^4\}^{1/4},
\chi_n:=\frac{\gamma_n n}{T},
\delta_n:=\frac{\chi_n \varepsilon_n^4}{8(T^2\|b\|_{\infty}^4+ 2^7 \overline{\sigma}^4)}=\frac{T}{n}.$ For each $k=0,\ldots,n-1$, we define $$\begin{aligned}
\Omega_{k,n,\varepsilon_n}
:=\left\{ \omega \in \Omega \bigg| \sup_{t_k^{(n)} \leq s \leq t_{k+1}^{(n)}}|X_s^{(n)}(\omega)-X_{t_k^{(n)}}^{(n)}(\omega)| \geq \varepsilon_n \right\}.
\end{aligned}$$ Then it holds that ${\mathbb{P}}(\Omega_{k,n,\varepsilon_n}) \leq \delta_n \chi_n = \gamma_n$.
Now we state the a key lemma of our demonstration.
\[key\_sigma\_0\] Let Assumption \[Ass\_1\]-(ii) hold and the drift coefficient $b$ be bounded and measurable. Let $f \in H^{\beta,\kappa}$ for some $\beta \in (0,1]$. Then for any $p\geq 1$ and $0< \alpha < \frac{p\beta}{2} \wedge \frac{2\kappa}{\kappa+4}$, there exists a positive constant $C_p^*(f)= C^*(p,\alpha, \beta, \kappa,T,x_0,\|f\|_\beta, C_{\beta,\kappa}, \|b\|_\infty, \overline{\sigma}, \underline{\sigma})$ which does not depend on $n$ such that for each $n \geq 3$, $$\begin{aligned}
\label{eqnL0}
\int_{0}^{T}{\mathbb{E}}\left[\left|f(X_s^{(n)})-f(X_{\eta_n(s)}^{(n)})\right|^p \right]ds \leq \frac{C_p^*(f)}{n^\alpha \log n}.\end{aligned}$$
From Lemma \[tight\_3\] and the boundedness of $f$, it holds that $$\begin{aligned}
\label{key_sigma_1}
&\int_{0}^{T}{\mathbb{E}}\left[\left|f(X_s^{(n)})-f(X_{\eta_n(s)}^{(n)}) \right|^p\right]ds \notag
\\
&=\sum_{k=0}^{n-1}\int_{t_{k}^{(n)}}^{t_{k+1}^{(n)}}{\mathbb{E}}\left[\left|f(X_s^{(n)})-f(X_{t_{k}^{(n)}}^{(n)}) \right|^p \left({{\bf 1}}_{\Omega_{k,n,\varepsilon_n}}+{{\bf 1}}_{\Omega_{k,n,\varepsilon_n}^c} \right)\right]ds \notag\\
&\leq 2^p\|f\|_{\infty}^pT \gamma_n +\sum_{k=0}^{n-1}\int_{t_{k}^{(n)}}^{t_{k+1}^{(n)}}{\mathbb{E}}\left[\left|f(X_s^{(n)})-f(X_{t_{k}^{(n)}}^{(n)}) \right|^p {{\bf 1}}_{\Omega_{k,n,\varepsilon_n}^c} \right]ds.
\end{aligned}$$ We estimate the second term of as follows $$\begin{aligned}
&\sum_{k=0}^{n-1}\int_{t_{k}^{(n)}}^{t_{k+1}^{(n)}}{\mathbb{E}}\left[\left|f(X_s^{(n)})-f(X_{t_{k}^{(n)}}^{(n)}) \right|^p {{\bf 1}}_{\Omega_{k,n,\varepsilon_n}^c} \right]ds \notag\\
=& \sum_{k=0}^{n-1}\int_{t_{k}^{(n)}}^{t_{k+1}^{(n)}}{\mathbb{E}}\left[\left|f(X_s^{(n)})-f(X_{t_{k}^{(n)}}^{(n)}) \right|^p {{\bf 1}}_{\Omega_{k,n,\varepsilon_n}^c} {{\bf 1}}_{X^{(n)}_s\in S^{\varepsilon_n}(f)}\right]ds \notag \\
& \qquad + \sum_{k=0}^{n-1}\int_{t_{k}^{(n)}}^{t_{k+1}^{(n)}}{\mathbb{E}}\left[\left|f(X_s^{(n)})-f(X_{t_{k}^{(n)}}^{(n)}) \right|^p {{\bf 1}}_{\Omega_{k,n,\varepsilon_n}^c} {{\bf 1}}_{X^{(n)}_s \not \in S^{\varepsilon_n}(f)}\right]ds. \label{eqnL1} \end{aligned}$$ On the set $\Omega_{k,n,\varepsilon_n}^c \cap \big\{ X^{(n)}_s \not \in S^{\varepsilon_n}(f)\big\}$, it holds that $S(f) \cap [X^{(n)}_s \wedge X^{(n)}_{t^{(n)}_k}, X^{(n)}_s \vee X^{(n)}_{t^{(n)}_k}] =\emptyset$, thus, $$\left|f(X_s^{(n)})-f(X_{t_{k}^{(n)}}^{(n)}) \right|^p {{\bf 1}}_{\Omega_{k,n,\varepsilon_n}^c} {{\bf 1}}_{X^{(n)}_s \not \in S^{\varepsilon_n}(f)}
\leq \|f\|_\beta^p \left|X_s^{(n)}- X_{t_{k}^{(n)}}^{(n)} \right|^{p\beta}.$$ This implies the second term of is bounded by $$\begin{aligned}
\label{eqnL2}
\|f\|_\beta^p \sum_{k=0}^{n-1}\int_{t_{k}^{(n)}}^{t_{k+1}^{(n)}}{\mathbb{E}}\left[ \left|X_s^{(n)}- X_{t_{k}^{(n)}}^{(n)} \right|^{p\beta} \right]ds
\leq \|f\|_\beta^p T C_{p\beta}n^{-p\beta/2},\end{aligned}$$ where the last inequality follows from Lemma \[Lem\_1\]. For each constant $K_n \geq 1\vee (|x_0|+ T\|b\|_\infty)$, the first term of is bounded by $$\begin{aligned}
&2^p\|f\|_\infty^p \sum_{k=0}^{n-1}\int_{t_{k}^{(n)}}^{t_{k+1}^{(n)}} \Big( {\mathbb{E}}\left[ {{\bf 1}}_{X^{(n)}_s\in S^{\varepsilon_n}(f)\cap[-K_n,K_n]}\right]+ {\mathbb{E}}\left[ {{\bf 1}}_{X^{(n)}_s\in S^{\varepsilon_n}(f)\backslash[-K_n,K_n]}\right]\Big)ds \notag \\
\leq & 2^p\|f\|_\infty^p \int_0^T {\mathbb{E}}\left[ {{\bf 1}}_{X^{(n)}_s\in S^{\varepsilon_n}(f)\cap[-K_n,K_n]}\right]ds + 2^p\|f\|_{\infty}^p\int_0^T {\mathbb{E}}\left[ {{\bf 1}}_{|X^{(n)}_s| \geq K_n}\right]ds. \label{eqnL3}\end{aligned}$$ Since $\sigma$ is uniformly elliptic, $ \langle X^{(n)}\rangle_t \geq \underline{\sigma}^2 t$, we obtain $$\begin{aligned}
\int_0^T {\mathbb{E}}\left[ {{\bf 1}}_{X^{(n)}_s\in S^{\varepsilon_n}(f)\cap[-K_n,K_n]}\right]ds &\leq \underline{\sigma}^{-2} {\mathbb{E}}\left[ \int_0^T {{\bf 1}}_{X^{(n)}_s\in S^{\varepsilon_n}(f)\cap[-K_n,K_n]} d\langle X^{(n)}\rangle_s\right] \\
& = \underline{\sigma}^{-2} {\mathbb{E}}\left[ \int_{\mathbb{R}}{{\bf 1}}_{ S^{\varepsilon_n}(f)\cap[-K_n,K_n]}(x)L_T^{x}(X^{(n)})dx \right],\end{aligned}$$ where the last equation follows from the occupation time formula. Moreover, it follows from Lemma \[local\_time\] that $$\begin{aligned}
{\mathbb{E}}\left[ \int_{\mathbb{R}}{{\bf 1}}_{S^{\varepsilon_n}(f)\cap[-K_n,K_n]}(x) L_T^{x}(X^{(n)})dx \right] &\leq \int_{\mathbb{R}}{{\bf 1}}_{S^{\varepsilon_n}(f)\cap[-K_n,K_n]}(x) {\mathbb{E}}[L_T^{x}(X^{(n)})]dx \\
&\leq \sup_{x \in {\mathbb{R}}} {\mathbb{E}}[L_T^{x}(X^{(n)})] \lambda\Big(S^{\varepsilon_n}(f)\cap[-K_n,K_n]\Big)\\
&\leq \{12\|b\|_{\infty}^2T^2+ 6 \overline{\sigma}^2 T\}^{1/2} C_{\beta,\kappa} K_n\varepsilon_n^\kappa.\end{aligned}$$ Now we consider the second term of . For each $s \in [0,T]$, $$\begin{aligned}
{\mathbb{E}}\left[ {{\bf 1}}_{|X^{(n)}_s| \geq K_n}\right]
&\leq {\mathbb{P}}\Big( \Big| \int_0^s \sigma(X^{(n)}_{\eta_n(u)}) dW_u\Big| \geq K_n - \Big| x_0+ \int_0^s b(X^{(n)}_{\eta_n(u)}) du\Big|\Big)\\
&\leq {\mathbb{P}}\Big( \Big| \int_0^s \sigma(X^{(n)}_{\eta_n(u)}) dW_u\Big| \geq K_n - \|b\|_\infty T - |x_0|\Big).\end{aligned}$$ Since $\langle \int_{0}^{\cdot} \sigma(X_{\eta_n(s)}^{(n)})dW_s \rangle_t \leq \overline{\sigma}^2 T$ almost surely, from Proposition 6.8 of [@Shigekawa] and the inequality $(a-b)^2 \geq a^2/2-b^2$ for any $a,b \in {\mathbb{R}}$, we have $$\begin{aligned}
&{\mathbb{P}}\left(\sup_{0 \leq t \leq T} \left| \int_{0}^{t} \sigma(X_{\eta_n(s)}^{(n)})dW_s \right|\geq K_n -\|b\|_{\infty}T - |x_0|\right) \notag\\
&\leq 2\exp \left(-\frac{(K_n-|x_0|-\|b\|_{\infty}T)^2}{2\overline{\sigma}^2T}\right)
\leq 2\exp \left(\frac{(|x_0|+\|b\|_{\infty}T)^2}{2\overline{\sigma}^2T}\right) \exp\left(-\frac{K_n^2}{4\overline{\sigma}^2T}\right).
\end{aligned}$$ This implies $$\begin{aligned}
\label{key_sigma_13}
\int_0^T {\mathbb{E}}\left[ {{\bf 1}}_{|X^{(n)}_s| \geq K_n}\right]ds \leq 2T\exp \left(\frac{(|x_0|+\|b\|_{\infty}T)^2}{2\overline{\sigma}^2T}\right) \exp\left(-\frac{K_n^2}{4\overline{\sigma}^2T}\right).\end{aligned}$$ Gathering together the estimates –, we get $$\begin{aligned}
\int_{0}^{T}{\mathbb{E}}\left[\left|f(X_s^{(n)})-f(X_{\eta_n(s)}^{(n)})\right|^p \right]ds
\leq & 2^p\|f\|_{\infty}^pT \gamma_n + \|f\|_\beta^p TC_{p\beta}n^{-p\beta/2} \notag\\
& + 2^p \|f\|_{\infty}^p \underline{\sigma}^{-2} \{12\|b\|_{\infty}^2T^2+6 \overline{\sigma}^2 T\}^{1/2} C_{\beta,\kappa} K_n\varepsilon_n^\kappa \notag\\
& + 2^{p+1} \|f\|_{\infty}^p T\exp \left(\frac{(|x_0|+\|b\|_{\infty}T)^2}{2\overline{\sigma}^2T}\right) \exp\left(-\frac{K_n^2}{4\overline{\sigma}^2T}\right). \label{eqnL6}\end{aligned}$$ For each $0< \alpha < \frac{p\beta}{2} \wedge \frac{2\kappa}{\kappa+4}$, by choosing $K_n = (1+|x_0|+T\|b\|_\infty + 2\overline{\sigma}\sqrt{T\alpha}) \sqrt{\log n}$ and $\gamma_n = \frac{1}{n^\alpha \log n}$, we obtain from .
Method of removal of drift
--------------------------
The following removal of drift transformation plays a crucial role in our argument. Suppose that $b \in L^1({\mathbb{R}})$. The function $\varphi (x) := \int_0^x \exp\Big(-2 \int_0^y \frac{b(z)}{\sigma^2(z)} dz \Big) dy$ is well-defined since $\sigma^2$ is uniformly elliptic. Define $Y_t:=\varphi(X_t)$ and $Y_t^{(n)}:=\varphi(X_t^{(n)})$. Then by Itô’s formula we have $$\begin{aligned}
Y_t
= \varphi(x_0) + \int_0^t \varphi'(X_s) \sigma(X_s)dW_s,
$$ and $$\begin{aligned}
Y_t^{(n)}
= \varphi(x_0)
+\int_0^t \left( \varphi'(X_s^{(n)}) b(X_{\eta_n(s)}^{(n)})+\frac{1}{2}\varphi''(X_s^{(n)}) \sigma^2(X_{\eta_n(s)}^{(n)}) \right) ds
+ \int_0^t \varphi'(X_s^{(n)}) \sigma(X_{\eta_n(s)}^{(n)})dW_s.\end{aligned}$$ To simplify the notation, we denote $K_\sigma = \overline{\sigma} \vee \underline{\sigma}^{-1}$ and $C_0 = e^{2{K_\sigma}^2 \|b\|_{L^1({\mathbb{R}})}}$. We will make repeated use of the following elementary lemma.
([@NT_2015_2]) \[PDE\_2\] Suppose that $b \in L^1({\mathbb{R}})$ and Assumption \[Ass\_1\]-(ii) holds.
- For any $x \in {\mathbb{R}}$, $ C_0^{-1} \leq \varphi'(x)=\exp\Big(-2 \int_0^x \frac{b(z)}{\sigma^2(z)} dz \Big) \leq C_0.$
- For any $x \in {\mathbb{R}}$, $|\varphi''(x)| \leq 2{K_\sigma}^2 \|b\|_{\infty} \|\varphi'\|_{\infty} \leq 2\|b\|_\infty {K_\sigma}^2 C_0.$
- For any $z,w \in Dom(\varphi^{-1})$, $$\begin{aligned}
\label{PDE_4}
|\varphi^{-1}(z)-\varphi^{-1}(w)|
\leq C_0 |z-w|.
\end{aligned}$$
Yamada and Watanabe approximation technique
-------------------------------------------
Under the Assumption 2.2, by using the Yamada-Watanabe approximation technique, Le Gall [@LeGall] show that the pathwise uniequness holds for SDE (1). We also use this technique to prove the main result (see [@Yamada] or [@Gyongy]). For each $\delta \in (1,\infty)$ and $\varepsilon \in (0,1)$, we define a continuous function $\psi _{\delta, \varepsilon}: {\mathbb{R}}\to {\mathbb{R}}^+$ with $\text{supp}\: \psi _{\delta, \varepsilon} \subset [\varepsilon/\delta, \varepsilon]$ such that $\int_{\varepsilon/\delta}^{\varepsilon} \psi _{\delta, \varepsilon}(z) dz
= 1 \text{ and } 0 \leq \psi _{\delta, \varepsilon}(z) \leq \frac{2}{z \log \delta}, \:\:\:z > 0.$ Since $\int_{\varepsilon/\delta}^{\varepsilon} \frac{2}{z \log \delta} dz=2$, there exists such a function $\psi_{\delta, \varepsilon}$. We define a function $\phi_{\delta, \varepsilon} \in C^2({\mathbb{R}};{\mathbb{R}})$ by $\phi_{\delta, \varepsilon}(x):=\int_0^{|x|}\int_0^y \psi _{\delta,
\varepsilon}(z)dzdy.$ It is easy to verify that $\phi_{\delta, \varepsilon}$ has the following useful properties: $$\begin{aligned}
&|x| \leq \varepsilon + \phi_{\delta, \varepsilon}(x), \text{ for any $x \in {\mathbb{R}}$}, \label{phi3}\\
&0 \leq |\phi'_{\delta, \varepsilon}(x)| \leq 1, \text{ for any $x \in {\mathbb{R}}$} \label{phi2}, \\
\phi''_{\delta, \varepsilon}(\pm|x|)&=\psi_{\delta, \varepsilon}(|x|)
\leq \frac{2}{|x|\log \delta}{\bf 1}_{[\varepsilon/\delta, \varepsilon]}(|x|),
\text{ for any $x \in {\mathbb{R}}\setminus\{0\}$}. \label{phi4}\end{aligned}$$ From and , for any $t \in [0,T]$, we have $$\begin{aligned}
\label{esti_X1}
|X_t-X_t^{(n)}|
\leq C_0 |Y_t-Y_t^{(n)}|
\leq C_0 \left( \varepsilon + \phi_{\delta,\varepsilon}(Y_t-Y_t^{(n)}) \right).\end{aligned}$$ Using Itô’s formula, we have $$\begin{aligned}
\label{esti_X2}
\phi_{\delta,\varepsilon}(Y_t-Y_t^{(n)})
=M_t^{n,\delta,\varepsilon}
+I_t^{(n)}
+J_t^{(n)},\end{aligned}$$ where $$\begin{aligned}
M_t^{n,\delta,\varepsilon}
&:=\int_0^t \phi'_{\delta,\varepsilon}(Y_s-Y_s^{(n)}) \left\{ \varphi'(X_s)\sigma(X_s) - \varphi'(X_s^{(n)}) \sigma(X_{\eta_n(s)}^{(n)}) \right\}dW_s,\\
I_t^{(n)}
&:=-\int_0^t \phi'_{\delta,\varepsilon}(Y_s-Y_s^{(n)})
\left\{ \varphi'(X_s^{(n)})b(X_{\eta_n(s)}^{(n)}) +\frac{1}{2} \varphi''(X_s^{(n)}) \sigma^2(X_{\eta_n(s)}^{(n)}) \right\} ds,\\
J_t^{(n)}
&:=\frac{1}{2}\int_0^t \phi''_{\delta,\varepsilon}(Y_s-Y_s^{(n)})
\left| \varphi'(X_s) \sigma(X_s) - \varphi'(X_s^{(n)}) \sigma(X_{\eta_n(s)}^{(n)}) \right|^2 ds.\end{aligned}$$
Proof of Theorem \[Main\_1\]
----------------------------
We will only present the detail proof for the case that $b \in L^1({\mathbb{R}})$. The proof for the case $b \not \in L^1({\mathbb{R}})$ is based on the localisation technique given in [@NT_2015_2] and it will be omitted.
We fix $n \geq 3$ and a constant $0< \alpha < \frac{\beta}{2} \wedge \frac{2\kappa}{\kappa+4}$. We first consider $I_t^{(n)}$. Since $\varphi'' = - \frac{2b\varphi'}{\sigma^2}$, $$\begin{aligned}
|I_t^{(n)}|
&\leq \int_0^T \left|\phi'_{\delta,\varepsilon}(Y_t-Y_t^{(n)}) \varphi'(X_s^{(n)}) \right|
\left|b(X_{\eta_n(s)}^{(n)}) - \frac{b(X_s^{(n)}) \sigma^2(X_{\eta_n(s)}^{(n)})}{\sigma^2(X_s^{(n)})} \right| ds. \notag\end{aligned}$$ Thanks to Lemma \[PDE\_2\] and estimate , we have $$\begin{aligned}
|I_t^{(n)}|
&\leq {K_\sigma}^2 C_0 \int_0^T
\left|b(X_{\eta_n(s)}^{(n)}) \sigma^2(X_s^{(n)}) - b(X_s^{(n)}) \sigma^2(X_{\eta_n(s)}^{(n)}) \right| ds \notag\\
&\leq {K_\sigma}^2 C_0
\int_0^T \left\{
{K_\sigma}^2 \left|b(X_s^{(n)}) - b(X_{\eta_n(s)}^{(n)}) \right|
+\|b\|_{\infty} \left| \sigma^2(X_s^{(n)}) - \sigma^2(X_{\eta_n(s)}^{(n)}) \right|
\right\}ds. \notag\end{aligned}$$ It follows from Lemma \[key\_sigma\_0\] that $$\label{eqnL7}
{\mathbb{E}}[|I_t^{(n)}|] \leq \frac{C_I}{n^\alpha \log n},$$ where $C_I:=K_{\sigma}^2 C_0\{K_{\sigma}^2 C_1^*(b)+2\|b\|_{\infty} \overline{\sigma} C_1^*(\sigma) \}$. Now we estimate $J_t^{(n)}$. From , we have $$\begin{aligned}
J_t^{(n)}
&\leq \int_0^T \frac{{{\bf 1}}_{[\varepsilon/\delta,\varepsilon]}(|Y_s-Y_s^{(n)}|)}{|Y_s-Y_s^{(n)}| \log \delta}
\left| \varphi'(X_s) \sigma(X_s) - \varphi'(X_s^{(n)}) \sigma(X_{\eta_n(s)}^{(n)}) \right|^2 ds\\
&\leq 3(J_T^{1,n}+J_T^{2,n}+J_T^{3,n}),\end{aligned}$$ where $$\begin{aligned}
J_t^{1,n}
&:= \int_0^t \frac{{{\bf 1}}_{[\varepsilon/\delta,\varepsilon]}(|Y_s-Y_s^{(n)}|)}{|Y_s-Y_s^{(n)}| \log \delta}
|\sigma(X_s)|^2 \left| \varphi'(X_s) - \varphi'(X_s^{(n)}) \right|^2 ds,\\
J_t^{2,n}
&:=\int_0^t \frac{{{\bf 1}}_{[\varepsilon/\delta,\varepsilon]}(|Y_s-Y_s^{(n)}|)}{|Y_s-Y_s^{(n)}| \log \delta}
|\varphi'(X_s^{(n)})|^2 \left| \sigma(X_s) - \sigma(X_s^{(n)}) \right|^2 ds, \\
J_t^{3,n}
&:=\int_0^t \frac{{{\bf 1}}_{[\varepsilon/\delta,\varepsilon]}(|Y_s-Y_s^{(n)}|)}{|Y_s-Y_s^{(n)}| \log \delta}
|\varphi'(X_s^{(n)})|^2 \left| \sigma(X_s^{(n)}) - \sigma(X_{\eta_n(s)}^{(n)}) \right|^2 ds.\end{aligned}$$ From Lemma \[PDE\_2\]-(ii), $\varphi'$ is Lipschitz continuous with Lipschitz constant $\|\varphi''\|_{\infty}$. Hence, we have $$\begin{aligned}
\label{esti_J1}
J_T^{1,n}
&\leq \frac{{K_\sigma}^2 \|\varphi''\|_{\infty}^2}{\log \delta} \int_0^T \frac{{{\bf 1}}_{[\varepsilon/\delta,\varepsilon]}(|Y_s-Y_s^{(n)}|)}{|Y_s-Y_s^{(n)}|}
\left| X_s - X_s^{(n)} \right|^2 ds \notag\\
&\leq \frac{{K_\sigma}^2 \|\varphi''\|_{\infty}^2 C_0^2}{\log \delta}
\int_0^T{{\bf 1}}_{[\varepsilon/\delta,\varepsilon]}(|Y_s-Y_s^{(n)}|) \left| Y_s - Y_s^{(n)} \right| ds \notag\\
&\leq \frac{C_{J,1} \varepsilon}{\log \delta},\end{aligned}$$ where $C_{J,1}:=4{K_\sigma}^6 C_0^4 \|b\|_\infty^2 T$. Next we consider $J_T^{2,n}$. We first note that by , $$J_T^{2,n} \leq \frac{C^3_0}{\log \delta} \int_0^T \frac{\left| \sigma(X_s) - \sigma(X_s^{(n)}) \right|^2}{|X_s-X_s^{(n)}|} {{\bf 1}}_{|X_s- X^{(n)}_s|\geq \varepsilon/(C_0\delta)} ds.$$ Recall that by Assumption \[Ass\_1\]-(i), there exists a bounded and strictly increasing function $f_{\sigma} : {\mathbb{R}}\to {\mathbb{R}}$ such that for any $x,y \in {\mathbb{R}}$, $$\begin{aligned}
|\sigma(x)-\sigma(y)|^2
\leq |f_{\sigma}(x)-f_{\sigma}(y)|.\end{aligned}$$ We consider approximation $f_{\sigma,\ell} \in C^1({\mathbb{R}})$ of $f_{\sigma}$ which is also strictly increasing function and satisfies $\|f_{\sigma, \ell}\|_{\infty} \leq \|f_{\sigma}\|_{\infty}$ and $f_{\sigma,\ell} \uparrow f_{\sigma}$ as $\ell \to \infty$ on ${\mathbb{R}}$. Then by using Fatou’s lemma and the mean value theorem, we have $$\begin{aligned}
\label{pr_1_5}
J_T^{2,n}
&\leq \frac{C^3_0}{\log \delta}
\int_0^T
\frac{|f_{\sigma}(X_s)-f_{\sigma}(X_s^{(n)})|}{|X_s-X_s^{(n)}|}
{{\bf 1}}_{|X_s- X^{(n)}_s|> \varepsilon/(C_0\delta)} ds \notag\\
&\leq \liminf_{\ell \to \infty}
\frac{C^3_0}{\log \delta}
\int_0^T
\frac{|f_{\sigma, \ell}(X_s)-f_{\sigma, \ell}(X_s^{(n)})| }{|X_s-X_s^{(n)}|} {{\bf 1}}_{|X_s- X^{(n)}_s|> \varepsilon/(C_0\delta)} ds \notag\\
&\leq \liminf_{\ell \to \infty}
\frac{C_0^3}{\log \delta}
\int_0^T ds \int_0^1 d\theta f'_{\sigma, \ell}(V_s^{(n)}(\theta)),\end{aligned}$$ where $V^{(n)}(\theta)=(V_t^{(n)}(\theta))_{0 \leq t \leq T}$ is defined in Lemma \[local\_time\]. Since $\sigma \geq \underline{\sigma}$, the quadratic variation of $V^{(n)}(\theta)$ satisfies $$\begin{aligned}
\langle V^{(n)}(\theta) \rangle_t
= \int_{0}^{t} \left\{ (1-\theta)\sigma(X_s) + \theta \sigma(X_{\eta_n(s)}^{(n)}) \right\}^2 ds
\geq \underline{\sigma}^2 t,\end{aligned}$$ which implies $$\begin{aligned}
\int_0^T ds \int_0^1 d \theta f'_{\sigma, \ell}(V_s^{(n)}(\theta))
&\leq
\underline{\sigma}^{-2} \int_0^1 d \theta \int_0^T d \langle V^{(n)}(\theta) \rangle_{s} f'_{\sigma, \ell}(V_s^{(n)}(\theta)) \notag\\
&= \underline{\sigma}^{-2} \int_{{\mathbb{R}}} dx f'_{\sigma, \ell}(x) \int_0^1 d \theta L_T^x(V^{(n)}(\theta)),\end{aligned}$$ where the last equality is implied from the occupation time formula. Using Lemma \[local\_time\] and the estimate $\|f'_{\sigma, \ell}\|_{L^1({\mathbb{R}})} \leq 2 \|f_{\sigma, \ell}\|_{\infty} \leq 2 \|f_{\sigma} \|_{\infty}$ we have $$\begin{aligned}
{\mathbb{E}}\left[
\int_0^T ds \int_0^1 d \theta f'_{\sigma, \ell}(V_s^{(n)}(\theta))
\right]
&\leq \underline{\sigma}^{-2} \int_{{\mathbb{R}}} dx f'_{\sigma, \ell}(x) \int_0^1 d \theta
{\mathbb{E}}[L_T^x(V^{(n)}(\theta))] \\
&\leq \underline{\sigma}^{-2} \|f'_{\sigma, \ell}\|_{L^{1}({\mathbb{R}})}
\sup_{\theta \in [0,1], x \in {\mathbb{R}}}
{\mathbb{E}}[|L_T^x(V^{(n)}(\theta))|^2]^{1/2}\\
&\leq 2 \underline{\sigma}^{-2} \|f_{\sigma}\|_{\infty}
\{ 12\|b\|_{\infty}^2T^2+6 \overline{\sigma}^2 T\}^{1/2}.\end{aligned}$$ By plugging this estimate to and using Fatou’s lemma, we get the following estimate for the expectation of $J^{2,n}_T$, $$\begin{aligned}
\label{pr_1_7}
{\mathbb{E}}[J^{2,n}_T]
&\leq \frac{C_{J,2}}{\log \delta},\end{aligned}$$ where $C_{J,2}:=2C_0^3\underline{\sigma}^{-2} \|f_{\sigma}\|_{\infty} \{ 12\|b\|_{\infty}^2T^2+6 \overline{\sigma}^2 T\}^{1/2}$. Finally, we estimate $J^{3,n}_T$ as follows $$\begin{aligned}
{\mathbb{E}}[J^{3,n}_T] \leq \frac{C_0^2 \delta}{\varepsilon \log \delta} \int_0^T {\mathbb{E}}\Big[\left|\sigma(X^{n}_s)- \sigma(X^{(n)}_{\eta_n(s)})\right|^2\Big]ds. \end{aligned}$$ Applying Lemma \[key\_sigma\_0\], we get $$\label{eqnL8}
{\mathbb{E}}[J^{3,n}_t] \leq \frac{\delta}{\varepsilon \log \delta} \frac{C_{J,3}}{n^\alpha \log n},$$ where $C_{J,3}:=C_0^2 C_2^*(\sigma)$. Since ${\mathbb{E}}[M_t^{n,\delta, \varepsilon}] = 0$, it follows from – that there exists a positive constant $C$ which do not depend on $n$ such that $$\begin{aligned}
\sup_{0\leq t \leq T}{\mathbb{E}}[|X_t - X^{(n)}_t|] \leq C \Big( \varepsilon + \frac{1}{n^\alpha \log n} + \frac{\varepsilon}{\log \delta} + \frac{1}{\log \delta} + \frac{\delta}{\varepsilon \log \delta} \frac{1}{n^\alpha \log n}\Big).\end{aligned}$$ By choosing $\varepsilon = \frac{1}{\log n}$ and $\delta = n^\alpha$, we obtain the desired result.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank Arturo Kohatsu-Higa, Miguel Martinez and Toshio Yamada for their helpful comments. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED). The second author was supported by JSPS KAKENHI Grant Number 16J00894.
[99]{} Akahori, J. and Imamura, Y.: On a symmetrization of diffusion processes. Quantitative Finance 14(7) 1211–1216 (2014).
Ankirchner, S., Kruse, T. and Urusov, M.: Numerical approximation of irregular SDEs via Skorokhod embeddings. Journal of Mathematical Analysis and Applications 440.2, 692–715 (2016). Chan, K.S. and Stramer, O.: Weak Consistency of the Euler Method for Numerically Solving Stochastic Differential Equations with Discontinuous Coefficient. Stochastic Process. Appl. 76, 33–44 (1998).
Cherny, A. and Engelbert, H-J.: Singular Stochastic Differential Equations. Lecture Notes in Math. Vol. 1858. Springer (2005).
Gyöngy, I.: A note on Euler’s approximations. Potential Anal. 8, 205–216 (1998)
Gyöngy, I. and Rásonyi, M.: A note on Euler approximations for SDEs with Hölder continuous diffusion coefficients. Stochastic. Process. Appl. 121, 2189–2200 (2011).
Hairer, M., Hutzenthaler, M. and Jentzen, A.: Loss of regularity for Kolmogorov equation. Ann. Probab. 43(2), 468–527 (2015).
Halidias, N. and Kloeden, P.E.: A note on the Euler-Maruyama scheme for stochastic differential equations with a discontinuous monotone drift coefficient. BIT 48(1) 51–59 (2008).
Hashimoto, H. and Tsuchiya, T. : Convergence rate of stability problems of SDEs with (Dis-)continuous coefficients. Preprint arXiv:1401.4542v4 (2014).
Hutzenthaler, M., Jentzen, A. and Kloeden, P. E.: Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients. The Annals of Applied Probability, 22(4), 1611-1641 (2012).
Jentzen, A., Müller-Gronbach, T. and Yaroslavtseva, L.: On stochastic differential equations with arbitrary slow convergence rates for strong approximation. arXiv:1506.02828v1 (2015).
Kloeden, P. and Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer (1995).
Kohatsu-Higa. A., Lejay, A. and Yasuda, K.: Weak approximation errors for stochastic differential equations with non-regular drift. Preprint (2013).
Le Gall, JF.: One-dimensional stochastic differential equations involving the local times of the unknown process. In Stochastic analysis and applications 1984 (pp. 51-82). Springer Berlin Heidelberg.
Lemaire, V. and Menozzi, S.: On some Non-Asymptotic Bounds for the Euler Scheme. Electron J. Probab., 15, 1645-1681 (2010)
Leobacher, G. and Szölgyenyi M.: A numerical method for SDEs with discontinuous drift. BIT Numer. Math. (2015).
Martinez, M., and Talay, D.: One-dimensional parabolic diffraction equations: pointwise estimates and discretization of related stochastic differential equations with weighted local times. Electron. J. Probab. 17, no. 27, 1–32 (2012).
Menoukeu Pamen, O. and Taguchi, D.: Strong rate of convergence for the Euler-Maruyama approximation of SDEs with Hölder continuous drift coefficient. Preprint, arXiv:1508.07513 (2015).
Nakao, S.: On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations. Osaka J. Math. 9, 513-518 (1972).
Ngo, H-L., and Taguchi, D.: Strong rate of convergence for the Euler-Maruyama approximation of stochastic differential equations with irregular coefficients. Math. Comp. 85(300), 1793–1819 (2016).
Ngo, H-L., and Taguchi, D.: Approximation for non-smooth functionals of stochastic differential equations with irregular drift. Preprint, arXiv:1505.03600. (2015)
Ngo, H-L., and Taguchi, D.: On the Euler-Maruyama approximation for one-dimensional stochastic differential equations with irregular coefficients. arXiv:1509.06532 (2015).
Shigekawa, I.: Stochastic analysis. Translations of Mathematical Monographs, 224. Iwanami Series in Modern Mathematics. American Mathematical Society, Providence, RI, (2004).
Yamada, T. and Watanabe, S.: On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11, 155-167 (1971).
Yan, B. L.: The Euler scheme with irregular coefficients. Ann. Probab. 30, no. 3, 1172–1194 (2002).
[^1]: Hanoi National University of Education, 136 Xuan Thuy - Cau Giay - Hanoi - Vietnam, email: ngolong@hnue.edu.vn
[^2]: Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu, Shiga, 525-8577, Japan, email: dai.taguchi.dai@gmail.com
|
---
abstract: 'Research-based assessment instruments (RBAIs) are ubiquitous throughout both physics instruction and physics education research. The vast majority of analyses involving student responses to RBAI questions have focused on whether or not a student selects correct answers and using correctness to measure growth. This approach often undervalues the rich information that may be obtained by examining students’ particular choices of incorrect answers. In the present study, we aim to reveal some of this valuable information by quantitatively determining the relative correctness of various incorrect responses. To accomplish this, we propose an assumption that allow us to define relative correctness: students who have a high understanding of Newtonian physics are likely to answer more questions correctly and also more likely to choose better incorrect responses, than students who have a low understanding. Analyses using item response theory align with this assumption, and Bock’s nominal response model allows us to uniquely rank each incorrect response. We present results from over 7,000 students’ responses to the Force and Motion Conceptual Evaluation.'
author:
- 'Trevor I. Smith'
- 'Kyle J. Louis'
- 'Bartholomew J. Ricci, IV'
- Nasrine Bendjilali
bibliography:
- 'TIS.bib'
title: 'Quantitatively ranking incorrect responses to multiple-choice questions using item response theory'
---
Introduction {#sec:intro}
============
Many instructional and research questions over the past three decades have been answered by examining student responses to multiple-choice Research-based Assessment Instruments (RBAIs) [@Madsen2017; @VonKorff2016]. Tens of thousands of students have provided responses to questions on the Force Concept Inventory (FCI [@Hestenes1992]), the Force and Motion Conceptual Evaluation (FMCE [@Thornton1998]), and many others [@physport], and dozens of analyses have been published that use these results to measure student learning (See Refs. [@Docktor2014] and [@Madsen2017]). A common factor throughout most of these analyses is that students’ responses are typically scored as being correct or incorrect; very little attention has been paid to which incorrect answers students choose. This dichotomous scoring scheme is very beneficial for simplifying student performance on a RBAI or growth in learning to a single number that may be compared between students or across populations. The simplicity of this analysis and the ability for instructors and researchers to compare their results with other data sets has contributed to the proliferation of RBAIs, to the benefit of the physics education research (PER) community; however, the dichotomous scoring scheme implicitly ignores any information about students’ choices that are not correct. All incorrect answers are treated equally, regardless of how similar or different they may be to the correct answer.
RBAIs are so powerful because their questions help to elicit students’ core beliefs about how the world works in ways that mathematical or problem-solving questions often do not. Many of the incorrect “distractor” response choices correspond with deeply held intuitive understandings that fit well with everyday experiences (and correspond with historically accurate models) but conflict with the principles of Newtonian physics [@Madsen2017]. The ability to deeply probe students’ conceptual understanding of physics and represent this understanding with a single numerical value is very powerful. The authors of the FMCE, in fact, argue against using a single numerical score to represent student understanding [@Thornton2009] instead favoring the examination of student performance on individual or small groups of questions [@Thornton1998], but the common practice persists. Moreover, the common practice of reporting normalized gain as a measure of student learning has been shown to be biased against students with little prior exposure to formal physics instruction [@Nissen2018a].
Other analyses of student responses to RBAI questions examine specific choices that students make and relate these choices to various mental models [@Bao2006], misconceptions [@Hestenes1992], views [@Thornton1997], or pieces of knowledge [@Smith2008; @Smith2014] that students may have or use when answering particular questions. These analyses provide a lot of rich information about students’ ideas, but the processes of conducting these analyses are often quite time intensive, and the presentation and visualization of the results can be conceptually dense and difficult to interpret [@Smith2015c; @Griffin2016]. As such, these analyses are not nearly as common as reporting a single numeric score.
Our ultimate goal is to define a single numeric score that represents student knowledge or understanding as measured by a RBAI by incorporating both correct responses and the good ideas that may be expressed in some incorrect responses. The first part of defining such a metric is to determine whether or not some incorrect answers may be considered better than others, where “better” means closer to correct or indicating a higher level of understanding. Considering one incorrect answer to be better than another can be a tricky business, and we want to make sure that we are not introducing personal bias into our definitions. As such, we carefully articulate the assumption for defining what makes one response better than another, and we choose an analysis method that correspond to that assumption to quantitatively rank incorrect responses based on students’ response patterns.
> **Assumption: Students who choose correct responses on most questions are more likely to choose better incorrect answers than students who choose few correct responses.**
This assumption is based on the premise that students who understand more about Newtonian physics are more likely to choose better incorrect answers than students who understand less physics, and these students are also more likely to choose a greater number of correct responses. This assumption is consistent with previous work that has used item response curves (IRCs) to examine and rank incorrect responses on both the FCI and the FMCE [@Morris2012; @Walter2016; @Ishimoto2017; @Smith2017]. We expand on this prior work by using a nested-logit item response theory (IRT) model to simultaneously estimate students’ overall understanding of Newtonian mechanics (the IRT latent trait, or person parameter) and determine how closely each response choice correlates with a high level of understanding using the estimated parameters of the model [@Bock1972; @Thissen2010; @Suh2010; @Chalmers2012; @mirt; @Louis2018]. Based on this assumption we would claim, for example, that a student who only incorrectly answers one question is more likely to choose a response that’s almost correct than a student who answers 20 questions incorrectly.
To illustrate the applicability of our assumption, we analyzed more than 7,000 students’ matched pre-/post-test responses to the FMCE to demonstrate how quantitative analyses can provide information about which response choices may be better than others. We present a ranking of incorrect responses for all FMCE questions as well as the parameter values used to make these determinations.
Data Sources and Preparation {#sec:data}
============================
Our data come from two primary sources:
- sets of student responses to the FMCE provided to one author (TIS) by colleagues from four different colleges or universities, Schools 1–4,[^1] as part of current and previous research projects ($N=952$), and
- student responses uploaded to PhysPort’s Data Explorer ($N=6,336$) [@physportde].
Some information is known about the instructional settings at Schools 1–4 (all of which used research-based instructional materials of some sort), but this information is not available for the PhysPort data. For the purposes of the current analysis, we combine all data into one set of $N=7,288$ students. We are not interested in how instructional factors impact student learning for this analysis, or whether or not student responses are different before or after instruction. As such, we have combined all pretest and posttest responses into a data set of $N=14,576$ response sets.
To prepare the data for analysis, we omit any responses that are inappropriate for a given question (e.g., a given response of E on question 45, which only includes options A, B, C, and D). We also omit response J (None of these answers is correct) from interpretations of our analyses because it does not represent a well-defined indication of what each student would consider correct: two students who choose answer A agree on what they consider to be correct, but two students who choose answer J may have very different ideas of what would be a correct answer, so we cannot claim similarities between the responses of students who choose J. We also removed response sets with three or more blank or unscorable responses. This gave us a usable data set of $N=12,388$ response sets.
The structure of the FMCE makes it an interesting focus for this work. Unlike many other RBAIs, the FMCE contains several questions for each physical scenario presented (e.g., a toy car moving horizontally), and all questions in each set have the same set of response choices. This is particularly interesting because a response choice that corresponds with the most common intuitive answer to one question, may not relate to any documented reasoning for another question.
The 2PL-NRM Nested Logit Model {#sec:irt}
==============================
Item Response Theory (IRT) uses students’ responses to multiple-choice questions to simultaneously estimate each student’s overall understanding of the material (a.k.a. the latent trait or person parameter, $\theta$) and determine the probability that a student will be correct on each question given his/her understanding [@Baker2001; @deAyala2008]. The latent trait is normalized such that the average value is $\langle\theta\rangle=0$ and the standard deviation is $\sigma_\theta=1$. In the two-parameter logistic (2PL) IRT model, the probability of a student answering a specific question correctly is given by, $$\begin{aligned}
P\left(\theta\right)&=\frac{1}{1+e^{-a\left(\theta-b\right)}}
\label{2pl}
\end{aligned}$$ where $a$ is the discrimination parameter and $b$ is the difficulty parameter. Some previous work has used the three-parameter logistic model to analyze RBAI data [@Wang2010], but we feel that the inclusion of the third “guessing” parameter is inappropriate for our analyses given that student responses to the FMCE are concentrated in a small subset of responses for each question: they are not, in fact, guessing [@Thornton2009; @Smith2008].
The interpretation of the parameters in the 2PL model may be understood by examining plots of $P(\theta)$ vs. $\theta$: Fig. \[2plPlots\] shows examples from several questions. The difficulty $b$ is the value of $\theta$ at which $P(b) = 0.5)$, and the discrimination $a$ is proportional to the slope of the curve at $\theta = b$: $\mathrm{d}P/\mathrm{d}\theta|_{b} = a/4$. Questions 1 and 14 (Fig. \[2plPlots\] (a) and (b), respectively) have similar difficulty parameters (the $b$ value differs by less than 0.1 standard deviations of the latent ability $\theta$), but Q14’s higher discrimination parameter $a$ shows up as a sharper transition from most likely incorrect to most likely correct, and a steeper slope at the midpoint of the curve. Question 22 in Fig. \[2plPlots\](c) has a similar discrimination to Q1 (similar slope at $P(\theta)=0.5$), but the difficulty is much lower (shown by a shift to the left compared to Fig. \[2plPlots\](a)), with many below-average students ($\theta<0$) being fairly likely to answer correctly. Question 47 in Fig. \[2plPlots\](d) has a difficulty parameter that is about average (close to zero), but the discrimination is relatively small, as shown by a shallow slope, and a more gradual transition from probably incorrect to probably correct than any of the other three. Higher values of discrimination $a$ mean a sharp transition and steeper slope; lower values mean a gradual transition and shallower slope. Higher values of difficulty $b$ mean a graph that is shifted to the right; lower values mean a graph that is shifted to the left.
-------------------- -------------------
\(a) Question 1 \(b) Question 14
$a=2.5$, $b=0.45$ $a=3.7$, $b=0.54$
\[5ex\]
\(c) Question 22 \(d) Question 47
$a=2.8$, $b=-0.27$ $a=1.1$, $b=0.10$
-------------------- -------------------
Bock’s nominal response model (NRM) provides the probability of a person choosing each possible response based on $\theta$ [@Bock1972], $$\begin{aligned}
P_k(\theta)&= \frac{e^{a_k\left(\theta-b_k\right)}}{\sum\limits_{i=1}^{N} e^{a_i\left(\theta-b_i\right)}},
\label{nrm}\end{aligned}$$ where $k$ indicates the particular response choice, and the summation is performed over all $N$ response choices. According to Bock and Moustaki, the value of the $a_k$ parameter may be used to rank the incorrect responses, with a higher value indicating a response that is more closely correlated with the latent trait and, therefore, better than a response with a lower value [@Bock2007]; however, the meaning of the $a_k$ and $b_k$ parameters is not as easily interpreted as the $a$ and $b$ of the 2PL model [@Thissen2010]. One shortcoming of the NRM is that the parameters are not uniquely defined, and a normalization constraint is required. This is often accomplished by setting the value of both parameters associated with one particular response to be fixed (at 0 or 1) and determining all other parameters relative to those. The NRM is excellent for analyzing data for which no prior information is available regarding the relative correctness of any of the responses; however, we have found that for our FMCE data, the parameters occasionally become reversed with choosing the correct response being associated with having a low value of $\theta$ (i.e., a poor overall understanding of Newtonian mechanics).
Ranking Incorrect Responses {#sec:irtrank}
===========================
In order to rank incorrect responses while properly accounting for the correct response, we use the 2PL-NRM nested logit model developed by Suh and Bolt [@Suh2010]. In this model, the probability of a student choosing a specific incorrect response $k$ is given by $$\begin{aligned}
P_k(\theta)&=\left(1-\frac{1}{1+e^{-a\left(\theta-b\right)}}\right)\frac{e^{a_k\left(\theta-b_k\right)}}{\sum\limits_{i\neq\textrm{correct}} e^{a_i\left(\theta-b_i\right)}}
\label{2plnrm}\end{aligned}$$ where the parenthetical term is the probability of being not correct from the 2PL model, and the second term is Bock’s NRM with the summation being over only the *in*correct responses. In this model, the values of all $\theta$, $a$, and $b$ parameters are calculated using the 2PL model, and all $a_k$ and $b_k$ parameters are determined using the NRM, given the 2PL results. We used the A Multidimensional Item Response Theory (mirt) package in the R programming language to perform all IRT analyses [@r; @Chalmers2012; @mirt].
To determine the ranking of incorrect responses, we calculated the values of $a$ and $b$ for every question, and $a_k$ and $b_k$ for each incorrect response choice. According to de Ayala, a data set must have at least 10 times as many response sets as the number of parameters to be calculated for an IRT model to have good convergence [@deAyala2008]; with our data set of $N=12,388$ response sets we are more than able to determine the 722 necessary parameters.
We choose to omit response J from our ranking of incorrect responses because it does not correspond to a unique choice that students make about what they think is correct. Two students who choose response A (for example) agree that the information associated with A is correct, but two students who choose response J may or may not agree with each other regarding what a correct response would be. As such, we do not think it is valid to suggest that choosing J represents a unique level of correctness.
As a result of using the mirt package to apply the 2PL-NRM nested logit model, every response choice within each question has a unique $a_k$ value, implying that all answers are meaningfully different from each other. The question is then whether or not any of the $a_k$ values may be considered approximately equal to others, indicating approximately equal correlations with the $\theta$ parameter (i.e., response choices that are equally correct). To determine whether or not response choices are different from each other, we calculated the sampling distribution of values for each $a_k$ parameter by selecting random sample of 7,300 respondents using the sample function in R, and we used the mirt package to calculate each parameter [^2]. We repeated this process over 100,000 times to create a set of values for each parameter.
The mirt package uniquely determines each value of $a_k$ by setting one parameter equal to 1 for each question [@mirt]. In order to ensure that we obtained a distribution of values for all parameters of interest, we chose to include one set of responses that included a “dummy” response and set $a_{0} = 0$ for this response. All other $a_k$ parameters are determined relative to $a_{0}$; as such, the $a_k$ values are only meaningful when compared within the same question, and there are no thresholds for determining whether a particular value of $a_k$ is high or low in and of itself.
Using the effsize package, we calculated a Hedges’ $g$ effect size to quantify the magnitude of the difference between the $a_k$ values for each pair [@effsize]. In our full data set, every response to every question is selected in at least one response set. The $a_k$ values reported in Tables \[rankTable126\] and \[rankTable2747\] are those determined from the full data set, with the dummy response set included to uniquely determine each parameter. Given that the values of $a_k$ parameters are only meaningful in relation to other values for the same question, the inclusion of the dummy response set has minimal impact on the overall results.
Figure \[Q19ak\] shows a graphical representation of the parameter distributions for question 19. There are several key features to notice about these distributions:
- there is no distribution for the correct response because the 2PL-NRM does not calculate an $a_k$ parameter for the correct response; it is automatically assumed to be the best,
- the distribution for $a_A$ is higher than any other value of $a_k$, with only minimal overlap with $a_E$, indicating that A has the highest parameter, and is thus the best incorrect response, and
- the distributions $a_C$ and $a_G$ are practically identical, indicating that these parameters have very similar values; thus, we would interpret them as being equally correct.
Other comparisons between various responses are a bit more ambiguous. The $a_D$ and $a_F$ distributions look quite similar, but not as similar as $a_C$ and $a_G$. The $a_H$ distribution is noticeably shifted to the left of $a_C$ and $a_G$, but there is still quite a bit of overlap. We use Hedges’ $g$ to quantify the magnitude of the difference between each pair of distributions of the $a_k$ values: if $g$ is small ($g<0.5$) we conclude that the parameter values are effectively equal and the responses are equally correct, if $g$ is very large ($g\geq 1.3$) we conclude that the parameters are significantly different and that the responses represent different levels of understanding, and if $0.5\leq g < 1.3$ then we cannot make a conclusive determination.
[rcc\*[12]{}[c@]{}c@rcc\*[12]{}[c@]{}c]{} Q1: &B&$>$&D&$>$&A&$\geq$&C&$=$&E&$=$&F&$\geq$&G&&&Q14: &E&$>$&H&$>$&F&$>$&G&$>$&A&$\geq$&D&$=$&C&$=$&B\
$a_k$:&&&0.77&$>$&-0.18&$\geq$&-0.48&$\approx$&-0.62&$\approx$&-0.77&$\geq$&-1.39&&&$a_k$:&&&2.07&$>$&1.01&$>$&0.3&$>$&-0.55&$\geq$&-0.84&$\approx$&-0.97&$\approx$&-0.99\
$g$:&&&&2.75&&1.25&&0.34&&0.28&&1.21&&&&$g$:&&&&2.7&&1.34&&1.89&&0.86&&0.32&&0.05&\
%:&32&&0&&66&&1&&0&&0&&0&&&%:&27&&2&&0&&0&&62&&1&&5&&1\
Q2: &D&$>$&C&$>$&E&$\geq$&G&$\geq$&A&$=$&B&$>$&F&&&Q15: &E&$>$&G&$\geq$&D&$\geq$&H&$=$&C&$=$&F&$>$&A&$\geq$&B\
$a_k$:&&&1.62&$>$&0.54&$\geq$&-0.14&$\geq$&-0.46&$\approx$&-0.51&$>$&-1.46&&&$a_k$:&&&0.94&$\geq$&0.43&$\geq$&0.19&$\approx$&0.17&$\approx$&0.04&$>$&-0.35&$\geq$&-0.62\
$g$:&&&&2.69&&1.04&&0.54&&0.16&&2.86&&&&$g$:&&&&1.27&&0.69&&0.03&&0.44&&1.37&&1.07&\
%:&29&&4&&0&&0&&1&&65&&0&&&%:&95&&0&&0&&0&&0&&0&&1&&2\
Q3: &F&$>$&B&$>$&E&$\geq$&G&$>$&A&$=$&C&$=$&D&&&Q16: &A&$>$&E&$>$&B&$>$&F&$=$&C&$>$&D&$=$&G&$\geq$&H\
$a_k$:&&&0.59&$>$&0.29&$\geq$&0.13&$>$&-0.49&$\approx$&-0.61&$\approx$&-0.71&&&$a_k$:&&&0.94&$>$&0.53&$>$&-0.26&$\approx$&-0.36&$>$&-0.95&$\approx$&-0.96&$\geq$&-1.3\
$g$:&&&&1.32&&0.73&&2.71&&0.45&&0.47&&&&$g$:&&&&1.5&&2.49&&0.34&&2.83&&0.05&&0.99&\
%:&41&&1&&6&&7&&1&&41&&3&&&%:&31&&0&&1&&0&&66&&1&&0&&0\
Q4: &F&$>$&D&$>$&B&$=$&E&$\geq$&C&$\geq$&G&$>$&A&&&Q17: &E&$>$&F&$=$&B&$>$&D&$\geq$&G&$=$&C&$\geq$&H&$\geq$&A\
$a_k$:&&&0.76&$>$&0.27&$\approx$&0.15&$\geq$&-0.02&$\geq$&-0.3&$>$&-0.7&&&$a_k$:&&&0.06&$\approx$&-0.02&$>$&-0.48&$\geq$&-0.64&$\approx$&-0.68&$\geq$&-0.81&$\geq$&-1.07\
$g$:&&&&2.05&&0.5&&0.78&&1.28&&1.66&&&&$g$:&&&&0.25&&2.36&&0.76&&0.22&&0.58&&1.23&\
%:&31&&1&&1&&3&&0&&63&&1&&&%:&24&&0&&53&&9&&2&&2&&5&&4\
Q5: &D&$>$&C&$>$&E&$>$&A&$\geq$&B&$=$&G&$=$&F&&&Q18: &B&$>$&A&$\geq$&E&$>$&D&$=$&G&$=$&H&$>$&C&$>$&F\
$a_k$:&&&1.31&$>$&0.18&$>$&-0.45&$\geq$&-0.61&$\approx$&-0.62&$\approx$&-0.73&&&$a_k$:&&&1.07&$\geq$&0.77&$>$&-0.26&$\approx$&-0.35&$\approx$&-0.35&$>$&-0.67&$>$&-0.97\
$g$:&&&&4.84&&2.56&&0.67&&0.06&&0.45&&&&$g$:&&&&1.26&&4.32&&0.39&&0&&1.46&&1.31&\
%:&48&&4&&1&&3&&36&&1&&7&&&%:&27&&2&&1&&8&&9&&49&&1&&2\
Q6: &B&$>$&F&$>$&A&$=$&E&$\geq$&G&$>$&C&$>$&D&&&Q19: &B&$>$&A&$>$&E&$>$&D&$=$&F&$>$&C&$=$&G&$\geq$&H\
$a_k$:&&&0.91&$>$&0.17&$\approx$&0.16&$\geq$&0.05&$>$&-0.36&$>$&-0.87&&&$a_k$:&&&2.31&$>$&0.76&$>$&-0.16&$\approx$&-0.19&$>$&-1.05&$\approx$&-1.06&$\geq$&-1.41\
$g$:&&&&3.62&&0.06&&0.53&&1.98&&2.46&&&&$g$:&&&&4.98&&2.83&&0.12&&2.85&&0.01&&1.19&\
%:&19&&12&&7&&16&&6&&34&&5&&&%:&27&&3&&0&&52&&1&&6&&5&&3\
Q7: &B&$>$&F&$=$&C&$=$&A&$>$&G&$>$&D&$=$&E&&&Q20: &G&$>$&B&$>$&A&$=$&D&$=$&C&$=$&E&$=$&H&$>$&F\
$a_k$:&&&0.22&$\approx$&0.14&$\approx$&0.1&$>$&-0.39&$>$&-0.68&$\approx$&-0.78&&&$a_k$:&&&0.59&$>$&0.27&$\approx$&0.22&$\approx$&0.21&$\approx$&0.21&$\approx$&0.19&$>$&-0.23\
$g$:&&&&0.34&&0.18&&2.23&&1.33&&0.43&&&&$g$:&&&&1.57&&0.25&&0.04&&0.03&&0.1&&2.31&\
%:&42&&4&&6&&8&&2&&4&&32&&&%:&29&&0&&1&&1&&0&&1&&2&&64\
Q8: &A&$>$&D&$>$&B&$=$&E&$=$&C&$>$&G&$>$&F&&&Q21: &E&$>$&B&$>$&A&$>$&D&$=$&H&$=$&F&$\geq$&C&$>$&G\
$a_k$:&&&1.1&$>$&0.48&$\approx$&0.37&$\approx$&0.35&$>$&-0.17&$>$&-0.94&&&$a_k$:&&&0.56&$>$&-0.01&$>$&-0.38&$\approx$&-0.47&$\approx$&-0.5&$\geq$&-0.69&$>$&-1\
$g$:&&&&2.5&&0.43&&0.08&&2.08&&3.14&&&&$g$:&&&&2.86&&1.77&&0.43&&0.19&&0.87&&1.52&\
%:&23&&1&&4&&9&&2&&44&&16&&&%:&36&&2&&8&&2&&25&&15&&2&&7\
Q9: &A&$>$&C&$=$&B&$>$&D&$\geq$&G&$=$&E&$=$&F&&&Q22: &A&$>$&C&$\geq$&B&$>$&D&$>$&G&$>$&F&$=$&E&&\
$a_k$:&&&0.29&$\approx$&0.27&$>$&-0.17&$\geq$&-0.5&$\approx$&-0.55&$\approx$&-0.63&&&$a_k$:&&&0.76&$\geq$&0.57&$>$&0.17&$>$&-0.4&$>$&-1.1&$\approx$&-1.16&&\
$g$:&&&&0.07&&1.85&&1.27&&0.16&&0.34&&&&$g$:&&&&0.91&&1.6&&1.93&&2.12&&0.09&&&\
%:&26&&1&&3&&67&&1&&3&&1&&&%:&55&&1&&1&&0&&0&&0&&42&&\
Q10: &A&$>$&E&$\geq$&D&$>$&G&$\geq$&B&$=$&C&$=$&F&&&Q23: &B&$>$&A&$=$&C&$>$&F&$=$&G&$=$&E&$\geq$&D&&\
$a_k$:&&&0.76&$\geq$&0.45&$>$&-0.2&$\geq$&-0.37&$\approx$&-0.4&$\approx$&-0.42&&&$a_k$:&&&0.59&$\approx$&0.58&$>$&-0.61&$\approx$&-0.69&$\approx$&-0.74&$\geq$&-0.99&&\
$g$:&&&&1.2&&2.53&&0.65&&0.11&&0.08&&&&$g$:&&&&0.05&&5.67&&0.38&&0.27&&1.03&&&\
%:&35&&1&&2&&1&&54&&6&&1&&&%:&43&&2&&1&&8&&41&&1&&4&&\
Q11: &A&$>$&D&$>$&B&$\geq$&C&$>$&E&$>$&G&$>$&F&&&Q24: &C&$>$&D&$>$&E&$=$&F&$>$&B&$>$&G&$>$&A&&\
$a_k$:&&&0.53&$>$&0.22&$\geq$&0.11&$>$&-0.19&$>$&-0.46&$>$&-1.64&&&$a_k$:&&&-0.2&$>$&-0.67&$\approx$&-0.68&$>$&-0.91&$>$&-1.17&$>$&-1.61&&\
$g$:&&&&1.5&&0.58&&1.56&&1.38&&6.23&&&&$g$:&&&&2.22&&0&&1.41&&1.51&&2.34&&&\
%:&34&&1&&4&&3&&4&&48&&7&&&%:&53&&1&&1&&4&&33&&4&&3&&\
Q12: &A&$>$&B&$=$&C&$>$&F&$>$&D&$>$&E&$\geq$&G&&&Q25: &B&$>$&A&$>$&C&$>$&D&$=$&F&$>$&E&$\geq$&G&&\
$a_k$:&&&0.23&$\approx$&0.16&$>$&-0.18&$>$&-0.48&$>$&-0.81&$\geq$&-1.14&&&$a_k$:&&&1.08&$>$&0.5&$>$&-0.65&$\approx$&-0.69&$>$&-1.14&$\geq$&-1.45&&\
$g$:&&&&0.32&&1.37&&1.41&&1.6&&1.15&&&&$g$:&&&&2.47&&4.37&&0.13&&1.97&&1.24&&&\
%:&36&&2&&1&&0&&60&&1&&0&&&%:&40&&6&&1&&1&&41&&5&&4&&\
Q13: &A&$>$&D&$=$&E&$>$&C&$\geq$&B&$\geq$&G&$\geq$&F&&&Q26: &C&$>$&G&$\geq$&D&$>$&F&$\geq$&B&$>$&E&$>$&A&&\
$a_k$:&&&0.45&$\approx$&0.42&$>$&-0.63&$\geq$&-0.75&$\geq$&-0.87&$\geq$&-1.06&&&$a_k$:&&&0.54&$\geq$&0.42&$>$&-0.12&$\geq$&-0.35&$>$&-0.78&$>$&-1.32&&\
$g$:&&&&0.11&&4.8&&0.57&&0.57&&0.84&&&&$g$:&&&&0.55&&2.31&&0.9&&1.79&&2.37&&&\
%:&44&&1&&1&&3&&50&&1&&1&&&%:&55&&1&&1&&0&&2&&3&&37&&\
\[rankTable126\]
[rcc\*[12]{}[c@]{}c@rcc\*[12]{}[c@]{}c]{} Q27: &A&$>$&D&$\geq$&B&$\geq$&E&$\geq$&C&$>$&G&$>$&F&&&Q38: &A&$>$&E&$=$&B&$\geq$&C&$\geq$&D&&&&&&\
$a_k$:&&&0.3&$\geq$&0.06&$\geq$&-0.11&$\geq$&-0.31&$>$&-0.92&$>$&-2.1&&&$a_k$:&&&0.27&$\approx$&0.16&$\geq$&0.05&$\geq$&-0.2&&&&&&\
$g$:&&&&1.17&&0.92&&1.06&&3.3&&6.13&&&&$g$:&&&&0.47&&0.53&&1.09&&&&&&&\
%:&42&&1&&5&&4&&7&&36&&5&&&%:&23&&2&&66&&5&&3&&&&&&\
Q28: &A&$>$&E&$\geq$&B&$=$&C&$>$&F&$>$&D&$>$&G&&&Q39: &E&$>$&A&$\geq$&C&$=$&D&$>$&B&&&&&&\
$a_k$:&&&0.31&$\geq$&0.16&$\approx$&0.15&$>$&-0.23&$>$&-0.61&$>$&-0.98&&&$a_k$:&&&-0.12&$\geq$&-0.4&$\approx$&-0.42&$>$&-1.28&&&&&&\
$g$:&&&&0.64&&0.07&&1.55&&1.52&&1.4&&&&$g$:&&&&1.25&&0.07&&3.96&&&&&&&\
%:&38&&2&&2&&1&&0&&57&&0&&&%:&51&&3&&4&&35&&6&&&&&&\
Q29: &A&$>$&E&$>$&D&$>$&C&$>$&F&$=$&G&$=$&B&&&Q40: &A&$>$&E&$\geq$&B&$=$&F&$\geq$&C&$\geq$&G&$\geq$&H&$\geq$&D\
$a_k$:&&&1.07&$>$&0.17&$>$&-0.46&$>$&-0.81&$\approx$&-0.85&$\approx$&-0.9&&&$a_k$:&&&0.29&$\geq$&0.12&$\approx$&0.08&$\geq$&-0.18&$\geq$&-0.4&$\geq$&-0.6&$\geq$&-0.84\
$g$:&&&&3.62&&2.34&&1.37&&0.16&&0.16&&&&$g$:&&&&0.77&&0.22&&1.09&&0.84&&0.55&&0.72&\
%:&45&&5&&1&&3&&6&&1&&40&&&%:&88&&2&&2&&0&&1&&0&&0&&7\
Q30: &E&$>$&B&$\geq$&F&$\geq$&C&$=$&A&$\geq$&D&&&&&Q41: &F&$>$&E&$>$&B&$>$&D&$=$&A&$\geq$&C&$>$&G&$\geq$&H\
$a_k$:&&&0.25&$\geq$&0.1&$\geq$&-0.1&$\approx$&-0.22&$\geq$&-0.38&&&&&$a_k$:&&&0.46&$>$&0.06&$>$&-0.28&$\approx$&-0.36&$\geq$&-0.48&$>$&-0.87&$\geq$&-0.99\
$g$:&&&&0.58&&0.74&&0.44&&0.62&&&&&&$g$:&&&&2.05&&1.69&&0.39&&0.55&&1.98&&0.56&\
%:&42&&1&&1&&1&&54&&1&&&&&%:&72&&1&&7&&1&&1&&5&&7&&5\
Q31: &E&$>$&D&$=$&F&$=$&C&$\geq$&A&$\geq$&B&&&&&Q42: &B&$>$&E&$>$&G&$=$&F&$=$&D&$=$&H&$\geq$&A&$=$&C\
$a_k$:&&&-0.01&$\approx$&-0.04&$\approx$&-0.14&$\geq$&-0.39&$\geq$&-0.61&&&&&$a_k$:&&&0.06&$>$&-0.37&$\approx$&-0.38&$\approx$&-0.43&$\approx$&-0.44&$\geq$&-0.61&$\approx$&-0.69\
$g$:&&&&0.09&&0.41&&0.95&&0.85&&&&&&$g$:&&&&1.9&&0.02&&0.26&&0.06&&0.79&&0.42&\
%:&47&&1&&31&&1&&5&&14&&&&&%:&78&&1&&0&&1&&2&&7&&2&&7\
Q32: &E&$>$&F&$=$&A&$=$&D&$=$&B&$\geq$&C&&&&&Q43: &D&$>$&B&$\geq$&G&$=$&C&$=$&F&$=$&A&$=$&H&$>$&E\
$a_k$:&&&-0.06&$\approx$&-0.16&$\approx$&-0.27&$\approx$&-0.37&$\geq$&-0.5&&&&&$a_k$:&&&0.03&$\geq$&-0.09&$\approx$&-0.12&$\approx$&-0.21&$\approx$&-0.26&$\approx$&-0.27&$>$&-0.77\
$g$:&&&&0.41&&0.44&&0.39&&0.55&&&&&&$g$:&&&&0.54&&0.14&&0.41&&0.19&&0.06&&2.1&\
%:&44&&7&&5&&2&&39&&2&&&&&%:&89&&1&&0&&1&&0&&4&&1&&1\
Q33: &E&$>$&D&$\geq$&F&$\geq$&B&$\geq$&C&$\geq$&A&&&&&Q44: &B&$>$&D&$=$&C&$\geq$&A&&&&&&&&\
$a_k$:&&&0.2&$\geq$&0.08&$\geq$&-0.14&$\geq$&-0.29&$\geq$&-0.45&&&&&$a_k$:&&&-0.03&$\approx$&-0.04&$\geq$&-0.25&&&&&&&&\
$g$:&&&&0.56&&1.03&&0.73&&0.7&&&&&&$g$:&&&&0.06&&0.81&&&&&&&&&\
%:&93&&1&&1&&1&&2&&2&&&&&%:&44&&2&&4&&49&&&&&&&&\
Q34: &E&$>$&F&$=$&A&$=$&D&$=$&C&$=$&B&&&&&Q45: &B&$>$&C&$=$&A&$=$&D&&&&&&&&\
$a_k$:&&&-0.13&$\approx$&-0.14&$\approx$&-0.23&$\approx$&-0.24&$\approx$&-0.35&&&&&$a_k$:&&&-0.17&$\approx$&-0.21&$\approx$&-0.24&&&&&&&&\
$g$:&&&&0.01&&0.37&&0.06&&0.42&&&&&&$g$:&&&&0.14&&0.11&&&&&&&&&\
%:&41&&1&&3&&2&&1&&50&&&&&%:&54&&6&&37&&3&&&&&&&&\
Q35: &A&$>$&E&$>$&B&$=$&C&$\geq$&D&&&&&&&Q46: &A&$>$&B&$=$&D&$\geq$&C&&&&&&&&\
$a_k$:&&&0.77&$>$&-0.13&$\approx$&-0.25&$\geq$&-0.57&&&&&&&$a_k$:&&&0.04&$\approx$&-0.02&$\geq$&-0.3&&&&&&&&\
$g$:&&&&3.6&&0.47&&1.27&&&&&&&&$g$:&&&&0.24&&1.11&&&&&&&&&\
%:&56&&1&&38&&2&&2&&&&&&&%:&44&&16&&17&&22&&&&&&&&\
Q36: &A&$>$&E&$>$&C&$\geq$&B&$>$&D&&&&&&&Q47: &A&$>$&D&$=$&B&$\geq$&C&&&&&&&&\
$a_k$:&&&0.51&$>$&0.16&$\geq$&-0.01&$>$&-0.54&&&&&&&$a_k$:&&&0.01&$\approx$&-0.01&$\geq$&-0.26&&&&&&&&\
$g$:&&&&1.51&&0.72&&2.32&&&&&&&&$g$:&&&&0.08&&1.03&&&&&&&&&\
%:&23&&1&&68&&3&&3&&&&&&&%:&46&&13&&22&&19&&&&&&&&\
Q37: &A&$>$&E&$>$&B&$>$&C&$\geq$&D&&&&&&&\
$a_k$:&&&0.29&$>$&-0.09&$>$&-0.56&$\geq$&-0.78&&&&&&&\
$g$:&&&&1.84&&2.27&&1.07&&&&&&&&\
%:&65&&4&&3&&22&&4&&&&&&&\
\[rankTable2747\]
Tables \[rankTable126\] and \[rankTable2747\] show the IRT ranking results for each question on the FMCE, including the $a_k$ value for each response, the value of Hedges’ $g$ for each nearest-neighbor comparison, and the percentage of the data set that chose each response. Consider the ranking shown for question 19 as it relates to Fig. \[Q19ak\]. The $a_C$ and $a_G$ parameters are nearly identical (when rounded to two decimal places), and the effect size between their distributions is negligibly small ($g=0.01$). The effect size between D and F is also quite small ($g=0.12$), suggesting that choosing either of these two responses indicates a similar level of understanding, and the effect size between G and H is large, but not above our threshold for different responses ($g=1.19$) [^3]. All other effect sizes for question 19 are very large, indicating considerably different values of $a_k$ that correspond to different levels of understanding.
Question 19 on the FMCE presents students with a situation in which a toy car “moves toward the left and is speeding up at a steady rate (constant acceleration)” and asks them to choose an appropriate graph of force vs. time. The correct response to question 19 is B: a graph with a constant negative value (zero slope). According to these results, the best incorrect answer is A: a graph with a constant positive value (zero slope). The second-best incorrect response is E: a graph with a constant zero value (zero slope). All other responses are graphs with nonzero slope. This suggests that realizing that a constant acceleration indicates a constant force is indicative of an above-average understanding of basic Newtonian mechanics. This result alone may not be revolutionary to anyone who has taught introductory mechanics, but the implication that claiming that zero force is required to make an object speed up is a better answer than selecting a graph showing a changing force may be more surprising. Response E is chosen by fewer than 1% of the data set, but these students seem to otherwise have a fairly strong understanding of Newton’s laws as measured by the FMCE.
[l\*[15]{}[c]{}@l\*[15]{}[c]{}]{} Q1: &B&$>$&A&&&&&&&&&&&&&Q25: &B&$>$&A&$>$&D&$=$&F&$>$&E&$\geq$&G&&&&\
Q2: &D&$>$&C&$>$&A&$=$&B&&&&&&&&&Q26: &C&$>$&G&$>$&B&$>$&E&$>$&A&&&&&&\
Q3: &F&$>$&E&$\geq$&G&$>$&C&$=$&D&&&&&&&Q27: &A&$>$&B&$\geq$&E&$\geq$&C&$>$&G&$>$&F&&&&\
Q4: &F&$>$&E&$>$&G&&&&&&&&&&&Q28: &A&$>$&E&$\geq$&B&$>$&D&&&&&&&&\
Q5: &D&$>$&C&$>$&E&$>$&A&$\geq$&B&$\geq$&F&&&&&Q29: &A&$>$&E&$>$&C&$>$&F&$=$&B&&&&&&\
Q6: &B&$>$&F&$>$&A&$=$&E&$\geq$&G&$>$&C&$>$&D&&&Q30: &E&$>$&B&$\geq$&F&$\geq$&C&$=$&A&$\geq$&D&&&&\
Q7: &B&$>$&F&$=$&C&$=$&A&$>$&G&$>$&D&$=$&E&&&Q31: &E&$>$&D&$=$&F&$=$&C&$\geq$&A&$\geq$&B&&&&\
Q8: &A&$>$&D&$>$&B&$=$&E&$=$&C&$>$&G&$>$&F&&&Q32: &E&$>$&F&$=$&A&$=$&D&$=$&B&$\geq$&C&&&&\
Q9: &A&$>$&B&$>$&D&$>$&E&&&&&&&&&Q33: &E&$>$&D&$\geq$&F&$>$&C&$\geq$&A&&&&&&\
Q10: &A&$>$&D&$>$&B&$=$&C&&&&&&&&&Q34: &E&$>$&F&$=$&A&$=$&D&$=$&C&$=$&B&&&&\
Q11: &A&$>$&B&$\geq$&C&$>$&E&$>$&G&$>$&F&&&&&Q35: &A&$>$&E&$>$&B&$=$&C&$\geq$&D&&&&&&\
Q12: &A&$>$&B&$>$&D&&&&&&&&&&&Q36: &A&$>$&E&$>$&C&$\geq$&B&$>$&D&&&&&&\
Q13: &A&$>$&C&$\geq$&B&$>$&F&&&&&&&&&Q37: &A&$>$&E&$>$&B&$>$&C&$\geq$&D&&&&&&\
Q14: &E&$>$&H&$>$&A&$>$&C&$=$&B&&&&&&&Q38: &A&$>$&E&$=$&B&$\geq$&C&$\geq$&D&&&&&&\
Q15: &E&$>$&A&$\geq$&B&&&&&&&&&&&Q39: &E&$>$&A&$\geq$&C&$=$&D&$>$&B&&&&&&\
Q16: &A&$>$&C&&&&&&&&&&&&&Q40: &A&$>$&E&$\geq$&B&$>$&C&$>$&D&&&&&&\
Q17: &E&$>$&B&$>$&D&$\geq$&G&$=$&C&$\geq$&H&$\geq$&A&&&Q41: &F&$>$&E&$>$&B&$>$&D&$\geq$&C&$>$&G&$\geq$&H&&\
Q18: &B&$>$&A&$>$&D&$=$&G&$=$&H&$>$&C&$>$&F&&&Q42: &B&$>$&E&$>$&D&$=$&H&$\geq$&A&$=$&C&&&&\
Q19: &B&$>$&A&$>$&D&$>$&C&$=$&G&$\geq$&H&&&&&Q43: &D&$>$&B&$\geq$&C&$\geq$&A&$=$&H&&&&&&\
Q20: &G&$>$&H&$>$&F&&&&&&&&&&&Q44: &B&$>$&D&$=$&C&$\geq$&A&&&&&&&&\
Q21: &E&$>$&B&$>$&A&$>$&D&$=$&H&$=$&F&$\geq$&C&$>$&G&Q45: &B&$>$&C&$=$&A&$=$&D&&&&&&&&\
Q22: &A&$>$&E&&&&&&&&&&&&&Q46: &A&$>$&B&$=$&D&$\geq$&C&&&&&&&&\
Q23: &B&$>$&A&$=$&C&$>$&F&$=$&G&$>$&D&&&&&Q47: &A&$>$&D&$=$&B&$\geq$&C&&&&&&&&\
Q24: &C&$>$&F&$>$&B&$>$&G&$>$&A&&&&&&&\
\[rankTableFiltered\]
Table \[rankTableFiltered\] shows IRT rankings that have been filtered to only include responses given by at least 1.00% of the population. For some questions (such as 1, 16, and 22) the difference is quite stark, with only the correct and one incorrect answer choice remaining. For many of the questions (such as 2, 14, and 26) the rankings remain the same, but many of the responses that are seen as equivalent to others (or ambiguously ranked) have been eliminated. Moreover, a smaller fraction of the rankings in Table \[rankTableFiltered\] are “$\geq$” as compared to Tables \[rankTable126\] and \[rankTable2747\] (29% vs. 35%), and a greater fraction of rankings are “$>$” (56% vs. 36%). This suggests that many of the ambiguities in rankings may be attributed to the relatively low probability of choosing those responses at all levels of physics understanding, which could result in relatively broad distributions of parameter values generated by randomly selecting subsets of data.
Relating Response Rankings to IRT Plots {#sec:rankplot}
=======================================
We can use plots of the IRT curves to better understand these rankings. Figure \[nrmPlots\] shows the 2PL-NRM curves for every response to several question. (The 2PL-NRM IRT plots for all other questions are included in the appendix.) In the filtered ranking, Question 1 only includes two dominant responses. This is consistent with Fig. \[nrmPlots\](a) in which responses A and B dominate at all values of $\theta$ (overall understanding), and all other responses have near zero probability of being chosen. Figure \[nrmPlots\](c) shows that Q14 is a bit more interesting: there are still two dominant responses (the correct E and a single incorrect A), but less-common incorrect responses are chosen differently by students with different $\theta$ values. Students who choose response H are likely to have an above-average understanding ($\theta>0$) with the H curve having a roughly symmetric probability distribution centered around $\theta=b\approx0.75$, while response C is mostly chosen by students with below-average understanding and is more and more likely with lower values of $\theta$. This is consistent with the ranking in Table \[rankTableFiltered\] with H being a better response than the most common A, and C being a worse response. We can also see in Fig. \[nrmPlots\](c) that responses B and C on Q14 have a similar shape, with the highest probability of choosing each being at the low end of the $\theta$-axis; this is consistent with these responses being considered equivalent in Table \[rankTableFiltered\] even though the value of the probability is quite different for each. It is also important to notice that the curves for the correct responses (B for Q1 and E for Q14) are identical to the 2PL curves for the questions shown in Fig. \[2plPlots\].
[ccc]{} & &\
(a) Question 1 & (b) Question 8 & (c) Question 14\
& &\
(d) Question 18 & (e) Question 19 & (f) Question 47\
Question 19 in Fig. \[nrmPlots\](e) is very similar to Q14 in that there is a single most-common-incorrect response (D), a better response (A) that has a relatively narrow symmetric probability distribution centered around $\theta\approx b>0$, and some worse responses (C, G, H) that have their highest probabilities at the low end of the $\theta$-axis. The major differences between the plots for Q19 and Q14 are that the most-common-incorrect response has a much broader range of values in the $\theta<0$ regime for Q19. In fact, response D is approximately equally probable to responses C, G, and H around $\theta\approx-3$. Moreover, the probability curves for the equivalent responses C and G are basically on top of each other for the entire range of $\theta$. This is strong evidence that students choose these responses in roughly the same proportions, and it may indicate that students are choosing them for the same reasons. Response H also has similar probabilities to C and G, but has a distinctly more negative slope than either, indicating a greater likelihood of being chosen by students with lower $\theta$ values.
The plot for question 18 in Fig. \[nrmPlots\](d) is even more complex than that of Q14, but there are several similarities between them. Once again there is a single dominant incorrect response (H), but it is less dominant than the most common response to Q1 or Q14. This is largely due to G and D being equivalent to H (according to Tables \[rankTable126\] and \[rankTableFiltered\]), and all three sharing a similar shape in which the probability is relatively uniform for $\theta<0$ (but has a slight peak around $-1<\theta<-0.5$) and decreases to zero for $0<\theta< 2$. Responses D and G each have probabilities between about 0.1 and 0.15 for $\theta<0$, which accounts for the probability of the most common response H never rising above 0.75. Also in Fig. \[nrmPlots\](d) we can see a small bump in probability for the better-than-common response A around $\theta\approx0.75$, and the worst responses (C and F) are most common at the lowest values of $\theta$. Questions 14, 18, and 19 all indicate that students with above-average understanding ($\theta>0$) are more likely to choose both the correct response and the highest-ranked incorrect response than students with $\theta<0$.
The 2PL-NRM plot for question 47 in Fig. \[nrmPlots\](f) shows an example of a question on the FMCE for which there is not a single dominant incorrect response. Once again we can see the ranking from Table \[rankTableFiltered\] in the shape of the curves: A is correct, D and B have similar shapes with maximum probabilities around $\theta \approx -1.5$, and C with the highest probability at the low end of the $\theta$-axis with a distinctly negative slope. What makes Q47 really interesting is that none of the responses is uniformly zero over a broad range of $\theta$ values. All incorrect responses have probabilities above 0.15 for $-3<\theta<-1$, and the probabilities for two of the incorrect responses (B and C) are roughly equal to the correct response around $\theta\approx-0.5$ (with response D only about 0.10 lower). The relatively high probabilities of all incorrect answer choices on Q47 may contribute to the low value of the discrimination parameter $a$ (see Fig. \[2plPlots\]).
Question 8 (Fig. \[nrmPlots\](b)) shows an interesting example of a case where there are multiple responses (B, C, D, and E) ranked higher than the most common incorrect response G. Only response F is ranked lower than G (with $g\approx16$). On the FMCE, question 8 is the first in a set of three that asks students about the forces on a toy car as it rolls up and down a ramp. In question 8, the car is moving up and slowing down: response A is correct that the net force on the car is constant and down the ramp, response G is that the force is up the ramp and decreasing, while response F is that the force is up the ramp and increasing. Responses B and C both indicate a force down the ramp (increasing and decreasing, respectively), response D indicates zero net force, and response E is that the force is up the ramp and constant. All of these better-than-most-common responses agree with the correct answer in one way (either the direction of the force or the fact that it is constant), while G and F have nothing in common with the correct response.
From a visual perspective, the approximate value of $\theta$ for which the probability curve for a particular response is maximized indicates the ranking for that response [^4]. The curve for the correct answer is always a monotonically increasing function of $\theta$; answers that are better than a most-common incorrect answer (like H on Q14 in Fig. \[nrmPlots\](c) and A on Q18 in Fig. \[nrmPlots\](f)) have relatively narrow probability distributions with peaks around $\theta=b$; the worst response in each rank has its highest value at the low end of the the displayed $\theta$-axis with a distinctly negative slope, indicating that lower values of $\theta$ would yield even higher probabilities of choosing those responses. Plots of IRT probability curves for all FMCE questions are included in Figs. \[fig:2pl-nrm120\]–\[fig:2pl-nrm4147\] below.
Anomalous Results {#sec:anomalous}
=================
We used Rowan University’s high-performance computing cluster to generate the distribution of values for each parameter. In examining the results, we noticed some anomalous values for each parameter as shown by the small bumps on the right side of Fig. \[Q19akAll\]. These bumps represent about 1% of the results. Figure \[Q19akOther\] shows an enlarged version of the smaller distribution [^5].
Comparing Figs. \[Q19ak\] and \[Q19akOther\], we see that the overall shapes and locations of the parameter distributions are similar. The distributions in Fig. \[Q19akOther\] are not as smooth or as broad as those in Fig. \[Q19ak\], but this may be attributed to the sampling method. Possibly the most interesting feature of these distributions is that there is not overlap between the left and right distributions of any parameter (which is not clear in any of the figures). For example, consider the distribution of the parameter for response A in Fig. \[Q19akAll\]: the left distribution for the parameter $a_A$ (also shown as the black curve in Fig. \[Q19ak\]) contains values between $-4.2$ and $-1.9$, while the right distribution (also shown as the black curve in Fig. \[Q19akOther\]) contains values between $+1.3$ and $+2.3$. This leaves a gap of $3.2$ that contains no results.
We found the same anomalous results in the distributions for all parameters in all questions. In 1,003 out of the 105,600 analyses all of the $a_k$ parameters are about 2–5 units larger than in the other 104,597 analyses. The same 1,003 analyses provide the higher results for all parameters, and for all questions the distributions of the higher values are similar in shape and relative location to the distributions for the lower values. Given the similarities in the distributions, and the consistency of the same 1% of analyses providing the differences for all parameters, we feel comfortable using the results from the 104,597 main analyses to make claims about the differences and similarities between the $a_k$ parameters for each response choice as we did in Sections \[sec:rankplot\] and \[sec:irtrank\].
Summary and Future Directions
=============================
The 2PL-NRM nested logit IRT model may be used to rank incorrect responses to FMCE questions by considering the calculated $a_k$ value as a measure of the correlation between choosing a particular response and the value of the $\theta$ parameter representing overall understanding of Newtonian mechanics (as measured by the FMCE). We have shown that using random samples of a large data set can generate distributions of values for each $a_k$ that allow us to determine whether or not these parameters are meaningfully different, and we used Hedges’ $g$ as a statistical measure to quantify these differences. We made particular choices regarding the values of $g$ that we consider to represent parameters that are approximately equal, those that are definitely different, and those that could go either way, and we have reported the value of $g$ for all comparisons to allow the reader to evaluate the validity of our claims or determine a ranking based on other choices of thresholds.
In many cases the responses that could not be determined to be definitely different or approximately the same as others are those that are rarely chosen by students. Future research will focus on clarifying these comparisons and trying to determine a robust ranking for all responses to every question. One way to accomplish this will be to perform similar analyses on other large data sets. Online data collection and analysis tools such as PhysPort’s Data Explorer [@physportde] and the Learning About STEM Student Outcomes (LASSO) [@lasso] make this task much more achievable than it would been even a decade ago.
The results presented here are based on the assumption that students who understand more about physics will answer more questions correctly on the FMCE, and will also select better incorrect responses than students who understand less overall physics. We also ignored whether data were collected before or after instruction: we didn’t care how students obtained their understanding, just that they had some level of understanding when they chose their responses. In future work, we will explore the implications of different assumptions for what makes one response better than another.
One such assumption is that students will choose better responses after instruction than before instruction. This assumption is supported by the fact that (on average) students are more likely to choose correct responses after instruction than before: even low class-averaged gains from traditional instruction tend to be positive [@Hake1998; @Thornton2009; @VonKorff2016]. A method consistent with this assumption would be to look for asymmetric transitions between response choices in matched pre/posttest data using a McNemar-Bowker chi-square test [@McNemar1947; @Bowker1948].
Another assumption that could be made is that students are more likely to choose correct responses after instruction if they chose better incorrect responses before instruction. Using conditional probabilities would allow us to identify a progression of responses to each question, with students moving up the progression being considered getting closer to correct. This is consistent with Thornton’s conceptual dynamics in which students move between various views as they progress toward the correct response [@Thornton1997].
Each of these methods could be used to test the rankings presented above and help clarify the ambiguously ranked responses. These methods and assumptions may also be applied to other research-based assessments to determine a robust ranking of incorrect responses for any multiple-choice question.
We thank Sam McKagan and Ellie Sayre for providing access to data from PhysPort’s Data Explorer and all of the instructors who were willing to share their students’ FMCE responses. We also thank Kerry Gray, Nicholas Wright, Ian Griffin, and Ryan Moyer for their previous contributions as members of the research team. This project was supported by the National Science Foundation through grant DUE-1836470.
Additional 2PL-NRM Plots
========================
Figures \[fig:2pl-nrm120\]–\[fig:2pl-nrm4147\] are included to allow the reader to see how the rankings in Tables \[rankTable126\] and \[rankTable2747\] relate to the probability of choosing each response, given the value of a student’s $\theta$ parameter. Each curve is labeled near the maximum value (with slight adjustments to avoid overlapping labels and curves), so the horizontal location of the label provides an approximate ranking of the responses (right is better, left is worse). Ambiguously ranked responses often show up as lines near zero probability for all values of $\theta$.
-- -- -- --
-- -- -- --
-- -- -- --
-- -- -- --
-- -- -- --
-- -- -- --
[^1]: Analyses of data from School 1–2 was published in Ref. [@Smith2014].
[^2]: We chose 7,300 response sets to ensure we had at least 10 times as many response sets as parameters, as recommended in Ref. [@deAyala2008]. Samples were created without replacement.
[^3]: The effect size between C and H is also in the ambiguous range: $g=1.20$.
[^4]: The label for each curve is placed as close to the peak as possible without overlapping with either another label or another curve.
[^5]: The vertical axis in Figs. \[Q19ak\], \[Q19akAll\], and \[Q19akOther\] is probability density, which is normalized based on the visible distribution; therefore, the values on the vertical axis should not be compared between plots.
|
---
abstract: 'This paper gives a partial description of the homotopy type of ${{\mathcal{K}}}$, the space of long knots in ${{\Bbb R}}^3$. The primary result is the construction of a homotopy equivalence ${{\mathcal{K}}}\simeq {{\mathcal{C}}}_2({{\mathcal{P}}}\sqcup \{*\})$ where ${{\mathcal{C}}}_2({{\mathcal{P}}}\sqcup \{*\})$ is the free little $2$-cubes object on the pointed space ${{\mathcal{P}}}\sqcup \{*\}$, where ${{\mathcal{P}}}\subset {{\mathcal{K}}}$ is the subspace of prime knots, and $*$ is a disjoint base-point. In proving the freeness result, a close correspondence is discovered between the Jaco-Shalen-Johannson decomposition of knot complements and the little cubes action on ${{\mathcal{K}}}$. Beyond studying long knots in ${{\Bbb R}}^3$ we show that for any compact manifold $M$ the space of embeddings of ${{\Bbb R}}^n \times M$ in ${{\Bbb R}}^n \times M$ with support in ${{\bf{I}}}^n \times M$ admits an action of the operad of little $(n+1)$-cubes. If $M=D^k$ this embedding space is the space of framed long $n$-knots in ${{\Bbb R}}^{n+k}$, and the action of the little cubes operad is an enrichment of the monoid structure given by the connected-sum operation.'
address:
- 'IHÉS, Le Bois-Marie, 35, route de Chartres, F-91440, Bures-sur-Yvette, France'
- 'Mathematics and Statistics, University of Victoria, PO BOX 3045 STN CSC, Victoria, B.C., Canada V8W 3P4'
author:
- Ryan Budney
title: Little cubes and long knots
---
spaces of knots; little cubes; operad; embedding; diffeomorphism
Introduction {#INTRODUCTION}
============
A theorem of Morlet’s [@Mor] states that the topological group ${{\mathrm{Diff}}}(D^n)$ of boundary-fixing, smooth diffeomorphisms of the unit $n$-dimensional closed disc is homotopy equivalent to the $(n+1)$-fold loop space $\Omega^{n+1}\left( PL_n/O_n \right)$. Morlet’s method did not involve the techniques invented by Boardman, Vogt and May [@BV; @May1] for recognizing iterated loop spaces, little cubes actions. This paper begins by defining little cubes operad actions on spaces of diffeomorphisms and embeddings, thus making the loop space structure explicit. In Theorem \[littlecthm\] it’s proved the embedding space $${{\mathrm{EC}({k,M})}} = \{ f \in {{{\mathrm{Emb}}({{\Bbb R}}^{k} \times {M},{{\Bbb R}}^{k} \times {M})}}, supp(f) \subset {{\bf{I}}}^k \times M\}$$ admits an action of the operad of little $(k+1)$-cubes. Here the support of $f$, $supp(f) = \overline{\{ x \in {{\Bbb R}}^k \times M : f(x)\neq x \}}$ and ${{\bf{I}}}= [-1,1]$.
The case $k=1$ and $M=D^2$ is of primary interest in this paper as ${{\mathrm{EC}({1,D^2})}}$ is the space of framed long knots in ${{\Bbb R}}^3$. In section \[longk\] the structure of ${{\mathrm{EC}({1,D^2})}}$ as a little $2$-cubes object is determined. It is shown in Proposition \[modpro\] that the little $2$-cubes action on ${{\mathrm{EC}({1,D^2})}}$ restricts to a subspace ${{\mathcal{\hat K}}}$ which is homotopy equivalent to ${{\mathcal{K}}}$, the space of long knots in ${{\Bbb R}}^3$. Moreover it is shown that as little $2$-cubes objects, ${{\mathrm{EC}({1,D^2})}} \simeq {{\mathcal{\hat K}}}\times {{\Bbb Z}}$. In Theorem \[freeness\] it is shown that ${{\mathcal{\hat K}}}$ is a free little $2$-cubes object on the subspace of prime long knots ${{\mathcal{\hat K}}}\simeq {{\mathcal{C}}}_2 \left( {{\mathcal{P}}}\sqcup \{*\} \right)$. Theorems \[freeness\] and \[littlecthm\] are the main theorems of this paper.
\[fig1\]
\[tl\]\[tl\]\[1.2\]\[0\][$f \in {{\mathrm{EC}({1,D^2})}}$]{} \[tl\]\[tl\]\[0.8\]\[0\][$1$]{} \[tl\]\[tl\]\[0.8\]\[0\][$-1$]{} $$\includegraphics[width=10cm]{modelspace.eps}$$
Figure 1
The homotopy-theoretic content of Theorem \[freeness\] is that ${{\mathcal{K}}}\simeq {{\mathcal{C}}}_2 \left( {{\mathcal{P}}}\sqcup \{*\} \right) \simeq \sqcup_{n=0}^{\infty} ({{\mathcal{C}}}_2(n) \times {{\mathcal{P}}}^n)/S_n$ where ${{\mathcal{C}}}_2(n)$ the space of $n$ little $2$-cubes. ${{\mathcal{C}}}_2(n)$ as an $S_n$-space has the same homotopy type as the configuration space of $n$ labeled points in the plane $C_n ({{\Bbb R}}^2)$. ${{\mathcal{P}}}\subset {{\mathcal{K}}}$ is the space of prime long knots, thus it is the union of all the components of ${{\mathcal{K}}}$ which consist of prime knots. $S_n$ is the symmetric group on $n$ elements, acting diagonally on the product. One interpretation of Theorem \[freeness\] is that it refines Schubert’s Theorem [@Sch] which states that $\pi_0 {{\mathcal{K}}}$ is a free commutative monoid with respect to the connected-sum operation $\pi_0 {{\mathcal{K}}}\simeq \bigoplus_{\infty} {{\Bbb N}}$. The refinement is a space-level theorem about ${{\mathcal{K}}}$ where the cubes action on ${{\mathcal{K}}}$ replaces the connected-sum operation on $\pi_0 {{\mathcal{K}}}$. The novelty of this interpretation is that the connected-sum is not a unique decomposition in ${{\mathcal{K}}}$, as it is parametrized by a configuration space. Perhaps the most interesting aspect of Theorem \[freeness\] is that it states that the homotopy type of ${{\mathcal{K}}}$ is a functor in the homotopy type of the space of prime long knots ${{\mathcal{P}}}$. In Section \[endsec\] we mention how the results in this paper combine with results of Hatcher [@Hatcher4] and other results of the author’s [@topknot] to determine the full homotopy-type of ${{\mathcal{K}}}$.
There are elementary consequences of the little cubes actions defined in Section \[littlec\] that are of interest. In Corollary \[cor1\] we mention how the cubes action on ${{\mathrm{EC}({n,\{*\}})}}$ endows ${{\mathrm{Diff}}}(D^n) \simeq {{\mathrm{EC}({n,\{*\}})}}$ with the structure of an $(n+1)$-fold loop space. This corollary is part of Morlet’s ‘Comparison’ Theorem [@Mor]. To my knowledge, it is the first explicit demonstration of the $(n+1)$-cubes acting on groups homotopy equivalent to ${{\mathrm{Diff}}}(D^n)$. In Corollary \[cor2\] the loop space recognition theorem together with the cubes action on ${{\mathrm{EC}({k,D^m})}}$ and some elementary differential topology tell us that ${{\mathrm{EC}({k,D^m})}}$ is a $(k+1)$-fold loop space provided $m>2$. This last result, to the best of my knowledge, is new. Since these results appeared, Dev Sinha [@Dev2] has constructed an action of the operad of $2$-cubes on the homotopy fiber of the map ${{\mathrm{Emb}}}({{\Bbb R}},{{\Bbb R}}^n) \to {{\mathrm{Imm}}}({{\Bbb R}},{{\Bbb R}}^n)$ for $n \geq 4$. Sinha’s result has recently been extended by Paolo Salvatore [@salvatore], to construct actions of the operad of $2$-cubes on both the full embedding space ${{\mathrm{Emb}}}({{\Bbb R}},{{\Bbb R}}^n)$ and the ‘framed’ long knot space ${{\mathrm{EC}({1,D^{n-1}})}}$ for $n \geq 4$, thus allowing for a comparison with the cubes actions constructed in this paper. Both the methods of Salvatore and Sinha use the Goodwillie Calculus of Embeddings [@Good; @Dev; @DevKevin; @Volic; @Bud] together with the techniques of McClure and Smith [@McClure].
The existence of cubes actions on the space of long knots in ${{\Bbb R}}^3$ was conjectured by Turchin [@Tour], who discovered a bracket on the $E^2$-page of the Vassiliev spectral sequence for the homology of ${{\mathcal{K}}}$ [@Vass]. Given the existence of a little $2$-cubes action on ${{\mathrm{EC}({1,D^k})}}$ one might expect a co-bracket in the Chern-Simons approach to the de Rham theory of spaces of knots [@Bott; @Kont; @Kohno; @Cat1] but at present only a co-multiplication is known [@Cat2]. This paper could also be viewed as an extension of the work of Gramain [@GramainPi1] who discovered subgroups of the fundamental group of certain components of ${{\mathcal{K}}}$ which are isomorphic to pure braid groups.
Actions of operads of little cubes on embedding spaces {#littlec}
======================================================
In this section we define actions of operads of little cubes on various embedding spaces. An invention of Peter May’s, operads are designed to parametrize the multiplicity of ways in which objects can be ‘multiplied’. In the case of iterated loop spaces, the relevant operad is the operad of little $n$-cubes, essentially defined by Boardman and Vogt [@BV] as ‘categories of operators in standard form,’ and later recast into the language of operads by May [@May1].
The space of long knots in ${{\Bbb R}}^n$ is defined to be ${{\mathrm{Emb}}}({{\Bbb R}},{{\Bbb R}}^n) = \{f : {{\Bbb R}}\to {{\Bbb R}}^n : \text{ where } f \text{ is a } C^\infty\text{-smooth embedding and } f(t)=(t,0,0,\cdots,0)$ for $|t|>1 \}$. We give ${{\mathrm{Emb}}}({{\Bbb R}},{{\Bbb R}}^n)$ the weak $C^\infty$ function space topology (see Hirsch [@Hirsch] §2.1). ${{\mathrm{Emb}}}({{\Bbb R}},{{\Bbb R}}^n)$ is considered a pointed space with base-point given by ${{\mathcal{I}}}: {{\Bbb R}}\to {{\Bbb R}}^n$ where ${{\mathcal{I}}}(t)=(t,0,0,\cdots,0)$. Any knot isotopic to ${{\mathcal{I}}}$ is called an unknot. We reserve the notation ${{\mathcal{K}}}$ for the space of long knots in ${{\Bbb R}}^3$, ie: ${{\mathcal{K}}}= {{\mathrm{Emb}}}({{\Bbb R}}, {{\Bbb R}}^3)$.
The connected-sum operation $\#$ gives a homotopy-associative pairing $$\# : {{\mathrm{Emb}}}({{\Bbb R}},{{\Bbb R}}^n) \times {{\mathrm{Emb}}}({{\Bbb R}},{{\Bbb R}}^n) \to {{\mathrm{Emb}}}({{\Bbb R}},{{\Bbb R}}^n)$$ As shown in Schubert’s work [@Sch], this pairing turns $\pi_0 {{\mathcal{K}}}$ (the path-components of ${{\mathcal{K}}}$) into a free commutative monoid with a countable number of generators (corresponding to the isotopy classes of prime long knots). Schubert’s argument that $\pi_0 {{\mathcal{K}}}$ is commutative comes from the idea of ‘pulling one knot through another,’ illustrated in Figure 2.
\[fig2\]
\[tl\]\[tl\]\[1\]\[0\][$f \# g$]{} \[tl\]\[tl\]\[1\]\[0\][$g \# f$]{} \[tl\]\[tl\]\[1\]\[0\][$f$]{} \[tl\]\[tl\]\[1\]\[0\][$g$]{} $$\includegraphics[width=12cm]{csloop.eps}$$
Figure 2
Figure 2 suggests the existence of a map $\iota: S^1 \times {{\mathcal{K}}}^2 \to {{\mathcal{K}}}$ such that $\iota(1,f,g) = f \# g$ and $\iota(-1,f,g)=g \# f$. Such a map would exist if the connected sum operation on ${{\mathcal{K}}}$ was induced by a $2$-cubes action. Turchin’s conjecture states that such a $2$-cubes action exists.
When first constructing the little $2$-cubes action on the space of long knots, it was observed that it is necessary to ‘fatten’ the space ${{\mathcal{K}}}$ into a homotopy equivalent space ${{\mathcal{\hat K}}}$ where the little cubes act. The problem with directly defining a little cubes action on ${{\mathcal{K}}}$ is that little cubes actions are very rigid. Certain diagrams must commute [@May1; @MSS]. A homotopy commutative diagram is not enough in the sense that one can not in general promote such diagrams to a genuine cubes action. All known candidates for little cubes actions on ${{\mathcal{K}}}$ that one might naively put forward have, at best, homotopy-commutative diagrams. Definition \[firstbigdef\] provides us a ‘knot space’ ${{\mathrm{EC}({k,M})}}$ where the connect-sum operation is given by composition of functions. The benefit of this construction is that connect-sum becomes a strictly associative function, allowing us to satisfy the rigid axioms of a cubes action.
\[fig3\]
$f \in {{\mathrm{EC}({1,D^2})}}$ and $L \in {{\mathrm{CAut}}}_1$
\[tl\]\[tl\]\[0.8\]\[0\][$-1$]{} \[tl\]\[tl\]\[0.8\]\[0\][$1$]{} \[tl\]\[tl\]\[1\]\[0\][$\mu$]{} \[tl\]\[tl\]\[1\]\[0\][$f$]{} \[tl\]\[tl\]\[1\]\[0\][$L$]{} \[tl\]\[tl\]\[1\]\[0\][$L.f$]{} \[tl\]\[tl\]\[1\]\[0\][$,$]{} $$\includegraphics[width=12cm]{caut_action.eps}$$
Figure 3
\[firstbigdef\]
- $D^n := \{ x \in {{\Bbb R}}^n : |x|\leq 1 \}$, where $\partial D^n = S^{n-1}$.
- A (single) little $n$-cube is a function $L : {{\bf{I}}}^n \to {{\bf{I}}}^n$ such that $L=l_1\times \cdots \times l_n$ where each $l_i : {{\bf{I}}}\to {{\bf{I}}}$ is affine-linear and increasing ie: $l_i(t)=a_i t + b_i$ for some $a_i >0 $ and $b_i \in {{\Bbb R}}$.
- Let ${{\mathrm{CAut}}}_n$ denote the monoid of affine-linear automorphisms of ${{\Bbb R}}^n$ of the form $L=l_1 \times \cdots \times l_n$ where $l_i$ is affine-linear and increasing for all $i \in \{1,2,\cdots,n\}$.
- Given a little $n$-cube $L$, we sometimes abuse notation and consider $L \in {{\mathrm{CAut}}}_n$ by taking the unique affine-linear extension of $L$ to ${{\Bbb R}}^n$.
- The space of $j$ little $k$-cubes ${{\mathcal{C}}}_k(j)$ is the space of maps $L : \sqcup_{i=1}^j {{\bf{I}}}^{k} \to {{\bf{I}}}^{k}$ such that the restriction of $L$ to the interior of its domain is an embedding, and the restriction of $L$ to any connected component of its domain is a little $k$-cube. Given $L \in {{\mathcal{C}}}_{k}(j)$, denote the restriction of $L$ to the $i$-th copy of ${{\bf{I}}}^k$ by $L_i$. By convention ${{\mathcal{C}}}_k(0)$ is taken to be a point. This makes the union $\sqcup_{j=0}^\infty {{\mathcal{C}}}_k(j)$ into an operad, called the operad of little $k$-cubes ${{\mathcal{C}}}_k$ [@May1; @MSS].
- Given a compact manifold $M$, let ${{{\mathrm{Emb}}({{\Bbb R}}^{k} \times {M},{{\Bbb R}}^{k} \times {M})}}$ denote the space of $C^\infty$-smooth embeddings of ${{\Bbb R}}^k \times M$ in ${{\Bbb R}}^k \times M$. We do not demand the embeddings to be proper ie: if $f \in {{{\mathrm{Emb}}({{\Bbb R}}^{k} \times {M},{{\Bbb R}}^{k} \times {M})}}$ then the image of the boundary of ${{\Bbb R}}^k \times M$ need not lay in the boundary of ${{\Bbb R}}^k \times M$. We give this space the weak $C^\infty$-topology (See [@Hirsch] §2.1).
- ${{\mathrm{EC}({k,M})}}$ is defined to be the subspace of ${{{\mathrm{Emb}}({{\Bbb R}}^{k} \times {M},{{\Bbb R}}^{k} \times {M})}}$ consisting of embeddings $f : {{\Bbb R}}^k \times M \to {{\Bbb R}}^k \times M$ whose support is contained in ${{\bf{I}}}^k \times M$ ie: they are required to restrict to the identity function outside of ${{\bf{I}}}^k \times M$. We consider ${{\mathrm{EC}({k,M})}}$ to be a based space, with base-point given by the identity function $Id_{{{\Bbb R}}^k \times M}$. Any knot in the path component of $Id_{{{\Bbb R}}^k \times M}$ is typically called an unknot.
We will show that the operad of little $(k+1)$-cubes acts on ${{\mathrm{EC}({k,M})}}$, but first we define an action of the monoid ${{\mathrm{CAut}}}_k$ on ${{{\mathrm{Emb}}({{\Bbb R}}^{k} \times {M},{{\Bbb R}}^{k} \times {M})}}$. $$\mu : {{\mathrm{CAut}}}_k \times {{{\mathrm{Emb}}({{\Bbb R}}^{k} \times {M},{{\Bbb R}}^{k} \times {M})}} \to {{{\mathrm{Emb}}({{\Bbb R}}^{k} \times {M},{{\Bbb R}}^{k} \times {M})}}$$ $$\mu(L,f) = (L \times Id_M)\circ f \circ (L^{-1} \times Id_M)$$ In the above formula, we consider both $L$ and $L^{-1}$ to be elements of ${{\mathrm{CAut}}}_n$. We write the above action as $\mu(L,f)=L.f$ (see Figure 3).
\[contprop\] The two maps $$\mu: {{\mathrm{CAut}}}_k \times {{{\mathrm{Emb}}({{\Bbb R}}^{k} \times {M},{{\Bbb R}}^{k} \times {M})}} \to {{{\mathrm{Emb}}({{\Bbb R}}^{k} \times {M},{{\Bbb R}}^{k} \times {M})}}$$ $$\circ : {{{\mathrm{Emb}}({{\Bbb R}}^{k} \times {M},{{\Bbb R}}^{k} \times {M})}} \times {{{\mathrm{Emb}}({{\Bbb R}}^{k} \times {M},{{\Bbb R}}^{k} \times {M})}} \to {{{\mathrm{Emb}}({{\Bbb R}}^{k} \times {M},{{\Bbb R}}^{k} \times {M})}}$$ are continuous, where $\circ$ is composition.
The continuity of $\circ$ is an elementary consequence of the weak topology. The continuity of $\mu$ follows immediately.
\[littlecdef\]
- Given $j$ little $(k+1)$-cubes, $L=(L_1,\cdots, L_j)\in {{\mathcal{C}}}_{k+1}(j)$ define the $j$-tuple of (non-disjoint) little $k$-cubes $L^\pi = (L_1^\pi,\cdots, L_j^\pi)$ by the rule $L_i^\pi =l_{i,1} \times \cdots \times l_{i,k}$ where $L_i= l_{i,1} \times \cdots \times l_{i,k+1}$. Similarly define $L^t \in {{\bf{I}}}^j$ by $L^t=(L_1^t,\cdots, L_j^t)$ where $L_i^t = l_{i,k+1}(-1)$ (see Figure 4).
\[fig4\]
\[tl\]\[tl\]\[1\]\[0\][$\{0\}^n\times {{\Bbb R}}$]{} \[tl\]\[tl\]\[0.9\]\[0\][$L^t$]{} \[tl\]\[tl\]\[0.9\]\[0\][$L^\pi$]{} \[tl\]\[tl\]\[1\]\[0\][${{\Bbb R}}^n\times\{0\}$]{} \[tl\]\[tl\]\[0.9\]\[0\][$L$]{} $$\includegraphics[width=5cm]{proj.eps}$$
Figure 4
- The action of the operad of little $(k+1)$-cubes on the space ${{\mathrm{EC}({k,M})}}$ is given by the maps $\kappa_j : {{\mathcal{C}}}_{k+1}(j) \times {{\mathrm{EC}({k,M})}}^j \to {{\mathrm{EC}({k,M})}}$ for $j \in \{1,2,\cdots\}$ defined by $$\kappa_j(L_1,\cdots,L_j,f_1,\cdots,f_j) =
L^\pi_{\sigma(1)}.f_{\sigma(1)}\circ L^\pi_{\sigma(2)}.f_{\sigma(2)}\circ\cdots \circ L^\pi_{\sigma(j)}.f_{\sigma(j)}$$ where $\sigma : \{1,\cdots, j\} \to \{1,\cdots,j\}$ is any permutation such that $L^t_{\sigma(1)} \leq L^t_{\sigma(2)} \leq \cdots \leq L^t_{\sigma(j)}$. The map $\kappa_0 : {{\mathcal{C}}}_{k+1}(0) \times {{\mathrm{EC}({k,M})}}^0 \to {{\mathrm{EC}({k,M})}}$ is the inclusion of a point $*$ in ${{\mathrm{EC}({k,M})}}$, defined so that $\kappa_0(*) = Id_{{{\Bbb R}}^k \times M}$ (see Figures 5 and 7).
\[fig5\]
$L^t_1 < L^t_2$ so $\sigma$ is the identity and $\kappa_2(L_1,L_2,f_1,f_2)=L^\pi_1.f_1 \circ L^\pi_2.f_2$.
\[tl\]\[tl\]\[1\]\[0\][$L_1$]{} \[tl\]\[tl\]\[1\]\[0\][$L_2$]{} \[tl\]\[tl\]\[0.7\]\[0\][$1$]{} \[tl\]\[tl\]\[0.7\]\[0\][$-1$]{} \[tl\]\[tl\]\[1\]\[0\][$L_1^t$]{} \[tl\]\[tl\]\[1\]\[0\][$L_2^t$]{} \[tl\]\[tl\]\[1\]\[0\][,]{} \[tl\]\[tl\]\[1\]\[0\][$\kappa_2$]{} $$\includegraphics[width=13cm]{action_2ndeg.eps}$$
Figure 5
\[littlecthm\] For any compact manifold $M$ and any integer $k \geq 0$ the maps $\kappa_j$ for $j \in \{0, 1,2,\cdots\}$ define an action of the operad of little $(k+1)$-cubes on ${{\mathrm{EC}({k,M})}}$.
First we show the map $\kappa_j$ is well-defined. The only ambiguity in the definition is the choice of the permutation $\sigma$. If there is an ambiguity in the choice of $\sigma$ this means that a pair of coordinates $L^t_p$ and $L^t_q$ in $j$-tuple $L^t=(L^t_1,\cdots,L^t_j)$ must be equal. Since $L=(L_1,\cdots, L_j)$ are disjoint cubes, if a pair $L_p$ and $L_q$ have projections $L^t_p=L^t_q$, then $L^\pi_p$ and $L^\pi_q$ are disjoint. Since $supp(L^\pi_p.f_p)=(L^\pi_p\times Id_M)(supp(f_p))$ and $supp(L^\pi_{q}.f_q)=(L^\pi_{q}\times Id_M)(supp(f_q))$, $L^\pi_{p}.f_p$ and $L^\pi_{q}.f_q$ must have disjoint support. So the order of composition of $L^\pi_p.f_p$ and $L^\pi_q.f_q$ is irrelevant. This proves the maps $\kappa_j$ are well-defined.
We prove the continuity of the maps $\kappa_j$. Given a permutation $\sigma$ of the set $\{1, \cdots, j\}$ consider the function $$\kappa_\sigma : {{\mathcal{C}}}_{k+1}(j) \times {{\mathrm{EC}({k,M})}}^j \to {{\mathrm{EC}({k,M})}}$$ defined by $$(L_1,\cdots,L_j,f_1,\cdots,f_j)
\longmapsto L^\pi_{\sigma(1)}.f_{\sigma(1)}\circ \cdots
\circ L^\pi_{\sigma(j)}.f_{\sigma(j)}$$ This function is continuous, since the composition operation and the action of ${{\mathrm{CAut}}}_k$ is continuous by Proposition \[contprop\]. Given a permutation $\sigma$, consider the subspace $W_\sigma$ of ${{\mathcal{C}}}_{k+1}(j) \times {{\mathrm{EC}({k,M})}}^j$ where $L^t_{\sigma(1)} \leq \cdots \leq L^t_{\sigma(j)}$. Notice that our map $\kappa_j$ when restricted to $W_\sigma$ agrees with $\kappa_\sigma$. Thus the map $\kappa_j$ is the union of finitely many continuous functions $\kappa_\sigma$ whose definitions agree where their domains $W_\sigma$ overlap, so $\kappa_j$ is a continuous function by the pasting lemma.
We need to show the maps $\kappa_j$ satisfy the axioms of a little cubes action as described in sections 1 and 4 of [@May1] (or II §1.4 of [@MSS]). There are three conditions that must be satisfied: the identity criterion, symmetry and associativity. The identity criterion is tautological, since if $Id_{{{\bf{I}}}^{k+1}}$ is the identity little $(k+1)$-cube, its projection is the identity cube, which acts trivially on ${{\mathrm{EC}({k,M})}}$. Symmetry is similarly tautological. The associativity condition demands that the diagram in Figure 6 commutes.
\[fig6\] \[tl\]\[tl\]\[0.9\]\[0\][${{\mathcal{C}}}_{k+1}(n) \hskip 2mm \times$]{} \[tl\]\[tl\]\[0.9\]\[0\][${{\mathcal{C}}}_{k+1}(n) \times {{\mathrm{EC}({k,M})}}^n$]{} \[tl\]\[tl\]\[0.9\]\[0\][${{\mathrm{EC}({k,M})}}$]{} \[tl\]\[tl\]\[0.9\]\[0\][${{\mathcal{C}}}_{k+1}(j_1+\cdots+j_n) \times {{\mathrm{EC}({k,M})}}^{j_1+\cdots+j_n}$]{} \[tl\]\[tl\]\[0.9\]\[0\][${{\mathcal{C}}}_{k+1}(j_1) \times {{\mathrm{EC}({k,M})}}^{j_1} \times \cdots \times {{\mathcal{C}}}_{k+1}(j_n) \times {{\mathrm{EC}({k,M})}}^{j_n}$]{} $$\includegraphics[width=12cm]{crampedCD.eps}$$
Figure 6
The commutativity of this diagram follows from the same argument given that shows that the maps are well-defined. If one chases the arrows around the diagram both ways, the two objects that you get in ${{\mathrm{EC}({k,M})}}$ are composites of the same embeddings, perhaps in a different order. Any pair of embeddings that have their order permuted must have disjoint supports, so the change in order of composition is irrelevant.
\[fig7\] $L^t_1 < L^t_3 < L^t_2$ so $\sigma=(23)$ and $\kappa_3(L_1,L_2,L_3,f_1,f_2,f_3)=L^\pi_1.f_1\circ L^\pi_3.f_3 \circ L^\pi_2.f_2$, which explains why we see the figure-8 knot ‘inside’ of the trefoil.
\[tl\]\[tl\]\[1\]\[0\][$L_1$]{} \[tl\]\[tl\]\[0.7\]\[0\][$L_2$]{} \[tl\]\[tl\]\[1\]\[0\][$L_3$]{} \[tl\]\[tl\]\[0.7\]\[0\][$1$]{} \[tl\]\[tl\]\[0.7\]\[0\][$-1$]{} \[tl\]\[tl\]\[1\]\[0\][,]{} \[tl\]\[tl\]\[1\]\[0\][$\kappa_3$]{} \[tl\]\[tl\]\[1\]\[0\][$L_3^t$]{} \[tl\]\[tl\]\[1\]\[0\][$L_2^t$]{} \[tl\]\[tl\]\[1\]\[0\][$L_1^t$]{} \[tl\]\[tl\]\[1\]\[0\][$f_3$]{} \[tl\]\[tl\]\[1\]\[0\][$f_1$]{} \[tl\]\[tl\]\[1\]\[0\][$f_2$]{} \[tl\]\[tl\]\[1\]\[0\][$f_3$]{} $$\includegraphics[width=13cm]{action_eg.eps}$$
Figure 7
\[cor1\] The group of boundary-fixing diffeomorphisms of the compact $n$-dimensional ball, ${{\mathrm{Diff}}}(D^n)$ is homotopy-equivalent to an $(n+1)$-fold loop space.
Peter May’s loop space recognition theorem [@May2] states that a little $(n+1)$-cubes object $X$ is (weakly) homotopy equivalent to an $(n+1)$-fold loop space if and only if the induced monoid structure on $\pi_0 X$ is a group.
Consider the monoid structure on $\pi_0 {{\mathrm{EC}({n,\{*\}})}}$. Let $L=(L_1,L_2) \in {{\mathcal{C}}}_{n+1}(2)$ be two little $(n+1)$-cubes such that $L^\pi=(L^\pi_1,L^\pi_2)=(Id_{{{\bf{I}}}^n},Id_{{{\bf{I}}}^n})$. Suppose $L^t=(L^t_1,L^t_2)$ with $L^t_1 < L^t_2$, then $\kappa_2(L_1,L_2,f_1,f_2)=f_1 \circ f_2$. This means the induced monoid structure on $\pi_0 {{\mathrm{EC}({n,\{*\}})}}$ is given by composition. ${{\mathrm{EC}({n,\{*\}})}}$ is a group under composition since it is the group of diffeomorphisms ${{\Bbb R}}^n$ with support contained in ${{\bf{I}}}^n$. Thus, $\pi_0 {{\mathrm{EC}({n,\{*\}})}}$ is also a group, and so ${{\mathrm{EC}({n,\{*\}})}}$ is weakly homotopy equivalent to an $(n+1)$-fold loop space. Since ${{\mathrm{EC}({n,\{*\}})}}$ has the weak (compact-open) topology (see Hirsch [@Hirsch] §2.1) it satisfies the first axiom of countability and so the topology on ${{\mathrm{EC}({n,\{*\}})}}$ is compactly-generated in the sense of Steenrod [@Steenrod]. Thus by the loop space recognition theorem, the ${{\mathrm{EC}({n,\{*\}})}}$ is homotopy-equivalent to an $(n+1)$-fold loop space.
Provided we show that ${{\mathrm{Diff}}}(D^n) \simeq {{\mathrm{EC}({n,\{*\}})}}$ we are done. Fix a collar neighborhood of $S^{n-1}$. There is a restriction map from ${{\mathrm{Diff}}}(D^n)$ to the space of collar neighborhoods of $S^{n-1}$ in $D^n$. This restriction map is a fibration [@Pal] and the space of collar neighborhoods of $S^{n-1}$ in $D^n$ is contractible (see [@Hirsch] §4.5.3). The above argument is not sufficient, because the fiber of this fibration is not ${{\mathrm{EC}({n,\{*\}})}}$. Replace the smooth collar neighborhood of $S^{n-1}$ in $D^n$ with a manifold-with-corners neighborhood of $S^{n-1}$ which is the complement of an open cube in $D^n$. In this case we get a fibration whose fiber we can identify with ${{\mathrm{EC}({n,*})}}$. The argument that the space of cubical collar neighborhoods is contractible is analogous to the proof in Hirsch’s text (see [@Hirsch] §4.5.3).
May’s recognition theorem applies equally-well to spaces that have actions of the operad of (unframed) little balls [@MSS]. Thus we could have simply adapted Definition \[littlecdef\] to give an action of the space of unframed $(n+1)$-balls directly on the the space ${{\mathrm{Diff}}}(D^n)$ and deduced the result without recourse to the intermediate homotopy-equivalence ${{\mathrm{Diff}}}(D^n) \simeq {{\mathrm{Diff}}}({{\bf{I}}}^n)$.
The above corollary is also a corollary of Morlet’s ‘Comparison Theorem’ [@Mor]. Morlet’s manuscript was not widely distributed. A proof of Morlet’s Theorem can be found in Burghelea and Lashof’s paper [@BL], as well as in Kirby and Siebenmann’s book [@KS]. As Siebenmann points out, the Morlet Comparison Theorem was first observed by Cerf.
\[cor2\] ${{\mathrm{EC}({k,D^n})}}$ is homotopy equivalent to a $(k+1)$-fold loop space, provided $n>2$.
This follows from the loop space recognition theorem [@May2] since we will show that $\pi_0 {{\mathrm{EC}({k,D^n})}}$ is a group. Consider the fibration ${{\mathrm{EC}({k,D^n})}} \to {{\mathrm{Emb}}}({{\Bbb R}}^k,{{\Bbb R}}^{k+n})$ where ${{\mathrm{Emb}}}({{\Bbb R}}^k,{{\Bbb R}}^{k+n})$ is the space $\{ f : {{\Bbb R}}^k \to {{\Bbb R}}^{k+n}$ : $f(t_1,t_2,$ $\cdots,t_k)=(t_1,t_2,\cdots,t_k,0,0,\cdots,0)$ if $|t_i| \geq 1$ for any $i \in \{1,2,\cdots,k\} \}$. Haefliger proved [@Haefliger2] that $\pi_0 {{\mathrm{Emb}}}({{\Bbb R}}^k,{{\Bbb R}}^{k+n})$ is a group provided $n>2$ where the group structure is induced by concatenation, thus $\pi_0 {{\mathrm{EC}({k,D^n})}}$ is a group as it is a monoid which is an extension of two groups (see [@family] for an alternative proof of Haefliger’s theorem).
Our preferred model for ${{\mathcal{K}}}$ will be a subspace ${{\mathcal{\hat K}}}$ of ${{\mathrm{EC}({1,D^2})}}$, which we will relate back to the standard model ${{\mathcal{K}}}$. Given an embedding $f \in {{\mathrm{EC}({1,D^2})}}$, define $\omega(f) \in {{\Bbb Z}}$ to be the linking number of $f_{|{{\Bbb R}}\times \{(0,0)\}}$ with $f_{|{{\Bbb R}}\times \{(0,1)\}}$. One concrete way to define this integer is as the transverse intersection number of the map ${{\Bbb R}}^2 \ni (t_1,t_2) \longmapsto f(t_1,0,1)-f(t_2,0,0) \in
{{\Bbb R}}^3 - \{(0,0,0)\}$ with the ray $\{(0,t,0) : t>0\} \subset {{\Bbb R}}^3 - \{(0,0,0)\}$. $\omega(f)$ is called the framing number of $f$. $\omega : {{\mathrm{EC}({1,D^2})}} \to {{\Bbb Z}}$ is a $2$-cubes equivariant fibration, and the framing number is additive $\omega(f_1\circ f_2)=\omega(f_1)+\omega(f_2)$. We consider ${{\Bbb Z}}$ to be an abelian group, and thus a little $2$-cubes object.
\[fatdef\] ${{\mathcal{\hat K}}}$, the space of ‘fat’ long knots in ${{\Bbb R}}^3$ is defined to be the kernel of $\omega$, ${{\mathcal{\hat K}}}= \omega^{-1}\{0\}$.
\[modpro\] The two spaces ${{\mathcal{\hat K}}}$ and ${{\mathcal{K}}}$ are homotopy equivalent.
Consider the fibration ${{\mathrm{EC}({1,D^2})}} \to {{\mathrm{Emb}}}({{\Bbb R}},{{\Bbb R}}^3)$ given by restriction $f \longmapsto f_{|{{\Bbb R}}\times \{(0,0)\}}$ [@Pal]. Let $X$ denote the fiber of this fibration. By definition, $X$ is the space of tubular neighborhoods of the unknot which are standard outside of ${{\bf{I}}}\times D^2$. By the classification of tubular neighborhoods theorem (see for example [@Hirsch] §4.5.3), $X$ is homotopy equivalent to the space of fibrewise-linear automorphisms of ${{\Bbb R}}\times D^2$ with support in ${{\bf{I}}}\times D^2$, ie: $X \simeq \Omega SO_2 \simeq {{\Bbb Z}}$. Thus $\omega$ defines a splitting of the fibration $X \to {{\mathrm{EC}({1,D^2})}} \to {{\mathrm{Emb}}}({{\Bbb R}},{{\Bbb R}}^3)$, giving the two homotopy equivalences $${{\mathrm{EC}({1,D^2})}} \simeq {{\mathrm{Emb}}}({{\Bbb R}},{{\Bbb R}}^3) \times {{\Bbb Z}}\hskip 10mm \xymatrix@R=5pt{{{\mathcal{\hat K}}}\ar@{}[d]|{{{ \xy
*{\xy (0,-3);(0,-2) **\dir{-} \endxy}
*\cir<3pt>{d^u}
\endxy}}} \ar[r]^-{\simeq} &
{{\mathcal{K}}}\ar@{}[d]|{{{ \xy
*{\xy (0,-3);(0,-2) **\dir{-} \endxy}
*\cir<3pt>{d^u}
\endxy}}} \\
f \ar@{|->}[r] & f_{|{{\Bbb R}}\times \{(0,0)\}} }$$
Combining Proposition \[modpro\] with the proof of Corollary \[cor2\] we get the following observation.
There is an action of the operad of $(k+1)$-cubes on spaces homotopy-equivalent to the ‘long embedding spaces’ ${{\mathrm{Emb}}}({{\Bbb R}}^k,{{\Bbb R}}^{k+n})$ for all $k \in {{\Bbb N}}$ and $n \leq 2$.
As mentioned in the introduction, Salvatore [@salvatore] has removed the bound $n \leq 2$ in the above corollary, provided $k=1$.
The freeness of the $2$-cubes action on ${{\mathcal{\hat K}}}$ {#longk}
==============================================================
The goal of this section is to prove that ${{\mathcal{\hat K}}}\simeq {{\mathcal{C}}}_2({{\mathcal{P}}}\sqcup \{*\})$, where ${{\mathcal{P}}}\subset {{\mathcal{\hat K}}}$ is the subspace of prime knots. ${{\mathcal{P}}}= \{ f \in {{\mathcal{\hat K}}}: f$ is nontrivial and not a connected-sum of 2 or more nontrivial knots$\}$.
If $X$ is a pointed space with base-point $* \in X$ the free little $2$-cubes object on $X$ [@May1] is the space ${{\mathcal{C}}}_2(X) = \left( \left(\sqcup_{n=0}^\infty {{\mathcal{C}}}_2(n) \times X^n\right)\kern-0.3em/S_n\right)\kern-0.3em/\kern-0.5em\sim$. $S_n$ is the symmetric group, acting diagonally on the product in the standard way, and the equivalence relation $\sim$ is generated by the relations $$\left((f_1,\cdots,f_{i-1},f_i,f_{i+1},\cdots,f_n),(x_1,\cdots,x_{i-1},*,x_{i+1},\cdots,x_n)\right)$$ $$\sim ((f_1,\cdots,f_{i-1},f_{i+1},\cdots,f_n),(x_1,\cdots,x_{i-1},x_{i+1},\cdots,x_n))$$ If we give an arbitrary unpointed space $X$ a disjoint base-point $*$, then there is the identity ${{\mathcal{C}}}_2(X \sqcup \{*\})\equiv \sqcup_{n=0}^\infty ({{\mathcal{C}}}_2(n) \times X^n)/S_n$. Thus, we will prove ${{\mathcal{\hat K}}}\simeq \sqcup_{n=0}^\infty {{\mathcal{C}}}_2(n) \times_{S_n} {{\mathcal{P}}}^n$.
\[freeness\] ${{\mathcal{\hat K}}}\simeq {{\mathcal{C}}}_2({{\mathcal{P}}}\sqcup \{*\})$, moreover the map $\sqcup_{n=0}^\infty \kappa_n : \sqcup_{n=0}^\infty {{\mathcal{C}}}_2(n) \times_{S_n} {{\mathcal{\hat K}}}^n \to {{\mathcal{\hat K}}}$ restricts to a homotopy equivalence $$\sqcup_{n=0}^\infty {{\mathcal{C}}}_2(n) \times_{S_n} {{\mathcal{P}}}^n \to {{\mathcal{\hat K}}}$$
To prove Theorem \[freeness\] we first build up a close correspondence between the little cubes action and the satellite decomposition of knots, or to be more precise, the JSJ-decomposition [@JacoShalen] of knot complements (also sometimes also known as the splice decomposition [@EN]) . We then use techniques of Hatcher’s to reduce the proof of Theorem \[freeness\] to a problem about a diagram of mapping class groups of $2$ and $3$-dimensional manifolds.
\[sumdef\]
- Given a long knot $f \in {{\mathcal{\hat K}}}$, we denote the component of ${{\mathcal{\hat K}}}$ containing $f$ by ${{\mathcal{\hat K}}}_f$.
- We say $f$ is a connected-sum of $f_1, \cdots, f_n$ if there exists $\tilde f \in {{\mathcal{\hat K}}}_f$ with $\tilde f = \kappa_n (L_1,L_2,\cdots,L_n,f_1,f_2,\cdots,f_n)$, for some $n$, $(L_1,L_2,\cdots,L_n)\in {{\mathcal{C}}}_2(n)$ and $f_i \in {{\mathcal{\hat K}}}$ for all $i \in \{1,2,\cdots,n\}$. Denote this by $f \sim f_1 \# f_2 \# \cdots \# f_n$ and call the long knots $\{f_i : i \in \{1,2,\cdots,n\}\}$ summands of $f$.
- For any long knot $f \sim f \# Id_{{{\Bbb R}}\times D^2}$. If $f \sim f_1 \# f_2 \# \cdots \# f_n$ we call the connected-sum trivial if $(n-1)$ of the long knots $\{f_1,f_2,\cdots,f_n\}$ are in ${{\mathcal{\hat K}}}_{Id_{{{\Bbb R}}\times D^2}}$. A long knot is prime if is not in the component of the unknot, and if all connected-sum decompositions of it are trivial.
Let $Q_i$ denote the $2$-cube $[-1+\frac{4i-2}{2n+1},-1+\frac{4i}{2n+1}]\times [0,\frac{2}{2n+1}]$. Choose the base-point $*$ for ${{\mathcal{C}}}_2(n)$, $*=(Q_1,Q_2,\cdots,Q_n)$ as in Figure 8.
\[fig8\]
\[tl\]\[tl\]\[1\]\[0\][$-1$]{} \[tl\]\[tl\]\[0.7\]\[0\][$-1+\frac{2}{2n+1}$]{} \[tl\]\[tl\]\[0.7\]\[0\][$-1+\frac{4}{2n+1}$]{} \[tl\]\[tl\]\[0.7\]\[0\][$-1+\frac{6}{2n+1}$]{} \[tl\]\[tl\]\[0.7\]\[0\][$-1+\frac{8}{2n+1}$]{} \[tl\]\[tl\]\[0.7\]\[0\][$-1+\frac{4n-2}{2n+1}$]{} \[tl\]\[tl\]\[0.7\]\[0\][$-1+\frac{4n}{2n+1}$]{} \[tl\]\[tl\]\[1\]\[0\][$1$]{} \[tl\]\[tl\]\[1\]\[0\][$Q_1$]{} \[tl\]\[tl\]\[1\]\[0\][$Q_2$]{} \[tl\]\[tl\]\[1\]\[0\][$Q_n$]{} \[tl\]\[tl\]\[1\]\[0\][$\frac{2}{2n+1}$]{} \[tl\]\[tl\]\[1\]\[0\][$\cdots$]{} \[tl\]\[tl\]\[1\]\[0\][${{\Bbb R}}\times \{0\}$]{} $$\includegraphics[width=13cm]{basec2.eps}$$
Figure 8
Since ${{\mathcal{C}}}_2(n)$ is connected, we can choose the $n$ little $2$-cubes $(L_1,L_2,\cdots,L_n)\in {{\mathcal{C}}}_2(n)$ in Definition \[sumdef\] to be $*$. As in Proposition \[modpro\] we can associate to $f \in {{\mathcal{\hat K}}}$ the long knot $g \in {{\mathrm{Emb}}}({{\Bbb R}},{{\Bbb R}}^3)$ where $g=f_{|{{\Bbb R}}\times \{(0,0)\}}$. Define $B_i = \{ x \in {{\Bbb R}}^3 : |x-(\frac{4i-2n-2}{2n+1},0,0)| \leq \frac{1}{2n+1} \}$ and $S_i = \partial B_i$ (see Figure 9). We will provide an equivalent definition for $f$ to be a connected-sum in terms of $g$ (see Figure 9).
We say $g$ is a connected-sum if $g$ is isotopic to $g' \in {{\mathrm{Emb}}}({{\Bbb R}},{{\Bbb R}}^3)$ such that:
- $supp(g')\subset (\sqcup_{i=1}^n B_i)\cap ({{\Bbb R}}\times \{0\}^2)$
- $img(g') \cap S_i = ({{\Bbb R}}\times \{0\}^2 )\cap S_i$ for all $i \in \{1,2,\cdots,n\}$.
- There exists long knots (the summands of $g$) $g_i \in {{\mathrm{Emb}}}({{\Bbb R}},{{\Bbb R}}^3)$ for $i \in \{1,2,\cdots,n\}$ such that $supp(g_i) \subset B_i \cap ({{\Bbb R}}\times \{0\}^2)$ and $g_{i|B_i \cap ({{\Bbb R}}\times \{0\}^2)} = g'_{|B_i \cap ({{\Bbb R}}\times \{0\}^2)}$
Non-trivial connected-sums and prime knots are defined analogously. A theorem of Schubert [@Sch] states that up to isotopy, every non-trivial $g$ can be written uniquely up to a re-ordering of the terms, as a connected-sum of prime knots $g=g_1 \# \cdots \# g_n$.
We review the Jaco-Shalen-Johannson decomposition of 3-manifolds [@JacoShalen]. This is a standard decomposition of 3-manifolds along spheres and tori, given by the connected-sum decomposition [@Kneser] followed by the torus decomposition of the prime summands [@JacoShalen] (see for example [@Hatcher3] or [@Neumann]). For us, all our $3$-manifolds will be compact, and they are allowed to have a boundary. For a more exhaustive treatment of JSJ-decompositions of knot and link complements in $S^3$, see [@bjsj].
\[fig9\]
\[tl\]\[tl\]\[1\]\[0\][$-1$]{} \[tl\]\[tl\]\[0.7\]\[0\][$-1+\frac{2}{2n+1}$]{} \[tl\]\[tl\]\[0.7\]\[0\][$-1+\frac{4}{2n+1}$]{} \[tl\]\[tl\]\[0.7\]\[0\][$-1+\frac{6}{2n+1}$]{} \[tl\]\[tl\]\[0.7\]\[0\][$-1+\frac{8}{2n+1}$]{} \[tl\]\[tl\]\[0.7\]\[0\][$-1+\frac{4n-2}{2n+1}$]{} \[tl\]\[tl\]\[0.7\]\[0\][$-1+\frac{4n}{2n+1}$]{} \[tl\]\[tl\]\[1\]\[0\][$1$]{} \[tl\]\[tl\]\[1\]\[0\][$B_1$]{} \[tl\]\[tl\]\[1\]\[0\][$B_2$]{} \[tl\]\[tl\]\[1\]\[0\][$B_n$]{} \[tl\]\[tl\]\[1\]\[0\][$\cdots$]{} \[tl\]\[tl\]\[1\]\[0\][${{\Bbb R}}\times \{0\}^2$]{} $$\includegraphics[width=13cm]{bi.eps}$$
Figure 9
A $3$-manifold $M$ is a connected-sum $M=M_1 \# M_2$ if surgery along an embedded $2$-sphere produces manifolds $M_1$ and $M_2$, called the summands of $M$. Provided neither $M_1$ nor $M_2$ are $3$-spheres we say the connect-sum is non-trivial. If a $3$-manifold $M$ is not $S^3$ and if all connect-sum decompositions of $M$ are trivial, $M$ is called prime. Kneser’s Theorem [@Kneser] states that every compact, orientable $3$-manifold is a connected-sum of a unique collection of prime $3$-manifolds $M=M_1 \# M_2 \# \cdots \# M_n$, where uniqueness is up to a re-ordering of the terms.
The torus decomposition of a prime $3$-manifold $M$ consists of a minimal collection of embedded incompressible tori $\sqcup_{i=1}^n T_i \subset M$ such that the complement $M - \sqcup_{i=1}^{n} \nu T_i$ is a disjoint union of atoroidal and Seifert-fibered manifolds, where $\nu T_i$ is an open tubular neighborhood of $T_i\subset M$. A torus $T_i$ is incompressible if the induced map $\pi_1 T_i \to \pi_1 M$ is injective. A torus in a $3$-manifold is peripheral if it is isotopic to a boundary torus. A $3$-manifold is atoroidal if all incompressible tori are peripheral. The theorem of Jaco, Shalen and Johannson states that such a collection of tori $\{T_1, T_2 \cdots, T_n\}$ always exists and they are unique up to isotopy [@JacoShalen]. Given an arbitrary prime $3$-manifold, there is an associated graph called the JSJ-graph of $M$. The vertices of the JSJ-graph are the components of the manifold $M - \sqcup_{i=1}^{n} \nu T_i$. The edges of the graph are the tori $T_i$ for $i \in \{1,2,\cdots,n\}$.
Given a long knot $f \in {{\mathcal{\hat K}}}$, consider the compact $3$-manifold $B-N'$ where $B\subset {{\Bbb R}}^3$ is a closed $3$-ball containing ${{\bf{I}}}\times D^2$, and $N'$ is the interior of the image of $f$. We will call $C=B-N'$ the knot complement. Define $T=\partial C$. We review JSJ-splittings of knot complements. Every sphere in ${{\Bbb R}}^3$ bounds a $3$-ball by the Alexander-Schoenflies Theorem (see for example [@Hatcher3]), thus knot complements are prime $3$-manifolds, and the Jaco-Shalen-Johannson decomposition of a knot complement is simply the torus decomposition. The Generalized Jordan Curve Theorem (see for example [@Pollack]) tells us a knot complement’s associated graph is a tree. The tree is rooted, as only one component of $C-\sqcup_{i=1}^n \nu T_i$ contains $T$. The component of $C-\sqcup_{i=1}^n \nu T_i$ containing $T$ will be called the root manifold of the JSJ-splitting.
\[pndef\] Fix an embedding $b: \sqcup_{i \in \{1,2,\cdots,n\}} D^2 \to D^2$ such that $\partial D^2 \cap img(b)=\phi$. Let $D^2_i$ denote the image of the $i$-th copy of $D^2$ under $b$. Choose $b$ so that $D^2_i$ is the disc of radius $\frac{1}{2n+1}$ centered around the point $(\frac{4i-2n-2}{2n+1},0)$. Define $P_n$ to be $D^2 - int(\sqcup_{i=1}^n D^2_i)$. $P_n$ will be called the $n$-times punctured disc. $\partial D^2$ is the external boundary and $\partial (img(b))$ the internal boundary of $P_n$ (see Figure 10).
\[fig10\]
\[tl\]\[tl\]\[1\]\[0\][$-1$]{} \[tl\]\[tl\]\[0.7\]\[0\][$-1+\frac{2}{2n+1}$]{} \[tl\]\[tl\]\[0.7\]\[0\][$-1+\frac{4}{2n+1}$]{} \[tl\]\[tl\]\[0.7\]\[0\][$-1+\frac{6}{2n+1}$]{} \[tl\]\[tl\]\[0.7\]\[0\][$-1+\frac{8}{2n+1}$]{} \[tl\]\[tl\]\[0.7\]\[0\][$-1+\frac{4n-2}{2n+1}$]{} \[tl\]\[tl\]\[0.7\]\[0\][$-1+\frac{4n}{2n+1}$]{} \[tl\]\[tl\]\[1\]\[0\][$1$]{} \[tl\]\[tl\]\[1\]\[0\][$D^2$]{} \[tl\]\[tl\]\[1\]\[0\][$D^2_1$]{} \[tl\]\[tl\]\[1\]\[0\][$D^2_2$]{} \[tl\]\[tl\]\[1\]\[0\][$D^2_n$]{} \[tl\]\[tl\]\[1\]\[0\][$\cdots$]{} \[tl\]\[tl\]\[1\]\[0\][${{\Bbb R}}\times \{0\}$]{} $$\includegraphics[width=13cm]{pn.eps}$$
Figure 10
There are a few elementary facts that we will need about JSJ-splittings of knot complements and diffeomorphism groups of $2$ and $3$-dimensional manifolds. We assemble these facts in the following lemmas, all which are widely ‘known’ yet published proofs are elusive. A more detailed study of JSJ-decompositions of knot and link complements in $S^3$ has recently appeared [@bjsj] and could be used in place of several of these lemmas. An essential reference for the following arguments is Hatcher’s notes on $3$-dimensional manifolds [@Hatcher3].
\[sfl\] If $M$ is sub-manifold of $S^3$ whose boundary consists of a non-empty collection of tori, then either $M$ is a solid torus $S^1\times D^2$ or a component of the complement of $M$ in $S^3$ is a solid torus.
Let $C=S^3 - int(M)$ be the complement. Since $\partial M$ consists of a disjoint union of tori, every component of $\partial M$ contains an essential curve $\alpha$ which bounds a disc $D$ in $S^3$. Isotope $D$ so that it intersects $\partial M$ transversely in essential curves. Then $\partial M \cap D \subset D$ consists of a finite collection of circles, and these circles bound a nested collection of discs in $D$. Take an innermost disc $D'$. If $D' \subset M$ then $M$ is a solid torus. If $D' \subset C$ then the component of $C$ containing $D'$ is a solid torus.
\[sf2\] If a Seifert-fibred $3$-manifold is a component of a knot complement (in $S^3$) split along its JSJ-decomposition, then it is diffeomorphic to one of the following:
- A solid torus (unknot complement).
- The complement of a non-trivial torus knot. Such a manifold is Seifert-fibred over a disc with two singular fibres.
- $S^1 \times P_n$ for $n \geq 2$ (trivially fibred over a $n$-times punctured disc).
- Fibred over an annulus with one singular fibre (The complement of a regular and singular fibre in a Seifert-fibring of $S^3$).
Seifert-fibered manifolds that fiber over a non-orientable surface do not embed in $S^3$ since a non-orientable, embedded closed curve in the base lifts to a Klein bottle, which does not embed in $S^3$ by the Generalized Jordan Curve Theorem [@Pollack]. Similarly, a Seifert-fibered manifold that fibers over a surface of genus $g>0$ does not embed in $S^3$ since the base manifold contains two curves that intersect transversely at a point. If we lift one of these curves to a torus in $S^3$, it must be non-separating. This again contradicts the Generalized Jordan Curve Theorem.
Consider a Seifert-fibered manifold $M$ over an $n$-times punctured disc with $n>0$ and with perhaps multiple singular fibers. By Lemma \[sfl\], either $M$ is a solid torus or some component $Y$ of $\overline{S^3 \setminus M}$ is a solid torus. Consider the latter case. There are two possibilities.
1. The meridians of $Y$ are fibres of $M$. If there is a singular fibre in $M$, let $\beta$ be an embedded arc in the base surface associated to the Seifert-fibring of $M$ which starts at the singular point in the base and ends at the boundary component corresponding to $\partial Y$. $\beta$ lifts to a $2$-dimensional CW-complex in $M$, and the endpoint of $\beta$ lifts to a meridian of $Y$, thus it bounds a disc. If we append this disc to the lift of $\beta$, we get a CW-complex $X$ which consists of a $2$-disc attached to a circle. The attaching map for the $2$-cell is multiplication by $\beta$ where $\frac{\alpha}{\beta}$ is the slope associated to the singular fibre. The boundary of a regular neighbourhood of $X$ is a $2$-sphere, so we have decomposed $S^3$ into a connected sum $S^3 = L_{\frac{\gamma}{\beta}} \# Z$ where $L_{\frac{\gamma}{\beta}}$ is a lens space with $H_1 L_{\frac{\gamma}{\beta}} = {{\Bbb Z}}_\beta$. Since $S^3$ is irreducible, $\beta=1$. Thus $M\simeq S^1 \times P_{n-1}$ for some $n \geq 1$.
2. The meridians of $Y$ are not fibres of $M$. In this case, we can extend the Seifert fibring of $M$ to a Seifert fibring of $M \cup Y$. Either $M \cup Y = S^3$, or $M \cup Y$ has boundary.
- If $M \cup Y = S^3$ then we know by the classification of Seifert fibrings of $S^3$ that any fibring of $S^3$ has at most two singular fibres. If $M$ is the complement of a regular fibre of a Seifert fibring of $S^3$, then $M$ is a torus knot complement. Otherwise, $M$ is the complement of a singular fibre, meaning that $M$ is a solid torus.
- If $M \cup Y$ has boundary, we can repeat the above argument. Either $M \cup Y$ is a solid torus, or a component of $S^3 \setminus \overline{M \cup Y}$ is a solid torus, so we obtain $M$ from the above manifolds by removing a Seifert fibre. By induction, we obtain $M$ from either a Seifert fibring of a solid torus, or a Seifert fibring of $S^3$ by removing fibres.
\[fig11\]
\[tl\]\[tl\]\[0.9\]\[0\][ cabling]{} \[tl\]\[tl\]\[0.8\]\[0\][JSJ]{} \[tl\]\[tl\]\[0.8\]\[0\][tree]{} $$\includegraphics[width=13cm]{cabling.eps}$$
Figure 11
\[cablelem\] Given a long knot $f \in {{\mathcal{\hat K}}}$ with complement $C$, if the root manifold of the JSJ-splitting of the knot complement is Seifert fibered with one singular fiber, then $f$ is a cabling of another long knot. Another way to say this is that the one-point compactification $\tilde f : S^1 \to S^3$ of $f_{|{{\Bbb R}}\times \{0\}^2} : {{\Bbb R}}\to {{\Bbb R}}^3$ is an essential curve in the boundary of a tubular neighborhood of some embedding $g : S^1 \to S^3$ (see Figure 11).
Thurston [@Thurston] has proved that the non-Seifert-fibered manifolds in the JSJ-splitting of a knot complement are finite-volume hyperbolic manifolds. These hyperbolic manifolds can have arbitrarily many boundary components [@bjsj]. Figure 12 demonstrates a hyperbolic satellite knot (a knot such that the root manifold in the JSJ-decomposition is hyperbolic) which contains the Borromean rings complement in its JSJ-decomposition. In general, one can prove that if the root manifold is a hyperbolic manifold with $n+1$ boundary components, then it is the complement of an $(n+1)$-component hyperbolic link in $S^3$ which contains an $n$-component sublink which is the unlink.
\[fig12\]
\[tl\]\[tl\]\[0.8\]\[0\][A hyperbolic satellite operation]{} \[tl\]\[tl\]\[0.8\]\[0\][JSJ]{} \[tl\]\[tl\]\[0.8\]\[0\][tree]{} $$\includegraphics[width=10cm]{kc6.eps}$$
Figure 12
\[pnlem\] A knot is a non-trivial connected-sum if and only if the root manifold of the associated JSJ-tree is diffeomorphic to $S^1 \times P_n$ for some $n \geq 2$. In this case, $n$ is the number of prime summands of $f$.
If $f \in {{\mathcal{\hat K}}}$ is a non-trivial connected-sum, let $n$ be the number of prime summands of $f$, and isotope $f$ so that $f_{|{{\Bbb R}}\times \{0\}^2}$ satisfies Definition \[sumdef\].
Let $L \subset {{\Bbb R}}^2$ be the closed disc of radius $\frac{1}{2}$ centered about the origin. Let $N'=img(f_{|{{\Bbb R}}\times L})$, $N=img(f)$ and define $C=B-int(N')$ where $B$ is a closed, convex ball neighborhood of ${{\bf{I}}}\times D^2$ in ${{\Bbb R}}^3$. Let $B_1, B_2, \cdots B_n$ be the closed $3$-balls from Definition \[sumdef\], with $S_i = \partial B_i$ and $S_i$ intersecting $img(f)$ in two discs for all $i \in \{1,2,\cdots,n\}$. Define $C_i = B_i - int(N)$ and $T_i = \partial C_i$.
Let $\nu T_i$ be a small open tubular neighborhood of $T_i$, then $C-\sqcup_{i=1}^n \nu T_i$ consists of $n+1$ components. One component contains $T=\partial C$ and the other $n$ components are the knot complements of the prime summands of $f$, $C_1,C_2,\cdots,C_n$. The component containing $T$ we will denote $V$. $V$ is diffeomorphic to $S^1 \times P_n$.
By Dehn’s Lemma the tori $\{T_i : i\in \{1,2,\cdots,n\}\}$ are incompressible in $C$. If $\{T_{n+1},\cdots,T_{n+m}\}$ are the tori of the JSJ-decomposition for $\sqcup_{i=1}^n C_i$, the collection $\{T_1, T_2, \cdots, T_n, T_{n+1}, \cdots, T_{n+m}\}$ is therefore the JSJ-decomposition of $C$. Thus, $V \simeq S^1 \times P_n$ is the root manifold in the JSJ-tree associated to $C$.
To prove the converse, let $V$ be the root manifold of the JSJ-splitting of $C$. Observe that $\partial V \simeq \partial (S^1 \times P_n)$ divides ${{\Bbb R}}^3$ into $n+2$ components, only one containing the knot. Let $T$ denote the boundary of the component which contains the knot. By Lemma \[sf2\] the fibers of $S^1 \times P_n$ are meridians of the knot. Let $L_1,\cdots,L_n$ be properly embedded intervals in $P_n$ which cut $P_n$ into the union of a disc with $n$ once-punctured discs. Then $\sqcup_{i=1}^{n} (S^1 \times L_i)$ can be extended to $n$ disjoint, embedded $2$-spheres $S_i \subset {{\Bbb R}}^3$ such that $S_i \cap (S^1 \times P_n) = S^1 \times L_i$, and $S_i \cap img(f_{|{{\Bbb R}}\times \{0\}^2})$ consists of two points. Thus we have decomposed the long knot $f$ into a connected-sum.
\[topm\] In the above lemma, we call the tori $T_1, \cdots, T_n$ the base level of the Jaco-Shalen-Johannson decomposition of the knot complement.
Lemmas \[pnlem\], \[cablelem\] and Thurston’s Hyperbolisation Theorem [@Thurston] gives us a canonical decomposition of knots into simpler knots via cablings, connected-sums and hyperbolic satellite operations commonly referred to as the satellite or splice decomposition of knots. This is worked out in detail in [@bjsj].
Figure 13 shows a knot with its JSJ-tori, and the associated JSJ-tree. In the standard terminology of knot theory, this knot would be described as a connect-sum of three prime knots: the left-handed trefoil, the figure-8 knot and the Whitehead double of the figure-8 knot. $V=S^1 \times P_3$ is the root manifold, $T_1,T_2,T_3$ are the base-level of the JSJ-decomposition of $C$, and $T_4$ is the remaining torus in the JSJ-decomposition of $C$. The leftmost summand is the trefoil knot. The center summand is a figure-$8$ knot, whose complement is hyperbolic. The rightmost summand is the Whitehead double of the figure-$8$ knot, it’s complement is $C_3$. $C_3$ is the union of $C'_3$ (the Whitehead link complement, which is hyperbolic) and $C''_3$ (a figure-$8$ knot complement) where $\partial C'_3=T_3 \sqcup T_4$, and $\partial C''_3=T_4$, $C''_3 \cap C'_3 = T_4$. The interior of $C'_3$ is also a hyperbolic $3$-manifold of finite volume.
\[fig13\] [ \[tl\]\[tl\]\[0.8\]\[0\][$T_1$]{} \[tl\]\[tl\]\[0.8\]\[0\][$T_2$]{} \[tl\]\[tl\]\[0.8\]\[0\][$T_3$]{} \[tl\]\[tl\]\[0.8\]\[0\][$T_4$]{} $$\includegraphics[width=13cm]{jsj_knot.eps}$$ ]{}
\[tl\]\[tl\]\[1.2\]\[0\][$V$]{} \[tl\]\[tl\]\[1\]\[0\][$T_1$]{} \[tl\]\[tl\]\[1\]\[0\][$T_2$]{} \[tl\]\[tl\]\[1\]\[0\][$T_3$]{} \[tl\]\[tl\]\[1\]\[0\][$T_4$]{} \[tl\]\[tl\]\[1\]\[0\][$C'_3$]{} \[tl\]\[tl\]\[1\]\[0\][$C_2$]{} \[tl\]\[tl\]\[1\]\[0\][$C_1$]{} \[tl\]\[tl\]\[1\]\[0\][$C''_3$]{} \[tl\]\[tl\]\[1\]\[0\][the JSJ]{} \[tl\]\[tl\]\[1\]\[0\][tree]{} \[tl\]\[tl\]\[1\]\[0\][$C'_3$]{} $$\includegraphics[width=6cm]{jsj_eg.eps} \hskip 1cm \includegraphics[width=3.4cm]{hyplink.eps}$$
Figure 13
\[bdiff\]The component ${{\mathcal{\hat K}}}_f$ of ${{\mathcal{\hat K}}}$ containing the long knot $f$ is the classifying space of ${{\mathrm{Diff}}}(C,T)$, the group of diffeomorphisms of the knot complement which fix the boundary torus $T=\partial C$ point-wise. Moreover, ${{\mathcal{\hat K}}}_f$ is a $K(\pi,1)$.
Let $B = {{\bf{I}}}\times D^2$ and let ${{\mathrm{Diff}}}(B)$ be the group of diffeomorphisms of ${{\Bbb R}}^3$ with support contained in $B$. The map ${{\mathrm{Diff}}}(B) \to {{\mathcal{\hat K}}}_f$ defined by restriction to $img(f)$ induces a fibration $${{\mathrm{Diff}}}(C,T) \to {{\mathrm{Diff}}}(B) \to {{\mathcal{\hat K}}}_f$$ where $C=B-int(img(f))$, $T=\partial C$, and ${{\mathrm{Diff}}}(C,T)$ is the group of diffeomorphisms of $C$ that fix $T$ point-wise. Since ${{\mathrm{Diff}}}(B)$ is contractible [@Hatcher2], $B{{\mathrm{Diff}}}(C,T) \simeq {{\mathcal{\hat K}}}_f$, where ${{\mathcal{\hat K}}}_f$ is the component of ${{\mathcal{\hat K}}}$ containing $f$. The fact that ${{\mathrm{Diff}}}(C,T)$ has contractible components is due to Hatcher [@Hatcher1].
In the above lemma, $BG=EG/G$ is the classifying space of a topological group $G={{\mathrm{Diff}}}(C,T)$ and $EG={{\mathrm{Diff}}}(B)$. Using Smale’s Theorem ${{\mathrm{Diff}}}(D^2) \simeq \{*\}$ [@Smale], an argument analogous to the above gives ${{\mathcal{C}}}_2(n)/S_n \simeq B{{\mathrm{Diff}}}(P_n)$ where ${{\mathrm{Diff}}}(P_n)$ is the group of diffeomorphisms of $P_n$ that fix the external boundary of $P_n$ point-wise.
Let ${{\mathrm{PDiff}}}(P_n)$ denote the subgroup of ${{\mathrm{Diff}}}(P_n)$ consisting of diffeomorphisms whose restrictions to $\partial P_n$ are isotopic to the identity map $Id_{\partial P_n} : \partial P_n \to \partial P_n$. Then similarly, by Smale’s Theorem ${{\mathcal{C}}}_2(n) \simeq B{{\mathrm{PDiff}}}(P_n)$.
Let ${{\mathrm{PFDiff}}}(P_n)$ be the subgroup of ${{\mathrm{PDiff}}}(P_n)$ consisting of diffeomorphisms whose restrictions to $\partial P_n$ are equal to the identity $Id_{\partial P_n}$. $\pi_0 {{\mathrm{Diff}}}(P_n)$ is called the braid group on $n$-strands. $\pi_0 {{\mathrm{PDiff}}}(P_n)$ is called the pure braid group on $n$-strands, and $\pi_0 {{\mathrm{PFDiff}}}(P_n)$ is called the pure framed braid group on $n$ strands. Observe that ${{\mathrm{PFDiff}}}(P_n)$ is homotopy equivalent to the subgroup ${{\mathrm{PFDiff}}}^+(P_n)$ of ${{\mathrm{PFDiff}}}(P_n)$ consisting of diffeomorphisms which restrict to the identity in an $\epsilon$-neighborhood $N$ of the internal boundary of $P_n$. This follows from the fact that the space of collar neighborhoods of $\partial P_n$ in $P_n$ is contractible.
\[windef\] This definition will use the notation of Definition \[pndef\] and the previous paragraph. Every diffeomorphism in ${{\mathrm{PFDiff}}}^+(P_n)$ can be canonically extended to a diffeomorphism of the once-punctured disc $D^2 - int(D^2_i)$ simply by taking the union with $Id_{D^2_j}$ for $j \neq i$. Thus, for each $i \in \{1,2,\cdots,n\}$ there is a homomorphism $w_i : {{\mathrm{PFDiff}}}^+(P_n) \to \pi_0 {{\mathrm{PFDiff}}}(S^1 \times {{\bf{I}}}) \simeq {{\Bbb Z}}$ given by the above extension together with an identification $D^2 - int(D^2_i) \equiv S^1 \times {{\bf{I}}}$. Here ${{\mathrm{PFDiff}}}(S^1 \times {{\bf{I}}})$ denotes the group of boundary-fixing diffeomorphisms of $S^1 \times {{\bf{I}}}$. The generator of $\pi_0 {{\mathrm{PFDiff}}}(S^1 \times {{\bf{I}}})\simeq {{\Bbb Z}}$ is a Dehn twist about a boundary-parallel curve [@Gramain]. Let ${{\mathrm{DN}}}_n$ denote a free abelian subgroup of ${{\mathrm{PFDiff}}}^+(P_n)$ having rank $n$, all whose elements have support in $N$, generated by Dehn twists about $n$ curves in $N$, the $i$-th curve parallel to $\partial D^2_i$.
\[pp\]There is an isomorphism of groups $$\pi_0 {{\mathrm{PDiff}}}(P_n) \times {{\Bbb Z}}^n \simeq \pi_0 {{\mathrm{PFDiff}}}(P_n)$$ Moreover, the subgroups $\cap_{i=1}^n ker(w_i)$ and ${{\mathrm{DN}}}_n$ satisfy:
- Inclusion $\cap_{i=1}^n ker(w_i) \to {{\mathrm{PDiff}}}(P_n)$ is a homotopy equivalence.
- The elements of ${{\mathrm{DN}}}_n$ and $\cap_{i=1}^n ker(w_i)$ commute with each other, and ${{\mathrm{DN}}}_n \cap \left(\cap_{i=1}^n ker(w_i) \right)$ is the trivial group.
- The homomorphism $\cap_{i=1}^n ker(w_i) \times {{\mathrm{DN}}}_n \to {{\mathrm{PFDiff}}}(P_n)$ is a homotopy equivalence.
Take ${{\mathrm{Diff}}}(S^1)$ to be the group of orientation preserving diffeomorphisms of a circle, and consider the fibration ${{\mathrm{PFDiff}}}(P_n) \to {{\mathrm{PDiff}}}(P_n) \to \prod_{i=1}^n {{\mathrm{Diff}}}(S^1)$ given by restriction to the internal boundary of $P_n$. This gives us the short exact sequence
$$0 \to \prod_{i=1}^n \pi_1 {{\mathrm{Diff}}}(S^1) \to \pi_0 {{\mathrm{PFDiff}}}(P_n) \to \pi_0 {{\mathrm{PDiff}}}(P_n) \to 0$$
but $\prod_{i=1}^n \pi_1 {{\mathrm{Diff}}}(S^1) \simeq {{\Bbb Z}}^n$, which is the subgroup ${{\mathrm{DN}}}_n \subset \pi_0 {{\mathrm{PFDiff}}}(P_n)$. The map $\prod_{i=1}^n w_i : \pi_0 {{\mathrm{PFDiff}}}(P_n) \to {{\Bbb Z}}^n \simeq \prod_{i=1}^n \pi_1 {{\mathrm{Diff}}}(S^1)$ is a splitting of the above short exact sequence. The kernel of $\prod_{i=1}^n w_i$ is $\pi_0 \cap_{i=1}^n ker(w_i)$. By definition, elements in $\cap_{i=1}^n ker(w_i)$ and ${{\mathrm{DN}}}_n$ commute with each other, and so the result follows.
We will also need a mild variation on Lemma \[pp\]. Let $*=(0,-1)$ be the base-point of $D^2$ and let $\gamma_i : [0,1] \to P_n$ for $i \in \{1,2,\cdots,n\}$ be the affine-linear map starting at $*$ and ending at $(\frac{4i-2n-2}{2n+1}, -\frac{1}{2n+1})$
\[fig14\]
\[tl\]\[tl\]\[1\]\[0\][$$]{} \[tl\]\[tl\]\[1\]\[0\][$P_n$]{} \[tl\]\[tl\]\[1\]\[0\][$\gamma_1$]{} \[tl\]\[tl\]\[1\]\[0\][$\gamma_2$]{} \[tl\]\[tl\]\[1\]\[0\][$\gamma_n$]{} \[tl\]\[tl\]\[1\]\[0\][$*=(0,-1)$]{} \[tl\]\[tl\]\[1\]\[0\][${{\Bbb R}}\times \{0\}$]{} \[tl\]\[tl\]\[1\]\[0\][$\cdots$]{} $$\includegraphics[width=5cm]{pnstuff.eps}$$
Figure 14
Define ${{\mathrm{KDiff}}}(P_n)$ to be $\cap_{i=1}^n ker(w_i)$. Define ${{\mathrm{FDiff}}}(P_n)$ to be the subgroup of ${{\mathrm{Diff}}}(P_n)$ such that each diffeomorphism $f \in {{\mathrm{FDiff}}}(P_n)$
- restricts to a diffeomorphism of $N$, ie: $f_{|N} : N \to N$.
- the restriction of $f_{|N}$ to any connected component of $N$ is a translation in the plane.
Observe, there is an epi-morphism ${{\mathrm{FDiff}}}(P_n) \to S_n \ltimes {{\Bbb Z}}^n$ given by $f \longmapsto (\sigma_f,\omega_1(f),\cdots,\omega_n(f))$ where
- $\sigma_f \in S_n$ is the permutation of $\{1,2,\cdots,n\}$ defined by $\sigma_f(i)=j$ if $f(\partial D^2_i)=\partial D^2_j$.
- $\omega_i(f) \in {{\Bbb Z}}$ is the linking number of $\overline{\gamma_j}\cdot (f\circ \gamma_i)$ with $D^2_j$ where $\sigma(i)=j$. Here $\overline{\gamma_j}(t)=\gamma_j (1-t)$ and concatenation is by convention right-to-left, ie: if $\gamma, \eta : [0,1] \to X$ satisfy $\eta(1)=\gamma(0)$ then $\gamma \cdot \eta(t)=\eta(2t)$ for $0 \leq t \leq \frac{1}{2}$ and $\gamma\cdot\eta(t)=\gamma(2t-1)$ for $\frac{1}{2} \leq t \leq 1$.
- $S_n \ltimes {{\Bbb Z}}^n$ is the semi-direct product of $S_n$ and ${{\Bbb Z}}^n$ where $S_n$ acts on ${{\Bbb Z}}^n$ by the regular representation ie: $\sigma.(i_1,i_2,\cdots,i_n)=(i_{\sigma^{-1}(1)},i_{\sigma^{-1}(2)},\cdots,i_{\sigma^{-1}(n)})$
Call the above epi-morphism $W : {{\mathrm{FDiff}}}(P_n) \to S_n \ltimes {{\Bbb Z}}^n$, and define $\widetilde{{{\mathrm{KDiff}}}}(P_n) = W^{-1}(S_n \times \{0\}^n)$.
\[wtkdiff\] There is a fiber-homotopy equivalence $$\xymatrix{
{{\mathrm{PDiff}}}(P_n) \ar[r] & {{\mathrm{Diff}}}(P_n) \ar[r] & S_n \\
{{\mathrm{KDiff}}}(P_n) \ar[r] \ar[u] & \widetilde{{{\mathrm{KDiff}}}}(P_n) \ar[r] \ar[u] & S_n \ar[u]
}$$ where all vertical arrows are inclusions.
The above lemma follows immediately from Lemma \[pp\].
Abstractly there is a homotopy equivalence between $B{{\mathrm{KDiff}}}(P_n)$ and ${{\mathcal{C}}}_2(n)$ given by the proof of Lemma \[bdiff\]. Since the properties of this homotopy equivalence will be important later, we define it precisely here.
\[pd\_def\] Given $f \in {{\mathrm{Diff}}}(D^2)$, let $\zeta(f)=(L_1,L_2,\cdots,L_n) \in {{\mathcal{C}}}_2(n)$ be $n$ little $2$-cubes such that the center of $L_i$ is $f(\frac{4i-2n-2}{2n+1},0)$. For $\zeta(f)$ to be well-defined (and continuous) we need to choose the the side lengths of $L_i$ equal to the minimum of these two numbers: $\frac{1}{2n+1}$ and the largest number $w$ so that the little cubes with centers $f(\frac{4i-2n-2}{2n+1},0)$ with width and height equal to $w$ for $i \in \{1,2,\cdots,n\}$ have disjoint interiors. Then $\phi : {{\mathrm{Diff}}}(D^2) \to {{\mathcal{C}}}_2(n)$ factors to a map $B{{\mathrm{KDiff}}}(P_n) \to {{\mathcal{C}}}_2(n)$ which is a homotopy-equivalence.
The definition below will use the conventions of Definition \[pndef\], in particular we will call $S^1 \times \partial D^2 \subset S^1 \times P_n$ the external boundary of $S^1 \times P_n$, and $\partial (S^1 \times P_n) - S^1 \times \partial D^2$ the internal boundary of $S^1 \times P_n$.
\[s1xpndiff\] Let $\eta_i : S^1 \to \partial D^2_i$ be a clockwise parametrization of $\partial D^2_i$ starting and ending at $\gamma_i(1)$. Notice that $\lambda_i = \overline{\gamma_i}\eta_i\gamma_i$ for $i \in \{1,2,\cdots,n\}$ are generators for $\pi_1 P_n$. Let $\{*\}\times \lambda_i$ and $S^1 \times \{*\}$ denote generators of $\pi_1 (S^1 \times \partial D^2_i)$. Let ${{\mathrm{Diff}}}(S^1 \times P_n)$ be the group of diffeomorphisms of $S^1 \times P_n$ whose restriction to the external boundary are equal to the identity $Id_{S^1 \times \partial D^2}$ and whose restriction to the internal boundary $S^1 \times \partial(img(b))$ sends $\{1\}\times \eta_i$ to a curve isotopic to $\{1\} \times \eta_{\sigma(i)}$ for all $i \in \{1,2,\cdots, n\}$ where $\sigma : \{1,2,\cdots,n\} \to \{1,2,\cdots,n\}$ is a permutation of $\{1,2,\cdots,n\}$. Let ${{\mathrm{PDiff}}}(S^1 \times P_n)$ denote the group of diffeomorphisms of $S^1 \times P_n$ whose restrictions to the internal boundary are isotopic to the identity and whose restrictions to the external boundary are equal to the identity $Id_{S^1 \times \partial D^2}$. Similarly, define ${{\mathrm{PFDiff}}}(S^1 \times P_n)$ to be the group of diffeomorphisms of $S^1 \times P_n$ which restrict to the identity $Id_{S^1\times \partial P_n}$. Let ${{\mathrm{KDiff}}}(S^1 \times P_n)$ be the subgroup of ${{\mathrm{PFDiff}}}(S^1 \times P_n)$ consisting of diffeomorphisms having the form $Id_{S^1} \times f$ where $f \in {{\mathrm{KDiff}}}(P_n)$, and let $\widetilde{{{\mathrm{KDiff}}}}(S^1 \times P_n)$ denote the subgroup of ${{\mathrm{Diff}}}(S^1 \times P_n)$ consisting of diffeomorphisms of the form $f=Id_{S^1} \times g$ for $g \in \widetilde{{{\mathrm{KDiff}}}}(P_n)$.
\[diffs1pn\] There is a fiber-homotopy equivalence $$\xymatrix{
{{\mathrm{PDiff}}}(S^1 \times P_n) \ar[r] & {{\mathrm{Diff}}}(S^1 \times P_n) \ar[r] & S_n \\
{{\mathrm{KDiff}}}(S^1 \times P_n) \ar[r] \ar[u] & \widetilde{{{\mathrm{KDiff}}}}(S^1 \times P_n) \ar[r] \ar[u] & S_n \ar[u] }$$
where all vertical arrows are inclusions (and homotopy equivalences).
We consider $S^1 \times P_n$ to be a Seifert fibered manifold. Hatcher [@Hatcher1] proves that the full group of diffeomorphism of $S^1 \times P_n$ is homotopy equivalent to the fiber-preserving subgroup. Let $G$ denote the fiber-preserving subgroup of ${{\mathrm{PDiff}}}(S^1 \times P_n)$. Thus, the inclusion $G \to {{\mathrm{PDiff}}}(S^1 \times P_n)$ is a homotopy equivalence. Since the group of orientation preserving diffeomorphisms of $S^1$ is homotopy equivalent to $SO_2$, $G$ is homotopy equivalent to the subgroup $G' \subset G$ of fibrewise-linear diffeomorphisms of $S^1 \times P_n$. Since every diffeomorphism in ${{\mathrm{PDiff}}}(S^1 \times P_n)$ restricts to a diffeomorphism of $\partial (S^1 \times P_n)$ which is isotopic to the identity, $G'$ is homotopy equivalent to the subgroup of diffeomorphisms of the form $Id_{S^1} \times f$ where $f \in {{\mathrm{PDiff}}}(S^1 \times P_n)$. The key consideration in the above argument is whether or not $f$ could be a Dehn twist along a vertical annulus. By Lemma \[pp\], ${{\mathrm{PDiff}}}(P_n)$ is homotopy equivalent to ${{\mathrm{KDiff}}}(P_n)$. The remaining results follow from Lemmas \[wtkdiff\] and \[pp\].
As a historical note, some of Hatcher’s results on diffeomorphism groups of Haken manifolds were independently discovered by Ivanov [@Ivanov1; @Ivanov2].
The following lemma is used to simplify the proof of Theorem \[freeness\]. It is a standard variation of a construction of Borel [@Borel] (chapter IV, §3).
\[cohenlemma\] If $G$ is a topological group with $H$ a closed normal subgroup such that $ G/H $ is a finite group, then there exists a canonical normal, finite-sheeted covering space $$G/H \to BH \to BG$$ where the map $BH \to BG$ is given by the projection $EG/H \to EG/G$ where we make the identification $BH=EG/H$.
First, we sketch the proof of Theorem \[freeness\]. The fact that the map $\sqcup_{n=0}^\infty \kappa_n$ induces a bijection $$\sqcup_{n \in \{ 0,1,2,3,\cdots \}}
\pi_0 \left( \left( {{\mathcal{C}}}_2(n) \times {{\mathcal{P}}}^n\right)/S_n \right) \to \pi_0 {{\mathcal{\hat K}}}$$ is due to Schubert [@Sch]. His theorem states that every long knot decomposes uniquely into a connected-sum of prime knots, up to a re-ordering of the terms. Since the map $\sqcup_{n=0}^\infty \kappa_n$ is bijective on components, we need only to verify that it is a homotopy equivalence when restricted to any single connected component. By Lemma \[bdiff\], the components of both the domain and range are $K(\pi,1)$’s. So we have reduced the theorem to checking that the induced map is an isomorphism of fundamental groups for every component. The inspiration for the proof of this is the fibration below, which we call the little cubes fibration.
$$S_n \to {{\mathcal{C}}}_2(n) \times {{\mathcal{P}}}^n \to ({{\mathcal{C}}}_2(n) \times {{\mathcal{P}}}^n)/S_n$$
Let $f \in {{\mathcal{\hat K}}}$ with $f=f_1\# f_2 \# \cdots \# f_n$, where $(f_1,\cdots,f_n)\in {{\mathcal{P}}}^n$ are the prime summands of $f$. Let ${{\mathcal{\hat K}}}_f$ denote the component of ${{\mathcal{\hat K}}}$ containing $f$, similarly define ${{\mathcal{\hat K}}}_{f_i}$. Thus the above fibration, when restricted to the connected component ${{\mathcal{C}}}_2(n) \times \prod_{i=1}^n {{\mathcal{\hat K}}}_{f_i}$ of ${{\mathcal{C}}}_2(n) \times {{\mathcal{P}}}^n$, has the form:
$$\Sigma_f \to {{\mathcal{C}}}_2(n) \times \prod_{i=1}^n {{\mathcal{\hat K}}}_{f_i} \to
({{\mathcal{C}}}_2(n) \times \prod_{i=1}^n {{\mathcal{\hat K}}}_{f_i})/\Sigma_f$$
where $\Sigma_f \subset S_n$ is the subgroup which preserves the partition $\sim $ of $\{1,2,\cdots,n\}$ with $i \sim j \Leftrightarrow {{\mathcal{\hat K}}}_{f_i} = {{\mathcal{\hat K}}}_{f_j}$.
By Lemma \[bdiff\] the little cubes fibration gives the short exact sequence below.
$$0 \to \pi_1 {{\mathcal{C}}}_2(n) \times \prod_{i=1}^n \pi_1 {{\mathcal{\hat K}}}_{f_i} \to
\pi_1 (({{\mathcal{C}}}_2(n) \times \prod_{i=1}^n {{\mathcal{\hat K}}}_{f_i})/\Sigma_f) \to \Sigma_f \to 0$$
$\pi_1 {{\mathcal{\hat K}}}_f \simeq \pi_0 {{\mathrm{Diff}}}(C,T)$ by Lemma \[bdiff\]. So the idea of the proof is to find an analogous fibration for ${{\mathcal{\hat K}}}_f$. So we are looking for an epimorphism $\pi_0 {{\mathrm{Diff}}}(C,T) \to \Sigma_f$.
Since the tori in the JSJ-splitting of $C$ are unique up to isotopy, define a permutation $\sigma_g : \{1,2,\cdots,n\} \to \{1,2,\cdots,n\}$ by the condition that $\sigma_g(i)=j$ if $g(T_i)$ is isotopic to $T_j$ where $T_1,T_2, \cdots,T_n$ are the base-level of the JSJ-decomposition of $C$. This is well-defined since $g$ fixes $T = \partial C$ and the JSJ-decomposition is unique up to isotopy. The homomorphism $\sigma : \pi_0 {{\mathrm{Diff}}}(C,T) \to S_n$ is onto $\Sigma_f$ since two long knots $f_i$ and $f_j$ are isotopic if and only if $C_i$ and $C_j$ admit orientation preserving diffeomorphisms which also preserve the (oriented) meridians of $C_i$ and $C_j$.
The kernel of $\sigma$ one would expect to be the mapping class group of diffeomorphisms of $C$ which do not permute the base-level of the JSJ-splitting of $C$. Such a diffeomorphism $g$, when restricted to $V \simeq S^1 \times P_n$ can isotoped to be in ${{\mathrm{KDiff}}}(S^1 \times P_n)$. Thus $g$ restricts to diffeomorphisms $g_{|C_i} \in {{\mathrm{Diff}}}(C_i,T_i)$ for all $i \in \{1,2,\cdots, n\}$, leading us to expect the kernel of $\sigma$ is $\pi_0 {{\mathrm{PDiff}}}(P_n) \times \prod_{i=1}^n \pi_0 {{\mathrm{Diff}}}(C_i,T_i)$. By Lemma \[bdiff\] $\pi_0 {{\mathrm{Diff}}}(P_n) \simeq \pi_1 {{\mathcal{C}}}_2(n)$ and $\pi_0 {{\mathrm{Diff}}}(C_i,T_i)\simeq \pi_1 {{\mathcal{\hat K}}}_{f_i}$ where $f_i$ denotes the $i$-th summand of $f$. So we have constructed a SES $$0 \to \pi_1 {{\mathcal{C}}}_2(n)\times \prod_{i=1}^n \pi_1 {{\mathcal{\hat K}}}_{f_i} \to \pi_1 {{\mathcal{\hat K}}}_f \to \Sigma_f \to 0$$ which is the analogue of the SES coming from the little cubes fibration.
In the argument below, we rigorously redo the above sketch at the space-level. We construct a fibration of diffeomorphism groups whose long exact sequence is the SES given above. We then use Lemma \[cohenlemma\] to convert this fibration of diffeomorphism groups into a fibration which describes ${{\mathcal{\hat K}}}_f$, and this we will show is equivalent to the little cubes fibration.
(of Theorem \[freeness\]) We will show that $\sqcup_{n=0}^\infty \kappa_n$ is a homotopy equivalence, component by component. Let $f \in {{\mathcal{\hat K}}}$ be a knot specifying a connected component ${{\mathcal{\hat K}}}_f$ of ${{\mathcal{\hat K}}}$.
In the case of the unknot $f = Id_{{{\Bbb R}}\times D^2}$, we know from the proof of the Smale conjecture [@Hatcher2] that the component of ${{\mathcal{\hat K}}}$ containing $f$ is contractible. ${{\mathcal{C}}}_2(0) \times {{\mathcal{P}}}^0$ is a point thus the map $\kappa_0$ is a homotopy equivalence between these two components.
If $f$ is a prime knot, $n=1$ and the little cubes fibration $S_1 \to {{\mathcal{C}}}_2(1)\times {{\mathcal{P}}}^1 \to ({{\mathcal{C}}}_2(1)\times {{\mathcal{P}}}^1)/S_1$ is trivial, thus ${{\mathcal{\hat K}}}_f$ is a component of ${{\mathcal{P}}}$. In this case, our map $\kappa_1 : {{\mathcal{C}}}_2(1) \times {{\mathcal{P}}}\to {{\mathcal{\hat K}}}$ is mapping from ${{\mathcal{C}}}_2(1) \times {{\mathcal{P}}}$ to ${{\mathcal{\hat K}}}$. Since ${{\mathcal{C}}}_2(1)$ is contractible and our action satisfies the identity axiom, $\kappa_1$ is homotopic to the composite of the projection map ${{\mathcal{C}}}_2(1) \times {{\mathcal{P}}}\to {{\mathcal{P}}}$ with the inclusion map ${{\mathcal{P}}}\to {{\mathcal{\hat K}}}$, and so $\kappa_1$ is a homotopy equivalence between $({{\mathcal{C}}}_2(1) \times {{\mathcal{P}}})/S_1$ and ${{\mathcal{P}}}$.
Consider the case of a composite knot $f=f_1 \# f_2 \# \cdots \# f_n \in {{\mathcal{\hat K}}}$ for $n \geq 2$ with $f_i$ prime for all $i \in \{1,2,\cdots,n\}$. Let $C = B - N'$ denote the knot complement, as in Lemma \[pnlem\]. Let $T=\partial C$, let $V \simeq S^1 \times P_n$ denote the root manifold of the associated tree to the JSJ-decomposition of $C$ and let $T_1,\cdots,T_n$ denote base-level of the JSJ-decomposition of $C$ (see Lemma \[pnlem\], Definition \[topm\]). Similarly, let $V \simeq S^1 \times P_n$, $B_i$ and $C_i$ for $i \in \{1,2,\cdots,n\}$ be as in Lemma \[pnlem\]. Let ${{\mathrm{Diff}}}(C,T)$ be the group of diffeomorphisms of $C$ that fix $T$ point-wise. Let ${{\mathrm{Diff}}}^V(C,T)$ denote the subgroup of ${{\mathrm{Diff}}}(C,T)$ consisting of diffeomorphisms which restrict to diffeomorphisms of $V$. Let ${{\mathrm{PDiff}}}^V(C,T)$ denote the subgroup of ${{\mathrm{Diff}}}^V(C,T)$ consisting of diffeomorphisms whose restrictions to $\partial V$ are isotopic to $Id_{\partial V}$. Let ${{\mathrm{Emb}}}(\sqcup_{i=1}^n T_i,C)$ denote the space of embeddings of $\sqcup_{i=1}^n T_i$ in $C$. If we restrict a diffeomorphism in ${{\mathrm{Diff}}}(C,T)$ to $\sqcup_{i=1}^n T_i$ and mod-out by the parametrization of the individual tori, we get a fibration (which is not necessarily onto)
$${{\mathrm{PDiff}}}^V(C,T) \to {{\mathrm{Diff}}}(C,T) \to {{\mathrm{Emb}}}(\sqcup_{i=1}^n T_i,C)/\prod_{i=1}^n {{\mathrm{Diff}}}(T_i)$$
Since $T_i$ is incompressible in $C$, this fibration is mapping to embeddings which are also incompressible. The tori $\sqcup_{i=1}^n T_i$ are part of the JSJ-splitting of $C$, and the JSJ-splitting is unique up to isotopy. This means that a diffeomorphism in ${{\mathrm{Diff}}}(C,T)$ must send $T_i$ to another torus in the JSJ-splitting (up to isotopy), but more importantly that torus must be in the base-level of the JSJ-splitting since the diffeomorphism is required to preserve $T$.
A component of ${{\mathrm{Emb}}}(\sqcup_{i=1}^n T_i,C)/\prod_{i=1}^n {{\mathrm{Diff}}}(T_i)$ is an isotopy class of $n$ embedded, labeled tori. Provided the tori are incompressible, such a component must be contractible [@Hatcher1]. Consider the union $X$ of all the components of ${{\mathrm{Emb}}}(\sqcup_{i=1}^n T_i,C)/\prod_{i=1}^n {{\mathrm{Diff}}}(T_i)$ which correspond to embeddings whose image are the base-level of the JSJ-splitting of $C$. $X$ must have the homotopy type of the symmetric group $S_n$. Consider $S_n$ to be the subspace $S_n \equiv {{\mathrm{Diff}}}(\sqcup_{i=1}^n T_i)/\prod_{i=1}^n {{\mathrm{Diff}}}(T_i)
\simeq X \subset {{\mathrm{Emb}}}(\sqcup_{i=1}^n T_i,C)/\prod_{i=1}^n {{\mathrm{Diff}}}(T_i)$.
The above argument proves that there is a fiber-homotopy equivalence, where all the vertical arrows are given by inclusion.
$$\xymatrix{{{\mathrm{PDiff}}}^V(C,T) \ar[r] & {{\mathrm{Diff}}}(C,T) \ar[r] & X \\
{{\mathrm{PDiff}}}^V(C,T) \ar[r] \ar[u] & {{\mathrm{Diff}}}^V(C,T) \ar[r] \ar[u] & S_n \ar[u] }$$
Typically it is demanded that fibrations are onto. Since the long knot $f$ is a connected-sum, and some of the summands $\{f_i : i \in \{1,2,\cdots,n\}\}$ may be repeated, define the equivalence relation $\sim$ on $\{1,2,\cdots, n\}$ by $i \sim j \Leftrightarrow
f_i$ is isotopic to $f_j$. Let $\Sigma_f \subset S_n$ be the partition-preserving subgroup of $S_n$. Thus the above fibration is onto $\Sigma_f \subset S_n$.
Since every diffeomorphism $g \in {{\mathrm{PDiff}}}^V(C,T)$ restricts to a diffeomorphism of $V$, consider the restriction to $V \simeq S^1 \times P_n$. Since the $g$ extends to a diffeomorphism of ${{\Bbb R}}^3$, $g_{|V} : V \to V$ must preserve (up to isotopy) the longitudes and meridians of each $T_i$. To be precise, a meridian of $T_i$ is an oriented closed essential curve in $T_i$ which bounds a disc in ${{\Bbb R}}^3 - int(C_i)$. The orientation of the meridian is chosen so that the linking number of the meridian with the knot is $+1$. A longitude in $T_i$ is an essential oriented curve in $T_i$ which bounds a Seifert surface in $C_i$. The orientation of the curve is chosen to agree with the orientation of $f_i$.
Thus, if we identify $V$ with $S^1 \times P_n$ in a way that sends knot meridians to fibers of $S^1 \times P_n$ and the longitude of $f_i$ to $\{1\}\times \eta_i\subset S^1 \times P_n$ for all $i \in \{1,2,\cdots,n\}$ then (by a slight abuse of notation) $g_{|S^1 \times P_n} \in {{\mathrm{PDiff}}}(S^1 \times P_n)$.
Define ${{\mathrm{KDiff}}}^V(C,T) \subset {{\mathrm{PDiff}}}^V(C,T)$ and $\widetilde{{{\mathrm{KDiff}}}}^V(C,T) \subset {{\mathrm{Diff}}}^V(C,T)$ to be the subgroups such that each diffeomorphism $g$ restricts to a diffeomorphism of $V \equiv S^1 \times P_n$, $g_{|S^1 \times P_n} \in {{\mathrm{KDiff}}}(S^1 \times P_n)$ and $g_{|S^1 \times P_n} \in \widetilde{{{\mathrm{KDiff}}}}(S^1 \times P_n)$ respectively. By Lemma \[diffs1pn\], the vertical inclusion maps in the diagram below give a fiber-homotopy equivalence
$$\xymatrix{{{\mathrm{PDiff}}}^V(C,T) \ar[r] & {{\mathrm{Diff}}}^V(C,T) \ar[r] & \Sigma_f \\
{{\mathrm{KDiff}}}^V(C,T) \ar[u]^\simeq
\ar[r] \ar[u]^\simeq & \widetilde{{{\mathrm{KDiff}}}}^V(C,T) \ar[r]
\ar[u]^\simeq & \Sigma_f \ar[u]^\simeq }$$
Analogously to Lemma \[pp\], the inclusion ${{\mathrm{KDiff}}}(S^1 \times P_n) \times \prod_{i=1}^n {{\mathrm{Diff}}}(C_i,T_i) \to {{\mathrm{KDiff}}}^V(C,T)$ is a homotopy equivalence.
If we apply Lemma \[cohenlemma\] to the above fibration, we get the normal covering space
$$\xymatrix{\Sigma_f \ar[r] &
B{{\mathrm{KDiff}}}(S^1 \times P_n) \times \prod_{i=1}^n B{{\mathrm{Diff}}}(C_i,T_i) \ar[r] \ar[d]_\simeq &
B\widetilde{{{\mathrm{KDiff}}}}^V(C,T) \ar[d]_\simeq \\
& {{\mathcal{C}}}_2(n)\times \prod_{i=1}^n {{\mathcal{\hat K}}}_{f_i} & {{\mathcal{\hat K}}}_f}$$
where the two vertical homotopy equivalences come from Lemma \[bdiff\] and the identification ${{\mathrm{KDiff}}}(S^1 \times P_n) \equiv {{\mathrm{KDiff}}}(P_n)$
Consider ${{\mathcal{C}}}_2(n) \times \prod_{i=1}^n {{\mathcal{\hat K}}}_{f_i}$ as a $\Sigma_f$-space, where the $\Sigma_f$ action is simply the restriction of the diagonal $S_n$ action $S_n \times ({{\mathcal{C}}}_2(n) \times {{\mathcal{\hat K}}}^n) \to {{\mathcal{C}}}_2(n) \times {{\mathcal{\hat K}}}^n$ to $\Sigma_f \times ({{\mathcal{C}}}_2(n) \times \prod_{i=1}^n {{\mathcal{\hat K}}}_{f_i}) \to {{\mathcal{C}}}_2(n) \times \prod_{i=1}^n {{\mathcal{\hat K}}}_{f_i}$. By design, the homotopy equivalence $B{{\mathrm{KDiff}}}^V(S^1 \times P_n) \times \prod_{i=1}^n B{{\mathrm{Diff}}}(C_i,T_i) \to
{{\mathcal{C}}}_2(n) \times \prod_{i=1}^n {{\mathcal{\hat K}}}_{f_i}$ is $\Sigma_f$-equivariant (see Definition \[pd\_def\]).
Thus we know abstractly that there exists a homotopy equivalence between $({{\mathcal{C}}}_2(n)\times \prod_{i=1}^n {{\mathcal{\hat K}}}_{f_i})/\Sigma_f$ and ${{\mathcal{\hat K}}}_f$. To finish the proof, we show $\kappa_n :({{\mathcal{C}}}_2(n)\times \prod_{i=1}^n {{\mathcal{\hat K}}}_{f_i})/\Sigma_f \to {{\mathcal{\hat K}}}_f$ is such a homotopy equivalence. Since both the domain and range of $\kappa_n$ are $K(\pi,1)$’s, it suffices to show that the diagram below commutes.
$$\xymatrix{\pi_1 B{{\mathrm{KDiff}}}(S^1 \times P_n) \times \prod_{i=1}^n \pi_1 B{{\mathrm{Diff}}}(C_i,T_i) \ar[r] \ar[d]_\simeq &
\pi_1 B\widetilde{{{\mathrm{KDiff}}}}^V(C,T) \ar[d]_\simeq \\
\pi_1 {{\mathcal{C}}}_2(n)\times \prod_{i=1}^n \pi_1 {{\mathcal{\hat K}}}_{f_i} \ar[r]^-{\pi_1 \kappa_n} & \pi_1 {{\mathcal{\hat K}}}_f}$$
Fix $i \in \{1,2,\cdots,n\}$ and $\phi \in \pi_0 {{\mathrm{Diff}}}(C_i,T_i)$. Consider $\phi$ to be an element of $\pi_1 B{{\mathrm{KDiff}}}(S^1 \times P_n) \times \prod_{i=1}^n \pi_1 B{{\mathrm{Diff}}}(C_i,T_i)$ by the standard inclusion. If one chases $\phi$ along the clockwise route around the diagram to $\pi_1 {{\mathcal{\hat K}}}_f$, one is simply converting $\phi$ into an element $\overline \phi \in \pi_1 {{\mathcal{\hat K}}}_f$ using Lemma \[bdiff\]. This means that one is applying an isotopy to the $i$-th knot summand $f_i$ of $f$, and the isotopy has support in $B_i$ (see Lemma \[pnlem\]). If one chases $\phi$ along the counter-clockwise route around the diagram, one converts $\phi$ into a loop in $\pi_1 {{\mathcal{\hat K}}}_{f_i}$ using Lemma \[bdiff\], then the little cubes construction is applied to this loop creating a second loop $\tilde \phi \in \pi_1 {{\mathcal{\hat K}}}_f$. The loop produced via the little cubes construction $\tilde \phi$ is the same loop in $\pi_1 {{\mathcal{\hat K}}}_f$ as $\overline \phi$ since the little cubes and other knot summands remain fixed through the isotopy, keeping the support of the isotopy in $B_i$.
Given $\theta \in \pi_0 {{\mathrm{KDiff}}}(S^1 \times P_n)$ consider it as an element of $\pi_1 B{{\mathrm{KDiff}}}(S^1 \times P_n) \times \prod_{i=1}^n \pi_1 B{{\mathrm{Diff}}}(C_i,T_i)$ by the standard inclusion. We will chase $\theta$ around the diagram. This chase is a little more involved than the previous one, as it involves the little cubes action on ${{\mathcal{\hat K}}}$ in a non-trivial manner.
Our strategy for the proof is to chase $\theta$ around the diagram in a counter-clockwise manner to get an element in $\pi_0 \widetilde{{{\mathrm{KDiff}}}}^V(C,T)$. We denote this diffeomorphism by $C_\theta$. We need to show that $C_\theta$ is the identity on $\sqcup_{i=1}^n C_i$ and when restricted to $V$, $C_{\theta | V}\equiv \theta$ under our identification $V \equiv S^1 \times P_n$. We will do this via an explicit computation. First, notice that we can simplify the problem. $\theta$ determines a loop $\widetilde{\theta} \in \pi_1 {{\mathcal{C}}}_2(n)$ which in turn defines an isotopy $\kappa_n(\widetilde{\theta},f_1,f_2,\cdots,f_n)$ of $f$, which by Lemma \[bdiff\] determines the diffeomorphism $C_\theta$ of $C$. Recall how $C_\theta$ is constructed. Given an isotopy $F_\theta: [0,1] \times B \to B$ such that
- $F_\theta(0,x)=x$ for all $x \in B$
- $F_\theta(t,x)=x$ for all $x \in T = \partial B$ and $t \in [0,1]$
- $F_\theta(t,x)=\kappa_n(\widetilde{\theta}(t),f_1,f_2,\cdots,f_n)(x)$ for all $(t,x) \in [0,1] \times B$.
Then $C_\theta(x)=F_\theta(1,x)$ for $x \in C$.
Define $T_\theta : B \to B$ by $T_\theta(x)=F_\theta(1,x)$ for $x \in B$. $\pi_0 {{\mathrm{KDiff}}}(S^1 \times P_n) \simeq \pi_0 {{\mathrm{KDiff}}}(P_n)$ is the pure braid group which can be in turn thought of as a subgroup of the full braid group, $\pi_0 \widetilde{{{\mathrm{KDiff}}}}(S^1 \times P_n) \simeq
\pi_0 \widetilde{{{\mathrm{KDiff}}}}(P_n) \simeq \pi_0 {{\mathrm{Diff}}}(P_n)$. In $\pi_0 {{\mathrm{Diff}}}(P_n)$ every element can be written as a product of Artin generators $\{\sigma_i : i \in \{1,2,\cdots,n-1\}\}$ (see for example [@Birman]), these are the half Dehn twists about curves bounding the $i$-th and $(i+1)$-st punctures of $P_n$. Let $\theta=\alpha_j \circ \alpha_{j-1} \circ \cdots \circ \alpha_1$ where $\alpha_i \in {{\mathrm{Diff}}}(P_n)$ are either Artin generators or their inverses, thus $T_\theta = T_{\alpha_j} \circ T_{\alpha_{j-1}} \circ \cdots \circ T_{\alpha_1}$. This in principle reduces our problem to studying $T_{\sigma_i}$ for $i \in \{1,2,\cdots,n-1\}$.
\[fig15\]
\[tl\]\[tl\]\[1\]\[0\][$1$]{} \[tl\]\[tl\]\[1\]\[0\][$n$]{} \[tl\]\[tl\]\[1\]\[0\][$i$]{} \[tl\]\[tl\]\[1\]\[0\][$i+1$]{} \[tl\]\[tl\]\[1\]\[0\][$\sigma_i$]{} $$\includegraphics[width=10cm]{sigmai.eps}$$
Figure 15
\[fig16\]
\[tl\]\[tl\]\[0.9\]\[0\][$1$]{} \[tl\]\[tl\]\[0.9\]\[0\][$i$]{} \[tl\]\[tl\]\[0.9\]\[0\][$i+1$]{} \[tl\]\[tl\]\[0.9\]\[0\][$n$]{} \[tl\]\[tl\]\[0.9\]\[0\][$\cdots$]{} \[tl\]\[tl\]\[1\]\[0\][$\widetilde{\sigma_{i}}$]{} $$\includegraphics[width=10cm]{cusig.eps}$$
Figure 16
By the definition of $\kappa_n$, $T_{\sigma_i}$ is the identity on the balls $B_k$ for $k \notin \{i,i+1\}$, and $T_{\sigma_i}$ permutes the two balls $B_i$ and $B_{i+1}$, acting by translation. Thus $T_\theta$ must restrict to be the identity on $\sqcup_{i=1}^n C_i$.
Let $*=(0,-1,0) \in \partial B$ be the base-point of $B$. Let $\xi_i : [0,1] \to B$ be the unique affine-linear function so that $\xi_i(0)=*$ and $\xi_i(1)=(\frac{4i-2n-2}{2n+1},-\frac{1}{2n+1},0)\in \partial B_i$. Let $p_i : S^1 \to C_i$ be a longitude of $C_i$ starting and ending at $\xi_i(1)$. Since $T_{\sigma_i}$ acts by translation on the balls $\{B_i : i \in \{1,2,\cdots,n\}\}$, for all $k \in \{0,1,2,\cdots,j\}$ define the $i$-th longitude $p_i^k$ of $(\alpha_k \circ \alpha_{k-1} \circ \cdots \circ \alpha_1)(C)$ to be the restriction of $\sqcup_{s=1}^n \left(\alpha_k \circ \alpha_{k-1} \circ \cdots
\circ \alpha_1 \circ p_s\right) :
\sqcup_{s=1}^n S^1 \to B$ to $(\sqcup_{s=1}^n \alpha_k \circ \alpha_{k-1} \circ \cdots \circ \alpha_1 \circ p_s)^{-1}(B_i)$. Define $l_i = \overline{\xi_i} \cdot p_i \cdot \xi_i$ and similarly $l_i^k=\overline{\xi_i} \cdot p_i^k \cdot \xi_i$, so $l_i^0=l_i=l_i^j$ for all $i \in \{1,2,\cdots,n\}$.
$\pi_1 \left( (\alpha_k \circ \alpha_{k-1} \circ \cdots \circ \alpha_1)(C)\right)$ therefore has a natural identification with ${{\Bbb Z}}\times (*_{i=1}^n {{\Bbb Z}})$ which has presentation $\langle m,l^k_1,l^k_2,\cdots,l^k_n : [m,l^k_1],[m,l^k_2],\cdots,[m,l^k_n]\rangle$. Here $m$ is a knot meridian, or equivalently a fiber of the Seifert fibering of the base-manifold of the JSJ-splitting of $C$.
\[fig17\]
\[tl\]\[tl\]\[0.9\]\[0\][$l_1$]{} \[tl\]\[tl\]\[0.9\]\[0\][$l_2$]{} \[tl\]\[tl\]\[0.9\]\[0\][$l_n$]{} \[tl\]\[tl\]\[0.9\]\[0\][$B_1$]{} \[tl\]\[tl\]\[0.9\]\[0\][$B_2$]{} \[tl\]\[tl\]\[0.9\]\[0\][$B_n$]{} \[tl\]\[tl\]\[0.9\]\[0\][$*$]{} \[tl\]\[tl\]\[0.9\]\[0\][$\cdots$]{} $$\includegraphics[width=10cm]{longeg.eps}$$
Figure 17
Call the above identification $\phi_k : \pi_1 \left( (\alpha_k \circ \alpha_{k-1} \circ \cdots \circ \alpha_1)(C)\right)
\to {{\Bbb Z}}\times (*_{i=1}^n {{\Bbb Z}})$. $\phi_k$ determines a diffeomorphism $\widetilde{\phi_k} : (\alpha_k \circ \alpha_{k-1} \circ \cdots \circ \alpha_1)(C)
\to S^1 \times P_n$ defined by the condition that $\widetilde{\phi_k}(l^k_i)=\{1\} \times \lambda_i$, $\widetilde{\phi_k}(m)=S^1 \times \{*\}$.
Recall the Dehn-Nielsen theorem [@Nielsen] (see [@Zie] for a modern proof). It states that the map $\pi_0 {{\mathrm{Diff}}}(P_n) \to {{\mathrm{Aut}}}(\pi_1 P_n)$ is injective. We compute the induced automorphism on ${{\Bbb Z}}\times (*_{i=1}^n {{\Bbb Z}})$ given by the composite $\widetilde{\phi_{k+1}} \circ T_{\alpha_{k+1}} \circ \widetilde{\phi_k^{-1}}$. Without loss of generality, assume $\alpha_{k+1} = \sigma_q$ for some $q \in \{1,2,\cdots,n-1\}$, therefore $\kappa_n (\widetilde{\alpha_{k+1}},f_1,f_2,\cdots,f_n)$ represents an isotopy which pulls the knot summand in the ball $B_{q+1}$ through the knot summand in the ball $B_q$. Therefore, $\pi_1 \left(\widetilde{\phi_{k+1}} \circ T_{\alpha_{q}} \circ \widetilde{\phi_k^{-1}}\right)$ fixes $m$ and fixes $\lambda_i$ unless $i \in \{q,q+1\}$, in which case $(\widetilde{\phi_{k+1}} \circ T_{\alpha_{q}} \circ \widetilde{\phi_k^{-1}})(\lambda_{q+1})=
\lambda_{q+1}\lambda_{q}\lambda_{q+1}^{-1}$ and $(\widetilde{\phi_{k+1}} \circ T_{\alpha_{q}} \circ \widetilde{\phi_k^{-1}})(\lambda_{q+1})=
\lambda_q$.
Thus, via our identifications, $C_\theta \in {{\mathrm{KDiff}}}^V(C,T)$ induces the same automorphism of $\pi_1 V \equiv \pi_1 (S^1 \times P_n)$ as does $\theta \in {{\mathrm{KDiff}}}(S^1 \times P_n)$, which proves the theorem.
There is a little $2$-cubes equivariant homotopy equivalence $${{\mathrm{EC}({1,D^2})}} \simeq {{\mathcal{C}}}_2 ({{\mathcal{P}}}\sqcup \{*\}) \times \Omega^2\Bbb CP^\infty$$ where $\Bbb CP^\infty=BS^1=B^2{{\Bbb Z}}$.
Where from here? {#endsec}
================
There are several directions one could go from here. One direction would be to ask, what is the homotopy type of the full space ${{\mathcal{K}}}$? By Theorem \[freeness\] this is equivalent to asking what is the homotopy type of ${{\mathcal{P}}}$ but Theorem \[freeness\] can be used to refine this question further.
Starting with the unknot, one can produce new knots by: using hyperbolic satellite operations, cablings, or taking the connected-sum of knots. If these procedures are iterated, one produces all knots [@Thurston; @JacoShalen; @bjsj]. Theorem \[freeness\] tells us the homotopy-type of a component corresponding to a knot which is a connect-sum. If $f \sim f_1 \# \cdots \# f_n$ is the prime decomposition of $f$, then ${{\mathcal{K}}}_f \simeq ({{\mathcal{C}}}_2(n) \times_{S_n} \prod_{i=1}^n {{\mathcal{K}}}_{f_i})$. To complete our understanding of ${{\mathcal{K}}}$ all we need to understand is:
1. How the homotopy type of ${{\mathcal{K}}}_f$ is related to the homotopy type of ${{\mathcal{K}}}_g$ if $f$ is a cabling of $g$.
2. If $f$ is obtained from knots $\{f_i : i \in \{1,2,\cdots,n\}\}$ via a hyperbolic satellite operation, how is the homotopy type of ${{\mathcal{K}}}_f$ related to ${{\mathcal{K}}}_{f_i}$ for $i \in \{1,2,\cdots, n\}$.
Hatcher has answered question 1.
\[htc\] (Hatcher) [@Hatcher4] If a knot $f$ is a cabling of a knot $g$ then ${{\mathcal{K}}}_f \simeq S^1 \times {{\mathcal{K}}}_g$
More recently, a solution to question 2 has appeared in [@topknot]. Roughly, if a knot $f$ is obtained from knots $\{f_i : i \in \{1,2,\cdots,n\}\}$ by a hyperbolic satellite operation then there is a fibration
$$\prod_{i=1}^n {{\mathcal{K}}}_{f_i} \to {{\mathcal{K}}}_f \to S^1 \times S^1$$
and the monodromy of this fibration depends on both the knots $f_i$, their symmetry properties, and the symmetry properties of the hyperbolic manifold that is the root of the JSJ-tree of $f$. For brevity, we skip the full statement of the result. A key theorem of Sakuma’s is used to compute the monodromy of this fibration – allowing us to show the fibration is split at the base, thus the fundamental group of any component of ${{\mathcal{K}}}$ is an iterated semi-direct product of finite-index subgroups of braid groups.
More generally, one could ask, what is the homotopy type of other spaces of knots?
Perhaps the next simplest case is the space of embeddings of a circle in a sphere ${{\mathrm{Emb}}}(S^1,S^n)$. As is shown in [@BudCoh], there is a homotopy equivalence ${{\mathrm{Emb}}}(S^1,S^n) \simeq {{\mathrm{Emb}}}({{\Bbb R}},{{\Bbb R}}^n) \times_{SO_{n-1}} SO_{n+1}$. Thus, if one knows the homotopy type of ${{\mathrm{Emb}}}({{\Bbb R}},{{\Bbb R}}^n)$ as an $SO_{n-1}$-space, one knows the homotopy type of ${{\mathrm{Emb}}}(S^1,S^n)$. The homotopy-type of ${{\mathcal{K}}}$ as an $SO_2$-space is determined in [@topknot].
Another interesting question is ‘what is the homotopy type of the space of closed, connected, $1$-dimensional submanifolds of $S^n$’? This space is naturally homeomorphic to ${{\mathrm{Emb}}}(S^1,S^n)/{{\mathrm{Diff}}}(S^1)$ and has been studied recently by Hatcher [@Hatcher4] in the $n=3$ case. Studying the homotopy type of these spaces appears to have more complications due to the delicate extension problems involved. An interesting point of Hatcher’s work is that one needs to know the answer to the Linearization Conjecture in order to understand even the homotopy type of the component of a knot as simple as a hyperbolic knot. One could go further and ask, what is the homotopy-type of the double-coset space $SO_{n+1}\backslash {{\mathrm{Emb}}}(S^1,S^n)/{{\mathrm{Diff}}}(S^1)$? This is a particularly delicate problem as the action of $SO_{n+1}\times {{\mathrm{Diff}}}(S^1)$ on ${{\mathrm{Emb}}}(S^1,S^n)$ is not free. A nice example of the kinds of problems that can arrise is the paper of Kodama and Michor [@komi], where they prove that the figure-8 component of ${{\mathrm{Imm}}}(S^1,{{\Bbb R}}^2)/{{\mathrm{Diff}}}(S^1)$ has the homotopy-type of $\Bbb CP^\infty$.
It would be very interesting to know more about the homotopy-type of the embedding spaces ${{\mathrm{Emb}}}({{\Bbb R}}^j,{{\Bbb R}}^n)$ or ${{\mathrm{Emb}}}(S^j,S^n)$. Unfortunately the techniques of this paper are of limited use since it is still unknown whether or not a smooth embedded $3$-sphere in ${{\Bbb R}}^4$ bounds a smooth ball [@Kirby], and very little is known about the homotopy type of ${{\mathrm{Diff}}}(D^4)$ other than Morlet’s ‘Comparison’ Theorem [@Mor; @BL; @KS].
There are however some results known in dimension $4$. Sinha and Scannell prove have computed many rational homotopy-groups of the long knot space ${{\mathrm{Emb}}}({{\Bbb R}},{{\Bbb R}}^4)$ and the corresponding framed long knot space ${{\mathrm{EC}({1,D^3})}}$, showing non-triviality in dimensions $\{2,4,5,6\}$. The fibration ${{\mathrm{Diff}}}(D^4) \to {{\mathrm{EC}({1,D^3})}}$ has a fiber which is homotopy equivalent to ${{\mathrm{Diff}}}(S^2 \times D^2)$ (diffeomorphisms fixing the boundary). The homotopy LES of this fibration splits into short exact sequences $0 \to \pi_{i+1} {{\mathrm{EC}({1,D^3})}} \to \pi_i {{\mathrm{Diff}}}(S^2\times D^2) \to \pi_i {{\mathrm{Diff}}}(D^4) \to 0$. We can deduce from this that $\pi_i {{\mathrm{Diff}}}(S^2 \times D^2)$ has non-torsion elements for $i\in \{1,3,4,5\}$. By Theorem \[littlecthm\] we know that ${{\mathrm{Diff}}}(S^2 \times D^2) \simeq {{\mathrm{EC}({2,S^2})}}$ is a $3$-fold loop space. Three-dimensional instincts might lead one to suspect that the inclusion $\Omega^2 SO_3 \subset {{\mathrm{Diff}}}(S^2 \times D^2)$ is a homotopy equivalence, where $\Omega^2 SO_3$ is thought of as the subgroup of fiber-preserving (fibrewise-linear) diffeomorphisms of $S^2 \times D^2$. These instincts would be wrong! We have just seen that although the inclusion $\Omega^2 SO_3 \to {{\mathrm{Diff}}}(S^2 \times D^2)$ admits a $3$-fold de-looping, it can not be a homotopy equivalence since the homotopy groups of the domain and range are not the same.
A possible application of the Sinha, Scannell result would be the study of ‘spun’ knots. Given $f \in \pi_i {{\mathrm{Emb}}}({{\Bbb R}},{{\Bbb R}}^n)$ one constructs a smooth embedding $S^{i+1} \to {{\Bbb R}}^{n+i}$ by ‘spinning’ $f$ about an $(n-1)$-dimensional linear subspace of ${{\Bbb R}}^{n+i}$ (this is a slight generalization of Litherland’s notion of deform twist-spun knots [@Lith], see Figure 18). In the spirit of Markov’s Theorem [@Birman], it would seem natural to conjecture that for some co-dimensions the ‘spinning map’ $$\pi_i {{\mathrm{Emb}}}({{\Bbb R}},{{\Bbb R}}^n) \to \pi_0 {{\mathrm{Emb}}}(S^{i+1},{{\Bbb R}}^{n+i})$$ is an isomorphism.
\[fig18\]
\[tl\]\[tl\]\[1\]\[0\] \[tl\]\[tl\]\[1\]\[0\][${{\Bbb R}}^{i+n}$]{} \[tl\]\[tl\]\[1\]\[0\][${{\Bbb R}}^{n-1}$]{} \[tl\]\[tl\]\[1\]\[0\] $$\includegraphics[width=8cm]{spun.eps}$$
Figure 18
In the time between this article being accepted and published, some progress has been made on this problem. It turns out that, provided $2n-3j-3 \geq 0$ the first non-trivial homotopy group of ${{\mathrm{Emb}}}({{\Bbb R}}^j,{{\Bbb R}}^n)$ is cyclic and in dimension $2n-3j-3$. Moreover in these cases, a spinning construction $\Omega {{\mathrm{Emb}}}({{\Bbb R}}^j,{{\Bbb R}}^n) \to {{\mathrm{Emb}}}({{\Bbb R}}^{j+1},{{\Bbb R}}^{n+1})$ induces an epi-morphism on the first non-trivial homotopy groups of the spaces. In particular, the spinning map $\pi_2 {{\mathrm{Emb}}}({{\Bbb R}},{{\Bbb R}}^4) \to \pi_0 {{\mathrm{Emb}}}(S^3,{{\Bbb R}}^6)$ is an isomorphism – both groups are infinite-cyclic in this case [@family].
I would like to thank Fred Cohen for teaching me about little cubes actions and pushing me to prove the existence of a little $2$-cubes action on spaces of long knots and ‘some kind of more general theorem,’ which turned out to be Theorem \[littlecthm\]. Allen Hatcher’s visit to Rochester in the spring of 2003, and his preprint [@Hatcher4] on the homotopy type of ${{\mathcal{K}}}$, were of immense help when it came to formulating Theorem \[freeness\]. I would like to thank Dev Sinha, whose work [@Dev] on spaces of knots has been highly inspiring. Allen Hatcher, Dev Sinha, and Victor Turchin’s comments on the first few iterations of this paper were immensely valuable. I would like to thank the Mathematics Department at the University of Rochester for their hospitality during my visit.
[999]{}
J. Birman, *Braids, links, and mapping class groups,* Annals of Mathematics Studies, No. **82**. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974.
J.M. Boardman, R.M. Vogt. *Homotopy-everything H-spaces,* Bull. Amer. Math. Soc. [**74**]{} (1968), 1117–1122.
A. Borel, *Seminar on transformation groups,* Ann. Math. Stud. [**46**]{} Princeton University Press, Princeton, NJ. (1960)
R. Bott, C. Taubes, *On the self-linking of knots,* J. Math. Phys. [**35**]{}, 5247–5287 (1994)
R. Budney, J. Conant, K. Scannell, D. Sinha. *New perspectives on self-linking,* Advances in Mathematics. [**191**]{} (2005) 78–113.
R. Budney, F.R. Cohen, *On the homology of the space of long knots in ${{\Bbb R}}^3$*, preprint.
R. Budney, *JSJ-decompositions of knot and link complements in $S^3$*, to appear in l’Enseignement Mathematique.
R. Budney, *Topology of spaces of knots in dimension $3$,* preprint.
R. Budney, *A family of embedding spaces,* preprint.
D. Burghelea, R. Lashof, *The homotopy type of spaces of diffeomorphisms. I,* Trans. Amer. Math. Soc. **196** (1974) 1–36.
A. Cattaneo, P. Cotta-Ramusino, R. Longoni. *Configuration spaces and Vassiliev classes in any dimension,* Algebr. and Geom. Top. [**2**]{} (2002), 949–1000.
A. Cattaneo, P. Cotta-Ramusino, R. Longoni. *Algebraic structures on graph cohomology,* Journal of Knot Theory and Its Ramifications, Vol. 14, No. 5 (2005) 627-640.
D. Eisenbud, W. Neumann, [*Three-dimensional link theory and invariants of plane curve singularities*]{}, Ann. Math. Stud. [**110**]{}. (1985)
T. Goodwillie, M. Weiss, *Embeddings from the point of view of immersion theory: Part II,* Geom. Topol. **3** (1999), 103-118
A. Gramain, *Le type d’homotopie du groupe des difféomorphismes d’une surface compacte,* Ann. Scient. Éc. Norm. Sup. $4^e$ serie, t. 6, 1973, p. 53 à 66.
A. Gramain, *Sur le groupe fondamental de l’espace des noeuds,* Ann. Inst. Fourier, Grenoble. **27,3** (1977), 29–44.
V. Guillemin, A. Pollack, *Differential Topology,* Prentice-Hall, 1974.
A. Haefliger, *Differentiable embeddings of $S^n$ in $S^{n+q}$ for $q>2$,* Ann. of Math., (1966)
A. Hatcher, *Homeomorphisms of sufficiently-large $P^2$-irreducible $3$-manifolds,* Topology. **15** (1976)
A. Hatcher, *A proof of the Smale conjecture,* Ann. of Math. **177** (1983)
A. Hatcher, *Basic Topology of 3-Manifolds,* *\[http://www.math.cornell.edu/0.5mmhatcher/3M/3Mdownloads.html\]*
A. Hatcher, *Topological Moduli Spaces of Knots,* *\[http://front.math.ucdavis.edu/math.GT/9909095\]*
M.W. Hirsch, *Differential Topology,* Springer-Verlag. (1976)
N.V. Ivanov, *Diffeomorphism groups of Waldhausen manifolds,* Research in Topology. II. Notes of LOMI scientific seminars, V. 66 (1976), 172–176. J. Soviet Math., V. 12, No. 1 (1979), 115–118.
N.V. Ivanov, *Homotopy of spaces of automorphisms of some three-dimensional manifolds,* DAN SSSR, V. 244, No. 2 (1979), 274–277. Soviet Mathematics-Doklady, V. 20, No. 1 (1979), 47–50.
W. Jaco, P. Shalen, *A new decomposition theorem for $3$-manifolds,* Proc. Sympos. Pure Math [**32**]{} (1978) 71–84.
R. Kirby, *Problems in low-dimensional topology,* Geometric topology (Athens, GA, 1993), 35–473, AMS/IP Stud. Adv. Math., 2.2, Amer. Math. Soc., Providence, RI, 1997.
R. Kirby, L. Siebenmann, *Foundational Essays on topological manifolds, smoothings, and triangulations,* Annals of math. Stud. **88** Princeton University Press. (1977)
H. Kneser, *Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten,* Jahresbericht der Deut. Math. Verein., [**38**]{} (1929), 248–260.
H. Kodama, P. Michor, *The homotopy type of the space of degree 0 immersed plane curves,* Revista Matemática Complutense [**19**]{} (2006), no. 1, 227-234.
T. Kohno, *Linear representations of braid groups and classical Yang-Baxter equations,* Cont. Math. [**78**]{} (1988), 339–363.
M. Kontsevich, *Vassiliev’s knot invariants,* I. M. Gelfand Seminar (S. Gelfand and S. Gindikin, eds.) Adv. Soviet. Math., [**16**]{}, Amer. Math. Soc., Providence, 1993, pp. 137–150.
R.A. Litherland, *Deforming twist-spun knots,* Trans. Amer. Math. Soc. [**250**]{} (1979), 311–331.
M. Markl, S. Shnider, J. Stasheff, *Operads in algebra, topology and physics,* AMS Mathematical surveys and monographs. Vol [**96**]{}.(2002)
J.P. May, *The Geometry of Iterated Loop Spaces,* Lecture Notes in Mathematics. **271** (1972)
J.P. May, *$E_{\infty}$ spaces, group completions, and permutative categories,* London Math. Soc. Lecture Notes Series **11**, 1974, 61–93.
J. McClure, J. Smith. *Cosimplicial objects and little $n$-cubes. I,* *\[http://front.math.ucdavis.edu/math.QA/0211368\]*
C. Morlet, *Plongement et automorphismes de variétés,* Notes multigraphiées, Collège de France, Cours Peccot (1969).
W. Neumann, *Notes on geometry and $3$-manifolds,* Low-dimensional Topology. Bolyai Society Mathematical Studies [**8**]{} (1999), 191–267.
J. Nielsen, *Untersuchungen zur theorie der geschlossenen zweiseitigen Flächen I,* Acta Math. [**50**]{} (1927), 189–358.
R. Palais, *Local triviality of the restriction map for embeddings,* Comment. Math. Helv. [**34**]{} (1960) 305–312.
P. Salvatore, *Knots, operads and double loop spaces,* preprint.
K. Scannell, D. Sinha, *A one-dimensional embedding complex,* Journal of Pure and Applied Algebra [**170**]{} (2002), No. 1, 93–107.
H. Schubert, *Die eindeutige Zerlegbarkeit eines Knoten in Primknoten,* Sitzungsber. Akad. Wiss. Heidelberg, math.-nat. KI., **3:57-167**. (1949)
D. Sinha, *The topology of spaces of knots,* preprint, *\[http://front.math.ucdavis.edu/math.AT/0202287\]*
D.Sinha, *Operads and knot spaces.* J. Amer. Math. Soc. [**19**]{} (2006), no. 2, 461–486
S. Smale, *Diffeomorphisms of the $2$-sphere,* Proc. Amer. Math. Soc. **10** (1959) 621–626.
N. Steenrod, *A convenient category of topological spaces,* Mich. Math. J. **14** no. 2, (1967) 133–152.
W. Thurston, *Hyperbolic structures on 3-manifolds, I. Deformation of acylindrical manifolds,* Ann. of Math. **124** (1986), 203–246.
V. Turchin, *On the homology of the spaces of long knots,* Advances in Topological Quantum Field Theory. NATO Sciences series by Kluwer 2005, pp 23-52.
V. Vassiliev, *Complements of discriminants of smooth maps: topology and applications,* Translations of Mathematical Monographs, [**98**]{}. American Mathematical Society, Providence, RI, 1992.
I. Volic, *Finite-type knot invariants and calculus of functors,* Brown University. (2003)
H. Zieschang, E. Vogt, H. Coldeway, *Surfaces and planar discontinuous groups,* Lecture Notes in Math., no. 835, Springer, 1980.
|
---
abstract: 'Fog computing extends the cloud computing paradigm by allocating substantial portions of computations and services towards the edge of a network, and is, therefore, particularly suitable for large-scale, geo-distributed, and data-intensive applications. As the popularity of fog applications increases, there is a demand for the development of smart data analytic tools, which can process massive data streams in an efficient manner. To satisfy such requirements, we propose a system in which data streams generated from distributed sources are digested almost locally, whereas a relatively small amount of distilled information is converged to a center. The center extracts knowledge from the collected information, and shares it across all subordinates to boost their performances. Upon the proposed system, we devise a distributed machine learning algorithm using the online learning approach, which is well known for its high efficiency and innate ability to cope with streaming data. An asynchronous update strategy with rigorous theoretical support is applied to enhance the system robustness. Experimental results demonstrate that the proposed method is comparable with a model trained over a centralized platform in terms of the classification accuracy, whereas the efficiency and scalability of the overall system are improved.'
author:
-
title: |
Data Analytics for Fog Computing by Distributed Online Learning with Asynchronous Update\
[^1]
---
edge computing, distributed computing, online learning, real-time analytics, stream processing
Introduction {#sec1}
============
Recent decades have witnessed an expansion of cloud computing in terms of both service models and applications. In its simplest form, cloud computing comprises the centralization of computing services using a network of remote servers to enable the sharing of infrastructure resources while achieving economies of scale. However, this centralization also leads to certain side effects. Security risks resulting from the exposure of private user data to cloud providers [@lu2018cyber; @lu2018managing; @lu2018cyber2; @niyato2015performance] and service latency incurred by data transmissions represent the two most significant issues. For privacy- and latency-sensitive applications that require the processing of data in the vicinity of their sources, the cloud’s centralized architecture exhibits defects.
Subsequently, an alternative called fog computing has been introduced [@BonomiMZA12; @iorga2018fog]. In contrast to the cloud, which sends data to a remote location for processing, fog computing allocates substantial amounts of computation, storage, and services toward the edge of a network, i.e., on smart end-devices. It comprises a large number of fog nodes residing between end-devices and centralized (cloud) services. Because fog nodes have awareness of their logical locations in the context of the entire system, they are capable of allocating data to apposite locations for processing. For example, time-sensitive data are analyzed close to their source, and laborious tasks are performed on the cloud. Fog computing thus can reduce latency, conserve network bandwidth, and to some extent alleviate security problems (because data are less centralized compared with the cloud). It is suitable for a breed of distributed, latency-aware services and applications, such as Internet-of-things (IoT) [@niyato2015economics], sensor networks [@niyato2016distributed; @lu2016sensor], smart grid systems [@LuXiaoPower; @niyato2012adaptive; @korki2011mac], cloud systems [@li2019data], communication systems [@lu2019intelligent; @niyato2015game; @lu2019coverage; @lu2018ambient], corporate networks [@lu2011payoff; @lu2015hierarchical; @lu2019ambient; @lu2018performance], and mobile social networks [@zhang2015optimizing].
With the advent of fog applications, there is a demand for smart data analytic systems with intrinsic distributed real-time processing support. However, most existing solutions are a stack of off-the-shelf tools, lacking a particular design that caters for the characteristics of fog computing [@YiLL15]. In this study, we tackle the problem from the core, by generalizing fog data analytics as performing classification over multiple data sources using machine learning. We present a system in which data streams generated from distributed sources are digested almost locally, whereas a relatively small amount of distilled information is converged to a center. The center extracts knowledge from the gathered information, and shares it across subordinates. The subordinates can consist of any computing units, while in a realistic setting, they involve a mass of miniature devices with limited energy, computing power, and communication capacity.
An ideal data analytic algorithm for such a system should have low complexity, high scalability, and a light communication overhead. We herein adopt an online learning approach for its simplicity and efficiency. The proposed distributed online multitask learning method employs a master/slave architecture, in which locally calculated gradients and globally updated model vectors are exchanged over the network. An asynchronous update strategy with rigorous theoretical support is also applied to enhance the system robustness. Experimental results demonstrate that the classification accuracy of the proposed method is comparable with those of classical models trained in a centralized manner, while the communication overhead is controlled at a reasonable level. Our approach is suitable for any classification task, and can be ported to any device with moderate computing power to perform data analytics under the fog computing paradigm.
Related Work {#sec2}
============
Fog computing has been adopted in a broad range of applications since first being proposed by Cisco in 2012 [@BonomiMZA12]. Augmented reality, online gaming, and real-time video surveillance applications that must process large volumes of data with tight latency constraints are the primary targets for fog computing [@YiLL15]. Mobile applications running on resource-constrained devices but requiring fast response time, such as wearable assistants [@DubeyYCAYM16] and smart connected vehicles [@HouLCWJC16], represent an additional use case for fog computing. Finally, geographically distributed systems represented by wireless sensor networks in general, and smart grids in particular, are compatible with fog computing too. Additional applications of fog computing, especially those involving big data analytics, are described in [@tang2015hierarchical; @ZhuCPNHB13; @hong2013mobile].
Online learning represents a family of efficient algorithms that can construct a prediction model incrementally by processing the training data in a sequential manner, as opposed to batch learning algorithms, which train the predictor by learning the entire dataset at once [@abs-1802-02871]. On each round, the learner receives an input, makes a prediction using an internal hypothesis that is retained in memory, and subsequently learns the true label. It then utilizes the new sample to modify its hypothesis according to some predefined rules. The goal is to minimize the total number of rounds with incorrect predictions. In general, online learning algorithms are fast, simple, and require few statistical assumptions. They scale well to a large amount of data, and are particularly suitable for real-world applications in which data arrive continuously.
Existing distributed online learning algorithms can generally be classified into two categories: delayed gradient [@AgarwalD11; @recht2011hogwild], and minibatch gradient [@dekel2012optimal] methods. The main idea of delayed gradient methods is that workers are allowed to pull the latest model from the master to compute gradients and then send them back, while the master can utilize these gradients to update the model if they are not delayed by too long or sparse enough. It has been proven that if the number of delayed iterations is not too significant, then the delayed gradient can still converge in line with the standard online gradient method [@AgarwalD11]. Alternatively, if the gradient is extremely sparse then the delayed gradient method can still converge very effectively [@recht2011hogwild]. In addition, the main idea behind minibatch gradient methods is to utilize the minibatch technique to reduce the variance of the stochastic gradient estimator, which can in turn improve the convergence rate [@dekel2012optimal].
System Overview {#sec3}
===============
We provide an overview of the proposed system from a fog computing perspective. As shown in Figure \[fig1\], fog computing employs a hierarchical architecture consisting of at least three layers. Small devices with cost-effective computing powers are located at the edge of the network. They either act as data sources, by generating data streams of their own, or as data sinks by collecting data from subordinate devices. Besides gathering data and controlling actuators, they can also perform preparatory data analytics in a timely manner. The next layer consists of a number of intermediate computing devices, namely fog nodes, each of which is connected to a group of edge devices in the first layer. These are typically focused on aggregating edge-device data and converting the collected data into knowledge. The cloud computing data center is in the top layer, providing system-wide monitoring and centralized control. Such a hierarchy enables the fog to allocate computing resource according to the task scale, thus striking a balance between quick response time and bulk processing power.
In the context of the fog computing architecture, we propose a system that can facilitate data processing in the fog. It also employs a hierarchical layout, consisting of a generic virtualized device that we call the *Master*, which is dedicated to serving the centralized applications, and numerous client devices that we call *Workers*. It is assumed that *Workers* are distributed over smart end-devices in different physical locations. The *Workers* located at the network edge ingest data generated by various sensors, and then transmit the processed information to the *Master*. Meanwhile, the *Master* sends the global model vector to the *Workers*. The information flow between the *Master* and *Workers* is bi-directional: *Workers* send locally calculated gradients to the *Master*, and the *Master* sends the global model to the *Workers*. As with the fog computing paradigm, there are no communications among *Workers*. It is worth noting that the proposed system resides in layers 1 and 2 of the fog computing architecture.
![Architecture of fog computing.[]{data-label="fig1"}](fog_arch.eps)
There are two important factors to consider when designing such a system. One is to reduce the data exchange over the network, and the other is to make the system robust when dealing with the inevitable network latency. As described below, our solution is well-suited to meet these requirements.
Proposed Algorithm {#sec4}
==================
This section presents a distributed online learning algorithm for binary classification, which serves as the core of the proposed system. The task for binary classification is to assign new observations into one of the two categories, on the basis of rules that are learned from a training set containing observations whose category memberships are known. We consider a scenario in which data streams generated from geo-distributed edge devices have to be processed as a coherent whole. For clarity, it is assumed that a dataset $D$ is distributed over $K$ different devices, and each device is associated with one of $N$ *Workers*, as described above. As the dataset $D$ is divided into $K$ partitions, we require that the data contained in each partition are homogeneous, e.g., for a sensing application they must be related to a well-defined physical entity that is sensed across different locations. Therefore, the data from all partitions can be represented in the same global feature space, and it is possible to utilize the shared information between partitions to enhance the overall learning process. To this end, we can restate our problem as learning from $K$ data sources (or tasks) using $N$ *Workers* under the supervision of one *Master*.
In the following, we employ the notation $I$ to denote the identity matrix. Given two matrices $M \in \mathbb{R}^{m \times n}$ and $N \in \mathbb{R}^{p \times q}$, we denote the Kronecker product of $M$ and $N$ by $M \otimes N$. We use $A_\otimes$ as a shorthand for $A \otimes I$. We will describe the algorithm from the viewpoint of a *Worker* and *Master*, respectively.
Worker {#sec4.1}
------
In the online learning setting, every *Worker* node observes data in a sequential manner. Formally speaking, at each step $t$, the $n$-th *Worker* receives a piece of data $(\x_{i_t,t}, y_{i_t,t})$, where $\x_{i_t,t} \in \mathbb{R}^d$ is a $d$-dimensional vector representing the sample, $y_{i_t,t} \in \{-1,1\}$ refers to its class label, and $i_t \in \{1,\ldots,K\}$ denotes the task index (i.e., the index of the task that generated this data). The classification model for each task is parameterized by a weight vector $\w_{i_t} \in \mathbb{R}^d$. As there are $K$ tasks involved during learning, we choose to update their weight vectors in a coherent manner. Specifically, we appoint the *Master* node to maintain a compound vector $\w_t$, which is formed by concatenating $K$ task weights. That is, $$\w_t^\top = (\w_{1,t}^\top, \ldots, \w_{K,t}^\top)
\label{eq4.1}$$
The model we aim to learn is now parameterized by $\w_t \in \mathbb{R}^{Kd}$. It is periodically updated on the *Master* side, and distributed to the *Workers* on demand. Note that we could designate a *Worker* to process a particular task’s data the whole time, but this is not compulsory. Any *Worker* can interact with any task, and vice versa. We will closely examine how the *Master* updates $\w_t$ later on.
Let us now focus on a single *Worker*. At time $t$, the *Worker* receives data $(\x_{i_t,t}, y_{i_t,t})$ from the task $i_t$, and the weight vector $\w_t$ from the *Master*. For ease of presentation, we introduce a compound representation for $\x_{i_t,t}$, and denote the following vector by $\phi_t \in \mathbb{R}^{Kd}$: $$\phi_t^\top = \left(0, \ldots, 0, \x_{i_t,t}^\top, 0, \ldots, 0 \right)
\label{eq4.2}$$
We can formulate the learning process as a regularized risk minimization problem. To devise the objective function, we first introduce a reproducing kernel Hilbert space (RKHS) $H=\R^{Kd}$ with an inner product $\langle u, v \rangle_H = u^ \top \Ap v$, where $A \in \mathbb{R}^{K \times K}$ is a predefined interaction matrix, which encodes our belief concerning the relationship between the $K$ learning tasks. Different choices for this interaction matrix result in different geometrical assumptions on the tasks, which will be explained later.
Specifically, for an instance $\x_{i_t,t}$ from the $i_t$-th task, we define the feature map as $$\Psi(\x_{i_t,t}) = \Ap^{-1} \phi_t
\label{eq4.3}$$
Therefore, the kernel product between two instances can be computed as $$\kappa(\x_{i_s,s},\x_{i_t,t}) = \langle \Psi(\x_{i_s,s}), \Psi(\x_{i_t,t}) \rangle = \phi_s^\top \Ap^{-1}\phi_t
\label{eq4.4}$$
If all the training data are provided in advance, then we can formulate the objective as an empirical risk minimization problem in the above RKHS. That is, $$\min_{\w} \frac{1}{T} \sum^{T}_{t=1} \log(1 + \exp(-y_{i_t,t} \langle \w, \Psi(\x_{i_t,t}) \rangle_H)) + \frac{\lambda}{2}\| \w \|_H^2
\label{eq4.5}$$
However, under the online learning setting, we can only access the $i_t$-th task sample at the $t$-th learning iteration, which can in turn be used to formulate the $t$-th loss: $$\ell_t(\w_t) = \log(1 + \exp(-y_{i_t,t} \langle \w_t, \Psi(\x_{i_t,t}) \rangle_H)) + \frac{\lambda}{2}\| \w_t \|_H^2
\label{eq4.6}$$
For the above loss, we can calculate its gradient with respect to $\w_t$ as follows: $$\begin{aligned}
\nabla \ell_t(\w_t)
& = \frac{-y_{i_t,t} \Psi(\x_{i_t,t}) \exp(-y_{i_t,t} \langle \w_t, \Psi(\x_{i_t,t}) \rangle_H}
{1 + \exp(-y_{i_t,t} \langle \w_t, \Psi(\x_{i_t,t}) \rangle_H)}
+ \lambda \w_t \\
& = \frac{-y_{i_t,t} \Ap^{-1} \phi_t \exp(-y_{i_t,t} \w_t^\top \phi_t)}
{1 + \exp(-y_{i_t,t} \w_t^\top \phi_t)}
+ \lambda \w_t
\end{aligned}
\label{eq4.7}$$
For the interaction matrix $A$ that encodes our beliefs concerning the relevance between learning tasks, we set it as $$\begin{aligned}
A = \frac{1}{K}
\left[\begin{matrix}
a & -b & \cdots & -b \\
-b & a & \cdots & -b \\
\vdots & \vdots & \ddots & \vdots \\
-b & -b & \cdots & a
\end{matrix} \right]
\end{aligned}
\label{eq4.8}$$ where $a = K + b(K-1)$ and $b$ is a user-defined parameter.
It is then easy to verify that $$\begin{aligned}
A^{-1} = \frac{1}{(1+b)K}
\left[\begin{matrix}
b+K & b & \cdots & b \\
b & b+K & \cdots & b \\
\vdots & \vdots & \ddots & \vdots \\
b & b & \cdots & b+K
\end{matrix} \right]
\end{aligned}
\label{eq4.9}$$
Plugging into and performing some calculations yields $$\nabla\ell_t(\w_t) = (\g_1, \ldots, \g_j, \ldots, \g_K)
\label{eq4.10}$$ with $$\g_j \!=\!
\begin{cases}
\frac{b+K}{(1+b)K} \frac{-y_{i_t,t} \x_{i_t,t} \exp(-y_{i_t,t} \w_{i_t,t}^\top \x_{i_t,t})}{1 + \exp(-y_{i_t,t} \w_{i_t,t}^\top \x_{i_t,t})} \!+\! \lambda \w_{i_t,t} \hspace{5pt} \textrm{if $j\!=\!i_t$} \\
\frac{b}{(1+b)K} \frac{-y_{i_t,t} \x_{i_t,t} \exp(-y_{i_t,t} \w_{i_t,t}^\top \x_{i_t,t})}{1 + \exp(-y_{i_t,t} \w_{i_t,t}^\top \x_{i_t,t})} \!+\! \lambda \w_{j,t} \hspace{6pt} \textrm{otherwise}
\end{cases}
\label{eq4.11}$$
It can be observed from and that the gradient $\nabla\ell_t(\w_t)$ can be computed on a task-wise basis. The gradient under a multitask setting is composed of the gradients for single tasks with different weights. Regarding the weights, we can observe the following: 1) The weight for the $i_t$-th task is the largest, while the weights for the other tasks are the same. 2) The parameter $b$ is employed to trade off the differences between the weights.
So far, we have described how a *Worker* derives the gradient using the latest $\phi_t$ (or equivalently, $\x_{i_t,t}$), $y_{i_t,t}$, $\w_t$, and $A$. Once we have obtained the latest gradient, it seems natural to transmit it to the *Master* immediately to update the model. However, to reduce network traffic and computational cost incurred by rapid updates, we choose to perform the transmission periodically. We allocate every *Worker* a buffer of size $m$, to record up to the $m$ latest data samples, and calculate the average gradient whenever the buffer is full. Specifically, the average gradient of the $n$-th *Worker* is calculated as $$\frac{1}{m} \sum_{s \in B} \nabla \ell_s(\w_t)
\label{eq4.12}$$ where $m$ is the user-defined buffer size and $B$ is the set of indexes for the $m$ buffered examples. We can control the degree of lazy update by tuning $m$.
In practice, however, we choose not to transmit the result of directly over the network. Referring to , we can decompose as $$\frac{1}{m} \sum_{s \in B} \nabla\ell_s(\w_t) =
\Ap^{-1}\bar{\g} + \lambda \w_t
\label{eq4.13}$$ where $$\bar{\g} =
\frac{1}{m}\sum_{s \in B} \frac{-y_{s_t,t} \phi_s \exp(-y_{s_t,t} \w_t^\top \phi_s)}{1 + \exp(-y_{s_t,t} \w_t^\top \phi_s)}
\label{eq4.14}$$
The $\bar{\g}$ in is what the *Worker* actually computes and transmits to the *Master*. The *Master* will utilize the received $\bar{\g}$ and the task-relationship matrix $A$ to construct the average gradient $\frac{1}{m} \sum_{s\in B} \nabla \ell_s(\w_t)$. The reason for this is that $\bar{\g}$ can be more sparse than $\frac{1}{m} \sum_{s\in B} \nabla \ell_s(\w_t)$, especially when $K$ is large. Transmitting a sparse vector rather than a dense one can reduce the network cost. Note that the sparsity results from two factors: 1) most blocks of $\phi_s$ are zero, and 2) we choose to shift the Kronecker product operation, which can reduce the sparsity of resulting vector, to the *Master* side.
Master {#sec4.2}
------
The *Master* node employs the gradient information provided periodically by the *Workers* to update $\w_t$, and then sends the updated $\w_t$ to the *Workers* whenever requested. Specifically, the $n$-th *Worker* transmits the $\bar{\g}$ in to the *Master*. The *Master* utilizes the received $\bar{\g}$ to compute the average gradient as in .
To counter the network latency, we let the *Master* to record the outage durations of each *Worker*, i.e., $\tau_n, n\in\{1,\ldots,N\}$, where $\tau_n$ denotes the number of learning rounds in which the $n$-th *Worker’s* $\bar{\g}$ has not been utilized for the model update. At the beginning of each learning round, the *Master* will first check whether the largest outage value $\max \tau_n$ exceeds the allowed threshold $\tau$. If so, then the *Master* will choose that $\bar{\g}$ to update the model (it may have to wait a short time for the corresponding *Worker* to response). Otherwise, the *Master* will use the latest $\bar{\g}$ from any *Worker* to update the model. This strategy is known as the delayed gradient descent approach [@AgarwalD11]. It can help to improve the convergence rate of a distributed online learning algorithm.
Finally, we summarize the pseudocode for *Worker* and *Master* in Algorithm \[alg1\] and \[alg2\], respectively.
: a sequence of instances $(\x_{i_t,t},y_{i_t,t})$, $i_t\in[K]$, $t\in[T]$; a parameter $m$ specifying the buffer size : a vector $\bar{\g}$ conveying the average gradient information : $\w_0 = \mathbf{0}$
: a regularization parameter $R$; a parameter $\tau$ specifying the maximum outage allowed; a $K \times K$ interaction matrix $A$; a number of gradients $\bar{\g}_n, n=1,\ldots,N$ provided by the Workers : a vector $\w$ conveying the learned model : set $\tau_n=0$ for $n=1,\ldots,N$, $\w_0 = \mathbf{0}$
Experimental Results {#sec5}
====================
We evaluate the proposed algorithm using a synthetic dataset, first introduced in [@sheldon2008graphical]. The problem is to discriminate two classes in a two-dimensional plane with nonlinear decision boundaries. By $\mathbf{x}=(x_1,x_2)$, we denote a point in two-dimensional space. The basic classification boundaries are generated according to the rule $g(\mathbf{x};\mathbf{a}) = \mathrm{sign}(x_2 - h(x_1;\mathbf{a}))$, where $h(x;\mathbf{a})$ is a family of nonlinear functions consisting of the first few terms of a Fourier series, defined as $h(x;\mathbf{a}) = a_1 \sin(x - a_0) + a_2 \sin(2(x - a_0)) + a_3 \cos(x - a_0) + a_4 \cos(2(x - a_0))$. A rotation is applied to the decision boundary to create multiple tasks. Let $R_\theta$ denote the operator that rotates a vector by $\theta$ radians in a counterclockwise direction about the origin. The final family of classifiers is $f(\mathbf{x};\mathbf{a},\theta) = g({R_{\theta}}\mathbf{x};\mathbf{a})$ with $\theta$ as an additional parameter.
A total of 64 tasks are involved in the experiment. Their parameters are generated via a random walk in a parameter space with Gaussian increments. The initial values are set as $\mathbf{a}^{(1)} = (0,1,1,1,1)$ and $\theta^{(1)} = 0$. For $t=2,\ldots,64$, $\mathbf{a}^{(t)} = \mathbf{a}^{(t-1)} + {\epsilon}_t, {\epsilon}_t \sim N(0,\sigma^2 I)$ and $\theta^{(t)} = \theta^{(t-1)} + \delta_t, \delta_t \sim N(0,\sigma^2(\pi/4)^2)$. The parameter $\sigma$ controls the step size, and hence the task similarity. We set it to 0.3. A training sample is generated by choosing an input $\mathbf{x}$ uniformly at random from the square $x_1, x_2 \in [-3, 3]$, and then labeling it according to $f(\mathbf{x};\mathbf{a},\theta)$ (see Figure \[fig2\] for an example). Because the problem is not linearly separable, we add seven additional features, which are derived from the original $x_1$ and $x_2$ via a mapping $(x_1,x_2) \mapsto (x_1,x_2,x_1x_2,x_1^2,x_2^2,x_1^3,x_2^3,x_1x_2^2,x_1^2x_2)$.
We construct a simulation system in accordance with the fog computing paradigm. Its underlying implementation consists of a set of PC-hosted programs communicating with each other in an asynchronous, full-duplex mode. We employ *asyncoro*, a Python library for asynchronous, concurrent, and distributed programming, as the framework [@WEB:http://asyncoro.sourceforge.net]. As illustrated in Figure \[fig3\], the system imitates a streaming data source using a *Spout* node. It continues to produce the aforementioned synthetic data, and sends them to a randomly selected *Worker*. However, in reality, the data source could be Twitter status updates, a branch of temperature sensors, or any entities that continuously generate data. Situations are commonly encountered in real-world applications in which there exist thousands of smart devices (e.g., sensors or smartphones), which are analogous to the *Spout* node in this simulation.
Our experiment involves 64 tasks (data sources), eight *Workers*, and a single *Master*. We set the learning rate $\eta=0.01$, regularization parameter $\lambda=0.001$, and interaction matrix parameter $b=6$, by referring to a small validation set. We employ the *cumulative error rate* as an evaluation metric, which is given by the ratio of the number of mistakes made by the online learner to the number of samples received to date. Besides the proposed distributed online multitask learning (DOML) algorithm, we include two existing methods for comparison. The first is online multitask learning (OML), which adopts a similar multitask learning approach to DOML, but runs on a single machine. The other is a vanilla online learning (OL) method, which maintains a single model for all tasks. Figure \[fig4\] depicts the variations in the cumulative error rate averaged over 64 tasks along the entire online learning process. Note that although online learning algorithms are capable of dealing with infinite samples, we truncate the result by 15,000 samples per task, as the curve will become flat before that, indicating that the model has reached a stable state.
![Architecture of the simulation system. The *Spout* acts as a streaming data source, by sending task-related data $(\x,y)$ to a randomly selected *Worker*. The *Worker* delivers the computed $\bar{\g}$ to the *Master*, and receives the updated model $\w$ in return.[]{data-label="fig3"}](simulation_system_arch.eps)
![Variations of the cumulative error rate, averaged over 64 tasks along the entire online learning process.[]{data-label="fig4"}](cumulative_error_rates.eps)
It can be observed from Figure \[fig4\] that the two multitask learning methods (DOML and OML) achieve the lowest cumulative error rates, demonstrating that they are effective for learning problems with a commonly shared representation across multiple related tasks. The difference between DOML and OML is marginal (24.22% vs. 24.67% in terms of the cumulative error rate evaluated at the 15,000-*th* epoch). However, owing to the distributed architecture, DOML enjoys more efficiency and almost unlimited horizontal scalability compared to the standalone OML. In our experimental setting with an Intel Core i7 2.4 GHz CPU and 8 GB RAM, DOML configured with eight *Workers* is able to process hundreds of thousands of samples within a few seconds. Furthermore, it is obvious that such processing power can easily be increased by introducing more *Workers* into the system.
Next, we analyze the communication cost of DOML. Regarding the cost related to data sources and *Workers* (i.e., the data emitted from the *Spout* in this experiment), it is clear that any dataset will be divided into $N$ chunks and distributed to $N$ *Workers*. This makes every *Worker’s* load equal to $1/N$ of the original problem load. This is especially helpful when devices with moderate computing power encounter a massive dataset that exceeds any of their processing capacities. Regarding the information exchange between *Workers* and the *Master*, a straightforward implementation would involve $N$ *Workers* periodically sending the *Master* their gradient information, calculated by averaging the gradients for $m$ samples. However, as described in Section \[sec4.1\], we choose to defer the calculation of the average gradient to the *Master* side, so that we can utilize the sparsity of $\bar{\g}$, as in , to save bandwidth. Given $N$ *Workers* learning from $K$ data sources (or tasks), with the buffer size set as $m$, the *Master* has to maintain $N$ communication channels, each of which conveys a sparse vector with only $m/K$ entities with nonzero values. As illustration, we depict the occurrences of nonzero elements of the delivered gradient vector $\bar{\g}$ corresponding to the first 100 learning epochs in Figure \[fig5\].
![Illustration of the sparsity of the gradient vector $\bar{\g}$ delivered from a *Worker* to the *Master* during the first 100 learning epochs, with the buffer size set to 10. A black spot denotes a nonzero element, whereas blank areas are all zeros.[]{data-label="fig5"}](doml_gradient_updates.eps)
It is noteworthy that the performance of the vanilla OL algorithm is inferior to those of DOML and OML (28.96% in Figure \[fig4\]). Our intuition is that learning related tasks via a single model is inappropriate, as this ignores the individual task characteristics. To verify this, we adjust the parameter $\sigma$ to generate a set of more similar tasks and a set of less similar tasks. For a dataset with $\sigma$ set as 0.1, the 64 tasks are more similar to each other, making a single model adequate for all of them. This is verified by the experimental results: 18.71% (OL) vs. 21.34% (DOML) in terms of the cumulative error rate. In contrast, by setting $\sigma$ to 0.5, the increased inconsistencies between tasks cause the OL error rate increase to 41.93%, whereas DOML achieves a lower value of 27.05%. Thus, it is obvious that compared with OL, DOML is more suitable for real-world applications where data are not strictly homogeneous.
Conclusion {#sec6}
==========
In this paper, we have proposed the use of online machine learning to classify streaming data in a manner that is compatible with the fog computing paradigm. To cope with a large number of edge devices and large volumes of data for real-time low-latency applications, we devised a distributed online multitask learning algorithm, which fits well with the architecture of fog systems. The experimental results demonstrated that jointly learning multiple related tasks is superior to a single model working in a standalone mode. More importantly, the classification accuracy of the proposed method is comparable with that of a centralized algorithm trained over the entire dataset, while the efficiency is enhanced and the network overhead is reduced. For future work, we aim to extend our experiments to a more substantial scale and additional applications. In conclusion, our work serves as an initial attempt to develop low-latency, real-time and online data analytic tools for fog computing.
Acknowledgment {#acknowledgment .unnumbered}
==============
The work was supported by the National Natural Science Foundation of China (Grant No.: 61602356, U1536202), the Key Research and Development Program of Shaanxi Province, China (Grant No.: 2018GY-002), and the Shaanxi Science & Technology Coordination & Innovation Project (Grant No.: 2016KTZDGY05-07-01).
[99]{}
X. Lu, D. Niyato, H. Jiang, P. Wang, and H. V. Poor, “Cyber insurance for heterogeneous wireless networks,” IEEE Communications Magazine, vol. 56, no. 6, pp. 21–27, 2018.
X. Lu, D. Niyato, N. Privault, H. Jiang, and P. Wang, “Managing physical layer security in wireless cellular networks: A cyber insurance approach,” IEEE Journal on Selected Areas in Communications, vol. 36, no. 7, pp. 1648–1661, 2018.
X. Lu, D. Niyato, N. Privault, H. Jiang, and S. S. Wang, “A cyber insurance approach to manage physical layer secrecy for massive mimo cellular networks,” in 2018 IEEE International Conference on Communications (ICC). IEEE, 2018, pp. 1–6.
D. Niyato, P. Wang, D. I. Kim, Z. Han, and L. Xiao, “Performance analysis of delay-constrained wireless energy harvesting communication networks under jamming attacks,” in 2015 IEEE Wireless Communications and Networking Conference (WCNC). IEEE, 2015, pp. 1823–1828.
F. Bonomi, R. A. Milito, J. Zhu, and S. Addepalli, “Fog computing and its role in the internet of things,” in Proceedings of the first edition of the MCC workshop on Mobile cloud computing, MCC@SIGCOMM 2012, 2012, pp. 13–16.
M. Iorga, L. Feldman, R. Barton, M. J. Martin, N. S. Goren, and C. Mahmoudi, “Fog computing conceptual model,” Tech. Rep. 325, 2018.
D. Niyato, X. Lu, P. Wang, D. I. Kim, and Z. Han, “Economics of internet of things (iot): An information market approach,” arXiv preprint arXiv:1510.06837, 2015.
——, “Distributed wireless energy scheduling for wireless powered sensor networks,” in 2016 IEEE International Conference on Communications (ICC). IEEE, 2016, pp. 1–6.
X. Lu, “Sensor networks with wireless energy harvesting.” 2016.
X. Lu, D. Niyato, and P. Wang, “1 power management for wireless base station in smart grid environment: Modeling and optimization.”
D. Niyato, X. Lu, and P. Wang, “Adaptive power management for wireless base stations in a smart grid environment,” IEEE Wireless Communications, vol. 19, no. 6, pp. 44–51, 2012.
M. Korki, H. L. Vu, C. H. Foh, X. Lu, and N. Hosseinzadeh, “Mac performance evaluation in low voltage plc networks,” ENERGY, pp. 135– 140, 2011.
G. Li, P. Zhao, X. Lu, J. Liu, and Y. Shen, “Data analytics for fog computing by distributed online learning with asynchronous update,” in ICC 2019- 2019 IEEE International Conference on Communications (ICC). IEEE, 2019, pp. 1–6.
X. Lu, E. Hossain, T. Shafique, S. Feng, H. Jiang, and D. Niyato, “Intelligent reflecting surface (IRS)-enabled covert communications in wireless networks,” arXiv preprint arXiv:1911.00986, 2019.
D. Niyato, P. Wang, D. I. Kim, Z. Han, and L. Xiao, “Game theoretic modeling of jamming attack in wireless powered communication networks,” in 2015 IEEE International Conference on Communications (ICC). IEEE, 2015, pp. 6018–6023.
X. Lu, E. Hossain, H. Jiang, and G. Li, “On coverage probability with type-ii harq in large-scale uplink cellular networks,” IEEE Wireless Communications Letters, 2019.
X. Lu, D. Niyato, H. Jiang, D. I. Kim, Y. Xiao, and Z. Han, “Ambient backscatter assisted wireless powered communications,” IEEE Wireless Communications, vol. 25, no. 2, pp. 170–177, 2018.
X. Lu, P. Wang, and D. Niyato, “Payoff allocation of service coalition in wireless mesh network: A cooperative game perspective,” in 2011 IEEE Global Telecommunications Conference-GLOBECOM 2011. IEEE, 2011, pp. 1–5.
——, “Hierarchical cooperation for operator-controlled device-to-device communications: A layered coalitional game approach,” in 2015 IEEE Wireless Communications and Networking Conference (WCNC). IEEE, 2015, pp. 2056–2061.
X. Lu, D. Niyato, H. Jiang, E. Hossain, and P. Wang, “Ambient backscatter assisted wireless-powered relaying,” IEEE Transactions on Green Communications and Networking, 2019.
X. Lu, G. Li, H. Jiang, D. Niyato, and P. Wang, “Performance analysis of wireless-powered relaying with ambient backscattering,” in 2018 IEEE International Conference on Communications (ICC). IEEE, 2018, pp. 1–6.
Y. Zhang, D. Niyato, P. Wang, and X. Lu, “Optimizing content relay policy in publish-subscribe mobile social networks,” in 2015 IEEE Wireless Communications and Networking Conference (WCNC). IEEE, 2015, pp. 2167–2172.
S. Yi, C. Li, and Q. Li, “A survey of fog computing: Concepts, applications and issues,” in Proceedings of the 2015 Workshop on Mobile Big Data, Mobidata@MobiHoc 2015, 2015, pp. 37–42.
H. Dubey, J. Yang, N. Constant, A. M. Amiri, Q. Yang, and K. Mankodiya, “Fog data: Enhancing telehealth big data through fog computing,” CoRR, vol. abs/1605.09437, 2016.
X. Hou, Y. Li, M. Chen, D. Wu, D. Jin, and S. Chen, “Vehicular fog computing: A viewpoint of vehicles as the infrastructures,” IEEE Trans. Vehicular Technology, vol. 65, no. 6, pp. 3860–3873, 2016.
B. Tang, Z. Chen, G. Hefferman, T. Wei, H. He, and Q. Yang, “A hierarchical distributed fog computing architecture for big data analysis in smart cities,” in Proceedings of the ASE BigData & SocialInformatics 2015. ACM, 2015, p. 28.
J. Zhu, D. S. Chan, M. S. Prabhu, P. Natarajan, H. Hu, and F. Bonomi, “Improving web sites performance using edge servers in fog computing architecture,” in Seventh IEEE International Symposium on Service-Oriented System Engineering, 2013, pp. 320–323.
K. Hong, D. Lillethun, U. Ramachandran, B. Ottenw¨alder, and B. Koldehofe, “Mobile fog: A programming model for large-scale applications on the internet of things,” in Proceedings of the second ACM SIGCOMM workshop on Mobile cloud computing. ACM, 2013, pp. 15–20.
S. C. H. Hoi, D. Sahoo, J. Lu, and P. Zhao, “Online learning: A comprehensive survey,” CoRR, vol. abs/1802.02871, 2018.
A. Agarwal and J. C. Duchi, “Distributed delayed stochastic optimization,” in Advances in Neural Information Processing Systems 24, 2011, pp. 873– 881.
B. Recht, C. Re, S. Wright, and F. Niu, “Hogwild: A lock-free approach to parallelizing stochastic gradient descent,” in Advances in neural information processing systems, 2011, pp. 693–701.
O. Dekel, R. Gilad-Bachrach, O. Shamir, and L. Xiao, “Optimal distributed online prediction using mini-batches,” Journal of Machine Learning Research, vol. 13, no. Jan, pp. 165–202, 2012.
D. Sheldon, “Graphical multi-task learning,” in NIPS’08 Workshop on Structured Input and Structured Output, 2008.
G. Pemmasani, asyncoro: Asynchronous, Concurrent, Distributed Programming with Python, 2016. \[Online\]. Available: http://asyncoro.sourceforge.net
[^1]: E-mail addresses: gxli@xidian.edu.cn (G. Li), masonzhao@tencent.com (P. Zhao), lu9@ualberta.ca (X. Lu), jialiu23@stu.xidian.edu.cn (J. Liu), ylshen@mail.xidian.edu.cn (Y. Shen).
|
---
abstract: 'In this paper, we address the problem of optimal power allocation at the relay in two-hop secure communications under practical conditions. To guarantee secure communication during the long-distance transmission, the massive MIMO (M-MIMO) relaying techniques are explored to significantly enhance wireless security. The focus of this paper is on the analysis and design of optimal power assignment for a decode-and-forward (DF) M-MIMO relay, so as to maximize the secrecy outage capacity and minimize the interception probability, respectively. Our study reveals the condition for a nonnegative the secrecy outage capacity, obtains closed-form expressions for optimal power, and presents the asymptotic characteristics of secrecy performance. Finally, simulation results validate the effectiveness of the proposed schemes.'
author:
-
title: Optimal Power Allocation for A Massive MIMO Relay Aided Secure Communication
---
Introduction
============
The open nature of the wireless channel facilitates a multiuser transmission, but also incurs the security problem. Recently, as a complement of traditional upper-layer encryption techniques, physical layer security (PHY-security) has been proposed to realize secure communications by making use of the characteristics of wireless channels, i.e., noise, fading and interference [@Wyner].
From an information-theoretic viewpoint, the performance of PHY-security is determined by the rate difference between the legitimate channel and the eavesdropper channel [@SC1] [@SC2]. Therefore, to enhance wireless security, it makes sense to simultaneously increase the legitimate channel rate and decrease the eavesdropper channel rate. Inspired by this, various physical layer techniques have been introduced to improve the secrecy performance. Wherein, MIMO relaying technique gains considerable attention due to the following two reasons. First, the relay can provide a diversity gain and shorten the accessing distance, and thus improve the secrecy performance [@Relay1]. Second, MIMO techniques, such as spatial beamforming, can reduce the information leakage to the eavesdropper [@Relay2]. The beamforming schemes at the MIMO relay based on amplify-and-forward (AF) and decode-and-forward (DF) relaying protocols were presented in [@AF] and [@DF], respectively. It is worth pointing out that the optimal beam design at the relay requires global channel state information (CSI) [@Beamforming]. Yet, the CSI, especially eavesdropper CSI is difficult to obtain, since the eavesdropper is usually passive and keeps silent. Therefore, it is impossible to realize absolutely secure communications over fading channels. In this context, statistical secrecy performance metrics, e.g., secrecy outage capacity and interception probability, are adopted to evaluate wireless security [@secrecyoutagecapacity].
Recently, an advanced MIMO relaying technology, namely massive MIMO (M-MIMO) relaying, is introduced into secure communications to further improve the secrecy performance [@LS-MIMORelaying]. Even without eavesdropper CSI, M-MIMO relaying can generate a very high-resolution spatial beam, and thus the information leakage to the eavesdropper is quite small. More importantly, the secrecy performance can be enhanced by simply adding more antennas. Hence, the challenging issue of short-distance interception in secure communications can be well solved. Note that in two-hop secure communications, the transmit power at the relay has a great impact on the secrecy performance, since it affects the signal quality at both the destination and the eavesdropper. An optimal power allocation scheme for a multi-carrier two-hop single-antenna relaying network was given in [@PowerAllocation] by maximizing the sum secrecy rate. However, the power allocation for a multi-antenna relay, especially an M-MIMO relay, is still an open issue. In this paper, we focus on power allocation for DF M-MIMO secure relaying systems under very practical assumptions, i.e., no eavesdropper CSI and imperfect legitimate CSI. The contributions of this paper are three-fold:
1. We reveal the relation between the secrecy outage capacity and a defined relative distance-dependent path loss, and then derive the condition for a nonnegative secrecy outage capacity and the constraint on the minimum number of antennas.
2. We derive closed-form expressions for the optimal power at the relay in the sense of maximizing the secrecy outage capacity and minimizing the interception probability, respectively.
3. We present clear insights into the secrecy performance through asymptotic analysis. We show that the maximum secrecy outage capacity is an increasing function of the source power while the minimum interception probability keeps constant.
The rest of this paper is organized as follows. We first give an overview of the DF M-MIMO secure relaying system in Section II, and then derive two optimal power allocation schemes for the relay in Section III. In Section IV, we present simulation results to validate the effectiveness of the proposed schemes. Finally, we conclude the paper in Section V.
System Model
============
We consider a time division duplex (TDD) two-hop massive MIMO (M-MIMO) relaying system, where a single antenna source transmits message to a single antenna destination with the aid of a relay with $N_R$ antennas, while a passive single antenna eavesdropper intends to intercept the information. Note that the number of antennas at the relay in such an M-MIMO relaying system is quite large, i.e., $N_R=100$ or even bigger. It is assumed that there is no direct transmission between the source and the destination due to a long propagation distance. The relay system works in a half-duplex mode, which means any successful transmission requires two time slots. Specifically, the source sends the signal to the relay in the first time slot, and then the relay forwards the post-processing signal to the destination in the second time slot. In this paper, we assume the eavesdropper is far from the source and close to the relay, since it was concluded that the signal comes from the relay directly. Therefore, it is reasonable to assume that the eavesdropper only monitors the transmission from the relay to the destination.
We use $\sqrt{\alpha_{S,R}}\textbf{h}_{S,R}$, $\sqrt{\alpha_{R,D}}\textbf{h}_{R,D}$ and $\sqrt{\alpha_
{R,E}}\textbf{h}_{R,E}$ to represent the channels from the source to the relay, the relay to the destination, and the relay to the eavesdropper, respectively, where $\alpha_{S,R}$, $\alpha_{R,D}$ and $\alpha_{R,E}$ are the distance-dependent path losses and $\textbf{h}_{S,R}$, $\textbf{h}_{R,D}$, and $\textbf{h}_{R,E}$ denote the channel fading coefficient vectors with independent and identically distributed (i.i.d.) zero mean and unit variance complex Gaussian entries. It is assumed that the channels remain constant during a time slot and fade independently over slots. Thus, the received signal at the relay in the first time slot can be expressed as $$\textbf{y}_R=\sqrt{P_S\alpha_{S,R}}\textbf{h}_{S,R}s+\textbf{n}_R,\label{eqn1}$$ where $s$ is the normalized Gaussian distributed transmit symbol, $P_S$ is the transmit power at the source, and $\textbf{n}_R$ stands for the additive white Gaussian noise with unit variance at the relay. We assume that the relay has perfect CSI about $\textbf{h}_{S,R}$ by channel estimation. Then, maximum ratio combination (MRC) decoding is performed to recover the information. Specifically, the received signal is multiplied by a vector $\textbf{w}_R=\textbf{h}_{S,R}^H/\|\textbf{h}_{S,R}\|$.
During the second time slot, the relay forwards the decoded signal $\hat{s}$ through maximum ratio transmission (MRT). We assume that the relay has partial CSI about $\textbf{h}_{R,D}$ due to channel reciprocity impairment in TDD systems. The relation between the estimated CSI $\hat{\textbf{h}}_{R,D}$ and the real CSI $\textbf{h}_{R,D}$ is given by $$\textbf{h}_{R,D}=\sqrt{\rho}\hat{\textbf{h}}_{R,D}+\sqrt{1-\rho}\textbf{e},\label{eqn2}$$ where $\textbf{e}$ is the error vector with i.i.d. zero mean and unit variance complex Gaussian entries, and is independent of $\hat{\textbf{h}}_{R,D}$. $\rho$, scaling from $0$ to $1$, is the correlation coefficient between $\hat{\textbf{h}}_{R,D}$ and $\textbf{h}_{R,D}$. Thus, the received signals at the destination and the eavesdropper are given by $$y_D=\sqrt{P_R\alpha_{R,D}}\textbf{h}_{R,D}^H\textbf{r}+n_D,\label{eqn3}$$ and $$y_{E}=\sqrt{P_R\alpha_{R,E}}\textbf{h}_{R,E}^H\textbf{r}+n_{E},\label{eqn4}$$ respectively, where $P_R$ is the transmit power of the relay, $\textbf{r}=\textbf{v}_R\hat{s}$ is the forwarded signal with $\textbf{v}_R=\hat{\textbf{h}}_{R,D}/\|\hat{\textbf{h}}_{R,D}\|$ being a MRT beamforming vector. $n_D$ and $n_{E}$ are additive white Gaussian noise (AWGN) samples with unit variance at the destination and the eavesdropper, respectively.
Optimal Power Allocation
========================
In this section, we aim to optimize the secrecy performance through allocating the relay power $P_R$, since it affects the signal quality at both the destination and the eavesdropper. In what follows, we analyze and design the power allocation schemes in the sense of maximizing the secrecy outage capacity and minimizing the interception probability, respectively.
Secrecy Outage Capacity Maximization Power Allocation
-----------------------------------------------------
Based on the received signal in (\[eqn1\]), when performing MRC decoding at the relay, the channel capacity from the source to the relay can be expressed as $$C_{S,R}=W\log_2(1+\gamma_R),\label{eqn7}$$ where $W$ is the spectral bandwidth and $\gamma_R=P_S\alpha_{S,R}\|\textbf{h}_{S,R}\|^2$ is the signal-to-noise ratio (SNR). Similarly, according to (\[eqn3\]) and (\[eqn4\]), channel capacities from the relay to the destination and from the relay to the eavesdropper are given by $$\begin{aligned}
C_{R,D}=W\log_2\left(1+\gamma_D\right),\label{eqn10}\end{aligned}$$ and $$\begin{aligned}
C_{R,E}=W\log_2\left(1+\gamma_E\right),\label{eqn11}\end{aligned}$$ respectively, where $\gamma_D=P_R\alpha_{R,D}\left|\textbf{h}_{R,D}^H\frac{\hat{\textbf{h}}_{R,D}}{\|\hat{\textbf{h}}_{R,D}\|}\right|^2$ and $\gamma_E=P_R\alpha_{R,E}\left|\textbf{h}_{R,E}^H\frac{\hat{\textbf{h}}_{R,D}}{\|\hat{\textbf{h}}_{R,D}\|}\right|^2$.
Then, for the secrecy outage capacity in a DF M-MIMO secrecy relaying system, we have the following theorem:
*Theorem 1*: Given an outage probability bound by $\varepsilon$, the secrecy outage capacity is approximated as $C_{soc}=W\log_2\left(1+\min(P_S\alpha_{S,R}N_R,P_R\alpha_{R,D}\rho
N_R)\right)-W\log_2(1-P_R\alpha_{R,E}\ln\varepsilon)$, when the number of relay antennas is large.
Please refer to Appendix I.
Note that the secrecy outage capacity may be negative from a pure mathematical view. Hence, it makes sense to find the condition that the secrecy outage capacity is nonnegative.
For notional simplicity, we let $\rho \alpha_{R,D}N_R=A,
-\alpha_{R,E}\ln\varepsilon=A\cdot r_{l}, P_S\alpha_{S,R}N_R=B$, where $r_{l}=-\frac{\alpha_{R,E}\ln\varepsilon}{\rho
\alpha_{R,D}N_R}$ is defined as the relative distance-dependent path loss. Then, the secrecy outage capacity can be rewritten as
$$C_{soc}=
\begin{cases}
W\log_2(1+B)-W\log_2(1+P_RAr_l), \!\!\!\!&\mbox{$B<P_RA$}\\
W\log_2(1+P_RA)-W\log_2(1+P_RAr_l),\!\!\!\!&\mbox{$B\geq P_RA$}\nonumber
\end{cases}$$
Observing the above secrecy outage capacity, we get the following theorem:
*Theorem 2*: If and only if $0<r_l<1$, the secrecy outage capacity in such a DF M-MIMO secure relaying system in presence of imperfect CSI is nonnegative.
Please refer to Appendix II.
Notice that $0<r_l<1$ is the precondition for power allocation in such a secure relaying system. Given channel conditions and outage probability requirements, in order to fulfill $ 0<r_l<1$, the number of antennas $N_R$ must be bigger than $-\frac{\alpha_{R,E}\ln\varepsilon}{\rho \alpha_{R,D}}$. For an M-MIMO relaying system, it is always possible to meet the condition of $0<r_l<1$ by adding more antennas, which is one of its main advantages. In what follows, we only consider the case of $0<r_l<1$.
Based on Theorem 1, the secrecy outage capacity maximization power allocation is equivalent to the following optimization problem: $$\begin{aligned}
J_1: \max_{P_R}\quad C_{soc}\nonumber\\
s.t.\quad P_R\leq P_{\max},\label{eqn12}\end{aligned}$$ where $P_{\max}$ is the transmit power constraint at the relay.
For the optimal solution of the above optimization problem, we have the following theorem:
*Theorem 3*: The optimal power at the relay is $P_R^{\star}=\min\left(\frac{P_S\alpha_{S,R}}{\rho
\alpha_{R,D}},P_{\max}\right)$, and the corresponding maximum secrecy outage capacity is $C_{soc}^{\max}=W\log_2\left(1+\min\left(\frac{P_S\alpha_{S,R}}{\rho
\alpha_{R,D}},P_{\max}\right)A\right)-W\log_2\left(1+\min\left(\frac{P_S\alpha_{S,R}}{\rho
\alpha_{R,D}},P_{\max}\right)Ar_l\right)$.
Please refer to Appendix III.
*Remark*: It is found that when eavesdropper CSI is unavailable, it is optimal for the DF secure relaying system to let the two hops have the same channel capacity, resulting in $P_R^{\star}=\min\left(\frac{P_S\alpha_{S,R}}{\rho
\alpha_{R,D}},P_{\max}\right)$.
Interception Probability Minimization Power Allocation
------------------------------------------------------
In this subsection, we analyze the optimal power allocation from the perspective of minimizing the interception probability. In general, interception probability is defined as the probability of information leakage, namely the probability of $C_D<C_E$. Then, interception probability is equivalent to the secrecy outage probability when $C_{soc}=0$ in (\[equ5\]). Thus, it can be computed as $$P_0=\exp\left(-\frac{2^{C_D/W}-1}{P_R\alpha_{R,E}}\right),\label{eqn13}$$ where $C_D=W\log_2\left(1+\min(P_S\alpha_{S,R}N_R,P_R\alpha_{R,D}\rho
N_R)\right)$ is the legitimate channel capacity.
Then, interception probability minimization power allocation can be described as the following optimization problem: $$\begin{aligned}
J_2: \min_{P_R}\quad P_0\nonumber\\
s.t.\quad P_R\leq P_{\max}.\label{eqn14}\end{aligned}$$ Since $\exp(x)$ is a monotonously increasing function of $x$ and $\min_{x}(-f(x))$ is equivalent to $\max_{x}(f(x))$, $J_2$ can be transformed to the following problem: $$J_3: \max_{P_R}\quad
\frac{\min(P_S\alpha_{S,R}N_R,P_R\alpha_{R,D}\rho
N_R)}{P_R\alpha_{R,E}}\nonumber$$ $$s.t.\quad P_R\leq P_{\max}.\label{eqn15}$$ By solving the above optimization problem, we have the following theorem:
*Theorem 4*: From the perspective of minimizing interception probability, the optimal transmit power at the relay $P_R^{\star}$ belongs to a region $\left(0,\min\left(\frac{P_S\alpha_{S,R}}{\rho
\alpha_{R,D}},P_{\max}\right)\right]$, and the corresponding minimum interception probability is $P_0^{\min}=\exp\left(-\frac{\rho\alpha_{R,D}
N_R}{\alpha_{R,E}}\right)$.
Please refer to Appendix IV.
*Remarks*: It is found that the optimal power minimizing the interception probability is not unique. However, from the perspective of maximizing the secrecy outage capacity, $P_R=\min\left(\frac{P_S\alpha_{S,R}}{\rho
\alpha_{R,D}},P_{\max}\right)$, namely the upper bound, is optimal. Thus, it is better to let $P_R^{\star}=\min\left(\frac{P_S\alpha_{S,R}}{\rho
\alpha_{R,D}},P_{\max}\right)$ in the sense of jointly optimizing interception probability and secrecy outage capacity.
Asymptotic Characteristic
-------------------------
In the above, we prove that the optimal relay power $P_R^{\star}$ in the sense of maximizing the secrecy outage capacity and minimizing the interception probability is a function of the source power $P_S$. Thus, the source power has a great impact on the secrecy performance, as described in Theorem 3 and 4. In order to get some clear insights, we carry out an asymptotic performance analysis with respect to the source power.
First, for the secrecy outage capacity in a DF M-MIMO secrecy relaying system, there are the following asymptotic characteristics:
*Proposition 1*: In the low $P_S$ regime, the optimal relay power $P_R^*$ and maximum secrecy outage capacity $C_{soc}^{\max}$ asymptotically approach zero. In the high $P_S$ regime, the maximum secrecy outage capacity will be saturated and is independent of $P_S$. Furthermore, $C_{soc}^{\max}$ is an increasing function of $P_S$.
In the low $P_S$ regime, the optimal relay power and the corresponding secrecy outage capacity are reduced as $P_R^{\star}=\frac{P_S\alpha_{S,R}}{\rho \alpha_{R,D}}$ and $C_{soc}^{\max}=W\log_2\left(\frac{1+P_S\alpha_{S,R}N_R}{1+P_S\alpha_{S,R}N_Rr_l}\right)$, respectively. Both of them asymptotically approach zero as $P_S$ tends to zero. Otherwise, in the high $P_S$ regime, the optimal relay power is limited by $P_{\max}$, then $C_{soc}^{\max}$ is constant. In other words, the secrecy outage capacity becomes saturated. In addition, because of $0<r_l<1$, $C_{soc}^{\max}$ is an increasing function of $P_S$.
Second, for the interception probability, we have the following asymptotic characteristics:
*Proposition 2*: The minimum interception probability is independent of $P_S$.
As proved in Theorem 4, although the optimal relay power is a function of the source power, the final interception probability is a constant independent of $P_S$ and $P_R$.
Simulation Results
==================
To examine the effectiveness of the proposed power allocation schemes for DF M-MIMO secure relaying systems, we present several simulation results for the following scenarios. We set $N_R=100$, $W=10$ kHz, $\rho=0.9$ and $\varepsilon=0.01$ without extra statements. For convenience, we normalize the path loss from the source to the relay as $\alpha_{S,R}=1$ and use $\alpha_{R,D}$ and $\alpha_{S,E}$ to denote the relative path loss. Specifically, $\alpha_{R,E}>\alpha_{R,D}$ means that the eavesdropper is closer to the relay than the destination. In addition, we use SNR$_S=10\log_{10}P_S$, SNR$_R=10\log_{10}P_R$ and SNR$_{\max}=10\log_{10}P_{\max}$ to represent the SNR in dB at the source, the relay and the constraint at the relay, respectively.
![Comparison of theoretical and simulation results with different outage probability requirements.[]{data-label="Fig2"}](Fig2){width="45.00000%"}
First, we verify the accuracy of the theoretical expression in Theorem 1 with SNR$_S=$ SNR$_R=10$ dB and $\alpha_{R,D}=1$. As seen in Fig. \[Fig2\], the theoretical results are well consistent with the simulations in the whole $\alpha_{R,E}$ region with different outage probability requirements, which proves the high accuracy of the derived results. Given an outage probability bound by $\varepsilon$, as $\alpha_{R,E}$ increases, the secrecy outage capacity decreases gradually. This is because the interception ability of the eavesdropper enhances due to the short interception distance. In addition, for a given $\alpha_{R,E}$, the secrecy outage capacity improves with the increase of $\varepsilon$, since the outage probability is an increasing function of the secrecy outage capacity.
![Secrecy outage capacity comparison of different schemes.[]{data-label="Fig3"}](Fig3){width="45.00000%"}
![Interception probability comparison of different schemes.[]{data-label="Fig4"}](Fig4){width="45.00000%"}
Next, we show the performance gain of the proposed optimal power allocation schemes compared to a fixed power allocation with SNR$_S=10$ dB, SNR$_{\max}=15$ dB and $\alpha_{R,E}=5$. It is worth pointing out that the fixed scheme uses a fixed power $P_R=15$ dB regardless of channel conditions and system parameters. As seen in Fig. \[Fig3\], the secrecy outage capacity maximization power allocation scheme performs better than the fixed power allocation scheme. Especially in the high $\alpha_{R,D}$ regime, the performance of the proposed scheme improves sharply, while that of the fixed allocation scheme nearly remains unchanged. This is because the legitimate channel capacity is limited by the source-relay channel capacity under this condition, but the fixed scheme is regardless of $\alpha_{S,R}$ and $P_S$. In the low $\alpha_{R,D}$ regime, the secrecy outage capacities of both schemes approach zero due to $r_l>1$, which verifies Theorem 2 again. In terms of interception probability, as shown in Fig. \[Fig4\], the proposed scheme also outperforms the fixed power allocation scheme. Consistent with the theoretical claims, the interception probability approaches zero when $\alpha_{R,D}$ is large enough.
![Asymptotic secrecy outage capacity with different $\alpha_{R,E}$.[]{data-label="Fig5"}](Fig5){width="45.00000%"}
![Asymptotic interception probability with different $\alpha_{R,E}$.[]{data-label="Fig6"}](Fig6){width="45.00000%"}
Finally, we check the asymptotic characteristics with $\alpha_{R,D}=1$. As shown in Fig. \[Fig5\], as $P_S$ tends to zero, the maximum secrecy capacities with different $\alpha_{R,E}$ approach zero. In the large $P_S$ regime, the maximum secrecy outage capacity will be saturated, which is in agreement with Proposition 1 again. From Fig. \[Fig6\], it is seen that the minimum interception probability is independent of $P_S$. Additionally, the interception probability floor becomes higher with the increase of $\alpha_{R,E}$.
Conclusion
==========
In this paper, we have first presented a secrecy performance analysis for a DF M-MIMO secure relaying system with imperfect CSI. We proved that in order to guarantee a nonnegative secrecy outage capacity, there is a constraint on the minimum number of antennas at the relay. Then, by maximizing the secrecy outage capacity and minimizing the interception probability, we proposed two optimal relay power allocation schemes. At last, we revealed the asymptotic characteristics of maximum secrecy outage capacity and minimum interception probability with respect to the source power.
Proof of Theorem 1
==================
Based on channel capacities from the source to the relay in (\[eqn7\]) and from the relay to the destination in (\[eqn10\]), the legitimate channel capacity can be computed as $$\begin{aligned}
C_D&=&\min(C_{S,R},C_{R,D}),\label{equ1}\\
&=&\min\Bigg(W\log_2\left(1+P_S\alpha_{S,R}\|\textbf{h}_{S,R}\|^2\right),\nonumber\\
&&W\log_2\left(1+P_R\alpha_{R,D}\Big|\textbf{h}_{R,D}^H\frac{\hat{\textbf{h}}_{R,D}}{\|\hat{\textbf{h}}_{R,D}\|}\Big|^2\right)\Bigg)\nonumber\end{aligned}$$ $$\begin{aligned}
&=&\min\Bigg(W\log_2(1+P_S\alpha_{S,R}\|\textbf{h}_{S,R}\|^2),W\log_2\Big(1+\nonumber\\
&&P_R\alpha_{R,D}\Big|(\sqrt{\rho}\hat{\textbf{h}}_{R,D}^H+\sqrt{1-\rho}\textbf{e}^H)\frac{\hat{\textbf{h}}_{R,D}}{\|\hat{\textbf{h}}_{R,D}\|}\Big|^2\Big)\Bigg)\label{equ2}\\
&=&\min\Bigg(W\log_2(1+P_S\alpha_{S,R}\|\textbf{h}_{S,R}\|^2),W\log_2\Big(1+\nonumber\\
&&P_R\alpha_{R,D}(\rho\|\hat{\textbf{h}}_{R,D}\|^2+2\sqrt{(1-\rho)\rho}\mathcal{R}(\textbf{e}^H\hat{\textbf{h}}_{R,D})\nonumber\\
&&+(1-\rho)\|\textbf{e}^H\hat{\textbf{h}}_{R,D}\|/\|\hat{\textbf{h}}_{R,D}\|^2)\Big)\Bigg)\nonumber\\
&\approx&\min\bigg(W\log_2(1+P_S\alpha_{S,R}\|\textbf{h}_{S,R}\|^2),\nonumber\\
&&W\log_2(1+P_R\alpha_{R,D}\rho
\|\hat{\textbf{h}}_{R,D}\|^2)\bigg)\label{equ3}\\
%&\approx&\min\big(W\log_2(1+P_S\alpha_{S,R}N_R),\nonumber\\
%&&W\log_2(1+P_R\alpha_{R,D}\rho N_R)\big)\nonumber\\
&\approx&W\log_2(1+\min(P_S\alpha_{S,R}N_R,P_R\alpha_{R,D}\rho N_R)).\label{equ4}\end{aligned}$$ where $\mathcal{R}(x)$ denotes the real part of $x$. $\textbf{h}_{R,D}$ has been replaced with $\sqrt{\rho}\hat{\textbf{h}}_{R,D}+\sqrt{1-\rho}\textbf{e}$ in (\[equ2\]). Eq. (\[equ3\]) follows from the fact that $\rho\|\hat{\textbf{h}}_{R,D}\|^2$ scales with the order $\mathcal{O}(\rho N_R)$ as $N_R\rightarrow\infty$ while $2\sqrt{\rho(1-\rho)}\mathcal{R}(\textbf{e}^H\hat{\textbf{h}}_{R,D})
+(1-\rho)\|\textbf{e}^H\hat{\textbf{h}}_{R,D}^H\|^2/\|\hat{\textbf{h}}_{R,D}\|^2$ scales as the order $\mathcal{O}(1)$, which is negligible. Eq. (\[equ4\]) holds true because of $\lim\limits_{N_R\rightarrow\infty}\frac{\|\hat{\textbf{h}}_{R,D}\|^2}{N_R}=1$ and $\lim\limits_{N_R\rightarrow\infty}\frac{\|\textbf{h}_{S,R}\|^2}{N_R}=1$, namely channel hardening [@ChannelHardening].
Similarly, the eavesdropper channel capacity is given by $$C_E=W\log_2(1+\min(P_S\alpha_{S,R}N_R,P_R\alpha_{R,E}\Big|\textbf{h}_{R,E}^H\frac{\hat{\textbf{h}}_{R,D}}{\|\hat{\textbf{h}}_{R,D}\|}\Big|^2)).\label{app1}$$
Then, the secrecy outage probability $\varepsilon$ with respect to a secrecy outage capacity $C_{SOC}$ can be computed as (\[equ5\]) at the top of next page,
$$\begin{aligned}
\varepsilon&=&P_r(C_{soc}>C_D-C_E)\nonumber\\
&=&P_r\left(\min\left(P_S\alpha_{S,R}N_R,P_R\alpha_{R,E}\left|\textbf{h}_{R,E}^H\frac{\hat{\textbf{h}}_{R,D}}{\|\hat{\textbf{h}}_{R,D}\|}\right|^2\right)>2^{(C_D-C_{soc})/W}-1\right)\nonumber\\
&=&P_r\left(P_S\alpha_{S,R}N_R\leq
P_R\alpha_{R,E}\left|\textbf{h}_{R,E}^H\frac{\hat{\textbf{h}}_{R,D}}{\|\hat{\textbf{h}}_{R,D}\|}\right|^2\right)P_r\left(P_S\alpha_{S,R}N_R>2^{(C_D-C_{soc})/W}-1\right)\nonumber\\
&&+P_r\left(P_S\alpha_{S,R}N_R>
P_R\alpha_{R,E}\left|\textbf{h}_{R,E}^H\frac{\hat{\textbf{h}}_{R,D}}{\|\hat{\textbf{h}}_{R,D}\|}\right|^2\right)P_r\left(P_R\alpha_{R,E}\left|\textbf{h}_{R,E}^H\frac{\hat{\textbf{h}}_{R,D}}{\|\hat{\textbf{h}}_{R,D}\|}\right|^2>2^{(C_D-C_{soc})/W}-1\right)\nonumber\\
&=&\exp\left(-\frac{P_S\alpha_{S,R}N_R}{P_R\alpha_{R,E}}\right)+\left(1-\exp\left(-\frac{P_S\alpha_{S,R}N_R}{P_R\alpha_{R,E}}\right)\right)\exp\left(-\frac{2^{(C_D-C_{soc})/W}-1}{P_R\alpha_{R,E}}\right)\label{app2}\\
&\approx&\exp\left(-\frac{2^{(C_D-C_{soc})/W}-1}{P_R\alpha_{R,E}}\right).\label{equ5}\end{aligned}$$
where (\[app2\]) follows from the fact that $\Big|\textbf{h}_{R,E}^H\frac{\hat{\textbf{h}}_{R,D}}{\|\hat{\textbf{h}}_{R,D}\|}\Big|^2$ is $\chi^2$ distributed with 2 degrees of freedom, and (\[equ5\]) holds true since $\exp\left(-\frac{P_S\alpha_{S,R}N_R}{P_R\alpha_{R,E}}\right)$ approaches zero when $N_R$ is sufficient large. Based on (\[equ5\]), it is easy to get the Theorem 1.
Proof of Theorem 2
==================
Based on the secrecy outage capacity in Theorem 1, when $P_R\geq\frac{P_S\alpha_{S,R}}{\rho \alpha_{R,D}}$, we have $C_{soc}=W\log_2(1+P_S\alpha_{S,R}N_R)-W\log_2(1+P_R\rho
\alpha_{R,D}N_Rr_l)$. To guarantee $C_{soc}\geq0$, the following condition $P_S\alpha_{S,R}N_R\geq P_R\rho \alpha_{R,D}N_Rr_l$ must be fulfilled, which is equivalent to $0<r_l<1$ in the case of $P_R>\frac{P_S\alpha_{S,R}}{\rho \alpha_{R,D}}$.
Otherwise, when $P_R\leq \frac{P_S\alpha_{S,R}}{\rho \alpha_{R,D}}$, the secrecy outage capacity is changed as $C_{soc}=W\log_2(1+P_R\rho
\alpha_{R,D}N_R)-W\log_2\left(1+P_R\rho\alpha_{R,D}N_Rr_l\right)$. Only when $0<r_l<1$, $C_{soc}$ is nonnegative.
Above all, $0<r_l<1$ or $N_R>-\frac{\alpha_{R,E}\ln\varepsilon}{\rho
\alpha_{R,D}}$ is the precondition that the nonnegative secrecy outage capacity exists. Therefore, we get the Theorem 2.
proof of Theorem 3
==================
According to the Theorem 1, when $P_R\geq\frac{P_S\alpha_{S,R}}{\rho
\alpha_{R,D}}$, the secrecy outage capacity $C_{soc}=W\log_2(1+P_S\alpha_{S,R}N_R)-W\log_2(1+P_R\rho
\alpha_{R,D}N_Rr_l)$ is maximized when $P_R=\frac{P_S\alpha_{S,R}}{\rho \alpha_{R,D}}$, since $C_{soc}$ is a monotonously decreasing function of $P_R$.
When $P_R\leq \frac{P_S\alpha_{S,R}}{\rho \alpha_{R,D}}$, the secrecy outage capacity $C_{soc}=W\log_2(1+P_R\rho
\alpha_{R,D}N_R)-W\log_2(1+P_R\rho \alpha_{R,D}N_Rr_l)$ is an increasing function of $P_R$ under the condition $0<r_l<1$. Thus, $P_R=\frac{P_S\alpha_{S,R}}{\rho \alpha_{R,D}}$ is the optimal power at the relay.
Considering the constraint on the transmit power $P_{\max}$ at the relay, the optimal power at the relay is $P_R^{\star}=\min\left(\frac{P_S\alpha_{S,R}}{\rho
\alpha_{R,D}},P_{\max}\right)$. Furthermore, by substituting $P_R^{\star}=\min\left(\frac{P_S\alpha_{S,R}}{\rho
\alpha_{R,D}},P_{\max}\right)$ into the expression of secrecy outage capacity, we can obtain the maximum secrecy outage capacity as shown in Theorem 3.
proof of Theorem 4
==================
First, when $P_R\leq\frac{P_S\alpha_{S,R}}{\rho \alpha_{R,D}}$, the optimization problem $J_3$ is equivalent to $$G_1 : \max_{P_R}\quad \frac{\alpha_{R,D}\rho
N_R}{\alpha_{R,E}}\nonumber$$ $$s.t. \quad P_R\leq \min\left(\frac{P_S\alpha_{S,R}}{\rho
\alpha_{R,D}},P_{\max}\right).\label{equ9}$$ Interestingly, it is found that the objective function $\frac{\rho
\alpha_{R,D}N_R}{\alpha_{R,E}}$ is independent of $P_R$. Hence, the optimal solution of $G_1$ can be an arbitrary value belonging to $\left(0,\min\left(\frac{P_S\alpha_{S,R}}{\rho
\alpha_{R,D}},P_{\max}\right)\right]$.
Second, when $P_R\geq\frac{P_S\alpha_{S,R}}{\rho \alpha_{R,D}}$, the optimization problem $J_3$ is reduced as $$G_2 : \max_{P_R}
\quad\frac{P_S\alpha_{S,R}N_R}{P_R\alpha_{R,E}}\nonumber$$ $$s.t. \quad \frac{P_S\alpha_{S,R}}{\rho \alpha_{R,D}}\leq P_R\leq
P_{\max}.\label{equ10}$$ The optimal solution of $G_2$ is $\frac{P_S\alpha_{S,R}}{\rho
\alpha_{R,D}}$, since the objective function is a decreasing function of $P_R$.
Thus, the optimal transmit power at the relay is $P_R^{\star}=\left(0,\min\left(\frac{P_S\alpha_{S,R}}{\rho
\alpha_{R,D}},P_{\max}\right)\right]$. Hence, we get the Theorem 4.
[1]{} A. D. Wyner, “The wire-tap channel," *Bell Syst. Tech. J.*, vol. 54, pp. 1355-1387, Oct. 1975.
P. K. Gopala, L. Lai, and H. El. Gamal, “On the secrecy capacity of fading channels," *IEEE Trans. Inf. Theory*, vol. 54, no. 10, pp. 4687-4698, Oct. 2008.
J. Barros, and M. R. D. Rodrigues, “Secrecy capacity of wireless channels," in *Proc. IEEE ISIT*, pp. 356-360, July 2006.
L. Dong, Z. Han, A. P. Petropulu, and H. V. Poor, “Improving wireless physical layer security via cooperating relays," *IEEE Trans. Signal Process.*, vol. 58, no. 3, pp. 1875-1888, Mar. 2010.
X. Chen, C. Zhong, C. Yuen, and H-H. Chen, “Multi-antenna relay aided wireless physical layer security," *IEEE Commun. Mag.*, vol. 53, no. 7, Jul. 2015.
C. Jeong, I-M. Kim, and D. Kim, “Joint secure beamforming design at the source and the relay for an amplify-and-forward MIMO untrusted relay system," *IEEE Trans. Signal Process.*, vol. 60, no. 1, pp. 310-325, Jan. 2012.
J. Huang, and A. L. Swindlehurst, “Cooperative jamming for secure communications in MIMO relay networks," *IEEE Trans. Signal Process.*, vol. 39, no. 10, pp. 4871-4884, Oct. 2011.
X. Wang, K. Wang, and X. Zhang, “Secure relay beamforming with imperfect channel side information," *IEEE Trans. Veh. Technol.*, vol. 62, no. 5, pp. 2140-2155, May 2013.
X. Chen, and L. Lei, “Energy-efficient optimization for physical layer security in multi-antenna downlink networks with QoS guarantee," *IEEE Commun. Lett.*, vol. 17, no. 4, pp. 637-640, Apr. 2013.
X. Chen, L. Lei, H. Zhang, and C. Yuen, “Large-scale MIMO relaying techniques for physical layer security: AF or DF?" *IEEE Trans. Wireless Commun.*, 2015. \[DOI\]: 10.1109/TWC.2015.2433291.
C. Jeong, and I-M. Kim, “Optimal power allocation for secure multicarrier relay systems," *IEEE Trans. Signal Process.*, vol. 59, no. 11, pp. 5428-5442, Nov. 2011.
B. M. Hochwald, T. L. Marzetta, and V. Tarokh, “Multiple-antenna channel hardening and its implications for rate-feedback and scheduling," *IEEE Trans. Inf. Theory*, vol. 50, no. 9, pp. 1893-1909, Sept. 2004.
|
---
author:
- |
M.A.Braun$^{a,b}$ and C.Pajares$^a$\
$^a$ Dep. of Elementary Particles,\
Univ. of Santiago de Compostela, 15706, Santiago de Compostela, Spain,\
$^b$ Dep. of High Energy physics, University of S.Petersburg,\
198504 S.Petersburg, Russia
title: '**Rapidity and centrality dependence in the percolating colour strings scenario**'
---
epsf
-30pt plus 1pt minus 1pt
In AA collisions fusion and percolation of colour strings is studied at fixed rapidity $y$. Distribution of strings in rapidity is obtained from the observed rapidity spectra in pp collisions. For $y$-dependence of multiplicities in Au-Au collisions good agreement is obtained with the existing experimental data. Predictions for LHC energies coincide with the extrapolation of the data. Agreement with the data of the transverse momentum spectra requires introduction of quenching into the model.
Introduction
============
The color string model with fusion [@Ref1], [@Ref2] and percolation [@Ref3]-[@Ref6] has produced results on multiplicities of secondaries which are in general agreement with the existing experimental data. The string fusion model predicted a strong reduction of multiplicities both at RHIC and LHC energies. At the time of the ALICE Technical proposal of 95 [@Ref7], most of model predictions, (including VENUS [@Ref8], HIJING [@Ref9], SHAKER [@Ref10] and DPM [@Ref11]) for LHC energies were more than 4000 charged particles at central rapidity region for central $(b\leq 3\ fm)$ Pb-Pb collisions and only the prediction of the string fusion model was much lower. Since then many of the models have lowered their predictions introducing several mechanisms, such as the triple pomeron coupling in DPM [@Ref12], stronger shadowing in HIJING [@Ref13] or other modifications in VENUS [@Ref14]. In the parton saturation picture, predictions for the central rapidity density of charged particles per participant for central Pb-Pb at the LHC energy range from around 15 [@Ref15] to a lower value around 9 [@Ref16]. Assuming that the observed geometrical scaling for the saturation momentum in lepton-hadron scattering is also valid for the nucleus-nucleus scattering, the value around 9 is also obtained [@Ref17]. In percolation of strings the obtained values 7.3 [@Ref18] and 8.6 [@Ref19] are not far from the above ones, as expected, given the similarities between percolation of strings and saturation of partons [@Ref20]. In any case, these values lie above 6.4 which is obtained by extrapolation to the LHC energy of the values experimentally found at $\sqrt{s}=19.4, 62.8, 130$ and 200 GeV [@Ref21]. The values obtained in percolation have some uncertainty due to simplifications done in the calculations, mainly related to the dependence of the multiplicity on the energy of a simple string and on the rapidities of fused strings. One of the goals of this paper is to take into account both these dependencies, using as an input the rapidity and energy dependence of multiplicities in pp collisions. We try to answer whether the charge particle density per participant is compatible with the one obtained in the mentioned extrapolation. We assume that strings occupy different regions in the available rapidity interval. Then at different rapidities one will see different number of overlapping strings, depending on the rapidities of the string ends. As a result the string density and the process of percolation become dependent on rapidity together with all the following predictions for observable quantities. The central part of our derivation is the calculation of the number of strings at a fixed rapidity which follows from the known parton distributions in the projectile and target. To this aim we shall use the simple parton distributions employed in the review [@Ref11]. Also, we pay attention to the fragmentation rapidity region. As an input we take the observed limiting fragmentation scaling in pp collisions. We then obtain a similar scaling for A-A collisions, in agreement with the experimental data [@Ref22].
One of the most interesting features of the RHIC data is the suppression of high transverse momenta. The nuclear modification factor defined as the ratio between inclusive A-A cross section normalized to the number of collisions and the inclusive proton-proton cross-section is found to lie below unity, in disagreement with the perturbative QCD expectations.
In the previous paper [@Ref6], in the framework of percolation of strings, a reasonable agreement with the data was obtained, describing A-A collisions as an exchange of clusters of overlapping strings. In percolation each cluster behaves like a new string with a larger tension, its value depending on the number of strings fused into the cluster and the cluster’s transverse area. Fragmentation of each cluster was assumed to give rise to an exponential distribution in $p^2_T$. Superposition of different exponential distributions then builds up a power-like distribution $\sim p_T^{-\kappa}$ with $\kappa$ inversely related to the magnitude of the dispersion in the number of different clusters. At low density, there is no overlapping of strings, thus no fluctuations and $\kappa$ is large. As the density increases, so does the string overlapping and more clusters are formed with different number of strings. So the dispersion increases and $\kappa$ decreases. Finally, at very high density, above the percolation threshold, there remains a single large cluster of nearly all the strings. Therefore, there are no fluctuations and $\kappa$ increases again becoming large. In this way, the suppression of high $p_T$ at large density follows as a result of formation of large clusters of color strings. In [@Ref6] we assumed that the spectrum of a simple string was a single exponential in $p^2_T$. However, at low string density, as in pp collisions, when fusion of strings is insignificant, the experimental data clearly show a power-like tail for the $p_T$ distribution.
In this paper we study this point with more attention. Instead of the exponential distribution we take the standard power-like parameterization for pp collisions [@Ref23]. The resulting $p_T$ distribution for central A-A collisions is again found suppressed at high $p_T$ but not enough to agree with the data. In order to describe the data we need additional suppression which would physically correspond to the fact that the produced particle, passing through a large cluster and interacting with the strong chromoelectric field, looses a part of its energy. This result is not unexpected. In fact a version of HIJING [@Ref24], with a string junction and doubling the string tension to simulate stronger color-fields, is able to explain the difference between baryons and mesons in the low and mid $p_T$ range but at high $p_T$ some jet quenching mechanism is needed. In our framework quenching at high $p_T$ may be introduced in a simple phenomenological manner by taking the average $p_T^2$ of a cluster of $n$ strings to grow with $n$ more slowly than $\sqrt{n}$, as predicted for a single small cluster in absence of others. Choosing an appropriate $n$ dependence of the average $p_T^2$ for clusters at a given string density allows to obtain a reasonable agreement with the experimental data on $p_T$ dependence in A-A collisions.
pp collisions
=============
Multiplicities and numbers of strings
-------------------------------------
The starting point for the calculation of fusion and percolation of strings in heavy -ion collisions is the distribution of strings in proton-proton collisions, where effects of fusion and percolation are very small. Our strategy will be to extensively use the existing experimental data for the multiplicity per unit rapidity $d\mu^{pp}(y)/dy\equiv \mu^{pp}(y)$ in pp collisions to extract the necessary distribution of strings in $y$ from them.
We recall that in the original DPM model without string fusion the multiplicity is given by a sum of contribution from the strings formed in the collision. In particular, in a configuration with $n=2k$ strings formed, which corresponds to the exchange of $k$ pomerons [@Ref11], the multiplicity is given by \^[pp]{}\_[n]{}(y)=\_[-Y/2]{}\^[Y/2]{} \_[i=1]{}\^ndu\_idw\_i p(u\_1,...,u\_n) t(w\_1,...,w\_n)\_[j=1]{}\^n\_j(y,u\_j,w\_j). We work in the c.m system of colliding protons; $Y$ is the overall rapidity admissible for nearly massless quarks. It is related to the beam rapidity as Y=Y\_[beam]{}+, where $m$ is the nucleon mass and $\mu$ the quark average transverse mass. The strings are enumerated according to their flavour content, that is according to which quark they are attached. Number 1 corresponds to the quark-diquark (qd) string, number 2 to diquark-quark (dq) string and all the rest correspond to sea quarks, which include $s\bar{s}$ and $\bar{s}s$ strings. Ends of strings in rapidity are denoted by $u_i$ in the projectile and $w_i$ in the target. Distributions $p(u_1,...,u_n)$ and $t(w_i,...w_n)$ give the probability to find the relevant quarks in the projectile and target proton respectively. Note that in the assumed notation $p(u_1,...u_n)$ gives the probability to find in the proton the valence quark at rapidity $u_1$ the diquark at rapidity $u_2$ and the sea quarks at rapidities $u_3,...u_n$. Distribution $t(w_1,...w_n)$, on the other hand, gives the probability to find the diquark at rapidity $w_1$, the quark at rapidity $w_2$ and sea quarks at rapidities $w_3,...w_n$. Function $\mu_j(y,u,w)$ gives the multiplicity per unit rapidity from the $j$th string at rapidity $y$ provided its ends are at $u$ in the projectile and $w$ in the target. Obviously this probability is zero if the string lies outside rapidity $y$. So $\mu_j(y,u,w)$ has a form \_j(y,u,w)=(y,u,w)\_j(y,u,w), where (y,u,w)=(u-w-y\_0)(u-y)(y-w) +(w-u-y\_0)(w-y)(y-u). Here the two terms correspond to the two possibilities of the higher rapidity end of the string to lie on the projectile or on the target parton. The rapidity interval $y_0$ corresponds to the minimal extension of the string in rapidity. We take $y_0=2$.
The string density $dN^{pp}_{n}(y)/dy\equiv
N^{pp}_{n}(y)$ at a given rapidity is given by an expression similar to (1) but without $\tilde{\mu}$: N\^[pp]{}\_[n]{}(y)=\_[-Y/2]{}\^[Y/2]{}\_[i=1]{}\^ndu\_idw\_i p(u\_1,...,u\_n) t(w\_1,...,w\_n)\_[j=1]{}\^n(y,u\_j,w\_j). To analyse fusion probabilities in nuclear collisions we need to know the latter quantity.
In principle, knowledge of the distributions $p(u_1,...,u_n)$ and $t(w_1,...,w_n)$ and of string luminosities $\tilde{\mu}_j(y,u,w)$ allows to calculate both $\mu_n^{pp}(y)$ and $N_n^{pp}(y)$. This was done in the extensive calculations within the original DPM model [@Ref11]. However these input quantities are in fact poorly known and our idea is to directly relate $\mu^{pp}(y)$ and $N^{pp}(y)$ using the experimental data for the former. To do this we assume that string luminosities are approximately $y$-independent and the same for all type of strings: \_j(y,u,w)=\_0. This approximation has been widely used in analytical studies of string fusion. It can be justified for relatively long strings far from their ends, when particle production can be well described by the Schwinger mechanism of pair creation in a strong field. With strings of finite dimension it may be considered as a sort of averaging over their length and rapidity of emission. With this approximation we obtain a simple and direct relation between the multiplicity and number of strings per unit rapidity: \^[pp]{}\_[n]{}(y)=\_0N\^[pp]{}\_[n]{}(y). In fact this relation for the central region was extensively used in earlier studies of string percolation.
Relation (7) allows to find only the total number of strings per unit rapidity from the experimental data on multiplicities. However we need something more. In nucleus-nucleus collisions separately enter multiplicities coming from the valence strings and sea strings. Obviously one cannot find each of them from the experimental data. So we choose to calculate the contribution of valence strings from the theoretical formulas (5) and (7) and then, subtracting this contribution from the experimental multiplicities, find the contribution from sea strings. This procedure can be justified by the fact that distributions of valence quarks are much better known and less dependent on the overall energy than the sea contribution.
Total and sea strings from the experimental data
------------------------------------------------
We calculate the number of quark-diquark strings from (5) as $$N^{qd}_{n}(y)=
\int_{-Y/2}^{Y/2}dudwq_n^{(p)}(u)d_n^{(t)}(w)\rho(y,u,w)$$= \_[y]{}\^[Y/2]{}du\_[-Y/2]{}\^[w\_1]{}dw (q\_n \^[(p)]{}(u)d\_n\^[(t)]{}(w)+d\_n\^[(t)]{}(u)q\_n\^[(p)]{}(w)), w\_1={y,u-y\_0}. Here $q_n ^{(p)}(u)$ and $d_n^{(t)}(w)$ are inclusive probabilities to find a valence quark in the projectile and a diquark in the target at rapidities $u$ and $w$ respectively in a configuration with $n$ strings. The second term in (8) corresponds to inverse strings whose upper ends lie on the target diquark. In our symmetric case the number of diquark-quark strings is obviously the same, so that the total number of valence strings is just twice the expression (8).
The final number of valence strings at given $y$ is obtained after averaging over the number of formed strings: N\^[v]{}(y)=2\_[k=1]{}\_kN\^[qd]{}\_[(2k)]{}(y)2N\^[qd]{}\_[(n)]{}(y). Here $\omega_k$ is the probability for the exchange of $k$ pomerons, given by [@Ref11] \_k=, where $\sigma_k$ is the cross-section for $k$ inelastic collisions. It is standardly taken in the K.A.Ter-Martirosyan model [@Ref25] \_n(s)=2\_0\^bdbe\^[-2]{}, where the eikonal $\chi$ corresponds to the single pomeron exchange (s,b)=C(s)e\^[-b\^2/b\_0\^2(s)]{}, with b\_0\^2(s)=4R\_N\^2+4’(s-i), C(s)=(se\^[-i/2]{})\^[-1]{}, $\alpha$ and $\alpha'$ are the pomeron intercept and slope, $g$ is its coupling to the proton and $R_N$ the proton radius. Some improvement of these $\sigma_k$ to include the triple pomeron interaction and diffractive states may be found in [@Ref11].
To calculate the number of valence strings per unit rapidity we have to know the inclusive distributions of quarks and diquarks. Following [@Ref11] we choose the exclusive distribution $p(u_1,u_2,...u_n)$ for a projectile in a factorized form p\_n(u\_1,u\_2,...u\_n)=c\_n(1-\_[i=1]{}\^nx\_i)\_[i=1]{}\^n x\_i\^[\_i]{}, where for the quark $\mu_1=1/2$ and for the diquark $\mu_2=5/2$ For the sea quarks and antiquarks we take $\mu=1/|\ln x_c|$, where in accordance with [@Ref11] $x_c=m_c/\sqrt{s}$ with $m_c=0.1$ GeV is a cutoff at small $x$. Scaling variables are related to rapidities as x=e\^[-Y/2+u]{}, Note that the distributions $p_n(u_1,u_2,...u_n)$ are defined and normalized in the interval $0<x<1$, that is for $-\infty<u<Y/2$. The actual strings are formed only in the part of this interval with $u>-Y/2$. This circumstance is inessential for valence quarks whose distributions rapidly vanish towards small values of $x$. For the distributions in the target one has only to invert the rapidities $u\to -w$ in (15). Integration over the scaling variables of unobserved partons gives the desired inclusive distributions. For the valence quark we find q\_p\^[(n)]{}(x)= c\_vx\^[1/2]{}(1-x)\^[3/2+(n-2)]{}, c\_v=. For the diquark d\_p\^[(n)]{}(x)=c\_d x\^[5/2]{}(1-x)\^[-1/2+(n-2)]{}, c\_d= .
After the averaged valence string number is found according to (9) we have to transform it into the valence multiplicity using (7). The value of $\mu_0$ can be found from the observed plateau hight assuming that at $y=0$ all strings contribute. Their average number can be found from (9) as $N^{pp}=2\langle k\rangle$. As a result we find values of $\mu_0$ slowly rising with energy and visibly saturating at TeV energies. In Fig. 1 we show these values extracted from the data [@Ref23] together with their extrapolation to the LHC energies in the assumption that the plateau in the pp multiplicity distribution rises linearly with $\ln s$.
The obtained multiplicities from valence strings vanish in the fragmentation region too slowly as compared to the experimental data, and at $y-Y_{beam}>0$ become greater than the latter. This is obviously related to our assumption of a constant string luminosity throughout the string length, whereas it should go to zero at its ends. Put in other words, in our approach the total energy is conserved in its division between different strings due to the $\delta$-function in (14) but it is not conserved inside each separate string, since near the string end its luminosity should vanish. To cure this defect in a simple manner we just assume that as soon as the calculated valence contribution becomes larger that the data we substitute the former by the latter, assuming that in this deep fragmentation region sea strings do not contribute at all.
With thus obtained valence contribution to the multiplicities we find the sea contribution just as the difference between the total and valence one. Dividing it by $\mu_0$ we find the number of sea strings per unit rapidity in pp collisions $N^{s,pp}(y)$. In fact we need not exactly this number but the one in the assumption that all strings are of the sea type, which is obtained from it by rescaling N\^[s]{}(y)= N\^[s,pp]{}(y). This quantity is shown in Fig. 2 for different values of the collision energy $\sqrt{s}$.
hA and AA collisions
====================
Generalization to hA and AA collisions is straightforward and follows [@Ref11]. At fixed impact parameter $b$ one introduces the average numbers of participants $2\nu_A(b)$ and collisions $\nu(b)$. Then the number of strings in AA collisions at given $b$ and $y$ is N\_[AA]{}(b,y)=\_[par]{}(b) N\^[pp]{}(y)+(\_[col]{}(b)-\_[par]{}(b)) N\^s(y). Here $N^{pp}(y)$ is obtained from the observed multiplicity in pp collisions according to (7) and $N^s(y)$ is given by (18).
One can easily further generalize (19) to collisions of different nuclei (see [@Ref11]).
In the Glauber approach the numbers $\nu_{par}$ and $\nu_{col}$ for AA collisions are obtained as follows \_[par]{}(b)=A, \_[col]{}(b)=A\^2, where $\sigma$ is the total pp-cross-section, $T_A(b)$ is the nuclear profile function normalized to unity and T\_[AA]{}(b)=d\^2b’T\_A(b’)T\_A(b-b’). For hA collision, as mentioned, $\nu_{par}=1$ and \_[col]{}(b)=A. Note that for AA collisions in the above formulas the denominator is written in the so-called optical approximation [@opt]. As is well-known, it works reasonably well except close to the nucleus boundary, where the collision numbers obtained from (20) may be quite deceptive.
It is customary to take the profile function $T_A(b)$ generated by the Woods-Saxon nuclear density. However for our purpose it is more convenient to assume the nucleus to have a well-defined radius, which allows to determine the interaction area in the transverse plane as just the area of the overlap. For this reason we take the nucleus as a sphere of radius $R_A= A^{1/3}\cdot 1.2$ fm, which gives T\_A(b)=(R\_A-b), where $V_A$ is the nuclear volume. With this choice the most peripheral collisions occur at $b=2R_A$ with $\nu_{col}=\nu_{par}=1$.
Multiplicities and $p_T$ distributions
======================================
In our previous studies of string fusion we always stressed that it can only occur in the common rapidity interval. However our attention was mostly centered on the central rapidity region where all (or nearly all) strings contribute, so that the requirement of common rapidity interval was of no relevance and strings could be considered as of practically infinite length in rapidity. Now we study the fusion process in more detail. At a fixed rapidity $y$ only strings which pass through this rapidity can fuse. Formulas of the previous sections allow to find the original number of strings $N(b,y)$ stretched between the projectile and target at fixed rapidity layer $y$ and impact parameter $b$. According to the percolation colour strings scenario these strings in fact fuse into strings with higher colour. The intensity of fusion is determined by the dimensionless percolation parameter $\eta$ proportional to the string density in the interaction area (b,y)=, where $s_0=\pi r_0^2$ is the transverse area of the string and $S(b)$ is the interaction area, that is, the overlap area in case of AB collisions. Obviously in our case the percolation parameter depends both on $b$ and $y$. At $\eta\sim 1.2\div 1.3$ fusion of strings leads to their percolation and formation of macroscopic string clusters. This phenomenon will take part only in restricted intervals of $b$ and $y$, predominantly at central collisions and rapidities, where the effects of string fusion and percolation will be most noticeable
Considering the case of AA collisions at reasonably high energies we shall assume the total number of strings high enough to allow use of the thermodynamic limit, in which the total areas of $n$-fold fused strings $S_n$ become distributed according to the Poisson law with $\langle n\rangle =\eta(b,y)$: S\_n(b,y)=S(b)e\^[-(b,)]{}. Due to averaging of the direction of colour, the $n$-fold fused string emits the number of particles which is only $\sqrt{n}$ times greater that the simple string. So the total production rate at fixed $b$ and $y$ will be given by (b,y)=\_0e\^[-(b,)]{}\_[n]{} , where $\mu_0$ is the production rate from the single string, which, as stated above, we assume to be independent of $y$ but dependent on energy.
As to the $p_T$ distribution, we use a slightly generalized model introduced and discussed in [@Ref26], in which the normalized probability $w_n(p)$ to find a particle with transverse momentum $p$ emitted from the $n$-fold fused string is given by w\_n(p)= ()\^. Here for $n=1$ the parameters are determined by the experimental data on pp collisions: p\_1=2 [GeV/c]{}, \_1=19.7-0.86E\_[cm]{} and $E_{cm}$ is the c.m. energy in GeV. With $n>1$ from the string fusion scenario it follows that the average transverse momentum squared of the particles emitted from the $n$-fold fused string is $n^{1/2}$ greater than for a single string. This gives a relation between $p_n$ and $\kappa_n$ p\_n\^2=n\^[1/2]{}p\_1\^2. So the distribution from $n$-fold string is fully determined by the $n$-dependence of $\kappa_n$. In [@Ref26] the simplest choice of $n$-independent $\kappa_n$ was used. However this simple choice does not allow to obtain the $p_T$ dependence in agreement with the data at RHIC. To improve our description, we introduce corrections to the original string picture which correspond to non-linear phenomena in string clustering and influence both $\kappa_n$ and the behaviour of $p_n$
In fact the value of $\kappa$ controls the difference of the distribution from the purely exponential one, passing into the latter at very large $\kappa$. For a single string a finite $\kappa_1$ may be thought of as a result of fluctuations in the string tension (or equivalently its transverse area) [@Ref27]. One may expect this fluctuations to grow as many string fuse and so the value of $\kappa_n$ should fall with $n$. However there is another effect acting in the opposite direction. As many strings fuse into clusters, multiple interactions of emitted particle inside the clusters should lead to thermalization of the particle spectra making it closer to an exponential. Thus eventually at large $n$ parameter $\kappa_n$ should grow to large values. Naturally we cannot determine the exact form of the $n$ dependence of $\kappa_n$ from purely theoretical reasoning. We can only think that the change from fall to growth should occur in the vicinity of the percolation threshold and that in any case $\kappa$ cannot be smaller than 4 to have a convergent $<p^2>$. In practice we take $\kappa_n$ as \_n=\_1+a(n-1)+b(n-1)\^2 and try to adjust $a$ and $b$ to get a better agreement with the experimental data. In fact the results are not very sensitive to the choice of $a$ and $b$ provided they are taken to have a behaviour of $\kappa_n$ in agreement with the above general theoretical observations. However the generalization (30) is not sufficient to bring our predictions in agreement with the observed quenching of the ratios $R_{AA}$ at RHIC. To this aim we have to introduce some quenching also in the string picture at large values of $\eta$. It corresponds to the fact that passing through a large cluster volume and interacting with the strong chromoelectric field the produced particles loose a part of their energy [@Ref28]. On our phenomenological level it would correspond to the behaviour of the average transverse momentum squared as \_n=n\^[\_n]{}<p\^2>\_1 with the exponent $\alpha_n$ less than 1/2 and diminishing with $n$. Similarly to (30) we parameterize \_n=+c(n-1)+d(n-1)\^2. The comparison with the experimental data determines the optimal fit for the parameters $a$, $b$, $c$ and $d$. The resulting values for $\kappa_n$ and $\alpha_n$ are shown in Figs. 3 and 4.
Averaging with the distribution in $n$ we find the final distribution in $p$ from the fusing strings at given $b$ and $y$ as w(p,b,y)=a(b,y)\_[n=1]{}w\_n(p), a\^[-1]{}(b,y)=e\^[(b,y)]{}- 1 (the change in the normalization is due to the restriction $n\geq 1$).
Numerical results
=================
We studied Au-Au collisions at energies 19,4, 62.8, 130, 200 and 6000 GeV corresponding to the existing experimental data and expected at LHC. Using our results on the string numbers in pp collisions we calculated their numbers in nucleus-nucleus collisions at a given $y$. Knowing these numbers and also numbers of participants and collisions we then determined values of the percolation parameter $\eta(y,b)$ at different rapidities and impact parameters. We have taken the transverse radius of the single string 0.3 fm. In Fig. 5 and 6 we illustrate values of $\eta(y,b)$ as a function of $y$ for central collisions and as a function of $b$ at mid-rapidity.
As one can observe, at RHIC and LHC energies these values are quite large, far beyond the percolation threshold. As a result one finds a very substantial reduction in multiplicity calculated according to Eq. (26) with luminosities determined from pp collisions. In Fig. 7 and 8 we show multiplicities as a function of $y$ for central collisions and function of $b$ at midrapidity. They agree rather well with the existing data both in form and absolute values. In Fig. 8 a rather sharp change is seen in the periphery of the nuclei, between $b/2R_A=0.8$ and unity. The RHIC data do not exhibit such a sharp saturation. This behaviour is a direct consequence of using in our calculations the nuclear profile function (23) corresponding to the step function for the nuclear density. A more realistic profile function would lead to a smoother transition from the periphery to center, although requiring a more complicated definition of the interaction area.
Our prediction for the plateau at LHC energy (divided by $\nu_{par}$) is around 9 for the sum of charged and neutral particles (that is 6 for charged), which is lower than derived in other publications [@Ref13]-[@Ref17]. The plateau height for charged particles in central Au-Au collisions as a function of energy is illustrated in Fig. 9. Of course our results are directly related to the chosen string radius and go upward if it is lowered. However then we loose the agreement with the existing data.
Note that our calculations also reproduce quite well the behaviour in the fragmentation region (limiting fragmentation). For this the dependence of the string luminosity on energy proved to be quite important. Without it the nice linear dependence of multiplicities in the fragmentation region is spoiled and the line is widened into a band.
To clearly see the effect of string fusion in Fig 10 we show the multiplicities at $b=0$ without fusion. Their values are several times greater than with fusion and do not agree with the experimental data at all.
Passing to the $p_T$ distributions in Fig. 11 we show the ratios $R_{AA}(p_T)$ for central collisions at midrapidity. The curve for 200 GeV served to determine our parameters $a,b,c$ and $d$ in (30) and (32). Our predictions for the LHC energy show a behaviour similar to the RHIC energy with a still more pronounced quenching effect. In the fragmentation region we prefer to show the ratios $R^{par}_{AA}$ with normalization respective to the number of participants, since in this region the multiplicities are roughly proportional to $\nu_{par}$ due to low string densities. Fig. 12 shows that these ratios are close to unity and may only fall a little below unity at LHC energies.
Conclusions
===========
In the framework of percolation of strings we have obtained a strong reduction of multiplicities at LHC, much larger than the rest of models but in agreement with the extrapolation from the SPS and RHIC experimental data. Due to similarities between percolation of strings and saturation of partons, it would be interesting to explore the possibility for further reduction of multiplicities in the saturation approach. In order to describe the energy dependence of multiplicities in AA collisions we need a rather large transverse size of the elementary string 0.3 fm. This enhances the interaction of strings and so cluster formation, which leads to stronger reduction of multiplicities. In our calculations we have used the standard optical approximation to compute the numbers of participants and collisions. This approximation enhances the number of collisions for peripheral collisions in comparison with Monte-Carlo evaluations and leads to some uncertainties also for central collisions. For this reason, our results must be regarded to have an uncertainty in the range of 10%-15% . So the string transverse size can be lower if the number of strings is in fact lower.
Taking limiting fragmentation scaling for pp collisions as an input, we have found the same behaviour for AA collisions, which is confirmed experimentally up to the RHIC energies. Our calculations predict that limiting fragmetation scaling also remains approximately valid at the LHC energy (with a 5% suppression compared with to SPS or RHIC, see Fig. 7).
We have been able to describe reasonably well the high transverse momentum spectrum at different energies ranging from SPS to RHIC. A large suppression is predicted for LHC. In order to obtain such an agreement, in addition to the usual effects of string clustering, such as reduction of the effective number of independent color sources and suppression of transverse momentum fluctuations, we need a shift of the $p_T$ spectrum due to energy loss. Considering string fusion as an initial state effect (before particle production), a final state effect is needed to account for the observed suppression, similarly to jet quenching in the QCD picture.
Acknowledgments
===============
This work has been partially done under contracts FPA2005-01963of Spain, and PGIDIT03PX1 of Galicia, and also supported by the NATO grant PST.CLG.980287 and Education Ministry of Russia grant RNP 2.1.1.1112.
[99]{} M.A. Braun and C. Pajares, Phys. Lett. B [**287**]{}, 154 (1992); Nucl. Phys. [**B390**]{}, 542 (1993).
N. Amelin, M.A. Braun and C. Pajares, Z. Phys. [**C63**]{}, 507 (1994);\
H.J. Mohring, J. Ranft, C. Merino and C. Pajares, Phys. Rev. [**D47**]{}, 4142 (1993);\
N. Armesto, C. Pajares and D. Sousa, Phys. Lett. [**B257**]{}, 92 (2002).
N. Armesto, M. A. Braun, E. G. Ferreiro and C. Pajares, Phys.Rev. Lett. [**77**]{}, 3736 (1996);\
M. Nardi and H. Satz, Phys. Lett. [**B442**]{}, 14 (1998).
J. Dias de Deus and R. Ugoccioni, Phys. Lett. [**B491**]{}, 253 (2000); Phys. Lett. [**B494**]{}, 53 (2000).
M.A. Braun, F. del Moral and C. Pajares, Phys. Rev. [**C65**]{}, 024907 (2002).
J. Dias de Deus, E.G. Ferreiro, C. Pajares and R. Ugoccioni, Eur. Phys. J. [**C40**]{}, 229 (2005).
ALICE Collaboration. CERN/LHCC/95-71 (1995).
K. Werner, Phys. Rep. [**232**]{}, 87 (1993).
X.N. Wang and M. Gyulassy, Phys. Rev. [**D44**]{}, 3501 (1991); Comput. Phys. Commun [**83**]{} 307 (1994).
N. van Eijndhoven et al. Internal Note ALICE 95-32 (1995).
A. Capella, U.P. Sukhatme, C.I. Tan and J. Tran Thanh Van,Phys. Rep. [**236**]{}, 225 (1994);\
A. Capella, C. Pajares and A.V. Ramallo,Nucl. Phys. [**B241**]{}, 75 (1984).
A. Capella, A.B. Kaidalov and J. Tran Thanh Van, Heavy Ion Physics [**9**]{}, 169 (1999).
X. Wang and M. Gyulassy, Phys. Rev. Lett. [**86**]{}, 3496 (2001).
H.J. Drescher, M. Hlakik, S. Ostapchenko, T. Pierog and K.Werner, Phys. Rep. [**350**]{}, 93 (2001).
A. Accardi, Phys. Rev. [**C64**]{}, 064905 (2001).
D. Kharzeev, E. Levin and M. Nardi, Phys. Rev. [**C72**]{}, 044904 (2005).
N. Armesto, C.A. Salgado, U.A. Wiedemann, Phys. Rev. Lett. [**94**]{}, 022002 (2005).
J. Dias de Deus and R. Ugoccioni, Phys. Lett. [**B494**]{}, 53 (2000).
M.A. Braun, F. del Moral and C. Pajares, Phys. Rev. [**C65**]{}, 024907 (2002)\
C. Pajares, Eur. Phys. J. [**C43**]{}, 9 (2005).
M.A. Braun and C. Pajares, Phys. Lett. [**B603**]{}, 21 (2004).
W. Busza, Acta Phys. Hungarica [**A24**]{}, 3 (2005).
PHOBOS Collaboration, B.B. Back et al., Nucl. Phys. [**A757**]{}, 28 (2005).
A. Drees, Nucl. Phys. [**A698**]{}, 331 (2002).
V. Topor Pop, M. Gyulassy, J. Barette and C. Gale, Phys. Rev., 054901 (2005).
K.A. Ter Martirosyan, Phys. Lett. [**B44**]{}, 377 (1973).
A. Bialas, M. Bleszynski and W. Czyz, Nucl. Phys. [**B111**]{}, 461 (1976)\
C. Pajares and A.V. Ramallo, Phys. Rev. [**C16**]{}, 2800 (1985).
M.A. Braun and C. Pajares, Phys. Rev. Lett. [**85**]{}, 4864 (2000).
A. Bialas, Phys. Lett. [**B466**]{}, 301 (1999).
M. Gyulassy, I. Vitev, X.N. Wang and B.W. Zhang. Published in Quark Gluon Plasma 3, editors: R.C. Hwa and X.N. Wang (World Scientific, Singapore, 2004);\
R. Baier, D. Schiff and B.G. Zakharov, Ann. Rev. Nucl.Part. Sci. 50, 37 (2000).
|
---
abstract: 'Using two pairs of lattice equations \[resembling Huang’s equations for bulk crystals, Proc. Roy. Soc. A [**208**]{}, 352 (1951)\] deduced from a microscopic dipole lattice model taking into account electronic polarization (EP) of ions and local field effects (LFEs) self-consistently, in-plane and out-of-plane optical vibrations in two-dimensional (2D) hexagonal BN are studied theoretically. The three mutually independent coefficients of either pair of lattice equations are determined by a set of three generally known quantities such as the 2D electronic and static susceptibilities and phonon frequency, making the lattice equations very useful for calculating the lattice dynamical properties. Explicit expressions are obtained for lattice vibrational energy density, and phonon dispersion, group velocity and density of states. The transparent phonon dispersion relations describe the previous numerical calculations very well, and the longitudinal optical (LO) phonon dispersion relation is identical to the analytical expression of Sohier [*et al*]{}. \[Nano Lett. [**17**]{}, 3758 (2017)\], and it expresses the degeneracy of the LO and transverse optical (TO) modes at $\Gamma$ and their splitting at finite wavevectors due to the long-range macroscopic field. The out-of-plane phonon frequency is finite owing to ionic EP. A 2D lattice dielectric function $\epsilon(k,\omega)$ is derived–due solely to the LO vibrations–which also allows the LO phonon dispersion to be rederived simply from $\epsilon(k,\omega)=0$, similar to the bulk case. A 2D Lyddane–Sachs–Teller relation and a frequency–susceptibility relation are obtained for the in-plane and out-of-plane vibrations, respectively, connecting the phonon frequencies to the 2D dielectric functions or susceptibilities. Using three first-principles calculated parameters, the lattice dynamical properties are studied comprehensively, particular attention being paid to the EP and LFEs. The ionic EP and LFEs should be included simultaneously, but otherwise neglecting either or both causes large discrepancies to the calculated dynamical properties. The rigid-ion model cannot properly describe the optical vibrations, which, for instance, yields a 15%-19% larger phonon frequency and 77% smaller Born charge for in-plane motion, and an infinitely large phonon frequency and four times larger Born charge for out-of-plane motion. With no LFEs or EP, the LO modes display very small linear dispersion, nearly flat in the long wavelength region, which is distinct from the LO phonon dispersion calculated after including both LFEs and EP. Furthermore, with ionic EP included, the LFEs increase the 2D susceptibility and Born charge by two and three times for in-plane vibrations but reduce both significantly for out-of-plane vibrations.'
author:
- 'J.-Z. Zhang'
title: Long optical lattice vibrations and dielectric function of monolayer hexagonal boron nitride
---
Introduction
============
Since the discovery of graphene two-dimensional (2D) materials have become a subject of intense research due to their novel mechanical, electronic and optical properties [@Geim:2013]. Monolayer (ML) hexagonal Boron nitride (hBN) and transition metal dichalcogenides (TMDs) such as MoS$_2$ are two types of prominent 2D binary crystals with a similar honeycomb lattice structure to graphene. While graphene is a purely covalent material, ML hBN is a 2D polar crystal with mixed covalent and ionic bonding. Also distinct from graphene, a semimetal, 2D hBN is an insulator with a large bandgap $\sim$7 eV [@Rasmussen:2016; @Sahin:2009]. Further, piezoelectricity occurs in ML hBN owing to inversion symmetry breaking. In recent years ML hBN has attracted intense interest due to its strong piezoelectric response [@Naumov:2009; @Droth:2016; @Michel:2017], large in-plane stiffness and strong resistance to stretching [@Sahin:2009; @Songl:2010; @Andrew:2012; @Thomas:2016], and also its capability to combine with graphene due to lattice match as a key material in layered graphene/hBN heterostructure electronic devices [@Ponomarenko:2011; @Haigh:2012].
Optical phonons are key scattering partners of electrons [@Ridley:2013] and play crucial roles in carrier transport in electronic devices. In a polar crystal long optical lattice vibrations are closely connected to the electric fields [@Born:1954]. While the [*macroscopic*]{} electric field $\mathbf{E}$ is an average field, with the averaging being made to the total field of all the ions over a lattice cell [@Born:1954; @Kittel:2004], the [*local*]{} field on any particular ion $\mathbf{E}_l$, as was introduced by Lorentz, is the total electric field after deduction of the contribution due to the ion itself. In a [*bulk*]{} polar crystal the Lorentz relation connects the local and macroscopic fields with the [*macroscopic*]{} dielectric polarization $\mathbf{P}$, $\mathbf{E}_l=\mathbf{E}+4\pi\mathbf{P}/3$ [@Born:1954; @Kittel:2004], where the last term is the [*inner*]{} field [@Born:1954]. The electric field acting on each ion of the crystal polarizes it by inducing an electric dipole moment, thus causing electronic polarization (EP) to the ion, or concisely ionic polarization. A treatment neglecting (including) such EP is termed rigid (polarizable) ion model, i.e., RIM and PIM. Using the Lorentz relation whilst considering the ions to be polarizable with a microscopic model, Huang deduced a pair of equations describing the long optical vibrations of isotropic bulk crystals, $\ddot{\mathbf{w}}=b_{11}\mathbf{w}+b_{12}\mathbf{E}$, $\mathbf{P}=b_{21}\mathbf{w}+b_{22}\mathbf{E}$ ($b_{12}=b_{21}$) [@Huang:1951; @Huang:1951a; @Born:1954], where $\mathbf{w}$ describes the optical displacement of the unit cell, $\mathbf{w}=\sqrt{\bar{m}/v_a}(\mathbf{u}_1-\mathbf{u}_2)$, $\mathbf{u}_1$ and $\mathbf{u}_2$ being the displacements of the positive and negative ions, $\bar{m}$ being their reduced mass and $v_a$ the cell volume. The lattice dielectric function can be deduced directly from Huang’s equations, allowing one to conveniently express the $b$-coefficients in terms of experimentally measurable quantities such as the static and high-frequency dielectric constants $\epsilon_0$ $\epsilon_{\infty}$ and infrared dispersion frequency $\omega_0$ [@Born:1954]. Solving Huang’s equations and the equation of electrostatics yields the longitudinal and the transverse optical (LO and TO) modes, with TO phonon frequency $\omega_t$ equal to $\omega_0$. Further, there is a macroscopic electrostatic field associated with the LO modes making their frequency $\omega_l$ higher than $\omega_t$, with the frequency ratio given by the Lyddane–Sachs–Teller (LST) relation [@Lyddane:1941], $\omega_l/\omega_t=\sqrt{\epsilon_0/\epsilon_{\infty}}$, which can also be rederived from Huang’s equations [@Born:1954].
In experimental study [@Rokuta:1997], phonon spectra for ML hBN on Ni and Pt have been measured by electron energy loss spectroscopy. Phonon spectra of ML hBN are usually calculated by diagonalizing the dynamical matrix which is obtained by first-principles calculation, or a tight-binding or an empirical model. Phonon spectra have been studied for ML hBN within the local-density approximation in density functional theory (DFT) [@Miyamoto:1995; @Illera:2017]. As a ML of hBN can be rolled up to form a BN nanotube, first-principles [@Ferrabone:2011] and tight-binding [@Sanchez:2002] calculations have been performed to make a comparative study of dielectric polarizabilities [@Ferrabone:2011] as well as phonon spectra [@Sanchez:2002] of hBN MLs and nanotubes. Phonon modes of 2D hBN calculated from first principles [@Serrano:2007; @Topsakal:2009; @Sahin:2009; @Michel:2017; @Sohier:2017] or an empirical force constant model [@Michel:2009; @Michel:2011; @Michel:2012] have been compared with those of three-dimensional (3D) bulk hBN [@Serrano:2007; @Topsakal:2009; @Michel:2011; @Michel:2012] or other 2D honeycomb materials such as TMDs [@Michel:2017; @Sohier:2017] and group III nitrides [@Sahin:2009]. In these studies, there are several common features present in the phonon spectra for ML hBN [@Sanchez:2002; @Serrano:2007; @Topsakal:2009; @Sahin:2009; @Michel:2009; @Sohier:2017]: (i) the LO and TO modes are degenerate at the $\Gamma$ point but split up at a finite wavevector with the LO mode having a higher frequency; (ii) overbending occurs in the LO phonon dispersion curve so the LO modes have maximum frequency not at $\Gamma$, but at an intermediate point away from the Brillouin zone edges; (iii) both TO and ZO (i.e., out-of-plane optical) modes show a nondispersive character at long wavelengths with a nearly constant frequency. While most of these studies are performed by numerical methods, only few studies have used analytical approaches [@Sanchez:2002; @Michel:2009; @Sohier:2017]. The degeneracy has been proved to be due to the macroscopic field’s in-plane component vanishing at zero wavevector [@Sanchez:2002], and analytical expressions have been obtained later for the long-wavelength dispersion of LO and TO modes [@Michel:2009; @Sohier:2017]. In Ref.[@Sohier:2017] an LO phonon dispersion relation is derived using a simple model, in which the relationship between the squared LO and TO phonon frequencies for bulk materials [@Cochran:1962; @Giannozzi:1991] is generalized and used for the 2D materials, and the parameters in the dispersion relation are obtained from their first-principles calculation so both ionic EP and LFEs are taken into account. The analytical theory of Michel [*et al*]{}. [@Michel:2009] tackles the dynamical matrix, which is obtained based on a microscopic RIM, to find the vibrational modes; while the eigensolutions of the dynamical matrix are numerically calculated for an arbitrary wavevector, the analytical phonon dispersions are obtained for small wavevectors. In the study [@Michel:2009] ionic polarization has not been accounted for, as the inclusion of EP presents a challenge and makes it more difficult to obtain analytical solutions to the polar optical vibrations. Equally important are the [*local fields*]{} on the ions, which have been found to be very strong in 2D hBN [@Mikhailov:2013]. Further, the local fields and EP are interdependent [@Born:1954] and should be included in a self-consistent manner, and therefore the addition of local field effects (LFEs) makes the lattice-dynamical solutions for a general wavevector more complicated. For the long wavelengths, however one expects that the 2D lattice motion can be described on a macroscopic basis, i.e., using macroscopic quantities such as the macroscopic field and dielectric polarization, by lattice equations like Huang’s equations for bulk crystals. Such equations need to be deduced from a microscopic model so as to include the intricate EP and LFEs. So far as we know, there are no equations of motion for the macroscopic description of the 2D lattice vibrations, and ionic EP or LFEs on the lattice dynamical properties of the 2D crystals have not been studied systematically. From the viewpoint of basic research, ML hBN, the structurally simplest 2D polar crystal, provides a model system for analytically studying the 2D optical vibrations and further local field and polarizable ion effects.
In this paper, we study long wavelength optical lattice vibrations and local field and polarizable ion effects for ML hBN with an analytical approach. For this purpose, we deduce lattice equations for optical vibrations using a microscopic model that includes the ionic EP and LFEs. We make the deduction with Huang’s approach, by introducing the [*macroscopic field*]{} into the equation of motion whilst constructing an equation for the [*macroscopic*]{} dielectric polarization by adding up the two contributions due to the lattice displacement and the induced electric polarization. We solve the simultaneous lattice equations and equation of electrostatics, rather than solve the dynamical matrix equations as in previous studies, to obtain explicit expressions for the optical modes, which can describe the key features of the 2D phonon modes. We derive a 2D longitudinal lattice dielectric function $\epsilon(k,\omega)$ which also allows one to rederive the LO phonon dispersion simply from $\epsilon(k,\omega)=0$. We also deduce a LST relation, a 2D counterpart of the LST relation in bulk, for in-plane motion and a frequency–susceptibility relation for out-of-plane motion. Using first-principles calculated quantities we study the lattice dynamical properties for 2D hBN and discuss in great detail the local field and polarizable ion effects.
This paper is organized as follows. In Section II, a deduction of 2D lattice equations for in-plane and out-of-plane motion is made, through a 2D Lorentz relation, from a microscopic dipole lattice model including LFEs and EP. From the lattice equations, the dispersion relations of the optical modes are deduced, followed by a derivation of the phonon group velocity and density of states (DOS). Then the dynamical lattice dielectric susceptibilities are derived yielding the relations relating the static and electronic susceptibilities to the $a_{12}$ and $c_{12}$ coefficients of the lattice equations. Further, a 2D longitudinal lattice dielectric function is deduced after considering the general test charge distributions, and from it a 2D LST relation follows. In Section III, we present results of the in-plane and out-of-plane optical vibrations in ML BN. A comparison of microscopic quantities such as effective charges and spring force constants is made, which are calculated when knowing three quantities including the electronic and static susceptibilities from independent first-principles calculations. Then various lattice-dynamical quantities, such as the Born charge, the phonon dispersion and the static and electronic susceptibilities, are compared with those obtained from a RIM, with or without LFEs taken into account, to study the EP and LFEs on the lattice dynamical properties. Finally, Section V summarizes the main results obtained. In Appendix A, we show that using the 2D Clausius-Mossotti relation obtained, the unit-cell atomic polarizability falls in an interval for in-plane or out-of-plane polarization, which is used to evaluate the LFEs on the phonon dispersion and 2D dielectric susceptibilities. In Appendix B we show that in terms of macroscopic theory the relations $a_{12}=a_{21}$ and $c_{12}=c_{21}$, which connect the coefficients of the lattice equations, follow from the principle of energy conservation. Further we obtain a lattice-vibrational energy density as a function of the optical displacement and electric field for in-plane or out-of-plane vibrations.
Theory
======
Equations of motion and lattice polarization
--------------------------------------------
ML hBN, a 2D binary crystal with point group D$_{3h}$, is composed of two sublattices of B and N (as shown in Fig. \[fig1\]), labeled with $\kappa=1, 2$, respectively. Let $m_{\kappa}$ and $e_{\kappa}$ be the mass and charge of the type $\kappa$ ions , and let $e_1=-e_2=e_a$, where $e_a$ is the static effective charge [@Karch:1997] due to electron charge transfer $-e_a$ from B to N in 2D h-BN, $e_a>0$ [@Topsakal:2009; @Michel:2017]. The masses of the B and N atoms are $m_1$=10.81 Da and $m_2$=14.01 Da. In the dipole lattice model [@Born:1954] each ion site of type $\kappa$ is occupied by an electric dipole $\mathbf{p}_{\kappa}$ which arises due partly to the ionic displacement $\mathbf{u}_{\kappa}$ and partly to the induced electric moment $\boldsymbol{\mu}_{\kappa}$ on the ion. Associated with the long wavelength optical modes, there is macroscopic dielectric polarization $\mathbf{P}=(\mathbf{p}/s)e^{i\mathbf{k}\cdot\boldsymbol{\rho}}\delta(z)$, where $\mathbf{p}$ is the total dipole moment $\mathbf{p}=\mathbf{p}_1+\mathbf{p}_2$ of a unit cell with area $s=\sqrt{3}a^2/2$, $a$ being the lattice constant $a$=2.5 $\AA$ (see Fig. \[fig1\]). The $\delta$ function describes the dependence of the polarization on $z$ for a ML in the plane $z=0$. $\mathbf{k}$ is the 2D wave vector, and $\boldsymbol{\rho}=(x,y)$ is the position vector in the plane parallel to the ML.
We first consider in-plane optical vibrations; that is, the displacements $\mathbf{u}_{\kappa}$ and dipole moments $\mathbf{p}_{\kappa}$ lie in the ML plane. The macroscopic electric field $\mathbf{E}$ due to the charge density $-\nabla\cdot\mathbf{P}$ is given by the equation of electrostatics $\nabla\cdot(\mathbf{E}+4\pi\mathbf{P})=0$, the electric field $\mathbf{E}$ being an irrotational field, $\mathbf{E}=-\nabla\phi$ \[$\phi$ is the electrostatic potential, $\phi(\boldsymbol{\rho},z)=\varphi(z)e^{i\mathbf{k}\cdot\boldsymbol{\rho}}$\]. To solve the corresponding Poisson’s equation $\nabla^2\phi(\boldsymbol{\rho},z)=4\pi i\mathbf{p}\cdot \mathbf{k}e^{i\mathbf{k}\cdot\boldsymbol{\rho}}\delta(z)/s$, we expand $\varphi(z)$ and $\delta(z)$, $$\varphi(z)=\int_{-\infty}^{\infty}\hat{\varphi}(q)e^{iqz}dq,
\label{vphx}$$ $$\delta(z)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{iqz}dq.
\label{delx}$$
We find $\hat{\varphi}(q)=-2i\mathbf{p}\cdot \mathbf{k}/[s(k^2+q^2)]$ and from Eq. (\[vphx\]) $\varphi(z)=-2\pi i\mathbf{p}\cdot\mathbf{k}e^{-k\lvert z\rvert}/(sk)$, and then obtain for the macroscopic field its components in the directions parallel and perpendicular to the ML:
$$\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},z)=-\frac{2\pi}{s}\frac{\mathbf{k}}{k}~\mathbf{p}\cdot\mathbf{k}e^{-k\lvert z\rvert}e^{i\mathbf{k}\cdot\boldsymbol{\rho}},
\label{Ero1}$$
$$\mathbf{E}_z(\boldsymbol{\rho},z)=-\mathbf{e}_z\frac{2\pi i}{s}\mathbf{p}\cdot\mathbf{k}~\operatorname{sgn}(z)e^{-k\lvert z\rvert}e^{i\mathbf{k}\cdot\boldsymbol{\rho}}.
\label{Ez1}$$
Here $\operatorname{sgn}(z)$ is the sign function and in particular the $z$-component of the field is zero in the ML.
Recall that in bulk polar crystals the macroscopic field is strongly nonanalytical close to zero wavevector, i.e., its limiting value depending on the direction along which zero wavevector is approached [@Born:1954], and the LO vibrations have a finite field at small wavevectors. Distinct from the field in 3D crystals, the macroscopic field \[Eqs. (\[Ero1\]) and (\[Ez1\])\] of 2D hBN vanishes at a very small wavevector for in-plane motion [@Sanchez:2002], independent of the direction of the wavevector.
In a 2D dipole lattice a dipole $\mathbf{p}$ on the ion at a lattice point $\boldsymbol{\rho}_i=(x_i,y_i)$ gives rise to a field at the origin equal to $-\mathbf{p}/\rho_i^3+3\mathbf{p}\cdot\boldsymbol{\rho}_i\boldsymbol{\rho}_i/\rho_i^5$. The Lorentz local field $\mathbf{E}_{l}$, also called the exciting field [@Born:1954], is the electric field acting on an ion due to all the [*other*]{} dipoles oscillating in the lattice, and can be written as the macroscopic field $\mathbf{E}$ plus another field $\mathbf{E}_{in}$ (namely, the [*inner*]{} field) [@Born:1954], $\mathbf{E}_{l}=\mathbf{E}+\mathbf{E}_{in}$. In a [*long wavelength*]{} lattice wave, the local fields at the B and N lattice points are given by the following expressions [@Mikhailov:2013], respectively,
$$\mathbf{E}_{l,1}=\mathbf{E}+Q_0~\mathbf{p}_1+Q_1~\mathbf{p}_2,
\label{Ex1}$$
$$\mathbf{E}_{l,2}=\mathbf{E}+Q_1~\mathbf{p}_1+Q_0~\mathbf{p}_2.
\label{Ex2}$$
with the coefficients $Q_0$ and $Q_1$ [@Mikhailov:2013; @Della:2016]
$$Q_0=\sum_{(m,n)\neq (0,0)}\frac{1}{2(m^2+n^2+mn)^{3/2}a^3}\approx \frac{5.5171}{a^3}~,
\label{Q0}$$
$$Q_1=\sum_{m,n}\frac{1}{2(m^2+n^2+mn+n+1/3)^{3/2}a^3}\approx \frac{11.5753}{a^3}~.
\label{Q1}$$
Let dipole $\mathbf{p}_1$ or equally $\mathbf{p}_2$ have 0.01 $e\AA$, for instance, corresponding to a displacement of 0.01 $\AA$ of the ions with charge $e$, then from these expressions one finds very strong local fields with a magnitude 1000 kV/cm [@Mikhailov:2013].
The LFEs are included through coefficients $Q_0$ and $Q_1$. Note that $\mathbf{E}$ is the macroscopic field \[Eqs. (\[Ero1\]) and (\[Ez1\])\] rather than an external field in a simple sense [@Mikhailov:2013]. Such a simple relation between the local field and the macroscopic field is valid only for the long lattice waves, while the general expression for an arbitrary wavelength is quite complicated with both coefficients $Q_0$ and $Q_1$ dependent on the wavevector. In 3D iostropic polar crystals [@Born:1954] the difference between the local and macroscopic fields is proportional to the [*macroscopic polarization*]{} directly (the Lorentz relation), $\mathbf{E}_l=\mathbf{E}+4\pi\mathbf{P}/3$, as the $Q$ coefficients are equal to $4\pi/(3v_a)$ at $\mathbf{k}=0$ [@Born:1954]. When approximating $Q_0=Q_1$, Eqs. (\[Ex1\]) and (\[Ex2\]) can be transformed into a simple Lorentz relation involving macroscopic [*areal polarization*]{} $(\mathbf{p}_1+\mathbf{p}_2)/s$, and also a Clausius-Mossotti relation can be deduced for 2D BN (Appendix A). Expressions (\[Ex1\]) and (\[Ex2\]), the Lorentz relations for 2D BN, show that the finite local fields occur in ML hBN at $\mathbf{k}=0$, different from those in the 3D polar crystals where local fields vanish in the long wavelength limit [@Born:1954].
Apart from the macroscopic field and local fields, we also need to find the field change at the center of an ion of type $\kappa$ owing to its own displacement $\mathbf{u}_{\kappa}$ [@Born:1954]. The field is evidently equal to the field created at the ion $\kappa$ site by displacing all other ions by $-\mathbf{u}_{\kappa}$. Hence it is equal to the local field at that ion site in a dipole lattice with displacement dipoles $\mathbf{p}_{\kappa'}=-e_{\kappa'}\mathbf{u}_{\kappa}$, where type $\kappa'=1, 2$. Substituting this dipole expression into Eqs. (\[Ex1\]) and (\[Ex2\]) and putting $\mathbf{E}=0$ as wavevector $\mathbf{k}=0$, we find the field changes at the centers of the B and N ions due to their own displacements respectively,
$$\mathbf{E}_{u,1}=-\mathbf{u}_1(e_1Q_0+e_2Q_1),
\label{Eu1}$$
$$\mathbf{E}_{u,2}=-\mathbf{u}_2(e_1Q_1+e_2Q_0).
\label{Eu2}$$
The [*total*]{} Coulomb fields $\mathbf{E}_1$ and $\mathbf{E}_2$ at the centers of the B and N ions are simply the sums of $\mathbf{E}_{l,1}$ and $\mathbf{E}_{u,1}$ \[Eqs. (\[Ex1\]) and (\[Eu1\])\], and $\mathbf{E}_{l,2}$ and $\mathbf{E}_{u,2}$ \[Eqs. (\[Ex2\]) and (\[Eu2\])\], respectively,
$$\mathbf{E}_1=\mathbf{E}+Q_0~\mathbf{p}_1+Q_1~\mathbf{p}_2+e_a(Q_1-Q_0)\mathbf{u}_1,
\label{Eto1}$$
$$\mathbf{E}_2=\mathbf{E}+Q_1~\mathbf{p}_1+Q_0~\mathbf{p}_2-e_a(Q_1-Q_0)\mathbf{u}_2,
\label{Eto2}$$
where all the vectors are in the layer plane.
Assuming that the [*electronic*]{} polarization of an ion is equivalent to a point-dipole [@Born:1954], the electronic (i.e., [*induced*]{}) dipole moment of the ion $\kappa$ is then given by $\boldsymbol{\mu}_{\kappa}=\alpha_{\kappa}\mathbf{E}_{\kappa}$, where $\alpha_{\kappa}$ is the in-plane [*electronic*]{} polarizability of the ion $\kappa$. Then the total dipole moments on the B and N ions are
$$\mathbf{p}_1=e_a\mathbf{u}_1+\alpha_1~\mathbf{E}_1,
\label{pto1}$$
$$\mathbf{p}_2=-e_a\mathbf{u}_2+\alpha_2~\mathbf{E}_2.
\label{pto2}$$
Inserting the expressions for the total fields $\mathbf{E}_1$ and $\mathbf{E}_2$ into Eqs. (\[pto1\]) and (\[pto2\]) and then rearranging the terms, we find
$$(1-\alpha_1Q_0)\mathbf{p}_1-\alpha_1Q_1\mathbf{p}_2=e_a[1+\alpha_1(Q_1-Q_0)]\mathbf{u}_1+\alpha_1~\mathbf{E},
\label{pto1a}$$
$$-\alpha_2Q_1\mathbf{p}_1+(1-\alpha_2Q_0)\mathbf{p}_2=-e_a[1+\alpha_2(Q_1-Q_0)]\mathbf{u}_2+\alpha_2~\mathbf{E}.
\label{pto2a}$$
Solving Eqs. (\[pto1a\]) and (\[pto2a\]) then we can express $\mathbf{p}_1$ and $\mathbf{p}_2$ in terms of $\mathbf{u}_1$, $\mathbf{u}_2$ and $\mathbf{E}$ as follows:
$$\begin{aligned}
\mathbf{p}_1&=\frac{1}{D}\Big \{e_a(1-\alpha_2Q_0)\big[1+\alpha_1(Q_1-Q_0)\big]\mathbf{u}_1 \Big.
\nonumber \\
&\qquad {} -e_a\alpha_1Q_1\big[1+\alpha_2(Q_1-Q_0)\big]\mathbf{u}_2
\nonumber \\
&\qquad {} \Big. +\alpha_1\big[1+\alpha_2(Q_1-Q_0)\big]\mathbf{E} \Big \},
\label{pt1}\end{aligned}$$
$$\begin{aligned}
\mathbf{p}_2&=\frac{1}{D}\Big \{e_a\alpha_2Q_1\big[1+\alpha_1(Q_1-Q_0)\big]\mathbf{u}_1 \Big.
\nonumber \\
&\qquad {} -e_a(1-\alpha_1Q_0)\big[1+\alpha_2(Q_1-Q_0)\big]\mathbf{u}_2
\nonumber \\
&\qquad {} \Big. +\alpha_2\big[1+\alpha_1(Q_1-Q_0)\big]\mathbf{E} \Big \},
\label{pt2}\end{aligned}$$
where $$D=1-(\alpha_1+\alpha_2)Q_0-\alpha_1\alpha_2(Q_1^2-Q_0^2).
\label{bigD}$$
Define the areal polarization $\boldsymbol{\mathcal{P}}$ $$\boldsymbol{\mathcal{P}}=(\mathbf{p}_1+\mathbf{p}_2)/s,
\label{defP}$$ and introduce the optical displacement $\mathbf{w}$ $$\mathbf{w}=\sqrt{\frac{\bar{m}}{s}}(\mathbf{u}_1-\mathbf{u}_2),
\label{smw}$$ where $\bar{m}$ is the reduced mass, $\bar{m}=m_1m_2/(m_1+m_2)$.
When expressions (\[pt1\]) and (\[pt2\]) are substituted for $\mathbf{p}_1$ and $\mathbf{p}_2$, we obtain $$\boldsymbol{\mathcal{P}}=a_{21}\mathbf{w}+a_{22}\mathbf{E},
\label{bigP1}$$ where
$$a_{21}=\frac{e_a}{D\sqrt{\bar{m}s}}\big[1+\alpha_1(Q_1-Q_0)\big]\big[1+\alpha_2(Q_1-Q_0)\big],
\label{a21}$$
$$a_{22}=\frac{1}{sD}\big[(\alpha_1+\alpha_2)+2\alpha_1\alpha_2(Q_1-Q_0)\big].
\label{a22}$$
Eq. (\[bigP1\]) shows that the [*macroscopic*]{} quantity $\boldsymbol{\mathcal{P}}$ of the 2D crystal, which has a clear physical meaning as given by expressions (\[defP\]), is simplified to a sum of two contributions, one due to the optical displacement and the other due to the [*macroscopic*]{} field.
It is evident from Eq. (\[bigP1\]) that $a_{22}$ is the in-plane component $\chi_e$ of the [*electronic*]{} susceptibility of the 2D crystal, $$a_{22}=\chi_e.
\label{a22alf}$$
When the Born charge [@Gonze:1997] $$e_B=\frac{e_a}{D}\big[1+\alpha_1(Q_1-Q_0)\big]\big[1+\alpha_2(Q_1-Q_0)\big],
\label{eB}$$ is introduced, then the coefficient $a_{21}$ relates simply to the Born charge $e_B$ by $$a_{21}=\frac{e_B}{\sqrt{\bar{m}s}}.
\label{a21b}$$
From Eqs. (\[a22\]), (\[a22alf\]) and (\[eB\]) we find $$a_{22}=\chi_e=\frac{1}{sQ_1}\left(\frac{e_B}{e_a}-1\right),
\label{a22eB}$$ showing that apart from $a_{21}$, the coefficient $a_{22}$ is also related to the Born charge. $e_B \neq e_a$ owing to the electronic polarization of the ions, and further considering $Q_1 >0$, $e_B$ is greater than $e_a$.
Inserting expressions (\[pt1\]) and (\[pt2\]) for $\mathbf{p}_1$ and $\mathbf{p}_2$ into Eqs. (\[Eto1\]) and (\[Eto2\]), we express the total fields $\mathbf{E}_1$ and $\mathbf{E}_2$ in terms of the ionic displacements and macroscopic field. After collecting like terms we can simplify the total field expressions to
$$\mathbf{E}_1=\frac{1}{D}\big[1+\alpha_2(Q_1-Q_0)\big]\big[e_aQ_1(\mathbf{u}_1-\mathbf{u}_2)+\mathbf{E}\big],
\label{Eto1b}$$
$$\mathbf{E}_2=\frac{1}{D}\big[1+\alpha_1(Q_1-Q_0)\big]\big[e_aQ_1(\mathbf{u}_1-\mathbf{u}_2)+\mathbf{E}\big].
\label{Eto2b}$$
The Coulomb force acting on the ion $\kappa$ consists of two parts [@Born:1954], (i) the force exerted on the ionic charge $e_{\kappa}$ by the total field $\mathbf{E}_{\kappa}$, and (ii) the force exerted on the induced dipole $\boldsymbol{\mu}_{\kappa}$ by the field of all other ions. The latter electric force can be sought by using again the Lorentz relations (\[Ex1\]) and (\[Ex2\]) as follows. Imagine that we subject the dipole $\boldsymbol{\mu}_{\kappa}$ at a particular site of ion $\kappa$ a virtual displacement $\mathbf{u}$ while keeping all other ions in their undisplaced positions. The virtual energy is then simply the interaction energy between the dipole and the field at the ion site which is created equivalently by displacing all other ions by $-\mathbf{u}$ [@Born:1954], and which is thus the local field in a dipole lattice with dipole moments $\mathbf{p}_{\kappa^\prime}=-e_{\kappa^\prime}\mathbf{u}$, (type $\kappa^\prime=1, 2$). Therefore the virtual energy is $-\boldsymbol{\mu}_{\kappa}\cdot\mathbf{E}_{l,\kappa}$, and the force on the dipole is the negative gradient of this virtual energy with respect to virtual displacement $\mathbf{u}$, given by $\nabla _{\mathbf{u}}(\boldsymbol{\mu}_{\kappa}\cdot\mathbf{E}_{l,\kappa})$. Inserting the above dipole moment expression into Eqs. (\[Ex1\]) and (\[Ex2\]) and recalling the macroscopic field $\mathbf{E}=0$, we find the local fields and then obtain for the forces on the dipoles $\boldsymbol{\mu}_1$ and $\boldsymbol{\mu}_2$ the expressions $e_a(Q_1-Q_0)\boldsymbol{\mu}_1$ and $-e_a(Q_1-Q_0)\boldsymbol{\mu}_2$, respectively, where $\boldsymbol{\mu}_{\kappa}=\alpha_{\kappa}\mathbf{E}_{\kappa}$ ($\kappa=1, 2$).
Apart from the electric forces, there are also restoring forces caused by the overlap potential between the positive and negative ions in the 2D polar crystal. The restoring forces on the B and N ions are $-K(\mathbf{u}_1-\mathbf{u}_2)$, and $-K(\mathbf{u}_2-\mathbf{u}_1)$, respectively, where the spring force constant $K$ for in-plane motion is a simple scalar rather than a tensor due to the hexagonal symmetry.
Therefore the equations of motion for the B and N ions are given by
$$m_1\ddot{\mathbf{u}}_1=-K(\mathbf{u}_1-\mathbf{u}_2)+e_a\mathbf{E}_1+e_a\alpha_1(Q_1-Q_0)\mathbf{E}_1,
\label{eom1}$$
$$m_2\ddot{\mathbf{u}}_2=-K(\mathbf{u}_2-\mathbf{u}_1)-e_a\mathbf{E}_2-e_a\alpha_2(Q_1-Q_0)\mathbf{E}_2.
\label{eom2}$$
Substitute expressions (\[Eto1b\]) and (\[Eto2b\]) for $\mathbf{E}_1$ and $\mathbf{E}_2$ respectively in the two equations above. Using the Born charge $e_B$ \[expression (\[eB\])\] and introducing a force constant due to LFEs $K_e$, $$K_e=e_ae_BQ_1,
\label{Ke}$$ the equations of motion reduce to
$$m_1\ddot{\mathbf{u}}_1=(K_e-K)(\mathbf{u}_1-\mathbf{u}_2)+e_B\mathbf{E},
\label{eom1b}$$
$$m_2\ddot{\mathbf{u}}_2=(K-K_e)(\mathbf{u}_1-\mathbf{u}_2)-e_B\mathbf{E}.
\label{eom2b}$$
Multiplying Eqs. (\[eom1b\]) and (\[eom2b\]) by $m_2$ and $m_1$ respectively, subtracting and then dividing by $(m_1+m_2)$, we find $$\bar{m}(\ddot{\mathbf{u}}_1-\ddot{\mathbf{u}}_2)=(K_e-K)(\mathbf{u}_1-\mathbf{u}_2)+e_B\mathbf{E}.
\label{eom12b}$$
On expressing $(\mathbf{u}_1-\mathbf{u}_2)$ in terms of $\mathbf{w}$ \[Eq. (\[smw\])\], we obtain $$\ddot{\mathbf{w}}=a_{11}\mathbf{w}+a_{12}\mathbf{E},
\label{eomw1}$$ where
$$a_{11}=\frac{1}{\bar{m}}(K_e-K)=-\omega_0^2,
\label{a11}$$
$$a_{12}=\frac{e_B}{\sqrt{\bar{m}s}},
\label{a12}$$
$\omega_0$ being the intrinsic oscillator frequency, i.e., in the absence of macroscopic field $\mathbf{E}$.
Comparing Eqs. (\[a21b\]) and (\[a12\]) we find the relation $$a_{12}=a_{21}.
\label{a12a21}$$
The equation of motion (\[eomw1\]) and the polarization equation (\[bigP1\]) describe the in-plane polar optical vibrations of 2D BN, the frequencies of which will be derived in Sec. III below. The lattice vibrations we are dealing with are of long wavelengths, and the pair of equations (\[eomw1\]) and (\[bigP1\]) constitute a macroscopic description of the lattice motion. It is shown in Appendix B that, from the viewpoint of the [*macroscopic*]{} theory, the relation (\[a12a21\]) is due to the principle of energy conservation, and further this relation makes it possible to define an areal energy density \[expression (\[endenuh\])\] from which the lattice equations (\[eomw1\]) and (\[bigP1\]) can be rederived. It is evident from Eqs. (\[eom1b\]) and (\[eom2b\]) that the center of mass of the two atoms remains stationary (frequency $\omega=0$), yielding trivial nondynamical solutions. We note that for clearness $\mathbf{E}$ appearing in all the equations above is used to represent the in-plane component of the macroscopic field in the ML, i.e., $\mathbf{E}=\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},0)$.
Now we consider ionic motion perpendicular to the layer plane, in which case the displacements $\mathbf{u}_{\kappa}$ and dipole moments $\mathbf{p}_{\kappa}$ are parallel to $\mathbf{e}_z$. Let $\alpha_{\kappa}^\prime$ denote the electronic polarizability of the type $\kappa$ ions; note that in general $\alpha_{\kappa}^\prime\neq\alpha_{\kappa}$ (the latter is the polarizability for in-plane motion), as they are simply components of the polarizability tensor [@Born:1954]. Let $K^\prime$ be the force constant associated with the perpendicular motion. Considering the anisotropic 3D charge density distribution, another static [*effective*]{} charge $e_a^\prime$ exists likewise, which may differ from $e_a$ for in-plane motion (anisotropy), and is needed in the [*point-ion*]{} model for the perpendicular motion; thus the charges on the ions are $e_1^\prime=-e_2^\prime=e_a^\prime$.
Solving the Poisson equation yields the electrostatic potential $$\varphi(z)=2\pi \mathbf{p}\cdot\mathbf{e}_z\operatorname{sgn}(z)e^{-k\lvert z\rvert}/s,
\label{Pophez}$$ and then the $z$- and in-plane components of the macroscopic field follow,
$$\mathbf{E}_z(\boldsymbol{\rho},z)=-\frac{4\pi}{s}\mathbf{p}\left[\delta(z)-\frac{1}{2}ke^{-k\lvert z\rvert}\right]e^{i\mathbf{k}\cdot\boldsymbol{\rho}},
\label{Ez2}$$
$$\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},z)=-\frac{2\pi i}{s}\mathbf{k}~\mathbf{p}\cdot\mathbf{e}_z\operatorname{sgn}(z)e^{-k\lvert z\rvert}e^{i\mathbf{k}\cdot\boldsymbol{\rho}}.
\label{Ero2}$$
We note that (i) there is a $\delta(z)$ term in the $z$-component of the macroscopic field, which reflects the microscopic character of the polarization in terms of its atomic scale when the dipoles point to the $z$ direction. This term follows the dielectric polarization $\mathbf{P}$ according to $-4\pi\mathbf{P}$, which is also true when the $\delta(z)$ function of $\mathbf{P}$ is generalized to an arbitrary function $f(z)$; (ii) the in-plane component of the field is zero in the ML.
The local fields at the B and N sites are given by
$$\mathbf{E}_{l,1}=\mathbf{E}+Q_0^\prime~\mathbf{p}_1+Q_1^\prime~\mathbf{p}_2,
\label{Ex1z}$$
$$\mathbf{E}_{l,2}=\mathbf{E}+Q_1^\prime~\mathbf{p}_1+Q_0^\prime~\mathbf{p}_2,
\label{Ex2z}$$
respectively, where the coefficients $Q_0^\prime$ and $Q_1^\prime$ are negative, $Q_0^\prime=-2Q_0$, and $Q_1^\prime=-2Q_1$ [@Mikhailov:2013]. The long wavelength macroscopic field has a $\delta(z)$ form \[Eq. (\[Ez2\])\] but in calculating the field change experienced by a type $\kappa$ ion owing to its own displacement $\mathbf{u}_{\kappa}$, the contribution of this macroscopic field is zero, i.e., $4\pi\mathbf{u}_{\kappa}\sum_{\kappa^\prime}e_{\kappa^\prime}^\prime\delta(z)/s=4\pi\mathbf{u}_{\kappa}(e_1^\prime+e_2^\prime)\delta(z)/s=0$. Thus the field changes at the centers of B and N owing to their own displacements are
$$\mathbf{E}_{u,1}=-\mathbf{u}_1(e_1^\prime Q_0^\prime+e_2^\prime Q_1^\prime),
\label{Eu1z}$$
$$\mathbf{E}_{u,2}=-\mathbf{u}_2(e_1^\prime Q_1^\prime+e_2^\prime Q_0^\prime),
\label{Eu2z}$$
respectively. Similarly, in calculating the force exerted on the induced dipole $\boldsymbol{\mu}_{\kappa}$ at a type $\kappa$ ion by the field of all other ions, the macroscopic field makes no contribution to this force again because the net charge per cell vanishes. Then repeating the process as before, we find that with replacements $Q_0\rightarrow Q_0^\prime$, $Q_1\rightarrow Q_1^\prime$, $e_a\rightarrow e_a^\prime$, $\alpha_{\kappa}\rightarrow\alpha_{\kappa}^\prime$ and $ K\rightarrow K^\prime$, the equations and expressions above for the in-plane motion are applicable to the out-of-plane motion.
The Born charge is $$e_B^\prime=\frac{e_a^\prime}{D^\prime}\big[1+\alpha_1^\prime(Q_1^\prime-Q_0^\prime)\big]\big[1+\alpha_2^\prime(Q_1^\prime-Q_0^\prime)\big],
\label{eBz}$$ where $$D^\prime=1-(\alpha_1^\prime+\alpha_2^\prime)Q_0^\prime-\alpha_1^\prime\alpha_2^\prime(Q_1^{\prime 2}-Q_0^{\prime 2}).
\label{bigD1}$$
The polarization equation is $$\boldsymbol{\mathcal{P}}=c_{21}\mathbf{w}+c_{22}\mathbf{E},
\label{bigP1z}$$ where $\mathbf{w}$ is the optical displacement as given by Eq. (\[smw\]) and $c_{22}$ is the out-of-plane component $\chi_e^\prime$ of the electronic susceptibility of the 2D crystal, $$c_{22}=\chi_e^\prime,
\label{c22alf}$$ and $c_{21}$ and $c_{22}$ relate to the Born charge via
$$c_{21}=\frac{e_B^\prime}{\sqrt{\bar{m}s}},
\label{c21z}$$
$$c_{22}=\chi_e^\prime=\frac{1}{sQ_1^\prime}\left(\frac{e_B^\prime}{e_a^\prime}-1\right).
\label{c22z}$$
The equation of motion is given by $$\ddot{\mathbf{w}}=c_{11}\mathbf{w}+c_{12}\mathbf{E},
\label{eomw1z}$$ where
$$c_{11}=\frac{1}{\bar{m}}(K_e^\prime-K^\prime)=-\omega_0'^2 \quad (K_e^\prime=e_a^\prime e_B^\prime Q_1^\prime), \label{c11z}$$
$$c_{12}=c_{21}=\frac{e_B^\prime}{\sqrt{\bar{m}s}},
\label{c12z}$$
$\omega_0'$ being the intrinsic oscillator frequency.
In the lattice equations (\[eomw1z\]) and (\[bigP1z\]) $\mathbf{E}$ is the field in the ML, $\mathbf{E}(\boldsymbol{\rho},0)$, and evidently $\mathbf{E}(\boldsymbol{\rho},0)=\mathbf{E}_z(\boldsymbol{\rho},0)$ for the out-of-plane vibrations. The areal energy density associated with the out-of-plane optical vibrations is given by expression (\[endenuv\]).
When the equations for in-plane motion \[Eqs. (\[eomw1\]) and (\[bigP1\])\] and out-of-plane motion \[Eqs. (\[eomw1z\]) and (\[bigP1z\])\] are considered simultaneously, they can be rewritten for clarity as
$$\ddot{\mathbf{w}}_{\boldsymbol{\rho}}(\boldsymbol{\rho})=a_{11}\mathbf{w}_{\boldsymbol{\rho}}(\boldsymbol{\rho})+a_{12}\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},0),
\label{eomw1co}$$
$$\boldsymbol{\mathcal{P}}_{\boldsymbol{\rho}}(\boldsymbol{\rho})=a_{21}\mathbf{w}_{\boldsymbol{\rho}}(\boldsymbol{\rho})+a_{22}\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},0),
\label{bigP1co}$$
and
$$\ddot{\mathbf{w}}_z(\boldsymbol{\rho})=c_{11}\mathbf{w}_z(\boldsymbol{\rho})+c_{12}\mathbf{E}_z(\boldsymbol{\rho},0),
\label{eomw1zco}$$
$$\boldsymbol{\mathcal{P}}_z(\boldsymbol{\rho})=c_{21}\mathbf{w}_z(\boldsymbol{\rho})+c_{22}\mathbf{E}_z(\boldsymbol{\rho},0),
\label{bigP1zco}$$
respectively, where $a_{12}=a_{21}$ and $c_{12}=c_{21}$. These equations have similar forms to Huang’s equations for bulk crystals [@Huang:1951; @Born:1954].
In-plane and out-of-plane optical modes
---------------------------------------
The in-plane optical vibration modes can be obtained from Eqs. (\[eomw1\]) and (\[bigP1\]) in conjunction with the equation of electrostatics $\nabla\cdot(\mathbf{E}+4\pi\mathbf{P})=0$, where $\mathbf{P}$ is the dielectric polarization (namely, a dipole moment per unit volume), $\mathbf{P}=\boldsymbol{\mathcal{P}}\delta(z)$, and $\mathbf{E}$ is an irrotational field, $\mathbf{E}=-\nabla\phi$. Let $\mathbf{w}(\boldsymbol{\rho})=\mathbf{w}_0e^{i\mathbf{k}\cdot\boldsymbol{\rho}}$ and electrostatic potential $\phi(\boldsymbol{\rho},z)=\varphi(z)e^{i\mathbf{k}\cdot\boldsymbol{\rho}}$ (time dependence $e^{-i\omega t}$ is omitted for clearness). Expressing the field $\mathbf{E}=-\nabla\phi$ in terms of $\varphi$ gives the in-plane component $\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},z)=-i\mathbf{k}\varphi(z)e^{i\mathbf{k}\cdot\boldsymbol{\rho}}$ and further this field in the ML $\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},0)=-i\mathbf{k}\varphi(0)e^{i\mathbf{k}\cdot\boldsymbol{\rho}}$. Then apply divergence $\nabla_{\boldsymbol{\rho}}$ to Eq. (\[bigP1\]), and we have the polarization charge density $-\nabla\cdot\mathbf{P}=-\delta(z)\nabla_{\boldsymbol{\rho}}\cdot\boldsymbol{\mathcal{P}}=-\delta(z)[a_{21}i\mathbf{k}\cdot\mathbf{w}+a_{22}k^2\varphi(0)e^{i\mathbf{k}\cdot\boldsymbol{\rho}}]$ and obtain Poisson’s equation, $$\nabla^2\phi(\boldsymbol{\rho},z)=4\pi\delta(z)[a_{21}i\mathbf{k}\cdot\mathbf{w}(\boldsymbol{\rho})+a_{22}k^2\varphi(0)e^{i\mathbf{k}\cdot\boldsymbol{\rho}}].
\label{poi1}$$
To solve Eq. (\[poi1\]), we insert the expansions of $\varphi(z)$ \[Eq. (\[vphx\])\] and $\delta(z)$ \[Eq. (\[delx\])\], yielding $$\hat{\varphi}(q)=-2\left[a_{21}i\mathbf{k}\cdot\mathbf{w}_0+a_{22}k^2\varphi(0)\right]\frac{1}{k^2+q^2}.
\label{phiq}$$ When this expression is substituted for $\hat{\varphi}(q)$ of the following equation, $$\varphi(0)=\int_{-\infty}^{\infty}\hat{\varphi}(q)dq,
\label{vph0x}$$ we find $$\varphi(0)=-\frac{2\pi a_{21}i\mathbf{k}\cdot\mathbf{w}_0}{k(1+2\pi a_{22}k)},
\label{vph0a}$$ and then obtain the electric field in the ML, $$\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},0)=-\frac{2\pi a_{21}\mathbf{k}}{k(1+2\pi a_{22}k)}\mathbf{w}\cdot\mathbf{k}.
\label{Eroz0}$$
For a normal mode with wavevector $\mathbf{k}$, Eq. (\[eomw1\]) becomes $$(-\omega^2-a_{11})\mathbf{w}(\boldsymbol{\rho})=a_{12}\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},0).
\label{eow1b}$$
The expression (\[Eroz0\]) admits two possibilities for $\mathbf{w}\cdot\mathbf{k}$, namely, case (i) $\mathbf{w}\cdot\mathbf{k}=0$, or case (ii) $\mathbf{w}\cdot\mathbf{k}\ne 0$. In case (i), as $\mathbf{w}\cdot\mathbf{k}=0$ the normal modes are transverse waves. Eq. (\[vph0a\]) gives $\varphi(0)=0$, and on account of Eq. (\[phiq\]) we have $\hat{\varphi}(q)=0$. Then it follows from Eq. (\[vphx\]) that $\varphi(z)=0$, and therefore in the electrostatic approximation the macroscopic field vanishes identically, $\mathbf{E}(\mathbf{r})=0$. The frequency of the TO mode of wavevector $\mathbf{k}$ is given by the solution of Eq. (\[eow1b\]) with the field equal to zero, $$\omega_t=\omega_0=\sqrt{-a_{11}}=\sqrt{\frac{K-e_ae_BQ_1}{\bar{m}}},
\label{wto2a}$$ which is independent of wavevector; that is, the long-wavelength TO modes are dispersionless, consistent with previous tight-binding [@Sanchez:2002] and first-principles [@Wirtz:2003; @Topsakal:2009; @Sohier:2017] calculations.
In case (ii) the electric field $\mathbf{E}_{\boldsymbol{\rho}}(\mathbf{r})$ is nonzero, and from the equations evidently the vectors $\mathbf{w}(\boldsymbol{\rho})$, $\mathbf{E}_{\boldsymbol{\rho}}(\mathbf{r})$, $\mathbf{P}(\mathbf{r})$ associated with the mode are all longitudinal, i.e., parallel to wave vector $\mathbf{k}$, $\mathbf{w}(\boldsymbol{\rho})\parallel\mathbf{E}_{\boldsymbol{\rho}}(\mathbf{r})\parallel\mathbf{P}(\mathbf{r})\parallel\mathbf{k}$. Inserting expression (\[Eroz0\]) for $\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},0)$ into Eq. (\[eow1b\]) we find the frequency of the longitudinal optical (LO) mode, $$\omega_l(k)=\Big(-a_{11}+\frac{2\pi a_{21}^2k}{1+2\pi a_{22}k}\Big)^{1/2}.
\label{wlo2a}$$ Expressing the $a$-coefficients in terms of $e_B$, $\chi_e$, $\omega_0$ by Eqs. (\[a22alf\]), (\[a21b\]) and (\[wto2a\]), then $\omega_l(k)$ can be rewritten as $$\omega_l(k)=\Big[\omega_0^2+\frac{2\pi e_B^2k}{\bar{m}s(1+2\pi\chi_ek)}\Big]^{1/2}.
\label{wlo2a2}$$ Physically, Eq. (\[wlo2a2\]) is valid for small wavevectors; mathematically $\omega_l$ is a monotonically increasing function of $k$ and has an upper limit $\omega_M$ for very large $k$, $$\omega_M=\sqrt{\omega_0^2+e_B^2/(\bar{m}s\chi_e)}~.
\label{wloupp}$$
In the theoretical study by Sohier [*et al*]{}. [@Sohier:2017] the LO phonon dispersion is expressed as $\omega_l^2=\omega_0^2+\mathcal{S}k/(1+r_{eff}k)$. From their Eqs. (2) and (3)[^1] one finds that the parameter $\mathcal{S}$ relates to the Born charge via $\mathcal{S}=2\pi e_B^2/(\bar{m}s)$. $r_{eff}$ is an effective screening length, given by by $r_{eff}=\epsilon_pt/2$ with an effective medium model [@Sohier:2016; @Sohier:2017], where $\epsilon_p$ and $t$ are effective dielectric constant and effective thickness of the ML material. Both parameters $\mathcal{S}$ and $r_{eff}$ are computed by first-principles calculation [@Sohier:2017]. According to the defintion in Refs.[@Cudazzo:2011; @Berkelbach:2013], the effective screening length is given by $2\pi\chi_e$ or $2\pi\chi_0$ when including the vibrational contribution \[$\chi_0$ is the static susceptibility Eq. (\[chiwps0\])\]. As it is the [*high-frequency*]{} dielectric constant that determines the difference between the squared LO and TO phonon frequencies [@Cochran:1962; @Giannozzi:1991; @Gonze:1997] so $r_{eff}$ takes the former, i.e., $r_{eff}=2\pi\chi_e$. Therefore the LO phonon dispersion Eq. (\[wlo2a2\]) is identical to the analytical expression of Sohier [*et al*]{}..
Having $\varphi(0)$ \[Eq. (\[vph0a\])\] for the LO mode $\omega_l(k)$, we substitute expression (\[phiq\]) back into Eq. (\[vphx\]) to give the electric potential and then obtain the macroscopic field associated with the LO mode, expressed in terms of mode displacement $\mathbf{w}$ \[in a form consistent with Eqs. (\[Ero1\]) and (\[Ez1\])\],
$$\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},z)=-\frac{2\pi e_B\mathbf{k}}{\sqrt{\bar{m}s}k(1+2\pi \chi_ek)}\mathbf{w}\cdot\mathbf{k}e^{-k\lvert z\rvert},
\label{Erolo}$$
$$\mathbf{E}_z(\boldsymbol{\rho},z)=-\mathbf{e}_z\frac{2\pi ie_B}{\sqrt{\bar{m}s}(1+2\pi \chi_ek)}\mathbf{w}\cdot\mathbf{k}\operatorname{sgn}(z)e^{-k\lvert z\rvert}.
\label{Ezlo}$$
The long-range electrostatic interactions cause a higher LO frequency than the TO frequency for a finite $k$ [@Sanchez:2002] with the splitting determined by the latter term in the square brackets of Eq. (\[wlo2a2\]). This term also shows that there is no splitting in the limit $k\rightarrow 0$ as the macroscopic field vanishes \[Eq. (\[Erolo\])\], which is different from the situation in bulk BN where LO-TO splitting occurs also at the $\Gamma$ point [@Ohba:2001; @Topsakal:2009]. Therefore, the transparent expressions (\[wto2a\]) and (\[wlo2a2\]) show the degeneracy of the LO and TO modes at $\Gamma$ and their splitting at a finite wavevector, well-known phenomena of the 2D semiconductors [@Sanchez:2002; @Wirtz:2003; @Michel:2009; @Topsakal:2009; @Sohier:2017].
The out-of-plane optical vibrations, namely, the ZO modes, can be obtained from Eqs. (\[eomw1z\]) and (\[bigP1z\]) with the electrostatic approach as follows. Recall that the $z$-component of the field \[Eq. (\[Ez2\])\] has a $\delta(z)$ term, divergent at $z=0$, because the ML is treated as a geometric plane where the ionic charge distribution and polarization density $\mathbf{P}$ have a $\delta(z)$ form. We approximate $\delta(z)$ by a Gaussian distribution with a small thickness $\varepsilon$ ($\varepsilon\rightarrow 0$ ), $\delta_{\varepsilon}(z)=\frac{1}{\sqrt{\pi}\varepsilon}e^{-z^2/\varepsilon^2}$, as in the theoretical study [@Michel:2009]. A similar treatment is used also in first-principles calculations [@Sohier:2017]. The $z$-component of the field in the ML is $\mathbf{E}_z(\boldsymbol{\rho},0)=-\mathbf{e}_z\varphi'(0)e^{i\mathbf{k}\cdot\boldsymbol{\rho}}$, and after inserting this field into Eq. (\[bigP1z\]) the polarization charge density can be expressed as $-\nabla\cdot\mathbf{P}=-\delta_{\varepsilon}'(z)\mathbf{e}_z\cdot\boldsymbol{\mathcal{P}}=-[c_{21}\mathbf{w}\cdot\mathbf{e}_z-c_{22}\varphi'(0)e^{i\mathbf{k}\cdot\boldsymbol{\rho}}]\delta_{\varepsilon}'(z)$. Thus Poisson’s equation is given by $$\nabla^2\phi(\boldsymbol{\rho},z)=4\pi\delta_{\varepsilon}'(z)[c_{21}\mathbf{w}(\boldsymbol{\rho})\cdot\mathbf{e}_z-c_{22}\varphi'(0)e^{i\mathbf{k}\cdot\boldsymbol{\rho}}].
\label{poiz1}$$
Expanding $\varphi(z)$ \[Eq. (\[vphx\])\] and $\delta_{\varepsilon}(z)$ $$\delta_{\varepsilon}(z)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-(q\varepsilon/2)^2}e^{iqz}dq,
\label{deldx}$$ then we have $$\varphi'(z)=\int_{-\infty}^{\infty}iq\hat{\varphi}(q)e^{iqz}dq,
\label{vph1x}$$ $$\delta_{\varepsilon}'(z)=\frac{1}{2\pi}\int_{-\infty}^{\infty}iqe^{-(q\varepsilon/2)^2}e^{iqz}dq.
\label{deld1x}$$ Inserting the expansions (\[vphx\]) and (\[deld1x\]) into Poisson’s equation (\[poiz1\]) we find $\hat{\varphi}(q)$, $$\hat{\varphi}(q)=-2\left[c_{21}\mathbf{w}_0\cdot\mathbf{e}_z-c_{22}\varphi'(0)\right]\frac{iq}{k^2+q^2}e^{-(q\varepsilon/2)^2},
\label{phiqz}$$ and then carrying out the integration in Eq. (\[vph1x\]) we obtain $$\begin{aligned}
\varphi'(z) &=2\Big[c_{21}\mathbf{w}_0\cdot\mathbf{e}_z-c_{22}\varphi'(0)\Big]\left\{\frac{2\sqrt{\pi}}{\varepsilon}e^{-z^2/\varepsilon^2}\right.
\nonumber \\ &\qquad {} -\frac{\pi}{2}ke^{(k\varepsilon/2)^2}\Big[2\cosh(kz)-e^{-k\lvert z\rvert}\operatorname{erf}\big(\frac{k\varepsilon}{2}-\frac{\lvert z\rvert}{\varepsilon}\big)
\Big.
\nonumber \\ &\qquad {} \left. \Big.
-e^{k\lvert z\rvert}\operatorname{erf}\big(\frac{k\varepsilon}{2}+\frac{\lvert z\rvert}{\varepsilon}\big)\Big] \right\},
\label{phi1za}\end{aligned}$$ where $\operatorname{erf}(x)$ is the error function, $\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{-t^2}dt$.
Taking $z=0$ in Eq. (\[phi1za\]) we find $\varphi'(0)$, $$\varphi'(0)=\frac{c_{21}}{\varepsilon/\zeta_k+c_{22}}\mathbf{w}_0\cdot\mathbf{e}_z,
\label{phi1z0b}$$ where $$\zeta_k=4\sqrt{\pi}\Big[1-\sqrt{\pi}\frac{k\varepsilon}{2}\operatorname{erfc}(\frac{k\varepsilon}{2})e^{(k\varepsilon/2)^2}\Big],
\label{zetak}$$ $\operatorname{erfc}(x)$ being the complementary error function, $\operatorname{erfc}(x)=1-\operatorname{erf}(x)$.
The electric field in the ML $\mathbf{E}_z(\boldsymbol{\rho},0)$ follows, $$\mathbf{E}_z(\boldsymbol{\rho},0)=-\frac{c_{21}}{\varepsilon/\zeta_k+c_{22}}\mathbf{w}=-\frac{e_B^\prime}{\sqrt{\bar{m}s}(\varepsilon/\zeta_k+\chi_e^\prime)}\mathbf{w}.
\label{Ezro0b1}$$ When this field is substituted into the equation of motion (\[eomw1z\]), we obtain the frequency of the ZO mode, $$\omega_z(k)=\Big[\omega_0'^2+\frac{e_B^{\prime 2}}{\bar{m}s(\varepsilon/\zeta_k+\chi_e^\prime)}\Big]^{1/2}.
\label{wzo2a}$$
Considering that both $\varepsilon$ and $k$ are small quantities, expression (\[zetak\]) reduces to a constant $\zeta=4\sqrt{\pi}$, and the ZO phonon frequency becomes independent of wavevector, $$\omega_z=\Big\{\omega_0'^2+\frac{e_B^{\prime 2}}{\bar{m}s[\varepsilon/(4\sqrt{\pi})+\chi_e^\prime]}\Big\}^{1/2};
\label{wzo2a2}$$ for a planar material when taking $\varepsilon \ll 4\sqrt{\pi}\chi_e^\prime$ the frequency becomes $$\omega_z=\Big(\omega_0'^2+\frac{e_B^{\prime 2}}{\bar{m}s\chi_e^\prime}\Big)^{1/2}.
\label{wzo2b3}$$ The ionic polarization (i.e., $\chi_e^\prime\ne 0$) ensures a finite frequency; otherwise $\omega_z$ \[Eq. (\[wzo2a2\])\] becomes extremely large and even $\omega_z\rightarrow \infty$ \[Eq. (\[wzo2b3\])\] when $\chi_e^\prime$ is neglected in the RIM (also see Table \[table:4\]). The result here that the long wavelength ZO modes are dispersionless agrees with previous calculations which show a very flat dispersion curve at small wavevectors [@Sanchez:2002; @Wirtz:2003; @Michel:2009; @Topsakal:2009].
Having $\varphi'(0)$ \[Eq. (\[phi1z0b\])\] for a ZO mode we then get $\varphi'(z)$ from Eq. (\[phi1za\]) and also obtain $\varphi(z)$ after substituting the $\hat{\varphi}(q)$ expression (\[phiqz\]) into Eq. (\[vphx\]). It follows that the macroscopic field associated with the ZO mode is given by
$$\begin{aligned}
\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},z)&=\frac{i\pi e_B^\prime\mathbf{k}}{\sqrt{\bar{m}s}(1+\chi_e^\prime\zeta_k/\varepsilon)}\mathbf{w}\cdot\mathbf{e}_z\operatorname{sgn}(z)e^{(k\varepsilon/2)^2} \nonumber \\
&\qquad {} \times\Big[2\sinh(k\lvert z\rvert)+e^{-k\lvert z\rvert}\operatorname{erf}\big(\frac{k\varepsilon}{2}-\frac{\lvert z\rvert}{\varepsilon}\big) \Big.
\nonumber \\ &\qquad {} \Big. -e^{k\lvert z\rvert}\operatorname{erf}\big(\frac{k\varepsilon}{2}+\frac{\lvert z\rvert}{\varepsilon}\big)\Big],
\label{Erozo8}\end{aligned}$$
$$\begin{aligned}
\mathbf{E}_z(\boldsymbol{\rho},z)&=\frac{-2e_B^\prime}{\sqrt{\bar{m}s}(1+\chi_e^\prime\zeta_k/\varepsilon)}\mathbf{w}\left\{\frac{2\sqrt{\pi}}{\varepsilon}e^{-z^2/\varepsilon^2} \right. \nonumber \\
&\qquad {}
-\frac{\pi}{2}ke^{(k\varepsilon/2)^2}\Big[2\cosh(kz)-e^{-k\lvert z\rvert}\operatorname{erf}\big(\frac{k\varepsilon}{2}-\frac{\lvert z\rvert}{\varepsilon}\big) \Big.
\nonumber \\ &\qquad {} \left. \Big. -e^{k\lvert z\rvert}\operatorname{erf}\big(\frac{k\varepsilon}{2}+\frac{\lvert z\rvert}{\varepsilon}\big)\Big] \right\},
\label{Ezzo8}\end{aligned}$$
which for small $k$ and $\varepsilon$ reduce to the following expressions \[compare to Eqs. (\[Ez2\]) and (\[Ero2\])\],
$$\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},z)=\frac{-2i\pi e_B^\prime\mathbf{k}}{\sqrt{\bar{m}s}(1+\chi_e^\prime 4\sqrt{\pi}/\varepsilon)}\mathbf{w}\cdot\mathbf{e}_z\operatorname{sgn}(z)e^{-k\lvert z\rvert},
\label{Erozo9}$$
$$\mathbf{E}_z(\boldsymbol{\rho},z)=\frac{-4\pi e_B^\prime}{\sqrt{\bar{m}s}(1+\chi_e^\prime 4\sqrt{\pi}/\varepsilon)}\mathbf{w}\left(\delta(z)-\frac{1}{2}ke^{-k\lvert z\rvert}\right).
\label{Ezzo9}$$
In particular we note that the in-plane component is an odd function of $z$, and negligible at very small $k$ and $\varepsilon$.
Phonon group velocity and density of states
-------------------------------------------
Expanding $\omega_l(k)$ \[Eq. (\[wlo2a2\])\] to second order in wavevector $k$, the LO phonon frequency near the $\Gamma$ point is given by $$\omega_l(k)=\omega_0+c_lk-\frac{1}{2}c_l(\frac{c_l}{\omega_0}+4\pi \chi_e)k^2,
\label{wlok0}$$ where $$c_l=\frac{\pi e_B^2}{\bar{m}s\omega_0},
\label{cl}$$ which is identical to that given in Ref.[@Michel:2009].
From the dispersion Eq. (\[wlo2a2\]) we find straightforward the DOS of the LO modes, $$g_l(\omega)=\frac{c_l\omega_0}{4\pi^4 \chi_e^3}\frac{\omega(\omega^2-\omega_0^2)}{\big(\omega_M^2-\omega^2\big)^3}, \quad \omega_0 \leq \omega < \omega_M,
\label{doslo1}$$ where $\omega_M$ is the upper bound of LO phonon frequency \[expression (\[wloupp\])\].
Similarly, expanding $\omega_z(k)$ \[Eq. (\[wzo2a\])\] we find the ZO phonon frequency near $\Gamma$, $$\omega_z(k)=\omega_z-c_zk-\frac{1}{2}c_z\Big(\frac{c_z}{\omega_z}+ \frac{4\pi\varepsilon\chi_e^\prime}{\varepsilon+4\sqrt{\pi}\chi_e^\prime}-\frac{2\varepsilon}{\sqrt{\pi}}\Big)k^2,
\label{wzok0}$$ where $\omega_z$ is the ZO mode frequency given by Eq. (\[wzo2a2\]), and $$c_z=\frac{\pi e_B'^2}{\bar{m}s\omega_z}\Big(\frac{\varepsilon}{\varepsilon+4\sqrt{\pi}\chi_e^\prime}\Big)^2.
\label{cz}$$ Dropping the terms due to the EP of ions, expansions (\[wlok0\]) and (\[wzok0\]) then reduce to a form as given by the RIM of Ref.[@Michel:2009] \[Eqs. (57) and (59) therein\]. The nearly flat dispersion of the ZO modes allows their DOS to be approximated by [@Michel:2009] $$g_z(\omega)=\frac{\omega_z-\omega}{2\pi c_z^2}, \quad \omega < \omega_z.
\label{doszo1}$$ Evidently $c_z$ is close to zero as $\varepsilon \ll 4\sqrt{\pi}\chi_e^\prime$ for the 2D crystal, resulting in the ZO modes being nondispersive.
The group velocity is $\nabla_{\mathbf{k}}\omega_i(k)=\omega_i'(k)\mathbf{k}/k$, where $i$ indexes a phonon branch of LO, TO or ZO modes so the $c_l$ and $c_z$ above are simply the norms of the group velocities of the LO and ZO modes at $\Gamma$, respectively, corresponding to the slopes [@Sohier:2017] of the phonon dispersion curves. For the long lattice waves, as the wavevector $k$ increases, clearly $\omega_l$ increases and $\omega_z$ decreases, whereas $\omega_t$ stays flat near $\Gamma$ \[Eq. (\[wto2a\])\].
2D lattice dielectric susceptibility and dielectric function
------------------------------------------------------------
The lattice susceptibility of ML hBN for in-plane polarization can be deduced from Eqs. (\[eomw1\]) and (\[bigP1\]) by considering periodic solutions, $\mathbf{W}$, $\mathbf{E}$, $\boldsymbol{\mathcal{P}}$ $\propto e^{-i\omega t}$, due to an external charge or electromagnetic field with oscillation frequency $\omega$. Then Eq. (\[eomw1\]) reduces simply to Eq. (\[eow1b\]). Inserting Eq. (\[eow1b\]) into Eq. (\[bigP1\]) to eliminate $\mathbf{W}$, one finds $$\boldsymbol{\mathcal{P}}(\boldsymbol{\rho})=[a_{22}-\frac{a_{12}a_{21}}{\omega^2+a_{11}}]\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},0),
\label{bgPfroE}$$ and subsequently the 2D dielectric susceptibility, defined by $\chi=\boldsymbol{\mathcal{P}}(\boldsymbol{\rho})/\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},0)$, $$\chi(\omega)=a_{22}-\frac{a_{12}a_{21}}{\omega^2+a_{11}}=\chi_e+\frac{e_B^2}{\bar{m}s(\omega_0^2-\omega^2)},
\label{chiwp1}$$ where $\chi_e$ is the high-frequency (i.e., clamped-ion, or electronic) susceptibility. The static dielectric susceptibility $\chi_0$ follows from expression (\[chiwp1\]), $$\chi_0=\chi_e+\frac{e_B^2}{\bar{m}s\omega_0^2},
\label{chiwps0}$$ where the second term on the right-hand side is the vibrational (also called ionic) contribution. With Eq. (\[chiwps0\]), $a_{12}$ \[Eq. (\[a12\])\] can be expressed in terms of the 2D susceptibilities, $$a_{12}=a_{21}=\omega_0\sqrt{\chi_0-\chi_e}.
\label{a12chi09}$$
The 2D dynamical susceptibility $\chi(\omega)$ can be expressed in terms of three quantities that are usually used, namely, the intrinsic oscillator frequency $\omega_0$ and the high-frequency and static susceptibilities $\chi_e$, $\chi_0$, $$\chi(\omega)=\chi_e+\frac{\chi_0-\chi_e}{1-\omega^2/\omega_0^2}~,
\label{chiwp2}$$ which has a similar form to its counterpart of bulk polar crystals [@Born:1954].
Similarly, the 2D dielectric susceptibility for the vertical $z$-polarization $\chi'$ (the prime does not indicate a derivative), defined by $\chi'=\boldsymbol{\mathcal{P}}(\boldsymbol{\rho})/\mathbf{E}_z(\boldsymbol{\rho},0)$ for any particular $\omega$, is deduced from Eqs. (\[eomw1z\]) and (\[bigP1z\]), $$\chi'(\omega)=c_{22}-\frac{c_{12}c_{21}}{\omega^2+c_{11}}=\chi_e'+\frac{e_B'^2}{\bar{m}s(\omega_0'^2-\omega^2)},
\label{chiwz1}$$ with the static dielectric susceptibility $\chi_0'$, $$\chi_0'=\chi_e'+\frac{e_B'^2}{\bar{m}s\omega_0'^2}.
\label{chiwzs0}$$
Combining Eqs. (\[chiwzs0\]) and (\[c12z\]), one can express $c_{12}$ in terms of the 2D susceptibilities, $$c_{12}=c_{21}=\omega_0'\sqrt{\chi_0'-\chi_e'}.
\label{c12chi09}$$
The 2D dynamical susceptibility $\chi'(\omega)$ can also be expressed in terms of three quantities $\chi_e'$, $\chi_0'$, $\omega_0'$, $$\chi'(\omega)=\chi_e'+\frac{\chi_0'-\chi_e'}{1-\omega^2/\omega_0'^2}.
\label{chiwz2}$$
On eliminating $e_B'^2/(\bar{m}s)$ in Eq. (\[wzo2b3\]) with the help of Eq. (\[chiwzs0\]) we find a simple relation $$\frac{\omega_z^2}{\omega_0'^2}=\frac{\chi_0'}{\chi_e'}.
\label{wz02chz0}$$
The 2D high-frequency and static dielectric susceptibilities can be calculated from first principles, and knowing their values the susceptibility expressions (\[chiwps0\]) and (\[chiwzs0\]) obtained here will be used below in Sec. III to determine the model parameters such as the effective charges and spring force constants, and the $a$- or $c$-coefficients of the lattice equations \[Eqs. (\[eomw1co\]), (\[bigP1co\]), (\[eomw1zco\]) and (\[bigP1zco\])\].
The lattice dielectric function (DF) can be obtained by considering the response of the 2D lattice to a test charge in the electrostatic approximation. Now the equation of electrostatics is given by $\nabla\cdot(\mathbf{E}+4\pi\mathbf{P})=4\pi\sigma$, where $\sigma$ is the test charge density function dependent on $\mathbf{r}$, $t$, and $\mathbf{E}$ is the total field of the test charge plus the polarization charge. To find the dielectric response let $\sigma$ have the form $\sigma=\sigma_0e^{i\mathbf{k}\cdot\boldsymbol{\rho}}f(z)$, where $f(z)$ is an arbitrary function to describe the charge distribution in the $z$ direction, for the general case that the external charge is not necessarily confined in the ML, and time dependence $e^{-i\omega t}$ has been absorbed into $\sigma_0$ for clearness.
First, let us consider a charge distribution that is asymmetric with respect to the ML, i.e., $f(-z)\neq f(z)$, for instance, when the test charge is simply put above or below the ML. Then the field due to the charge is nonzero in the ML, $\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},0)\neq 0$, $\mathbf{E}_z(\boldsymbol{\rho},0)\neq 0$, and therefore the lattice responds creating both in-plane \[Eqs. (\[eomw1co\]) and (\[bigP1co\])\] and out-of-plane \[Eqs. (\[eomw1zco\]) and (\[bigP1zco\])\] vibrations, with all quantities such as $\mathbf{w}_{\boldsymbol{\rho}}$, $\mathbf{w}_z$, $\boldsymbol{\mathcal{P}}_{\boldsymbol{\rho}}$, $\boldsymbol{\mathcal{P}}_z$, $\mathbf{E}$ varying according to $e^{i(\mathbf{k}\cdot\boldsymbol{\rho}-\omega t)}$. Now the dielectric polarization is given by the sum of the in-plane and out-of-plane contributions, $\mathbf{P}=(\boldsymbol{\mathcal{P}}_{\boldsymbol{\rho}}+\boldsymbol{\mathcal{P}}_z)\delta(z)$, where $\boldsymbol{\mathcal{P}}_{\boldsymbol{\rho}}=\chi(\omega)\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},0)$ and $\boldsymbol{\mathcal{P}}_z=\chi'(\omega)\mathbf{E}_z(\boldsymbol{\rho},0)$, with $\chi$ and $\chi'$ being the 2D susceptibilities \[Eqs. (\[chiwp1\]) and (\[chiwz1\])\]. Writing $\mathbf{E}=-\nabla\phi$ with the total electrostatic potential $\phi(\boldsymbol{\rho},z)=\varphi(z)e^{i\mathbf{k}\cdot\boldsymbol{\rho}}$, one expresses the polarization charge density $-\nabla\cdot\mathbf{P}$ in terms of the derivatives of potential $\phi$, and then Poisson’s equation follows from the equation of electrostatics, $$\begin{aligned}
\nabla^2\phi(\boldsymbol{\rho},z)&=-4\pi\left[\sigma_0 e^{i\mathbf{k}\cdot\boldsymbol{\rho}}f(z)+\chi(\omega)\delta(z)
\nabla_{\boldsymbol{\rho}}^2\phi(\boldsymbol{\rho},0) \right.
\nonumber \\ &\qquad {}
\left. +\chi'(\omega)\varphi'(0)e^{i\mathbf{k}\cdot\boldsymbol{\rho}}\delta'(z)\right],
\label{poions1}\end{aligned}$$ where $\phi(\boldsymbol{\rho},0)$ is the total potential in the ML, $\phi(\boldsymbol{\rho},0)=\varphi(0)e^{i\mathbf{k}\cdot\boldsymbol{\rho}}$. Inserting the expansion of $f(z)$, $$f(z)=\int_{-\infty}^{\infty}f(q)e^{iqz}dq,
\label{fzex}$$ and expansions (\[vphx\]) and (\[delx\]) into Poisson’s equation (\[poions1\]) one finds $\hat{\varphi}(q)$, $$\hat{\varphi}(q)=\frac{2}{k^2+q^2}\left[ 2\pi\sigma_0f(q)-\chi(\omega)k^2\varphi(0)+i\chi'(\omega)\varphi'(0)q \right],
\label{vaphqdcp}$$ where the three terms on the right-hand side represent the respective contributions due to the external charge, in-plane and out-of-plane lattice polarization. Here $\hat{\varphi}(q)$ is expressed in terms of $\varphi(0)$ and $\varphi'(0)$ as they are proportional to the field components in the ML, i.e., $\varphi(0)\propto \mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},0)$ and $\varphi'(0)\propto \mathbf{E}_z(\boldsymbol{\rho},0)$. Integrating $\hat{\varphi}(q)$ over $q$ according to Eq. (\[vph0x\]), one then obtains $\varphi(0)$, $$\varphi(0)=\frac{\varphi_\sigma(0)}{1+2\pi k\chi(\omega)}~,
\label{vaphz0dc}$$ where $$\varphi_\sigma(0)=4\pi\sigma_0\int_{-\infty}^{\infty}\frac{f(q)}{k^2+q^2}dq~.
\label{vafexz0}$$
The out-of-plane (ZO) motion makes no contribution to $\varphi(0)$ as the integration value of the third term of $\hat{\varphi}(q)$ is zero. Clearly $\varphi_\sigma(0)e^{i\mathbf{k}\cdot\boldsymbol{\rho}}$ is the electrostatic potential in the ML due to the test charge alone, and therefore the dielectric function of the 2D lattice, which is defined as the ratio of this test charge potential to the total potential in the ML $\varphi(0)e^{i\mathbf{k}\cdot\boldsymbol{\rho}}$ (extension of the 2D [*wavevector*]{}-dependent dielectric function $\epsilon(k)$ of Ref.[@Cudazzo:2011]), is given by $$\epsilon(k,\omega)=1+2\pi k\chi(\omega)=1+2\pi k\left[\chi_e+\frac{e_B^2}{\bar{m}s(\omega_0^2-\omega^2)}\right],
\label{df2D}$$ showing a linear dependence on wavevector as the $\epsilon(k)$ of Ref.[@Cudazzo:2011]).
In expression (\[vaphqdcp\]) for $\hat{\varphi}(q)$, $\varphi(0)$ is known \[Eq. (\[vaphz0dc\])\], and $\varphi'(0)$ can be determined through expression (\[vph1x\]) for $\varphi'(z)$ as follows. Substituting expression (\[vaphqdcp\]) for $\hat{\varphi}(q)$ in Eq. (\[vph1x\]) one finds $$\begin{aligned}
\varphi'(z) &=4\pi\sigma_0\int_{-\infty}^{\infty}\frac{iqf(q)}{k^2+q^2}e^{iqz}dq \nonumber \\
&\qquad {} -2k^2\chi(\omega)\varphi(0)\int_{-\infty}^{\infty}\frac{iq}{k^2+q^2}e^{iqz}dq \nonumber \\
&\qquad {} -4\pi\chi'(\omega)\varphi'(0)\Big[\delta(z)-\frac{1}{2}ke^{-k\lvert z\rvert}\Big].
\label{vaph1zdfa}\end{aligned}$$ The square bracketed part is a familiar form that has appeared in Eq. (\[Ez2\]). To find $\varphi'(0)$ we introduce an effective thickness $\varepsilon$ to $\delta(z)$ to approach it with $\delta_{\varepsilon}(z)$ as was done above in Sec. III. Let $\Lambda=\delta_{\varepsilon}(0)$; $\Lambda=1/(\sqrt{\pi}\varepsilon)$, for instance, when $\delta(z)$ is approximated by a Gaussian [@Michel:2009], $\delta_{\varepsilon}(z)=e^{-z^2/\varepsilon^2}/(\sqrt{\pi}\varepsilon)$. Now taking $z=0$ in Eq. (\[vaph1zdfa\]), the term containing $\chi(\omega)\varphi(0)$ vanishes, i.e., the in-plane motion makes no contribution to $\varphi'(0)$, which is given by $$\varphi'(0)=\frac{4\pi\sigma_0}{1+4\pi\chi'(\omega)(\Lambda-k/2)}\int_{-\infty}^{\infty}\frac{iqf(q)}{k^2+q^2}dq~.
\label{vaph1z0dfb}$$
Having $\hat{\varphi}(q)$ \[expression (\[vaphqdcp\])\] then the total potential $\phi$ is known, and the total field $\mathbf{E}$ can be obtained straightforward,
$$\begin{aligned}
\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},z)&=-2\pi i\mathbf{k}\Big\{2\sigma_0\int_{-\infty}^{\infty}\frac{f(q)e^{iqz}}{k^2+q^2}dq -[ k\chi(\omega)\varphi(0) \Big.
\nonumber \\ &\qquad {} \Big.+\chi'(\omega)\varphi'(0)\operatorname{sgn}(z)]e^{-k\lvert z\rvert}\Big\}e^{i\mathbf{k}\cdot\boldsymbol{\rho}},
\label{fiedxydfc1}\end{aligned}$$
$$\begin{aligned}
\mathbf{E}_z(\boldsymbol{\rho},z) &=-2\pi\mathbf{e}_z\left\{2\sigma_0\int_{-\infty}^{\infty}\frac{iqf(q)}{k^2+q^2}e^{iqz}dq \right.
\nonumber \\ &\qquad {} +k^2\chi(\omega)\varphi(0)\operatorname{sgn}(z)e^{-k\lvert z\rvert}
\nonumber \\
&\qquad {} \left. -2\chi'(\omega)\varphi'(0)\Big[\delta(z)-\frac{1}{2}ke^{-k\lvert z\rvert}\Big]\right\}e^{i\mathbf{k}\cdot\boldsymbol{\rho}}~,
\label{fiedzdfc1}\end{aligned}$$
where $\varphi(0)$ and $\varphi'(0)$ are given by expressions (\[vaphz0dc\]) and (\[vaph1z0dfb\]), respectively.
Recalling $\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},z)=-i\mathbf{k}\phi(\mathbf{r})$ with Eq. (\[fiedxydfc1\]) for $\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},z)$, evidently the induced potential associated with the ZO polarization is proportional to $\chi'(\omega)\varphi'(0)\operatorname{sgn}(z)e^{-k\lvert z\rvert}$, which is zero in the ML plane and [*antisymmetric*]{} with respect to it, consistent with Eq. (\[Pophez\]) above. The in-plane component of the field associated with the ZO motion \[the term containing $\chi'(\omega)\varphi'(0)$ in Eq. (\[fiedxydfc1\])\] is zero in the ML and thus the ZO motion does not influence the LO motion; meanwhile the LO motion has zero $z$-component of its field \[the $\chi(\omega)\varphi(0)$ term in Eq. (\[fiedzdfc1\])\] in the ML and thus no influence on the ZO motion, making the LO and ZO vibrations driven by the exernal charge essentially independent of each other. For small thickness $\varepsilon$, $\varphi'(0)$ is small from Eq. (\[vaph1z0dfb\]) and there is only a small ZO component, which becomes negligible in the limit $\varepsilon\rightarrow 0$, in the driven lattice motion \[refer to Eq. (\[poions1\])\]. For a symmetric test charge distribution $f(-z)=f(z)$, $\mathbf{E}_z(\boldsymbol{\rho},0)=0$, and this causes only in-plane vibration, with displacement $\mathbf{w}$ and polarization $\mathbf{P}$ both parallel to the ML plane, and evidently yields the same DF $\epsilon(k,\omega)$ as given by expression (\[df2D\]) above.
As the polarization charge density associated with the in-plane motion is $-\delta(z)\nabla_{\boldsymbol{\rho}}\cdot\boldsymbol{\mathcal{P}}_{\boldsymbol{\rho}}(\boldsymbol{\rho})=-\delta(z)[a_{21}-a_{22}(\omega^2+a_{11})/a_{12}]i\mathbf{k}\cdot\mathbf{w}_{\boldsymbol{\rho}}\ne 0$, and $\mathbf{w}_{\boldsymbol{\rho}}\parallel\boldsymbol{\mathcal{P}}_{\boldsymbol{\rho}}\parallel\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},0)\parallel\mathbf{k}$, the DF $\epsilon(k,\omega)$ is due only to the LO vibrations and thus is a longitudinal DF. In the absence of a test charge ($\sigma=0$), there is still a finite electric field $\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},0)$ and potential $\phi(\boldsymbol{\rho},0)$ due to the LO vibrations, and therefore it follows from Eq. (\[vaphz0dc\]) that the LO modes are the solutions to $$\epsilon(k,\omega)=0.
\label{dfwl0}$$
From Eqs. (\[df2D\]) and (\[dfwl0\]) one indeed obtains the LO phonon dispersion $\omega_l(k)$ of expression (\[wlo2a2\]). In bulk crystals the lattice DF is zero at the bulk LO mode frequency [@Born:1954]. From expression (\[df2D\]) the static DF is $\epsilon_0(k)=1+2\pi\chi_0k$, while at high frequencies $\omega\gg\omega_0$ the vibrational contribution is negligible yielding the DF $\epsilon_{\infty}(k)=1+2\pi\chi_ek$, with $2\pi\chi_0$ and $2\pi\chi_e$ being the effective screening lengths [@Cudazzo:2011; @Berkelbach:2013]. Both static and high-frequency DFs have the same form as that deduced by Cudazzo [*et al*]{}. [@Cudazzo:2011].
In terms of the three key quantities $\omega_t$, $\omega_l(k)$, $\epsilon_{\infty}(k)$ the lattice DF of the 2D crystal can be expressed in the form $$\epsilon(k,\omega)=\epsilon_{\infty}(k)\frac{\omega^2-\omega_l^2(k)}{\omega^2-\omega_t^2},
\label{df2D2}$$ which is similar to the lattice DF of bulk polar crystals [@Born:1954], the difference being that here both the LO phonon frequency and DF are dependent on wavevector. In bulk crystals there is the LST relation [@Lyddane:1941], $\omega_l^2/\omega_t^2=\epsilon_0/\epsilon_{\infty}$; for the 2D crystal a similar relation can be deduced from expression (\[df2D2\]), $$\frac{\omega_l^2(k)}{\omega_t^2}=\frac{\epsilon_0(k)}{\epsilon_{\infty}(k)},
\label{LST2}$$ with the three quantities dependent on wavevector.
The extended LST relation (\[LST2\]) connects the phonon frequencies to the two DFs; given the former, then the ratio of the latter is known, and vice versa. Furthermore, since the difference between the LO and TO frequencies is due entirely to the macroscopic field, the relation can be used to measure the phonon frequency change caused by the field. Similarly, for out-of-plane motion the susceptibilities $\chi_0'$ and $\chi_e'$ are related to phonon frequencies $\omega_z$ and $\omega_0'$ via the frequency–susceptibility relation (\[wz02chz0\]), which conveniently quantifies the effect of the macroscopic field on the ZO phonon frequency.
Lattice dynamical properties: local field and polarizable ion effects
=====================================================================
In-plane optical modes
----------------------
The 2D clamped-ion susceptibility $\chi_e$, a key quantity entering the derived expressions for in-plane motion, can be obtained from first-principles calculation in four ways as follows. The 2D DF $\epsilon(q)$ was calculated within the random-phase approximation by a first-principles supercell approach for a number of 2D crystals such as h-BN [@Rasmussen:2016], phosphorene and TMDs [@Berkelbach:2013; @Huser:2013; @Rasmussen:2016]. In the first method, from the calculated DF the $\chi_e$ values of 2D TMDs were extracted in Ref.[@Berkelbach:2013] by employing the relation $\epsilon(L)=1+4\pi \chi_e/L$ [@Cudazzo:2011], where $L$ is the interlayer separation for a supercell containing two MLs of TMD, and $\epsilon(L)$ is the in-plane dielectric constant due to [*electronic*]{} polarization. For 2D MoS$_2$, the obtained $\chi_e$=6.6 $\AA$ [@Berkelbach:2013] is nearly equal to the susceptibility 6.5 $\AA$ extracted from [*bulk*]{} MoS$_2$ simply by taking $L=c/2$ ($c$ is the lattice constant of bulk MoS$_2$) whilst making $\epsilon(L)$ equal to the in-plane dielectric constant of bulk MoS$_2$. When the experimental lattice constant $c$=6.66 $\AA$ [@Solozhenko:1995] and in-plane high-frequency dielectric constant $\epsilon=4.95$ (Ref.[@Geick:1966]) of bulk h-BN are put into the above relation, the susceptibility of 2D BN is estimated to be $\chi_e$=1.0 $\AA$. The 2D susceptibility $\chi_e$ can also be obtained by employing $\epsilon(q)=1+2\pi\chi_eq$ [@Cudazzo:2011] as the slope of the $\epsilon(q)$ curve [@Huser:2013; @Rasmussen:2016] in the small-wavevector $q$ region equals the screening length $r_{eff}=2\pi\chi_e$. From Fig. 1 of Ref.[@Rasmussen:2016] one finds a susceptibility $\chi_e$=5.9 $\AA$ for 2D MoS$_2$ and $\chi_e$=0.8 $\AA$ for 2D hBN. The 2D susceptibility can also be calculated when knowing the screening length $r_{eff}$. For 2D hBN the calculated screening length is 7.64 $\AA$ [@Sohier:2017], thus corresponding to a susceptibility $\chi_e$ of 1.216 $\AA$. In the fourth method, the (clamped ion) electronic energy in the unit cell is calculated for crystals in an external electric field [@Ferrero:2008; @Ferrabone:2011], and then the first derivative of the electronic energy with respect to the field yields the induced dipole moment per cell, namely, the induced macroscopic polarization corresponding, for 2D hBN, to the last term in Eqs. (\[bigP1\]) and (\[bigP1z\]), and the second derivative of the energy with respect to the field yields the electronic polarizability per cell. For ML hBN an electronic polarizability 4.591 $\AA^3$ per cell (Table IV of Ref.[@Ferrabone:2011]) was obtained from a coupled perturbed Kohn-Sham (CPKS) calculation with the [*ab initio*]{} CRYSTAL09 code. Dividing this value by the area of the unit cell $s$ we find a 2D susceptibility $\chi_e$=0.848 $\AA$ which is very close to the value 0.8 $\AA$ of the second method. Similarly, using the static polarizability 7.111 $\AA^3$ per cell given in the same reference [@Ferrabone:2011] we find a static dielectric susceptibility (i.e., including the lattice contribution) $\chi_0$=1.314 $\AA$. Unless otherwise stated, these 2D susceptibilities $\chi_e$ and $\chi_0$ from Ref.[@Ferrabone:2011] are used in this study.
Through the three expressions (\[a22eB\]), (\[wto2a\]) and (\[chiwps0\]) we obtained, the three quantities of the 2D crystal, namely, the two [*macroscopic*]{} susceptibilities $\chi_e$, $\chi_0$ and the collective vibration frequency $\omega_0$ are related to the three [*microscopic*]{} quantities, $e_a$, $e_B$, $K$. Therefore, the values of the three microscopic quantities can be determined by using a set of three values $\chi_e$, $\chi_0$, $\omega_0$ calculated from first principles. Further, all the three mutually independent $a$-coefficients $a_{11}$, $a_{21}$ (or $a_{12}$), $a_{22}$ in the equation of motion (\[eomw1\]) and polarization equation (\[bigP1\]) can also be determined through Eqs. (\[a22alf\]), (\[a11\]) and (\[a12chi09\]). Of the four microscopic quantities $\alpha_1$, $\alpha_2$, $e_a$, $K$ on which the $a$-coefficients originally depend \[refer to Eqs. (\[a21\]), (\[a22\]), (\[a11\]), (\[Ke\]) and (\[eB\])\], the polarizabilities of the constituent atoms of ML hBN $\alpha_1$, $\alpha_2$ are unknown. Therefore, the adoption of the three known quantities $\chi_e$, $\chi_0$, $\omega_0$ facilitates the use of the deduced equations for in-plane motion by circumventing the two unknowns $\alpha_1$ and $\alpha_2$. We note that, although the values of $\alpha_1$, $\alpha_2$ are unknown, their sum $\alpha_1+\alpha_2$, the atomic poarizability of the unit cell, is found to fall in an interval, expressed in terms of $\chi_e$ in inequality (\[alfbu13\]), which is obtained with a generalized Clausius-Mossotti relation connecting the [*microscopic*]{} polarizabilities $\alpha_1$, $\alpha_2$ and the [*macroscopic*]{} dielectric susceptibility $\chi_e$ (refer to Appendix A). Taking $\chi_e=0.85 ~\AA$ [@Ferrabone:2011] yields $1.3084~\AA^3 \leq\alpha_1+\alpha_2\leq 1.7530~\AA^3$, which are smaller than the total free-atom polarizability 4.0 $\AA^3$ (Appendix A). This interval will be used below to evaluate the LFEs on the LO phonon frequency and 2D dielectric susceptibilities (Table \[table:2\] below).
Using the susceptibilities $\chi_e=0.848~\AA$ and $\chi_0=1.314~\AA$ from CPKS calculation (Ref.[@Ferrabone:2011]) and the phonon frequency $\omega_0$=1371 cm$^{-1}$ calculated from first principles by the same group (Ref.[@Erba:2013]), we calculated the three microscopic quantities $e_a$, $e_B$, $K$ from Eqs. (\[a22eB\]), (\[wto2a\]) and (\[chiwps0\]). We also calculated the force constant due to LFEs $K_e$ \[Eq. (\[Ke\])\] and the LO phonon group velocity $c_l$ \[Eq. (\[cl\])\] and present the result in Table \[table:1\] (upper row). In the theoretical study by Sohier [et al]{}. [@Sohier:2017] the $\mathcal{S}$ parameter, related to $e_B$ by $\mathcal{S}=2\pi e_B^2/(\bar{m}s)$, is calculated by density-functional perturbation theory (DFPT). For 2D hBN the value $\mathcal{S}=8.4\times 10^{-2}$ eV$^2\cdot$$\AA$ [@Sohier:2017] so we find $e_B=2.71e$. In that study, the calculated frequency $\omega_0$ is 1387.2 cm$^{-1}$ (Table 1 of Ref.[@Sohier:2017]), and recall that a susceptibility $\chi_e$=1.216 $\AA$ has been already calculated above through $r_{eff}$=7.64 $\AA$ [@Sohier:2017]. Now having these three values of $\chi_e$, $e_B$, $\omega_0$, again by using Eqs. (\[a22eB\]), (\[wto2a\]) and (\[chiwps0\]) we calculated the quantities $K_e$, $c_l$ as well as $\chi_0$, $e_a$, $K$, which are also listed in Table \[table:1\] (lower row) for comparison. First, we see two nearly equal values of Born charge, 2.70$e$ and 2.71$e$, the latter being also equal to another recent DFPT calculation [@Michel:2017] as well as the $e_B$ value of the bulk hBN [@Ohba:2001]. According to Eq. (\[a22eB\]), $e_B$ is greater than $e_a$ as $\chi_e >0$ due to EP of ions, and the numerical result shows that $e_B$ is four times larger than $e_a$. Also the two effective charge $e_a$ values are close to previous first-principles results 0.56$e$ [@Grad:2003] and 0.43$e$ [@Topsakal:2009] obtained respectively with Bader’s method and Löwdin’s analysis on charge transfer. Second, both susceptibilities $\chi_e$ and $\chi_0$ are larger in the lower row but the vibrational contributions to the static susceptibility namely the $\chi_0-\chi_e$ values are nearly equal in the two cases, which are 0.46 and 0.45 $\AA$ for the upper and lower rows, respectively. Third, the group velocity $c_l$ corresponds to the slope of the LO phonon dispersion curve at $\Gamma$; the two values of $c_l$ are very close with a difference of only 0.38% (Table \[table:1\]) but they are one order of magnitude larger than the value 1.94 km/s calculated without taking into account EP of ions [@Michel:2009]. Fourth, in Ref.[@Michel:2009] the force constant parameters are generated by making a percentage reduction of those of graphene, and from their table I we work out a spring force constant $K$=41.86 eV/$\AA^2$ ($K'$=14.48 eV/$\AA^2$ for out-of-plane motion), which is 26% and 30% smaller than the force constant $K$ values 56.5 and 59.8 eV/$\AA^2$ of the present Table \[table:1\]. Fifth, the intrinsic oscillator frequency $\omega_0$ is determined by $K-K_e$ \[Eq. (\[a11\])\] rather than by $K$ alone, i.e., $\omega_0^2=(K-K_e)/\bar{m}$; then the ratio $K_e/K$ expresses the percentage reduction due to the LFEs, which from Table \[table:1\] is $K_e/K$=24% and 29%. Sixth, the two quantities $c_l/\omega_0$ and $4\pi \chi_e$ form a factor of the $k^2$ term in the expansion of $\omega_l(k)$ \[Eq. (\[wlok0\])\], and from Table \[table:1\] the ration is $\frac{c_l}{\omega_0}/4\pi \chi_e$=13.5% and 9.3%, indicating the EP of ions makes the dominant contribution. Further, this result, together with the third point above, shows that neglecting ionic polarization such as RIM [@Michel:2009] cannot properly describe the LO phonon dispersion. Below we shall use the first row of parameters to evaluate the influence of EP and LFEs on the lattice vibrations (as is presented in Table \[table:2\]).
In Sec. II the electronic polarization of all the ions is included through $\alpha_{\kappa}$, and the model is referred to as PIM hereafter, in particular when comparing to the results calculated with the RIM ($\alpha_{\kappa}=0$). In the above calculations for Table \[table:1\], therefore both the EP and LFEs have been taken into account. We now look at what happens when EP or LFEs are not included. To do this, we compare various lattice-dynamical quantities such as $\chi_e$, $\chi_0$, $e_B$, $\omega_t$, $\omega_l(k)$, $c_l$ obtained with the RIM and PIM when the LFEs are neglected or taken into account, the expressions of which are listed in Table \[table:2\], for the four approaches in total. Given $\chi_e$, $\chi_0$, $\omega_0$, their expressions (\[a22eB\]), (\[wto2a\]) and (\[chiwps0\]), as noted in Table \[table:2\], are used to calculate the microscopic quantities such as $e_a$, $e_B$, $K$, $\alpha_1+\alpha_2$ (see Appendix A) needed in these approaches. Just below the expressions in the table the specific value-substituted LO phonon dispersion $\omega_l(k)$ and the values of the other five quantities are also listed for comparison: for the PIM including the LFEs (last row), the susceptibilities $\chi_e$, $\chi_0$ and frequency $\omega_t$ (equal to $\omega_0$) are input values taken from first-principles calculations [@Ferrabone:2011; @Erba:2013], while the others were calculated using this set of $\chi_e$, $\chi_0$, $\omega_0$ values, as we did in the above calculations for Table \[table:1\]; for the other three approaches, the quantities are transformed to depend on $e_a$ and/or $K$ as their expressions show, and are therefore calculated using the $e_a$ and $K$ parameters that are obtained with the same set of $\chi_e$, $\chi_0$, $\omega_0$ values (as was given in the first row of Table \[table:1\]). Several points can be made by comparing the results obtained with the four approaches. First, there is no electronic polarization in the RIM and therefore $\chi_e=0$. In the PIM with no LFEs the electronic susceptibility of ML hBN is simply $\chi_e=(\alpha_1+\alpha_2)/s$, and taking the interval we obtained above for $\alpha_1+\alpha_2$ gives $0.2417~\AA \leq\chi_e\leq 0.3239~\AA$, values that are even smaller than half the 0.85 $\AA$ obtained when the LFEs are taken into account. Second, in the RIM—for both cases of excluding and including the LFEs—and also in the PIM when neglecting LFEs, the Born charge reduces simply to the static charge $e_a=0.61e$, a value only 23% of the $e_B$ that we calculated with the PIM including the LFEs. Third, the ions in motion contribute $e_B^2/[s(K-e_ae_BQ_1)]$ to the static susceptibility $\chi_0$, and with the RIM a smaller Born charge of 0.61$e$ causes this contribution to be only $\sim$0.018 $\AA$, thus underestimating the vibrational contribution by 96% compared to the 0.46 $\AA$ which was calculated with the PIM including the LFEs. Fourth, the LO phonon group velocity $c_l$ evaluated with the three simpler approaches, 1.61-1.67 km/s, is similar to the 1.94 km/s of a RIM result [@Michel:2009] but is one order of magnitude smaller than the 37.24 km/s given by the PIM including the LFEs.
Furthermore, the LO phonon dispersion relations given by the four quantitative expressions of Table \[table:2\] are compared in Fig. \[fig2\], where wavevector $k$ is given in units of $\lvert \Gamma-K\rvert$ [@Sohier:2017], the distance between the $\Gamma$ and $K$ points in the Brillouin zone. For the PIM without LFEs, the parameter $\gamma$ is simply taken to be the lower bound 3.8172 as the numerical calculation indicates that the use of the upper bound 5.1144, for instance, reduces the LO phonon frequencies by less than 0.08% (not shown), a change indiscernible to the dispersion curve. When LFEs are neglected, the two models PIM and RIM yield very close phonon frequencies that fall in a very narrow range from 1631 to 1653 cm$^{-1}$ (upper two lines). The two curves touch each other at the $\Gamma$ point with the same tangent and slope $c_l$=1.61 km/s; the curve obtained by the RIM displays a linear dispersion relation through the long-wavelength region $k \leq 0.15\lvert \Gamma-K\rvert$, while the PIM curve becomes flatter at larger wavevectors in the same region of $k$. When the LFEs are taken into account, the phonon frequencies are reduced significantly. For the dispersion calculated with the RIM (middle line), the dependence of frequency on wavevector remains linear, with a small slope similar to the case with no LFEs. For the dispersion curve obtained with the PIM (solid line), in contrast, owing to the EP of ions, a steep slope appears on the small wavevector side, corresponding to an substantially increased phonon group velocity 37.24 km/s, and as the wavevector increases the increase of the phonon frequency becomes slower (i.e. with a smaller $c_l$), deviating significantly from its linear dependence near the $\Gamma$ point. The LO phonon dispersion of Ref.[@Sohier:2017], $\omega_l=[\omega_0^2+\mathcal{S}k/(1+r_{eff}k)]^{1/2}$, with first-principles calculated values $\omega_0$=1387.2 cm$^{-1}$, $\mathcal{S}=8.4\times 10^{-2}$ eV$^2\cdot$$\AA$, and $r_{eff}$=7.64 $\AA$, is also shown (dotted line) and is very close to the dispersion curve obtained with the our PIM. According to Eq. (\[wlo2a2\]), the upper limit of $\omega_l$ at a sufficiently large $k$ is $\omega_M$=1701.6 cm$^{-1}$; at wavevector $k=0.15\lvert \Gamma-K\rvert$, the LO phonon frequency $\omega_l$ is 1569 cm$^{-1}$, which is quite close to its limiting value (only 7.8% smaller). For wavevectors $k>0.02\lvert \Gamma-K\rvert$, as $\omega$ increases, the decrease of the group velocity with wavevector $k$ causes a rapid increase to the LO phonon DOS, as is clearly seen from Fig. \[fig3\] \[also refer to the DOS expression (\[doslo1\])\]. Overbending is a prominent feature of the numerically calculated LO phonon dispersion of ML hBN [@Miyamoto:1995; @Sanchez:2002; @Wirtz:2003; @Serrano:2007; @Michel:2009; @Topsakal:2009]. For instance, a maximum LO phonon frequency of 1533 cm$^{-1}$ (Ref.[@Wirtz:2003]) or 1570 cm$^{-1}$ (Ref.[@Topsakal:2009]) appears at a wavevector $\sim$0.30$\lvert \Gamma-K\rvert$. Evidently a positive slope $c_l$ is essential for the overbending of the dispersion curve $\omega_l(k)$. Therefore our result above indicates that due to the EP of ions, the increase of group velocity $c_l$ has enhanced the overbending substantially.
Out-of-plane optical modes
--------------------------
Values of phonon frequency $\omega_z$ [@Erba:2013; @Topsakal:2009; @Miyamoto:1995; @Wirtz:2003], 2D electronic susceptibility $\chi_e'$ and static susceptibility $\chi_0'$ [@Ferrabone:2011] associated with out-of-plane motion have been calculated from first principles. Having these then the intrinsic oscillator frequency $\omega_0'$ is known through the relation (\[wz02chz0\]). Given three quantities $\omega_0'$, $\chi_e'$, $\chi_0'$, the three microscopic quantities namely the static effective charge $e_a'$, the Born charge $e_B'$ and the effective force constant $K'$ follow from the three equations (\[c22z\]), (\[c11z\]) and (\[chiwzs0\]). Further, all the three mutually independent $c$-coefficients $c_{11}$, $c_{12}$ (or $c_{21}$), $c_{22}$ in the lattice equations (\[eomw1z\]) and (\[bigP1z\]) are also determined by Eqs. (\[c22z\]), (\[c11z\]) and (\[c12chi09\]).
From Ref.[@Ferrabone:2011] the 2D electronic susceptibility is given by $\chi_e'=0.815\AA^3/s=0.151 \AA$. Putting this into Eq. (\[c22z\]) gives a negative ratio $e_B^\prime/e_a^\prime=-0.21$; that is, the positive B ions carry a negative Born charge while the negative N ions move with a positive Born charge. As a result, the polarization induced by ionic vibrations $e_B^\prime(\mathbf{u}_1-\mathbf{u}_2)/s$ is antiparallel (parallel) to the displacements of the B (N) ions, and meanwhile the electric force exerted on the B (N) ions is antiparallel (parallel) to the macroscopic field. However the negative $e_B'$ value has no influence on the out-of-plane optical (ZO) phonon frequency as $e_B'$ enters the frequency expression via $e_B'^2$ \[Eq. (\[wzo2b3\])\].
Using the first-principles calculated values $\chi_e'=0.151~\AA$ and $\chi_0'=0.164~\AA$ (Ref.[@Ferrabone:2011]), we calculated $\omega_0'$, $e_a'$, $e_B'$, $K'$, $K_e'$ for two values of frequency $\omega_z$ taken from previous first-principles calculations, 836 cm$^{-1}$ (first-principles perturbation result of Ref.[@Erba:2013], by the same group of Ref.[@Ferrabone:2011], and also the direct method result of Ref.[@Miyamoto:1995]), 800 cm$^{-1}$ (DFPT value of Refs.[@Wirtz:2003; @Topsakal:2009]), and for one experimental $\omega_z$ of 734 cm$^{-1}$ (Ref.[@Rokuta:1997]) and also for a low frequency $\omega_z$=405 cm$^{-1}$ (see Table \[table:3\]). A larger static effective charge $e_a'$, greater than 1.0$e$, occurs for out-of-plane motion than in-plane motion (compare with in-plane effective charge $e_a$ values given in Table \[table:1\]). Charge transfer is quite complicated in semiconductors where covalent bonds and ionic bonds coexist [@Phillips:1973]. For 2D hBN there is a space allowing a more electron charge distribution outside the layer plane, probably causing a larger out-of-plane charge transfer from B to N. In fact, the calculation shows that a smaller $e_a'$ equal to the in-plane effective charge of $0.61e$ corresponds to a very low phonon frequency 405 cm$^{-1}$ (last row of Table \[table:3\]) that is unacceptably lower than the first-principles and experimental values. We see a negative Born charge $e_B'$ (on B ions) one order of magnitude smaller than the in-plane Born charge $e_B$, while the force constant due to LFEs $K_e'$ is one-third the effective spring force constant $K'$, similar to the in-plane motion case.
To examine the EP of ions and LFEs on the out-of-plane motion, we look at the lattice-dynamical properties $\chi_e'$, $\chi_0'$, $e_B'$, $\omega_0'$, $\omega_z$ obtained with the RIM and PIM each including or excluding the LFEs, whose expressions are listed in Table \[table:4\] and also whose values obtained with a set of three first-principles values, $\chi_e'=0.151$ $\AA$, $\chi_0'=0.164$ $\AA$, $\omega_z=836$ cm$^{-1}$ (Refs.[@Ferrabone:2011; @Erba:2013]), are given just below the corresponding expressions (the microscopic quantities $e_a'$, $e_B'$, $K'$ used in the models were calculated using the same method as above for Table \[table:3\]). For the adopted $\chi_e'$ value 0.151 $\AA$ the atomic polarizability of the unit cell is found to fall in the interval, $1.9329~\AA^3 \leq\alpha_1^\prime+\alpha_2^\prime\leq 2.5792~\AA^3$ (Appendix A). In the PIM without LFEs then the dielectric susceptibility of ML hBN is given by $\chi_e'=(\alpha_1'+\alpha_2')/s$, thus yielding $0.3571~\AA \leq\chi_e'\leq 0.4765~\AA$. When the LFEs are accounted for, the dielectric susceptibility $\chi_e'$ is reduced significantly (by over 55%)—different from the in-plane motion case where the dielectric susceptibility is increased—and the Born charge on the B ions becomes negative with a magnitude that is only 21% of their static effective charge. In fact, there is a very small vibrational contribution (only 0.013 $\AA$) to the static susceptibility $\chi_0'$ due to this small value of Born charge; in contrast, when LFEs are neglected, a large Born charge equal to the static effective charge leads to an overestimation of the vibrational contribution, which rises to a one order of magnitude higher value (0.196 $\AA$). The out-of-plane phonon frequency $\omega_z$ is 836 cm$^{-1}$ with the PIM including the LFEs, which is overestimated by over 39% when LFEs are not included. In the RIM there is no EP, i.e., $\alpha_1'=\alpha_2'=0$, which causes $\chi_e^\prime$ to vanish and consequently an extremely large out-of-plane phonon frequency, $\omega_z \rightarrow \infty$. In consequence the RIM fails to describe the out-of-plane vibrations.
Summary and Conclusions
=======================
We have studied long wavelength optical vibrations in ML hBN using two pairs of equations \[Eqs. (\[eomw1co\]), (\[bigP1co\]), (\[eomw1zco\]) and (\[bigP1zco\])\] to describe the in-plane and out-of-plane lattice vibrations, respectively. These lattice equations, which have similar forms to Huang’s equations for bulk crystals, are deduced from a microscopic dipole lattice model taking into account the LFEs and EP self-consistently. The 2D Lorentz relation connecting the macroscopic and local fields, and the use of the areal polarization, a macroscopic quantity to describe the 2D dielectric polarization, are fundamental to deducing the lattice equations from the atomic theory. From the lattice equations the expressions for the areal energy density are obtained for the in-plane and out-of-plane lattice vibrations, respectively. The three mutually independent $a$- or $c$-coefficients of the equations are expressed in terms of a set of three quantities such as the 2D electronic and static susceptibilities and the intrinsic oscillator frequency, calculated from first principles, thus making the lattice equations very useful for calculating the lattice dynamical properties.
The lattice equations are solved simultaneously with the equation of electrostatics to deduce the optical modes of the TO, LO and ZO vibrations. The explicit expressions have been obtained for phonon frequency, macroscopic field, and also phonon group velocity and density of states. The frequency expressions are found to describe the dispersion relations of previous first-principles calculations very well: while the TO and ZO modes are dispersionless, the LO modes have dispersion with the frequency increasing with the wavevector. In particular, our LO phonon dispersion relation is identical to the analytical expression of Sohier [*et al*]{}., and it evidently shows that the LO and TO modes are degenerate at $\Gamma$ and split up at a finite wavevector due to the long-range macroscopic field. It is also shown that the finite ZO phonon frequency exists owing to the electronic polarization of ions, without which the frequency becomes infinitely large in the rigid ion model.
From the lattice equations the frequency-dependent dielectric susceptibilities are deduced for in-plane and out-of-plane lattice motion. By considering the response of the lattice to a test charge with a charge distribution asymmetric or symmetric with respect to the ML, a 2D lattice dielectric function $\epsilon(k,\omega)$ is derived \[expression (\[df2D\])\], which is shown to be due solely to the LO vibrations, and the driven LO and ZO vibrations are further discussed. It is also shown that such a 2D longitudinal DF allows the LO phonon dispersion \[expression (\[wlo2a2\])\] to be rederived from $\epsilon(k,\omega)=0$, similar to the case of bulk crystals. The 2D LST relation (\[LST2\]) and frequency–susceptibility relation (\[wz02chz0\]) are obtained for in-plane and out-of-plane motion, respectively, connecting the phonon frequencies to the 2D dielectric functions or susceptibilities, which are very useful for quantifying the effects of the macroscopic field on the phonon frequencies.
The analytical expressions have been applied to study the lattice dynamical properties of ML hBN. For the in-plane motion, two sets of three quantities from two independent first-principles calculations, one set of $\chi_e$, $\chi_0$, $\omega_0$ from Ref.[@Ferrabone:2011] and the other set of $\chi_e$, $e_B$, $\omega_0$ from Ref.[@Sohier:2017], are used as parameters and very close results are obtained for the calculated properties such as the LO phonon dispersion relation, the effective spring force constant and the vibrational contribution to the static susceptibility. To evaluate the LFEs the unit-cell atomic polarizability is used, which, given a dielectric susceptibility, is found to be limited in an interval using the 2D Clausius-Mossotti relation. The LFEs and electronic polarization of ions should be included simultaneously, but otherwise neglecting either or both causes large discrepancies to the calculated properties: the phonon frequency at $\Gamma$ is overestimated by 15%-19%, whereas the Born charge, the LO phonon group velocity and the vibrational contribution to the static susceptibility are underestimated by 77%, 96%, 96%, respectively. With no ionic EP or LFEs, the LO modes display very small linear dispersion, almost flat in the long wavelength region, which is distinct from the LO phonon dispersion calculated after accounting for both EP and LFEs. For out-of-plane motion, using $\chi_e'$, $\chi_0'$, $\omega_z$ calculated mainly from Refs.[@Ferrabone:2011; @Erba:2013] as parameters, the static effective charge is found to be different from that for in-plane motion reflecting anisotropy of the 3D charge density distribution. Further, the positive B ions carry a negative Born charge while the negative N ions move with a positive Born charge. Similar to the in-plane motion case, the RIM can not properly describe the out-of-plane vibrations, which, for instance, yields a four times larger Born charge and infinitely large phonon frequency. When the EP of ions is included, the LFEs have significantly reduced the 2D dielectric susceptibility and the Born charge (by 60% and 80%, respectively), different from the in-plane motion case where the dielectric susceptibility and Born charge are both increased substantially (by two and three times, respectively).
We acknowledge support from the Natural Science Research Funds (Nos. 419080500175 & 419080500260) of Jilin University.
The generalized Clausius-Mossotti relations and intervals of the total atomic polarizability in ML hBN
======================================================================================================
In bulk diatomic crystals the electronic susceptibility $\chi$ relates to the atomic polarizabilities of constituent atoms $\alpha_1$ and $\alpha_2$ by the Clausius-Mossotti relation [@Clausius:1879; @Mossotti:1850], $$\chi=\frac{(\alpha_1+\alpha_2)/v_a}{1-4\pi(\alpha_1+\alpha_2)/(3v_a)}.
\label{clausius3d}$$
There is no such a simple relation for 2D hBN. Substituting Eqs. (\[bigD\]) and (\[a22alf\]) into Eq. (\[a22\]), $\chi_e$, the in-plane electronic susceptibility of ML hBN, is given by [@Mikhailov:2013; @Della:2016] $$\chi_e=\frac{(\alpha_1+\alpha_2)+2\alpha_1\alpha_2(Q_1-Q_0)}{s[1-(\alpha_1+\alpha_2)Q_0-\alpha_1\alpha_2(Q_1^2-Q_0^2)]}.
\label{clausius1}$$
If $Q_0=Q_1$, then the 2D susceptibility of the dielectric $\chi_e$ relates to the total atomic polarizability of the unit cell $\alpha_1+\alpha_2$ by a simple relation, $$\chi_e=\frac{(\alpha_1+\alpha_2)/s}{1-(\alpha_1+\alpha_2)Q_{0,1}},
\label{clausius2}$$ which is similar to the Clausius-Mossotti relation (\[clausius3d\]) for bulk materials. In fact in ML hBN $Q_0 \neq Q_1$ (Sec. II) so the sum $\alpha_1+\alpha_2$ can not be determined from the generalized Clausius-Mossotti relation (\[clausius1\]); adding the Born charge expression (\[eB\]) does not help because $e_B$ and $\chi_e$ are related through Eq. (\[a22eB\]) and not independent of each other.
Interestingly, when knowing $\chi_e$ the upper and lower bounds of the unit-cell atomic polarizability $\alpha_1+\alpha_2$ can be determined from expression (\[clausius1\]). Let $\alpha=\alpha_1+\alpha_2$. Then Eq. (\[clausius1\]) can be transformed into a quadratic equation in variable $\alpha_1$, $$(Q_1-Q_0)\big(Q_1+Q_0+\frac{2}{\chi_es}\big)(\alpha_1^2-\alpha\alpha_1)+1-\alpha\big(Q_0+\frac{1}{\chi_es}\big)=0,
\label{alfbo1}$$ with the discriminant $\Delta_{\alpha}$, $$\begin{aligned}
\Delta_{\alpha}&=(Q_1-Q_0)\big(Q_1+Q_0+\frac{2}{\chi_es}\big)\Big\{(Q_1-Q_0)\big(Q_1+Q_0 \big.\Big.
\nonumber \\ &\qquad {}
\Big.\big.+\frac{2}{\chi_es}\big)\alpha^2-4\big[1-\alpha(Q_0+\frac{1}{\chi_es})\big]\Big\} \nonumber \\
& =(Q_1-Q_0)^2\big(Q_1+Q_0+\frac{2}{\chi_es}\big)^2 \Big(\alpha+\frac{2}{Q_1-Q_0}\Big)
\nonumber \\ &\qquad {}
\Big(\alpha-\frac{2}{Q_1+Q_0+\frac{2}{\chi_es}}\Big).
\label{alfdel1}\end{aligned}$$
Eq. (\[alfbo1\]) has two real roots only when $\Delta_{\alpha}\geq 0$. Therefore it follows from the $\Delta_{\alpha}$ expression (\[alfdel1\]) that either $$\alpha\geq\frac{2}{Q_1+Q_0+\frac{2}{\chi_es}},
\label{alfbu1}$$ or $$\alpha\leq-\frac{2}{Q_1-Q_0}.
\label{alfbu2}$$ For ML hBN ($Q_1>Q_0$) the latter inequality leads to only negative $\alpha$ and is dropped. Now one finds that $\alpha_1$ and $\alpha_2$ are given by $$\alpha_{1,2}=\frac{1}{2}\Big[\alpha\pm\sqrt{\Big(\alpha+\frac{2}{Q_1-Q_0}\Big) \Big(\alpha-\frac{2}{Q_1+Q_0+\frac{2}{\chi_es}}\Big)}~\Big];
\label{alf1root}$$ if $\alpha_1$ takes the plus square root then $\alpha_2$ takes the minus square root, and vice versa. For $\alpha_1$ and $\alpha_2$ to be positive, one further requires that the above square root is not greater than $\alpha$ and obtains $$\alpha\leq\frac{1}{Q_0+\frac{1}{\chi_es}}.
\label{alfbu3}$$
Combining the inequalities (\[alfbu1\]) and (\[alfbu3\]) then restricts the atomic polarizability in the unit cell for in-plane polarization to the following interval, $$\frac{2}{Q_1+Q_0+\frac{2}{\chi_es}}\leq\alpha_1+\alpha_2\leq\frac{1}{Q_0+\frac{1}{\chi_es}}.
\label{alfbu13}$$ When the 2D susceptibility $\chi_e=0.85 ~\AA$ [@Ferrabone:2011] is used, for instance, one finds $1.3084~\AA^3 \leq\alpha_1+\alpha_2\leq 1.7530~\AA^3$, which is only a narrow range of 0.44 $\AA^3$.
For the vertical polarization (parallel to $\mathbf{e}_z$), the electronic susceptibility $\chi_e^\prime$ can be expressed in terms of the atomic polarizabilities $\alpha_1^\prime$, $\alpha_2^\prime$ as (refer to Sec. II above and also Ref.[@Mikhailov:2013]) $$\chi_e^\prime=\frac{(\alpha_1^\prime+\alpha_2^\prime)-4\alpha_1^\prime\alpha_2^\prime(Q_1-Q_0)}{s[1+2(\alpha_1^\prime+\alpha_2^\prime)Q_0-4\alpha_1^\prime\alpha_2^\prime(Q_1^2-Q_0^2)]}.
\label{clausiusz1}$$
Following the preceding derivation we find that the atomic polarizability of the unit cell falls in the interval, $$\frac{1}{\frac{1}{\chi_e^\prime s}-2Q_0}\leq\alpha_1^\prime+\alpha_2^\prime\leq\frac{1}{Q_1-Q_0}.
\label{alfbuz13}$$
Taking the 2D susceptibility $\chi_e^\prime=0.151 ~\AA$ [@Ferrabone:2011], one finds $1.9329~\AA^3 \leq\alpha_1^\prime+\alpha_2^\prime\leq 2.5792~\AA^3$. Mathematically, the solution $\alpha_1^\prime+\alpha_2^\prime \geq 1/(\frac{1}{\chi_e^\prime s}-Q_1-Q_0)$ is also permitted, but yielding $\alpha_1^\prime+\alpha_2^\prime \geq 7.3 ~\AA^3$ for ML hBN, which are much larger than the total polarizability 4.0 $\AA^3$ of the free atoms B and N (the polarizabilities of free atoms B and N are 3.038 and 1.097 $\AA^3$, respectively [@PScollect:2006]). Using the linear volume-polarizability relationship [@Brinck:1993], the polarizability of a constituent atom $\alpha_i$ in a crystal can be estimated by $\alpha_i=(v_i/v_i^0)\alpha_i^0$ [@Tkatchenko:2009; @Hod:2012], where $v_i$ is the effective volume of the atom in the crystal, and $v_i^0$ and $\alpha_i^0$ are the free-atom volume and polarizability, respectively. Numerical calculations have shown that the polarizabilities of the B and N atoms in bulk h-BN are reduced compared to their free-atom polarizabilities [@Tkatchenko:2009; @Hod:2012]. Anyhow the values $\alpha_1^\prime+\alpha_2^\prime \geq 7.3 ~\AA^3$ are unacceptably large to be considered.
Compared to the their free-atom polarizability 4.0 $\AA^3$, the atoms B and N have a significantly reduced total polarizability after forming the 2D crystal hBN. There is a slightly larger atomic polarizability in the vertical direction than in the layer plane, i.e., $\alpha_1^\prime+\alpha_2^\prime >\alpha_1+\alpha_2$ (with a 0.6 $\AA^3$ difference in upper bound), as the constituent B and N atoms are close packed on the plane causing the electron cloud to be more easily deformed and polarized along the vertical direction. If LFEs are neglected then the dielectric susceptibilities become $\chi_e=(\alpha_1+\alpha_2)/s$, and $\chi_e^\prime=(\alpha_1^\prime+\alpha_2^\prime)/s$, leading to $\chi_e^\prime > \chi_e$. Therefore the first-principles result $\chi_e > \chi_e^\prime$ as shown in Ref.[@Ferrabone:2011] is attributed to the LFEs.
We note that $\chi_e$ and $\chi_e^\prime$ are susceptibilities due to polarizable atoms or ions, and thus they also occur in non-ionic 2D crystals such as graphene. For graphene expressions (\[clausius1\]) and (\[clausiusz1\]) for $\chi_e$ and $\chi_e^\prime$ are still applicable but become simpler considering there are two C atoms in a unit cell, namely $\alpha_1=\alpha_2$ and $\alpha_1^\prime=\alpha_2^\prime$.
Proof of $a_{12}=a_{21}$ and $c_{12}=c_{21}$ from macroscopic theory and areal energy density in 2D hBN
=======================================================================================================
We first consider in-plane motion. Let the ions have a displacement in the layer plane, $\mathbf{w}(\boldsymbol{\rho})=\mathbf{w}_{\boldsymbol{\rho}}(\boldsymbol{\rho})$, in the presence of a point charge $e_{\nu}$ fixed at ($\boldsymbol{\rho}_0$,0), and let us find the electric field in the monolayer (in-plane component), $\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},0)$. The polarization $\mathbf{P}$ is $\mathbf{P(\mathbf{r})}=[a_{21}\mathbf{w}(\boldsymbol{\rho})+a_{22}\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},0)+c_{22}\mathbf{E}_z(\boldsymbol{\rho},0)]\delta(z)$ upon using Eqs. (\[bigP1co\]) and (\[bigP1zco\]), and the Poisson equation is given by $$\begin{aligned}
&\nabla^2\phi(\boldsymbol{\rho},z)=4\pi\{\nabla_{\boldsymbol{\rho}}\cdot[a_{21}\mathbf{w}(\boldsymbol{\rho})+a_{22}\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},0)]\delta(z)
\nonumber \\ &\qquad {}
+c_{22}E_z(\boldsymbol{\rho},0)\delta'(z)-e_{\nu}\delta(\boldsymbol{\rho}-\boldsymbol{\rho}_0)\delta(z)\}.
\label{poiapp1}\end{aligned}$$
Expanding $\phi(\boldsymbol{\rho},z)$, $\mathbf{w}(\boldsymbol{\rho})$ and $\delta(\boldsymbol{\rho}-\boldsymbol{\rho}_0)$, $$\phi(\boldsymbol{\rho},z)=\sum_{\mathbf{k}}\varphi_{\mathbf{k}}(z)e^{i\mathbf{k}\cdot\boldsymbol{\rho}},
\label{phixapp1}$$ $$\mathbf{w}(\boldsymbol{\rho})=\sum_{\mathbf{k}}\mathbf{w}_{\mathbf{k}}e^{i\mathbf{k}\cdot\boldsymbol{\rho}},
\label{wxapp1}$$ $$\delta(\boldsymbol{\rho}-\boldsymbol{\rho}_0)=\frac{1}{A}\sum_{\mathbf{k}}e^{i\mathbf{k}\cdot(\boldsymbol{\rho}-\boldsymbol{\rho}_0)},
\label{delroxapp1}$$ $A$ being the sample area, Eq. (\[poiapp1\]) is then transformed to $$\begin{aligned}
&(\nabla_{\boldsymbol{\rho}}^2+\frac{\partial^2}{\partial z^2})\sum_{\mathbf{k}}\varphi_{\mathbf{k}}(z)e^{i\mathbf{k}\cdot\boldsymbol{\rho}}=4\pi\sum_{\mathbf{k}}\left [\delta(z)(a_{21}i\mathbf{k}\cdot\mathbf{w}_{\mathbf{k}}
\right.
\nonumber \\ &\qquad {} \left. +a_{22}k^2\varphi_{\mathbf{k}}(0))
-\delta'(z)c_{22}\varphi'_{\mathbf{k}}(0)\right ] e^{i\mathbf{k}\cdot\boldsymbol{\rho}}
\nonumber \\ &\qquad {} -\frac{4\pi e_{\nu}}{A}\sum_{\mathbf{k}}e^{i\mathbf{k}\cdot(\boldsymbol{\rho}-\boldsymbol{\rho}_0)}\delta(z).
\label{poiapp2}\end{aligned}$$
Expanding $\varphi_{\mathbf{k}}(z)$ $$\varphi_{\mathbf{k}}(z)=\int_{-\infty}^{\infty}\hat{\varphi}_{\mathbf{k}}(q)e^{iqz}dq,
\label{vphxapp1}$$ and $\delta(z)$ as Eq. (\[delx\]), one finds $\hat{\varphi}_{\mathbf{k}}(q)$ from Eq. (\[poiapp2\]), $$\begin{aligned}
\hat{\varphi}_{\mathbf{k}}(q)&=\frac{2}{k^2+q^2}\left[-(a_{21}i\mathbf{k}\cdot\mathbf{w}_{\mathbf{k}}+a_{22}k^2\varphi_{\mathbf{k}}(0)) \right. \nonumber \\ &\qquad {}
\left.+\frac{e_{\nu}}{A}e^{-i\mathbf{k}\cdot\boldsymbol{\rho}_0}+c_{22}\varphi'_{\mathbf{k}}(0)iq\right].
\label{vphkqapp1}\end{aligned}$$
We note that from the above expansions one has reality conditions $\varphi_{-\mathbf{k}}(z)=\varphi^*_{\mathbf{k}}(z)$, $\mathbf{w}_{-\mathbf{k}}=\mathbf{w}^*_{\mathbf{k}}$ and $\hat{\varphi}_{-\mathbf{k}}(-q)=\hat{\varphi}^*_{\mathbf{k}}(q)$ considering that quantities such as $\phi(\boldsymbol{\rho},z)$, $\mathbf{w}(\boldsymbol{\rho})$ are real.
Taking $z=0$ in Eq. (\[vphxapp1\]) and integrating $\hat{\varphi}_{\mathbf{k}}(q)$ over $q$ one finds $$\varphi_{\mathbf{k}}(0)=\frac{2\pi}{k(1+2\pi a_{22}k)}\left(-a_{21}i\mathbf{k}\cdot\mathbf{w}_{\mathbf{k}}+\frac{e_{\nu}}{A}e^{-i\mathbf{k}\cdot\boldsymbol{\rho}_0}\right),
\label{vphz0app1}$$ and then the in-plane component of the electric field in the monolayer, $$\begin{aligned}
\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},0)&=-\sum_{\mathbf{k}}\frac{2\pi\mathbf{k}}{k(1+2\pi a_{22}k)}\left(a_{21}\mathbf{k}\cdot\mathbf{w}_{\mathbf{k}} \right. \nonumber \\ &\qquad {}
\left.+\frac{ie_{\nu}}{A}e^{-i\mathbf{k}\cdot\boldsymbol{\rho}_0}\right)e^{i\mathbf{k}\cdot\boldsymbol{\rho}},
\label{Eroz0app1}\end{aligned}$$ where the latter sum represents the field due to the point charge (including dielectric effects through $a_{22}$), which is zero at the charge site. Eq. (\[Eroz0app1\]) will be used below to calculate the work required to displace the ions or the charge.
We have obtained Eq. (\[a12a21\]), namely, $a_{12}=a_{21}$, from the microscopic dipole lattice model, and here we show that from the viewpoint of macroscopic theory it follows from the principle of energy conservation. Place the point charge $e_{\nu}$ at the origin, whilst setting the ions in the configuration $\mathbf{w}(\boldsymbol{\rho})=0$, and consider the following cycle [@Born:1954]: (a) keeping the charge at the origin, displace the ions horizontally into an irrotational configuration $\mathbf{w}(\boldsymbol{\rho})=\nabla_{\boldsymbol{\rho}}\psi(\boldsymbol{\rho})$, according to $$\mathbf{w}(\boldsymbol{\rho})=\xi\nabla_{\boldsymbol{\rho}}\psi(\boldsymbol{\rho}),
\label{wroxiap}$$ by increasing $\xi$ from 0 to 1, $\psi(\boldsymbol{\rho})$ being a nonzero arbitrary scalar field; (b) keeping the ions at $\mathbf{w}(\boldsymbol{\rho})=\nabla_{\boldsymbol{\rho}}\psi(\boldsymbol{\rho})$, displace the charge to $\Delta \boldsymbol{\rho}$; (c) fixing the charge at $\Delta \boldsymbol{\rho}$, reverse process (a), i.e., by decreasing $\xi$ from 1 to 0 according to Eq. (\[wroxiap\]); (d) move the charge back to the origin to complete the cycle.
$-[a_{11}\mathbf{w}(\boldsymbol{\rho})+a_{12}\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},0)]\cdot \Delta \mathbf{w}(\boldsymbol{\rho})$ is the work per unit area required to change $\mathbf{w}(\boldsymbol{\rho})$ to $\mathbf{w}(\boldsymbol{\rho})+\Delta \mathbf{w}(\boldsymbol{\rho})$, and total work expended on the ionic system for the configuration change is $$-\int [a_{11}\mathbf{w}(\boldsymbol{\rho})+a_{12}\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},0)]\cdot \Delta \mathbf{w}(\boldsymbol{\rho})d\boldsymbol{\rho}.
\label{dewpaap1}$$ From Eq. (\[wroxiap\]) $\Delta\mathbf{w}(\boldsymbol{\rho})=\Delta\xi\nabla_{\boldsymbol{\rho}}\psi(\boldsymbol{\rho})$ follows, and then insert expansion $\psi(\boldsymbol{\rho})=\sum_{\mathbf{k}}\psi_{\mathbf{k}}e^{i\mathbf{k}\cdot\boldsymbol{\rho}}$ into $\mathbf{w}(\boldsymbol{\rho})$ \[Eq. (\[wroxiap\])\] and $\Delta\mathbf{w}(\boldsymbol{\rho})$. Comparing the former expansion with Eq. (\[wxapp1\]) one finds $\mathbf{w}_{\mathbf{k}}$ for Eq. (\[Eroz0app1\]), $\mathbf{w}_{\mathbf{k}}=\xi i\mathbf{k}\psi_{\mathbf{k}}$. Inserting these expansions of $\mathbf{w}(\boldsymbol{\rho})$ and $\Delta\mathbf{w}(\boldsymbol{\rho})$ together with Eq. (\[Eroz0app1\]) into the above expression (\[dewpaap1\]), and integrating over $\xi$ from 0 to 1, one obtains the work done during process (a), $$\begin{aligned}
W_a&=\sum_{\mathbf{k}}\left[\frac{1}{2}A\left(\frac{2\pi a_{12}a_{21}k}{1+2\pi a_{22}k}-a_{11}\right)\psi_{\mathbf{k}} \right. \nonumber \\ &\qquad {}
\left. +\frac{2\pi a_{12}e_{\nu}}{k(1+2\pi a_{22}k)}\right]k^2\psi_{-\mathbf{k}}.
\label{workpaap1}\end{aligned}$$
The field acting on charge $e_{\nu}$ during process (b) is given by setting $\boldsymbol{\rho}=0$ (as $\Delta \boldsymbol{\rho}$ is small) and $\mathbf{w}_{\mathbf{k}}=i\mathbf{k}\psi_{\mathbf{k}}$ in the first term of $\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},0)$ \[Eq. (\[Eroz0app1\])\], and the work expended during the process is $$W_b=2\pi ie_{\nu}a_{21}\sum_{\mathbf{k}} \frac{k\psi_{\mathbf{k}}\mathbf{k}}{1+2\pi a_{22}k}\cdot \Delta \boldsymbol{\rho}.$$
Process (c) is the reverse of process (a) except for the altered position of charge $e_{\nu}$. Therefore after reversing the sign of expression (\[workpaap1\]) and multiplying its latter summands by a factor $e^{-i\mathbf{k}\cdot\Delta \boldsymbol{\rho}}$ we find the work expended during the process, $$\begin{aligned}
W_c&=-\sum_{\mathbf{k}}\left[\frac{1}{2}A\left(\frac{2\pi a_{12}a_{21}k}{1+2\pi a_{22}k}-a_{11}\right)\psi_{\mathbf{k}} \right. \nonumber \\ &\qquad {} \left.
+\frac{2\pi a_{12}e_{\nu}}{k(1+2\pi a_{22}k)}e^{-i\mathbf{k}\cdot\Delta \boldsymbol{\rho}}\right]k^2\psi_{-\mathbf{k}}.
\label{workpcap1}\end{aligned}$$
During process (d) $\mathbf{w}(\boldsymbol{\rho})=0$, and according to Eq. (\[Eroz0app1\]) no field acts on charge $e_{\nu}$, and therefore no work is needed to restore the charge to the origin, $W_d=0$.
As $W_a+W_b+W_c+W_d=0$, one finds $$\begin{aligned}
&2\pi a_{12}e_{\nu}\sum_{\mathbf{k}}\frac{k\psi_{-\mathbf{k}}}{1+2\pi a_{22}k}\left(1-e^{-i\mathbf{k}\cdot\Delta \boldsymbol{\rho}}\right)
\nonumber \\ &\qquad {}
+2\pi ia_{21}e_{\nu}\sum_{\mathbf{k}}\frac{k\psi_{\mathbf{k}}\mathbf{k}}{1+2\pi a_{22}k}\cdot \Delta \boldsymbol{\rho}=0.
\label{worksap1}\end{aligned}$$
Upon making $e^{-i\mathbf{k}\cdot\Delta \boldsymbol{\rho}}=1-i\mathbf{k}\cdot\Delta \boldsymbol{\rho}$, and changing the index of summation $\mathbf{k}$ to $-\mathbf{k}$ for the former summation, one immediately obtains $a_{12}=a_{21}$.
The relation $a_{12}=a_{21}$ allows us to define an energy density (energy per unit area) $u_p$ associated with the in-plane optical vibrations, as a function of $\mathbf{w}_{\boldsymbol{\rho}}(\boldsymbol{\rho})$ and $\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},0)$, $$u_h=-\frac{1}{2}\left[a_{11}\mathbf{w}^2_{\boldsymbol{\rho}}(\boldsymbol{\rho})+2a_{12}\mathbf{w}_{\boldsymbol{\rho}}(\boldsymbol{\rho})\cdot\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},0)+a_{22}\mathbf{E}^2_{\boldsymbol{\rho}}(\boldsymbol{\rho},0)\right].
\label{endenuh}$$ Inserting this $u_h$ expression into $\ddot{\mathbf{w}}_{\boldsymbol{\rho}}(\boldsymbol{\rho})=-\nabla _{\mathbf{w}}u_h$, and $\boldsymbol{\mathcal{P}}(\boldsymbol{\rho})=-\nabla _{\mathbf{E}}u_h$, where $\nabla _{\mathbf{w}}=(\frac{\partial}{\partial w_x},\frac{\partial}{\partial w_y})$, for instance, and the vector subscripts $\mathbf{w}$ and $\mathbf{E}$ of del $\nabla$ are shorthand notations for in-plane vectors $\mathbf{w}_{\boldsymbol{\rho}}(\boldsymbol{\rho})$ and $\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},0)$, one rederives the lattice equations (\[eomw1co\]) and (\[bigP1co\]). We note that, similar to the bulk polar crystal case [@Born:1954], it is not [*a priori*]{} obvious that an energy density of this simple form should exist for ML hBN, in particular when considering $\mathbf{E}$ is the macroscopic field – not simply an externally applied field.
Now we consider out-of-plane motion. Let the ions have a displacement perpendicular to the layer plane, $\mathbf{w}(\boldsymbol{\rho})=\mathbf{w}_z(\boldsymbol{\rho})$, in the presence of a point charge $e_{\nu}$ placed at (0,$z_0$), and let us find the $z$-component of the field, $\mathbf{E}_z(\boldsymbol{\rho},z)$. Now the polarization $\mathbf{P}$ is $\mathbf{P(\mathbf{r})}=[c_{21}\mathbf{w}(\boldsymbol{\rho})+c_{22}\mathbf{E}_z(\boldsymbol{\rho},0)+a_{22}\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},0)]\delta(z)$ upon using Eqs. (\[bigP1co\]) and (\[bigP1zco\])\], and the Poisson equation is given by $$\begin{aligned}
&\nabla^2\phi(\boldsymbol{\rho},z)=4\pi\{[c_{21}\mathbf{w}(\boldsymbol{\rho})+c_{22}\mathbf{E}_z(\boldsymbol{\rho},0)]\delta'(z) \nonumber \\ &\qquad {}
+ a_{22}\nabla_{\boldsymbol{\rho}}\cdot\mathbf{E}_{\boldsymbol{\rho}}(\boldsymbol{\rho},0)\delta(z)-e_{\nu}\delta(\boldsymbol{\rho})\delta(z-z_0)\}.
\label{poiapz1}\end{aligned}$$
Expanding $\delta(z)$ \[Eq. (\[delx\])\] and the other spatially varying quantities exactly as in the in-plane motion case \[Eqs. (\[phixapp1\]), (\[wxapp1\]), (\[delroxapp1\]) and (\[vphxapp1\])\], one finds from the Poisson equation $\hat{\varphi}_{\mathbf{k}}(q)$, $$\begin{aligned}
\hat{\varphi}_{\mathbf{k}}(q)&=\frac{2}{k^2+q^2}\left[-(c_{21}\mathbf{w}_{\mathbf{k}}\cdot\mathbf{e}_z-c_{22}\varphi'_{\mathbf{k}}(0))iq \right. \nonumber \\ &\qquad {}
\left. +\frac{e_{\nu}}{A}e^{-iqz_0}-a_{22}k^2\varphi_{\mathbf{k}}(0)\right].
\label{vphkqapz1}\end{aligned}$$
Taking $z=0$ in Eq. (\[vphxapp1\]) and integrating $\hat{\varphi}_{\mathbf{k}}(q)$ over $q$ one finds $$\varphi_{\mathbf{k}}(0)=\frac{2\pi e_{\nu}}{Ak(1+2\pi a_{22}k)}e^{-k\lvert z_0\rvert}.
\label{vphz0apz1}$$
Differentiating Eq. (\[vphxapp1\]) with respect to $z$ and substituting Eq. (\[vphkqapz1\]) for $\hat{\varphi}_{\mathbf{k}}(q)$ and then integrating over $q$, one obtains $$\begin{aligned}
\varphi'_{\mathbf{k}}(z)&=2\pi \Big\{2(c_{21}\mathbf{w}_{\mathbf{k}}\cdot\mathbf{e}_z-c_{22}\varphi'_{\mathbf{k}}(0))\Big[\delta(z)-\frac{1}{2}ke^{-k\lvert z\rvert}\Big] \Big .
\nonumber \\ &\qquad {} -\frac{e_{\nu}}{A}\operatorname{sgn}(z-z_0)e^{-k\lvert z-z_0\rvert}
\nonumber \\ &\qquad {} \Big. +a_{22}k^2\varphi_{\mathbf{k}}(0)\operatorname{sgn}(z)e^{-k\lvert z\rvert}\Big\}.
\label{vph1kzapz1}\end{aligned}$$ To find $\varphi'_{\mathbf{k}}(0)$ we approach $\delta(z)$ with $\delta_{\varepsilon}(z)$ where $\varepsilon$ is a small thickness [@Michel:2009; @Sohier:2017] and let $\Lambda=\delta_{\varepsilon}(0)$. Thus from Eq. (\[vph1kzapz1\]) one finds $$\begin{aligned}
\varphi'_{\mathbf{k}}(0)&=\frac{2\pi}{1+4\pi c_{22}(\Lambda-k/2)}\left[2c_{21}(\Lambda-\frac{k}{2})\mathbf{w}_{\mathbf{k}}\cdot\mathbf{e}_z \right.
\nonumber \\ &\qquad {} \left. +\frac{e_{\nu}}{A}\operatorname{sgn}(z_0)e^{-k\lvert z_0\rvert}\right].
\label{vph1kz0apz1}\end{aligned}$$
Having $\varphi_{\mathbf{k}}(0)$ and $\varphi'_{\mathbf{k}}(0)$ \[Eqs. (\[vphz0apz1\]) and (\[vph1kz0apz1\])\], now insert them into Eq. (\[vph1kzapz1\]) for $\varphi'_{\mathbf{k}}(z)$, from which one obtains $\mathbf{E}_z(\boldsymbol{\rho},z)$ via $\mathbf{E}_z(\boldsymbol{\rho},z)=-\mathbf{e}_z\sum_{\mathbf{k}}\varphi'_{\mathbf{k}}(z)e^{i\mathbf{k}\cdot\boldsymbol{\rho}}$, $$\begin{aligned}
\mathbf{E}_z(\boldsymbol{\rho},z)&= \sum_{\mathbf{k}}\Big\{ \frac{2\pi}{1+4\pi c_{22}(\Lambda-k/2)}\Big(-2c_{21}\mathbf{w}_{\mathbf{k}} \Big. \Big. \nonumber \\ &\qquad {}
\Big.+4\pi c_{22}\frac{e_{\nu}}{A}\operatorname{sgn}(z_0)e^{-k\lvert z_0\rvert}\mathbf{e}_z\Big)\big[\delta(z)-\frac{1}{2}ke^{-k\lvert z\rvert}\big]
\nonumber \\
&\qquad {} \Big. +\frac{2\pi e_{\nu}}{A}\big[-\frac{2\pi a_{22}k}{1+2\pi a_{22}k}\operatorname{sgn}(z)e^{-k(\lvert z\rvert +\lvert z_0\rvert)} \big.
\nonumber \\ &\qquad {}
\Big.\big.+\operatorname{sgn}(z-z_0)e^{-k\lvert z-z_0\rvert}\big]\mathbf{e}_z \Big\}e^{i\mathbf{k}\cdot\boldsymbol{\rho}}.
\label{vph1kzapz2}\end{aligned}$$
Let $z_0>0$ in what follows. Taking $z=0$ yields the $z$-component of the field in the ML, $$\begin{aligned}
\mathbf{E}_z(\boldsymbol{\rho},0)&=-2\pi\sum_{\mathbf{k}}\frac{1}{1+4\pi c_{22}(\Lambda-k/2)}\left[2c_{21}(\Lambda-\frac{k}{2})\mathbf{w}_{\mathbf{k}} \right. \nonumber \\ &\qquad {}
\left. +\frac{e_{\nu}}{A}e^{-k z_0}\mathbf{e}_z\right]e^{i\mathbf{k}\cdot\boldsymbol{\rho}},
\label{Ez0apz2}\end{aligned}$$ while taking $z=z_0$ and $\boldsymbol{\rho}=0$ one finds the $z$-component of the field acting on the point charge $e_{\nu}$, $$\begin{aligned}
&\mathbf{E}_z(0,z_0)=-2\pi \sum_{\mathbf{k}}\Big\{ \frac{k}{1+4\pi c_{22}(\Lambda-k/2)}\Big(-c_{21}\mathbf{w}_{\mathbf{k}} \Big. \Big .
\nonumber \\ &\qquad {} \Big. \Big. +2\pi c_{22}\frac{e_{\nu}}{A}e^{-k z_0}\mathbf{e}_z\Big)
+\frac{e_{\nu}}{A}\frac{2\pi a_{22}k}{1+2\pi a_{22}k}e^{-k z_0}\mathbf{e}_z \Big\}e^{-k z_0},
\label{Ezz0apz3}\end{aligned}$$ where the sum of the terms containing $e_{\nu}$ ($\mathbf{w}_{\mathbf{k}}$) represents the field due to the point charge (ionic displacements), denoted by $\mathbf{E}_{z,e_{\nu}}(0,z_0)$ ($\mathbf{E}_{z,w}(0,z_0)$) for simplicity.
Now we show that $c_{12}=c_{21}$ \[Eq. (\[c12z\])\] follows from the principle of energy conservation. Place the point charge $e_{\nu}$ above the ML at a point $P$ ($\boldsymbol{\rho}_0=0$,$z_0$), while keeping the ions in the configuration $\mathbf{w}(\boldsymbol{\rho})=0$, and consider the following cycle: (a) keeping the charge at the point $P$, displace the ions vertically into the configuration $\mathbf{w}(\boldsymbol{\rho})=\boldsymbol{\psi}(\boldsymbol{\rho})=\psi(\boldsymbol{\rho})\mathbf{e}_z$, according to $$\mathbf{w}(\boldsymbol{\rho})=\xi\psi(\boldsymbol{\rho})\mathbf{e}_z,
\label{wroxiapz2}$$ by increasing $\xi$ from 0 to 1; (b) keeping the ions at $\mathbf{w}(\boldsymbol{\rho})=\boldsymbol{\psi}(\boldsymbol{\rho})$, displace the charge vertically by a small $\Delta z$ to point $P'$ ($\boldsymbol{\rho}_0=0$,$z_0+\Delta z$); (c) fixing the charge at the point $P'$, reverse process (a), i.e., by reducing $\xi$ from 1 to 0 according to Eq. (\[wroxiapz2\]); (d) move the charge back to point $P$ to complete the cycle.
The work per unit area required to change $\mathbf{w}(\boldsymbol{\rho})$ to $\mathbf{w}(\boldsymbol{\rho})+\Delta \mathbf{w}(\boldsymbol{\rho})$ is $-[c_{11}\mathbf{w}(\boldsymbol{\rho})+c_{12}\mathbf{E}_z(\boldsymbol{\rho},0)]\cdot \Delta \mathbf{w}(\boldsymbol{\rho})$, and thus total work expended on the ionic system for the configuration change is $$-\int [c_{11}\mathbf{w}(\boldsymbol{\rho})+c_{12}\mathbf{E}_z(\boldsymbol{\rho},0)]\cdot \Delta \mathbf{w}(\boldsymbol{\rho})d\boldsymbol{\rho}.
\label{dewpaap1z}$$
From Eq. (\[wroxiapz2\]) one has $\Delta\mathbf{w}(\boldsymbol{\rho})=\Delta\xi\psi(\boldsymbol{\rho})\mathbf{e}_z$. Expanding $\mathbf{w}(\boldsymbol{\rho})$ and $\psi(\boldsymbol{\rho})$ as in-plane motion above, one finds $\mathbf{w}_{\mathbf{k}}$ in Eq. (\[Ez0apz2\]) is given by $\mathbf{w}_{\mathbf{k}}=\xi \boldsymbol{\psi}_{\mathbf{k}}$. Substituting these expansions in terms of $\psi_{\mathbf{k}}$ together with Eq. (\[Ez0apz2\]) into the above expression (\[dewpaap1z\]), and integrating over $\xi$ from 0 to 1, one obtains for the work expended during process (a), $$\begin{aligned}
W_a&=\sum_{\mathbf{k}}\left[\frac{1}{2}A\left(\frac{4\pi c_{12}c_{21}(\Lambda-k/2)}{1+4\pi c_{22}(\Lambda-k/2)}-c_{11}\right)\psi_{\mathbf{k}} \right. \nonumber \\ &\qquad {}
\left. +\frac{2\pi c_{12}e_{\nu}}{1+4\pi c_{22}(\Lambda-k/2)}e^{-k z_0}\right]\psi_{-\mathbf{k}}.
\label{workpaap1z}\end{aligned}$$
During process (b) the field acting on charge $e_{\nu}$ due to the ionic displacements is $\mathbf{E}_{z,w}(0,z_0)$, namely, the sum of the terms containing $\mathbf{w}_{\mathbf{k}}$ of Eq. (\[Ezz0apz3\]), with $\mathbf{w}_{\mathbf{k}}= \boldsymbol{\psi}_{\mathbf{k}}$, and the work expended for this field of the ionic displacements is $$W_{b,w}=-2\pi e_{\nu}c_{21}\sum_{\mathbf{k}} \frac{k\psi_{\mathbf{k}}}{1+4\pi c_{22}(\Lambda-k/2)}e^{-k z_0}\Delta z.
\label{workpbwap1z}$$ The field acting on charge $e_{\nu}$ due to the charge itself is $\mathbf{E}_{z,e_{\nu}}(0,z_0)$, i.e., the sum of the terms containing $e_{\nu}$ of Eq. (\[Ezz0apz3\]), and the work done for this field can be simply written as $W_{b,e_{\nu}}=-e_{\nu}E_{z,e_{\nu}}(0,z_0)\Delta z$.
Process (c) is the reverse of process (a) except for the displacement of charge $e_{\nu}$. Therefore upon reversing the sign of expression (\[workpaap1z\]) and multiplying its latter summands by a factor $e^{-k \Delta z}$, we readily find the work expended during the process, $$\begin{aligned}
W_c&=-\sum_{\mathbf{k}}\left[\frac{1}{2}A\left(\frac{4\pi c_{12}c_{21}(\Lambda-k/2)}{1+4\pi c_{22}(\Lambda-k/2)}-c_{11}\right)\psi_{\mathbf{k}} \right. \nonumber \\ &\qquad {}
\left. +\frac{2\pi c_{12}e_{\nu}}{1+4\pi c_{22}(\Lambda-k/2)}e^{-k(z_0+\Delta z)}\right]\psi_{-\mathbf{k}}.
\label{workpcap1z}\end{aligned}$$
During process (d) $\mathbf{w}(\boldsymbol{\rho})=0$, and according to Eq. (\[Ezz0apz3\]) no field is associated with displacement $\mathbf{w}(\boldsymbol{\rho})$, and accordingly no work is needed for the displacement part, $W_{d,w}=0$. For the field due to the point charge itself, the work done is given by reversing the sign of $W_{b,e_{\nu}}$, i.e., $W_{d,e_{\nu}}=e_{\nu}E_{z,e_{\nu}}(0,z_0)\Delta z$.
As $W_a+W_{b,w}+W_{b,e_{\nu}}+W_c+W_{d,w}+W_{d,e_{\nu}}=0$, one finds $$\begin{aligned}
&2\pi e_{\nu}\left[c_{12}\sum_{\mathbf{k}}\frac{\psi_{-\mathbf{k}}e^{-k z_0}}{1+4\pi c_{22}(\Lambda-k/2)}(1-e^{-k \Delta z}) \right. \nonumber \\ &\qquad {}
\left. - c_{21}\sum_{\mathbf{k}}\frac{k\psi_{\mathbf{k}}e^{-k z_0}}{1+4\pi c_{22}(\Lambda-k/2)}\Delta z\right]=0.
\label{worksap2z}\end{aligned}$$ Using $e^{-k \Delta z}=1-k \Delta z$, and changing the index of summation $\mathbf{k}$ to $-\mathbf{k}$ for the former summation, one readily finds $c_{12}=c_{21}$.
Having $c_{12}=c_{21}$, now an areal energy density associated with the out-of-plane optical vibrations can be introduced, $$u_v=-\frac{1}{2}\left[c_{11}\mathbf{w}^2_z(\boldsymbol{\rho})+2c_{12}\mathbf{w}_z(\boldsymbol{\rho})\cdot\mathbf{E}_z(\boldsymbol{\rho},0)+c_{22}\mathbf{E}^2_z(\boldsymbol{\rho},0)\right],
\label{endenuv}$$ from which the lattice equations (\[eomw1zco\]) and (\[bigP1zco\]) can be rederived through $\ddot{w}_z=-\partial u_v/\partial w_z$, $\mathcal{P}_z=-\partial u_v/\partial E_z$ \[$w_z=w_z(\boldsymbol{\rho})$, and $E_z=E_z(\boldsymbol{\rho},0)$\].
[53]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\
12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [**]{} (, , ) @noop [**]{} (, , ) @noop [**]{} (, , ) @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [**]{} (, , ) @noop [**]{} (, , ) @noop [****, ()]{} in @noop [**]{}, (, ) Chap. , pp. , @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{}
$\omega_0$ (cm$^{-1}$) $\chi_e$ ($\AA$) $\chi_0$ ($\AA$) $e_B$ ($e$) $e_a$ ($e$) $K$ (eV/$\AA^2$) $K_e$ (eV/$\AA^2$) $c_l$ (km/s)
------------------------ ------------------ ------------------ ------------- ------------- ------------------ -------------------- --------------
1371$^a$ 0.85 1.31 2.70 0.61 59.796 17.613 37.24
1387$^b$ 1.22 1.67 2.71 0.46 56.495 13.317 37.10
$^a$ First-principles perturbation result of Ref.[@Erba:2013], equal to the experimental value [@Rokuta:1997] of ML hBN on substrate Ni and very close to DFPT value 1378 cm$^{-1}$ of Ref.[@Wirtz:2003].\
$^b$ DFPT value from Ref.[@Sohier:2017].
[p[0.6cm]{}< p[1.3cm]{}< p[1.9cm]{}< p[2.2cm]{}< p[1.4cm]{}< p[1.5cm]{}< p[3.3cm]{}< p[2.2cm]{}<]{} & $\chi_e$ & $\chi_0$ & $e_B$ & $\omega_t$ & $\omega_l(k)$ & $c_l$\
& & 0 & $\frac{e_a^2}{sK}$ & $e_a$ & $\sqrt{\frac{K}{\bar{m}}}$ & $\sqrt{\frac{K}{\bar{m}}+\frac{2\pi e_a^2k}{\bar{m}s}}$ & $\frac{\pi e_a^2}{s\sqrt{\bar{m}K}}$\
& & 0 & 0.017 & 0.61 & 1632 & $\sqrt{1632.4^2+837.4^2\tilde{k}}$ & 1.61\
& & 0 & $\frac{e_a^2}{s(K-e_a^2Q_1)}$ & $e_a$ & $\sqrt{\frac{K-e_a^2Q_1}{\bar{m}}}$ & $\sqrt{\frac{K-e_a^2Q_1}{\bar{m}}+\frac{2\pi e_a^2k}{\bar{m}s}}$ & $\frac{\pi e_a^2}{s\sqrt{\bar{m}(K-e_a^2Q_1)}}$\
& & 0 & 0.018 & 0.61 & 1577 & $\sqrt{1577^2+837.4^2\tilde{k}}$ & 1.67\
& & $\frac{\alpha_1+\alpha_2}{s}$ & $\frac{\alpha_1+\alpha_2}{s}+\frac{e_a^2}{sK}$ & $e_a$ & $\sqrt{\frac{K}{\bar{m}}}$ & $\sqrt{\frac{K}{\bar{m}}+\frac{2\pi e_a^2k}{\bar{m}[s+2\pi(\alpha_1+\alpha_2)k]}}$ & $\frac{\pi e_a^2}{s\sqrt{\bar{m}K}}$\
& & \[0.24,0.32\]$^\dagger$ & \[0.26,0.34\]$^\dagger$ & 0.61 & 1632 & $\sqrt{1632.4^2+\frac{837.4^2\tilde{k}}{1+\gamma\tilde{k}}}^\oplus$ & 1.61\
& & Eq. (\[a22eB\]) & Eq. (\[chiwps0\]) & & Eq. (\[wto2a\]) & Eq. (\[wlo2a2\]) & Eq. (\[cl\])\
& & 0.85 & 1.31 & 2.70 & 1371 & $\sqrt{1371^2+\frac{3691.5^2\tilde{k}}{1+13.42\tilde{k}}}$ & 37.24\
$\tilde{k}$ is the normalized wavevector (dimensionless), $\tilde{k}=k/\frac{2\pi}{a}$.\
$^\dagger$ An interval (see text). $^\oplus$ $\gamma$ is dimensionless, $3.8172 \leq \gamma \leq 5.1144$.\
Dispersion relations given by the specific value-substituted expressions $\omega_l(k)$ are also shown in Fig. \[fig2\].
$\omega_z$ (cm$^{-1}$) $\omega_0'$ (cm$^{-1}$) $e_a'$ ($e$) $e_B'$ ($e$) $K_e'$ (eV/$\AA^2$) $K'$ (eV/$\AA^2$)
------------------------ ------------------------- -------------- -------------- --------------------- -------------------
836$^a$ 802 1.26 -0.27 7.136 21.575
800$^b$ 768 1.20 -0.25 6.535 19.758
734$^c$ 704 1.11 -0.23 5.501 16.630
405$^d$ 389 0.61 -0.13 1.675 5.063
$^a$ First-principles perturbation result of Ref.[@Erba:2013] and direct method result of Ref.[@Miyamoto:1995]. $^b$ DFPT value of Refs.[@Wirtz:2003; @Topsakal:2009]. $^c$ Experimental value of ML hBN on substrate Ni of Ref.[@Rokuta:1997].\
$^d$ This frequency yields a static effective charge 0.61$e$ equal to the $e_a$ value for in-plane motion.
[p[0.6cm]{}< p[1.3cm]{}< p[1.8cm]{}< p[2.2cm]{}< p[1.3cm]{}< p[1.9cm]{}< p[3.3cm]{}<]{} & $\chi_e'$ & $\chi_0'$ & $e_B'$ & $\omega_0'$ & $\omega_z$\
& & 0 & $\frac{e_a'^2}{sK'}$ & $e_a'$ & $\sqrt{\frac{K'}{\bar{m}}}$ & $\infty$\
& & 0 & 0.196 & 1.26 & 981 & $\infty$\
& & 0 & $\frac{e_a'^2}{s(K'+2e_a'^2Q_1)}$ & $e_a'$ & $\sqrt{\frac{K'+2e_a'^2Q_1}{\bar{m}}}$ & $\infty$\
& & 0 & 0.076 & 1.26 & 1571 & $\infty$\
& & $\frac{\alpha_1'+\alpha_2'}{s}$ & $\frac{\alpha_1'+\alpha_2'}{s}+\frac{e_a'^2}{sK'}$ & $e_a'$ & $\sqrt{\frac{K'}{\bar{m}}}$ & $\sqrt{\frac{K'}{\bar{m}}+\frac{e_a'^2}{\bar{m}(\alpha_1'+\alpha_2')}}$\
& & \[0.36,0.48\]$^\dagger$ & \[0.55,0.67\]$^\dagger$ & 1.26 & 981 & \[1165,1220\]$^\dagger$\
& & Eq. (\[c22z\]) & Eq. (\[chiwzs0\]) & & Eq. (\[c11z\]) & $\sqrt{\frac{K'+2e_a'e_B'Q_1}{\bar{m}}+\frac{e_B'^2}{\bar{m}s\chi_e'}}$\
& & 0.151 & 0.164 & -0.27 & 802 & 836\
$^\dagger$ An interval.
{width="12cm"}
{width="15cm"}
{width="15cm"}
[^1]: There is a misprint in Eq. (2) of Ref.[@Sohier:2017]: the square should be for each summand rather than for the total sum.
|
---
abstract: |
Understanding the physics of neutrinos is of paramount relevance for the development of high energy physics, cosmology and astrophysics, thanks to their characteristics and phenomenology. In particular, the property of changing flavor while neutrinos travel, the so-called neutrino oscillation phenomenon, provides us with valuable information about their behavior and their impact on the standard model of particles and the evolution of the universe, for instance.\
Here I present an overview of the most recent results as reported by relevant experiments studying neutrinos produced by accelerator facilities and detected after traveling long distances: the so-called Long-Baseline neutrino experiments.
author:
- |
Mario A. Acero$\thanks{e-mail: marioacero@mail.uniatlantico.edu.co}$, (for the NOvA Collaboration)\
Programa de Física, Universidad del Atlántico,\
Carrera 30 8-49 Puerto Colombia, Colombia
title: 'Recent results from Long-Baseline Neutrino experiments'
---
Introduction
============
It is a very well stablished fact that neutrinos are massive particles (contrary to what the Standard Model –SM– suggests) and that mix: neutrino-flavor eigenstates are related to neutrino-mass eigenstates though the mixing (also known as the PMNS) matrix, as $$\label{eq_nuMix}
\left(
\begin{array}{c}
\nu_e \\
\nu_{\mu} \\
\nu_{\tau} \\
\end{array}
\right)
= U_{\rm{PMNS}}
\left(
\begin{array}{c}
\nu_1 \\
\nu_2 \\
\nu_3 \\
\end{array}
\right).$$ Here, the vector on the left represents the three flavor neutrinos ($\nu_{\alpha}, \alpha=e,\mu,\tau$) which are actually created and detected (through weak interaction processes), while the vector on the right includes the definite-mass neutrinos ($\nu_i, i=1,2,3$), which propagate through vacuum or matter.
The mixing matrix in (\[eq\_nuMix\]) is usually parametrized in terms of three orthogonal rotation matrices, each one depending upon the three so-called mixing angles ($\theta_{ij}, i\neq j=1,2,3$), and a phase[^1] which parametrizes the CP violation in the lepton sector, ($\delta_{CP}$): $$\left(
\begin{array}{c}
\nu_e \\
\nu_{\mu} \\
\nu_{\tau} \\
\end{array}
\right)
= R(\theta_{23}) \cdot R(\theta_{13},\delta_{CP}) \cdot R(\theta_{12})
\left(
\begin{array}{c}
\nu_1 \\
\nu_2 \\
\nu_3 \\
\end{array}
\right).$$ The mixing angles and the CP-violating phase, together with two mass-squared difference ($\Delta m^2_{21}$, $\Delta m^2_{31}$; $\Delta m^2_{jk} \equiv m_j^2 - m_k^2$), define the change of flavor that neutrinos can undergo while traveling an specific distance (from the source to the detector, for instance), a phenomenon known as neutrino oscillations.
A number experiments have studied neutrino oscillations using different neutrino sources (the sun [@Ahmad:2002jz], the atmosphere [@Fukuda:1998mi], nuclear reactors [@Eguchi:2002dm; @An:2012eh; @Ahn:2012nd], accelerators [@Abe:2011sj; @Adamson:2017gxd]) and detection techniques, measuring five out of six of the parameters with great precision. However, there are still important open questions regarding this phenomenon, having implications in other areas of particle physics, cosmology and astrophysics: it is not yet clear wether there is maximal mixing in the $\mu-\tau$ sector or not (i.e., is $\theta_{23} = \pi/4$?); we do not know the value of $\delta_{CP}$ (i.e. is there a violation of CP symmetry in the lepton sector?); the neutrino mass pattern (order or hierarchy) is unknown (i.e. is $\Delta m^2_{32}>0$ –normal order– or $\Delta m^2_{32}>0$ –inverted order–?).
There are other important and interesting questions[^2], but answer to those just exposed are currently under investigation by some of the long-baseline (LBL) neutrino experiments. In the following sections, a short review of the recent results from accelerator-based oscillation neutrino experiments is presented.
LBL neutrino experiments
========================
Results from four LBL accelerator-based neutrino experiments are shown here: NOvA, T2K, MINOS and OPERA. All of them use a neutrino beam created by a similar mechanism: high energy protons are fired against a fixed target (made of graphite or beryllium, for instance), producing $\pi^+$ ($\pi^{-}$) which, after decaying, generate a beam mainly composed by muon (anti)neutrinos. So produced muon-(anti)neutrinos, traveling across the earth, may change flavor with a probability which depends on the mixing angles and the mass-squared differences. In this way LBL experiments are able to study neutrino physics from the observation of $\nu_{\mu}$ disappearance, $\nu_{e}$ appearance, and $\nu_{\tau}$ appearance, allowing them to measure the oscillation parameters (mainly $\theta_{13}, \theta_{23}, \Delta m^2_{32}, \delta_{CP}$, and and to study possible differences between neutrinos and antineutrinos.
NOvA
----
The NuMI[^3] Off-axis $\nu_e$ Appearance (NOvA) experiment [@Ayres:2007TDR; @Adamson:2017gxd; @NOvA:2018gge] is a two-detector accelerator-based neutrino experiment designed to study the appearance of electron-(anti)neutrinos from a beam of muon-(anti)neutrinos. The $\nu_{\mu}$ beam travels through the earth from the Near Detector (ND) (100 m underground) at Fermilab, to the 14 kton Far Detector (FD) in Ash River, Minnesota, around 810 km apart. The FD is located 14 mrad off the centerline of the neutrino beam coming from Fermilab, so that the flux of neutrinos has a narrow peak at an energy of 2 GeV, the energy at which oscillation from muon neutrinos to electron neutrinos is expected to be at a maximum. After collecting data from neutrinos and antineutrinos beams, NOvA has observed 58 neutrino (15 background) and 18 (5 background) antineutrino events, while studying the neutrino $\nu_e$ appearance, thus providing a larger than $4\sigma$ evidence of electron anti-neutrino appearance. From the $\nu_{\mu}$ disappearance analysis, NOvA observed 113 neutrino and 65 antineutrino events, when 730 and 266 events were expected in absence of oscillations, respectively [@Sanchez:2018Neutrino; @Vahle:2018Nufact].
![\[fig\_NOvA\_contours\] Regions of $\Delta m^2_{32}$ vs. $\sin^2\theta_{23}$ (*left*) and $\sin^2\theta_{23}$ vs. $\delta_{CP}$ parameter spaces obtained from the $\nu_e$-appearance and $\nu_{\mu}$-disappearance data analysis at various levels of significance. The top panels correspond to normal mass ordering while the bottom panels to inverted ordering [@Sanchez:2018Neutrino; @Vahle:2018Nufact].](NOvA_contour_joint_realData_onlyNH_deltaM32_th23.pdf "fig:") ![\[fig\_NOvA\_contours\] Regions of $\Delta m^2_{32}$ vs. $\sin^2\theta_{23}$ (*left*) and $\sin^2\theta_{23}$ vs. $\delta_{CP}$ parameter spaces obtained from the $\nu_e$-appearance and $\nu_{\mu}$-disappearance data analysis at various levels of significance. The top panels correspond to normal mass ordering while the bottom panels to inverted ordering [@Sanchez:2018Neutrino; @Vahle:2018Nufact].](NOvA_contour_joint_realData_onlyNH_deltaCP_th23.pdf "fig:") ![\[fig\_NOvA\_contours\] Regions of $\Delta m^2_{32}$ vs. $\sin^2\theta_{23}$ (*left*) and $\sin^2\theta_{23}$ vs. $\delta_{CP}$ parameter spaces obtained from the $\nu_e$-appearance and $\nu_{\mu}$-disappearance data analysis at various levels of significance. The top panels correspond to normal mass ordering while the bottom panels to inverted ordering [@Sanchez:2018Neutrino; @Vahle:2018Nufact].](NOvA_contour_joint_realData_onlyIH_deltaM32_th23.pdf "fig:") ![\[fig\_NOvA\_contours\] Regions of $\Delta m^2_{32}$ vs. $\sin^2\theta_{23}$ (*left*) and $\sin^2\theta_{23}$ vs. $\delta_{CP}$ parameter spaces obtained from the $\nu_e$-appearance and $\nu_{\mu}$-disappearance data analysis at various levels of significance. The top panels correspond to normal mass ordering while the bottom panels to inverted ordering [@Sanchez:2018Neutrino; @Vahle:2018Nufact].](NOvA_contour_joint_realData_onlyIH_deltaCP_th23.pdf "fig:")
A joint $\nu_{\mu}$-disappearance–$\nu_e$-appearance data analysis allows NOvA to constraint the oscillation parameters as depicted in Fig. \[fig\_NOvA\_contours\], where 1,2 and 3$\sigma$ C.L. allowed regions for both, normal (top panels) and inverted (bottom panels) mass orderings are shown for comparison. The best fit values of the oscillation parameters are $$\Delta m^2_{32} = 2.51^{+0.13}_{-0.08} \times 10^{-3} \rm{ eV}^2,
\qquad
\sin^2\theta_{23} = 0.58 \pm 0.03,
\qquad
\delta_{PC} = 0.17\pi.$$ In addition, NOvA data favor the normal mass ordering, non-maximal mixing with $\theta_{23} > 45^{\circ}$, and excludes $\delta_{CP} = \pi/2$ at more than $3\sigma$ C.L. for the inverted mass ordering.
T2K
---
T2K[^4] is a LBL neutrino experiment [@Abe:2011ks] studying neutrino oscillations using muon (anti)neutrinos produced at the Japan Proton Accelerator Research Center (JPARC). The neutrino beam is directed towards a two detector system: one located 280 m from the production point, and the other, far detector, located 295 km away, at the Kamioka Observatory, 2.5$^{\circ}$ off-axis with respect to the neutrino beam [@Khabibullin:2018bmg] (neutrino energy spectra peaked at 800 MeV).
The T2K analysis was performed using data from neutrinos and antineutrinos and studying $\nu_{\mu}$-disappearance as well as $\nu_{e}$-appearance channels. After comparing the observed rates at the FD to predictions under oscillation hypothesis, they found that data is consistent with that model, for any value of $\delta_{CP}$. However, regarding the $\bar{\nu}_e$-appearance, T2K observed fewer events than expected for any value of $\delta_{CP}$ (9 events observed while 11.8 events were expected with oscillations and 6.5 without oscillations), preventing them to arrive to a robust statistical conclusion.
![\[fig\_T2K\_contours\] *Left*. Regions of $\sin^2\theta_{23}$ vs. $\delta_{CP}$ parameter space obtained by T2K data only (top panel) and combined with reactors (bottom panel). *Right*. Regions of $\Delta m^2_{32}$ vs. $\sin^2\theta_{23}$ parameter space (top) and confidence intervals for $\sin^2\theta_{23}$ (bottom) obtained by T2K combined with reactor constraint [@Wascko:2018Neutrino].](T2K_contour_joint_deltaCP_th23.pdf "fig:") ![\[fig\_T2K\_contours\] *Left*. Regions of $\sin^2\theta_{23}$ vs. $\delta_{CP}$ parameter space obtained by T2K data only (top panel) and combined with reactors (bottom panel). *Right*. Regions of $\Delta m^2_{32}$ vs. $\sin^2\theta_{23}$ parameter space (top) and confidence intervals for $\sin^2\theta_{23}$ (bottom) obtained by T2K combined with reactor constraint [@Wascko:2018Neutrino].](T2K_1D_joint_th23.pdf "fig:") ![\[fig\_T2K\_contours\] *Left*. Regions of $\sin^2\theta_{23}$ vs. $\delta_{CP}$ parameter space obtained by T2K data only (top panel) and combined with reactors (bottom panel). *Right*. Regions of $\Delta m^2_{32}$ vs. $\sin^2\theta_{23}$ parameter space (top) and confidence intervals for $\sin^2\theta_{23}$ (bottom) obtained by T2K combined with reactor constraint [@Wascko:2018Neutrino].](T2K_contour_joint_deltaM32_th23.pdf "fig:")
Their data fit analysis was done considering both channels to find constraints on $\Delta m_{23}^2$, $\sin^2\theta_{23}$ and $\delta_{CP}$, but T2K also considered the constraints coming from reactor experiments, and their results are shown in Fig. \[fig\_T2K\_contours\]. The best fit values for the oscillation parameters are $$\begin{aligned}
\Delta |m^2_{32}| &= 2.434\pm0.064 \times 10^{-3} \, \rm{ eV}^2,
\qquad
& \sin^2\theta_{23} &= 0.536^{+0.031}_{-0.046},
\qquad
& \rm{Normal \, Ordering}; \\
\Delta |m^2_{32}| &= 2.410^{+0.062}_{-0.063} \times 10^{-3} \, \rm{ eV}^2,
\qquad
&\sin^2\theta_{23} &= 0.536^{+0.031}_{-0.041},
\qquad
& \rm{Inverted \, Ordering}.\end{aligned}$$ Notice that CP conserving values are outside of 2$\sigma$ region for both mass orderings and that normal ordering is favored by data.
MINOs and MINOS+
----------------
With its 735 km baseline, MINOS (originally planned to perform research on atmospheric neutrinos in the FD) and MINOS+ were designed to observe muon (anti)neutrino flavor changing from a beam produced at Fermilab and directed towards a two-detector system (the Far detector located at the Soudan Underground Laboratory in Minnesota) [@Evans:2017brt]. Thanks to improvements implemented on the NuMI beam, the neutrinos energy peak increased from 3 GeV for MINOS to 7 GeV for MINOS+ [@Aurisano:2018Neutrino].
Using neutrinos from the NuMI beam, MINOS+ found that FD data are consistent with three flavor prediction, imposing tightly constrains on alternate oscillations hypotheses [@DeRijck:2017ynh; @Aurisano:2018Neutrino]. The combination of atmospheric and beam neutrinos (in the appearance and disappearance channels), using data from neutrinos and antineutrinos, allows the collaboration to constraint the oscillation parameters as shown in Fig. \[fig\_MINOS\_contours\], finding the best fit at $$\Delta m^2_{32} = 2.42\times 10^{-3}\,\,\rm{eV}^2,\quad\quad \sin^2\theta_{23} = 0.42.$$ Their results present a 1.1$\sigma$ exclusion of maximal mixing and a 0.8$\sigma$ preference for the lower octant. They also point to a normal mass order preference with a significance of 0.2$\sigma$ [@Aurisano:2018Neutrino].
![\[fig\_MINOS\_contours\] Regions of $\sin^2\theta_{23}$ vs. $\delta_{CP}$ parameter space (left) and one-dimensional significance plots for each oscillation parameter (right), obtained by MINOS and MINOS+ data for the Normal and inverted mass orderings [@Aurisano:2018Neutrino].](MINOS_contour_joint_deltaM32_th23.pdf)
OPERA
-----
![\[fig\_OPERA\_data\] Energy distribution of OPERA data compared with the expectation [@Agafonova:2018auq].](OPERA_events_P_sum.pdf)
Using a muon-neutrino beam generated at CERN and directed towards a detector located 730 km away, at the LNGS[^5], OPERA [@Guler:2000bd] was designed to detect the appearance of tau-neutrinos through the $\nu_{\mu} \to \nu_{\tau}$ oscillation. With a total of 10 $\nu_{\tau}$ candidate events, their final analysis of the full data sample confirms the appearance on tau-neutrinos with a significance of 6.1$\sigma$ (Fig \[fig\_OPERA\_data\]), and the statistical analysis allowed them to report the first measurement of $\Delta m^2_{32}$ from the $\nu_{\tau}$ appearance mode [@Agafonova:2018auq]: $$\Delta m^2_{32} = 2.7^{+0.7}_{-0.6} \times 10^{-3} \,\, {\rm eV}^2,$$ which is consistent with the results by other experiments in the disappearance mode.
The Future
==========
In addition to the expected results coming from the currently active experiments, neutrino oscillations will be extensively end deeply studied by two impressive LBL experiments: Hyper-Kamiokande and DUNE.
Hyper-Kamiokande
----------------
![\[fig\_HK\_exp\] The Hyper-Kamiokande experimental layout [@Shiozawa:2018Neutrino].](hk_graphic.pdf)
Hyper-Kamiokande (HK), located in Japan, is the successor of and take advantage of all the technological success from the very well known Super-Kamiokande (SK) experiment. This water Cherenkov detector will be placed in the Tochibora mine, about 295 km away from the J-PARC proton accelerator research complex in Tokai [@Abe:2018uyc] (Figure \[fig\_HK\_exp\]).
![\[fig\_HK\_cpv\] Expected significance to exclude $\delta_{CP} = 0$ for the normal mass order (*Left panel*) and neutrino Mass Order sensitivity as a function of the true value of $\sin^2\theta_{23}$ (*Right panel*) in the HK experiment [@Abe:2018uyc; @Shiozawa:2018Neutrino].](HK_cpv.pdf "fig:") ![\[fig\_HK\_cpv\] Expected significance to exclude $\delta_{CP} = 0$ for the normal mass order (*Left panel*) and neutrino Mass Order sensitivity as a function of the true value of $\sin^2\theta_{23}$ (*Right panel*) in the HK experiment [@Abe:2018uyc; @Shiozawa:2018Neutrino].](HK_mh.pdf "fig:")
As can be seen in Fig. \[fig\_HK\_cpv\] (left panel), HK has the potential to exclude CP conservation ($\delta_{CP} = 0$) for the normal mass ordering, with a significance larger than $5\sigma$ after 10 years of data-taking. With enough time, HK will also reach a large sensitivity for the determination of the correct neutrino mass ordering (depending on the value of $\delta_{CP}$ (right panel of Fig. \[fig\_HK\_cpv\]).
However, HK has a rich scientific program which goes beyond the study od neutrino oscillations, including the search of nucleon decays, neutrinos emitted by supernova and from other astrophysical sources (dark matter annihilation, gamma ray burst jets, and pulsar winds) [@Abe:2018uyc].
DUNE
----
![\[fig\_DUNE\_exp\] The DUNE experimental layout.](lbnf_graphic_metric.pdf)
The Deep Underground Neutrino Experiment (Figure \[fig\_DUNE\_exp\]) [@Acciarri:2016crz] will be based at Fermilab, from where an intense neutrino beam is going to be fired towards a system of two Liquid Argon detectors, 1300 km apart: the Near Detector (ND), placed at Fermilab and the Far Detector (FD) at the Sanford Underground Research Facility.
![\[fig\_DUNE\_cpv\] Significance with which CP violation (*Left panel*) and the neutrino Mass Order (*Right panel*) can be determined as a function of the true value of $\delta_{CP}$, for exposures of seven (green) or 10 (orange) years, and for the Normal Mass Ordering case in the DUNE experiment. The width of the band represents the range of sensitivities for the 90% C.L. range of $\theta_{23}$ values [@Brailsford:2018dzn].](DUNE_cpv_two_exps_th23band_no_2017.pdf "fig:") ![\[fig\_DUNE\_cpv\] Significance with which CP violation (*Left panel*) and the neutrino Mass Order (*Right panel*) can be determined as a function of the true value of $\delta_{CP}$, for exposures of seven (green) or 10 (orange) years, and for the Normal Mass Ordering case in the DUNE experiment. The width of the band represents the range of sensitivities for the 90% C.L. range of $\theta_{23}$ values [@Brailsford:2018dzn].](DUNE_mh_two_exps_th23band_no_2017.pdf "fig:")
DUNE is expected to explore the neutrino oscillation phenomenon with an unprecedented precision, aiming to determine the mass ordering and the value of the CP-violating phase, $\delta_{CP}$. As depicted in Fig. \[fig\_DUNE\_cpv\], DUNE should be capable of determining the CP violation (i.e. measuring $\delta_{CP} \neq 0,\pi$)with a significance around $5\sigma$ after 7 years of data-taking, and even larger significance after 10 years. For the neutrino Mass Ordering (right panel of Fig. \[fig\_DUNE\_cpv\]), after 7 years, DUNE could reach a $5\sigma$ sensitivity for all possibe values of $\delta_{CP}$ [@Brailsford:2018dzn; @Worcester:2018Neutrino].
In addition, DUNE will be able to search for signals of proton decay, neutrinos coming form supernovae and some exotic physics related to sterile neutrinos, non-standard neutrino interactions and Dark Matter, for instance [@Acciarri:2015uup].
Conclusions
===========
Research on neutrino physics, and specially on neutrino oscillations, has been extensive and we are currently living a very exciting time with very important observations and with increasing precision measurements: strong evidence of $\bar{\nu}_e$ and $\nu_{\tau}$ appearance, CP-conserving values excluded at 2$\sigma$, data preference of normal mass ordering. All these confirming the 3 flavor oscillation hypothesis.
On the other hand, as there are yet opened questions to be resolved (precise measurement of $\delta_{CP}$; existence of sterile neutrinos; the Dirac/Majorana nature and the absolute mass of neutrinos, among others), a number of proposed experiments are starting to become real as they are at their commissioning and/or building stages, allowing us to foresee a bright future for the neutrino physics scientific community and beyond.
Acknowledgment {#acknowledgment .unnumbered}
==============
I am deeply grateful to the Organizers of the PIC2018 symposium for the kind invitation to participate in this event, and to the NOvA Collaboration for their scientific support. I also thank the *Vicerrectoría de Investigaciones, Extensión y Proyección Social* of the Universidad del Atlántico for their financial support.
[99]{} Q. R. Ahmad [*et al.*]{} \[SNO Collaboration\], “Direct evidence for neutrino flavor transformation from neutral current interactions in the Sudbury Neutrino Observatory,” Phys. Rev. Lett. [**89**]{}, 011301 (2002) \[nucl-ex/0204008\]. Y. Fukuda [*et al.*]{} \[Super-Kamiokande Collaboration\], “Evidence for oscillation of atmospheric neutrinos,” Phys. Rev. Lett. [**81**]{}, 1562 (1998) \[hep-ex/9807003\]. K. Eguchi [*et al.*]{} \[KamLAND Collaboration\], “First results from KamLAND: Evidence for reactor anti-neutrino disappearance,” Phys. Rev. Lett. [**90**]{}, 021802 (2003) \[hep-ex/0212021\]. F. P. An [*et al.*]{} \[Daya Bay Collaboration\], “Observation of electron-antineutrino disappearance at Daya Bay,” Phys. Rev. Lett. [**108**]{}, 171803 (2012) \[arXiv:1203.1669 \[hep-ex\]\]. J. K. Ahn [*et al.*]{} \[RENO Collaboration\], “Observation of Reactor Electron Antineutrino Disappearance in the RENO Experiment,” Phys. Rev. Lett. [**108**]{}, 191802 (2012) \[arXiv:1204.0626 \[hep-ex\]\]. K. Abe [*et al.*]{} \[T2K Collaboration\], “Indication of Electron Neutrino Appearance from an Accelerator-produced Off-axis Muon Neutrino Beam,” Phys. Rev. Lett. [**107**]{}, 041801 (2011) \[arXiv:1106.2822 \[hep-ex\]\]. D. S. Ayres, et al., The NOvA Technical Design Report, 396 FERMILAB-DESIGN-2007-01. P. Adamson [*et al.*]{} \[NOvA Collaboration\], “Constraints on Oscillation Parameters from $\nu_e$ Appearance and $\nu_\mu$ Disappearance in NOvA”, Phys. Rev. Lett. [**118**]{}, no. 23, 231801 (2017) \[arXiv:1703.03328 \[hep-ex\]\]. M. A. Acero [*et al.*]{} \[NOvA Collaboration\], “New constraints on oscillation parameters from $\nu_e$ appearance and $\nu_\mu$ disappearance in the NOvA experiment”, Phys. Rev. D [**98**]{}, 032012 (2018) \[arXiv:1806.00096 \[hep-ex\]\]. M. Sanchez, “NOvA Results and Prospects”, Talk at XXVIII International Conference on Neutrino Physics and Astrophysics, 4-9 June 2018, Heidelberg, Germany, DOI: 10.5281/zenodo.1286758, URL: https://doi.org/10.5281/zenodo.1286758 P. Vahle, “Accelerator Neutrino Physics Overview”, Talk at 20th Workshop on Neutrinos from Accelerators, NuFACT, 12-18 August 2018, Blacksburg, Virginia, USA, URL: https://nufact2018.phys.vt.edu. K. Abe [*et al.*]{} \[T2K Collaboration\], “The T2K Experiment”, Nucl. Instrum. Meth. A [**659**]{}, 106 (2011) \[arXiv:1106.1238 \[physics.ins-det\]\]. M. Khabibullin, “Recent results from the T2K experiment”, EPJ Web Conf. [**191**]{}, 03001 (2018). M. Wascko, “T2K Status, Results and Plans”, Talk at XXVIII International Conference on Neutrino Physics and Astrophysics, 4-9 June 2018, Heidelberg, Germany, DOI: 10.5281/zenodo.1286751, URL: https://doi.org/10.5281/zenodo.1286751. J. Evans \[MINOS and MINOS+ Collaborations\], “New results from MINOS and MINOS+”, J. Phys. Conf. Ser. [**888**]{}, no. 1, 012017 (2017). A. Aurisano, “Recent results from MINOS and MINOS+”, Talk at XXVIII International Conference on Neutrino Physics and Astrophysics, 4-9 June 2018, Heidelberg, Germany, DOI: 10.5281/zenodo.1286759, URL: https://doi.org/10.5281/zenodo.1286759. S. De Rijck \[MINOS and MINOS+ Collaborations\], “Latest Results from MINOS and MINOS+”, J. Phys. Conf. Ser. [**873**]{}, no. 1, 012032 (2017). M. Guler [*et al.*]{} \[OPERA Collaboration\], “OPERA: An appearance experiment to search for $\nu_{\mu} \leftrightarrow \nu_{\tau}$ oscillations in the CNGS beam. Experimental proposal”, CERN-SPSC-2000-028, CERN-SPSC-P-318, LNGS-P25-00. N. Agafonova [*et al.*]{} \[OPERA Collaboration\], “Final Results of the OPERA Experiment on $\nu_\tau$ Appearance in the CNGS Neutrino Beam”, Phys. Rev. Lett. [**120**]{}, no. 21, 211801 (2018), Erratum: \[Phys. Rev. Lett. [**121**]{}, no. 13, 139901 (2018)\], \[arXiv:1804.04912 \[hep-ex\]\]. K. Abe [*et al.*]{} \[Hyper-Kamiokande Collaboration\], “Hyper-Kamiokande Design Report”, arXiv:1805.04163 \[physics.ins-det\]. M. Shiozawa, “Hyper-Kamiokande”, Talk at XXVIII International Conference on Neutrino Physics and Astrophysics, 4-9 June 2018, Heidelberg, Germany, DOI: 10.5281/zenodo.1286768, URL: https://doi.org/10.5281/zenodo.1286768 R. Acciarri [*et al.*]{} \[DUNE Collaboration\], “Long-Baseline Neutrino Facility (LBNF) and Deep Underground Neutrino Experiment (DUNE) : Conceptual Design Report, Volume 1: The LBNF and DUNE Projects”, arXiv:1601.05471 \[physics.ins-det\]. D. Brailsford \[DUNE Collaboration\], “DUNE: Status and Perspectives”, arXiv:1804.04979 \[physics.ins-det\]. E. Worcester, “DUNE: Status and Science”, Talk at XXVIII International Conference on Neutrino Physics and Astrophysics, 4-9 June 2018, Heidelberg, Germany, DOI: 10.5281/zenodo.1286764, URL: https://doi.org/10.5281/zenodo.1286764. R. Acciarri [*et al.*]{} \[DUNE Collaboration\], “Long-Baseline Neutrino Facility (LBNF) and Deep Underground Neutrino Experiment (DUNE) : Conceptual Design Report, Volume 2: The Physics Program for DUNE at LBNF”, arXiv:1512.06148 \[physics.ins-det\].
[^1]: Here, neutrinos are assumed to be Dirac particles.
[^2]: Are there only three neutrinos? What is the absolute mass of the neutrinos? Is the neutrino its own antiparticle?
[^3]: Neutrinos at the Main Injector
[^4]: Tokai-to-Kamioka
[^5]: Gran Sasso National Laboratory, in Italy
|
---
abstract: |
Exceptional sequences of vector bundles over a variety $X$ are special generators of the triangulated category $D^b(Coh\,X)$. Kapranov proved the existence of tilting bundles over homogeneous varieties for the general linear group. King conjectured the existence of tilting sequences of vector bundles on projective varieties which are obtained as quotients of Zariski open subsets of affine spaces.
The goal of this paper is to give further examples of strong exceptional sequences of vector bundles on certain projective varieties. These are obtained as geometric invariant quotients of affine spaces by linear actions of reductive groups, as appears in King’s conjecture.
author:
- Mihai Halic
title: Strong exceptional sequences of vector bundles on certain Fano varieties
---
Introduction {#introduction .unnumbered}
============
The concept of derived categories has been introduced by Grothendieck and developed further by Verdier. However, their work remained within a very general and abstract setting, and people wished to have concrete examples which arise from geometry. In algebraic geometry one of the essential objects associated to a projective variety is the (bounded) derived category of coherent sheaves over it. Its knowledge allows to recover all the cohomological data of the variety.
Beilinson made the first major step by proving that the line bundles $
\mathcal O_{\mathbb P^n},\mathcal O_{\mathbb P^n}(1),\!...,
\mathcal O_{\mathbb P^n}(n)
$ generate $D^b({\rm Coh}\,\mathbb P^n)$, and actually form a tilting sequence. Afterwards have appeared several other examples of varieties admitting (strong and complete) exceptional sequences of vector bundles. One of the most notable results in this direction has been obtained by Kapranov [@ka]. He explicitly constructed tilting sequences of vector bundles over homogeneous varieties for ${\rm Gl}(n)$, that is over Grassmannians and flag manifolds. Further examples, which are based on Kapranov’s result, have been obtained in [@cm].
In the unpublished preprint [@ki], King conjectured that there are tilting bundles over projective varieties which are obtained as invariant quotients of affine spaces for linear actions of reductive groups. Observe that flag varieties for $Gl(n,\mathbb C)$, and toric varieties are special cases of such quotient varieties.
The answer to King’s conjecture is negative in general. Hille and Perling gave in [@hp] an example of a toric variety ($\mbb P^2$ blown-up successively three times) with the property that it does not admit a tilting object formed by line bundles. However it is still a very interesting problem to find classes of examples for which the conjecture holds. In the paper [@ah], Altmann and Hille proved the existence of (partial) strong exceptional sequences on toric varieties arising from thin representations of quivers, but their construction gives sequences of very short length.
The goal of this paper is to give further examples of strong exceptional sequences of vector bundles over certain Fano varieties. The varieties considered in this paper are obtained as geometric quotients of open subsets of affine spaces by linear actions of a reductive groups.
For the comfort of the reader, we recall that a sequence of vector bundles $(\cal F_1,\ldots,\cal F_z)$ over a variety $Y$ is called [*strongly exceptional*]{} if the following two conditions are fulfilled:
1. $H^0\bigl(Y,{\mathop{\rm Hom}\nolimits}(\cal F_j,\cal F_i)\bigr)=0,
\;\forall\,1\les i< j\les z$;
2. $H^q\bigl(Y,{\mathop{\rm Hom}\nolimits}(\cal F_j,\cal F_i)\bigr)=0,
\;\forall\,i,j=1,\ldots,z$, and $\forall\,q>0$.
3. A [*tilting sequence*]{} is a strongly exceptional sequence $(\cal F_1,\ldots,\cal F_z)$ with the property that $\cal F_1,\ldots,\cal F_z$ generate $D^b({\rm Coh}\,Y)$.
Consider an algebraically closed field $K$ of characteristic zero, a connected, reductive group $G$ over $K$, and a representation $\rho:G\rar{{\rm Gl}}(V)$. Let ${{\mbb V}}:={\mathop{\rm Spec}\nolimits}\bigl({\mathop{\rm Sym}\nolimits}^\bullet V^\vee\bigr)$ be the affine space corresponding to $V$. We denote $\chi_{{\rm ac}}=\chi_{{\rm ac}}(G,V)$ the weight of the $G$-module ${{\rm det}}V$. We make the following assumptions:
1. the ring of invariants $K[{{\mbb V}}]^T\!=\!K$, where $T$ is the maximal torus of $G$;
2. ${\rm codim}_{{{\mbb V}}}{{\mbb V}}^{{\rm us}}(G,\chi_{{\rm ac}})\ges 2$, and $G$ acts freely on the semi-stable locus ${{\mbb V}}^{{\rm ss}}(G,\chi_{{\rm ac}})$.
We denote $Y:={{\mbb V}}{{\slash\kern-0.65ex\slash}}_{\chi_{{\rm ac}}}G$ the invariant quotient. The main ingredient that we use for constructing exceptional sequences over $Y$ is the set ${\cal E_1,\ldots,\cal E_N}$ of ‘extremal’ nef vector bundles over $Y$ (see section \[sct:nef-vb\]). They enjoy good cohomology vanishing properties which are required by the definition of exceptional sequences. The first main result of this paper is the following:
The estimates appearing in this theorem are not strong enough to recover Kapranov’s construction for partial flag varieties. We have to go on, and exploit the fibre bundle structure. The optimal result would be the following:
- Consider a fibre bundle $Y\srel{\phi}{\rar}X$. Suppose that $(\cal F_i)_{i\in I}$ is a strong exceptional sequence of vector bundles on $X$, and that $(\cal E_j)_{j\in J}$ is a sequence of vector bundles on $Y$ whose restriction to the fibres of $\phi$ give rise to strong exceptional sequences relative to $\phi$. Then $(\phi^*\cal F_i\otimes\cal E_j)_{(i,j)\in I\times J}$ is a strong exceptional sequence on $Y$.
Unfortunately such a statement is overoptimistic in general. The content of our second main result is that the statement above becomes true under suitable restrictive hypotheses on the fibration $\phi$. More precisely, we place ourselves in the following framework:
1. There is a quotient group $H$ of $G$ with kernel $G_0$, and a quotient $H$-module $W$ of $V$ with kernel $V_0$, such that the natural projection ${\mathop{\rm pr}\nolimits}^{{{\mbb V}}}_{\mbb W}:{{\mbb V}}\rar\mbb W$ has the following property:
${\mathop{\rm pr}\nolimits}^{{{\mbb V}}}_{\mbb W}\bigl(\;{{\mbb V}}^{{\rm ss}}\bigl(G,\chi_{{\rm ac}}(G,V)\bigr)\;\bigr)
\subseteq\mbb W^{{\rm ss}}\bigl(G,{\chi_{{\rm ac}}}(H,W)\bigr).$
We denote by $Y\srel{\phi}{\rar}X$ the induced morphism at the quotient level.
2. The unstable loci have codimension at least two, and both quotients
${{\mbb V}}^{{\rm ss}}(G,\chi_{{\rm ac}}(G,V))\rar Y\;$ and $\;\mbb W^{{\rm ss}}(H,\chi_{{\rm ac}}(H,W))\rar X$
are principal bundles.
3. The nef cone of the total space $Y$ is the sum of the nef cones of the base $X$, and that of the fibre: $\crl N(G,V)=\crl N(H,W)+\crl N(G_0,V_0)$. Denote ${{\cal V\kern-.41ex\euf B}}^+(X)$ and ${{\cal V\kern-.41ex\euf B}}^+_0$ the corresponding sets of extremal nef vector bundles.
4. The maximal torus $T_0\subset G_0$ has exactly $\dim T_0$ weights on $V_0$.
Our main result in the relative case is the following:
We point out that in both cases it [*remains open*]{} the question under which hypothesis these sequences are/extend to [*tilting*]{} objects. However, we remark that, taking into account the example constructed in [@hp], a [*general answer*]{} concerning the (non-)existence of tilting vector bundles over quotients of affine spaces must be involved.
The definition of an exceptional set involves two conditions. Accordingly, the paper is divided in two main parts, each focusing on one of the two conditions:
– The sections \[sct:stability\] and \[sct:conseq-stab\] form the first part: we prove a stability result for associated vector bundles, and define an order on the set of irreducible $G$-modules for which there are no homomorphisms from a ‘larger’ vector bundle into a ‘smaller’ one (see theorem \[thm:h000\]).
– The sections \[numer-crit\], \[sct:nef-vb\] and \[cohom-nef\], have a preparatory character: we introduce the ‘extremal’ nef vector bundles, and study their cohomological properties.
– The second part of the article consists of the sections \[sct:main\] and \[sct:main2\]: they contain the proofs of the main results. The main tool used for proving the vanishing of the higher cohomology groups is a result due to Manivel (see [@ma]), and Arapura (see [@ar]). However, this general result is not sufficient to address the relative case, and we have to dwell on our particular context. In theorem \[thm:direct-image\] we prove the following nefness property, which is an essential ingredient in the proof of Theorem B.
– Finally, in section \[sct:expl\], we illustrate the general theory. On one hand, we recover Kapranov’s construction for the Grassmannian and for flag varieties, by using our results. On the other hand, we give further examples of strong exceptional sequences over quiver varieties. The very pleasant feature is that we obtain these example by an almost algorithmic procedure, which applies to any quiver variety.
Some of the results have been presented at the HOCAT 2008 Conference, held at Centre de Recerca Matemàtica, Bellaterra, Spain.
A stability property
====================
[\[sct:stability\]]{}
The symbol $\mbb Q$ will always denote the field of rational numbers, and $K$ will be an algebraically closed field $K$ of characteristic zero. Throughout the paper, $G$ will always denote a connected, reductive group over $K$, and $T$ will be the maximal torus of $G$. We consider a faithful representation $\rho:G\rar{{\rm Gl}}(V)$, and denote by ${{\mbb V}}:={\mathop{\rm Spec}\nolimits}({\mathop{\rm Sym}\nolimits}^\bullet V^\vee)$ the corresponding affine space. We shall assume that the ring of invariants $K[{{\mbb V}}]^T=K$; it follows automatically that $K[{{\mbb V}}]^G=K$.
[\[lm:Zneq0\]]{} Let $V$ be a non-zero $G$-module such that $K[{{\mbb V}}]^T=K$. Then:
1. There is a 1-PS $\l\in\cal X_*(T)$ such that all its weights on $V$ are strictly positive.
2. $G$ is not semi-simple.
We fix once for all ${{\mfrak l}}\in\cal X_*(T)\otimes_{\mbb Z}\mbb R$ such that its weights on $V$ are all positive, and moreover it has ‘irrational slope’, that is ${{\rm Ker}}({{\mfrak l}}:\cal X^*(T)\rar\mbb R)=\{0\}.$
\(i) Let $\Phi$ denote the set of weights of the $T$-module $V$. Then the set of weights of the $T$ on $K[{{\mbb V}}]$ is the ‘cone’ $\underset{\eta\in\Phi}\sum\mbb N\eta$. Since $K[{{\mbb V}}]^T=K$, this cone is strictly convex. Otherwise we can construct a non-trivial $T$-invariant monomial. It follows that there is $\l\in\cal X_*(T)$ with $\langle\eta,\l\rangle>0$ for all $\eta\in\Phi$.
\(ii) Assume that $G$ is a semi-simple group. The previous step implies that $K[{{\mbb V}}^m]^T=K$, hence $K[{{\mbb V}}^m]^G=K$ for all $m\ges 1$. Since $G$ is semi-simple, it has an open orbit in ${{\mbb V}}^m$. For large $m$ we get a contradiction.
Let $\th\in\cal X^*(G)$ be a character. We denote: $${\label{G-sst}}
\begin{array}{l}
K[{{\mbb V}}]^G_\th:=\{f\in K[{{\mbb V}}]\mid f(g\times y)=\th(g)\cdot f(y),
\,\forall y\in{{\mbb V}}\}
\\[2ex]
K[{{\mbb V}}]^{G,\th}:=K\oplus\underset{n\ges 1}\bigoplus K[{{\mbb V}}]^G_{\th^n},
\\[2ex]
{{\mbb V}}^{{\rm ss}}(G,\th):=\{y\in{{\mbb V}}\mid\exists n\ges 1\text{ and }
f\in K[{{\mbb V}}]^G_{\th^n}\text{ s.t. }f(y)\neq 0\}.
\end{array}$$ We say that $\th$ is [*effective*]{} if there is $n\ges 1$ such that $K[{{\mbb V}}]^G_{\th^n}\neq 0$, that is ${{\mbb V}}^{{\rm ss}}(G,\th)\neq\emptyset$.
We define the [*anti-canonical character*]{} of the $G$-module $V$ to be the character of the $G$-module ${{\rm det}}V$.
Explicitly: decompose $V=\underset{\og\in\cal X}\bigoplus M_\omega^{\oplus m_\og}$ into its $G$-isotypical components. Let $\chi_\og$ be the character by which $Z(G)^\circ$ acts on $M_\og$, and denote $d_\og:=\dim M_\og$. Then $\;
\chi_{{\rm ac}}(G,V):=
\mbox{$\underset{\og\in\cal X}\sum$}
m_\og d_\og\chi_\og\in\cal X^*(G).
$ For shorthand, we will write $\chi_{{\rm ac}}=\chi_{{\rm ac}}(G,V)$.
[\[lm:effective\]]{} Assume that $m_\og \ges d_\og$. Then the character $\chi_\og$ is effective. Moreover, if $m_\og > d_\og$ for all $\og$, then $\chi_{{\rm ac}}$ is effective, and the $\chi_{{\rm ac}}$-unstable locus has codimension at least two.
We view $V$ as $\bigoplus_{\og\in\cal X} {\mathop{\rm Hom}\nolimits}(K^{m_\og }, M_\og)$. Since $m_\og \ges d_\og$, we can associate to an element ${\mathop{\rm Hom}\nolimits}(K^{m_\og }, M_\og)$ the $d_\og\times d_\og$-minor corresponding to the first $d_\og$ columns. This defines a regular function $f_\og$ which is $d_\og\chi_\og$-equivariant; moreover, $f_\og$ does not vanish on surjective homomorphisms. It follows that $d_\og\chi_\og$, and therefore $\chi_\og$, is effective for all $\og$.
If a point belongs to the unstable locus, then all the minors $f_\og$ have to vanish. Since $m_\og \ges d_\og+1$, this implies the vanishing of at least two independent minors.
Now we prove a general stability result of independent interest. It is well known that the tangent bundle of the projective space is stable, and more generally the tautological bundles over Grassmannians are stable. Our goal is to generalize these facts.
We denote ${\{G_j\}}_{j\in J}$ the simple factors of $G$, and let $\gamma_j:G\rar G_j$, be the corresponding quotient morphisms. Using the $\gamma_j$’s we extend the structural group of $\O\rar Y$, and obtain the principal $G_j$-bundles $\O(G_j)\rar Y$. The main result of this section is:
[\[thm:stab-bdl\]]{} Assume that $G$ acts freely on $\O\!:=\!{{\mbb V}}^{{\rm ss}}(G,\th)$, for some $\th\in\cal X^*(G)$, and let $Y$ be the quotient. Assume that $m_\omega >\dim M_\og$ holds for all $\omega\in\cal X$. Then the principal $G_j$-bundles $\O(G_j)\rar Y\!$, $j\in J$, obtained by extending the structural group are semi-stable.
We fix $j\in J$, and a maximal parabolic subgroup $P_j\subset G_j$; denote $P:=\gamma_j^{-1}P_j$: it is a maximal parabolic subgroup of $G$. We observe that the associated homogeneous bundles $\bigl(\O(G_j)\bigr)(G_j/P_j)$ and $\O\bigl(G/P\bigr)$ are isomorphic.
We denote $H=\underset{\omega }{\prod}\,H_\omega :=
\underset{\og }{\prod}\,{{\rm Gl}}_K(m_\omega )$: it acts naturally on ${{\mbb V}}$; the $G$- and $H$-actions on ${{\mbb V}}$ commute. It follows that $H$ still acts on $\O(G/P)$ by
$
H\times\O(G/P)\rar\O(G/P),\; h\times[y,gP]:=[hy,gP].
$
We will prove that whenever there is a reduction of the structural group
$
s:Y^o\rar\bigl(\O(G_j)\bigr)(G_j/P_j)=\O\bigl(G/P\bigr),
$ with $Y^o\subset Y$ open and ${{\rm codim}}_Y(Y\sm Y^o)\ges 2,$
holds $\deg_{Y}\big(s^*{\sf T}_{\O(G/P)/Y}\big)\ges 0.$ Equivalently, the reduction $s$ can be viewed as a $G$-equivariant morphism $S:\O^o=q^{-1}(Y^o)\rar G/P$.
The idea is to move $s$ using the $H$-action on $\O(G/P)$. Let $\hat y\in Y$ be a generic point, and consider $y\in\O$ over $\hat y$. We define the following subgroups of $H$: $\quad K_{\hat y}:={\rm Stab}_H(y)$, and $$H_{\hat y}:=\{h\in H\mid \exists\,g_h\in G\text{ s.t. }
h\times y=\rho(g_h^{-1})y\}
=\mbox{$\underset{\og}\prod$} H_{\og ,\hat y}.$$ We observe that $K_{\hat y}$ does not depend on the choice of $y\in q^{-1}(\hat y)$. Since $G$ acts freely on $\O$, the assignment $h\mt g_h$ defines a group homomorphism $\rho_{\hat y}:H_{\hat y}\rar G$ whose kernel is $K_{\hat y}$. We move the section $s$ using the action of $H_{\hat y}$. For $h\in H_{\hat y}$ define a new section $s_h$ as follows: $$s_h(\hat x):=[x,S(h^{-1}\times x)]
\quad
\text{(equivalently, $S_h(x):=S(h^{-1}\times x)$).}$$ Observe that as $h\in H_{\hat y}$ varies, $s_h(\hat y)=h\times s(\hat y)$ moves in the vertical direction.
$H_{\hat y}/K_{\hat y}\rar G/Z(G)^\circ$ is surjective. Write $y={(y_\omega )}_\omega $ w.r.t. the direct sum decomposition of $V$; for each $\omega\in\cal X$, $y_\omega =(y_{\omega 1},\ldots,y_{\omega m_\omega})$. Since $y\in\O$ is chosen generically, and $m_\omega>\dim M_\og=:d_\omega$, we may assume that for each $\omega\in\cal X$ the vectors $y_{\omega 1},\ldots,y_{\omega m_\omega}$ span $M_\og$. Equivalently, we may view $y_\omega$ as a surjective homomorphism $K^{m_\omega}\rar M_\og$.
For $g\in G$ holds $\rho(g)y={(\rho_\omega (g)y_\omega)}_\omega$. Using that $m_\omega> d_\omega$, we deduce that for each $\omega\in\cal X$ there is $h_\omega\in{{\rm Gl}}_K(m_\omega)$ such that $h_\omega y_\omega=\rho_\omega(g^{-1})y_\omega$. For $h:={(h_\omega)}_\omega$ we have $hy=\rho(g^{-1})y$, that is $g\in{\rm Image}\bigl(H_{\hat y}/K_{\hat y}\rar G\bigr)$.
Back to our proof: the infinitesimal action of $H_{\hat y}$ preserves the restriction to the fibre $q^{-1}(\hat y)=\{[y,gP]\mid g\in G\}\cong G/P$ of the relative tangent bundle ${\sf T}_{\O(G/P)/Y}$. By this isomorphism the relative tangent bundle corresponds to ${\sf T}_{G/P}\rar G/P$. The claim implies that the infinitesimal action ${\mathop{{\cal L}ie}\nolimits}(H_{\hat y})\rar{\sf T}_{\O(G/P)/Y, s(\hat y)}$ is surjective. Hence there is a section $\si\in H^0(Y^o,s^*{{\rm det}}{\sf T}_{\O(G/P)/Y})$ which does not vanish at $\hat y$. It follows $\deg_Y\big(s^*{\sf T}_{\O(G/P)/Y}\big)\ges 0$.
[\[cor:stab-bdl\]]{} Assume $\th\in\cal X^*(G)$ has the property that $G$ acts freely on $\O:={{\mbb V}}^{{\rm ss}}(G,\chi)$, and let $Y$ be the quotient. Let $E$ be an irreducible $G$-module, and denote by $\cal E:=\O(E)$ the associated vector bundle over $Y$. Assume that $m_\omega>\dim M_\og$ holds for all $\omega\in\cal X$. Then $\cal E\rar Y$ is slope semi-stable with respect to the polarization induced by the character $\th$.
We may assume that $G=Z(G)^\circ\times\bigl(\underset{j\in J}{\times}G_j\bigr)$. Since each $\O(G_j)$ is semi-stable, $\O\rar Y$ itself is semi-stable. The homomorphism $\rho_\omega:G\rar{{\rm Gl}}(E)$ maps $Z(G)^\circ$ into the centre of ${{\rm Gl}}(E)$. By using [@rr theorem 3.18], we deduce that $\cal E=\O(E)\rar Y$ is semi-stable.
The $H^0$ spaces
================
[\[sct:conseq-stab\]]{}
Assume that $E$ is a $G$-module. We will denote by $\cal E$ the vector bundle over $Y$ associated to it. More precisely, $\cal E$ corresponds to the module of covariants $\bigl(K[{{\mbb V}}]\otimes_K E^\vee\bigr)^G$.
The classical Schur lemma says that for two irreducible $G$-modules $E$ and $F$, the space ${\mathop{\rm Hom}\nolimits}(E,F)$ consists either of scalars (if $E=F$), or vanishes (if $E\neq F$). In this section we will prove that a similar result holds for the associated vector bundles $\cal E$ and $\cal F$.
For warming-up, we start with a special case. We have proved in corollary \[cor:stab-bdl\] that $\cal E\rar Y$ is a semi-stable vector bundle w.r.t. any polarization on $Y$, as soon as the multiplicities $m_\og> d_\og$ for all $\og$. Its first Chern class equals $\dim(E)\cdot\chi_E$, where $\chi_E$ denotes the character of $Z(G)^\circ$ on $E$. Let $\th\in\cal X^*(G)$ be an ample class on $Y$; the slope of $\cal E$ w.r.t. $\th$ equals $$\mu_\th(\cal E)=
\frac{\deg_\th\cal E}{\dim E}=
\langle \chi_\og\cdot\th^{\dim Y-1},[Y] \rangle.$$
[\[defn:order1\]]{} Let $\th$ be a polarization of $Y$. We define the order $<_\th$ on $\cal X^*\bigl(Z(G)^\circ\bigr)$ as follows: we declare that $\chi<_\th\eta$ if holds: $$\mu_\th(\chi):=\langle\chi\cdot\th^{\dim Y-1},[Y]\rangle
<
\mu_\th(\eta):=\langle\eta\cdot\th^{\dim Y-1},[Y]\rangle.$$ Observe that, by the hard Lefschetz property, we can choose $\th$ in such a way that $\chi=\eta\Leftrightarrow\mu_\th(\chi)=\mu_\th(\eta)$.
[\[prop:h00\]]{} We assume that $m_\og> d_\og$ holds for all $\og$. Let $E$ and $F$ be two distinct irreducible $G$-modules, such that $Z(G)^\circ$ acts on them by two different characters $\chi_E$ and $\chi_F$ respectively, such that $\mu_\th(\cal E)<\mu_\th(\cal F)$. Then $H^0\bigl(Y,{\mathop{\rm Hom}\nolimits}(\cal F,\cal E)\bigr)=0$.
This is an immediate consequence of the semi-stability property of $\cal E$ and $\cal F$.
The proposition has two shortcomings: first, we have imposed the condition on the multiplicities; second, there are distinct representations $E$ and $F$ such that the characters $\chi_E$ and $\chi_F$ coincide. So we need to sharpen our result.
[\[not-eff\]]{} Assume that ${\rm codim}_{{\mbb V}}{{\mbb V}}^{{\rm us}}(G,\chi_{{\rm ac}})\ges 2$. Let $E$ be an irreducible $G$-module, and let $\cal E\rar Y$ be the associated vector bundle. Suppose that there is a weight $\veps$ of $T$ on $E$ which is not $T$-effective (that is ${{\mbb V}}^{{\rm ss}}(T,\veps)=\emptyset$). Then $H^0(Y,\cal E)=0$.
Recall that $H^0(Y,\cal E)={\mathop{\rm Mor}\nolimits}(\mbb V\rar E)^G$, where $$(g\times S)(y)=g\times S(g^{-1}\times y),\quad\forall\,g\in G
\text{ and }{{\mbb V}}\srel{S}{\rar}E.$$ Assume that there is a non-zero $G$-equivariant morphism $S:\mbb V\rar E$. Then the linear span $\langle S\rangle:=\langle S(y), y\in\mbb V\rangle$ is actually a $G$-submodule of $E$. Since $E$ is irreducible and $S\neq 0$, we deduce $\langle S\rangle=E$.
On the other hand, $\veps$ is a weight of $T$ on $E$ which is not effective. We choose a one dimensional $T$-submodule $E_\veps\subset E$, and consider [*the function*]{} $S_\veps:={\mathop{\rm pr}\nolimits}^E_{E_\veps}\circ\, S$. Then $
S_\veps(t\times y)=\veps(t)\cdot S_\veps(y),\;
\forall t\in T,\,y\in\mbb V.
$
Since $\veps$ is not effective, the function $S_\veps$ must vanish. This implies that the image of the morphism $S$, and consequently its linear span $\langle S\rangle$, is contained in the complement $E'$ of $E_\veps$. The contradiction shows that $\langle S\rangle=E$.
In order to check that a sequence of vector bundles forms an exceptional sequence, one has to prove that there are no non-trivial homomorphisms from ‘larger’ bundles into ‘smaller’ ones. Now we define the total order required for this property.
[\[defn:order2\]]{} Consider ${{\mfrak l}}\in\cal X_*(T)$ as in lemma \[lm:Zneq0\].
1. For any irreducible $G$-module, we define $$\mfrak l(E):=
\max\{\langle\eta,\mfrak l\rangle\mid\eta\text{ is a weight of $T$ on $E$}\}.$$ Equivalently:\
$\mfrak l(E)=\langle\eta_E,\mfrak l\rangle$, where $\eta_E$ is the dominant weight of $E$ (with respect to $\mfrak l$).
2. Let $E$ and $F$ be two irreducible $G$-modules. We say that $E<_{{{\mfrak l}}}\,F$ if $\mfrak l(E)\,<\,\mfrak l(F)$.
Since $\mfrak l$ has irrational slope, for any two irreducible $G$-modules $E$ and $F$ holds:
$\mfrak l(E)=\mfrak l(F)\Rightarrow E=F$.
Hence $<_{{{\mfrak l}}}$ is a total order relation.
The following result can be viewed as a generalization of Schur’s lemma.
[\[thm:h000\]]{} Assume that ${\rm codim}_{{\mbb V}}{{\mbb V}}^{{\rm us}}(G,\chi_{{\rm ac}})\ges 2$.
1. Let $E$ be an irreducible $G$-module. Then $H^0\bigl(Y,{\mathop{\rm End}\nolimits}(\cal E)\bigr)=K.$
2. Let $E$ and $F$ be two irreducible $G$-modules such that $E<_{{{\mfrak l}}}\,F$. Then
$H^0\bigl(Y,{\mathop{\rm Hom}\nolimits}(\cal F,\cal E)\bigr)=0.$
\(i) A section $s\!\in\! H^0\bigl(Y,{\mathop{\rm End}\nolimits}(\cal E)\bigr)$ corresponds to a $G$-equivariant morphism $S:{{\mbb V}}\rar{\mathop{\rm End}\nolimits}(E)$, where the action on ${\mathop{\rm End}\nolimits}(E)$ is by conjugation. (Here we use the hypothesis on the codimension of the unstable set: regular maps defined on the semi-stable locus extend to the whole affine space.) We will prove that the morphism $S$ is a scalar multiple of the identity.
The origin $0\in{{\mbb V}}$ is fixed under $G$. Since $S$ is $G$-equivariant, the homomorphism $S_0\in{\mathop{\rm End}\nolimits}(E)$ is ${{\rm Ad}}_G$-invariant. Schur’s lemma implies that $S_0=c\cdot{{1\kern-0.57ex\rm l}}_{E}$, with $c\in K$. By lemma \[lm:Zneq0\], there is a 1-PS $\l\in\cal X_*(T)$ such that all its weights on $V$ are strictly positive. In particular $\disp\lim_{t\rar 0}\l(t)y=0$ for all $y\in{{\mbb V}}$. The $G$-equivariance implies $S_{\l(t)y}={{\rm Ad}}_{\l(t)}\circ S_y$, hence $\,\disp\lim_{t\rar 0}{{\rm Ad}}_{\l(t)}\circ S_y =S_0=c{{1\kern-0.57ex\rm l}}_{E}.$
The $\l(t)$-action on $E$ can be diagonalized in an appropriate basis formed by weight vectors. We denote ${\{E_i\}}_{i\in I}$ the weight spaces of $E$. We order the elements of $I$ in decreasing order, and consider the corresponding basis in $E$. Then w.r.t. this basis, $S_y$ has the following block-matrix shape: $$S_y
=
\left(
\begin{array}{c|c|c}
c{{1\kern-0.57ex\rm l}}& *&*\\ \hline
0&c{{1\kern-0.57ex\rm l}}&*\\ \hline
0&0&c{{1\kern-0.57ex\rm l}}\end{array}
\right)
\;\text{or equivalently}\;
S_y-c{{1\kern-0.57ex\rm l}}=
\left(
\begin{array}{c|c|c}
0 & *&*\\ \hline
0&0&*\\ \hline
0&0&0
\end{array}
\right),
\;\forall\,y\in{{\mbb V}}$$ Let $\mfrak N_\l$ be the vector space which is formed by matrices having this shape ($\mfrak N_\l$ is actually a nilpotent Lie algebra). Intrinsically, $$\disp\mfrak N_\l=\{A\in{\mathop{\rm End}\nolimits}(E)\mid\lim_{t\rar 0}{{\rm Ad}}_{\l(t)}\circ A=0\}.$$ We denote $
{{\rm Ker}}(\mfrak N_\l):=\kern-1ex
\mbox{$\underset{N\in\mfrak N_\l}\bigcap$}\kern-1ex{{\rm Ker}}(N).
$ By Engel’s theorem, ${{\rm Ker}}(\mfrak N_\l)$ is a non-zero vector subspace of $E$. Applying the $G$-equivariance once more, we deduce that for any $g\in G$ holds: $$Ad_{g^{-1}}\circ
\bigl(
S_y-c{{1\kern-0.57ex\rm l}}\bigr)
=S_{g^{-1}y}-c{{1\kern-0.57ex\rm l}}\in\mfrak N_\l.$$ It follows that for all $g\in G$, $${{\rm Ker}}\bigl(
S_y-c{{1\kern-0.57ex\rm l}}\bigr)
\supset
g\cdot{{\rm Ker}}(\mfrak N_\l)
\;\Longrightarrow\;
{{\rm Ker}}\bigl(
S_y-c{{1\kern-0.57ex\rm l}}\bigr)
\!\supset\kern-.7ex
\mbox{$\underset{g\in G}\sum$}
g\cdot{{\rm Ker}}(\mfrak N_\l).$$ Note that the right-hand-side is a non-zero $G$-submodule of $E$. Since $E$ is irreducible, it follows taht ${{\rm Ker}}\bigl(S_y-c{{1\kern-0.57ex\rm l}}\bigr)=E$, that is $S_y=c{{1\kern-0.57ex\rm l}}$ for all $y\in{{\mbb V}}$.
\(ii) The $G$-module ${\mathop{\rm Hom}\nolimits}(F,E)=F^\vee\otimes E$ contains the difference $\veps:=\eta_E-\eta_F$ of the corresponding dominant characters. Since $E<_{{\mfrak l}}\,F$, ${{\mfrak l}}(E)-{{\mfrak l}}(F)<0$, the weight $\veps$ is not $T$-effective. The conclusion follows from theorem \[not-eff\].
Numerical criteria for semi-stability
=====================================
[\[numer-crit\]]{}
In this section we are reviewing some numerical criteria for semi-stability, needed later on. The following convention is used throughout this section: the letters $E, V, W$ denote $G$-modules, while the symbols $\mbb{E, V, W}$ will denote the corresponding affine spaces: [*e.g.*]{} $\mbb E:={\mathop{\rm Spec}\nolimits}\bigl({\mathop{\rm Sym}\nolimits}^\bullet E^\vee\bigr)$.
For a $G$-module $W$, let $\eta_1,\ldots,\eta_R$ be the weights of the maximal torus $T\subset G$. We define: $\;m:\mbb W\times \cal X_*(G)_{\mbb R}\rar \mbb R,$ $$\begin{array}{l}
m(w,\l):=
\min
\biggl\{\;
j\;\biggl|\;
\begin{array}{l}
\text{the $t^j$-isotypical component of
$w$ w.r.t. $\,\l$}
\\
\text{does not vanish}
\end{array}
\biggr.\biggr\}.
\end{array}$$ Observe that for $\l\in\cal X_*(T)$ holds: $$m(w,\l):=
\min
\biggl\{
\langle\eta_j,\l\rangle
\;\biggl|\;
\begin{array}{l}
\text{the $\eta_j$-isotypical component of $w$}
\\
\text{does not vanish}
\end{array}
\biggr\}.$$ We fix a norm $|\cdot|$ on $\cal X_*(T)$, invariant under the Weyl group of $G$. For a character $\th\in\cal X^*(G)$, the Hilbert-Mumford criterion for $(G,\th)$-(semi-)stability reads: $$\begin{aligned}
{\label{fct-m}}
\begin{array}{rl}
w\!\in\!\mbb W^{{{\rm s}}\,{\rm (resp. }\,{{\rm ss}}\rm)}(G,\th)
&\Leftrightarrow
m(w)\!:=\!
\inf
\left\{
\frac{\langle \th,\l\rangle}{|\l|}
\,\biggl|\, m(w,\l)\ges 0\biggr.
\right\}
\underset{(\ges)}{>} 0
\\[2ex]
&\Leftrightarrow
\biggl[\,
m(w,\l)\ges 0
\,\Rightarrow\,
\langle \th,\l\rangle\underset{(\ges)}{>} 0
\,\biggr].
\end{array}\end{aligned}$$ For $w\in\mbb W$ we define: $$\begin{array}{rl}
S(w):=&
\{\eta_j\mid
\text{the $\eta_j$-isotypical component of $w$ does not vanish}
\}
\\[1.5ex]
\crl C_w=&
\underset{\eta\in S(w)}\sum\mbb R_{\ges 0}\eta
\\[1.5ex]
\L^G_w:=&
\{\l\in\cal X_*(G)\mid m(w,\l)\ges 0\}
\\[1.5ex]
\L^T_w:=&
\{\l\in\cal X_*(T)\mid m(w,\l)\ges 0\}
\\[1.5ex]
=&
\{\l\in\cal X_*(T)\mid
\langle\eta,\l\rangle\ges 0,\;\forall\eta\in\crl C_w
\}
=\crl C_w^\vee.
\end{array}$$ Note that $\crl C_w$ and $\L^T_w$ are convex, polyhedral cones. Since there are finitely many $\eta$’s, only finitely many cones $\crl C_w$ and $\L_w^T$ occur as $w$ varies in $\mbb W^s(G,\th)$. We are interested in the [*minimal*]{} cones $\crl C_w$.
Let $\th$ be a character of $G$. A subset $S\subset\{\eta_1,\ldots,\eta_R\}$ is [*minimal for $\th$*]{} if $
\th\in
\mbox{$\underset{\eta\in S}\sum$}
\mbb R_{\ges 0}\eta
$ and $
\th\not\in
\mbox{$\underset{\eta\in S\sm\{\eta_0\}}\sum$}
\mbb R_{\ges 0}\eta
$ for all $\eta_0\in S$.
We denote $S_1,\ldots,S_z$ the (finitely many) minimal sets for $\th$, and the corresponding cones by $\crl C_j$ and $\L_j:=\crl C_j^\vee$, $j=1,\ldots,z$, respectively. The Weyl group of $G$ operates by permutations on them.
Observe that $\L^G_w=
\underset{g\in G}\bigcup{{\rm Ad}}_{g^{-1}}\bigl(\L^T_{gw}\bigr)$. As $\th$ is ${{\rm Ad}}_G$-invariant, the numerical criterion can be reformulated as follows: $${\label{intersect}}
\th\in\cal X^*(G)\cap{\rm int.}
\biggl(\,
\bigcap_{w\in\mbb W^{{\rm s}}(G,\th)}
\kern-2ex\crl C_w
\biggl)
=
{\rm int.}\biggl(
\cal X^*(G)\cap\crl C_1\cap\ldots\cap\crl C_z
\biggr).$$
For two $G$-modules $V, E$, we define the $K^\times\times G$-module $W_m:=E\times V^{m}$, $m\ges 1$, with the module structure given by
$
(t,g)\times\bigl(\vphi,(v_j)_j\bigr)
:=
\bigl(t\cdot(g\times\vphi),(g\times v_j)_j\bigr).
$
Consider $l>0$, and define $\th_m:=l\chi_t+m\chi_{{\rm ac}}\in\cal X^*(K^\times\times G)$. The numerical functions on $\mbb V^m$ and $\mbb W_m$ are the following: $$\begin{array}{l}
\disp
m(\unl v,\l)
\,=\,
\min_j m(v_j,\l),\quad \forall\,\unl v=(v_j)_j\in\mbb V^{m},
\\[2.5ex]
\disp
m((\vphi,\unl v),t^\veps\l)
=\min_{1\les j\les m}\{\veps+m(\vphi,\l),m(v_j,\l)\},
\quad \forall\,(\vphi,\unl v)\in\mbb W_{m}.
\end{array}$$ The stability criterion for $\mbb W_m$ reads: a point $w=(\vphi,\unl v)$ is stable w.r.t. $(K^\times\times G,\,\th_m)$ if and only if $${\label{abc}}
\kern-.5ex\left\{
\begin{array}{lrlr}
(A)&
m(\vphi,\l)\ges 0,\, m(\unl v,\l)\ges 0
&\Rightarrow&
\langle \chi_{{\rm ac}},\l\rangle > 0;
\\[1ex]
(B)&
1+m(\vphi,\l)\ges 0,\, m(\unl v,\l)\ges 0
&\Rightarrow&
l+m\cdot \langle \chi_{{\rm ac}},\l\rangle > 0;
\\[1ex]
(C)&
-1+m(\vphi,\l)\ges 0,\, m(\unl v,\l)\ges 0
&\Rightarrow&
-l+m\cdot\langle \chi_{{\rm ac}},\l\rangle > 0.
\end{array}
\right.$$ Note that $\crl C_{(\vphi,\unl v)}
=\crl C_\vphi+\bigl(\mbb R\times\crl C_{\unl v}\bigr)$ for all $(\vphi,\unl v)\in\mbb W_m$; moreover, for $\unl v=(v_1,\ldots,v_m)$, then $\crl C_{\unl v}=\crl C_{v_1}+\ldots+\crl C_{v_m}$. We deduce that as [*both*]{} $m$ and $(\vphi,\unl v)\in\mbb W_m$ vary, there will be only [*finitely many*]{} dual cones: $${\label{fcs}}
\L_{(\vphi,\unl v)}
=
\L_\vphi\cap\bigl(\mbb R\times\L_{\unl v}\bigr)
=
\L_\vphi\cap\bigl(\mbb R\times(\L_{v_1}\cap\ldots\cap\L_{v_m})\bigr).$$ We denote by $\L_1',...\,,\L_Z'$ the various intersections of $\L_1,...\,,\L_z$ defined above, corresponding to the [*fixed*]{} representation $G\rar{{\rm Gl}}(V)$.
[\[prop:large-m\]]{} Let us assume that the $G$-module $V$ has the property:\
$({{\mbb V}}^m)^{{\rm ss}}(G,\chi_{{\rm ac}})=({{\mbb V}}^m)^{{\rm s}}(G,\chi_{{\rm ac}})\;$ for all $\;m\ges 1$.
Then there is a constant $a_0(E)$ depending on $E$ such that for $\frac{m}{l}>a_0(E)$:
$
\bigl(\mbb E\times\mbb V^{m}\bigr)^{{\rm s}}(K^\times\times G,l\chi_t+m\chi_{{\rm ac}})
=\bigl(\mbb E\sm\{0\}\bigr)\times \bigl(\mbb V^{m}\bigr)^{{\rm s}}(G,\chi_{{\rm ac}}).
$
Equivalently, $\chi_t+r\chi_{{\rm ac}}$ is an ample class on $\mbb P(\cal E)$ for $r>a_0(E)$.
‘$\supset$’ Let $(\vphi,\unl v)\in (\mbb E\sm\{0\})\times (\mbb V^m)^{{\rm s}}(G,\chi_{{\rm ac}})$. By definition, this means:$\;$ $
m(\unl v,\l)\ges 0
\;\Rightarrow\;
\langle\chi_{{\rm ac}},\l\rangle > 0.
$\
The conditions $(A)$ and $(B)$ in are automatically fulfilled. We prove that for large $m$ the condition $(C)$ holds too. Let $\l_0$ be such that $m(\vphi,\l_0)\ges 1$ and $m(\unl v,\l_0)\ges 0$.
Recall that only finitely many cones $\L_{\unl v}$ will appear when both $m$ and $\unl v\in(\mbb V^m)^{{\rm s}}$ vary. On each such cone, the linear function $\langle\chi_{{\rm ac}},\,\cdot\,\rangle$ is strictly positive. We choose $a_1>0$ such that $
\langle\chi_{{\rm ac}},\l\rangle\ges a_1|\l|,\; \forall\,
\l\in\L_1'+\ldots+\L_Z'.
$
For fixed $\vphi$, the function $m(\vphi,\cdot)$ is piecewise linear. As $\vphi$ varies, $m(\vphi,\cdot)$ depends only on the weights of $T$ on $E$. Overall we find a constant $a_2(E)>0$ [*independent of*]{} $\vphi$ such that $|m(\vphi,\l)|\les a_2(E)\cdot|\l|$ for all $\l\in\cal X_*(T)$.
Back to the proposition: $$\begin{array}{lr}
&a_2(E)\cdot|\l_0|\ges m(\vphi,\l_0)\ges 1
\;\Rightarrow\;
|\l_0|\ges\frac{1}{a_2(E)}.
\\[2ex]
\text{Hence:}\,
&
-l+m\cdot \langle \chi_{{\rm ac}},\l_0\rangle
\ges
-l+m\cdot a_1|\l_0|
\ges
-l+m\cdot \frac{a_1}{a_2(E)}.
\end{array}$$ We conclude that for $\frac{m}{l}>\frac{a_2(E)}{a_1}$ the condition $(C)$ is satisfied.
‘$\subset$’ We prove that $$\bigl(\mbb E\times\mbb V^{m}\bigr)^{{\rm us}}(K^\times\times G,l\chi_t+m\chi_{{\rm ac}})
\supset
\bigl(\mbb E\sm\{0\}\bigr)\times \bigl(\mbb V^{m}\bigr)^{{\rm us}}(G,\chi_{{\rm ac}})
\;\text{ for }m\gg0.$$ The conclusion follows from the hypothesis $(\mbb V^{m})^{{\rm ss}}(G,\chi_{{\rm ac}})
=(\mbb V^{m})^{{\rm s}}(G,\chi_{{\rm ac}})$.
Recall from (\[fct-m\]) that $\unl v\!\in\!(\mbb V^{m})^{{\rm us}}(G,\chi_{{\rm ac}})$ if and only if $m(\unl v)\!<\!0$. The value $m(\unl v)$ is reached at the ‘worst’ destabilizing $\l\in\cal X_*(G)$ (see [@ke2]). For variable $m$, there are only finitely many combinatorial strata in $(\mbb V^{m})^{{\rm us}}(G,\chi_{{\rm ac}})$ ([*c.f.*]{} ), hence only finitely many possible values for $m(\unl v)$. It follows that $$-\mu:=\max\bigl\{
m(\unl v)\mid m\ges 1,\,\unl v\in(\mbb V^{m})^{{\rm us}}(G,\chi_{{\rm ac}})
\bigr\}<0.$$ Now consider $(\vphi,\unl v)\in
\bigl(\mbb E\sm\{0\}\bigr)\times \bigl(\mbb V^{m}\bigr)^{{\rm us}}(G,\chi_{{\rm ac}})$, and its worst destabilizing $\l\in\cal X_*(G)$. After possibly moving $\unl v$ by an element in $G$, we may assume that $\l\in\cal X_*(T)$. Then holds: $m(\unl v,\l)\ges 0$, and $\frac{\langle\chi_{{\rm ac}},\l\rangle}{|\l|}=m(\unl v)\les-\mu$.
We distinguish the following cases:
– If $m(\vphi,\l)=0\text{ resp.}>0$, then (A) and (B) imply that $(\vphi,\unl v)$ is $l\chi_t+m\chi_{{\rm ac}}$ unstable.
– If $m(\vphi,\l)<0$, we normalize $\l$ such that $m(\vphi,\l)=-1$. We claim that $l+m\langle\chi_{{\rm ac}},\l\rangle\les 0$ for $m$ large enough. Otherwise we deduce: $$\phantom{xxxxxxxxxx}
\begin{array}{l}
\mu|\l|\les|\langle\chi_{{\rm ac}},\l\rangle|<l/m
\\[1ex]
1=|m(\vphi,\l)|\les a_2(E)\cdot|\l|
\end{array}
\biggr\}\,\Rightarrow\,
\frac{m}{l}<\frac{a_2(E)}{\mu}.
\phantom{xxxxxxxxxx}$$
The nef vector bundles
======================
[\[sct:nef-vb\]]{}
In this section we define a finite set of ‘extremal’ nef vector bundles, which will be the building blocks of the exceptional sequences. We continue the notations of the previous section. Consider the following Weyl group invariant cone: $${\label{nef-cone}}
\crl N=\crl N(G,V):=\crl C_1\cap\ldots\cap\crl C_z
={(\L_1+\ldots+\L_z)}^\vee.$$ When $G$ is a torus, $\crl N$ is the nef cone of the quotient, which is a toric variety. In our context, $\crl N$ can be viewed as the nef cone of $\mbb V^{{\rm ss}}(T,\chi_{{\rm ac}})/T$. Its importance relies on the following:
[\[prop:nef-vb\]]{} We consider a $G$-module $V$ which has the following property: ${{\mbb V}}^{{\rm ss}}(G,\chi_{{\rm ac}})={{\mbb V}}^{{\rm s}}(G,\chi_{{\rm ac}})$. Let $E$ be a $G$-module, and $\cal E\rar Y$ be its associated vector bundle.
Then $\cal E$ is nef if and only if all the weights of $T\!$ on $E\!$ belong to the cone $\crl N\!$. We call a module with this property [*a nef module*]{}.
($\Leftarrow$) Let us assume that the weights of $E$ belong to $\crl N$. We prove that, on $\mbb P(\cal E^\vee)$, the class $\chi_t$ is nef, it means $\chi_t+r\chi_{{\rm ac}}$ is ample for all $r>0$. This translates into the following condition: $$\bigl(\mbb E^\vee\times\mbb V\bigr)^{{\rm s}}(K^\times\times G,\chi_t+r\chi_{{\rm ac}})
=\bigl(\mbb E^\vee\sm\{0\}\bigr)\times
\mbb V^{{\rm s}}(G,\chi_{{\rm ac}}),
\quad\forall r>0.$$ ‘$\supset$’ The conditions (A) and (B) are trivially satisfied. We show that the case (C) does not occur.
Take $(\psi,v)\in(\mbb E^\vee\sm\{0\})\times\mbb V^{{\rm s}}(G,\chi_{{\rm ac}})$, and suppose that there is $\l_0$ with $m(\psi,\l_0)\!\ges 1$ and $m(v,\l_0)\!\ges 0$. Then $
\l_0\!\in{\rm int.}(\crl C_\psi)^\vee
\!\subset\!
-{\rm int.}\crl N^\vee
$ and also $\l_0\!\in\crl C_v^\vee\!\subset\!\crl N^\vee$. Contradiction.
‘$\subset$’ For shorthand, we denote $\cal S_L$ resp. $\cal S_R$ the left- and the right-hand-side above. Note that the quotient $\cal S_R\bigr/K^\times\kern-.7ex\times G$ exists, and equals $\mbb P(\cal E^\vee)$; let $Z:=\cal S_L/(K^\times\!\times G)$ be the quotient.
By previous step, there is a morphism $\phi:\mbb P(\cal E^\vee)\rar Z$. Since $\phi$ is open and $\mbb P(\cal E^\vee)$ is projective, $\phi$ is surjective. Recall from [@mfk Theorem 1.10], that $K^\times\!\times G$ acts with closed orbits on $\cal S_L$, and the quotient $\cal S_L\rar Z$ is geometric. Since $\mbb P(\cal E^\vee)\rar Z$ is surjective, the inclusion $\cal S_L\supset\cal S_R$ must be an equality. Otherwise we find closed orbits in $\cal S_L$, which are not contained in $\cal S_R$.
($\Rightarrow$) Assume that $\cal E\rar Y$ is nef, that means $\chi_t$ is a nef class on $\mbb P(\cal E^\vee)$. By inspecting the conditions we deduce: $$\raise.15ex\hbox{$\not$}\exists\,\psi\in\mbb E^\vee\sm\{0\},\;
v\in{{\mbb V}}^{{\rm s}}(G,\chi_{{\rm ac}}),\;
\l\in\cal X_*(T)\text{ s.t. }
\left\{\begin{array}{l}
m(\psi,\l)\ges 1,\\
m(v,\l)\ges 0.
\end{array}\right.$$ We choose $\psi=\vphi^\vee$, with $\vphi\in\mbb E$ of weight $\veps$. The previous condition implies: $\;\raise.15ex\hbox{$\not$}\exists\,
\l\in\cal X_*(T)$ such that $\langle\veps,\l\rangle<0$, and $\l\in\bigl(-\mbb R_+\veps+\crl N\bigr)^\vee.$ This happens only for $\veps\in\crl N$.
There is also an effective procedure to produce ‘the smallest’ such modules. Let us consider the set of weights: $${\label{n1}}
\crl N_1=\crl N_1(G,V):=
\left\{
\xi\,\biggl|\,
\begin{array}{l}
\mbb R_+\xi\text{ is an extremal ray of }\crl N,\\
\xi\text{ generates }\mbb R_+\xi\cap\cal X^*(T)
\text{ over }\mbb Z_{\ges0}
\end{array}
\right\}.$$ It is a Weyl-invariant set, and therefore it makes sense considering the irreducible $G$-modules whose dominant weights belong to $\crl N_1$. These modules will be the building blocks for constructing exceptional sequences. We denote $${\label{eqn:nef-vb}}
{{\cal V\kern-.41ex\euf B}}^+(Y):=
\left\{
E
\;\biggl|\;
\begin{array}{l}
\text{the dominant weight of the $G$-module $E$}
\\
\text{belongs to }\crl N_1
\end{array}\biggr.
\right\}.$$ Equivalently, denote $\mfrak W_G^+$ the closure of the positive Weyl chamber of $G$. Then ${{\cal V\kern-.41ex\euf B}}^+(Y)$ can be identified with $$\crl N_1^+(G,V):=\mfrak W_G^+\cap\crl N_1(G,V).$$
The set ${{\cal V\kern-.41ex\euf B}}^+(Y)$ is finite. For any $E\in{{\cal V\kern-.41ex\euf B}}^+(Y)$, the weights of $T$ on $E$ belong to the cone $\crl N$.
As $\crl N_1$ is finite, ${{\cal V\kern-.41ex\euf B}}^+(Y)$ is the same. Let $\xi$ be the dominant weight of $E$. The weights of $T$ on $E$ belong to the convex hull of the images of $\xi$ under the Weyl group. But all of them generate rays of $\crl N\!$. Hence the convex hull of the images of $\xi$ is contained in $\crl N\!$.
[\[prop:+comb\]]{} Let $M$ be an irreducible, nef $G$-module. Then there are $E_1,\ldots,E_n\in{{\cal V\kern-.41ex\euf B}}^+(Y)$, and $c_1,\ldots,c_n\ges 1$ such that $M\!\subset\!
\mbox{$\overset{n}{\underset{j=1}\bigotimes}$}{\mathop{\rm Sym}\nolimits}^{c_j}E_j.$ We say that $M$ is [*a positive combination*]{} of extremal nef modules.
Since the $G$-module $M$ is nef, its highest weight $\xi_M$ belongs to the cone $\crl N$. Then $\xi_M$ is a positive combination of $\xi_1,\ldots,\xi_n\in\crl N_1\,$:\
$\xi_M=\mbox{$\overset{n}{\underset{j=1}\sum}$}c_j\xi_j,\;c_j\ges 1.$
\
Each $\xi_j$ is conjugated to some $\xi^+_j\in\crl N_1^+$, since the Weyl group acts transitively on the Weyl chambers. The irreducible $G$-module $E_j$ with highest weight $\xi^+_j$ belongs to ${{\cal V\kern-.41ex\euf B}}^+(Y)$. Now observe that $\xi_M$ appears among the weights of $\overset{n}{\underset{j=1}\bigotimes}{\mathop{\rm Sym}\nolimits}^{c_j}E_j$. Hence the whole module $M$ is contained in it.
[\[lm:chi\]]{} Consider the set ${{\cal V\kern-.41ex\euf B}}^+(Y)$ of extremal nef vector bundles on $Y$, defined in . Then the anti-canonical character $\chi_{{\rm ac}}(G,V)$ is a positive linear combination of ${{\rm det}}E$, with $E\in{{\cal V\kern-.41ex\euf B}}^+(Y)$: $$\chi_{{\rm ac}}=\mbox{$\underset{E\in{{\cal V\kern-.41ex\euf B}}^+(Y)}\sum$}\kern-1ex
m_{E}\!\cdot{{\rm det}}(E),\quad\text{with }m_E\ges 0.$$
Let $\{\xi_j\}_j$ be the elements of $\crl N_1$. Since $\chi_{{\rm ac}}$ belongs to the interior of $\crl N$, there are positive numbers $c_j$ such that $
\chi_{{\rm ac}}\!=\!\underset{j}\sum c_j\xi_j
\!=\!\underset{j}\sum c_j\xi_j^\circ
+\underset{j}\sum c_j\xi_j'.
$ We decompose $\cal X^*(T)_{\mbb Q}
\!=\!\cal X^*(Z(G)^\circ)_{\mbb Q}\oplus\cal X^*(T')_{\mbb Q}$. Accordingly, each $\xi_j$ decomposes into $\xi_j=\xi_j^\circ+\xi_j'$, and each $\xi_j$ is conjugated to some $\xi_j^+\in\crl N_1^+$. Let $E_j\in{{\cal V\kern-.41ex\euf B}}^+(Y)$ be the irreducible $G$-module with highest weight $\xi^+_j$. Note that $Z(G)^\circ$ acts on $E_j$ by the character $\xi_j^\circ$. Since $\chi_{{\rm ac}}$ is trivial on the semi-simple part of $G$, we deduce that $
\chi_{{\rm ac}}=\mbox{$\underset{j}\sum$} c_j\xi_j^\circ
=\mbox{$\underset{j}\sum$}
\frac{c_j}{\dim E_j}{{\rm det}}E_j.$
Cohomological properties of nef vector bundles
==============================================
[\[cohom-nef\]]{}
In section \[sct:nef-vb\] we have introduced the set of nef vector bundles associated to representations of $G$. In this section we are going to study their cohomological properties.
[\[thm:hq-nef\]]{} Let $E$ be a nef $G$-module. Then $H^q(Y,\cal E)=0$ for all $q>0$.
Using the projection formula, $H^q(Y,\cal E)=H^q(\mbb P(\cal E^\vee),\cal O_{\mbb P}(1))$, and $\cal O_{\mbb P}(1)\rar \mbb P(\cal E^\vee)$ is a nef line bundle. The vanishing of the latter cohomology group is a consequence of the Hochster-Roberts theorem (see [@ke]).
We place ourselves in the following framework: $${\label{rel-situation}}
\left\{\text{
\begin{minipage}{27.5em}
(i) There is a quotient group $H$ of $G$ with kernel $G_0$
(note that $G_0$ and $H$ are still reductive),
and a quotient $H$-module $W$ of $V$ with kernel $V_0$, such that the
natural projection
${\mathop{\rm pr}\nolimits}^{{{\mbb V}}}_{\mbb W}\!:\!{{\mbb V}}\!\rar\!\mbb W$ has the property
${\mathop{\rm pr}\nolimits}^{{{\mbb V}}}_{\mbb W}\bigl(\;{{\mbb V}}^{{\rm ss}}\bigl(G,\chi_{{\rm ac}}(G,V)\bigr)\;\bigr)
\subseteq\mbb W^{{\rm ss}}\bigl(H,{\chi_{{\rm ac}}}(H,W)\bigr).$\break
We denote $Y\srel{\phi}{\rar}X$ the induced morphism.
\\[1.5ex]
(ii) Both unstable loci have codimension at least two.
\\[1.5ex]
(iii) $G$ and $H$ act freely on ${{\mbb V}}^{{\rm ss}}\bigl(G,\chi_{{\rm ac}}(G,V)\bigr)$
and \phantom{MMM}
$\mbb W^{{\rm ss}}\bigl(H,{\chi_{{\rm ac}}}(H,W)\bigr)$ respectively.
\end{minipage}
}\right.$$ Now let us study the positivity properties of direct images of nef vector bundles.
[\[lm:phiE\]]{} Suppose that we are in the situation , and that $E$ is a $G$-module such that its associated vector bundle $\cal E\rar Y$ is nef. Then $\phi_*\cal E\rar X$ is a vector bundle, and it is associated to the $H$-module $\phi_*E:={\mathop{\rm Mor}\nolimits}({{\mbb V}}_0,E)^{G_0}=H^0({{\mbb V}}_0{{\slash\kern-0.65ex\slash}}_{\chi_{{\rm ac}}}G_0,\cal E)$.
The restriction of $\cal E$ to the fibres of $\phi$ are nef. By applying theorem \[thm:hq-nef\], we obtain that $R^q\phi_*\cal E=0$ for all $q>0$, and therefore $\phi_*\cal E\rar X$ is locally free. Observe that both $V_0$ and $H$ are actually $G$-modules, and $V=V_0\oplus W$; the kernel $G_0$ is acting trivially on $W$. For an $H$-invariant open set $O\subset\mbb W$, holds:
$
\!\begin{array}[b]{ll}
H^0(O{{\slash\kern-0.65ex\slash}}H,\phi_*\cal E)
&=
H^0\bigl((\mbb V_0\times O){{\slash\kern-0.65ex\slash}}G,\cal E\bigr)
={\mathop{\rm Mor}\nolimits}\bigl(\mbb V_0\times O,E\bigr)^G\!
\\
&=
\kern-.4ex{\bigl({\mathop{\rm Mor}\nolimits}(\mbb V_0\times O,E)^{G_0}\bigr)}^H
\kern-.8ex=\!
{{\mathop{\rm Mor}\nolimits}\bigl(O,{\mathop{\rm Mor}\nolimits}(\mbb V_0,E)^{G_0}\bigr)}^H
\kern-.5ex.
\end{array}
$
[\[thm:direct-image\]]{} Assume that [($\!$]{}\[rel-situation\][)]{} holds, and let $E$ be a nef $G$-module. Then the $H$-module $\phi_*E$ is still nef. (The direct image $\phi_*\cal E\rar X$ is a nef vector bundle.)
Mourougane proves in [@mou] a similar statement for adjoint bundles. The proof below follows [*ad litteram*]{} his proof ([*loc.cit.*]{} section 3), with the necessary changes.
By lemma \[lm:phiE\], $\phi_*\cal E\rar X$ is locally free.
: Construct the tensor powers $(\phi_*\cal E)^{\otimes n}$.\
Let $Y^{(n)}=Y\times_X\ldots\times_XY$ be the fibre product, and $\phi^{(n)}:Y^{(n)}\rar X$ be the projection. Note that the vector bundle $\cal E^{(n)}:=\cal E\times_X\ldots\times_X\cal E$ on $Y^{(n)}$ is nef. Its direct image is $\phi^{(n)}_*\cal E^{(n)}=(\phi_*\cal E)^{\otimes n}$. Moreover, $Y^{(n)}$ is the quotient of the affine space $\mbb V^{(n)}$ by the action of the group $G^{(n)}$, and $\cal E^{(n)}$ is associated to the $G^{(n)}$-module $E^{\oplus n}$:
– $V^{(n)}:=
\{(v_1,\ldots,v_n)\in V^{\oplus n}\mid {\mathop{\rm pr}\nolimits}^V_W(v_1)=\ldots={\mathop{\rm pr}\nolimits}^V_W(v_n)\};
$
– The group $G^{(n)}:=G\times_H\ldots\times_HG$ is still reductive.
: Let $A\rar X$ be a very ample line bundle, associated to some character of $H$. Then $(\phi_*\cal E)^{\otimes n}\otimes A^{\dim X+1}$ is globally generated.\
We replace $Y$ by $Y':=Y^{(n)}$, $\phi$ by $\phi':=\phi^{(n)}$, and $\cal E$ by $\cal E':=\cal E^{(n)}$.
By the Castelnuovo-Mumford criterion, in order to prove that $\phi'_*\cal F\otimes A^{\dim X+1}$ is globally generated, it is enough to check that $H^q(X,\phi'_*\cal E'\otimes A^{\dim X+1-q})=0$ for all $q>0$. Since the higher direct images of $\cal E'$ vanish, the projection formula gives:
$
H^q(X,\phi'_*\cal E'\otimes A^{\dim X+1-q})=
H^q(Y',\cal E'\otimes {(\phi')}^*A^{\dim X+1-q}).
$
But $Y'$ is still a quotient of an affine space, $\cal E'$ is associated to a nef $G$-module, and ${(\phi')}^*A$ corresponds to a nef character of $G$. We apply theorem \[thm:hq-nef\] to $\cal E'\otimes {(\phi')}^*A^{\dim X+1-q}$, and deduce that its higher cohomology groups vanish.
: According to the previous step $(\phi_*\cal E)^{\otimes n}\otimes A^{\dim X+1}$ is globally generated for all $n>0$, and therefore $\phi_*\cal E$ is nef.
We use this result to describe more precisely the nef cone $\crl N(G,V)$. We consider the projective variety $${{\rm Flag}}(Y):=\O_G/B=\bigl(\O_G\times(G/B)\bigr)\bigr/G,$$ and denote $\pi:{{\rm Flag}}(Y)\rar Y$ the projection. It is a $G/B$-fibre bundle over $Y\!$, justifying the notation ${{\rm Flag}}(Y)$. For any $\xi\in\cal X^*(T)=\cal X^*(B)$, we denote by $\cal L_\xi\rar{{\rm Flag}}(Y)$ the line bundle $(\O_G\times K)/B$, where $B$ acts on $K$ by $\xi$.
[\[LE\]]{} Let $\xi\in\cal X^*(T)$ be a dominant character, and let $E_\xi$ be the corresponding irreducible $G$-module. Then holds:
1. $\cal E_\xi=\pi_*\cal L_\xi$;
2. $\cal E_\xi\rar Y$ is nef if and only if $\cal L_\xi\in{\mathop{\rm Pic}\nolimits}^+\bigl({{\rm Flag}}(Y)\bigr):=$ the nef cone of ${{\rm Flag}}(Y)$.
\(i) The equality is a direct consequence of the Borel-Weil theorem, which says that $H^0(G/B,\cal L_\xi)=E_\xi$.
\(ii) Assume that $\cal L_\xi$ is nef. The Borel-Weil theorem implies that the higher direct images $R^{>0}\pi_*\cal L_\xi=0$. By the same argument of the theorem \[thm:direct-image\], we deduce that $\cal E_\xi=\pi_*\cal L_\xi\rar Y$ is still nef.
Conversely, assume that $E_\xi$ is nef, hence ${{\mbb V}}^{{\rm ss}}(T,\chi_{{\rm ac}})\subset{{\mbb V}}^{{\rm ss}}(T,\xi)$. We claim that some tensor power of $\cal L_\xi$ is globally generated, and therefore $\cal L_\xi$ is nef. Let $B$ be the Borel subgroup of $G$ for which $\xi$ is dominant. Our hypothesis implies that $$\begin{array}{r}
{{\mbb V}}^{{\rm ss}}(G,\chi_{{\rm ac}})=
\mbox{$\underset{g\in G}\bigcap$}g{{\mbb V}}^{{\rm ss}}(T,\chi_{{\rm ac}})
\subset
\mbox{$\underset{b\in B}\bigcap$}b{{\mbb V}}^{{\rm ss}}(T,\chi_{{\rm ac}})
\subset
\mbox{$\underset{b\in B}\bigcap$}
b{{\mbb V}}^{{\rm ss}}(T,\xi)
\\=
{{\mbb V}}^{{\rm ss}}(B,\xi).
\end{array}$$ Observe that $B$ is solvable, [*not reductive*]{}, and therefore the standard invariant theory does not apply. The $B$-semi-stable locus ${{\mbb V}}^{{\rm ss}}(B,\xi)$ is defined exactly as in , in terms of the algebra $K[{{\mbb V}}]^{B,\xi}$. Its [*finite generacy*]{} has been proved by Grosshans (see [*e.g.*]{} [@gross Corollary 9.5]).
We deduce that for some $n>0$, ${{\mbb V}}^{{\rm ss}}(B,\xi)$ can be covered by a finite number of sets $\{y\mid f(y)\neq 0\}$, with $f\in K[{{\mbb V}}]^B_{\xi^n}$. Altogether, we find at each point $y\in{{\mbb V}}^{{\rm ss}}(G,\chi_{{\rm ac}})$ a function which is $(B,\xi^n)$-equivariant, and does not vanish at $y$. Hence $\cal L^n_{\xi}$ is globally generated.
[\[cor:+comb\]]{} Suppose that holds. Let $E$ be a nef $G$-module, and $M$ an irreducible $H$-submodule of $\phi_*E$. Then $M$ is a direct summand in a $H$-module of the form $\underset{F\in{{\cal V\kern-.41ex\euf B}}^+(X)}\bigotimes\kern-1.2ex{\mathop{\rm Sym}\nolimits}^{c_F}F.$
The push-forward $\phi_*\cal E\rar X$ is nef, and therefore all its weights belong to the cone $\crl N(H,W)$. We deduce that $M$ is nef too, and the conclusion follows from proposition \[prop:+comb\].
Consider the Grassmannian $X:={{\rm Grass}}(K^m,d)$ of $d$-dimensional quotients, and denote $\cal Q$ the tautological quotient on it. Note that the variety ${{\rm Flag}}(X)$ is the variety of full quotient flags of $\cal Q$. The cone $\mfrak W^+\cap{\mathop{\rm Pic}\nolimits}^+\bigl({{\rm Flag}}(X)\bigr)$ is generated by $d$ elements which correspond to the characters $\tau_1$, $\tau_1+\tau_2$,$\ldots$,$\tau_1+\ldots+\tau_d$ (here the $\tau_j$’s denote the obvious characters of the maximal torus in ${{\rm Gl}}(d)$).
We deduce that for any nef ${{\rm Gl}}(d)$-module $F$, its associated vector bundle $\cal F\rar {{\rm Grass}}(K^m,d)$ is a direct summand in a tensor product of the form
${\mathop{\rm Sym}\nolimits}^{c_1}(\cal Q)\otimes{\mathop{\rm Sym}\nolimits}^{c_2}(\overset{2}\bigwedge\cal Q)
\otimes\ldots\otimes{\mathop{\rm Sym}\nolimits}^{c_d}(\overset{d}\bigwedge\cal Q)$.
This is in agreement with the fact that this tensor product contains the Schur power $\mbb S^{\alpha}\cal Q$, where $\alpha=(\alpha_1\ges\ldots\ges\alpha_{d}\ges0)$, and the positive integers $c_j$ satisfy $\alpha_j=c_j+\ldots+c_d$ for $j=1,...,d$.
The main result: the absolute case
==================================
[\[sct:main\]]{}
In this section we prove our first main result. We consider a $G$-module $V$, and the character $\chi_{{\rm ac}}=\chi_{{\rm ac}}(G,V)$. Assume that the codimension of the $\chi_{{\rm ac}}$-unstable locus is at least two, and $G$ acts freely on the semi-stable locus. It follows that $Y:={{\mbb V}}{{\slash\kern-0.65ex\slash}}_{\chi_{{\rm ac}}}G={{\mbb V}}^{{\rm ss}}(G,\chi_{{\rm ac}})/G$ is a projective Fano variety. Observe that lemma \[lm:effective\] implies that $\chi_{{\rm ac}}=\chi_{{\rm ac}}(G,V)$ is effective as soon as $m_\og > d_\og$ for all $\og\in\cal X$ (the result below does not require this hypothesis).
We define a Young diagram $\l$ of length $d$ to be an array of decreasing integers $(\l_1\ges\!\ldots\!\ges\l_d)$. We denote $\l_{{\rm max}}:=\l_1$, $\l_{{\rm min}}:=\l_d$, ${\rm length}(\l):=d$. For arrays consisting of positive integers, we visualize the Young diagrams, and the parameters as in the figure:\
We introduce the following shorthand notation: for a Young diagram $\l$, let $\l\pm\fbox{$c$}$ be the diagram obtained by adding/subtracting the integer $c$ to/from the entries of $\l$. For a vector space $E$ and a Young diagram $\l$ of length $\dim E$, we will denote $\mbb S^\l E$ its usual Schur power (for $\l_{{\rm min}}\ges 0$), or $\mbb S^{\l-
\raise.3ex\hbox{\tiny$\begin{array}{|c|}
\hline\kern-1.5ex \l_{{\rm min}}\kern-1.5ex\\[.3ex] \hline
\end{array}$}
}\otimes({{\rm det}}E)^{\l_{{\rm min}}}$ (for arbitrary $\l$).
For two positive numbers $m,d$ we define the following sets: $$\begin{aligned}
\begin{array}{l}
\widetilde{\cal Y}_{d}:=
\text{ the set of Young diagrams $\l$ with }{\rm length}(\l)= d;
\\[2ex]
\cal Y_{m,d}:=
\bigl\{
\l\in\widetilde{\cal Y}_d\mid
0\les \l_{{\rm min}}\les\l_{{\rm max}}\les m
\bigr\};
\\[2ex]
\cal Y_{d}:=\underset{m\ges 0}\bigcup\cal Y_{m,d}\,;
\quad
\cal Y^+_{d}:=
\underset{m\ges 0}\bigcup
\bigl\{
\l\in\cal Y_{m,d}\mid\l_d\ges{\rm length}
\bigl(
\l-\fbox{$\l_d$}
\,\bigr)\,
\bigr\}.
\end{array}\end{aligned}$$ Roughly speaking, our main result is that certain Schur powers of the extremal nef bundles on $Y$ form a strong exceptional sequence.
The main technical tool that will be used is the following cohomology vanishing theorem, proved by Manivel for Kählerian varieties (see [@ma]), and Arapura for projective ones (see [@ar]).
Next comes our first main result.
[\[thm:main\]]{} Let $V$ be a $G$-module such that $K[\mbb V]^T=K$. Assume that the unstable locus has codimension at least two, and that $G$ acts freely on ${{\mbb V}}^{{\rm ss}}(G,\chi_{{\rm ac}})$; we denote by $Y:={{\mbb V}}^{{\rm ss}}(G,\chi_{{\rm ac}})/G$ the quotient. We consider the order $<_{\mfrak l}$ defined in \[defn:order2\].
Let $E_1,\ldots,\kern-.1exE_N\!$ be the elements of ${{\cal V\kern-.41ex\euf B}}^+\!(Y\kern-.1ex)$,$\kern-.2ex$ and denote $d_j\!:=\!\dim E_j$. We write $\chi_{{\rm ac}}=\overset{N}{\underset{j=1}\sum}m_j\cdot{{\rm det}}(E_j)$, with $m_j\ges 0$ as in lemma \[lm:chi\], and assume that all the numbers $m_j$ are integers. Consider the set $$\begin{array}[t]{ll}
\euf{ES}(Y):=
&
\text{the set of all irreducible $G$-modules contained in}
\\[1ex]
&
\mbb S^{\l^{\bullet}}E_\bullet
\!:=
\mbb S^{\l^{(1)}} E_1
\otimes\ldots\otimes
\mbb S^{\l^{(N)}} E_N,
\text{ where }\l^{(j)}\!\in\!\cal Y_{m_j-d_j,d_j}.
\end{array}$$ Then the vector bundles $\cal E\rar Y$ associated to the modules $E\in\euf{ES}(Y)$ form a strong exceptional sequence over $Y$ w.r.t. the order $<_{\mfrak l}$.
The condition on $H^0({\mathop{\rm Hom}\nolimits}(\cal U',\cal U''))$ for two elements $\cal U',\cal U''\in\euf{ES}(Y)$ is implied by theorem \[thm:h000\]. It remains to prove the vanishing of the higher cohomology groups. First of all we observe that, by definition, the vector bundles $\cal{U', U''}$ are direct summands of $\mbb S^{\l^\bullet}\cal E_\bullet$. Therefore it is enough to prove that vanishing of $H^q\bigl(Y,
{\mathop{\rm Hom}\nolimits}(\mbb S^{\l^\bullet}\cal E,\mbb S^{\mu^\bullet}\cal E)
\bigr)$, $q>0$. Using the Littlewood-Richardson rules, we decompose $${\mathop{\rm Hom}\nolimits}(\mbb S^{\l^\bullet}\cal E_\bullet,\mbb S^{\mu^\bullet}\cal E_\bullet)
=
\mbox{
${\underset{\alpha^\bullet=(\alpha^{(1)},\ldots,\alpha^{(N)})}\bigoplus}$
}
\kern-2ex\mbb S^{\alpha^\bullet}\cal E_\bullet\,,$$ and observe that $\alpha^{(j)}\!=\!\bigl(m_j-d_j\ges
\alpha^{(j)}_1\ges\ldots\ges\alpha^{(j)}_{d_j}
\ges -m_j+d_j\bigr)$. For each direct summand holds: $$\begin{array}{ll}
H^q\bigl(
Y,\mbb S^{\alpha^\bullet}\cal E_\bullet
\bigr)
&=
H^q\biggl(
Y,\kappa_Y\otimes\mbox{$\overset{N}{\underset{j=1}\bigotimes}$}
\bigl(
\mbb S^{\alpha^{(j)}}\cal E_j\otimes {{\rm det}}(\cal E_j)^{m_j}
\bigr)
\biggr)
\\&
=
H^q\biggl(
Y,\kappa_Y\otimes\mbox{$\overset{N}{\underset{j=1}\bigotimes}$} \,
\mbb S^{\alpha^{(j)}+
\tiny\begin{array}{|c|}
\hline\kern-1.5ex m_j\kern-1.5ex\\ \hline
\end{array}
}
\cal E_j
\biggr).
\end{array}$$ Note that $
\alpha^{(j)}+
\small\begin{array}{|c|}
\hline m_j\\ \hline
\end{array}
=
\underbrace{
\alpha^{(j)}+\small\begin{array}{|c|}
\hline -\alpha^{(j)}_{d_j}+d_j-1\\ \hline
\end{array}}_{:=\bar\alpha^{(j)}}
\,+\,
\small\begin{array}{|c|}
\hline \alpha^{(j)}_{d_j}+m_j-d_j+1\\ \hline
\end{array},$ and $$\left\{\begin{array}{ll}
\bar\alpha^{(j)}_{d_j}=d_j-1\ges{\rm length}
\bigl(\bar\alpha^{(j)}-\small\fbox{$d_j-1$}
\,\bigr)
&\text{ and}
\\[1.5ex]
\bar a_j:=\alpha^{(j)}_{d_j}+m_j-d_j+1\ges 1.
\end{array}\right.$$ Since $E_1,\ldots,E_N$ are [*all*]{} the extremal nef bundles, it follows that the $A:=\overset{N}{\underset{j=1}\bigotimes}{{\rm det}}(\cal E_j)^{\bar a_j}$ is an ample line bundle over $Y$. The theorem cited above implies that the higher cohomology of $S^{\alpha^\bullet}\cal E_\bullet$ vanishes.
Assume that the $G$-module $V$ has the property that the multiplicities $m_\og > d_\og$ for all $\og\in\cal X$. Then the exceptional sequence constructed above is formed by semi-stable vector bundles.
It is an immediate consequence of the corollary \[thm:stab-bdl\].
[\[rmk:length\]]{} It is important to observe that $\kappa_Y^{-1}$ is ample, and it becomes increasingly positive as we increase the multiplicities $m_\og $ of the isotypical components of $V$. It follows that the effect of increasing the $m_\og $’s is that of [*simultaneously*]{} increasing the dimension of the quotient, and that of the length of the exceptional sequence. In other words, for our construction we will always have a [*lower bound*]{} for $$\frac{\text{length of exceptional sequence on $Y$}}
{\text{Euler characteristic of } Y}.$$ Compare this construction with the one discussed in subsection \[ssect:AH\].
The main result: the relative case
==================================
[\[sct:main2\]]{}
Theorem \[thm:main\] is too weak for fibred varieties. By applying it directly, one looses many terms of the exceptional sequences (see subsections \[ssect:kapranov\] and \[ssect-A3\]). The goal of this section is to address the relative case described in . The additional hypothesis which will be imposed in may look overabundant, but in many concrete cases they are naturally fulfilled (especially for quiver representations).
Denote $T_0$ and $T_H$ the maximal tori of $G_0$ and $H$ respectively. The exact sequence $1\!\rar\! G_0\!\rar\! G\!\rar\! H\!\rar\!1$ induces a natural splitting $\cal X^*(T)_{\mbb Q}=\cal X^*(T_0)_{\mbb Q}\oplus\cal X^*(T_H)_{\mbb Q}$.
We will denote by $\crl N(G_0,V_0)$ respectively $\crl N(H,W)$ the nef cones of the $G_0$-module $V_0$ and $H$-module $W$, corresponding to $\chi_{{\rm ac}}(G_0,V_0)={\chi_{{\rm ac}}(G,V)|}_{G_0}$ and $\chi_{{\rm ac}}(H,W)$.
Throughout this section we will assume: $${\label{rel-situation2}}
\left\{\text{
\begin{minipage}{27.5em}
(i) The situation described in \eqref{rel-situation} holds.
\\[1ex]
(ii) $\crl N(G,V)=\crl N(G_0,V_0)+\crl N(H,W)$.\\
(We use the shorthand notation $\crl N=\crl N_0+\crl N_H$.)
\\[1ex]
(iii) The maximal torus $T_0\subset G_0$ has exactly $\dim T_0$
weights\\ on $V_0$.
\end{minipage}
}\right.$$
Let us make a few comments related to the assumptions:
– The condition (ii) means that there is a partition
${{\cal V\kern-.41ex\euf B}}^+(Y)={{\cal V\kern-.41ex\euf B}}^+(X)\,\dot\cup\,{{\cal V\kern-.41ex\euf B}}^+(\text{fibre}).$
The set ${{\cal V\kern-.41ex\euf B}}^+(X)$ can always be viewed as a subset of ${{\cal V\kern-.41ex\euf B}}^+(Y)$ via the pull-back ${{\mbb V}}\srel{\phi}{\rar}\mbb W$. What we assume is that the ‘extremal’ nef bundles on the fibres extend to ‘extremal’ nef bundles on the whole $Y$. For shorthand, we will write ${{\cal V\kern-.41ex\euf B}}^+_0:={{\cal V\kern-.41ex\euf B}}^+(\text{fibre})$.
– $T_0$ has always at least $\dim T_0$ linearly independent weights on $V_0$. The assumption (iii) is equivalent to any of the following:
(iii$'$) For any $\xi\in\cal X^*(T_0)$, $\xi$ is $T_0$-nef on $V_0$ if and only if $\xi$ is $T_0$-effective on $V_0$;
(iii$''$) The quotient ${{\mbb V}}_0{{\slash\kern-0.65ex\slash}}T_0$ is a product of projective spaces.
Observe that by lemma \[lm:chi\], we can express
$
\begin{array}{l}
\chi_{{\rm ac}}(H,W)=\kern-1ex
\mbox{$\underset{F\in{{\cal V\kern-.41ex\euf B}}^+(X)}\sum$}\kern-1.9ex
m_F\,\cdot\,{{\rm det}}F\;\;(m_F\ges 0),
\quad\text{and}
\\[2ex]
\chi_{{\rm ac}}(G_0,V_0)=\kern-1ex
\mbox{$\underset{E\in{{\cal V\kern-.41ex\euf B}}^+_0}\sum$}\kern-.4ex
m_E\,\cdot\,{{\rm det}}E\;\;(m_E\ges 0).
\end{array}
$
[\[prop:tech\]]{} Assume that holds, and denote $d_F:=\dim F$, and $d_E:=\dim E$.
Suppose that $(a_E)_{E\in{{\cal V\kern-.41ex\euf B}}^+_0}$ and $(b_F)_{F\in{{\cal V\kern-.41ex\euf B}}^+(X)}$ are integers having the following property: for all $q>0$, and all Young diagrams $\alpha^{E}\in\wtld{\cal Y}_{d_E}$ resp. $\beta^{F}\in\wtld{\cal Y}_{d_F}$, such that $\alpha^{E}_{{\rm min}}\ges -a_E$ and $\beta^{F}_{{\rm min}}\ges -b_F$, holds: $$\begin{aligned}
{\label{eqn:a}}
H^q\biggl(
V_0{{\slash\kern-0.65ex\slash}}_{\chi_{{\rm ac}}(G_0,V_0)}G_0,
\mbox{$\underset{E\in{{\cal V\kern-.41ex\euf B}}^+_0}\bigotimes$}
\mbb S^{\alpha^{E}}\cal E
\biggr)=0,
\\
{\label{eqn:b}}
H^q\biggl(
X,\mbox{$\underset{F\in{{\cal V\kern-.41ex\euf B}}^+(X)}\bigotimes$}\kern-1ex
\mbb S^{\beta^{F}}\cal F
\biggr)=0.\hspace{5em}\end{aligned}$$ Then $H^q\biggl(
Y,\underset{F\in{{\cal V\kern-.41ex\euf B}}^+(X)}\bigotimes\kern-1.7ex
\phi^*\,\mbb S^{\beta^{F}}\cal F
\;\otimes
\underset{E\in{{\cal V\kern-.41ex\euf B}}^+_0}\bigotimes\kern-.5ex
\mbb S^{\alpha^{E}}\cal E
\biggr)=0$ for all $q>0$, and for all Young diagrams $\beta^{F}\in\wtld{\cal Y}_{d_F}$ and $\alpha^{E}\in\wtld{\cal Y}_{d_E}$ with $\beta^{F}_{{\rm min}}\ges -b_F$ and $\alpha^{E}_{{\rm min}}\ges -a_E$ respectively.
The condition is fulfilled for $a_E:=m_E-d_E$, $\forall E\in{{\cal V\kern-.41ex\euf B}}^+_0$.
The condition is fulfilled for $b_F:=m_F-d_F$, $\forall F\in{{\cal V\kern-.41ex\euf B}}^+(X)$.
$\!$(i) The hypothesis implies that the higher direct images of $\!\underset{E\in{{\cal V\kern-.41ex\euf B}}^+_0}\bigotimes\kern-1ex\mbb S^{\alpha^{E}}\cal E$ vanish. By using the projection formula we deduce: $$\begin{aligned}
H^q\biggl(
Y,\mbox{$\underset{F\in{{\cal V\kern-.41ex\euf B}}^+(X)}\bigotimes$}\kern-1.7ex
\phi^*\mbb S^{\beta^{F}}\cal F
\;\otimes
\mbox{$\underset{E\in{{\cal V\kern-.41ex\euf B}}^+_0}\bigotimes$}\kern-.5ex
\mbb S^{\alpha^{E}}\cal E
\biggr)\hspace{10em}
\\
=H^q\biggl(
X,\mbox{$\underset{F\in{{\cal V\kern-.41ex\euf B}}^+(X)}\bigotimes$}\kern-1.7ex
\mbb S^{\beta^{F}}\cal F
\otimes\;
\phi_*\biggl(
\mbox{$\underset{\;E\in{{\cal V\kern-.41ex\euf B}}^+_0}\bigotimes$}\kern-.5ex
\mbb S^{\alpha^{E}}\cal E
\biggr)\biggr).\end{aligned}$$ Let us write $\cal V^0:=
\underset{E\in{{\cal V\kern-.41ex\euf B}}^+_0}\bigotimes\kern-.5ex
\mbb S^{\alpha^{E}}\cal E$, and decompose it into the direct sum corresponding to the irreducible $G$-modules appearing in the tensor product: $\cal V^0=\bigoplus\cal V^0_j$. The cohomology group breaks up into the direct sum of the ‘smaller’ cohomology groups. For each component $\cal V^0_j$ there are two possibilities:
There is a weight of $T_0$ on $V^0_j$ which is not effective. In this case $\phi_*\cal V^0_j\!=\!0$ ([*c.f.*]{} theorem \[not-eff\]), and we discard it from the direct sum.
All the weights of $T_0$ on $V^0_j$ are effective. In this case the hypotheses (ii)+(iii) imply that the weights of $V^0_j$ are nef, and therefore $\cal V^0_j\rar Y$ is nef itself. Using theorem \[thm:direct-image\] and proposition \[prop:+comb\], we deduce that $\phi_*\cal V^0_j\rar X$ is nef, and is actually contained in $\underset{F\in{{\cal V\kern-.41ex\euf B}}^+(X)}\bigotimes\kern-1.5ex{\mathop{\rm Sym}\nolimits}^{c_F}\cal F$, with $c_F\ges 0$.
The Littlewood-Richardson rules imply that the tensor product
$\mbox{$\underset{F\in{{\cal V\kern-.41ex\euf B}}^+(X)}\bigotimes$}\kern-1.5ex
\mbb S^{\beta^{F}}\cal F\,\otimes
\mbox{$\underset{F\in{{\cal V\kern-.41ex\euf B}}^+(X)}\bigotimes$}\kern-1.5ex
{\mathop{\rm Sym}\nolimits}^{c_F}\cal F$
breaks up into the direct sum of $\underset{F\in{{\cal V\kern-.41ex\euf B}}^+(X)}\bigotimes\kern-1.5ex
\mbb S^{\bar\beta^{F}}\cal F$, with $\bar\beta^F_{{\rm min}}\ges\beta^F_{{\rm min}}+c_F\ges b_F$. By the hypothesis, their higher cohomology vanishes.
(ii$_1$) Note that ${\kappa_{Y/X}^{-1}\bigr|}_{\rm fibre}
\!=\!\underset{E\in{{\cal V\kern-.41ex\euf B}}_0^+}\sum\! m_E\cdot{{\rm det}}E$. Consider Young diagrams $(\alpha^E)_{E\in{{\cal V\kern-.41ex\euf B}}^+_0}$ with $\alpha^E_{{\rm min}}\ges d_E-m_E$ for all $E$. It holds: $$\begin{array}{ll}
\mbox{$\underset{E\in{{\cal V\kern-.41ex\euf B}}^+_0}\bigotimes$}\kern-1ex
\mbb S^{\alpha^{E}}\cal E\otimes\,\kappa_{Y/X}^{-1}
\biggr|_{\rm fibre}
&=
\mbox{$\underset{\;E\in{{\cal V\kern-.41ex\euf B}}^+_0}\bigotimes$}\kern-1ex
\bigl(
\mbb S^{\alpha^{E}}\cal E\otimes({{\rm det}}\cal E)^{m_E}
\bigr)
\biggr|_{\rm fibre}
\\
&=
\mbox{$\underset{E\in{{\cal V\kern-.41ex\euf B}}^+_0}\bigotimes$}\kern-1ex
{\mbb S^{{\alpha^{E}+
\tiny\begin{array}{|c|}
\hline\kern-1.5ex m_E\kern-1.5ex\\ \hline
\end{array}
}}\,\cal E}
\biggr|_{\rm fibre},
\end{array}$$ and ${\alpha^{E}+
\small\begin{array}{|c|}
\hline m_E\\ \hline
\end{array}
}
=
\underbrace{
\alpha^{E}+\small\begin{array}{|c|}
\hline -\alpha^{E}_{{{\rm min}}}+d_E-1\\ \hline
\end{array}}_{:=\bar\alpha^{E}}
\,+\,
\small\begin{array}{|c|}
\hline \alpha^{E}_{{{\rm min}}}+m_E-d_E+1\\ \hline
\end{array}\,$ with $$\left\{\begin{array}{l}
\bar\alpha^{E}_{{{\rm min}}}=d_E-1\ges{\rm length}
\bigl(\bar\alpha^{E}-\small\fbox{$d_E-1$}
\,\bigr)_{\,\mbox{,}}
\\[1.5ex]
\bar a_E:=\alpha^{E}_{{{\rm min}}}+m_E-d_E+1\ges 1.
\end{array}\right.$$ Manivel and Arapura’s theorem implies that $R^q\phi_*(\mbb S^{\alpha^\bullet}\cal E_\bullet)=0$, for all $q>0$.
(ii$_2$) Consider Young diagrams $(\beta^F)_{F\in{{\cal V\kern-.41ex\euf B}}^+(X)}$ with $\beta^F_{{\rm min}}\ges d_F-m_F$ for all $F$. Then holds: $$\mbox{$\underset{F\in{{\cal V\kern-.41ex\euf B}}^+(X)}\bigotimes$}\kern-2ex
\mbb S^{\beta^{F}}\cal F\otimes\,\kappa_X^{-1}
=\!\!
\mbox{$\underset{F\in{{\cal V\kern-.41ex\euf B}}^+(X)}\bigotimes$}\kern-2ex
\bigl(
\mbb S^{\beta^{F}}\cal F\otimes({{\rm det}}\cal F)^{m_F}
\bigr)
=\!\!
\mbox{$\underset{F\in{{\cal V\kern-.41ex\euf B}}^+(X)}\bigotimes\,$}\kern-2ex
{\mbb S^{{\beta^{F}+
\tiny\begin{array}{|c|}
\hline\kern-1.5ex m_F\kern-1.5ex\\ \hline
\end{array}
}}\,\cal F}.$$ We deduce the vanishing of the higher cohomology as in (ii$_1$).
[\[thm:main2\]]{} Assume that the conditions are satisfied, and that there are integers $(b_F)_{F\in{{\cal V\kern-.41ex\euf B}}^+(X)}$ which fulfill the property . Then the elements of the set ${\euf{ES}}(Y)$ defined below form a strong exceptional sequence of vector bundles over $Y$: $$\begin{aligned}
\begin{array}{ll}
{\euf{ES}}(Y):=
&
\text{all the direct summands, corresponding to irreducible}
\\
&\text{$G$-modules contained in }
\\[0.5ex] &
\phi^*\bigl(\mbb S^{\l^{\bullet}}\cal F_\bullet\bigr)
\otimes
\mbb S^{\nu^{\bullet}}\cal E_\bullet
:=
\phi^*\bigl(
\mbox{\kern-2ex$\underset{\tiny F\in{{\cal V\kern-.41ex\euf B}}^+(X)}\bigotimes$}
\kern-1.7ex\mbb S^{\l^{F}}\cal F\,
\bigr)
\otimes\kern-.2ex
\mbox{$\underset{E\in{{\cal V\kern-.41ex\euf B}}^+_0}\bigotimes$\,}
\kern-.5ex\mbb S^{\nu^{E}}\cal E,
\end{array}\end{aligned}$$ with $\l^F\in\cal Y_{b_F,\,d_F},$ and $\nu^{E}\in\cal Y_{m_E-d_E,\,d_E}$.
Moreover, it holds: $H^q\biggl(
Y,\underset{F\in{{\cal V\kern-.41ex\euf B}}^+(X)}\bigotimes\kern-1.7ex
\phi^*\,\mbb S^{\beta^{F}}\cal F
\,\otimes
\underset{E\in{{\cal V\kern-.41ex\euf B}}^+_0}\bigotimes\kern-.5ex
\mbb S^{\alpha^{E}}\cal E
\biggr)\!=\!0$ for all $q>0$, and all Young diagrams $\beta^{F}\in\wtld{\cal Y}_{d_F}$ and $\alpha^{E}\in\wtld{\cal Y}_{d_E}$, with $\beta^{F}_{{\rm min}}\ges -b_F$ and $\alpha^{E}_{{\rm min}}\ges -(m_E-d_E)$ respectively.
Let $\cal U'$ and $\cal U''$ be two elements of ${\euf{ES}}(Y)$. The condition on the $H^0({\mathop{\rm Hom}\nolimits}(\cal U',\cal U''))$ follows again from theorem \[thm:h000\].
It remains to prove the vanishing of $H^q({\mathop{\rm Hom}\nolimits}(\cal U',\cal U''))$, for $q\ges 1$. By using the Littlewood-Richardson rules, we deduce that ${\mathop{\rm Hom}\nolimits}(\cal U',\cal U'')$ is direct summand in $\mbox{$\underset{\alpha^\bullet,\beta^\bullet}\bigoplus$}
\phi^*\bigl(\mbb S^{\beta^{\bullet}}\cal F_\bullet\bigr)
\otimes\mbb S^{\alpha^{\bullet}}\cal E_\bullet,$ with $$\left\{\begin{array}{l}
\hskip2.75em b_F\ges\beta^F_{{\rm max}}\ges\beta^F_{{\rm min}}\ges-b_F,
\\[1ex]
m_E-d_E\ges\alpha^E_{{\rm max}}\ges\alpha^E_{{\rm min}}\ges-m_E+d_E.
\end{array}\right.$$ The conclusion of the theorem follows from proposition \[prop:tech\](ii$_1$).
An immediate consequence of the previous theorem is the following:
[\[cor:tower\]]{} Assume the following assumptions hold:
1. There is a sequence of quotients $G\!\rar\! G_1\!\rar\!\ldots\!\rar G_k\!\rar\! 1$, with $\Gamma_{j}\!:=\!{{\rm Ker}}(G_{j}\!\rar\! G_{j+1})$.
2. $V=W_1\oplus\ldots\oplus W_k$, where $W_j$ is a $G_j$-module for all $j$. We define $V_j:=W_j\oplus\ldots W_k$ for all $j$.
3. The projections ${\mathop{\rm pr}\nolimits}_j:V_j\rar V_{j+1}$ satisfy the conditions . The induced morphisms are denoted by $$\phi_j\!:\!\mbb V_{j}{{\slash\kern-0.65ex\slash}}_{\!\chi_{{\rm ac}}(G_{j},V_{j})} G_{j}
\rar \mbb V_{j+1}{{\slash\kern-0.65ex\slash}}_{\!\chi_{{\rm ac}}(G_{j+1},V_{j+1})} G_{j+1},\;
\text{for all }\,1\les j\les k-1.$$
Let us write $\chi_{{\rm ac}}(\Gamma_j,W_j)
=\kern-.9ex
\underset{E\in{{\cal V\kern-.41ex\euf B}}^+(\mbb W_j/\!/ \Gamma_j)}\sum\kern-3ex
m_{j,E}\cdot{{\rm det}}E$ ([*c.f.*]{} \[lm:chi\]), and denote ${{\cal V\kern-.41ex\euf B}}^+_j:={{\cal V\kern-.41ex\euf B}}^+(\mbb W_j/\!/ \Gamma_j)$.
Then the elements of the set ${\euf{ES}}(Y)$ defined below form a strong exceptional sequence of vector bundles over ${{\mbb V}}{{\slash\kern-0.65ex\slash}}H$: $$\begin{array}{ll}
{\euf{ES}}(Y):=
&
\text{all the direct summands, corresponding to irreducible}
\\
&
\text{$G$-modules contained in }
\ouset{j=1}{k}{\bigotimes}
\biggl(
\mbox{$\underset{\tiny E\in{{\cal V\kern-.41ex\euf B}}^+_j}\bigotimes$}
\mbb S^{\alpha^{j,E}}\cal E
\biggr),
\\[1ex] &
\text{with }\;\alpha^{j,E}\in\cal Y_{m_{j,E}-d_E\,,\,d_E}.
\end{array}$$ Assume moreover that the multiplicity condition in corollary \[cor:stab-bdl\] is fulfilled. Then $\euf{ES}(Y)$ consists of semi-stable vector bundles over $Y$.
Examples
========
[\[sct:expl\]]{}
In this section we are going to present a few particular cases, in order to illustrate the general discussion. We concentrate on quiver varieties because they are a source of infinitely many examples, and are also very convenient: for generic choices of the dimension vector, the semi-stability and stability concepts agree. Therefore the quotients which will appear are geometric, as we wish. Even more remarkably, the procedure of constructing exceptional sequences of vector bundles over quiver varieties is [*almost algorithmic*]{}.
Let $Q=(Q_0,Q_1,h,t)$ be a quiver, and $\unl d=(d_q)_{q\in Q_0}$ be a dimension vector. We adopt the following convention: suppose that $q,q'$ are two vertices, and there is (at least) one arrow from $q$ to $q'$; then we draw [*only one*]{} arrow $a$, and we denote by $m_a$ its [*multiplicity*]{} (that is how many times the arrow is repeated). In other words, we consider the group $G=\kern-.5ex
\underset{q\in Q_0}{\hbox{\Large$\times$}}\kern-.5ex{{\rm Gl}}(d_q)$, and the $G$-module $V=\underset{a\in Q_1}\bigoplus\kern-.5ex
{{\mathop{\rm Hom}\nolimits}(K^{d_{t(a)}},K^{d_{h(a)}})}^{\oplus m_a}\!$. The construction of exceptional sequences involves the following steps:
Compute the anti-canonical character: $$\begin{array}{rl}
\chi_{{\rm ac}}=&
\mbox{$\underset{a\in Q_1}\sum$} m_a\cdot
\bigl(
d_{t(a)}{{\rm det}}_{h(a)}-d_{h(a)}{{\rm det}}_{t(a)}
\bigr)
\\[2ex]
=&
\mbox{$\underset{q\in Q_0}\sum$}
\biggl(
\mbox{$\underset{a\in Q_1^{\rm in}(q)}\sum$}
\kern-1ex m_ad_{t(a)}
-
\mbox{$\underset{a\in Q_1^{\rm out}(q)}\sum$}
\kern-1ex m_ad_{h(a)}
\biggr)\cdot{{\rm det}}_q.
\end{array}$$ Note that the multiplicative group, embedded diagonally in $G$, acts trivially on $V$, and the quotient $G/{(K^\times)}_{\rm diag}$ acts effectively on $V$. Moreover, for generic choices of the multiplicities $m_a$ (w.r.t. the dimension vector $\unl d$), the $\chi_{{\rm ac}}$-semi-stable locus of ${{\mbb V}}$ coincides with the stable locus (see [*e.g.*]{} [@king proposition 3.1]). For such a generic choice, there is a natural ‘Euler sequence’ over the quotient $Y$: $$\vspace{-1.5ex}
0
\lar
\cal O_Y^{^{\oplus\dim\hat G}}
\lar
\mbox{$\underset{a\in Q_1}\bigoplus$}
{\cal Hom\bigl(\cal E_{t(a)},\cal E_{h(a)}\bigr)}^{\oplus m_a}
\lar
T_Y
\lar 0.$$ It follows that the anti-canonical class of the quotient is $\kappa_Y^{-1}=\chi_{{\rm ac}}$.
It consists in determining the ‘extremal bundles’ in the set ${{\cal V\kern-.41ex\euf B}}^+(Y)$ (see ), and expressing $\chi_{{\rm ac}}$ as a positive combination of their determinants (see lemma \[lm:chi\]). Actually this step is responsible for the use of the word ‘almost’ above: the computation of the extremal nef bundles is algorithmic, but involves the maximal torus of $G$, and is therefore tedious.
Denote $\cal E_1,\ldots,\cal E_N$ the extremal bundles above, and take tensor products of their Schur powers $
\mbb S^{\l_1,\ldots,\l_N}\cal E\kern-.5ex
:=
\mbb S^{\l_1}\cal E_1\otimes\ldots\otimes\mbb S^{\l_N}\cal E_N.
$ The third step consists in determining the sizes of the Young diagrams $\l_1,\ldots,\l_N$ which fulfill the requirements of theorem \[thm:main\].
Search for fibrations coming from a sub-quiver. More precisely, we are looking for a sub-quiver $R\subset Q$ having the property: $$\begin{array}{rcr}
\forall\,(A_a)_{a\in Q_1}\in{{\mbb V}}^{{\rm ss}}\bigl(G,\chi_{{\rm ac}}(V)\bigr)
&\quad\Longrightarrow\quad&
(A_a)_{a\in R_1}\in\mbb W^{{\rm ss}}\bigl(H,\chi_{{\rm ac}}(W)\bigr),
\\[1.5ex]
G=\underset{v\in Q_0}\prod{{\rm Gl}}(v)
&&
H=\underset{v\in R_0}\prod{{\rm Gl}}(v).
\end{array}$$ Here $V$ and $W$ denote the representation spaces of $Q$ and $R$ respectively. In such a situation there is a natural projection map $Y\rar X$ between the corresponding quotients. Moreover, if $R$ is chosen appropriately, the numerous hypotheses in are naturally fulfilled.
Very often one obtains better bounds for the sizes of the Young diagrams involved in the Schur powers than those which are obtained by applying the step 3 directly (see subsections \[ssect:kapranov\] and \[ssect-A3\] below).
Kapranov’s construction
-----------------------
[\[ssect:kapranov\]]{} Let us start by reviewing Kapranov’s examples of tilting bundles over the Grassmannian, and over the flag variety for ${{\rm Gl}}(m)$. We show that by using our approach we automatically recover the vector bundles which appear in the tilting objects constructed by Kapranov over the Grassmannian, and over partial flag manifolds.
They are the quiver varieties associated respectively to: $$\entrymodifiers={++[o][F-]}
\xymatrix@+.7em@R=.9em{
m
\ar[r]^-{B}
&
*++[o][F=]{_{\phantom{i}}d_{\phantom{i}}}
&*\txt{}&*\txt{}
&*\txt{\kern-1ex with $m>d$.}
\\
m
\ar[r]^-{A_{k}}
&
*++[o][F=]{d_k}
\ar[r]^-{\!A_{{k-1}}}
&*\txt{$\;$\ldots}\ar[r]^-{A_{1}}&
*++[o][F=]{d_1}
&
*\txt{\kern-1ex with $m>d_k>\ldots>d_1$.}
}$$ A doubled circle means that the corresponding linear group acts at that entry (we have factored out the diagonal $K^\times$-action).
### The case of the Grassmannian
Let us consider the Grassmannian $Y:={{\rm Grass}}(\mbb C^m,d)$ of $d$-dimensional quotients of $K^m$. Its anti-canonical class is $\kappa_{{{\rm Grass}}(K^m,d)}^{-1}=({{\rm det}}\cal Q)^m$, where $\cal Q$ denotes the universal quotient bundle. The cone $\crl N$ is generated by the characters $t_1,\ldots,t_d$ of ${{\rm Gl}}(d)$, and $\crl N_1^+=\{t_1\}$. Hence the set ${{\cal V\kern-.41ex\euf B}}^+(Y)$ of extremal nef bundles ${{\cal V\kern-.41ex\euf B}}^+(Y)$ consists of $\cal Q$ only.
Theorem \[thm:main\] says that the elements of the set $\{\mbb S^\l\cal Q\mid \l\in\cal Y_{m-d,d}\}$ form a strong exceptional sequence of vector bundles on ${{\rm Grass}}(K^m\!,d)$. Indeed, this is what Kapranov proves in [*loc.cit.*]{}. Let us remark that he actually proves that they form a tilting sequence.
### The case of flag manifolds
We denote by $\mbb F_k:={{\rm Flag}}(K^m,d_k,\ldots,d_1)$ the variety of quotient $k$-flags of $K^m$. Let $\cal Q_1,\ldots,\cal Q_k$ be the tautological quotient bundles over $\mbb F_k$ with ${{\rm rank\,}}\cal Q_j\!=\!d_j$.
The anti-canonical class is $\kappa_{\mbb F_k}^{-1}
\!\!=\!\!
\hbox{$\overset{k}{\underset{j=1}\bigotimes}$}
{({{\rm det}}\cal Q_j)}^{d_{j+1}-\,d_{j-1}}\!\!.$ The cone $\crl N$ is generated by the characters $t^{(j)}_1,...,t^{(j)}_{d_j}$, $j=1,...,k$, and $\crl N_1^+=\{t^{(1)}_1,\ldots,t^{(k)}_1\}$. We deduce that ${{\cal V\kern-.41ex\euf B}}^+(\mbb F_k)=\{\cal Q_1,\ldots,\cal Q_k\}$. By applying theorem \[thm:main\] directly, we obtain that the elements of $$\left\{
\begin{array}{l}
\mbb S^{\l_\bullet}\cal Q_\bullet^\vee:=
\mbb S^{\l_k}\cal Q_k\otimes\ldots\otimes\mbb S^{\l_1}\cal Q_1,
\quad
\l_\bullet=(\l_k,\ldots,\l_1),
\\[1ex]
\text{with }\l_\bullet\in
\cal Y_{m-d_k-d_{k-1},d_k}
\times\ldots\times
\cal Y_{d_3-d_2-d_{1},d_2}\times\cal Y_{d_2-d_1,d_1}\!
\end{array}
\right\}$$ form a strong exceptional sequence over $\mbb F_k$. The problem is that these bounds are very weak, and this set can be empty!
At this point Step 4 becomes useful. There is a natural projection from the $k$-flag onto the $(k-1)$-flag variety $$\mbb F_k\srel{\phi}{\lar}\mbb F_{k-1},\quad
[A_k,\ldots,A_2,A_1]\lmt[A_k,\ldots,A_2].$$ One checks easily that all the conditions of are fulfilled. By applying corollary \[cor:tower\] we deduce that the elements of the set $$\left\{
\begin{array}{l}
\mbb S^{\l_\bullet}\cal Q_\bullet^\vee:=
\mbb S^{\l_k}\cal Q_k^\vee\otimes\ldots\otimes\mbb S^{\l_1}\cal Q_1^\vee
\\[1ex]
\text{with }\l_\bullet=(\l_k,\ldots,\l_1)
\in
\cal Y_{m-d_k,d_k}\times\ldots\times\cal Y_{d_2-d_1,d_1}
\end{array}
\right\}$$ form a strong exceptional sequence of vector bundles over $\mbb F_k$.
$A_3$-type quiver with multiple arrows
--------------------------------------
[\[ssect-A3\]]{} Interesting phenomena occur already for $A_3$-type quivers, as soon as we increase the multiplicities of the arrows. Consider the quiver $$\begin{array}{l}
\entrymodifiers={++[o][F-]}
\xymatrix@+.9em{
m
\ar[rr]^-{B}
&*\txt{}&
*++[o][F=]{d_2}
\ar[rr]^-{{\mbb A}=(A_1,\ldots,A_{n})}
&*\txt{}&
*++[o][F=]{d_1}
&
*\txt{with $m>d_2>d_1$,}
}
\\[2ex]
V={(K^{d_2})}^{\oplus m}\oplus{{\mathop{\rm Hom}\nolimits}(K^{d_2},K^{d_1})}^{\oplus n},
\quad
G={{\rm Gl}}(d_1)\times{{\rm Gl}}(d_2).
\end{array}$$ Let $Y:={{\mbb V}}{{\slash\kern-0.65ex\slash}}_{\chi_{{\rm ac}}}G$ be the corresponding quiver variety. The flag variety ${{\rm Flag}}(K^m,d_2,d_1)$ corresponds to the case $n=1$. We denote the vector bundles over $Y$ associated to the $G$-modules $K^{d_1}$ and $K^{d_2}$ by $\cal E_1$ and $\cal E_2$ respectively. The anti-canonical character is $$\chi_{{\rm ac}}=nd_2\cdot{{\rm det}}_1+(m-nd_1)\cdot{{\rm det}}_2
\!=\!
n\cdot\bigl[
d_2\cdot{{\rm det}}{{\cal E}}_1+(r-d_1)\cdot{{\rm det}}{{\cal E}}_2
\bigr],
\; r:=\!\frac{m}{n}.$$ We are going to see that the effect of introducing the parameter $n$ is that of obtaining several types of quotients. Observe that for generic choices of $m$ and $n$, the semi-stable and the stable loci coincide; this happens for $$\begin{array}{c}
\text{gcd}(nd_2,m-nd_1)=\text{gcd}(nd_2,m-nd_1,md_2)=1.
\end{array}$$ For details about semi-stability criteria for quiver representations, the reader may consult [@king].
### Case $r\!>\!d_1$
The $\chi_{{\rm ac}}$-semi-stability condition for $(B,\mbb A)\!\in\!\mbb V$ is: $$\begin{aligned}
\begin{array}{l}
\left\{
\begin{array}[c]{l}
U_2\subset K^{d_2}\text{ and }U_1\subset K^{d_1}\text{ s.t. }
\mbb A(U_2)\subset U_1
\\
\dim(U_2)=d_2'\text{ and }\dim(U_1)=d_1'
\end{array}
\right\}
\\[3ex]
\hspace{3ex}\Longrightarrow
d_2d_1'+(r-d_1)d_2'\ges rd_2\text{ for }(d_2',d_1')\neq(d_2,d_1).
\end{array}\end{aligned}$$ The set of extremal nef vector bundles is ${{\cal V\kern-.41ex\euf B}}^+(Y)=\{\cal E_1,\cal E_2\}$, and the anti-canonical class is $\kappa_Y^{-1}=({{\rm det}}\cal E_2)^{m-nd_1}\otimes({{\rm det}}\cal E_1)^{nd_2}$. Theorem \[thm:main\] implies that the elements of the set $$\{
\mbb S^\l\cal E_1\otimes
\mbb S^\mu\cal E_2
\mid
\l\in\cal Y_{nd_2-d_1,d_1}\text{ and }
\mu\in\cal Y_{m-nd_1-d_2,d_2}
\}$$ form a strong exceptional sequences of vector bundles over $Y$.
We illustrate again the role of Step 4 described at the beginning of this section: by using an appropriate fibre bundle structure on $Y$, we will increase the number of elements in the exceptional sequence.
Observe that both $B$ and $\mbb A\in{\mathop{\rm Hom}\nolimits}(K^{d_2}\otimes K^n,K^{d_1})$ are surjective, for any $\chi_{{\rm ac}}$-semi-stable point $(B,\mbb A)$. Indeed: by inserting $d_1'=d_1$ we obtain $d_2'\ges d_2$, and by inserting $d_2'=d_2$ we obtain $d_1'\ges d_1$. It follows that there is a natural projection $\phi:\!Y\!\rar{{\rm Grass}}(K^m,d_2)$, whose fibres are isomorphic to ${{\rm Grass}}(K^{nd_2},d_1)$. The group ${{\rm Gl}}(m)\times{{\rm Gl}}(n)$ acts on $Y$, and the projection is equivariant for the ${{\rm Gl}}(m)$-action. However $Y$ is not the $2$-flag variety.
We observe that the projection $V\rar{\mathop{\rm Hom}\nolimits}(K^m\!,K^{d_2})$ fulfills the conditions , and moreover ${{\cal V\kern-.41ex\euf B}}^+\bigl({{\rm Grass}}(K^m\!,d_2)\bigr)=\{\cal E_2\}$, and ${{\cal V\kern-.41ex\euf B}}^+_0=\{\cal E_1\}$. Applying corollary \[cor:tower\] to $\phi$ we deduce that the elements of the following set form a strong exceptional set of vector bundles over $Y$: $$\begin{aligned}
\{
\mbb S^\l\cal E_1\otimes
\mbb S^\mu\cal E_2
\mid
\l\in\cal Y_{nd_2-d_1,d_1}\text{ and }
\mu\in\cal Y_{m-d_2,d_2}
\}.\end{aligned}$$
### Case $r\!<\!d_1$
The $\chi_{{\rm ac}}$-semi-stability condition for $(B,\mbb A)\!\in\!\mbb V$ is: $$\begin{aligned}
{\label{ss2}}
\begin{array}{l}
\left\{\kern-.5ex
\begin{array}[c]{l}
U_2\subset K^{d_2}\text{ and }U_1\subset K^{d_1}\text{ s.t. }
\mbb A(U_2)\subset U_1
\\
\dim(U_2)=d_2'\text{ and }\dim(U_1)=d_1'
\end{array}\kern-.5ex
\right\}
\\[3ex]
\hspace{3ex}
\Longrightarrow
\left\{\begin{array}{rll}
\rm (i)\;&
d_2d_2'-(d_1-r)d_1'\ges 0&\text{for }(d_2',d_1')\neq(0,0),
\\
\rm (ii)\;&
d_2d_1'-(d_1-r)d_2'\ges rd_2&\text{for }(d_2',d_1')\neq(d_2,d_1).
\end{array}\right.
\end{array}\end{aligned}$$ Now we determine the extremal nef vector bundles. Since $r-d_1<0$, the situation differs from the previous case; now we will have ${{\cal V\kern-.41ex\euf B}}^+(Y)=\bigl\{\cal E_2, \cal H\bigr\}$, with $\cal H:={\mathop{\rm Hom}\nolimits}(\cal E_2,\cal E_1)$. We express the anti-canonical class as a positive combination of the extremal bundles: $\kappa_Y^{-1}=({{\rm det}}\cal E_2)^m\otimes({{\rm det}}\cal H)^n$. Theorem \[thm:main\] implies that $$\begin{aligned}
\{\mbb S^\l\cal E_2\otimes\mbb S^\mu\cal H
\mid
\l\in\cal Y_{m-d_2,d_2}\text{ and }
\mu\in\cal Y_{n-d_1d_2,d_1d_2}
\}\end{aligned}$$ is a strong exceptional sequence of vector bundles over $Y$.
Let us interpret the result. We consider the sub-quiver formed by the two rightmost vertices, and let $W:={\mathop{\rm Hom}\nolimits}(K^{d_2}\otimes K^n,K^{d_1})$ be its representation space. The anti-canonical character is $\chi_{{\rm ac}}(W)=d_2{{\rm det}}_1-d_1{{\rm det}}_2$. The symmetry group which is acting (effectively) is $G/{(K^\times)}_{\rm diag}$. The condition (i) implies that if $(B,\mbb A)$ is $\chi_{{\rm ac}}$-semi-stable, then $\mbb A$ is $\chi_{{\rm ac}}(W)$-semi-stable. Hence there is a natural morphism $$Y\srel{\phi}{\lar} X:=
{\mathop{\rm Hom}\nolimits}(K^{d_2}\otimes K^n\!,K^{d_1}){{\slash\kern-0.65ex\slash}}_{\chi_{{\rm ac}}(W)}\,G,$$ which is a projective bundle, with fibre isomorphic to $\mbb P(K^{md_2})$. The conditions are fulfilled, and we may apply corollary \[cor:tower\]. However, in this case we do not improve the previous bound.
Altman and Hille’s examples
---------------------------
[\[ssect:AH\]]{} In the article [@ah] the authors present the following construction: consider a quiver $Q$ without oriented cycles, and a thin and faithful representation space $V$ of it. This means that the dimension vector of the representation space is $\unl d={(1)}_{q\in Q_0}$, and the symmetry group which is acting is the torus $T=\underset{q\in Q_0}\prod K^\times\bigl/{(K^\times)}_{\rm diag}$.
(Altmann, Hille) [ *Assume that ${{\mbb V}}^{{\rm ss}}(T,\chi_{{\rm ac}})={{\mbb V}}_{(0)}^{{\rm s}}(T,\chi_{{\rm ac}})$. Then the tautological line bundles ${(\cal L_q)}_{q\in Q_0}$ form an exceptional sequence over the toric variety $Y:={{\mbb V}}^{{\rm ss}}(T,\chi_{{\rm ac}})/T$.* ]{}
We wish to remark that this construction fits into a more general framework: we consider a quiver $Q=(Q_0,Q_1,h,t)$ without oriented cycles, and we fix a dimension vector $\unl d=(d_q)_{q\in Q_0}$; we denote $V$ the corresponding representation space. For $m\ges 1$, we denote $Q^{(m)}$ the quiver obtained from $Q$ by multiplying each arrow $m$ times. The representation space of $Q^{(m)}$ with dimension vector $\unl d$ is $V^m$, and the symmetry group which is acting is $G=\underset{q\in Q_0}\prod {{\rm Gl}}(d_q)\bigl/K^\times_{\rm diagonal}$.
[\[prop:general-AH\]]{} Assume that ${({{\mbb V}}^m)}^{{{\rm ss}}}(G,\chi_{{\rm ac}})={({{\mbb V}}^m)}^{{{\rm s}}}(G,\chi_{{\rm ac}})$, and denote $Y_m$ the quotient by the $G$-action. For $q\in Q_0$, we denote $\cal E_q$ the tautological bundle over $Y_m$, associated to $G\rar{{\rm Gl}}(d_q)$.
Then there is a constant $m(Q)\ges 1$ such that for all $m> m(Q)$, the set $\{\cal E_q\}_{q\in Q_0}$ is a strong exceptional sequence of vector bundles over $Y_m$ (with respect to an appropriate ordering). Moreover, these vector bundles are semi-stable.
For two vertices $p,q\in Q_0$, let $E_{pq}:={\mathop{\rm Hom}\nolimits}(E_p, E_q)$, and $\cal E_{pq}$ the associated vector bundle over $Y_m$, and let $e_{pq}:=\dim E_{pq}=d_pd_q$.
The condition on $H^0(Y_m,\cal E_{pq})$ follows from theorem \[thm:h000\]. It remains to prove the vanishing of the higher cohomology. We compute $H^{n-i}\bigl(Y_m,\cal E_{pq}\bigr)$, $n=\dim Y$, by using the relative duality for $\mbb P(\cal E_{qp})\srel{{\mathop{\rm pr}\nolimits}}{\rar}Y_m$; it equals: $$H^{(e_{pq}-1)+i}\bigl(\mbb P({{\cal E}}_{qp}),
{\mathop{\rm pr}\nolimits}^*(\kappa_{Y_m}\otimes({{\rm det}}{{\cal E}}_{pq})^{-1})
\otimes{{\cal O}}_{\mbb P({{\cal E}}_{qp})}(-e_{pq}-1)
\bigr)^\vee.$$ The Kodaira vanishing theorem implies that $H^j(Y_m,\cal E_{pq})$ vanishes for all $j\ges 1$, as soon as ${\mathop{\rm pr}\nolimits}^*(\kappa_{Y_m}^{-1}\otimes({{\rm det}}{{\cal E}}_{pq}))
\otimes{{\cal O}}_{\mbb P({{\cal E}}_{qp})}(e_{pq}+1)$ is ample over $\mbb P({{\cal E}}_{qp})$. By proposition \[prop:large-m\], there is a number $m_{pq}$ such that this property holds for all $m>m_{pq}$. Consider now $m(Q):=\max\{m_{qp},e_{pq}\mid p,q\in Q_0\}$.
The isotypical components of $V^m$ are ${\mathop{\rm Hom}\nolimits}(E_{t(a)},E_{h(a)})$, $a\in Q_1$. Note that $m>m(Q)$ implies $m>\dim{\mathop{\rm Hom}\nolimits}(E_p,E_q)$, and the semi-stability of the tautological bundles $\cal E_q$ follows from corollary \[cor:stab-bdl\].
We wish to point out the following shortcoming: in this construction the length of the exceptional sequence equals the number of vertices of $Q$, which is independent of the multiplicity $m$. It follows that for large $m$ this sequence is certainly [*not*]{} a tilting object for $Y_{m}$ (compare this with remark \[rmk:length\]).
[00]{}
K. Altmann, L. Hille, [*Strong exceptional sequences provided by quivers*]{}, Algebr. Represent. Theory [**2**]{} (1999) 1-17.
D. Arapura, [*A class of sheaves satisfying Kodaira’s vanishing theorem*]{}, Math. Ann. [**318**]{} (2000) 235-253.
L. Costa, R.M. Miró-Roig, [*Tilting sheaves on toric varieties*]{}, Math. Z. [**248**]{} (2004) 849-865.
F. Grosshans, [*Algebraic Homogeneous Spaces and Invariant Theory*]{}, Lect. Notes Math. [**1673**]{}, Springer, 1997.
L. Hille, M. Perling, [*A counterexample to King’s conjecture*]{}, Compos. Math. [**142**]{} (2006) 1507-1521.
M. Kapranov, [*On the derived categories of coherent sheaves on some homogeneous spaces*]{}, Invent. Math. [**92**]{} (1988) 479-508.
G. Kempf, [*The Hochster-Roberts theorem of invariant theory*]{}, Michigan Math. J. [**26**]{} (1979) 19-32.
G. Kempf, [*Instability in invariant theory*]{}, Ann. Math. [**108**]{} (1978) 299-316.
A. King, [*Moduli of representations of finite dimensional algebras*]{}, Quarterly J. Math. [**45**]{} (1994) 515-530.
A. King, [*Tilting bundles on some rational surfaces*]{}, unpublished preprint available at: http://www.maths.bath.ac.uk/$\sim$masadk/papers/.
L. Manivel, [*Vanishing theorems for ample vector bundles*]{}, Invent. Math. [**127**]{} (1997) 401-416.
Ch. Mourougane, [*Images Directes de Fibres en Droites Adjoints*]{}, Publ. RIMS Kyoto Univ. [**33**]{} (1997) 893-916.
D. Mumford, J. Fogarty, F. Kirwan, [*Geometric Invariant Theory*]{}, 3$^{\rm rd}$ edition, Springer-Verlag Berlin New York, 1994.
S. Ramanan, A. Ramanathan, [*Some remarks on the instability flag*]{}, Tôhoku Math. J. [**36**]{} (1984) 269-291.
|
---
abstract: 'Let $K$ be a complete, algebraically closed nonarchimedean valued field, and let $\varphi(z) \in K(z)$ have degree $d \ge 1$. We provide explicit bounds for the Lipschitz constants ${\operatorname{Lip}}_\Berk(\varphi)$, ${\operatorname{Lip}}_{\PP^1(K)}(\varphi)$, in terms of algebraic and geometric invariants of $\varphi$.'
address:
- |
Robert Rumely\
Department of Mathematics\
University of Georgia\
Athens, Georgia 30602\
USA
- |
Stephen Winburn\
Ally Corporation\
440 S. Church Street\
Charlotte, N.C. 28202\
USA
author:
- Robert Rumely
- Stephen Winburn
date: 'December 3, 2015'
title: The Lipschitz Constant of a Nonarchimedean Rational Function
---
Let $K$ be a complete, algebraically closed nonarchimedean valued field with absolute value $| \cdot |$ and associated valuation ${\operatorname{ord}}(\cdot)$. Write $\cO$ for the ring of integers of $K$, $\cO^\times$ for its group of units, $\fM$ for its maximal ideal, and $\tk$ for its residue field.
Let $\varphi(z) \in K(z)$ be a rational function with ${{\operatorname{deg}}}(\varphi) = d \ge 1$. The action of $\varphi$ on $\PP^1(K)$ extends canonically to an action on Berkovich projective line ${\operatorname{{\bf P}}}^1_K$. In [@F-RL2] Favre and Rivera-Letelier define a metric $d(x,y)$ on ${\operatorname{{\bf P}}}^1_K$, which induces the strong topology on ${\operatorname{{\bf P}}}^1_K$; if $\|x,y\|$ denotes the spherical metric on $\PP^1(K)$, then $d(x,y)$ restricts to $2 \|x,y\|$ on $\PP^1(K)$. The Lipschitz constant of $\varphi$ with respect to $d(x,y)$ is $${\operatorname{Lip}}_\Berk(\varphi) \ := \ \sup_{\substack{ x,y \in\, {\operatorname{{\bf P}}}^1_K \\ x \ne y }} \frac{d(\varphi(x),\varphi(y))}{d(x,y)} \ .$$ It is the only inexplicit term in Favre and Rivera-Letelier’s quantitative equidistribution theorem for dynamical small points ([@F-RL2], Theorem 7). One may also be interested in the Lipschitz constant of $\varphi$ on classical points, $${\operatorname{Lip}}_{\PP^1(K)}(\varphi)
\ := \ \sup_{\substack{ x,y \in \PP^1(K) \\ x \ne y }} \frac{\|\varphi(x),\varphi(y)\|}{\|x,y\|} \ .$$
The purpose of this paper is to bound ${\operatorname{Lip}}_{\PP^1(K)}(\varphi)$ and ${\operatorname{Lip}}_\Berk(\varphi)$ using algebraic and geometric ${\operatorname{GL}}_2(\cO)$-invariants of $\varphi$. Let $(F,G)$ be a [*normalized representation*]{} for $\varphi$, a pair of homogeneous polynomials $F(X,Y), G(X,Y) \in \cO[X,Y]$ of degree $d$, with at least one coefficient of $F$ or $G$ in $\cO^{\times}$, such that $[F:G]$ gives the action of $\varphi$ on $\PP^1(K)$. The absolute value of the resultant $|{\operatorname{Res}}(\varphi)| := |{\operatorname{Res}}(F,G)|$ is independent of the choice of normalized representation.
\[FirstCor\] Let $K$ be a complete, algebraically closed nonarchimedean field, and let $\varphi(z) \in K(z)$ have degree $d \ge 1$. Then $$\label{FirstCorBound}
{\operatorname{Lip}}_{\PP^1(K)}(\varphi) \le \frac{1}{|{\operatorname{Res}}(\varphi)|} \ , \quad
{\operatorname{Lip}}_\Berk(\varphi) \le \max\Big( \frac{d}{|{\operatorname{Res}}(\varphi)|} \, , \frac{1}{|{\operatorname{Res}}(\varphi)|^d} \, \Big)
\ .$$
[In particular,]{} ${\operatorname{Lip}}_{\PP^1(K)}(\varphi)$ and ${\operatorname{Lip}}_\Berk(\varphi)$ are uniformly bounded in terms of the proximity of $\varphi$ to the boundary of the parameter space $\Rat_d$.
We prove Theorem \[FirstCor\] by first bounding ${\operatorname{Lip}}_\Berk(\varphi)$ and ${\operatorname{Lip}}_{\PP^1(K)}(\varphi)$ in terms of geometric invariants of $\varphi$. Suppose $$\label{varphi_Factorization}
\varphi(z) \ = \ C \cdot \frac{\prod_{i=1}^N (z-\alpha_i)}{\prod_{j=1}^M (z-\beta_j)} \ ,$$ where $d = {{\operatorname{deg}}}(\varphi) = \max(M,N)$.
If the $\alpha_i, \beta_j$ are fixed but $|C| \rightarrow \infty$, then ${\operatorname{Lip}}_{\PP^1(K)}(\varphi), {\operatorname{Lip}}_\Berk(\varphi) \rightarrow \infty$. We introduce the [*Gauss Image Radius*]{} $\GIR(\varphi)$ as a geometric replacement for $C$: given $x \in {\operatorname{{\bf P}}}^1_K$, let $0 \le {\operatorname{diam}}_G(x) \le 1$ be its diameter with respect to the Gauss point $\zeta_G$ (see §1), and put $$\GIR(\varphi) \ := \ {\operatorname{diam}}_G(\varphi(\zeta_G)) \ .$$ Likewise, if $C$ is fixed, but a root approaches a pole, then ${\operatorname{Lip}}_{\PP^1(K)}(\varphi), {\operatorname{Lip}}_\Berk(\varphi) \rightarrow \infty$. Let the Root-Pole number $\RP(\varphi)$ be the minimal spherical distance between a zero and a pole of $\varphi$. We introduce the [*Ball-Mapping radius*]{} $B_0(\varphi)$ as a geometric replacement for $\RP(\varphi)$: for each $a \in \PP^1(K)$ and each $0 < r \le 1$, let $B(a,r)^- = \{z \in \PP^1(K) : \|z,a\| < r\}$, and let $\cB(a,r)^-$ be the smallest open connected subset of ${\operatorname{{\bf P}}}^1_K$ containing $B(a,r)^-$. It is known that $\varphi(\cB(a,r)^-)$ is either an open ball $\cB_Q(\vv)^-$, or is all of ${\operatorname{{\bf P}}}^1_K$. Define $$B_0(\varphi)
\ := \ \sup \{ \ 0 < r \le 1 \ : \text{ for all $a \in \PP^1(K)$, $\varphi(\cB(a,r)^-)$ is a ball } \} \ .$$
The number $\GIR(\varphi)$ can be readily computed from the coefficients of $\varphi$ (see Proposition \[GIRProp\]). We do not know how to determine $B_0(\varphi)$ in general; this seems an interesting problem. By Proposition \[GaussInequalitiesProp\] and Corollary \[ResultantBoundCor\] one has $B_0(\varphi) \ge \RP(\varphi) \ge |{\operatorname{Res}}(\varphi)|$. In Proposition \[B\_0RationalityProp\] we show that $B_0(\varphi) \in |K^{\times}|$ and is achieved by some ball $\cB(a,r)^-$.
Our main result is
\[MainThm0\] Let $K$ be a complete, algebraically closed nonarchimedean field, and let $\varphi(z) \in K(z)$ have degree $d \ge 1$. Then $$\label{MainBound0}
{\operatorname{Lip}}_\Berk(\varphi) \ \le \ \max\Big( \frac{1}{\GIR(\varphi) \cdot B_0(\varphi)^d} \, ,
\frac{d}{\GIR(\varphi)^{1/d} \cdot B_0(\varphi)} \Big) \ .$$
Combined with the inequality $B_0(\varphi) \ge \RP(\varphi)$, Theorem \[MainThm0\] implies the following bounds, which may be useful when a factorization of $\varphi$ in the form (\[varphi\_Factorization\]) is known: $$\label{SecondCorBound}
{\operatorname{Lip}}_\Berk(\varphi) \ \le \ \max\Big( \frac{1}{\GIR(\varphi) \cdot \RP(\varphi)^d} \, ,
\frac{d}{\GIR(\varphi)^{1/d} \cdot \RP(\varphi)} \Big)
\ \le \ \frac{d}{\GIR(\varphi) \cdot \RP(\varphi)^d} \ .$$
The bound in Theorem \[MainThm0\] is sharp when $d=1$, that is, when $\varphi(z)$ is a linear fractional transformation (see Theorem \[MobiusCase\] below). When $d \ge 2$, for each triple $\big(d,\GIR(\varphi),B_0(\varphi)\big)$ we give examples where ${\operatorname{Lip}}_\Berk(\varphi)$ is within a factor $(d-1)/d$ of the right side of (\[MainBound0\]) (see §\[ExamplesSection\]).
Trivially ${\operatorname{Lip}}_{\PP^1(K)}(\varphi) \le {\operatorname{Lip}}_\Berk(\varphi)$, however, there is an explicit formula for ${\operatorname{Lip}}_{\PP^1(K)}(\varphi)$ which yields a better bound. The set $\varphi^{-1}(\{\zeta_G\}) \subset {\operatorname{{\bf P}}}^1_K \backslash \PP^1(K)$ is finite; define the [*Gauss Pre-Image radius*]{} $$\GPR(\varphi) \ = \ \min_{\varphi(\xi) = \zeta_G} {\operatorname{diam}}_G(\xi) \ .$$
\[Classical\_Lip\] Let $K$ be a complete, algebraically closed nonarchimedean field, and let $\varphi(z) \in K(z)$ have degree $d \ge 1$. Then $${\operatorname{Lip}}_{\PP^1(K)}(\varphi) \ = \ \frac{1}{\GPR(\varphi)} \ . $$
In Corollary \[ResultantBoundCor\], we show that $\GIR(\varphi)^d \cdot B_0(\varphi) \ge |{\operatorname{Res}}(\varphi)|$, and that $\GPR(\varphi) \ge |{\operatorname{Res}}(\varphi)|$. Combining this with Theorems \[MainThm0\] and \[Classical\_Lip\] yields Theorem \[FirstCor\].
For a Möbius transformation, the bounds in Theorems $\ref{FirstCor}$, $\ref{MainThm0}$, and $\ref{Classical_Lip}$ are sharp, and can be made much more explicit:
\[MobiusCase\] Let $K$ be a complete, algebraically closed nonarchimedean field, and let $\varphi(z) = (az+b)/(cz+d) \in K(z)$ have degree $1$. Then $B_0(\varphi) = 1$, and $$\begin{aligned}
{\operatorname{Lip}}_\Berk(\varphi) & = & {\operatorname{Lip}}_{\PP^1(K)}(\varphi) \ = \ \frac{1}{\GIR(\varphi)} \ = \ \frac{1}{\GPR(\varphi)} \\
& = & \frac{1}{|{\operatorname{Res}}(\varphi)|} \ = \ \frac{\max(\,|a|,|b|,|c|,|d|\,)}{|ad-bc|} \ .\end{aligned}$$
The plan of the paper is as follows. In $\S1$ we recall facts and notation concerning the Berkovich projective line. In $\S2$ we establish some preliminary lemmas, showing that to bound ${\operatorname{Lip}}_\Berk(\varphi)$ it suffices to bound it on a restricted class of segments $[x,y]$. In $\S3$ we prove Theorem \[MobiusCase\]. In $\S4$ we study the constants $\GIR(\varphi)$, $B_0(\varphi)$, $\RP(\varphi)$, and $\GPR(\varphi)$, and we give a formula for $|{\operatorname{Res}}(\varphi)|$ which may be of independent interest. In $\S5$ we prove Theorems \[FirstCor\], \[MainThm0\] and \[Classical\_Lip\]. Finally, in $\S6$ we provide examples showing that Theorem \[MainThm0\] cannot be significantly improved.
We thank Xander Faber and Kenneth Jacobs for useful discussions. In particular we thank Jacobs for pointing out that Theorem \[MainThm0\] could yield bounds of the form (\[FirstCorBound\]).
The Berkovich Projective Line {#PreliminariesSection}
=============================
The Berkovich projective line over $K$ is a locally ringed space, functorially constructed from $\PP^1/K$, whose sheaf of rings comes from rigid analysis and whose underlying point set is gotten by gluing the Gel’fand spectra of those rings (see [@Ber]). We will write ${\operatorname{{\bf P}}}^1_K$ for its point set, which is a uniquely path-connected Hausdorff space containing $\PP^1(K)$. For proofs of the properties of ${\operatorname{{\bf P}}}^1_K$ discussed below, see ([@B-R], Chapters 1 and 2); for additional facts about ${\operatorname{{\bf P}}}^1_K$, see ([@BIJL], [@Ber], [@Fab], [@F-RL2], [@FRLErgodic], [@R-L1]).
Berkovich’s classification theorem (see ([@Ber], p.18), or ([@B-R], p.5)) provides an elementary model for ${\operatorname{{\bf P}}}^1_K$: its points correspond to discs $D(a,r) = \{z \in K : |z-a| \le r\}$, where $a \in K$ and $0 \le r \in \RR$, or to cofinal equivalence classes of sequences of nested discs with empty intersection, or to the point $\infty \in \PP^1(K)$. There are four kinds of points. Type I points, which are the points of $\PP^1(K)$, correspond to degenerate discs of radius $0$ in $K$ and the point $\infty \in \PP^1(K)$. Type II points correspond to discs $D(a,r)$ with $r$ in the value group $|K^{\times}|$, and type III points correspond to discs $D(a,r)$ with $r \notin |K^{\times}|$. Type IV points correspond to (cofinal equivalence classes of) sequences of nested discs with empty intersection; they serve to complete ${\operatorname{{\bf P}}}^1_K$ but rarely need to be dealt with explicitly: they are usually handled by continuity arguments.
We call ${\operatorname{{\bf A}}}^1_K = {\operatorname{{\bf P}}}^1_K \backslash \{\infty\}$ the Berkovich Affine Line.
We write $\zeta_{a,r}$ for the point corresponding to $D(a,r)$. The point $\zeta_G := \zeta_{0,1}$ corresponding to $D(0,1)$ is called the [*Gauss point*]{}, and plays a particularly important role.
Paths in ${\operatorname{{\bf P}}}^1_K$ correspond to ascending or descending chains of discs, or concatenations of such chains sharing an endpoint. For example the path from $0$ to $1$ in ${\operatorname{{\bf P}}}^1_K$ corresponds to the concatenation of the chains $\{D(0,r) : 0 \le r \le 1\}$ and $\{D(1,r) : 0 \le r \le 1\}$; here $D(0,1) = D(1,1)$. The point $\infty$, and type IV points, can also be endpoints of chains. Topologically, ${\operatorname{{\bf P}}}^1_K$ is a tree: for any two points $x, y \in {\operatorname{{\bf P}}}^1_K$, there is a unique path $[x,y]$ between $x$ and $y$. We will write $(x,y)$, $[x,y)$, and $(x,y]$ for the corresponding open or half-open paths.
Fix a point $\xi \in {\operatorname{{\bf P}}}^1_K$. The fact that ${\operatorname{{\bf P}}}^1_K$ is uniquely path connected means that for any two points $x, y \in {\operatorname{{\bf P}}}^1_K$, we can define the join $x \vee_\xi y$ to be the first point where the paths $[x,\xi]$ and $[y,\xi]$ meet. We will be particularly interested in the cases where $\xi = \infty$ and $\xi = \zeta_G$; we denote the corresponding joins by $x \vee_\infty y$ and $x \vee_G y$.
The fact that ${\operatorname{{\bf P}}}^1_K$ is uniquely path-connected also means that for each $Q \in {\operatorname{{\bf P}}}^1_K$, the path-components of ${\operatorname{{\bf P}}}^1_K \backslash \{Q\}$ have the property that for any $P_1, P_2$ in the same component, the paths $[Q,P_1]$ and $[Q,P_2]$ share an initial segment. Because of this, the components are in $1-1$ correspondence with germs of paths emanating from $Q$, which we call [*tangent vectors*]{} $\vv$ at $Q$. We write $T_Q$ for the set of tangent vectors at $Q$. If $Q$ is of type I or type IV, then $T_Q$ has one element; if $Q$ is of type III, $T_Q$ has two elements. If $Q$ is of type II, then $T_Q$ is in $1-1$ correspondence with $\PP^1(\tk)$, for the residue field $\tk = \cO/\fM$.
\[BallDef\] For each $\vv \in T_Q$, we write $\cB_Q(\vv)^-$ for the component of ${\operatorname{{\bf P}}}^1_K \backslash \{Q\}$ containing points for which $[Q,P]$ prolongs $\vv$. If $P \in \cB_Q(\vv)^-$, we say that $P$ is in the direction $\vv$ at $Q$. Given $P \ne Q, $ we write $\vv_P$ for the direction in $T_Q$ such that $P \in B_Q(\vv_P)^-$.
Let ${\operatorname{ord}}(x)$ be the additive valuation on $K$ corresponding to $|x|$; there is a unique base $q > 1$ for which ${\operatorname{ord}}(x) = -\log_q(|x|)$. We will write $\log(z)$ for $\log_q(z)$. The set ${\operatorname{{\bf H}}}^1_K = {\operatorname{{\bf P}}}^1_K \backslash \PP^1(K)$ is called the [*Berkovich upper halfspace*]{}; it carries a metric $\rho(x,y)$ called the [*logarithmic path distance*]{}, for which the length of the path corresponding to the chain of discs $\{D(a,r) : R_1 \le r \le R_2\}$ is $\log(R_2/R_1)$.
There are two natural topologies on ${\operatorname{{\bf P}}}^1_K$, called the [*weak*]{} and [*strong*]{} topologies. The basic open sets for the weak topology are the path-components of ${\operatorname{{\bf P}}}^1_K \backslash \{P_1, \ldots, P_n\}$ as $\{P_1, \ldots, P_n\}$ ranges over finite subsets of ${\operatorname{{\bf H}}}^1_K$. Under the weak topology, ${\operatorname{{\bf P}}}^1_K$ is compact, and $\PP^1(K)$ is dense in it. The weak topology is not in general defined by a metric. The basic open sets for the strong topology are the $\rho(x,y)$-balls $$\cB_\rho(x,r)^- \ = \ \{z \in {\operatorname{{\bf H}}}^1_K : \rho(x,z) < r\}$$ for $x \in {\operatorname{{\bf H}}}^1_K$ and $r > 0$, together with the basic open sets from the weak topology. Under the strong topology, ${\operatorname{{\bf P}}}^1_K$ is complete but not compact. The strong topology is induced by the Favre-Rivera Letelier metric $d(x,y)$, defined at (\[FRLdMetricDef\]) below. Type II points are dense in ${\operatorname{{\bf P}}}^1_K$ for both topologies. For each $\varphi(z) \in K(z)$, the action of $\varphi(z)$ on $\PP^1(K)$ extends functorially to an action on ${\operatorname{{\bf P}}}^1_K$ which is continuous for both the weak and strong topologies. If $\varphi$ is nonconstant, the action is open, surjective, and preserves the type of each point. The action of $\varphi(z) \in K(z)$ on type II points corresponds to its ‘generic’ action on punctured discs, in the following sense. Let $D(a,r)^- = \{z \in K: |z-a| < r\}$. One has $\varphi(\zeta_{a,r}) = \zeta_{b,R}$ if and only if there are finitely many points $a_1, \ldots, a_m \in D(a,r)$ and finitely many points $b_1, \ldots, b_m \in D(b,R)$ such that $$\varphi\big(D(a,r) \backslash \bigcup_{i=1}^m D(a_i,r)^-\big) \ = \ D(b,R) \backslash \bigcup_{j=1}^n D(b_j,R)^-$$ (see [@B-R], Proposition 2.8). For example, the inversion map $\iota(z) = 1/z$ satisfies $$\left\{ \begin{array}{ll}
\iota(D(0,r) \backslash D(0,r)^-) \ = \ D(0,1/r) \backslash D(0,1/r)^- \ , & \text{ }\\
\iota(D(a,r) \backslash D(a,r)^-) \ = \ D(1/a,r/|a|^2) \backslash D(1/a,r/|a|^2)^-
& \text{if $|a| > r$ \ .}
\end{array} \right.$$ Since a disc can be written as $D(0,r)$ if and only if $|a| \le r$, one has $$\iota(\zeta_{a,r}) \ = \ \left\{ \begin{array}{ll}
\zeta_{0,1/r} & \text{if $|a| \le r$ \ , } \\
\zeta_{1/a,r/|a|^2} & \text{if $|a| > r$ \ . }
\end{array} \right.$$
In particular, the action of ${\operatorname{GL}}_2(K)$ on $\PP^1(K)$ though linear fractional transformations extends to an action on ${\operatorname{{\bf P}}}^1_K$, which is transitive on type I points and type II points. The action of ${\operatorname{GL}}_2(K)$ preserves the logarithmic path distance: $\rho(\gamma(x),\gamma(y)) = \rho(x,y)$ for all $x, y \in \HH^1_K$ and $\gamma \in {\operatorname{GL}}_2(K)$. The stabilizer of $\zeta_G$ in ${\operatorname{GL}}_2(K)$ is $K^{\times} \cdot {\operatorname{GL}}_2(\cO)$.
If we identify $\PP^1(K)$ with $K \cup \{\infty\}$ and make the usual conventions for arithmetic operations involving $\infty$, the spherical metric on $\PP^1(K)$ is given by $$\|x,y\| \ = \ \left\{ \begin{array}{ll} |x-y| & \text{if $|x|, |y| \le 1$ \ ,} \\
|1/x-1/y| & \text{if $|x|, |y| > 1$ \ ,} \\
1 & \text{otherwise \ .} \end{array} \right.$$ For $x,y \ne \infty$, one has $\|x,y\| = \ |x-y|/\big(\max(1,|x|) \max(1,|y|)\big)$. For all $x,y \in \PP^1(K)$, and all $\gamma \in {\operatorname{GL}}_2(\cO)$, one has $\|\gamma(x),\gamma(y)\| = \|x,y\|$.
We will use two “diameter” functions on ${\operatorname{{\bf P}}}^1_K$. For $x \in {\operatorname{{\bf P}}}^1_K$, the diameter of $x$ with respect to the point $\infty \in \PP^1(K)$ is given by $${\operatorname{diam}}_\infty(x) \ = \ \left\{ \begin{array}{ll}
0 & \text{if $x \in K$\ , } \\
r & \text{if $x = \zeta_{a,r}$ is of type II or III \ ,} \\
\inf\{ r : \zeta_{a,r} \in (x,\infty) \} & \text{if $x$ is of type IV \ ,} \\
\infty & \text{if $x = \infty \in \PP^1(K)$\ . }
\end{array} \right. \label{diam_infty}$$ The function ${\operatorname{diam}}_\infty(x)$ is preserved by translations: for any $b \in K$, if $\gamma(z) = z + b$, then ${\operatorname{diam}}_\infty(\gamma(x)) = {\operatorname{diam}}_\infty(x)$.
The diameter with respect to the Gauss point $\zeta_G$ is defined by $$\label{diam_G}
{\operatorname{diam}}_G(x) \ = \ q^{-\rho(\zeta_G,x)} \ .$$ If $x = \zeta_{a,r}$ is a point of type II or III, one has $${\operatorname{diam}}_G(x) \ = \ \left\{ \begin{array}{ll} r & \text{if $|a|, |r| \le 1$} \\
r/|a|^2 & \text{if $|a| > 1$ and $r < |a|$} \\
1/r & \text{if $r > 1$ and $|a| \le r$} \end{array} \right.
\label{diam_G1}\\$$ Evidently $0 \le {\operatorname{diam}}_G(x) \le 1$, with ${\operatorname{diam}}_G(x) < 1$ if $x \ne \zeta_G$, and ${\operatorname{diam}}_G(x) = 0$ if and only $x \in \PP^1(K)$. For each $\gamma \in {\operatorname{GL}}_2(\cO)$, one has ${\operatorname{diam}}_G(\gamma(x)) = {\operatorname{diam}}_G(x)$. Moreover, ${\operatorname{GL}}_2(\cO)$ acts transitively on type I points, and, for a given $r \in |K^\times|$, on type II points with ${\operatorname{diam}}_G(x) = r$. If $a, b \in \PP^1(K)$, then $${\operatorname{diam}}_G( a \vee_G b) \ = \ \|a,b\| \ .$$
The Favre-Rivera Letelier metric $d(x,y)$ is defined by $$\label{FRLdMetricDef}
d(x,y) \ = \ \big({\operatorname{diam}}_G(x \vee_G y) - {\operatorname{diam}}_G(x)\big) + \big({\operatorname{diam}}_G(x \vee_G y) - {\operatorname{diam}}_G(y)\big) \ .$$ One has $0 \le d(x,y) \le 2$ for all $x, y$. To see that $d(x,y)$ is a metric, note that it is positive if $x \ne y$, and is clearly symmetric. It satisfies the triangle inequality $d(x,z) \le d(x,y) + d(y,z)$ because it is additive on paths: if $Q$ is any point in $[x,y]$, then $d(x,y) = d(x,Q) + d(Q,y)$.
\[FRLd\_Properties\] The metric $d(x,y)$ has the following properties:
$(1)$ $d(\gamma(x),\gamma(y)) = d(x,y)$ for each $\gamma \in {\operatorname{GL}}_2(\cO);$
$(2)$ $d(a,b) = 2 \|a,b\|$ for all $a,b \in \PP^1(K)$.
Assertion (1) follows from the definition of $d(x,y)$ and the fact that $K^\times \cdot {\operatorname{GL}}_2(\cO)$ stabilizes $\zeta_G$ and preserves the metric $\rho(x,y)$. Assertion (2) follows from the fact that ${\operatorname{diam}}_G(a \vee_G b) = \|a,b\|$.
In addition to the balls $\cB_Q(\vv)^-$ and $\cB_\rho(x,r)^-$ introduced above, we will use several other kinds of balls and discs. In naming them, we make the convention that [*Roman letters*]{} will be used for sets in $\AA^1(K)$ or $\PP^1(K)$, and [*script letters*]{} for ones in ${\operatorname{{\bf A}}}^1_K$ or ${\operatorname{{\bf P}}}^1_K$. Also, we speak of [*discs*]{} in $\AA^1(K)$ and ${\operatorname{{\bf A}}}^1_K$, and [*balls*]{} in $\PP^1(K)$ and ${\operatorname{{\bf P}}}^1_K$.
For each $a \in K$ and $0 < r < \infty$ we have the classical discs $$D(a,r)^- = \{z \in \AA^1(K) : |z-a| < r \}, \quad D(a,r) = \{z \in \AA^1(K) : |z-a| \le r \}.$$ The associated Berkovich discs are $$\cD(a,r)^- = \{x \in {\operatorname{{\bf A}}}^1_K : \zeta_{a,r} \in (x,\infty] \}, \quad
\cD(a,r)\ = \{x \in {\operatorname{{\bf A}}}^1_K : \zeta_{a,r} \in [x,\infty] \} .$$ Note that $\cD(a,r)^-$ is the path-component of ${\operatorname{{\bf P}}}^1_K \backslash \{\zeta_{a,r}\}$ containing $D(a,r)^-$, and $\cD(a,r)$ is the union of $\{\zeta_{a,r}\}$ and the path components of ${\operatorname{{\bf P}}}^1_K \backslash \{\zeta_{a,r}\}$ which do not contain $\infty$. If $\vv_a \in T_{\zeta_{a,r}}$ points towards $a$, and $\vv_\infty \in T_{\zeta_{a,r}}$ points towards $\infty$, then $$\cD(a,r)^- \ = \ \cB_{\zeta_{a,r}}(\vv_a)^- \ , \quad \cD(a,r) = {\operatorname{{\bf P}}}^1_K \backslash \cB_{Q_{a,r}}(\vv_\infty)^- \ .$$ For either the weak or strong topology, $\cD(a,r)^-$ is open, and $\cD(a,r)$ is closed.
Given $a \in {\operatorname{{\bf P}}}^1_K$ and $0 < r < 1$, we write $$B(a,r)^- = \{z \in \PP^1(K) : \|z,a\| < r \} \ , \quad B(a,r) = \{z \in \PP^1(K) : \|z,a\| \le r \}.$$ There is a unique point $Q_{a,r} \in [a,\zeta_G]$ for which ${\operatorname{diam}}_G(Q_{a,r}) = r$. The associated Berkovich balls are $$\cB(a,r)^- = \{x \in {\operatorname{{\bf P}}}^1_K : Q_{a,r} \in (x,\zeta_G] \} , \quad
\cB(a,r) = \{z \in {\operatorname{{\bf P}}}^1_K : Q_{a,r} \in [x,\zeta_G] \}.$$ When $r = 1$, we define $B(a,1)^- = \bigcup_{r < 1} B(a,r)^-$ and $\cB(a,1)^- = \bigcup_{r < 1} \cB(a,r)^-$. Note that $\cB(a,r)^-$ is the path-component of ${\operatorname{{\bf P}}}^1_K \backslash \{Q_{a,r}\}$ containing $B(a,r)^-$, and $\cB(a,r)$ is the union of $\{Q_{a,r}\}$ and the path components of ${\operatorname{{\bf P}}}^1_K \backslash \{Q_{a,r}\}$ which do not contain $\zeta_G$. For either the weak or strong topology, $\cB(a,r)^-$ is open, and $\cB(a,r)$ is closed. If $\vv_a \in T_{Q_{a,r}}$ is the tangent vector pointing towards $a$, one has $$\cB(a,r)^- \ = \ \cB_{Q_{a,r}}(\vv_a)^- \ , \quad \cB(a,r) = {\operatorname{{\bf P}}}^1_K \backslash \cB_{Q_{a,r}}(\vv_{\zeta_G})^- \ .$$ For any $\gamma \in {\operatorname{GL}}_2(\cO)$, one has $\gamma(\cB(a,r)^-) = \cB(\gamma(a),r)^-$ and $\gamma(\cB(a,r)) = \cB(\gamma(a),r)$.
Given a nonconstant function $\varphi(z) \in K(z)$, we define its Berkovich Lipschitz constant (relative to the Favre-Rivera-Letelier metric $d(x,y)$), to be $$\label{LipschitzConstDef}
{\operatorname{Lip}}_\Berk(\varphi) \ = \ \sup_{\substack{x, y \in {\operatorname{{\bf P}}}^1_K \\ x \ne y}} \frac{d(\varphi(x), \varphi(y))}{d(x,y)} \ .$$
\[LipInvariance\] Let $\varphi(z) \in K(z)$ have degree $d \ge 1$. Then for any $\gamma_1, \gamma_2 \in {\operatorname{GL}}_2(\cO)$, one has ${\operatorname{Lip}}_\Berk(\gamma_1 \circ \varphi \circ \gamma_2) = {\operatorname{Lip}}_\Berk(\varphi)$.
This follows from the definition of ${\operatorname{Lip}}_\Berk(\varphi)$ and the fact that ${\operatorname{GL}}_2(\cO)$ preserves $d(x,y)$.
Preliminary Lemmas {#PreliminaryLabelSection}
==================
In this section we prove some lemmas which reduce bounding ${\operatorname{Lip}}_\Berk(\varphi)$ on ${\operatorname{{\bf P}}}^1_K$ to bounding it on a restricted class of segments $[x,y]$.
\[Radial-Limited\] Fix $0 < B_0 \le 1$. A segment $[b,c] \subset {\operatorname{{\bf P}}}^1_K$ will be called [*radial*]{} if it is contained in a segment $[\xi,\zeta_G]$, and it will be called [*$B_0$-limited*]{} if it is either contained in a segment $[\alpha,\xi]$ where $\alpha \in \PP^1(K)$ and ${\operatorname{diam}}_G(\xi) = B_0$, or in a segment $[\xi,\zeta_G]$, where ${\operatorname{diam}}_G(\xi) = B_0$.
\[DecompositionLemma\] Let $\varphi(z) \in K(z)$ have degree $d \ge 1$, put $B_0 = B_0(\varphi)$, and let $[b,c] \in {\operatorname{{\bf P}}}^1_K$ be a segment. Then there is a finite partition $\{a_1, \ldots, a_{n+1}\}$ of $[b,c]$ such that $a_1 = b$, $a_{n+1} = c$, and each of $a_2, \ldots, a_n$ is of type [II]{}, such that for each $i = 1, \ldots, n$,
1. $\varphi$ maps the segment $[a_i,a_{i+1}]$ homeomorphically onto $[\varphi(a_i),\varphi(a_{i+1})];$
2. $[a_i,a_{i+1}]$ and $[\varphi(a_i),\varphi(a_{i+1})]$ are both radial, and $[a_i,a_{i+1}]$ is $B_0$-limited$;$
3. there is an integer $1 \le \delta_i \le d$ such that ${{\operatorname{deg}}}_\varphi(x) = \delta_i$ for each $x \in (a_i,a_{i+1});$
4. $\rho(\varphi(x),\varphi(y)) = \delta_i \cdot \rho(x,y)$ for all $x,y \in [a_i,a_{i+1}];$ and
5. there are a constant $C_i > 0$ and an integer $k_i = \pm \delta_i$ such that for each $x \in [a_i,a_{i+1}]$, if we put $r = {\operatorname{diam}}_G(x)$ and $R = {\operatorname{diam}}_G(\varphi(x))$, then $R = C_i \cdot r^{k_i}$.
The existence of a partition $\{a_1, \ldots, a_{n+1}\}$ satisfying conditions $(1)$, $(3)$ and $(4)$ is due to Rivera-Letelier (see [@R-L2], Corollaries 4.7 and 4.8, or [@B-R], Theorem 9.33). To refine the partition so that it satisfies (2), successively carry out the following adjunctions:
- To assure that each segment $[a_i,a_{i+1}]$ is radial, adjoin $\zeta_G$ to the partition if $\zeta_G \in [b,c]$, and for each $[a_i,a_{i+1}]$ which is now not radial, let $t_i = a_i \wedge_G a_{i+1}$ be the nearest point in $[a_i,a_{i+1}]$ to $\zeta_G$, and adjoint it to the partition.
- To assure that each each segment $[a_i,a_{i+1}]$ is $B_0$-limited, for each $[a_i,a_{i+1}]$ which is not $B_0$-limited, let $\xi_i \in [a_i,a_{i+1}]$ be the unique point with ${\operatorname{diam}}_G(\xi_i) = B_0$, and adjoin it to the partition;
- To assure that each segment $[\varphi(a_i),\varphi(a_{i+1})]$ is radial, consider each of the finitely many pre-images of $\zeta_G$ under $\varphi$, and if it belongs to $[b,c]$, then adjoint it to the partition.
Assertion (5) is now immediate. There is a base $q > 1$ such that for each $x \in \HH^1_K$ one has $\rho(\zeta_G,x) = -\log_q({\operatorname{diam}}_G(x))$. Hence $\rho(\zeta_G,x) = -\log_q(r)$ and $\rho(\zeta_G,\varphi(x)) = -\log_q(R)$. By (2) and (4), for an appropriate choice of $k_i = \pm \delta_i$, for each $x \in [a_i,a_{i+1}]$, $$\rho(\zeta_G,\varphi(x))-\rho(\zeta_G,\varphi(a_i)) \ = \ k_i \cdot \big(\rho(\zeta_G,x)-\rho(\zeta_G,a_i)\big)$$ and (5) follows by exponentiating this.
\[SupCor\] Let $\varphi(z) \in K(z)$ have degree $d \ge 1$, and put $B_0 = B_0(\varphi)$. Let $\cI(B_0)$ be the collection of all radial segments of the form $[\alpha,\xi]$ or $[\xi,\zeta_G]$, where $\alpha \in \PP^1(K)$ and ${\operatorname{diam}}_G(\xi) = B_0$. If $L \in \RR$ is an upper bound for $\{{\operatorname{Lip}}_\Berk(\varphi\vert_I) : I \in \cI(B_0)\}$, then $L$ is an upper bound for ${\operatorname{Lip}}_\Berk(\varphi)$.
We must show that $d(\varphi(b),\varphi(c)) \le L \cdot d(b,c)$ for all $b, c \in {\operatorname{{\bf P}}}^1_K$. This is trivial if $b = c$, so we can assume $b \ne c$.
First suppose $b$ and $c$ are of type II, and take a partition $\{a_1, \ldots, a_{n+1}\}$ of $[b,c]$ satisfying the conditions of Lemma \[DecompositionLemma\]. Each subsegment $[a_i,a_{i+1}]$ is contained in some $I \in \cI(B_0)$, so ${\operatorname{Lip}}_\Berk(\varphi\vert_{[a_i,a_{i+1}]}) \le {\operatorname{Lip}}_\Berk(\varphi\vert_I) \le L$. Furthermore $d(b,c) = \sum_{i=1}^n d(a_i,a_{i+1})$, so $$\begin{aligned}
d(\varphi(b),\varphi(c)) & \le & \sum_{i=1}^n d\big(\varphi(a_i),\varphi(a_{i+1})\big) \ \le \
\sum_{i=1}^n {\operatorname{Lip}}_\Berk(\varphi\vert_{[a_i,a_{i+1}]}) \cdot d(a_i,a_{i+1}) \\
& \le & L \cdot \sum_{i=1}^n d(a_i,a_{i+1}) \ = \ L \cdot d(b,c) \ . \end{aligned}$$
Now let $b \ne c$ in ${\operatorname{{\bf P}}}^1_K$ be arbitrary. Choose an exhaustion of $(b,c)$ by segments $$[b^{(1)},c^{(1)}] \ \subset \ [b^{(2)},c^{(2)}] \ \subset \ \cdots \subset \ [b^{(j)},c^{(j)}]
\ \subset \ \cdots \subset \ (b,c) \ .$$ with type II endpoints. Then $$d\big(\varphi(b),\varphi(c)\big) \ = \ \lim_{j \rightarrow \infty} d\big(\varphi(b^{(j)}),\varphi(c^{(j)})\big)
\ \le \ \lim_{j \rightarrow \infty} L \cdot d\big(b^{(j)},c^{(j)}\big) \ = \ L \cdot d(b,c) \ .$$ Letting $b$ and $c$ range over ${\operatorname{{\bf P}}}^1_K$, we see that ${\operatorname{Lip}}_\Berk(\varphi) \le L$.
\[DerivCor\] Let $\varphi(z) \in K(z)$ have degree $d \ge 1$, and put $B_0 = B_0(\varphi)$. Let $I = [b,c]$ be a segment in ${\operatorname{{\bf P}}}^1_K$ with $b \ne c$, and let $\{a_1, \ldots, a_{n+1}\}$ be a partition of $[b,c]$ with the properties in Lemma [\[DecompositionLemma\]]{}. For each $i = 1, \ldots, n$, put $r_i = \min({\operatorname{diam}}_G(a_i),{\operatorname{diam}}_G(a_{i+1}))$, $s_i = \max({\operatorname{diam}}_G(a_i),{\operatorname{diam}}_G(a_{i+1}))$, and define $F_{\varphi,i} : [r_i,s_i] \rightarrow \RR$ by $F_{\varphi,i}(r) = C_i \cdot r^{k_i}$. Then $${\operatorname{Lip}}_\Berk(\varphi\vert_I) \ = \ \textstyle{\max_{1 \le i \le n}
\Big(\sup_{r \in (r_i,s_i)} \big|F_{\varphi,i}^{\prime}(r)\big| \Big)} \ .$$
It suffices to show that for each $i$, ${\operatorname{Lip}}_\Berk(\varphi\vert_{[a_i,a_{i+1}]}) = \sup_{r \in (r_i,s_i)} |F_{\varphi,i}^{\prime}(r)|$.
Take $x \ne y$ in $[a_i,a_{i+1}]$, and put $u = {\operatorname{diam}}_G(x)$, $v = {\operatorname{diam}}_G(y)$. Without loss we can assume that $r < s$. By the Mean Value Theorem there is an $r_* \in (u,v)$ such that $$\label{MVT}
F_{\varphi,i}(v) - F_{\varphi,i}(u) \ = \ F_{\varphi,i}^{\prime}(r_*) \cdot (v-u) \ ,$$ so $d(\varphi(x),\varphi(y)) = |F_{\varphi,i}^{\prime}(r_*)| \cdot d(x,y)$. Hence ${\operatorname{Lip}}_\Berk(\varphi\vert_{[a_i,a_{i+1}]}) \le \sup_{r \in (r_i,s_i)} |F_{\varphi,i}^{\prime}(r)|$.
The opposite inequality follows from the fact that $F_{\varphi,i}^{\prime}(r)$ is continuous: for each $r_\# \in (r_i,s_i)$ and each $\varepsilon > 0$, there is a $\delta > 0$ such that $[r_\#-\delta,r_\#+\delta] \subset (r_i,s_i)$, and $|F_{\varphi,i}^\prime(t) - F_{\varphi,i}^\prime(r_\#)| < \varepsilon$ for all $t \in [r_\#-\delta,r_\#+\delta]$. Take $x, y \in [a_i,a_{i+1}]$ with $r_\#-\delta < {\operatorname{diam}}_G(x) < {\operatorname{diam}}_G(y) < r_\#+\delta$, and let $r_*$ be as in (\[MVT\]) for this choice of $x, y$. Then $|F_{\varphi,i}^\prime(r_*) - F_{\varphi,i}^\prime(r_\#)| < \varepsilon$. It follows that ${\operatorname{Lip}}_\Berk(\varphi\vert_{[a_i,a_{i+1}]}) \ge \sup_{r \in (r_i,s_i)} |F_{\varphi,i}^{\prime}(r)|$.
\[fprimeMonoLemma\] Let $\Phi(z) \in K(z)$ have degree $d \ge 1$; write $B_0 = B_0(\Phi)$, put $c_0 = \Phi(0)$, and assume $\Phi(D(0,B_0)^-) = D(c_0,R)^-$. Expand $$\Phi(z) \ = \ c_0 + (c_1 z + c_2 z^2 + \cdots c_n z^n) \cdot U(z) \ ,$$ on $D(0,B_0)^-$, where $U(z)$ is a unit power series. Then we can partition $[0,B_0]$ into finitely many subintervals $[r_i,r_{i+1}]$, where $0 = r_1 < \cdots < r_{\ell+1} = B_0$, such that on $[r_i,r_{i+1}]$ we have $$f_\Phi(r) \ = \ f_i(r) \ := \ |c_{k(i)}| \cdot r^{k(i)}$$ for a suitable index $k(i)$. Let $f_\Phi^\prime(r) = \lim_{h \rightarrow 0^+} (f_\Phi(r+h)-f_\Phi(r))/h$ be the right-derivative of $f$ on $[0,B_0)$. Then $f_\Phi^\prime(r)$ is non-decreasing on $[0,B_0)$, and for each $i = 1, \ldots, \ell-1$ we have $k(i) \le k(i+1)$.
[**Remark.**]{} There is a minimal partition with the properties in Lemma \[fprimeMonoLemma\], which has the additional property that $k(i) < k(i+1)$ for $i = 1, \ldots, \ell-1$. However, in the applications the partition we use may not be minimal, so we only assume that $k(i) \le k(i+1)$.
We will regard each $f_i(r) = \ |c_{k(i)}| \cdot r^{k(i)}$ as defined for all $r \ge 0$. Clearly $f_i(r)$ is continuous and monotone increasing, and $f_i^\prime(r) = k(i) |c_{k(i)}| r^{k(i)-1}$ is continuous and nondecreasing.
Since $f_\Phi(r)$ is continuous and monotone increasing, at each break point $r_i$ we must have $f_{i-1}^\prime(r_i) \le f_i^\prime(r_i)$. Thus $f_\Phi^{\prime}(r)$ is non-decreasing. Furthermore at each such $r_i$ $$k(i-1) \cdot \frac{f_{i-1}(r_i)}{r_i} \ = \ f_{i-1}^\prime(r_i)
\ \le \ f_i^{\prime}(r_i) \ = \ k(i) \cdot \frac{f_i(r_i)}{r_i} \ ,$$ so since $f_{i-1}(r_i) = f_i(r_i)$ we must have $k(i-1) \le k(i)$.
Lipschitz constants for Linear Fractional Transformations {#LinearBoundSection}
=========================================================
When $\varphi \in {\operatorname{PGL}}_2(K)$, one can find its Lipschitz constants exactly:
\[LinearThm\] Let $K$ be a complete, algebraically closed nonarchimedean field, and let $\varphi(z) = (az+b)/(cz+d) \in K(z)$ have degree $d =1$. Then $B_0(\varphi) = 1$, and $$\begin{aligned}
{\operatorname{Lip}}_\Berk(\varphi) & = & {\operatorname{Lip}}_{\PP^1(K)}(\varphi) \ = \ \frac{1}{\GIR(\varphi)} \ = \ \frac{1}{\GPR(\varphi)} \\
& = & \frac{1}{|{\operatorname{Res}}(\varphi)|} \ = \ \frac{\max(\,|a|,|b|,|c|,|d|\,)}{|ad-bc|} \ .\end{aligned}$$
Theorem \[LinearThm\] is a restatement of Theorem \[MobiusCase\] in the Introduction. Write $[\varphi]$ for the matrix $$\left[\begin{array}{cc} a & b \\ c & d \end{array} \right] \ .$$ Since $\varphi$ is unchanged when $[\varphi]$ is scaled by an element of $K^{\times}$, we can assume that $[\varphi] \in M_2(\cO)$ and $\max(|a|,|b|,|c|,|d|) = 1$. Since $d(x,y)$ and $\|x,y\|$ are preservied by ${\operatorname{GL}}_2(\cO)$, we can pre- and post-compose $\varphi$ with elements of ${\operatorname{GL}}_2(\cO)$ without changing ${\operatorname{Lip}}_\Berk(\varphi)$ and ${\operatorname{Lip}}_{\PP^1(K)}(\varphi)$; such compositions also preserve $\GIR(\varphi)$, $\GPR(\varphi)$, the value of $|{\operatorname{Res}}(\varphi)| = |ad-bc|$, and the fact that $\max(|a|,|b|,|c|,|d|) = 1$. Choosing $\gamma_1, \gamma_2 \in {\operatorname{GL}}_2(\cO)$ to carry out appropriate combinations of elementary row and column operations, and setting $\Phi = \gamma_1 \circ \varphi \circ \gamma_2$, we can arrange that $$[\Phi] \ = \ \left[\begin{array}{cc} 1 & 0 \\ 0 & D \end{array} \right]$$ where $D \in \cO \backslash \{0\}$. Note that $\Phi(\zeta_G) = \zeta_{0,1/|D|}$, so $\GIR(\varphi) = \GIR(\Phi) = |D| = |{\operatorname{Res}}(\varphi)|$. Similarly $\Phi(\zeta_{0,|D|}) = \zeta_G$, so $\GPR(\varphi) = \GPR(\Phi) = |D|$. Trivially $B_0(\varphi) = B_0(\Phi) = 1$.
We will now show that ${\operatorname{Lip}}_\Berk(\Phi) = 1/|D|$. Recall that $${\operatorname{diam}}_G(\zeta_{a,r}) \ = \ \left\{ \begin{array}{ll} r & \text{ if $|a|, r \le 1$ \ ,} \\
r/|a|^2 & \text{ if $|a| > 1$ and $r < |a|$ \ ,} \\
1/r & \text{ if $|a| \le 1$ and $r \ge 1$, or if $1 < |a| \le r$ \ .}
\end{array} \right.$$ For future use, note that the three formulas on the right can be combined the a single expression $$\label{diamGformula}
{\operatorname{diam}}_G(\zeta_{a,r}) \ = \ \frac{r}{\max(1,|a|,r)^2} \ .$$ Given a point $\zeta_{a,r} \in \AA^1_\Berk$, we have $\Phi(\zeta_{a,r}) = \zeta_{a/D,r/|D|}$. It follows that $${\operatorname{diam}}_G(\Phi(\zeta_{a,r})) \ = \ \left\{ \begin{array}{ll} r/|D| & \text{ if $|a|, r \le |D|$ \ ,} \\
r |D| / |a|^2 & \text{ if $|a| > |D|$ and $r < |a|$ \ ,} \\
|D|/r & \text{ if $|a| \le |D|$ and $r \ge |D|$, or if $|D| < |a| \le r$ \ .} \\
\end{array} \right.$$
Since points of type II and III are dense in ${\operatorname{{\bf P}}}^1_K$ for the strong topology, it suffices to bound ${\operatorname{Lip}}_\Berk(\varphi)$ on paths $[a,\infty]$ where $a \in K$. The remainder of the argument is a case by case verification.
Fix $a \in K$, and consider a point $\zeta_{a,r}$. If $|a| \le |D|$ and $s < r \le |D|$ then $$\frac{d(\Phi(\zeta_{a,r}),\Phi(\zeta_{a,s}))}{d(\zeta_{a,r},\zeta_{a,s})}
\ = \ \frac{r/|D| - s/|D|}{r-s} \ = \ \frac{1}{|D|} \ .$$ If $|a| \le |D|$ and $|D| \le s < r$, then $|a/D| \le 1$ while $ 1 \le s/|D| < r/|D|$, so $$\frac{d(\Phi(\zeta_{a,r}),\Phi(\zeta_{a,s}))}{d(\zeta_{a,r},\zeta_{a,s})}
\ = \ \frac{|D|/s - |D|/r}{r-s} \ = \ \frac{|D|}{rs} \ < \ \frac{1}{|D|}\ .$$ If $|D| < |a| \le 1$ and $0 < s < r \le |a|$, then $$\frac{d(\Phi(\zeta_{a,r}),\Phi(\zeta_{a,s}))}{d(\zeta_{a,r},\zeta_{a,s})}
\ = \ \frac{(r/|D|)/(|a/D|)^2 - (s/|D|)/(|a/D|)^2)}{r-s} \ = \ \frac{|D|}{|a|^2} \ < \ \frac{1}{|D|}\ .$$ If $|D| < |a| \le 1$ and $|a| \le s < r$, then $1 < |a/D| \le s/|D| < r/|D|$ so again $$\frac{d(\Phi(\zeta_{a,r}),\Phi(\zeta_{a,s}))}{d(\zeta_{a,r},\zeta_{a,s})}
\ = \ \frac{|D|/s - |D|/r}{r-s} \ = \ \frac{|D|}{rs} \ < \ \frac{1}{|D|}\ .$$ If $|a| > 1$ and $0 < s < r \le |a|$, then $$\frac{d(\Phi(\zeta_{a,r}),\Phi(\zeta_{a,s}))}{d(\zeta_{a,r},\zeta_{a,s})}
\ = \ \frac{(r/|D|)/(|a|/|D|)^2 - (s/|D|)/(|a|/|D|)^2)}{r/|a|^2-s/|a|^2} \ = \ |D| \ \le \ \frac{1}{|D|}\ .$$ Finally, if $|a| > 1$ and $|a| \le s < r$, then $\zeta_{a,s} = \zeta_{0,s}$ and $\zeta_{a,r} = \zeta_{0,r}$ so $$\frac{d(\Phi(\zeta_{a,r}),\Phi(\zeta_{a,s}))}{d(\zeta_{a,r},\zeta_{a,s})}
\ = \ \frac{|D|/s - |D|/r}{1/s-1/r} \ = \ |D| \ \le \ \frac{1}{|D|}\ .$$ Thus ${\operatorname{Lip}}_\Berk(\varphi) = {\operatorname{Lip}}_\Berk(\Phi) = 1/|D|$.
Clearly ${\operatorname{Lip}}_{\PP^1(K)}(\Phi) \le {\operatorname{Lip}}_\Berk(\Phi) = 1/|D|$. To prove that ${\operatorname{Lip}}_{\PP^1(K)}(\Phi) = 1/|D|$, it suffices to show that ${\operatorname{Lip}}_{\PP^1(K)}(\Phi) \ge 1/|D|$. This is trivial, since if $x = 0$ and $y = D$, then $\|x,y\|=\|0,D\|=|D|$ and $\|\Phi(x),\Phi(y)\|=\|0,1\|=1$.
Some Auxiliary Constants {#AuxSection}
========================
In this section, we study the four constants associated to $\varphi$ in the Introduction: the Gauss Pre-Image radius, the Root-Pole number, the Ball-Mapping radius, and the Gauss Image radius.
\[FourDefs\] Let $\varphi(z) \in K(z)$ have degree $d \ge 1$.
$(A)$ The Gauss Image radius of $\varphi$ is $\GIR(\varphi) = {\operatorname{diam}}_G(\varphi(\zeta_G))$.
$(B)$ The Root-Pole number of $\varphi$ is $$\RP(\varphi) \ = \ \min \{\|\alpha, \beta\| : \alpha, \beta \in \PP^1(K), \varphi(\alpha) = 0, \varphi(\beta) = \infty \} \ .$$
$(C)$ The Ball-Mapping radius of $\varphi$ is $$B_0(\varphi) = \sup \{0 < r \le 1 : \text{for all $a \in \PP^1(K)$, $\varphi(\cB(a,r)^-) \ne {\operatorname{{\bf P}}}^1_K$}\} \ .$$
$(D)$ The Gauss Pre-Image radius of $\varphi$ is $$\GPR(\varphi) \ = \ \min \{ {\operatorname{diam}}_G(x) : x \in {\operatorname{{\bf P}}}^1_K, \varphi(x) = \zeta_G \} \ .$$
Clearly the Ball-Mapping radius, the Gauss Image radius, and the Gauss Pre-Image radius are invariant under pre- and post- composition of $\varphi$ with elements of ${\operatorname{GL}}_2(\cO)$; the Ball-Mapping radius is also invariant under post-composition of $\varphi$ with elements of ${\operatorname{GL}}_2(K)$. The Root-Pole number is not invariant under either pre- or post- composition by ${\operatorname{GL}}_2(\cO)$, but it lies between the Gauss Pre-Image radius and the Ball-Mapping radius:
\[GaussInequalitiesProp\] Let $\varphi(z) \in K(z)$ have degree $d \ge 1$. Then $$0 \ < \ \GPR(\varphi) \ \le \ \RP(\varphi) \ \le \ B_0(\varphi) \ \le \ 1 \ .$$
Since there are most $d$ pre-images of $\zeta_G$ under $\varphi$, which all lie in $\HH^1_K$, clearly $\GPR(\varphi) > 0$. Also, by the definition of $B_0(\varphi)$, we trivially have $B_0(\varphi) \le 1$.
It is also easy to see that $\GPR(\varphi) \le \RP(\varphi)$. Indeed, if $r = \RP(\varphi)$, then there are a root $\alpha$ and a pole $\beta$ of $\varphi$ with $\|\alpha,\beta\| = r$. The image of the path $[\alpha,\beta]$ under $\varphi$ is connected and contains $0$ and $\infty$, so it contains the path $[0,\infty]$. Hence it contains $\zeta_G$, and there is a point $x$ of $\varphi^{-1}(\zeta_G)$ in $[\alpha,\beta]$. It follows that $r = {\operatorname{diam}}_G(\alpha \wedge_G \beta) \ge {\operatorname{diam}}_G(x) \ge \GPR(\varphi)$.
Finally, we show that $\RP(\varphi) \le B_0(\varphi)$. If $B_0(\varphi) = 1$, then trivially $\RP(\varphi) \le B_0(\varphi)$, since $\|\alpha,\beta\| \le 1$ for any pair of elements $\alpha, \beta \in \PP^1(K)$. Suppose $B_0(\varphi) < 1$, and take any $r$ with $B_0(\varphi) < r \le 1$. Since $r > B_0(\varphi)$, there is a ball $\cB(a,r)^-$ with $\varphi(\cB(a,r)^-) = {\operatorname{{\bf P}}}^1_K$. Hence there are $\alpha, \beta \in \PP^1(K) \cap \cB(a,r)^-$ such that $\varphi(\alpha) = 0$, $\varphi(\beta) = \infty$. It follows that $r > \|\alpha,\beta\| \ge \RP(\varphi)$. Since $B_0(\varphi)$ is the infimum of all such $r$, we must have $B_0(\varphi) \ge \RP(\varphi)$.
The inequalities $\GPR(\varphi) \le \RP(\varphi) \le B_0(\varphi)$ in Proposition \[GaussInequalitiesProp\] can both be strict. For example, consider the polynomial $\varphi(z) = z^2 - 1/p^2 \in \CC_p[z]$, where $p$ is an odd prime. One sees easily that $\varphi^{-1}(\zeta_G) = \{ \zeta_{1/p,1/p}, \zeta_{-1/p,1/p} \}$ so $\GPR(\varphi) = p^{-3}$. The zeros of $\varphi$ are $\{\pm 1/p\}$ and the only pole is $\{\infty\}$, so $\RP(\varphi) = p^{-1}$. Finally, the only solution to $\varphi(z) = -1/p^2$ is $z = 0$. It follows that if $\varphi(\cB(a,r)^-) = {\operatorname{{\bf P}}}^1_K$ for some ball, then both $0, \infty \in \cB(a,r)^-$. This is impossible with $r < 1$, so $B_0(\varphi) = 1$.
Our next proposition says that $B_0(\varphi) \in |K^\times|$, and there is a ball which realizes it.
\[B\_0RationalityProp\] Let $\varphi(z) \in K(z)$ have degree $d \ge 1$, and put $B_0 = B_0(\varphi)$. Then $B_0 \in |K^{\times}|$. Moreover, if $B_0 < 1$ $($so necessarily $d \ge 2)$, there is an $\alpha \in \PP^1(K)$ for which $\varphi(\cB(a,B_0)^-)$ is a ball, but $\varphi(\cB(\alpha,B_0)) = {\operatorname{{\bf P}}}^1_K$.
The proof uses the theory of the “crucial set” from ([@RR-GMRL], [@RR-NEND]).
Write $B_0 = B_0(\varphi)$. If $d=1$, then $B_0 = 1$, and the assertions are trivial. Assume $d \ge 2$. If $B_0 = 1$, the assertions are again trivial, so we can assume $0 < B_0 < 1$. Choose a sequence of numbers $1 > R_1 > R_2 > \cdots > B_0$ in $|K^{\times}|$ with $\lim_{i \rightarrow \infty} R_i = B_0$, and a sequence of balls $\cB(a_i,R_i)^-$ such that $\varphi(\cB(a_i,R_i)^-) = {\operatorname{{\bf P}}}^1_K$ for each $i$. Each of the balls $\cB(a_i,R_i)^-$ can be written in the form $\cB_{P_i}(\vv_i)^-$ where $P_i$ is a type II point and $\vv_i \in T_{P_i}$ is a suitable tangent vector.
By the proof of Theorem 4.6 of [@RR-NEND] (alternately see Theorem 4.6 of [@RR-GMRL]), each $\cB(a_i,R_i)^-$ contains either a classical fixed point of $\varphi$ (that is, a fixed point in $\PP^1(K)$), or a repelling fixed point of $\varphi$ in ${\operatorname{{\bf H}}}^1_K$ of a special type, a [*focused repelling fixed point*]{}. A focused repelling fixed point is a type II point $Q$ with $\varphi(Q) = Q$, such that ${{\operatorname{deg}}}_\varphi(Q) \ge 2$ and there is a unique $\vv_\# \in T_Q$ for which $\varphi_*(\vv_\#) = \vv_\#$. (We are using the case of ([@RR-GMRL], [@RR-NEND], Theorem 4.6) concerning a ball with a type II boundary point: each such ball is dealt with by one of Lemmas 2.1, 2.2, and 4.5 of ([@RR-GMRL], [@RR-NEND]). Lemmas 2.1 and 2.2 produce classical fixed points in $\cB(a_i,R_i)^-$, while Lemma 4.5 produces either a classical fixed point or a focused repelling fixed point.) By ([@RR-GMRL], [@RR-NEND], Proposition 3.1) $\cB_Q(\vv_\#)^-$ contains all the classical fixed points of $\varphi$, and $\varphi({\operatorname{{\bf P}}}^1_K \backslash \cB_Q(\vv_\#)^-) = {\operatorname{{\bf P}}}^1_K$. There are at most $d+1$ classical fixed points of $\varphi$, and by ([@RR-GMRL], [@RR-NEND], Corollary 6.3) there are at most $d-1$ repelling fixed points of $\varphi$ in ${\operatorname{{\bf H}}}^1_K$. Thus we can apply the Pigeon-hole Principle to the balls and fixed points.
First suppose there is a classical fixed point $\alpha$ which is contained in infinitely many balls $\cB(a_i,R_i)^-$. By replacing the sequence of balls with a subsequence, we can assume $\alpha \in \cB(a_i,R_i)^-$ for each $i$. After conjugating $\varphi$ by a suitable element of ${\operatorname{GL}}_2(\cO)$, we can assume that $\alpha = 0$, and that the sequence of balls is $\{\cB(0,R_i)^-\}_{i \ge 1}$. Suppose $B_0 \notin |K^{\times}|$. Then the point $P = \zeta_{0,B_0}$ is of type III. The tangent space $T_P$ consists of two directions $\vv_0,\vv_\infty$, and $\cB(0,B_0)^- = \cB_P(\vv_0)^-$. Put $Q = \varphi(P)$, $\vw_1 = \varphi_*(\vv_0)$ and $\vw_2 = \varphi_*(\vv_\infty)$. Necessarily $Q$ is of type III, and $\vw_1, \vw_2$ are the two tangent directions in $T_Q$ (see [@B-R], Corollary 9.20). By the definition of the ball mapping radius, $\varphi(\cB_P(\vv_0)^-)$ is a ball, hence necessarily $\varphi(\cB_P(\vv_0))^-) = \cB_Q(\vw_1)^-$. By Rivera-Letelier’s Annulus Mapping Theorem (see [@B-R], Lemma 9.45), there is a point $P_1 \in \cB_P(\vv_\infty)^-$ for which $\varphi({\operatorname{Ann}}(P,P_1))$ is an annulus ${\operatorname{Ann}}(Q,Q_1) \subset \cB_Q(\vw_2)^-$. Without loss we can suppose $P_1 = \zeta_{0,R}$ for some $R > B_0$. Since $\cB(0,R)^- = \cB(0,B_0)^- \cup \{P\} \cup {\operatorname{Ann}}(P,P_1)$, it follows that $$\varphi(\cB(0,R)^-) \ = \ \cB_Q(\vw_1)^- \cup \{Q\} \cup {\operatorname{Ann}}(Q,Q_1) \ \ne \ {\operatorname{{\bf P}}}^1_K \ .$$ This contradicts that $\varphi(\cB(0,R_i)^-) = {\operatorname{{\bf P}}}^1_K$ when $B_0 < R_i < R$, hence $B_0 \in |K^{\times}|$.
By definition $\varphi(\cB(0,B_0)^-)$ is a ball; we claim that $\varphi(\cB(0,B_0)) = {\operatorname{{\bf P}}}^1_K$. Suppose this were not the case; write $P = \zeta_{0,B_0}$ and put $Q = \varphi(P)$. Let $\vv_\infty \in T_P$ be the direction containing $\infty$, and put $\vw_\infty = \varphi_*(\vv_\infty) \in T_Q$. The map $\varphi_* : T_P \rightarrow T_Q$ is surjective, so for each $\vw \in T_Q$ with $\vw \ne \vw_\infty$ there is some $\vv \in T_P$ with $\varphi_*(\vv) = \vw$. Since $\varphi(\cB_P(\vv)^-)$ contains $\cB_Q(\vw)^-$, we see that $$\varphi(\cB(0,B_0))
\ \supseteq \ \{Q\} \cup \bigcup_{\vw \ne \vw_\infty} \cB_Q(\vw)^- \ \supseteq \ {\operatorname{{\bf P}}}^1_K \backslash \cB_Q(\vw_\infty)^- \ .$$ Moreover, for each $\vv \in T_P$, the image $\varphi(\cB_P(\vv)^-)$ is either a ball or all of ${\operatorname{{\bf P}}}^1_K$. (If there were some $\vv \in T_P$ with $\vv \ne \vv_\infty$ for which $\varphi_*(\vv) = \vw_\infty$, then $\varphi(\cB(0,B_0))$ would contain $\cB_Q(\vw_\infty)^-$, hence would be ${\operatorname{{\bf P}}}^1_K$.) Thus $\vv_\infty$ is the only direction in $T_P$ with $\varphi_*(\vv) = \vv_\infty$. It follows for each $\vv \ne \vv_\infty$, the image $\varphi(\cB_P(\vv)^-)$ is a ball $\cB_Q(\vw)^-$ with $\vw \ne \vv_\infty$, and that $\varphi(\cB(0,B_0)) = {\operatorname{{\bf P}}}^1_K \backslash \cB_Q(\vw_\infty)^-$. However, now Rivera-Letelier’s Annulus mapping theorem shows there is a point $P_1 = \zeta_{0,S_1} \in \cB_P(\vv_\infty)^-$ for which $\varphi_*({\operatorname{Ann}}(P,P_1))$ is the annulus ${\operatorname{Ann}}(Q,\varphi(P_1)) \subset \cB_Q(\vw_\infty)^-$. This would mean that for each $R$ with $B_0 < R < S_1$ $\varphi(\cB(0,R)^-) \ne {\operatorname{{\bf P}}}^1_K$, which contradicts the fact that $\varphi(\cB(0,R_i)^-) = {\operatorname{{\bf P}}}^1_K$ for all $i$. Hence it must be that $\varphi(\cB(0,B_0)) = {\operatorname{{\bf P}}}^1_K$.
Next consider the case where no classical fixed point is contained in infinitely many $\cB(a_i,R_i)^-$. In this situation there must be a focused repelling fixed point $\xi$ which belongs to infinitely many $\cB(a_i,R_i)^-$. After passing to a subsequence of the balls, if necessary, we can assume that $\xi \in \cB(a_i,R_i)^-$ for each $i$, and that no classical fixed point is contained in any $\cB(a_i,R_i)^-$. After conjugating $\varphi$ by a suitable element of ${\operatorname{GL}}_2(\cO)$ if necessary, we can assume that $\xi = \zeta_{0,S_1}$ for some $S_1 \le B_0$, and that the sequence of balls is $\{\cB(0,R_i)^-\}_{i \ge 1}$. Since $\xi$ is of type II, necessarily $S_1 \in |K^{\times}|$. Since no $\cB(0,R_i)^-$ contains classical fixed points of $\varphi$, the distinguished direction $\vv_\# \in T_\xi$ must be $\vv_\# = \vv_\infty$. It follows that ${\operatorname{{\bf P}}}^1_K \backslash \cB_\xi(\vv_\#)^- = \cB(0,S_1)$, and $\varphi(\cB(0,S_1)) = {\operatorname{{\bf P}}}^1_K$. If $B_0 > S_1$, then $\varphi(\cB(0,R)^-) = {\operatorname{{\bf P}}}^1_K$ for each $R$ with $B_0 > R > S_1$. This contradicts the definition of the ball mapping radius, so $B_0 = S_1 \in |K^{\times}|$. The equality $B_0 = S_1$ also shows that $\varphi(\cB(0,B_0)^-)$ is a ball, but $\varphi(\cB(0,B_0)) = {\operatorname{{\bf P}}}^1_K$.
There is a simple formula for $\GIR(\varphi)$ in terms of the coefficients of a normalized representation of $\varphi$:
\[GIRProp\] Let $\varphi(z) \in K(z)$ have degree $d \ge 1$, and let $(F,G)$ be a normalized representation of $\varphi$. Write $F(X,Y) = a_d X^d + \ldots + a_1 X Y^{d-1} + a_0 Y^d$, $G(X,Y) = b_d X^d + \ldots + b_1 X Y^{d-1} + b_0 Y^d$. Then $$\label{GIRformula}
\GIR(\varphi) \ = \ \max_{i \ne j} \Big(
\Big| \det \Big[ \begin{array}{cc} a_i & a_j \\ b_i & b_j \end{array} \Big] \Big| \Big) \ .$$
Suppose $\varphi(\zeta_G) = Q$. After replacing $\varphi$ with $\gamma \circ \varphi$ for a suitable $\gamma \in {\operatorname{GL}}_2(\cO)$ we can assume that $Q = \zeta_{0,R}$, where $R = \GIR(\varphi)$. Since $\gamma$ preserves ${\operatorname{diam}}_G(\cdot)$ and $$\Big| \det \Big( \gamma \circ \Big[ \begin{array}{cc} a_i & a_j \\ b_i & b_j \end{array} \Big] \Big) \Big| \ = \
\Big| \det \Big[ \begin{array}{cc} a_i & a_j \\ b_i & b_j \end{array} \Big] \Big| \ ,$$ this does not affect (\[GIRformula\]). Since $(F,G)$ is normalized, for generic $z \in \cO_K$ we must have $|F(z,1)| = R$ and $|G(z,1)| = 1$. This means that all coefficients of $F$ must satisfy $|a_i| \le R$ and at least one coefficient of $G$ must satisfy $|b_j| = 1$.
Next take $C \in K$ with $|C|=R$, and put $\Phi(z) = (1/C) \varphi(z)$. Then $\Phi(\zeta_G) = \zeta_G$, so $\GIR(\Phi) = 1$, and $(F_0(X,Y),G_0(X,Y)) := ((1/C)F(X,Y),G(X,Y))$ is a normalized representation of $\Phi$. Writing $F_0(X,Y) = A_d X^d + \ldots + A_1 X Y^{d-1} + A_0 Y^d$, $G_0(X,Y) = B_d X^d + \ldots + B_1 X Y^{d-1} + B_0 Y^d$, it suffices to show that $$\max_{i \ne j} \Big(
\Big| \det \Big[ \begin{array}{cc} A_i & A_j \\ B_i & B_j \end{array} \Big] \Big| \Big) \ = \ 1 \ .$$ If this were not the case, then $\tA_i \tB_j - \tA_j \tB_i = \widetilde{0} \pmod{\fM}$ for all $i \ne j$, so one of the vectors $\tA = (\tA_d, \ldots, \tA_0)$, $\tB = (\tB_d, \ldots, \tB_0)$ gotten by reducing the coefficients of $F_0, G_0 \pmod{\fM}$ would be a multiple of the other. Hence $\Phi$ would have constant reduction at $\zeta_G$. However, this contradicts ([@B-R], Lemma 2.17), which says that $\Phi(\zeta_G) = \zeta_G$ if and only if $\Phi$ has nonconstant reduction.
We next seek lower bounds for $B_0(\varphi)$, $\GIR(\varphi)$, and $\GPR(\varphi)$ in terms of $|{\operatorname{Res}}(\varphi)|$. For this, we will need the following proposition, which is a projective version of the classical formula for the resultant of two polynomials.
Let the zeros $\alpha_1, \ldots, \alpha_d$ of $\varphi$ in $\PP^1(K)$ (listed with multiplicity) have homogeneous coordinates $$(1:\sigma_1), \ldots, (1:\sigma_m), (\delta_{m+1}:1), \ldots, (\delta_d:1) \ ,$$ where $|\sigma_1|, \ldots, |\sigma_m| \le 1$ and $|\delta_{m+1}|, \ldots, |\delta_d| < 1$. Likewise, let the poles $\beta_1, \ldots, \beta_d$ of $\varphi$ (listed with multiplicity) have homogeneous coordinates $$(1:\tau_1), \ldots, (1:\tau_n), (\eta_{n+1}:1), \ldots, (\eta_d:1) \ ,$$ where $|\tau_1|, \ldots, |\tau_n| \le 1$ and $|\eta_{n+1}|, \ldots, |\eta_d| < 1$.
Let $(F,G)$ be a normalized representation of $\varphi$. Then we can write $$\begin{aligned}
F(X,Y) & = & C_0 \cdot \prod_{i=1}^m (X-\sigma_i Y) \cdot \prod_{i=m+1}^d (\delta_i X - Y) \ , \\
G(X,Y) & = & C_1 \cdot \prod_{j=1}^n (X-\tau_i Y) \cdot \prod_{j=n+1}^d (\eta_i X - Y) \ ,\end{aligned}$$ where $C_0, C_1 \in K^{\times}$ satisfy $0 < |C_0|, |C_1| \le 1$ and $\max(|C_0|, |C_1|) = 1$.
\[ResultantProp\] Let $\varphi \in K(z)$ have degree $d \ge 1$. With notations as above, we have $$|{\operatorname{Res}}(\varphi)| \ = \ |C_0|^d |C_1|^d \cdot \prod_{i, j = 1}^d \|\alpha_i, \beta_j\| \ .$$
By perturbing $\varphi$ slightly we can assume that none of its zeros or poles are the point $\infty = (0:1)$, while preserving the distances $\|\alpha_i,\beta_j\|$ and the absolute values $|C_0|$, $|C_1|$. (For instance, we can replace $\varphi$ with $\varphi \circ \gamma$ for a suitable $\gamma \in {\operatorname{GL}}_2(\cO)$, sufficiently close to the identity). If we expand $$\begin{aligned}
F(X,Y) & = & a_d X^d + a_{d-1} X^{d-1} Y + \cdots + a_0 Y^d \ , \\
G(X,Y) & = & b_d X^d + b_{d-1} X^{d-1} Y + \cdots + b_0 Y^d \ ,\end{aligned}$$ then $|{\operatorname{Res}}(\varphi)| = |{\operatorname{Res}}(F,G)|$ where $$\label{ResultantFormula}
{\operatorname{Res}}(F,G) \ = \ \det\Bigg( \ \left[ \begin{array}{cccccccc}
a_d & a_{d-1} & \cdots & a_1 & a_0 & & & \\
& a_d & a_{d-1} & \cdots & a_1 & a_0 & & \\
& & & & \vdots & & & \\
& & & a_d & a_{d-1} & \cdots & a_1 & a_0 \\
b_d & b_{d-1} & \cdots & b_1 & b_0 & & & \\
& b_d & b_{d-1} & \cdots & b_1 & b_0 & & \\
& & & & \vdots & & & \\
& & & b_d & b_{d-1} & \cdots & b_1 & b_0
\end{array} \right] \ \Bigg) \ ,$$ Here, $a_d = C_0 \cdot \prod_{i=m+1}^d \delta_i$ and $b_d = C_1 \cdot \prod_{j=n+1}^d \eta_j$.
Now dehomogenize $F(X,Y)$ and $G(X,Y)$, setting $z = X/Y$, obtaining $$\begin{aligned}
f(z) & = & a_d z^d + a_{d-1} z^{d-1} + \cdots + a_0 \ = \ a_d \cdot \prod_{i=1}^d (z-\alpha_i) \ , \\
g(z) & = & b_d z^d + b_{d-1} z^{d-1} + \cdots + b_0 \ = \ b_d \cdot \prod_{j=1}^d (z-\beta_j) \ .\end{aligned}$$ where $z = X/Y$ and now $\alpha_1, \ldots, \alpha_d, \beta_1, \ldots, \beta_d \in K$. Evidently $$\begin{array}{ll}
\alpha_1 = \sigma_1, \ldots, \alpha_m = \sigma_m, & \quad \alpha_{m+1} = 1/\delta_{m+1} \ldots, \alpha_d = 1/\delta_d \ , \\
\beta_1 = \tau_1, \ldots, \beta_n = \tau_n, & \quad \beta_{n+1} = 1/\eta_{n+1} \ldots, \beta_d = 1/\eta_d \ .
\end{array}$$
The resultant ${\operatorname{Res}}(f,g)$ is given by the same determinant (\[ResultantFormula\]) as ${\operatorname{Res}}(F,G)$. Since $a_d, b_d \ne 0$, a well-known formula for the resultant (see \[L\], Proposition 10.3) gives $${\operatorname{Res}}(f,g) \ = \ (a_d)^d \cdot (b_d)^d \cdot \prod_{i,j = 1}^d (\alpha_i-\beta_j) \ .$$ Inserting the above values for $a_d$, $b_d$ and the $\alpha_i$, $\beta_j$, then simplifying, we see that $$\begin{aligned}
|{\operatorname{Res}}(F,G)| \ = \ |{\operatorname{Res}}(f,g)| & = & |C_0|^d \cdot |C_1|^d \cdot \prod_{i=1}^m \prod_{j=1}^n |\sigma_i - \tau_j|
\cdot \prod_{i=m+1}^d \prod_{j=1}^n | 1 - \delta_i \tau_j|\\
& & \qquad \quad
\cdot \prod_{i=1}^m \prod_{j=n+1}^d | \sigma_i \eta_j - 1|
\cdot \prod_{i=m+1}^d \prod_{j=n+1}^d | \eta_j - \delta_i| \ .\end{aligned}$$ Here $$\left\{ \begin{array}{ll}
|\sigma_i - \tau_j| = \|\alpha_i, \beta_j\| & \text{ for $i = 1, \ldots, m$, $j= 1, \ldots, n$;} \\
|1 - \delta_i \tau_j| = 1 = \|\alpha_i, \beta_j\| & \text{ for $i = m+1, \ldots, d$, $j= 1, \ldots, n$;} \\
|\sigma_i \eta_j - 1| = 1 = \|\alpha_i, \beta_j\| & \text{ for $i = 1, \ldots, m$, $j= n+1, \ldots, d$; } \\
|\eta_j - \delta_i| = \|\alpha_i, \beta_j\| & \text{ for $i = m+1, \ldots, d$, $j= n+1, \ldots, d$.}
\end{array} \right.$$ Thus $|{\operatorname{Res}}(\varphi)| = |{\operatorname{Res}}(F,G)| = |C_0|^d \cdot |C_1|^d \cdot \prod_{i,j= 1}^d \|\alpha_i,\beta_j\|$.
\[ResultantBoundCor\] Let $\varphi(z) \in K(z)$ have degree $d \ge 1$. Then $\GPR(\varphi) \ge |{\operatorname{Res}}(\varphi)|$ and $\GIR(\varphi)^d \cdot B_0(\varphi) \ge |{\operatorname{Res}}(\varphi)|$. In particular $\GIR(\varphi)\ge |{\operatorname{Res}}(\varphi)|^{1/d}$ and $B_0(\varphi) \ge |{\operatorname{Res}}(\varphi)|$.
Recall that $\GPR(\varphi)$, $\GIR(\varphi)$, $B_0(\varphi)$, and $|{\operatorname{Res}}(\varphi)|$ are invariant under pre- and post- composition of $\varphi$ with elements of ${\operatorname{GL}}_2(\cO)$.
To show that $\GPR(\varphi) \ge |{\operatorname{Res}}(\varphi)|$, put $R = \GPR(\varphi)$ and fix $Q \in \varphi^{-1}(\{\zeta_G\})$ with ${\operatorname{diam}}_G(Q) = R$. Choose $\gamma_1 \in {\operatorname{GL}}_2(\cO)$ so that $\gamma_1(\zeta_{0,R}) = Q$; after replacing $\varphi$ with $\varphi \circ \gamma_1$ we can assume that $Q = \zeta_{0,R}$. There are at most $d$ directions $\vv \in T_P$ for which $\varphi(\cB_Q(\vv)^-) = {\operatorname{{\bf P}}}^1_K$, since such a direction must contain a solution to $\varphi(\alpha)) = 0$. Similarly, for each $\vw \in T_{\zeta_G}$, there are at most $d$ directions $\vv \in T_Q$ for which $\varphi_*(\vv) = \vw$. Since the map $\varphi_*: T_Q \rightarrow T_{\zeta_G}$ is surjective, we can find directions $\vv_1, \vv_2 \in T_Q$ satisfying the following conditions:
1. $\vv_1, \vv_2 \in T_Q \backslash \{\vv_\infty\}$;
2. $\vv_1 \ne \vv_2$ and $\varphi_*(\vv_1) \ne \varphi_*(\vv_2)$;
3. $\varphi(\cB_Q(\vv_1)^-) = \cB_{\zeta_G}(\varphi_*(\vv_1))^-$ and $\varphi(\cB_Q(\vv_2)^-) = \cB_{\zeta_G}(\varphi_*(\vv_2))^-$ are balls.
Fix $\alpha \in \PP^1(K) \cap \cB_Q(\vv_1)^-$ and $\beta \in \PP^1(K) \cap \cB_Q(\vv_2)^-$, and note that $\|\alpha,\beta\| = R$. Put $A = \varphi(a)$, $B = \varphi(\beta)$.
Since $A$ and $B$ belong to distinct tangent directions at $\zeta_G$, by ([@B-R], Corollary 2.13(B)) there is a $\gamma_2 \in {\operatorname{GL}}_2(K)$ which takes the triple $(A,\zeta_G,B)$ to $(0,\zeta_G,\infty)$. Since $\gamma_2(\zeta_G) = \zeta_G$, and the stabilizer of $\zeta_G$ is $K^\times \cdot {\operatorname{GL}}_2(\cO)$, we can scale $\gamma_2$ so that it belongs to ${\operatorname{GL}}_2(\cO)$. After replacing $\varphi$ with $\gamma_2 \circ \varphi$, we can assume that $\varphi(\alpha) = 0$ and $\varphi(\beta) = \infty$. By Proposition \[ResultantProp\] $$\GPR(\varphi) \ = \ R \ = \ \|\alpha,\beta\| \ \ge \ |{\operatorname{Res}}(\varphi)| \ .$$
To show that $\GIR(\varphi)^d \cdot B_0(\varphi) \ge |{\operatorname{Res}}(\varphi)|$, put $r = \GIR(\varphi)$ and choose $\gamma \in {\operatorname{GL}}_2(\cO)$ with $\gamma(\varphi(\zeta_G)) = \zeta_{0,r}$. After replacing $\varphi$ with $\gamma \circ \varphi$ we can assume that $\varphi(\zeta_G) = \zeta_{0,r}$.
In this setting, if $(F,G)$ is a normalized representation of $\varphi$, and notations are as in Proposition \[ResultantProp\], then $|C_0| = r = \GIR(\varphi)$ and $|C_1| = 1$. From the inequality $B_0(\varphi) \ge \RP(\varphi)$ it follows that $B_0(\varphi) \ge \min_{i,j} \|\alpha_i, \beta_j\|$. Furthermore, $\|\alpha_i, \beta_j\| \le 1$ for all $i, j$, and $|\GIR(\varphi)| \le 1$.
Thus, by Proposition \[ResultantProp\], we have $\GIR(\varphi)^d \cdot B_0(\varphi) \ge |C_0|^d \cdot \min_{i,j} \|\alpha_i, \beta_j\| \ \ge \ |{\operatorname{Res}}(\varphi)|$. Since $\GIR(\varphi) \le 1$ and $B_0(\varphi) \le 1$, the last two inequalities in the Corollary are immediate.
Proofs of Theorems \[FirstCor\], \[MainThm0\], and \[Classical\_Lip\] {#MainThmSection}
=====================================================================
Our main result is
\[MainThm\] Let $K$ be a complete, algebraically closed nonarchimedean field, and let $\varphi(z) \in K(z)$ have degree $d \ge 1$. Then $$\label{MainBound}
{\operatorname{Lip}}_\Berk(\varphi) \ \le \ \max\Big( \frac{1}{\GIR(\varphi) \cdot B_0(\varphi)^d} \, ,
\frac{d}{\GIR(\varphi)^{1/d} \cdot B_0(\varphi)} \Big) \ .$$
This is a restatement of Theorem \[MainThm0\] in the Introduction. Before giving the proof of Theorem \[MainThm\], we will make some reductions. Put $B_0 = B_0(\varphi)$, and consider a ball $\cB(a,B_0)^-$. By the definition of $B_0$, the image $\varphi(\cB(a,B_0)^-)$ is a ball. In particular there is an $\alpha \in \PP^1(K)$ with $\alpha \notin \varphi(\cB(a,B_0)^-)$. By choosing $\gamma_1, \gamma_2 \in {\operatorname{GL}}_2(\cO)$ with $\gamma_1(\alpha) = \infty$, $\gamma_2(0) = a$, and replacing $\varphi$ with $\Phi = \gamma_1 \circ \varphi \circ \gamma_2$, we can arrange that that $a = 0$ and that $\Phi(D(0,B_0)^-)$ omits $\infty$.
This means that $\Phi(D(0,B_0)^-)$ is a disc $D(c_0,R)^-$, where $\Phi(0) = c_0$. By the Weierstrass Preparation theorem, we can expand $\Phi(z)$ on $D(0,B_0)^-$ in the form $$\Phi(z) \ = \ c_0 + (c_1 z + c_2 z^2 + \cdots + c_n z^n) \cdot U(z)$$ where $1 \le n \le d$ is the number of solutions to $\Phi(z) = c_0$ in $B(0,B_0)^-$, and $U(z) = 1 + u_1 z + u_2 z^2 + \cdots$ is a unit power series converging on $D(0,B_0)^-$; here $|u_i| \le 1/B_0^i$ for each $i$.
Put $$f_\Phi(r) \ = \ \max_{1 \le k \le n} |c_k| r^k \ .$$ By the theory of Newton polygons, for each $r \in |K^{\times}|$ with $0 < r < B_0$, we have $\Phi(D(a,r)) = D(c_0,f(r))$, so $$\label{MapFormula}
\Phi(\zeta_{0,r}) \ = \ \zeta_{c_0,f_\Phi(r)}$$ By the continuity of the action of $\Phi$ on ${\operatorname{{\bf P}}}^1_K$, (\[MapFormula\]) holds for all $r \in [0,B_0]$, and $f_\Phi(B_0) = R$.
We will prove Theorem \[MainThm\] by applying Corollary \[SupCor\], and dealing with five cases: four cases dealing with radial paths $[0,B_0]$ contained in balls $\cB(0,B_0)^-$ according as $$\left\{\begin{array}{l}
\text{$|c_0| \le 1$ and $f_\Phi(B_0) \le 1$ ,} \\
\text{$|c_0| \le 1$ and $f_\Phi(B_0) > 1$ ,} \\
\text{$|c_0| > 1$ and $f_\Phi(B_0) \le |c_0|$ ,} \\
\text{$|c_0| > 1$ and $f_\Phi(B_0) > |c_0|$ ,}
\end{array} \right.$$ and one case dealing with radial paths $[\xi,\zeta_G]$ in the central ball $$\cB_\rho(\zeta_G,-\log(B_0))^- \ = \ \{x \in {\operatorname{{\bf P}}}^1_K : {\operatorname{diam}}_G(x) \ge B_0 \} \ .$$
Case 1 is covered by the following Proposition:
\[Case1\] Let $\Phi(z) \in K(z)$ have degree $d \ge 1$; write $B_0 = B_0(\Phi)$, put $c_0 = \Phi(0)$, and assume $\Phi$ has no poles in $D(0,B_0)^-$, so $\Phi(D(0,B_0)^-) = D(c_0,R)^-$ is a disc. Suppose $|c_0| \le 1$ and $R \le 1$. Then the restriction of $\Phi$ to $[0,\zeta_{0,B_0}]$ satisfies $${\operatorname{Lip}}_\Berk\big(\Phi\vert_{[0,\zeta_{0,B_0}]}\big) \ \le \ \frac{d}{B_0} \ .$$
As in Lemma \[fprimeMonoLemma\] we can partition $[0,B_0]$ into subintervals $[r_i,r_{i+1}]$ where $0 = r_1 < \cdots < r_{\ell+1} = B_0$, such that on $[r_{i-1},r_i]$ we have $$f_\Phi(r) \ = \ f_i(r) \ = \ |c_{k(i)}| \cdot r^{k(i)} \ .$$ Write $f_\Phi^{\prime}(r)$ for the right-derivative of $f_\Phi(r)$ on $[0,B_0)$. By Lemma \[fprimeMonoLemma\], $f_\Phi^{\prime}(r)$ is non-decreasing. Hence $$\label{fprimeLimit1}
\sup_{r \in [0,B_0)} f_\Phi^\prime(r) \ = \ \lim_{r \rightarrow B_0^-} f_\ell^\prime(r)
\ = \ k(\ell) \cdot |c_{k(\ell)}| \cdot B_0^{k(\ell) - 1} \ = \ \frac{k(\ell) \cdot f_\Phi(B_0)}{B_0} \ .$$
Since $|c_0| \le 1$ and $f_\Phi(B_0) \le 1$, we have $\Phi(D(0,B_0)^-) \subset D(0,1)$, so $F_\Phi(r) = f_\Phi(r)$ for all $r \in [0,B_0]$. Using the inequalities $k(\ell) \le n \le d$ and $f_\Phi(B_0) \le 1$ we conclude from (\[fprimeLimit1\]) that $${\operatorname{Lip}}_\Berk\big(\Phi\vert_{[0,B_0)}\big) \ = \ \sup_{r \in [0,B_0)} f_\Phi^\prime(r)\ \le \ \frac{d}{B_0} \ .$$
To deal with Case 2, we will need several lemmas. The first is an elementary maximization bound from Calculus:
\[CalculusLemma\] Let $H \ge 1$, and put $g(x) = x \cdot H^{1/x}$ for $x > 0$. Then for each closed interval $[a,b] \subset (0,\infty)$, $$\max_{x \in [a,b]} g(x) \ = \ \max\big(g(a),g(b)\big) \ .$$
If $H = 1$ then $g(x) = x$ and the result is trivial. If $H > 1$, then $g^{\prime}(x) = (1-\ln(H)/x) \cdot H^{1/x}$ and $g^{\prime \prime}(x) = (\ln(H))^2/x^3 \cdot H^{1/x}$, so $g(x)$ is convex up for $x > 0$, and its unique minimum is at $x = \ln(H)$. Thus the maximum value of $g(x)$ on $[a,b]$ is achieved at an endpoint.
The second is a bound for $\lim_{|z| \rightarrow 1^-} |\Phi(z)|$. Recall that if $P,Q$ are distinct points in ${\operatorname{{\bf P}}}^1_K$, the annulus ${\operatorname{Ann}}(P,Q)$ is the component of ${\operatorname{{\bf P}}}^1_K \backslash \{P,Q\}$ containing $(P,Q)$.
\[LimitLemma\] Let $\Phi(z) \in K(z)$ have degree $d \ge 2$; write $B_0 = B_0(\Phi)$. If $\Phi(\zeta_G) = \zeta_{a,R}$, then $$\label{Want}
\lim_{\substack{|z| \rightarrow 1^- \\ z \in K}} |\Phi(z)| \ = \ \max(|a|,R) \ .$$
Let $\vv_0 \in T_{\zeta_G}$ be the tangent direction towards $0$, and put $\vw = \Phi_*(\vv_0) \in T_{\zeta_{a,R}}$. Fix a point $b \in B_{\zeta_{a,R}}(\vw)^- \cap \PP^1(K)$. By ([@B-R], Corollary 9.21 and Lemma 9.45), there are points $X \in (\zeta_G,0)$, $Y \in (\zeta_{a,R},b)$ such that $\Phi$ maps $[\zeta_G,X]$ homeomorphically onto $[\zeta_{a,R},Y]$, and for each $x \in [\zeta_G,X]$, $\Phi$ maps ${\operatorname{Ann}}(\zeta_G,x)$ onto ${\operatorname{Ann}}(\zeta_{a,R},\Phi(x))$. Put $r = {\operatorname{diam}}_\infty(x)$, $S = {\operatorname{diam}}_\infty(\Phi(x))$. When $r \rightarrow 1^-$, then $\Phi(x) \rightarrow \zeta_{a,R}$ and $S \rightarrow R$. Note that $$K \cap {\operatorname{Ann}}(\zeta_G,x) \ = \ \{ z \in K : r < |z| < 1 \} \ .$$
Consider the possibilities for $\Phi\big(K \cap {\operatorname{Ann}}(\zeta_G,x)\big)$. If $|a| \le R$, then $\zeta_{a,R} = \zeta_{0,R}$. In this situation, if $\vw \in T_{\zeta_{0,R}}$ points towards $0$, then for $r$ near enough $1$, $\Phi\big(K \cap {\operatorname{Ann}}(\zeta_G,x)\big) = \{ z \in K : S < |z| < R \}$. If $\vw$ points towards $\infty$, then for $r$ near enough $1$, $\Phi\big(K \cap {\operatorname{Ann}}(\zeta_G,x)\big) = \{ z \in K : R < |z| < S \}$. Otherwise, $\Phi\big(K \cap {\operatorname{Ann}}(\zeta_G,x)\big) \subset D(b,R)^- \subset \{ z \in K : |z| = R \}$. In any case, $$\lim_{\substack{|z| \rightarrow 1^- \\ z \in K}} |\Phi(z)| \ = \ R \ = \ \max(|a|,R) \ .$$ If $|a| > R$, put $R_0 = |\,a|$. Regardless of the direction $\vw$, when $r$ is close enough to $1$ we will have $\Phi\big(K \cap {\operatorname{Ann}}(\zeta_G,x)\big) \subset D(a,R_0)^- \subset \{z \in K : |z| = |a| \}$. Thus $$\lim_{\substack{|z| \rightarrow 1^- \\ z \in K}} |\Phi(z)| \ = \ |a| \ = \ \max(|a|,R) \ .$$ Hence (\[Want\]) holds in all cases.
The third is a bound for $f_\Phi(B_0)$:
\[GrowthBound2\] Let $\Phi(z) \in K(z)$ have degree $d \ge 2;$ write $B_0 = B_0(\Phi)$. Assume that $\Phi(0) = 0$, and that $\Phi$ has no poles in $D(0,B_0)^-$, so $\Phi(D(0,B_0)^-) = D(0,f_\Phi(B_0))^-$ is a disc, and $f_\Phi(r) = {\operatorname{diam}}_\infty\big(\Phi(\zeta_{0,r})\big)$ is increasing for $0 \le r \le B_0$. Suppose $\Phi$ has $n \ge 1$ zeros in $D(0,B_0)^-$, and $m \ge 0$ poles in $D(0,1)^- \backslash D(0,B_0)^-$ $($counting multiplicities$)$. Then $$f_\Phi(B_0) \ \le \ \frac{B_0^{n-m}} {\GIR(\Phi)} \ .$$
Since $\Phi(0) = 0$ and $\Phi$ has no poles in $D(0,B_0)^-$, for each $0 < r \le B_0$, $\Phi(D(0,r)^-)$ is a disc $D(0,f_\Phi(r))^-$; clearly $f_\Phi(r)$ is increasing with $r$.
We can write $$\label{FacF1}
\Phi(z) \ = \ C \cdot \frac{\prod_{i=1}^N (z-\alpha_i)}{\prod_{j=1}^M (z-\beta_j)}$$ where $C \ne 0$ is a constant, $\alpha_1, \ldots, \alpha_N$ are the zeros of $\Phi$ in $K$ (listed with multiplicity), and $\beta_1, \ldots, \beta_M$ are the poles of $\Phi$ in $K$ (listed with multiplicity). Since ${{\operatorname{deg}}}(\Phi) = d$, $\max(N,M) = d$. Without loss, we can assume that $0 = |\alpha_1| \le |\alpha_2| \le \cdots \le |\alpha_N|$. Since $\Phi$ has $n$ zeros in $D(0,B_0)^-$ and $\Phi(D(0,B_0)^-) = D(0,f_\Phi(B_0))^-$, (\[FacF1\]) gives $$\label{CValue21}
f_\Phi(B_0) \ = \ \lim_{|z| \rightarrow B_0^-} |\Phi(z)|
\ = \ |C| \cdot \frac{B_0^n \cdot \prod_{i=n+1}^N \max(B_0,|\alpha_i|)}{\prod_{j=1}^M \max(B_0,|\beta_j|)} \ .$$ Also, writing $\Phi(\zeta_G) = \zeta_{a,R}$, by Lemma \[LimitLemma\] and formula (\[diamGformula\]) we have $$\label{LimF11}
\lim_{\substack{|z| \rightarrow 1^- \\ z \in K}} |\Phi(z)| \ = \ \max(|a|,R)
\ \le \ \frac{\max(1,|a|,R)^2}{R} \ = \ \frac{1}{\GIR(\Phi)} \ .$$ Using (\[FacF1\]) to evaluate $\lim_{|z| \rightarrow 1^-}|\Phi(z)|$ in (\[LimF11\]), we see that $$\label{CValue11}
|C| \cdot \frac{\prod_{i=1}^M \max(1,|\alpha_i|)}{\prod_{j=1}^N \max(1,|\beta_j|)} \ \le \ \frac{1}{\GIR(\Phi)} \ .$$
Using (\[CValue11\]) to eliminate $|C|$ in (\[CValue21\]), and recalling that $\Phi$ has no poles in $B(0,B_0)^-$ and $m$ poles in $B(0,1)^- \backslash B(0,B_0)^-$, we obtain $$f_\Phi(B_0) \ \le \ \frac{1}{\GIR(\Phi)} \cdot \frac{B_0^n \cdot \prod_{B_0 \le |\alpha_i| < 1} |\alpha_i|}{\prod_{B_0 \le |\beta_j| < 1} |\beta_j| }
\ \le \ \frac{B_0^{n-m}}{\GIR(\Phi)} \ .$$
Case 2 is covered by the following Proposition:
\[Case2\] Let $\Phi(z) \in K(z)$ have degree $d \ge 1$; write $B_0 = B_0(\Phi)$, and put $c_0 = \Phi(0)$. Assume $\Phi$ has no poles in $D(0,B_0)^-$, so $\Phi(D(0,B_0)^-) = D(c_0,R)^-$ is a disc. Suppose $|c_0| \le 1$ and $R > 1$. Then the restriction of $\Phi$ to $[0,\zeta_{0,B_0}]$ satisfies $$\label{Case2Formula}
{\operatorname{Lip}}_\Berk\big(\Phi\vert_{[0,\zeta_{0,B_0}]}\big) \ \le \
\max\big( \frac{1}{\GIR(\Phi) \cdot B_0^d}, \frac{d}{\GIR(\Phi)^{1/d} \cdot B_0} \big) \ .$$
Since $\gamma_{c_0}(z) := z-c_0 \in {\operatorname{GL}}_2(\cO)$, after replacing $\Phi(z)$ with $\gamma_{c_0} \circ \Phi(z) = \Phi(z) - c_0$, we can assume that $\Phi(0) = 0$. By the Weierstrass Preparation theorem, we can expand $\Phi(z)$ in $D(0,B_0)^-$ as $$\Phi(z) \ = \ (c_1 z + \cdots + c_n z^n) \cdot U(z)$$ where $U(z)$ is a unit power series. Hence $$f_\Phi(r) \ = \ \max_{1 \le k \le n}( |c_k| r^k) \ , \qquad
F_\Phi(r) \ = \ \left\{ \begin{array}{ll} f_\Phi(r) & \text{if $f_\Phi(r) \le 1$ \ , } \\
1/f_\Phi(r) & \text{if $f_\Phi(r) \ge 1$ \ .}
\end{array} \right.$$ By Lemma \[fprimeMonoLemma\], there is a partition $0 = r_1 < \cdots < r_{\ell+1} = B_0$ of $[0,B_0]$ such that for each subinterval $[r_i,r_{i+1}]$ there is an index $k(i)$ for which $f_\Phi(r) = |c_{k(i)}| r^{k(i)}$. After inserting an extra partition point if necessary, we can assume there is an $i_0$ for which $f_\Phi(r_{i_0}) = 1$. The right-derivative $f_\Phi^\prime(r)$ is non-decreasing on $[0,B_0]$, and $k(i-1) \le k(i)$ for each $i =2, \ldots, \ell$.
We will now bound the absolute value of the right-derivative $F_\Phi^{\prime}(r)$ on $[0,B_0)$. Since $F_\Phi(r) = f_\Phi(r)$ on $[0,r_{i_0})$, $$\sup_{r \in [0,r_{i_0})} |F_\Phi^\prime(r)|
\ = \ \lim_{r \rightarrow r_{i_0}^-} f_\Phi^{\prime}(r) \ \le \ f_\Phi^\prime(r_{i_0})
\ = \ \frac{k(i_0)}{r_{i_0}} \ .$$ For each $i \ge i_0$, on the interval $[r_i,r_{i+1})$ $$|F_\Phi^\prime(r)| \ = \ \left|-\frac{f_\Phi^{\prime}(r)}{f_\Phi(r)^2}\right|
\ = \ \frac{k(i)}{r} \cdot \frac{1}{f_\Phi(r)}
\ \le \ \frac{k(i)}{r_i \cdot f_\Phi(r_i)} \ \le \ \frac{k(i)}{r_i}\ .$$ Thus by Corollary \[DerivCor\], $$\begin{aligned}
{\operatorname{Lip}}_\Berk(\Phi\vert_{[0,B_0]}) \ = \ \sup_{r \in [0,B_0)} |F_\Phi^{\prime}(r)|
& = & \max_{i_0 \le i \le \ell} \ \frac{k(i)}{r_i f_\Phi(r_i)} \label{FprimeBasicIneq1} \\
& \le & \max_{i_0 \le i \le \ell} \frac{k(i)}{r_i} \ . \label{FprimeBasicIneq2}\end{aligned}$$ Here the second and third expressions in (\[FprimeBasicIneq1\]) are equal by the right-continuity of $f_\Phi^{\prime}(r)$.
For each $k = 1, \ldots, n$ with $c_k \ne 0$, let $u_k > 0$ be the unique solution to $|c_k| r^k = 1$. For brevity, write $G_0 = \GIR(\Phi)$. By the nonarchimedean Maximum Modulus principle and Lemma \[GrowthBound2\], we have $|c_k| B_0^k \le f_\Phi(B_0) \le B_0^{n-m}/G_0$, where $n$ is the number of zeros of $\Phi$ in $D(0,B_0)^-$ and $m$ is the number of poles of $\Phi$ in $D(0,1)^- \backslash D(0,B_0)^-$. Hence $|c_k| \le B_0^{n-m-k}/G_0$, so $1 = |c_k| (u_k)^k \le B_0^{n-m-k} \cdot (u_k)^k/G_0$, and $$\label{uIneq}
\frac{1}{u_k} \ \le \ B_0^{-1} \cdot \Big( \frac{B_0^{n-m}}{G_0} \Big)^{1/k} \ .$$ For each $i \ge i_0$, we have $u_{k(i)} \le r_i$. Using Corollary \[DerivCor\] and (\[FprimeBasicIneq2\]), (\[uIneq\]), we see that $$\begin{aligned}
{\operatorname{Lip}}_\Berk\big(\Phi\vert_{[0,\zeta_{0,B_0}]}\big) & = & \sup_{r \in [0,B_0)} |F_\Phi^{\prime}(r)|
\ \le \ \max_{i_0 \le i \le \ell} \frac{k(i)}{r_i}
\ \le \ \max_{i_0 \le i \le \ell} k(i) \cdot B_0^{-1} \cdot \Big( \frac{B_0^{n-m}}{G_0} \Big)^{1/k(i)} \notag \\
& \le & B_0^{-1} \cdot\max_{1 \le k \le n} k \cdot \Big( \frac{B_0^{n-m}}{G_0} \Big)^{1/k} \ .
\label{fIntermediate}\end{aligned}$$
However, we want a bound for ${\operatorname{Lip}}_\Berk\big(\Phi\vert_{[0,\zeta_{0,B_0}]}\big)$ independent of $n$ and $m$. By the discussion above $1 \le k \le n \le d$ and $0 \le m \le d$. We need only consider pairs $(n,m)$ for which $B_0^{n-m}/G_0 > 1$, since by assumption $1 < f_\Phi(B_0)$ and Lemma \[GrowthBound2\] gives $f_\Phi(B_0) \le B_0^{n-m}/G_0$. Letting $(k,n,m)$ range over all triples of integers satisfying these conditions we see that $$\label{kmnFormula}
{\operatorname{Lip}}_\Berk\big(\Phi\vert_{[0,\zeta_{0,B_0}]}\big)
\ \le \ \max_{\substack{ 0 \le m \le d \\ 1 \le n \le d \\ B_0^{n-m}/G_0 > 1}}
\Big( \ B_0^{-1} \cdot \max_{1 \le k \le n} k \cdot \Big( \frac{B_0^{n-m}}{G_0} \Big)^{1/k} \ \Big) \ .$$
We will now bound the right side of (\[kmnFormula\]). Fixing $n$ and $m$ with $B_0^{n-m}/G_0 > 1$, and taking $H = B_0^{n-m}/G_0$ in Lemma \[CalculusLemma\], shows that $$B_0^{-1} \cdot \max_{1 \le k \le n} k \cdot \Big( \frac{B_0^{n-m}}{G_0} \Big)^{1/k}
\ = \ \max \Big( \ \frac{B_0^{n-m-1}}{G_0}, n \Big(\frac{B_0^{-m}}{G_0} \Big)^{1/n} \ \Big) \ .$$ Inserting this in (\[kmnFormula\]), interchanging the order of the maxima, and dropping the condition $B_0^{n-m}/G_0 > 1$ gives $$\label{kmnFormula2}
{\operatorname{Lip}}_\Berk\big(\Phi\vert_{[0,\zeta_{0,B_0}]}\big) \ \le \
\max \Big( \ \max_{\substack{ 0 \le m \le d \\ 1 \le n \le d}} \frac{B_0^{n-m-1}}{G_0} ,
\max_{\substack{ 0 \le m \le d \\ 1 \le n \le d }}
n \Big(\frac{B_0^{-m}}{G_0} \Big)^{1/n} \ \Big) \ .$$ The first inner maximum in (\[kmnFormula2\]) is $B_0^{-d}/G_0$, achieved when $n = 1$ and $m = d$. For the second inner maximum, fixing $m$ and taking $H = B_0^{-m}/G_0$ in Lemma \[CalculusLemma\] gives $$\max_{ 1 \le n \le d } n \Big(\frac{B_0^{-m}}{G_0} \Big)^{1/n}
\ = \ \max \Big( \frac{B_0^{-m}}{G_0}\ , d \Big(\frac{B_0^{-m}}{G_0} \Big)^{1/d} \Big)$$ The maximum of this for $0 \le m \le d$ is attained when $m = d$, and is $$\max \Big( \frac{1}{ G_0 \cdot B_0^d} \ , \frac{d}{G_0^{1/d} \cdot B_0} \Big)$$ Combining these results gives (\[Case2Formula\]).
Case 3 is covered by the following result:
\[Case3\] Let $\Phi(z) \in K(z)$ have degree $d \ge 1$; write $B_0 = B_0(\Phi)$, put $c_0 = \Phi(0)$, and assume that $\Phi(D(0,B_0)^-) = D(c_0,R)^-$. Suppose $|c_0| > 1$ and $R \le |c_0|$. Then the restriction of $\Phi$ to $[0,\zeta_{0,B_0}]$ satisfies $${\operatorname{Lip}}_\Berk\big(\Phi\vert_{[0,\zeta_{0,B_0}]}\big) \ \le \ \frac{d}{B_0} \ .$$
Partition $[0,B_0]$ by taking $0 = r_1 < \cdots < r_{\ell+1} = B_0$ so that $$f_\Phi(r) \ = \ f_i(r) \ := \ |c_{k(i)}| \cdot r^{k(i)}$$ on $[r_i,r_{i+1})$. Write $f_\Phi^{\prime}(r)$ for the right-derivative of $f_\Phi(r)$. Just as in Proposition \[Case1\], $$\label{fprimeLimit2}
\sup_{r \in [0,B_0)} f_\Phi^\prime(r) \ = \ \lim_{r \rightarrow B_0^-} f_\ell^\prime(r)
\ = \ k(\ell) \cdot |c_{k(\ell)}| \cdot B_0^{k(\ell) - 1} \ = \ \frac{k(\ell) \cdot f_\Phi(B_0)}{B_0} \ .$$
Since $|c_0| > 1$ and $f_\Phi(B_0) \le |c_0|$, we have $\Phi(D(0,B_0)^-) \subset D(c_0,|c_0|)^-$, hence $F_\Phi(r) = f_\Phi(r)/|c_0|$ and $F_\Phi^{\prime}(r) = f_\Phi^{\prime}(r)/|c_0|$ for all $r \in [0,B_0)$. Using that $k(\ell) \le n \le d$ and $f_\Phi(B_0) \le |c_0|$ we conclude from (\[fprimeLimit2\]) that $${\operatorname{Lip}}_\Berk\big(\Phi\vert_{[0,B_0)}\big) \ = \ \sup_{r \in [0,B_0)} F_\Phi^\prime(r)\ \le \ \frac{d}{B_0} \ .$$
Case 4 reduces to Case 2 by a trick:
\[Case4\] Let $\Phi(z) \in K(z)$ have degree $d \ge 1$; write $B_0 = B_0(\Phi)$, put $c_0 = \Phi(0)$, and assume $\Phi(D(0,B_0)^-) = D(c_0,R)^-$. Suppose $|c_0| > 1$ and $R > |c_0|$. Then the restriction of $\Phi$ to $[0,\zeta_{0,B_0}]$ satisfies $${\operatorname{Lip}}_\Berk\big(\Phi\vert_{[0,\zeta_{0,B_0}]}\big) \ \le \
\max\big( \frac{1}{\GIR(\Phi) \cdot B_0^d}, \frac{d}{\GIR(\Phi)^{1/d} \cdot B_0} \big) \ .$$
Since $f_\Phi(r)$ is continuous and monotonic, with $f_\Phi(0) = 0$ and $f_\Phi(B_0) > |c_0| > 1$, there is a unique $0 < R < B_0$ with $f_\Phi(R) = |c_0|$. For this $R$, the theory of Newton polygons shows that $\Phi(D(0,R)) = D(c_0,|c_0|) = D(0,|c_0|)$, so there is an $\alpha \in D(0,R)$ for which $\Phi(\alpha) = 0$. Write $\gamma_\alpha(z) = z + \alpha \in {\operatorname{GL}}_2(\cO)$, and put $$\Psi(z) \ = \ \Phi(z+\alpha) \ = \ (\Phi \circ \gamma_\alpha)(z) \ .$$ By construction $\Psi(0) = 0$, so $\Psi$ satisfies the conditions of Proposition \[Case2\]. Since $\gamma_\alpha(z) \in {\operatorname{GL}}_2(\cO)$ we have $\GIR(\Psi) = \GIR(\Phi)$ and $B_0(\Psi) = B_0(\Phi) = B_0$.
We will prove Proposition \[Case4\] by showing that ${\operatorname{Lip}}_\Berk(\Phi_{[0,B_0]}) \le {\operatorname{Lip}}_\Berk(\Psi_{[0,B_0]})$ and applying Proposition \[Case2\] to $\Psi$.
For each $r$ with $R < r < B_0$, we have $\Psi(\cD(0,r)^-) = \Phi(\cD(0,r)^-)$, so $f_\Psi(r) = f_\Phi(r)$ for $R \le r \le B_0$. As usual, we can write $\Phi(z) = c_0 + (c_1 z + \cdots + c_n z^n) \cdot U(z)$, where $U(z)$ is a unit power series converging on $\cD(0,B_0)^-$; then $$f_\Phi(r) \ = \ \max_{1 \le k \le n} |c_k| r^k$$ for each $r \in [0,B_0]$. Likewise we can write $\Psi(z) = (C_1 z + \cdots + C_N z^N) \cdot W(z)$, where $W(z)$ is a unit power series converging on $D(0,B_0)^-$; and $$f_\Psi(r) \ = \ \max_{1 \le k \le N} |C_k| r^k$$ for each $r \in [0,B_0]$.
Partition $[0,B_0]$ simultaneously for $\Phi$ and $\Psi$, choosing $0 = r_1 < \cdots < r_{\ell+1} = B_0$ so that for each $i = 1, \ldots, \ell$ there are indices $1 \le j(i) \le n$, $1 \le k(i) \le N$ such that on $[r_i,r_{i+1})$ we have $$f_\Phi(r) \ = \ |c_{j(i)}| \cdot r^{j(i)} \ , \qquad f_\Psi(r) \ = \ |C_{k(i)}| \cdot r^{k(i)} \ .$$ After refining the partition if necessary, we can assume there are indices $i_0$, $i_1$ such that $f_\Psi(r_{i_0}) = 1$ and $f_\Psi(r_{i_1}) = f_\Phi(r_{i_1}) = |c_0|$ (evidently $r_{i_1} = R$). Clearly $i_0 < i_1$, since $f_\Psi$ is monotonic.
We claim that $j(i) = k(i)$ for $i = i_1, \cdots, \ell$. To see this, note first that for each $r$ with $r_{i_1} \le r \le B_0$, we have $f_\Phi(r) = f_\Psi(r)$. For each $r \in |K^{\times}|$ with $r_i < r < r_{i+1}$, and each $w \in K$ with $|w| \le f_{\Psi}(r)$, the theory of Newton polygons shows that $\Phi(z) = w$ has $j(i)$ solutions in $D(0,r)$, counting multiplicities. Similarly $\Psi(z) = w$ has $k(i)$ solutions in $D(0,r)$ counting multiplicities. But $\Phi(z) = w$ if and only if $\Psi(z-\alpha) = w$. Since $|\alpha| = r_{i_1} \le r$, we have $|z - \alpha| \le r$ if and only if $|z| \le r$. Hence $j(i) = k(i)$.
We have $$F_\Phi(r) \ = \ \left\{ \begin{array}{ll} f_\Phi(r)/|c_0|^2 & \text{if $r \in [0,r_{i_1})$ \ , } \\
1/f_\Phi(r) & \text{if $r \in [r_{i_1},B_0)$ \ ,}
\end{array} \right.$$ and $$F_\Psi(r) \ = \ \left\{ \begin{array}{ll} f_\Psi(r) & \text{if $r \in [0,r_{i_0})$ \ , } \\
1/f_\Psi(r) & \text{if $r \in [r_{i_0},B_0)$ \ .}
\end{array} \right.$$ Write $f_\Phi^{\prime}(r)$ for the right-derivative of $f_\Phi(r)$ on $[0,B_0)$, and $F_\Phi^{\prime}(r)$ for the right-derivative of $F_\Phi(r)$. Noting that $|c_0| = f_\Phi(r_{i_1}) = |c_{k(i_1)}| (r_{i_1})^{j(i_1)}$ and recalling from Lemma \[fprimeMonoLemma\] that $f_\Phi^{\prime}(r)$ is non-decreasing with $r$, we see that $$\sup_{r \in [0,r_{i_1})} |F_\Phi^{\prime}(r)| \ \le \ \frac{f_\Phi^{\prime}(r_{i_1})}{|c_0|^2}
\ = \ \frac{k(i_1) |c_{j(i_1)}| (r_{i_1})^{j(i_1)-1}}{|c_0|^2} \ = \ \frac{j(i_1)}{r_{i_1}f_\Phi(r_{i_1})} \ .$$ For each $i = i_1, \cdots, \ell$, on $[r_i,r_{i+1})$ we have $F_\Phi^{\prime}(r) = -f_\varphi^{\prime}(r)/(f_\varphi(r))^2$, so $$\sup_{r \in [r_i,r_{i+1})} |F_\Phi^{\prime}(r)| \ =
|F_\Phi^{\prime}(r_i)| \ = \ \left| \frac{-f_\Phi^{\prime}(r_i)} {(f_{\Phi}(r_i))^2} \right|
\ = \ \frac{j(i)}{r_i f_\Phi(r_i)} \ .$$ Thus $${\operatorname{Lip}}_\Berk(\Phi\vert_{[0,B_0]}) \ = \ \sup_{r \in [0,B_0)} |F_\Phi^{\prime}(r)|
\ = \ \max_{i_1 \le i \le \ell} \Big( \frac{j(i)}{r_i f_\Phi(r_i)} \Big) \ .$$ Since $i_0 \le i_1$, and $k(i) = j(i)$ and $f_\Phi(r_i) = f_\Psi(r_i)$ for each $i \ge i_1$, by applying (\[FprimeBasicIneq1\]) with $\Phi$ replaced by $\Psi$ and then using the bound for ${\operatorname{Lip}}_\Berk(\Psi\vert_{[0,B_0]})$ from Proposition \[Case2\], we get $$\begin{aligned}
{\operatorname{Lip}}_\Berk(\Phi|_{[0,B_0]})
& = & \max_{i_1 \le i \le \ell} \Big( \frac{j(i)}{r_i f_\Phi(r_i)} \Big)
\ \le \ \max_{i_0 \le i \le \ell} \Big( \frac{k(i)}{r_i f_\Psi(r_i)} \Big) \\
& = & {\operatorname{Lip}}_\Berk(\Psi|_{[0,B_0]})
\ \le \ \max\Big(\ \frac{1}{\GIR(\Phi) \cdot B_0^d}\ , \frac{d}{\GIR(\Phi)^{1/d} \cdot B_0} \ \Big) \ .\end{aligned}$$
Case 5 (the central ball) is dealt with by the following Proposition:
\[Case5\] Let $\Phi(z) \in K(z)$ have degree $d \ge 1$; write $B_0 = B_0(\Phi)$. Then $${\operatorname{Lip}}_\Berk\big(\Phi\vert_{[\zeta_{0,B_0},\zeta_G]}\big) \ \le \ \frac{d}{B_0} \ .$$
We use the fact that in the $\rho$-metric, along a given segment $\Phi$ locally scales distances by an integer $1 \le m \le d$. To obtain the Lipschitz bound for the $d$-metric, we conjugate this between the $d$- and $\rho$-metrics. Fix a base $q > 1$ such that for each $0 < r \le 1$ we have $\rho(\zeta_G,\zeta_{0,r}) = -\log_q(r)$, and put $E(z) = q^z$, $L(r) = \log_q(r)$. Define $F_\Phi : (B_0,1] \rightarrow (0,1]$ by $$F_\Phi(r) \ = \ {\operatorname{diam}}_G\big(\Phi(\zeta_{0,r})\big) \ .$$
Note that $\zeta_G = \zeta_{0,1}$. Choose a partition $B_0 = r_1 < \cdots < r_{\ell+1} = 1$ of $[B_0,1]$ such that on each subinterval $[r_i,r_{i+1}]$, $\Phi$ has the following properties:
1. $\Phi$ maps the segment $[\zeta_{0,r_i},\zeta_{0,r_{i_1}}]$ homeomorphically onto some radial segment;
2. there is an integer $1 \le m_i \le d$ such that ${{\operatorname{deg}}}_\Phi(P) = m_i$ for all $P \in (\zeta_{0,r_i},\zeta_{0,r_{i+1}})$.
To prove the Proposition it suffices to show that ${\operatorname{Lip}}_\Berk\big(\Phi\vert_{[\zeta_{0,r_i},\zeta_{0,r_{i+1}}]}\big) \le d/B_0$ for each $i$.
Fix $i$. By (1) and (2) there is an affine function $M_i(y) = a_i y + b_i$, where $a_i = \pm m_i$ and $b_i \in \RR$, such that $\rho(\zeta_G,\Phi(\zeta_{0,r})) = M_i(L(r))$. Hence for each $r \in [r_i,r_{i+1}]$, $$F_\Phi(r) \ = \ E \circ M_i \circ L(r) \ .$$ In particular, $F_\Phi$ is differentiable on $(r_i,r_{i+1})$. By the Mean Value Theorem, for each $r, s$ with $r_i \le r < s \le r_{i+1}$ there is an $r_* \in (r,s)$ such that $$\frac{d(\Phi(\zeta_{0,r}),\Phi(\zeta_{0,s}))}{d(\zeta_{0,r},\zeta_{0,s})}
\ = \ \left| \frac{F_\Phi(r)-F_\Phi(s)}{r-s} \right|\ = \ |F_\Phi^\prime(r_*)| \ ,$$ so it will be enough to show that $|F_\Phi^{\prime}(r)| \le d/B_0$ on $(r_i,r_{i+1})$. However, this follows easily from the Chain rule: for each $r \in (r_i,r_{i+1})$ $$|F_\Phi^\prime(r)| \ = \ \big(q^{M_i(L(r))} \cdot \ln(q)\big) \cdot |a_i| \cdot \frac{1}{r \cdot \ln(q)}
\ = \ F_\Phi(r) \cdot \frac{m_i}{r} \ \le \ \frac{m_i}{r_i} \ \le \ \frac{d}{B_0} \ .$$
By Corollary \[SupCor\], it suffices to show that for each radial segment $I$ of the form $[\alpha,\xi]$ or $[\xi,\zeta_G]$, where $\alpha \in \PP^1(K)$ and ${\operatorname{diam}}_G(\xi) = B_0(\varphi)$, one has $$\label{DesiredBound}
{\operatorname{Lip}}_\Berk(\varphi\vert_I) \ \le \ \max\Big( \frac{1}{\GIR(\varphi) \cdot B_0(\varphi)^d} \, ,
\frac{d}{\GIR(\varphi)^{1/d} \cdot B_0(\varphi)} \Big) \ .$$
First suppose $I = [\alpha,\xi]$. By the definition of the ball-mapping radius $B_0 = B_0(\varphi)$, $\varphi(\cB(\alpha,B_0)^-)$ is a ball, and hence omits some $\beta \in \PP^1(K)$. Take any $\gamma_1 \in {\operatorname{GL}}_2(\cO)$ with $\gamma_2(\beta) = \infty$, and take any $\gamma_2 \in {\operatorname{GL}}_2(\cO)$ with $\gamma_2(0) = \alpha$. Put $\Phi = \gamma_1 \circ \varphi \circ \gamma_2$. Then $[\alpha,\xi] = \gamma_2([0,\zeta_{0,B_0}])$, ${\operatorname{Lip}}_\Berk(\varphi\vert_{[\alpha,\xi]}) ={\operatorname{Lip}}_\Berk(\Phi\vert_{[0,\zeta_{0,B_0}]})$, and $\Phi(\cD(0,B_0)^-)$ is a disc $\cD(c_0,R)^-$ for some $c_0 \in K$ and some $0 < R < \infty$. Propositions \[Case1\], \[Case2\], \[Case3\], and \[Case4\] cover all possibilities for $|c_0|$ and $R$, and they show that (\[DesiredBound\]) holds.
Next suppose $I = [\xi,\zeta_G]$. Take any type II point $\xi_0 \in (\xi,\zeta_G)$, and let $\alpha_0 \in \PP^1(K)$ be such that $\xi_0 \in [\alpha_0,\zeta_G]$. Choose any $\gamma_2 \in {\operatorname{GL}}_2(\cO)$ with $\gamma_2(0) = \alpha_0$; then $[\xi_0,\zeta_G] \subset \gamma_2([\zeta_{0,B_0},\zeta_G])$. Put $\Phi = \varphi \circ \gamma_2$. Then ${\operatorname{Lip}}_\Berk(\varphi\vert_{[\xi_0,\zeta_G]}) \le {\operatorname{Lip}}_\Berk(\Phi\vert_{[\zeta_{0,B_0},\zeta_G]})$, and Proposition \[Case5\] shows that ${\operatorname{Lip}}_\Berk(\Phi\vert_{[\zeta_{0,B_0},\zeta_G]})$ satisfies (\[DesiredBound\]). Since we can choose $\xi_0$ as close to $\xi$ as desired, ${\operatorname{Lip}}_\Berk(\varphi\vert_{[\xi,\zeta_G]})$ satisfies (\[DesiredBound\]) as well.
Given $\varphi$, we first show that $$\label{GIRsup}
\sup_{\substack{x, y \in \PP^1(K) \\ x \ne y}} \frac{\|\varphi(x),\varphi(y)\|}{\|x,y\|} \ \le \ \frac{1}{\GPR(\varphi)} \ .$$
Fix $x, y \in \PP^1(K)$ with $x \ne y$. We claim that $\|\varphi(x),\varphi(y)\|/\|x,y\| \le 1/\GPR(\varphi)$. If $\|x,y\| \ge \GPR(\varphi)$, the inequality is trivial since $\|\varphi(x),\varphi(y)\| \le 1$. Suppose $\|x,y\| < \GPR(\varphi)$. After pre-composing and post-composing $\varphi$ with suitable elements of ${\operatorname{GL}}_2(\cO)$, we can assume that $y = 0$ and $\varphi(y) = 0$. Put $R = \GPR(\varphi)$. By the definition of $\GPR(\varphi)$, the image $\varphi(\cD(0,R)^-)$ omits $\zeta_G$. In particular, $\varphi$ has no poles in $D(0,R)^-$ and $|\varphi(z)| < 1$ for all $z \in D(0,R)^-$. Thus we can expand $\varphi(z)$ as a power series converging in $D(0,R)^-$, $$\varphi(z) \ = \ \sum_{i=0}^\infty c_i z^i \ ,$$ where $c_0 = 0$ and $|c_i| \le 1/R^i$ for $i \ge 1$. Note that $\|x,y\| = |x-0| = |x|$, and that $$\|\varphi(x),\varphi(y)\| \ = \ |\varphi(x)-0| \ = \ |\sum_{i=1}^\infty c_i x^i|
\ \le \ \max_{i \ge 1} (|x|/R)^i \ = \ |x|/R \ .$$ It follows that $\|\varphi(x),\varphi(y)\|/\|x,y\| \le 1/R = 1/\GPR(\varphi)$.
To complete the proof, we will show that there exist $x, y \in \PP^1(K)$ with $$\frac{\|\varphi(x),\varphi(y)\|}{\|x,y\|} \ = \ \frac{1}{\GPR(\varphi)} \ .$$ Let $Q \in \varphi^{-1}(\zeta_G)$ be a point (necessarily of type II) for which ${\operatorname{diam}}_G(Q) = \GPR(\varphi)$. After post-composing $\varphi$ with a suitable element of ${\operatorname{GL}}_2(\cO)$, we can assume $Q = \zeta_{0,R}$; by construction, $\varphi(Q) = \zeta_G$. Consider the tangent space $T_Q$. For any $\vw \in T_{\zeta_G}$, there are at most $d$ directions $\vv \in T_Q$ with $\varphi_*(\vv) = \vw$. Also, there are only finitely many directions $\vv \in T_Q$ for which $\varphi(\cB(Q,\vv)^-) = {\operatorname{{\bf P}}}^1_K$. Hence, we can choose $\vv_1, \vv_2 \in T_Q \backslash \{\vv_\infty\}$ with $\varphi_*(\vv_1) \ne \varphi_*(\vv_2)$, such that $\varphi(\cB(Q,\vv_1)^-)$ and $\varphi(\cB(Q,\vv_2)^-)$ are balls. Take any $x \in \PP^1(K) \cap \cB(Q,\vv_1)^-$, $y \in \PP^1(K) \cap \cB(Q,\vv_2)^-$. Then $\|x,y\| = R$ and $\|\varphi(x),\varphi(y)\| = 1$.
In Theorem \[Classical\_Lip\] we have shown that ${\operatorname{Lip}}_{\PP^1(K)}(\varphi) \le 1/\GPR(\varphi)$. It follows from Corollary \[ResultantBoundCor\] that $1/|\GPR(\varphi)| \le 1/|{\operatorname{Res}}(\varphi)|$, so ${\operatorname{Lip}}_{\PP^1(K)}(\varphi) \le 1/|{\operatorname{Res}}(\varphi)|$.
In Theorem \[MainThm0\] we have shown that $$\label{MainBound1}
{\operatorname{Lip}}_\Berk(\varphi) \ \le \ \max\Big( \frac{d}{\GIR(\varphi)^{1/d} \cdot B_0(\varphi)} \, ,
\frac{1}{\GIR(\varphi) \cdot B_0(\varphi)^d} \Big) \ .$$ By Corollary \[ResultantBoundCor\] we have $\GIR(\varphi)^d \cdot B_0(\varphi) \ge |{\operatorname{Res}}(\varphi)|$. Since $1 \ge \GIR(\varphi) > 0$ it follows that $\GIR(\varphi)^{1/d} \ge \GIR(\varphi)^d$, which yields $\GIR(\varphi)^{1/d} \cdot B_0(\varphi) \ge |{\operatorname{Res}}(\varphi)|$. Similarly $\GIR(\varphi)\cdot B_0(\varphi) \ge |{\operatorname{Res}}(\varphi)|$ and $B_0(\varphi) \ge |{\operatorname{Res}}(\varphi)|$, so $\GIR(\varphi) \cdot B_0(\varphi)^d \ge |{\operatorname{Res}}(\varphi)|^d$. Thus ${\operatorname{Lip}}_\Berk(\varphi) \ \le \ \max\big( d/|{\operatorname{Res}}(\varphi)|, 1/|{\operatorname{Res}}(\varphi)|^d\big)$.
Examples {#ExamplesSection}
========
An analysis of the proof of Theorem \[MainThm\] leads to the following examples, which show the bound (\[MainBound\]) in Theorem \[MainThm\] is nearly optimal.
[**Example 1.**]{} Let $2 \le d \in \ZZ$, and fix $S \in |K^{\times}|$ with $0 < S \le 1$. Choose $\beta_1, \ldots, \beta_d \in K$ with $|\beta_i| = S$ for all $i$ and $|\beta_i - \beta_j| = S$ for all $i \ne j$. Fix an integer $1 \le k \le d-1$ and a constant $C \in K$ with $|C| \ge 1$, and put $$\varphi(z) \ = \ \frac{Cz^k}{(z-\beta_1) \cdots (z-\beta_d)} \ .$$ One sees easily that $\varphi(\zeta_G) = \zeta_{0,|C|}$, so $\GIR(\varphi) = 1/|C|$. Using the theory of Newton polygons, one sees that for each $a \in \PP^1(K)$, the image of $B(a,S)^-$ omits at least one point of $\PP^1(K)$, and that $\varphi(\cD(0,S)^-) = \cD(0,|C|/S^d)^-$ but $\varphi(\cD(0,S)) = \PP^1(K)$. Thus $B_0(\varphi) = S$. For $0 \le r \le S$ one has $$f_\varphi(r) \ := \ {\operatorname{diam}}_\infty(\varphi(\zeta_{0,r})) \ = \ |C| r^k/S^d \ .$$ Put $r_1 = (S^d/|C|)^{1/k} \le S$; then $f_\varphi(r_1) = 1$, and $${\operatorname{Lip}}_\Berk(\varphi) \ \ge \ f_\varphi^{\prime}(r_1) \ = \ k \cdot |C| r_1^{k-1}
\ = \ \frac{k}{r_1} \ = \ \frac{k}{\GIR(\varphi)^{1/k} \cdot B_0(\varphi)^{d/k}} \ .$$ Taking $k = 1$, one sees that the first term in (\[MainBound\]) cannot be improved. Taking $k = d-1$, one obtains a quantity which differs from the second term $d/\big(\GIR(\varphi)^{1/d} \cdot B_0(\varphi)\big)$ by a factor $\Delta$ satisfying $(d-1)/d < \Delta < 1$.
[**Example 2.**]{} With $d$, $S$, $C$ and the $\beta_i$ as in Example 1, put $$\varphi(z) \ = \ \frac{C z^d}{(z-\beta_1) \cdots (z-\beta_{d-1})} \ .$$ As before, one has $\GIR(\varphi) = 1/|C|$ and $B_0(\varphi) = S$. For $0 \le r \le S$ one has $$f_\varphi(r) \ := \ {\operatorname{diam}}_\infty(\varphi(\zeta_{0,r})) \ = \ |C| r^d/S^{d-1} \ .$$ Put $r_1 = (S^{d-1}/|C|)^{1/d} \le S$; then $f_\varphi(r_1) = 1$, and $${\operatorname{Lip}}_\Berk(\varphi) \ \ge \ f_\varphi^{\prime}(r_1) \ = \ d \cdot |C| r_1^{d-1}
\ = \ \frac{d}{r_1} \ = \ \frac{d}{\GIR(\varphi)^{1/d} \cdot B_0(\varphi)^{(d-1)/d}} \ ,$$ which differs from $d/\big(\GIR(\varphi)^{1/d} \cdot B_0(\varphi)\big)$ by the factor $\Delta = B_0(\varphi)^{1/d}$.
Thus the second term in $(\ref{MainBound})$ cannot be greatly improved, and when $B_0(\varphi) = 1$ it is sharp.
[999]{}
M. Baker and R. Rumely, Potential Theory and Dynamics on the Berkovich Projective Line, AMS Surveys and Monographs 159, Providence, 2010.
R. L. Benedetto, P. Ingram, R. Jones, and A. Levy, [*Critical orbits and attracting cycles in $p$-adic dynamics*]{}, Online preprint arXiv:12011605v2 (September 2012).
V. G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs 33, American Mathematical Society, Providence, RI, 1990.
X. Faber, [*Topology and Geometry of the Berkovich Ramification Locus I, II*]{}, Online preprints arXiv:1102:1432 and arXiv:1104:0943 (May 2011). I: to appear in Manuscripta Math.; II: to appear in Math. Annalen.
C. Favre and J. Rivera-Letelier, [*Equidistribution des points de petite hauteur*]{}, Math. Ann. 335(2), 2006, 311-361;. Online preprint arXiv:math/0407471.
C. Favre and J. Rivera-Letelier, [em Théorie ergodique des fractions rationelles sur un corps ultramétrique]{}, Proc. London Math. Soc. 100(1) (2010), 116-154.
S. Lang, Algebra (2nd edition), Addison-Wesley, Reading, 1984.
J. Rivera-Letelier, [*Espace hyperbolique $p$-adique et dynamique des fonctions rationelles*]{}, Compositio Math. 138(2) (2003), 199-231.
J. Rivera-Letelier, [*Pointes périodiques des fonctions rationelles dans l’esoace hyperbolique $p$-adique*]{}, Comment. Math. Helv., (2005), 80(3):593–629.
R. Rumely, [*The Minimal Resultant Locus*]{}, Online preprint arXiv:1304.1201 (April 2013).
R. Rumely, [*The Geometry of the Minimal Resultant Locus*]{}, Online preprint arXiv:1402.6017 (February 2014).
R. Rumely, [*A New Equivariant in Nonarchimedean Dynamics*]{}, submitted to the Duke Mathematical Journal.
J. Silverman, The Arithmetic of Dynamical Systems, GTM 241, Springer-Verlag, New York 2007.
|
---
abstract: 'The effects of introducing a harmonic spatial inhomogeneity into the Kalb-Ramond field, interacting with the Maxwell field according to a ‘string-inspired’ proposal made in earlier work are investigated. We examine in particular the effects on the polarization of synchrotron radiation from cosmologically distant (i.e. of redshift greater than 2) galaxies, as well as the relation between the electric and magnetic components of the radiation field. The rotation of the polarization plane of linearly polarized radiation is seen to acquire an additional contribution proportional to the square of the frequency of the dual Kalb-Ramond axion wave, assuming that it is far smaller compared to the frequency of the radiation field.'
address:
- '[$^1$]{}Centre for Theoretical Studies and Department of Physics, Indian Institute of Technology, Kharagpur 721302, India'
- '[$^2$]{}Institute of Mathematical Sciences, Chennai 600113, India'
- '[$^{3,4}$]{}Department of Physics, Jadavpur University, Calcutta 700 032, India'
author:
- 'Sayan Kar[^1]$^1$, Parthasarathi Majumdar[^2]$^2$, Soumitra SenGupta[^3]$^3$, and Saurabh Sur[^4]$^4$'
title: 'Cosmic optical activity from an inhomogeneous Kalb-Ramond field'
---
Introduction
============
The behaviour of electromagnetic waves in a curved background spacetime with torsion and its cosmological consequences, has been an area of some interest in recent years. This is in view of its prospective implications for the low energy approximation to string theory. One way to investigate these implications is by identifying spacetime torsion [@pmss], with the massless antisymmetric second rank tensor field, i.e., the Kalb-Ramond (KR) field, existing in most supergravity theories and as such in the massless sector of the most viable string theories [@gsw]. Certain physically observable phenomena result from the above analysis: a cosmic optical activity involving the rotation of the plane of polaization of linearly polarized synchrotron rotation from high redshift galaxies investigated recently [@skpm] being an example. The angle through which the polarization plane rotates is shown to be proportional, to leading order in inverse conformal time (which decreases with redshift), to the rate of change of the KR dual axion field. It is also independent of the frequency of radiation. This latter aspect is in contrast to the well-known Faraday rotation, and hence a new phenomenon.
Another point to note is that the KR field, argued to be responsible for the effect, has been treated in [@skpm] as a perturbation on the Maxwell equations in a standard Friedmann-Robertson-Walker (FRW) cosmological background with both matter and radiation domination. Thus, it is assumed to have a negligible effect on shaping cosmological background spacetime. One way of thinking about this is to imagine that the KR axion decouples from the radiation or matter (dust) fluid shaping cosmic geometry far prior to dust-photon decoupling, leaving behind a ‘Cosmic KR Background’ which affects incoming radiation from distant galaxies, albeit rather softly. As the universe expands further, this effect will no doubt gradually subside. The point made in [@skpm] is that the effect may yet be observable in this epoch. This viewpoint has received support from [@pdpj] where the the effects of a time-dependent dilaton field are additionally incorporated, while demonstrating that the earlier findings have a degree of robustness.
However, in these earlier assays, it is assumed that the axion and dilaton fields are spatially homogeneous, depending only on the conformal time coordinate $\eta$. This is possibly quite justified considering the overall homogeneity of the universe over cosmological distance scales.
In this paper, we generalize the scenario in [@skpm] by introducing a spatial inhomogeneity of a particular type into the KR axion field $H$: the massless Klein-Gordon equation obeyed by the axion quite naturally leads us to take the spatial dependence in the form of a plane wave propagating in space. However, taking into consideration the fact that the effects caused by the KR field should be confined to a very feeble disturbance on the overall homogeneity of space, we assume the the frequency of the axionic wave to be far smaller compared to that of the electromagnetic wave. This modification produces some interesting features, such as, it alters the mutual orthogonality of the electric and magnetic field vectors while inflicting a change on the Poynting equation as well. Furthermore, the inhomogeneity produces an additional rotation of the plane of polarization of radiation over that found in [@skpm]. This additional contribution turns out to be porportional to the square of the frequency of the axion field while being independent of the wavelength of the radiation.
The paper is organized as follows. For completeness and as a background, we briefly recapitulate in Section 2 the basic tenets of [@skpm], which lead to a modified set of Maxwell equations. The harmonic spatial dependence assumed for the axion field further modifies these equations as well as the one for the H field itself. Demanding that the waves retain their forms obtained in [@skpm] in the limit $H$ becomes space-independent, we obtain the equations representing circularly-polarized states. Carrying out a standard WKB type procedure, we solve these equations and calculate the rotation angle of the plane of polarization of the fields for a flat background spacetime in Section 3 and for a spatially flat spacetime in Section 4, considering separately the radiation and matter dominated cases therein. We make a few concluding remarks on our results in Section 5.
Gauge invariant Einstein-Cartan-Maxwell-Kalb-Ramond coupling
============================================================
Modified field equations
------------------------
The action for gauge-invariant Einstein-Cartan-Maxwell-Kalb-Ramond coupling is taken to be of the form [@pmss]:
$$S = \int~ d^{4}x \sqrt{-g} ~\left[~\frac{\tilde{R} (g,T)}{\kappa} -
\frac{1}{4} F_{\mu \nu} F^{\mu \nu} -
\frac{1}{2} \tilde{H}_{\mu \nu \lambda} \tilde{H}^{\mu \nu \lambda} +
\frac{1}{\sqrt{\kappa}} T^{\mu \nu \lambda} \tilde{H}_{\mu \nu
\lambda}~\right]$$
$\tilde{R}(g,T)$ being the scalar curvature for the Einstein-Cartan spacetime where the connection contains the torsion tensor $T_{\alpha \mu \nu}$ (supposed to be antisymmetric in all its indices) in addition to the Christoffel term; $\kappa = 16 \pi G$ is the coupling constant; and $\tilde{H}_{\mu \nu \lambda}$ is the KR field strength three-tensor modified by U(1) Chern-Simons term arising form the quantum consistency of an underlying string theory: $$\tilde{H}_{\mu \nu \lambda} = \partial_{[\mu} B_{\nu \lambda]} +
\frac{1}{3} \sqrt{\kappa} A_{[\mu} F_{\nu \lambda]},$$ with, $B_{\nu \lambda}$ being the antisymmetric KR potential which is considered to be the possible source of torsion. $\tilde{R}$ is related to the scalar curvature $R$ of purely Riemannian (torsion-free) space-time by $$\tilde{R}(g,T) = R(g) + T_{\mu \nu \lambda} T^{\mu \nu \lambda},$$
The fact that the augmented KR field strength three tensor plays the role of spin angular momentum density (which is the source of torsion [@hehl]) can be evidenced directly from Eq.(1) where the torsion tensor $T_{\mu \nu
\lambda}$, being an auxiliary field, obeys the constraint equation $$T_{\mu \nu \lambda} = \sqrt{\kappa} \tilde{H}_{\mu \nu \lambda}.$$ Substituting Eq.(4) in the action (1) and varying the latter with respect to $B_{\mu \nu}$ and $A_{\mu}$ respectively, two sets of field equations are obtained $$D_{\mu} \tilde{H}^{\mu \nu \lambda} \equiv
\frac{1}{\sqrt{- g}} \partial _{\mu}
(\sqrt{- g} \tilde{H}^{\mu \nu \lambda}) = 0$$ and $$D_{\mu} F^{\mu \nu} = \sqrt{\kappa} \tilde{H}^{\mu \nu \lambda}F_{\lambda \mu}.$$
In addition, there is also the Maxwell - Bianchi identity $$D_{\mu}~ ^{*}F^{\mu \nu} \equiv \frac{1}{\sqrt{- g}} \partial_{\mu}
(\sqrt{- g}~ ^{*}F^{\mu \nu}) = 0.$$ Note here that all covariant derivatives are defined with the Christoffel connection and the Maxwell field strength is the standard 2-form $F=dA$.
Now, expressing the KR field strength three tensor $H_{\mu \nu \lambda} \equiv
\partial_{[\mu} B_{\nu \lambda]}$ as the Hodge-dual to the derivative of the spinless pseudoscalar field $H$ (the axion): $$H_{\mu \nu \lambda} = \epsilon_{\mu \nu \lambda}^{\rho} D_{\rho} H.$$ and substituting in Eqs.(6) and (7), the modified generally covariant Maxwell’s equations are obtained in three-vectorial form $$\begin{aligned}
{\bf D\/}\cdot{\bf E\/} &=& 2 \sqrt{\kappa}~{\bf D\/} H\cdot{\bf B\/} \\
D_{0} {\bf E\/} ~-~ {\bf D\/} \times {\bf B\/} &=&
-~2 \sqrt{\kappa}~[~D_{0}H~{\bf B\/} ~-~ {\bf D\/}H \times {\bf E\/}~]
~+~ O(\kappa)\\
{\bf D\/}\cdot{\bf B\/} &=& 0 \\
D_{0} {\bf B\/} ~-~ {\bf D\/} \times {\bf E\/} &=& 0 \end{aligned}$$ where $D_{\mu}$ stands for the covariant derivative. On dropping all the higher order terms, as a first approximation, and retaining terms only of the order of $\sqrt{\kappa}$, in a spatially flat isotropic FRW background with metric $$ds^{2} = R^{2}(\eta) (d\eta^{2} - d{\bf x\/}^{2}),$$ the set of equations (9)-(12) take the form $$\begin{aligned}
\nabla \cdot {{\bf \tilde{E}\/}}&=& 2 \nabla H \cdot {{\bf \tilde{B}\/}}\\
\partial_{\eta} {{\bf \tilde{E}\/}}~-~ \nabla \times {{\bf \tilde{B}\/}}&=& -2[ ~\partial_{\eta} H {{\bf \tilde{B}\/}}- \nabla H \times {{\bf \tilde{E}\/}}~] \\
\nabla \cdot {{\bf \tilde{B}\/}}&=& 0 \\
\partial_{\eta} {{\bf \tilde{B}\/}}~+~ \nabla \times {{\bf \tilde{E}\/}}&=& 0 \end{aligned}$$ where $\eta$ the conformal time coordinate, defined by $d\eta = dt/R(t)$, $R$ is the cosmological scale factor; and ${{\bf \tilde{E}\/}}= R^{2}{\bf E\/}$ and ${{\bf \tilde{B}\/}}= R^{2}{\bf B\/}$. $H$ is redefined by absorbing the $\sqrt{\kappa}$ in it.
It is easy to show from the very form of the KR field strength, viz., $H_{\mu \nu \lambda} = \partial_{[\mu}B_{\nu \lambda]}$, that it satisfies the Bianchi identity $$\epsilon^{\mu \nu \lambda \rho} \partial_{\rho} H_{\mu \nu \lambda} = 0$$ which immediately implies that $H$ satisfies the wave equation $$D_{\rho} D^{\rho} H = 0$$
In isotropic spatially flat universe this equation reduces to $$(\partial_{\eta}^{2} ~-~ \nabla^{2}) H ~= ~-~2 \frac{\dot{R}}{R} \dot{H}$$ where $H$ is taken to be a general function of both space and time coordinates, and the over-dot implies partial differentiation with respect to $\eta$. Assuming a general wave solution for $H$, viz., $$H(\eta,{\bf x\/}) ~=~ H_{0}(\eta)~ \cos ~{{{\bf p \/}}\cdot {\bf x\/}}$$ Eq.(20) enables us to get an equation for $H$ $$\ddot{H} ~+~ 2 \frac{\dot{R}}{R} \dot{H} ~+~ p^{2} H ~=~ 0$$
We should point out here that as a consequence of our prior assumption that the overall homogeneity of the universe over long distance scales is not much disturbed by the inclusion of the spatial part in $H$, we are taking $p$ to be much less compared to the wave number for the electromagnetic radiation.
Modifications in electro-magnetic orthogonality and the Poynting equation
--------------------------------------------------------------------------
From the field equations (14) - (17) we derive the wave equations for the electric and magnetic fields $$\begin{aligned}
\Box {{\bf \tilde{B}\/}}~\equiv~ (~\partial_{\eta}^{2} ~-~ \nabla^{2}~)~ {{\bf \tilde{B}\/}}&=&
2~ \nabla \times (~\dot{H} {{\bf \tilde{B}\/}}) ~-~ 2~ \nabla \times (~\nabla H \times {{\bf \tilde{E}\/}}~) \\
\Box {{\bf \tilde{E}\/}}~\equiv~ (~\partial_{\eta}^{2} ~-~ \nabla^{2}~)~ {{\bf \tilde{E}\/}}&=&
2~ \nabla \times (~\dot{H} {{\bf \tilde{E}\/}}) ~+~ 2~ \nabla H \times \dot{{{\bf \tilde{E}\/}}} ~-~
\nabla (~\nabla \cdot {{\bf \tilde{E}\/}}~) ~-~ 2~ \ddot{H} {{\bf \tilde{B}\/}}\end{aligned}$$ These equations indeed reduce to the pure Maxwell equations in the limit $H ~\rightarrow ~0$, or a constant. Treating the axion $H$ as a [*tiny*]{} purturbation over the Maxwell equations we argue that the solutions should have a form not much departing from the usual plane wave structure with the wave vector ${{\bf k \/}}$ perpendicular to both the electric and magnetic vectors. Moreover, the field equations (14) and (15) enable us to derive $$\nabla \dot{H} \cdot {{\bf \tilde{B}\/}}= 0$$ which, in view of the specific form of $H$, viz., $H_{0}(\eta)~ \cos
({{{\bf p \/}}\cdot {\bf x\/}})$, implies $${{\bf p \/}}\cdot {{\bf \tilde{B}\/}}= 0$$ provided $\dot{H} \ne 0$, which is the general case we are handling.
This orthogonality of ${{\bf p \/}}$ and ${{\bf \tilde{B}\/}}$ \[Eq.(26)\] makes it easier to assume, for simplicity, that ${{\bf p \/}}$ can be taken to be orthogonal to ${{\bf \tilde{E}\/}}$ as well, i.e., ${{\bf p \/}}$ is either parallel or antiparallel to ${{\bf k \/}}$. This is fairly justified, as it looks, from the similar wave nature of the electric and magnetic fields, at least in the limiting pure Maxwellian case as we are treating the modification caused by the KR field as a small purturbation over the Maxwellian behaviour. In fact, we may choose to observe the electromagnetic radiation which is travelling in the direction of propagation of the KR field.
Considering the z-direction to be the propagation direction of the electromagnetic waves and as such for the axion (following the abovementioned assumption) we reduce the four-dimesional problem to a two-dimensional one with $\eta$ and $z$ being the only variables. The field equations now reduce to simpler forms $$\begin{aligned}
\Box {{\bf \tilde{B}\/}}&=& 2~ \dot{H} \nabla \times {{\bf \tilde{B}\/}}~+~ 2 \partial_{\eta}
(H' {\bf \hat{e}_{z}} \times {{\bf \tilde{B}\/}}) ~+~ 2~ H'' {{\bf \tilde{E}\/}}\\
\Box {{\bf \tilde{E}\/}}&=& 2~ \dot{H} \nabla \times {{\bf \tilde{E}\/}}~+~ 2 \partial_{\eta}
(H' {\bf \hat{e}_{z}} \times {{\bf \tilde{E}\/}}) ~-~ 2~ \ddot{H} {{\bf \tilde{B}\/}}\end{aligned}$$ where the over-dot and prime denote respectively the partial differentiations with respect to $\eta$ and $z$; and ${\bf \hat{e}_{z}}$ is the unit vector along the z-direction. Now, it is easy to show from the field equations (15) and (17) that $$\partial_{\eta}~({{\bf \tilde{E}\/}}\cdot {{\bf \tilde{B}\/}})~ = ~{{\bf \tilde{B}\/}}\cdot \nabla \times {{\bf \tilde{B}\/}}~-
{{\bf \tilde{E}\/}}\cdot \nabla \times {{\bf \tilde{E}\/}}~-~ 2~\dot{H}~{{\bf \tilde{B}\/}}^2 ~+~ 2~\nabla H \cdot
{{\bf \tilde{E}\/}}\times {{\bf \tilde{B}\/}}$$
Considering the limiting plane wave behaviour of the solutions of the electromagnetic wave equations as $~H ~\rightarrow ~0~$, the magnetic and the electric fields can be expressed $$\begin{aligned}
{{\bf \tilde{B}\/}}(\eta,z) &=& {{\bf \tilde{B}_{0}\/}}(\eta,z) ~ e^{- i k z} \\
{{\bf \tilde{E}\/}}(\eta,z) &=& {{\bf \tilde{E}_{0}\/}}(\eta,z) ~ e^{- i k z}\end{aligned}$$
When the KR field actually vanishes, ${{\bf \tilde{E}_{0}\/}}, {{\bf \tilde{B}_{0}\/}}~\equiv~$ constant vectors $\times~ e^{i k \eta}$ in the plane wave solutions of the pure Maxwell equations. Eq.(29) then gives $~{{\bf \tilde{E}\/}}\cdot \nabla
\times {{\bf \tilde{E}\/}}~=~ {{\bf \tilde{B}\/}}\cdot \nabla \times {{\bf \tilde{B}\/}}~=~ 0$, i.e., $\partial_{\eta}
({{\bf \tilde{E}\/}}\cdot {{\bf \tilde{B}\/}}) ~=~ 0$. Moreover, since ${{\bf \tilde{B}\/}}$ is in the direction of $\nabla \times {{\bf \tilde{E}\/}}$, as is evident from Eq.(17), it follows that ${{\bf \tilde{E}\/}}\cdot {{\bf \tilde{B}\/}}~=~ 0$. When the KR field is present, but is only time-dependent (the case in [@skpm]), the vectors ${{\bf \tilde{E}_{0}\/}}$ and ${{\bf \tilde{B}_{0}\/}}$ in the solutions (30) and (31) again depend on time only and Eq.(29) reduces to $$\partial_{\eta}~({{\bf \tilde{E}_{0}\/}}\cdot {{\bf \tilde{B}_{0}\/}})~ = ~-~ 2~\dot{H}~{{\bf \tilde{B}_{0}\/}}^2.$$
Clearly, $~{{\bf \tilde{E}_{0}\/}}\cdot {{\bf \tilde{B}_{0}\/}}~\ne~ 0$, which implies that the mutual orthogonality of ${{\bf \tilde{E}\/}}$ and ${{\bf \tilde{B}\/}}$ is lost. In the most general case where the KR field depends on both space and time coordinates, ${{\bf \tilde{E}_{0}\/}}$ and ${{\bf \tilde{B}_{0}\/}}$ are also spacetime-dependent and they satisfy the relation $$\partial_{\eta}~({{\bf \tilde{E}_{0}\/}}\cdot {{\bf \tilde{B}_{0}\/}})~ = ~{{\bf \tilde{B}_{0}\/}}\cdot \nabla \times {{\bf \tilde{B}_{0}\/}}~-~
{{\bf \tilde{E}_{0}\/}}\cdot \nabla \times {{\bf \tilde{E}_{0}\/}}~-~ 2~\dot{H}~{{\bf \tilde{B}_{0}\/}}^2 ~+~ 2~\nabla H \cdot
{{\bf \tilde{E}_{0}\/}}\times {{\bf \tilde{B}_{0}\/}}$$ Here we find, one cannot ascertain conclusively that ${{\bf \tilde{E}_{0}\/}}\cdot {{\bf \tilde{B}_{0}\/}}$ is manifestly zero. It is, in fact, more justified to think that ${{\bf \tilde{E}_{0}\/}}\cdot
{{\bf \tilde{B}_{0}\/}}$ is essentially non-zero in general, at least, by looking at the limiting behaviour when the KR field is stripped off the spatially depending part. The last term in Eq.(33), appearing due to the inclusion of the spatial dependence in $H$ cannot, in general, compensate for the term which actually renders ${{\bf \tilde{E}_{0}\/}}\cdot {{\bf \tilde{B}_{0}\/}}$ non-vanishing in Eq.(32), as the temporal and spatial components of the KR field are completely separate entities.
The Poynting equation in the present scenario can be obtained directly from the field equations (14) - (17). It is given by $$\nabla \cdot {\bf S} ~+~ \dot {\omega}_{em} ~=~ - 2 \dot{H}~ {{\bf \tilde{E}\/}}\cdot {{\bf \tilde{B}\/}}$$ where $ {\bf S} ~=~ ({{\bf \tilde{E}\/}}\times {{\bf \tilde{B}\/}}) $ is the Poynting vector and $ \omega_{em} ~=~ \frac 1 2 ({{\bf \tilde{E}\/}}^2 ~+~ {{\bf \tilde{B}\/}}^2) $ is the electromagnetic energy density. The distinction of this equation over the Poynting equation in the pure Maxwellian case is the presence of the term on the right hand side which is, in general, non-zero for reasons discussed above. In fact, this term vanishes in the limit $\dot{H} ~
\rightarrow~ 0$, i.e., when $H$ becomes purely space-dependent. This is quite obvious since spatial inhomogeneity alone cannot bring in any change in the power conservation equation.
Polarization States and Duality transformation
----------------------------------------------
Following [@cf] and rearranging terms of the components of the wave equations (27) and (28) (following the procedure in [@cf]), we obtain the following equations for the polarized states $$\begin{aligned}
\ddot{b}_{\pm} ~\mp~ 2 i H' \dot{b}_{\pm} ~\mp~ 2 i \dot{H}' b_{\pm} ~&=&~ ~b_{\pm}''
~\pm~ 2 i \dot{H} b_{\pm}' ~+~ 2 H'' e_{\pm} \\
\ddot{e}_{\pm} ~\mp~ 2 i H' \dot{e}_{\pm} ~\mp~ 2 i \dot{H}' e_{\pm} ~&=&~ ~e_{\pm}''
~\pm~ 2 i \dot{H} e_{\pm}' ~-~ 2 \ddot{H} b_{\pm}, \end{aligned}$$ where $$\begin{aligned}
b_{\pm}(\eta,z) &=& \tilde{B}_{x}(\eta,z) ~\pm~ i \tilde{B}_{y}(\eta,z) \nonumber\\
e_{\pm}(\eta,z) &=& \tilde{E}_{x}(\eta,z) ~\pm~ i \tilde{E}_{y}(\eta,z).\end{aligned}$$
Note that the Eqs.(35) and (36) are converted into each other by the transformation $e_{\pm} \rightarrow b_{\pm}$, $b_{\pm} \rightarrow
- e_{\pm}$ provided the equation $\Box H ~\equiv~ {\ddot H} ~-~ H''
~=~0$ is obeyed. This is the usual electro-magnetic [*duality*]{} symmetry of the Maxwell equations. The Maxwell-KR system indeed possess this invariance in a flat spacetime background with cosmplogical scale factor $R ~=~ 1$, as is evident from Eq.(20). In a curved spacetime background, however, one does not have this invariance any more.
Rewriting the Eqs.(35) and (36) as follows: $$\begin{aligned}
\ddot{b}_{\pm} ~\mp~ 2 i H' \dot{b}_{\pm} ~\mp~ 2 i \dot{H}' b_{\pm} ~&=&~ ~b_{\pm}''
~\pm~ 2 i \dot{H} b_{\pm}' ~+~ 2 H'' e_{\pm} ~=~ \alpha_{\pm}(\eta,z) \\
\ddot{e}_{\pm} ~\mp~ 2 i H' \dot{e}_{\pm} ~\mp~ 2 i \dot{H}' e_{\pm} ~&=&~ ~e_{\pm}''
~\pm~ 2 i \dot{H} e_{\pm}' ~-~ 2 \ddot{H} b_{\pm} ~=~ \beta_{\pm}(\eta,z), \end{aligned}$$ We seek the appropriate forms of the functions $\alpha(\eta,z)$ and $\beta(\eta,z)$ by examining the limiting forms of the above equations:
[**I. Limit $H' \rightarrow 0$ :** ]{}
In this limit, where $H$ becomes purely a function of $\eta$, the equations (37) and (38) reduce to $$\begin{aligned}
\ddot{b}_{\pm} &=& ~b_{\pm}'' ~\pm~ 2 i \dot{H} b_{\pm}' ~=~
\alpha_{\pm}(H' \rightarrow 0) \\
\ddot{e}_{\pm} &=& ~e_{\pm}'' ~\pm~ 2 i \dot{H} e_{\pm}' ~-~ 2 \ddot{H} b_{\pm} ~=~
\beta_{\pm}(H' \rightarrow 0) \end{aligned}$$
Assuming solutions of the form $$\begin{aligned}
b_{\pm}(\eta,z) &=& ~b_{0}^{\pm}(\eta)~ e^{- i k z} \\
e_{\pm}(\eta,z) &=& ~e_{0}^{\pm}(\eta)~ e^{- i k z}\end{aligned}$$ we find $$\begin{aligned}
\ddot{b}_{\pm} &=& ~ - (k^{2} \mp 2 k \dot{H}) b_{\pm} ~=~
\alpha_{\pm}(H' \rightarrow 0) \\
\ddot{e}_{\pm} &=& ~ - (k^{2} \mp 2 k \dot{H}) e_{\pm} ~-~ 2 \ddot{H} b_{\pm} ~=~
\beta_{\pm}(H' \rightarrow 0). \end{aligned}$$
[**II. Limit $\dot{H} \rightarrow 0$ :** ]{}
This limiting case implies $H$ to be a function of $z$ only but as is evident from Eq.(22) $H'$ is merely a constant. Therefore Eqs.(37) and (38) reduce to $$\begin{aligned}
\ddot{b}_{\pm} ~\mp~ 2 i H' \dot{b}_{\pm} &=& ~b_{\pm}'' ~=~
\alpha_{\pm}(\dot{H} \rightarrow 0) \\
\ddot{e}_{\pm} ~\mp~ 2 i H' \dot{e}_{\pm} &=& ~e_{\pm}'' ~=~
\beta_{\pm}(\dot{H} \rightarrow 0)\end{aligned}$$
Again assuming solutions of the form $$\begin{aligned}
b_{\pm}(\eta,z) &=& ~b_{1}^{\pm}(z)~ e^{ i k \eta} \\
e_{\pm}(\eta,z) &=& ~e_{1}^{\pm}(z)~ e^{ i k \eta}\end{aligned}$$ we get $$\begin{aligned}
b_{\pm}'' &=& ~ - (k^{2} \mp 2 k H') b_{\pm} ~=~
\alpha_{\pm}(\dot{H} \rightarrow 0) \\
e_{\pm}'' &=& ~ - (k^{2} \mp 2 k H') e_{\pm} ~=~
\beta_{\pm}(\dot{H} \rightarrow 0). \end{aligned}$$
By looking at these limiting forms of $\alpha_{\pm}$ and $\beta_{\pm}$ given in Eqs.(44),(45) and in Eqs.(50),(51) it seems reasonable to suggest the following simplest possible structures of these functions: $$\begin{aligned}
\alpha_{\pm}(\eta,z) &=& ~- \left[ k^{2} \mp 2 k ( \dot{H} + H' ) \right]
b_{\pm}(\eta,z) \\
\beta_{\pm}(\eta,z) &=& ~- \left[ k^{2} \mp 2 k ( \dot{H} + H' ) \right]
e_{\pm}(\eta,z) ~-~ 2 \ddot{H} b_{\pm}(\eta,z). \\\end{aligned}$$ Moreover, setting $$e_{\pm} (\eta,z) = a_{\pm} (\eta,z)~ b_{\pm} (\eta,z)$$ we write the equations (37) and (38) in a more elegant form $$\begin{aligned}
\ddot{b}_{\pm} ~\mp~ 2 i H' \dot{b}_{\pm} ~+~ \left[ k^{2} ~\mp~ 2 k
( \dot{H} + H' ) ~\mp~ 2 i \dot{H}' \right] b_{\pm}(\eta,z) &=& 0 \\
b_{\pm}'' ~\pm~ 2 i \dot{H} b_{\pm}' ~+~ \left[ k^{2} ~\mp~ 2 k
( \dot{H} + H' ) ~+~ 2 H'' a_{\pm} \right] b_{\pm}(\eta,z) &=& 0;\end{aligned}$$ and $$\begin{aligned}
\ddot{e}_{\pm} ~\mp~ 2 i H' \dot{e}_{\pm} ~+~ \left[ k^{2} ~\mp~ 2 k
( \dot{H} + H' ) ~\mp~ 2 i \dot{H}' ~+~ 2 \frac{\ddot{H}}{a_{\pm}} \right]
e_{\pm}(\eta,z) &=& 0 \\
e_{\pm}'' ~\pm~ 2 i \dot{H} e_{\pm}' ~+~ \left[ k^{2} ~\mp~ 2 k
( \dot{H} + H' ) \right] e_{\pm}(\eta,z) &=& 0.\end{aligned}$$
Flat Spacetime Background
=========================
In order to get a preliminary idea as to how the coupling of a spacetime dependent KR field to Einstein-Maxwell theory affects the electromagnetic waves, and thereby effects in an optical activity in the radiation coming from distant galactic sources, we consider the simplest situation — that is, of a flat universe with cosmological scale factor $R(\eta) =
1$. Admittedly, quantitative details of results of this section are cosmologically untenable for obvious reasons.
The equation of motion (22) for $H$ can be solved readily to obtain $$H(\eta,z) ~=~ (c_{1} \sin p \eta ~+~ c_{2} \cos p \eta)~ \cos p z$$ where $c_{1}$ and $c_{2}$ are arbitrary integration constants. Demanding that the above solution must reduce to the form $( h \eta ~+~ h_{0} )$ which is the solution of Eq.(22) in the limit $p \rightarrow 0$ we infer $c_{1} = h/p$ and $c_{2} = h_{0}$. Here $h$ and $h_{0}$ are the same arbitrary constants denoted in [@skpm]. Making a Taylor series expansion of the various functions appearing in Eq.(60) around $p = 0$ we write $$H(\eta,z) ~=~ (h \eta ~+~ h_{0}) ~-~ \frac{p^{2}}{2} \left(\frac{h \eta^{3}}{3}
~+~ h_{0} \eta^{2} + h \eta z^{2} + h_{0} z^{2}\right) ~+~ O(p^{4})$$
Substituting this $H$ in Eqs.(55) and (57) and assuming solution of the standard WKB type given by $$b_{\pm} (\eta,z) ~=~ \bar{b} ~ e^{i k~ S_{\pm} (\eta,z)}$$ with $$S_{\pm} ~=~ S_{0}^{\pm} ~+~ \frac{S_{1}^{\pm}}{k} ~+~ \frac{S_{2}^{\pm}}{k^2}
~+~ \cdots$$
we get after partial integrations of Eqs.(56) and (57) with respect to $\eta$ and $z$ respectively $$b_{\pm} (\eta,z) ~=~ \bar{b}~ \exp \left\{ i k \eta ~\mp~ i \left[ h \eta
~-~
\frac{ p^2}{2} \left( \frac{ h \eta^3}{3} ~+~ h_{0} \eta^2 ~+~ h z^2 \eta \right)
~+~ O(p^4) \right] ~+~ O\left(\frac 1 k \right) ~+~ i k f_{\pm}
(z) \right\}$$ and $$b_{\pm} (\eta,z) ~=~ \bar{b}~ \exp \left \{ i k g_{\pm} (\eta) ~-~ i k z
~\mp~
i \left[ \frac{ p^2}{2} ( h \eta ~+~ h_{0} ) z^2 ) ~+~ O(p^4) \right] ~+~
O\left(\frac 1 k \right) \right \}.$$
Comparing these two expressions we assert the forms of the arbitrary functions $f_{\pm} (z)$ and $g_{\pm} (\eta)$ and write $b_{\pm} (\eta,z)$ as follows: $$b_{\pm} (\eta,z) ~=~ \bar{b}~ \exp \left \{ i k ( \eta - z ) ~\mp~
i \left[ h \eta ~-~ \frac{ p^2}{2} \left( \frac{ h \eta^3}{3} ~+~ h_{0} \eta^2
~-~ h_{0} z^2 \right) ~+~ O(p^4) \right] ~+~ O\left(\frac 1 k
\right) \right \}$$
A similar approach for $e_{\pm} (\eta,z)$ yields $$e_{\pm} (\eta,z) ~=~ \bar{e}~ \exp \left \{ i k ( \eta - z ) ~\mp~
i \left[ h \eta ~-~ \frac{ p^2}{2} \left( \frac{ h \eta^3}{3} ~+~ h_{0} \eta^2
~-~ h_{0} z^2 \right) ~+~ O(p^4) \right] ~+~ O\left(\frac 1 k
\right) \right \}$$ which differs from Eq.(66) only in the constant coefficient $\bar{e}$ and in the higher order $O\left(\frac 1 k \right)$. However, it should be mentioned here that while using the WKB technique we are assuming that the function $a_{\pm}$ is not increasing rapidly as $k$ increases. In fact, in absence of the KR field, when we have the plane wave solutions of Maxwell’s equations, $a_{+} ~=~ a_{-} ~=~ \bar{e} / \bar{b} ~=~$ constant. The spacetime dependence of $a_{\pm}$ comes only in presence of a spacetime-dependent $H$. Since $H$ is being treated as a small purturbation over the Maxwell field, it is rather plausible to think $a_{\pm}$ to be not much different from the constant $\bar{e} / \bar{b}$ and a very slowly-varying function of spacetime. But the constant $\bar{e} / \bar{b}$ is arbitrary and cannot generically be argued as increasing with $k$. Therefore, the assumption that $a_{\pm}$ remains quite invariant as $k$ increases is fairly justified. Using WKB method, the solutions obtained above involves $a_{\pm}$ only in the higher order $O\left(\frac 1 k \right)$.
Now, the circular polarization states are defined by $b_{\pm}$ and $e_{\pm}$ and the extent of the optical birefringence due the presence of the KR field can be estimated directly by calculating the rotation angle of the plane of polarization of the electromagnetic wave, which is given by the phase difference $\phi_{mag}~
\equiv~ \frac 1 2 [arg ~b_{+} - arg ~b_{-}] $ for the magnetic field and $\phi_{elec}~
\equiv~ \frac 1 2 [arg ~e_{+} - arg ~e_{-}] $ for the electric field. The phase shift is given by $$\phi_{mag} (\eta,z) ~\approx~ \phi_{elec} (\eta,z) ~\approx~ - h \eta ~+~ \frac{p^2}{2}
\left( \frac{h \eta^3}{3} ~+~ h_{0} \eta^2 ~-~ h_{0} z^2 \right)$$ for $~ h,p << k$.
It is interesting to see that the change in the rotation angle, calculated here, over that found in [@skpm] for flat universe, is primarily given by the $p^2$-dependent part. But as $p$ is considered to be very small we infer that this change is rather insignificant.
Spatially Flat FRW Spacetime Background
=======================================
We now turn to less trivial background spacetimes. We consider a spatially flat expanding universe dominated by radiation and matter, in turn.
Radiation dominated Universe
----------------------------
The scale factor for this model, in real time, is given by $$~[R(\eta)]^{RD} ~=~ \frac{\eta}{\eta_{r}}$$ where $\eta_{r} = \left(8 \pi G \epsilon_{0} /3\right)^{-1/3}$, $\epsilon_{0}$ being the primordial radiant energy density.
Substituting this in the equation of motion (22) for $H$ we obtain $$\eta^{2} \ddot{H} + 2\eta \dot{H} + p^{2} \eta^{2} H ~=~ 0$$ which has the form of a transformed Bessel equation with solution $$H (\eta,z) ~=~ \eta^{- \frac 1 2}~\left[ \bar{c}_{1} ~ J_{1/2} (p \eta)
~+~ \bar{c}_{2} ~Y_{1/2} (p \eta) \right].$$
Simplifying the Bessel functions of first and second kinds, viz., $J$ and $Y$, the above solution can be written as $$H (\eta,z) ~=~ \frac 1 \eta ~( c_{1} ~\sin ~p \eta ~+~ c_{2} ~ \cos ~p \eta )
~ \cos~p z.$$ Imposing again the boundary condition that this must reduce to the limiting form $ ~\left[ -~ \frac {h \eta_{r}^2}{\eta} ~+~ h_{0} \right]~$ — the solution of Eq.(22) — as $~p \rightarrow 0~$, we set $~c_{1} ~=~ \frac
{h_{0}} p~$ and $~c_{2} ~=~ - h_{r}$, with $~h_{r} ~=~ h~\eta_{r}^2$, whence $$H (\eta,z) ~=~ \frac 1 \eta ~\left( \frac {h_{0}} p ~\sin ~p \eta ~-~ h_{r}
~ \cos ~p \eta \right)~ \cos~p z.$$
Plugging in the Taylor expanded form of this $H$ in Eqs.(56) - (59) and using the same WKB technique as for the flat universe, we obtain $$b_{\pm} (\eta,z) ~=~ \bar{b}~ \exp \left[ i k ( \eta - z ) ~\pm~
i \left[ \frac {h_{r}} \eta ~+~ \frac{p^{2}}{2} \left( \frac{h_{0} \eta^2}{3}
~-~ h_{r} \eta ~-~ h_{0} z^2 \right) ~+~ O(p^4) \right] ~+~ O\left(\frac 1 k \right)
\right]$$ and similar expression for $e_{\pm} (\eta,z)$.
The phase shift in this case is given by $$\phi_{mag} (\eta,z) ~\approx~ \phi_{elec} (\eta,z) ~\approx~ \frac {h_{r}} \eta
~-~ \frac{p^2}{2} \left( h_{r} \eta ~-~ \frac{h_{0} \eta^2}{3} ~+~ h_{0} z^2 \right)$$ for $ ~h,p ~<<~ k$ and $~h_{r} ~=~ h~ \eta_{r}^2$.
Matter dominated Universe
-------------------------
In this case, where the scale factor is governed by $$~[R(\eta)]^{MD} ~=~ \frac{\eta^2}{\eta_{m}^2}$$ with $\eta_{m} = \left(8 \pi G \rho_{0} /3\right)^{-1/3}$ ($\rho_{0}$ – the initial matter density), Eq.(22) reduces again to a transformed Bessel equation $$\eta^{2} \ddot{H} + 4\eta \dot{H} + p^{2} \eta^{2} H ~=~ 0$$ having simplified solution $$H (\eta,z) ~=~ \frac 1 {\eta^3} ~[ (c_{1} ~-~ c_{2}~p \eta) ~\sin ~p \eta ~-~
(c_{2} ~+~ c_{1}~p \eta) ~ \cos ~p \eta ]~ \cos~p z.$$
Determining the constants $c_{1}$ and $c_{2}$ using, as before, the boundary condition on $H$ in the limit $~p \rightarrow 0~$, we write $$H (\eta,z) ~=~ - ~\frac {h_{m}} {3 \eta^3} ~( \cos ~p \eta ~+~ p \eta ~\sin ~
p \eta)~ \cos~p z.$$ where $~h_{m} ~=~ h \eta^4$. With this $H$, we obtain using the WKB method $$b_{\pm} (\eta,z) ~=~ \bar{b}~ \exp \left[ i k ( \eta - z ) ~\pm~
i \left[ \frac {h_{m}} {3 \eta^3} ~+~ p^{2} h_{m} \left( \frac 1 \eta
~+~ \frac {z^2} 6 \right) ~+~ O(p^4) \right] ~+~ O\left(\frac 1 k \right)
\right]$$ and similar expression for $e_{\pm} (\eta,z)$. The phase shift can be calculated
$$\phi_{mag} (\eta,z) ~\approx~ \phi_{elec} (\eta,z) ~\approx~ \frac {h_{m}}{3 \eta^3}
~+~ p^2 h_{m} \left( \frac 1 \eta ~+~ \frac {z^2} 6 \right)$$
for $ ~h,p ~<<~ k$.
Conclusions
===========
One rather surprising aspect of our finding is the loss of orthogonality of the electric and magnetic vectors in the radiation field which exists even in the case of a spatially homogeneous axion field, so long as the axion is time-dependent. This will indeed affect the measurement of the precise rotation of the plane of polarization, although for large redshift sources to which we confine, this effect may be ignored for all practical purposes.
In general the electric and magnetic vectors seem to have solutions generically represented as $$(Electric/Magnetic ~field ~combinations) ~=~
(constant) ~e^{i(kz ~-~ \omega \eta)}~e^{i H(\eta, z)}$$
The additional contribution to the angle of rotation arising out of the spatial inhomogeneity introduced in this paper is actually quite small under our assumption that the wavelength of the axion wave is far larger than the radiation. Thus from an observational standpoint this new effect is not too significant, although it is still quite distinct in its behaviour from Faraday rotation.
We have used standard WKB type methods to arrive at some solution to the complicated set of equations which arise even at the lowest order. Among other things we have explored the properties of the electric and magnetic fields of the Maxwell–KR system. We also checked the validity of the standard electric–magnetic duality and pointed out the existence of a very simple solution to the complicated equations.
Acknowledgements
================
This work is supported by Project grant no. 98/37/16/BRNS cell/676 from The Board of Research in Nuclear Sciences, Department of Atomic Energy, Government of India, and the Council of Scientific and Industrial Research, Government of India.
.5in
P. Majumdar and S. SenGupta, Class. Quan. Grav. [**16**]{} L89 (1999). M. Green, J. Schwarz and E. Witten, Superstring Theory v.2, 1985 (Cambridge: Cambridge Univ. Press). S. Kar, P. Majumdar, S. SenGupta and A. Sinha, 2000, [*Preprint*]{} gr-qc/0006097. P. Jain and J. Ralston, Mod. Phys. Lett. [**A14**]{}, 417 (1999). B. Nodland and J. P. Ralston, Phys. Rev. Lett. [**78**]{}, 3043 (1997) ; [*ibid*]{} Phys. Rev. Lett. [**79**]{}, 1958 (1997); astro-ph/9708114;astro-ph/9706126. P. Das, P. Jain and S. Mukherji, 2000, [*Preprint*]{} hep-ph/ 0011279. F. Hehl, P. von der Heyde, G. Kerlick and J. Nester, Rev. Mod. Phys.[**48**]{} 393 (1976). S.M. Carroll and G.B.Field, Phys. Rev. [**D 43**]{}, 3789 (1991).
[^1]: Electronic address: [*sayan@cts.iitkgp.ernet.in*]{}
[^2]: Electronic address: [*partha@imsc.ernet.in*]{}
[^3]: Electronic address: [*soumitra@juphys.ernet.in*]{}
[^4]: Electronic address: [*saurabh@juphys.ernet.in*]{}
|
---
abstract: 'A self-contained description of algebraic structures, obtained by combinations of various limit procedures applied to vertex and face $sl(2)$ elliptic quantum affine algebras, is given. New double Yangians structures of dynamical type are in particular defined. Connections between these structures are established. A number of them take the form of twist-like actions. These are conjectured to be evaluations of universal twists.'
---
[**Cladistics[^1] of Double Yangians and Elliptic Algebras**]{}
[D. Arnaudon$^a$, J. Avan$^b$, L. Frappat$^a$, E. Ragoucy$^a$, M. Rossi$^c$]{}
*$^a$ Laboratoire d’Annecy-le-Vieux de Physique Th[é]{}orique LAPTH*
*CNRS, UMR 5108, associ[é]{}e [à]{} l’Universit[é]{} de Savoie*
*LAPP, BP 110, F-74941 Annecy-le-Vieux Cedex, France*
*$^b$ LPTHE, CNRS, UMR 7589, Universit[é]{}s Paris VI/VII, France*
*$^c$ Department of Mathematics, University of Durham\
South Road, Durham DH1 3LE, UK*
MSC number: 81R50, 17B37
Introduction
============
Overview
--------
The study of elliptic quantum algebras, defined with the help of elliptic $R$-matrices, has yielded a number of algebraic structures relevant to certain integrable systems in quantum mechanics and statistical mechanics (noticeably the $XYZ$ model [@JKKMW], RSOS models [@ABF; @Ko1] and Sine–Gordon theory [@Ko2; @KLCP]). More recently the definition and construction of some scaling limits has led to the notion of deformed double Yangian algebras. We will investigate and develop here in great detail the occurrence of these and other limit algebraic structures and the pattern of connection in between, in the simplest case of an underlying $sl(2)$ algebra.
Two classes of elliptic solutions to the Yang–Baxter equation have been identified, respectively associated with the vertex statistical models [@Ba; @Be] and the face-type statistical models [@ABF; @DJMO; @JMO]. The vertex elliptic $R$-matrix for $sl(2)$ was first used by Sklyanin [@Skl] to construct a two-parameter deformation of the enveloping algebra $\cU(sl(2))$. The central extension of this structure was proposed in [@FIJKMY] for $sl(2)$, and later extended to ${\cal A}_{q,p}{{(\widehat{sl(N)}_{c})}}$ in [@JKOS]. Its connection to $q$-deformed Virasoro and $\cW_N$ algebras [@AKOS; @FR; @FF] was established in [@AFRS3; @AFRS5].
The face-type $R$-matrices, depending on the extra parameters $\lambda$ belonging to the dual of the Cartan algebra in the underlying algebra, were first used by Felder [@Fe] to define the algebra ${{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}$ in the $R$-matrix approach. Enriquez and Felder [@EF] and Konno [@Ko1] introduced a current representation, although differences arise in the treatment of the central extension. A slightly different structure, also based upon face-type $R$-matrices but incorporating extra, Heisenberg algebra generators, was introduced as $\cU_{q,p}(sl(2))$ [@Ko1; @JKOS2]. This structure is relevant to the resolution of the quantum Calogero–Moser and Ruijsenaar–Schneider models [@ABB; @BBB; @JurcoSchupp]. Another dynamical elliptic algebra, denoted ${\cal A}_{q,p;\pi}{{(\widehat{sl(2)}_{c})}}$, was also defined and studied in [@HouYang]. It was then interpreted, at the level of representation, as a twist of ${{{\cal A}_{q,p}{{(\widehat{sl(2)}_{c})}}}}$.
Particular limits of the $\cA_{q,p}$-type algebras were subsequently defined and compared with previously known structures. The limit $p\rightarrow 0$ together with the renormalization of the generators by suitable powers of $p$ *before* taking the limit, leads to the quantum algebra ${{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}$ such as presented in [@RSTS; @FF]. It differs from the presentation in [@DF] by a scalar factor in the $R$-matrix. The scaling limit of the algebra ${\cal A}_{q,p;\pi}{{(\widehat{sl(2)}_{c})}}$ was also defined in [@HouYang].
A second limit was considered in [@JKM; @Ko2] ($R$-matrix formulation) and [@KLP] (current algebra formulation). It is defined by taking $p=q^{2r}$ (elliptic nome) and $z=q^{i\beta/\pi}$ (spectral parameter) with $q\rightarrow 1$. This algebra, denoted $\cA_{\hbar,\eta}{{(\widehat{sl(2)}_{c})}}$, where $\eta\equiv \frac1r$ and $q\simeq e^{\epsilon \hbar}$ with $\epsilon\rightarrow 0$, is relevant to the study of the $XXZ$ model in its gapless regime [@JKM]. It admits a further limit $r\rightarrow \infty$ ($\eta \rightarrow 0$) where its $R$-matrix becomes identical to the $R$-matrix defining the double Yangian ${{{\cal D}Y_{}^{}(sl(2))_{c}}}$ (centrally extended), defined in [@KT] (Yangian double), [@Kh] (central extension); alternative versions with a different normalization are given in [@IK] (for $sl(2)$) and [@Io] (for $sl(N)$). This difference in the normalization factors of the $R$-matrix, crucial in confronting the centrally extended versions, is the exact counterpart of the difference between the presentation of ${{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}$ in [@DF] and [@RSTS].
One must however be careful in this identification in terms of $R$-matrix structure since the generating functionals (Lax matrices) of these algebras admit different interpretations in terms of modes (generators of the enveloping algebra). In the context of $\cA_{\hbar,0}{{(\widehat{sl(2)}_{c})}}$ the expansion is done in terms of continuous-index Fourier modes of the spectral parameter (see [@Ko2; @KLP]); in the context of ${{{\cal D}Y_{}^{}(sl(2))_{c}}}$ the expansion is done in terms of powers of the spectral parameter (see [@KT; @Kh; @Io]).
It was shown recently that both vertex algebras $\cA_{q,p}{{(\widehat{sl(N)}_{c})}}$ and face-type algebras $\cB_{q,\lambda}{{(\widehat{sl(N)}_{c})}}$ were in fact Drinfel’d twists [@Dr] of the quantum group $\cU_q{{(\widehat{sl(N)}_{c})}}$. Originating with the proposition of [@BBB] on face-type algebras, the construction of the twist operators was undertaken in both cases by Fr[ø]{}nsdal [@Fro1; @Fro2] and finally achieved at the level of formal universal twists in [@JKOS; @ABRR]. In [@ABRR], the universal twist is obtained by solving a linear equation introduced in [@BR], this equation playing a fundamental r[ô]{}le for complex continuation of $6j$ symbols. Moreover in the case of finite (super)algebras, the convergence of the infinite products defining the twists was also proved in [@ABRR]. This led to a formal construction of universal $R$-matrices for the elliptic algebras $\cA_{q,p}$ and $\cB_{q,\lambda}$, of which the BB and ABF $4 \!\times\! 4$ matrices are respectively (spin $1/2$) evaluation representations.
General settings
----------------
Our strategy is to combine in as many patterns as possible the different limit procedures introduced previously in the literature; to apply them to cases not already considered, in particular the face type algebras ${{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}$; and thus to achieve as large as possible a self-contained network of algebraic structures extending from the elliptic quantum affine algebras to the affine Lie algebra ${{{\cal U}_{}{{(\widehat{sl(2)}_{c})}}}}$.
Before summarizing our investigations, we must first of all define precisely the concepts which we will use throughout this paper, so that no ambiguity arises in our statements.
We shall deal with formal algebraic structures defined by $R$-matrix exchange relations between formal $2 \!\times\! 2$ matrix-valued generating functionals denoted Lax operators, using the well-known $RLL$ formalism [@FRT]. Explicit $R$-matrices here are interpreted as evaluation representations of universal objects whenever they are known to exist, or conjectural universal objects when not. We shall not give any precise definition of the individual generators themselves, i.e. the specific expression of the individual generators in terms of spectral parameter dependent Lax operators. These definitions would eventually give rise to the fully explicit algebraic structure. For instance we shall not distinguish here between the double Yangian ${{{\cal D}Y_{}^{}(sl(2))_{c}}}$ and the scaled algebra $\cA_{\hbar,0}{{(\widehat{sl(2)}_{c})}}$. Definition of, and identification between algebraic structures will therefore be understood at the sole level of their $R$-matrix presentation, except in explicitly specified cases where we are able to state relations between the full (generator-described) exchange structures, or even the Hopf or quasi-Hopf algebraic structures. We consider that the existence of such relations is in any case an indication that similar connections exist at the level of universal algebras, to be explicitly formulated once the explicit algebra generators are defined.
Similarly we shall manipulate $R$-matrices at the level of their evaluation representation of spin $1/2$ ($4\!\times\!4$ matrices). Only when we shall use the term “universal”, will it mean the abstract algebraic object known as universal $R$-matrix. The same will apply to twist operators connecting (quasi)-Hopf algebraic structures [@Dr], and the $R$-matrices of the algebras. We recall that a twist operator $F$ lives in the square $\cA^{\otimes 2}$ of an algebraic structure; it connects two coproducts in $\cA$ as $\Delta_F(\cdot) = F\Delta(\cdot) F^{-1}$, and two universal $R$-matrices as $R_F=F^{\pi} R F^{-1}$. Its evaluation representation acts similarly on the evaluation representation of the universal $R$-matrices: $$R^F_{12}=F_{21} R_{12} F^{-1}_{12} \;.
\label{eq:twist_intro}$$
As in the previous case of identifications of algebras, we conjecture that occurrence of a relation of this form at the level of evaluated $R$-matrices is an indication that a similar relation exists at the level of universal algebras. We shall therefore denote any such relation between evaluated $R$-matrices as a “twist-like action” between two algebraic structures respectively characterized by $R$ and $R^F$, even when we do not have explicit proof that a universal twist exists between the universal $R$-matrices, or the respective coproduct structures.
A connection of the form (\[eq:twist\_intro\]) where $F$ will not depend on any parameter (spectral ($z$ or $\beta$, elliptic ($p$ or $r$) or dynamical ($w$ or $s$)) will be termed “rigid twist action”.
We must also introduce the notion of homothetical twist-like connection, whereby we mean the existence of an invertible matrix $F(z)$ such that two $R$-matrices are connected by $$\widetilde{R} = f(z,p,q) F_{21}(z^{-1}) R F_{12}(z)^{-1} \;,
\label{eq:homothetic}$$ where $f(z,p,q)$ is a $c$-number function.\
At this point, we do not have an interpretation of this kind of relation between algebraic structure. We shall come back to this point in the conclusion.
General properties of $R$-matrices and twists
---------------------------------------------
All evaluated $R$-matrices in this paper will obey one of the following equations, implying the associativity of the exchange algebra.
- Yang–Baxter equation: $$R_{12}(z) \, R_{13}(zz') \, R_{23}(z') =
R_{23}(z') \, R_{13}(zz') \, R_{12}(z) \;,
\label{eq:YBE}$$ $$R_{12}(\beta) \, R_{13}(\beta+\beta') \,
R_{23}(\beta') = R_{23}(\beta') \,
R_{13}(\beta+\beta') \, R_{12}(\beta) \;,
\label{eq:YBE2}$$
- Dynamical Yang–Baxter equation: $$R_{12}(z,\lambda+h^{(3)}) \, R_{13}(zz',\lambda) \,
R_{23}(z',\lambda+h^{(1)}) = R_{23}(z',\lambda) \,
R_{13}(zz',\lambda+h^{(2)}) \, R_{12}(z,\lambda) \;,
\label{eq:DYBE}$$ $$R_{12}(\beta,\lambda+h^{(3)}) \,
R_{13}(\beta+\beta',\lambda) \,
R_{23}(\beta',\lambda+h^{(1)}) =
R_{23}(\beta',\lambda) \,
R_{13}(\beta+\beta',\lambda+h^{(2)}) \,
R_{12}(\beta,\lambda) \;,
\label{eq:DYBE2}$$
depending upon the multiplicative or additive nature of the spectral parameter.
Among the algebraic structures which we consider here, some are known to have structure (for instance ${{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}$, ${{{\cal D}Y_{}^{}(sl(2))_{c}}}$) [@Dri], and others are [@Dr] (for instance ${{{\cal A}_{q,p}{{(\widehat{sl(2)}_{c})}}}}$, ${{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}$).
Their universal $R$-matrices obey the universal Yang–Baxter equation in the first case, $$\cR_{12} \cR_{13} \cR_{23} =
\cR_{23} \cR_{13} \cR_{12}
\label{eq:YBEuniv}$$ and a more complicated Yang–Baxter-type equation in the second case, involving a cocycle $\Phi\in
\fA\otimes\fA\otimes\fA$: $$\cR_{12} \Phi_{312} \cR_{13} \Phi_{132}^{-1} \cR_{23} \Phi_{123} =
\Phi_{321} \cR_{23} \Phi_{231}^{-1} \cR_{13} \Phi_{213} \cR_{12} \;.
\label{eq:quasiYBE}$$
However, in all the cases which are considered here, the $R$-matrices, once evaluated, obey the Yang–Baxter or dynamical Yang–Baxter equation.
We now recall the following contingent properties of evaluated $R$-matrices.
- Unitarity: $$\begin{aligned}
R_{12}(z) \, R_{21}(z^{-1}) &=& 1 \;,
\label{eq:unitarity}
\nonumber\\
R_{12}(\beta) \, R_{21}(-\beta) &=& 1 \;,
\end{aligned}$$
- Crossing-symmetry: $$\begin{aligned}
\Big(R_{12}(x)^{t_2}\Big)^{-1} &=& \Big(R_{12}(q^2x)^{-1}\Big)^{t_2} \;,
\label{eq:crossing}
\nonumber\\
\left( R_{12}(\beta)^{t_2}
\right)^{-1}
&=&
\left( R_{12}(\beta - 2i\pi)^{-1}
\right)^{t_2} \;,
\end{aligned}$$
depending upon the multiplicative or additive nature of the spectral parameter.
The unitarity relation is not satisfied in most cases: the already known evaluated $R$-matrices for ${{{\cal A}_{q,p}{{(\widehat{sl(2)}_{c})}}}}$, ${{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}$, $\cU_{q,\lambda}{{(\widehat{sl(2)}_{c})}}$ only obey the crossing relation (\[eq:crossing\]) [@IIJMNT; @JKOS]. We shall meet with $R$-matrices obeying unitarity relations at the end of the paper, but we have no proof that they do correspond to evaluations of universal objects. We shall comment on this in the conclusion.
We have indicated that Universal Twist Operators $\cal F$ transform a coproduct $\Delta$ into another one $\Delta^\cF(\cdot) = \cF \Delta(\cdot) \cF^{-1}$ and the $\cR$ matrix into $\cR^\cF = \cF_{21} \cR \cF_{12}^{-1}$. If now $(\fA,\Delta,\cR)$ defines a quasi-triangular Hopf algebra and $\cF$ satisfies the cocycle condition $${\cF}_{12} (\Delta \otimes id)
{\cF} = {\cF}_{23} (id \otimes \Delta)
{\cF} \;.
\label{eq:cocycle}$$ $(\fA,\Delta^\cF,{\cal R}^\cF)$ defines again a quasi-triangular Hopf algebra. If however $\cF$ satisfies no particular cocycle-like relation, $(\fA,\Delta^\cF,{\cR}^\cF)$ defines a : $\cR^\cF$ satisfies then the YB-type equation (\[eq:quasiYBE\]). An interesting intermediate structure arises when $\cF$ satisfies a so-called shifted cocycle condition, depending upon a parameter $\lambda$ such that [@BBB; @Fe]: $${\cF}_{12}(\lambda) (\Delta \otimes id) {\cF} =
{\cF}_{23}(\lambda+h^{(1)}) (id \otimes \Delta) {\cF}
\label{eq:cocycle_decale}$$ where $h\in\fA$. In this case, $\cR^\cF$ satisfies the dynamical Yang–Baxter equation (\[eq:DYBE\]).
Summary
-------
Our paper is divided into two parts.
We shall first of all describe the limit procedures whereby the number of parameters in the $R$-matrix description of the algebra (hence including the spectral parameter) is decreased, starting from either ${{{\cal A}_{q,p}{{(\widehat{sl(2)}_{c})}}}}$ or ${{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}$; we shall define the limit algebraic structures in both cases. These limit procedures may go in three (for ${{{\cal A}_{q,p}{{(\widehat{sl(2)}_{c})}}}}$) or four (for ${{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}$) directions:
- *non elliptic limit:* one sends $p$ to 0;
- *scaling limit:* one sends $q$ to 1, with $p=q^{2r}$, $z=q^{i\beta/\pi}$ ($z=q^{2i\beta/\pi}$ and $w=q^{2s}$ in the face case, where $w$ is related to $\lambda$, see below);
- *factorization:* one “eliminates” the spectral parameter by a Sklyanin-type factorization. At the level of the universal algebra this corresponds to a degeneracy homomorphism (see [@KLP]). This procedure is only known for vertex algebras at this point. Finite face type algebras however are known and shall be considered here, albeit without an established connection with the affine structures.
- *non dynamical limit:* in the face case the dynamical parameter $\lambda$ can also be eliminated by a procedure which we shall detail in the main body of the text.
These limit procedures, and combinations thereof, lead to the set of objects described by Figure \[fig:RCube\].
![$R$-matrix network[]{data-label="fig:RCube"}](cladisticsfig.ps)
Already known structures are of course present in the diagram: ${{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}$ is the face elliptic, centrally extended algebra; ${{{\cal A}_{q,p}{{(\widehat{sl(2)}_{c})}}}}$ is the vertex elliptic, centrally extended algebra; $\cU_q^F{{(\widehat{sl(2)}_{c})}}$ and $\cU_q^V{{(\widehat{sl(2)}_{c})}}$ are two presentations [@IIJMNT; @JKOS] of the quantum group ${{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}$ [@RSTS] connected by a conjugation and a twist-like action; $\cD Y_r^{V8}{{(\widehat{sl(2)}_{c})}}$ is the deformed double Yangian algebra $\cA_{\hbar,\eta}{{(\widehat{sl(2)}_{c})}}$ in [@KLP] with $\hbar=1$ and $\eta=1/r$; $\cD Y_r^{V6}{{(\widehat{sl(2)}_{c})}}$ is the deformed double Yangian algebra defined in [@Ko2], connected to the previous one by a rigid twist; ${{{\cal D}Y_{}^{}(sl(2))_{c}}}$ is the double Yangian defined in [@KT; @Kh]; $\cU_q(sl(2))$ is the $q$-deformed $sl(2)$ algebra; $\cS_{q,p}(sl(2))$ is Sklyanin’s elliptic “degenerate” algebra, and $\cU_r(sl(2))$ is the “degenerate” trigonometric algebra identified with $\cU_q(sl(2))$ by $q=e^{i\pi/r}$.
New algebraic structures also appear in this diagram, mostly due to the systematic application of the limit procedures to the face algebra ${{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}$: ${{{\cal D}Y_{r,s}^{}(sl(2))_{c}}}$ is the scaling limit of ${{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}$; ${{{\cal D}Y_{r,s}^{-\infty}(sl(2))_{c}}}$ is its $s\ll 0$ limit where the periodic behaviour in $s$ is nevertheless retained; ${{{\cal D}Y_{s}^{}(sl(2))_{c}}}$ is a dynamical deformation of the double Yangian; ${{{\cal U}_{q,\lambda}{{(\widehat{sl(2)}_{c})}}}}$ and $\cU_{q,\lambda}^\Gamma{{(\widehat{sl(2)}_{c})}}$ are dynamical deformations of ${{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}$, respectively homothetical to ${{{\cal D}Y_{r,s}^{-\infty}(sl(2))_{c}}}$ and ${{{\cal D}Y_{r,s}^{}(sl(2))_{c}}}$ by a suitable redefinition of the parameters; ${{{\cal D}Y_{r}^{F}(sl(2))_{c}}}$ is an “elliptic” non dynamical deformation of the double Yangian, connected to ${{{\cal D}Y_{r}^{V}(sl(2))_{c}}}$ by a twist-like action and homothetical to ${{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}$ by the same redefinition of the parameters. Finally $\cU_s(sl(2))$ and $\cB_{q,\lambda}(sl(2))$ are dynamical deformations of the factorized structures [à]{} la Sklyanin, although they themselves are not yet understood as originating from such a factorization. In addition, we also compare the structures resulting from ${{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}$ and the structures derived [@HouYang] in the analysis of ${\cal
A}_{q,p;\pi}{{(\widehat{sl(2)}_{c})}}$. These structures are in in fact connected by a TLA which we shall describe.
In order to avoid fastidious repetitions in the body of the text, we state immediately that *all* these new $R$-matrices have been explicitly checked to obey the Yang–Baxter equation (\[eq:YBE\])–(\[eq:YBE2\]) or dynamical Yang–Baxter equation (\[eq:DYBE\])–(\[eq:DYBE2\]). Such checks are indeed required since the computational procedures which yield these $R$-matrices may entail regularizations of infinite products. This fact in turn potentially invalidates a direct application of these computational procedures to the Yang–Baxter equation originally satisfied by the elliptic $R$-matrices.
In the second part we describe the connections which implement the *addition* of supplementary parameters. To be precise:
- implementation of the elliptic nome $p$ (or $r$);
- implementation of the dynamical parameter $w$ (or $s$);
- implementation of the quantum parameter $q$ along the scaling limit connections.
Three types of twist-like actions (TLA) appear:
1. TLA explicitely proved to be evaluation of universal twists, represented on the figures \[fig:twv\]–\[fig:ftwf\] by a triple arrow. Most of them have been previously established in the literature, particularly in [@Fro1; @Fro2; @Bab; @JKOS].
2. TLA conjectured to be evaluations of universal twists, represented on the figures \[fig:twv\]–\[fig:ftwf\] by a double arrow. All these objects are new. They are either deduced from previously known ones by limit procedures or combinations; or explicitly computed from scratch.
3. Homothetical TLA. These are also new; they connect either the affine Lie algebra ${{{\cal U}_{}{{(\widehat{sl(2)}_{c})}}}}$ with double Yangian or ${{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}$; or they act as reciprocal of the scaling transformations on the vertex or face side. By contrast, let us point out that the first two implementations (of $p$ and $w$ – or $r$ and $s$) are achieved in all cases by twist-like actions.
We finally give some indications on further possible investigations in the conclusion.
Vertex type algebras
====================
We will start from the elliptic algebra ${{{\cal A}_{q,p}{{(\widehat{sl(2)}_{c})}}}}$ and take the above described different limits to obtain various quantum algebras and deformed double Yangians.
Elliptic algebra ${{{\cal A}_{q,p}{{(\widehat{sl(2)}_{c})}}}}$
--------------------------------------------------------------
Let us consider the following $R$-matrix [@Ba; @FIJKMY]: $$R(z,q,p) = \frac{\tau(q^{1/2}z^{-1})}{\mu(z)} \left(
\begin{array}{cccc}
a(u) & 0 & 0 & d(u) \\
0 & b(u) & c(u) & 0 \\
0 & c(u) & b(u) & 0 \\
d(u) & 0 & 0 & a(u) \\
\end{array} \right)
\label{eq:elpa}$$ where $$\begin{aligned}
&& a(u) = \frac{\snh(v-u)}{\snh(v)} = z^{-1} \;
\frac{\Theta_{p^2}(q^2z^2) \; \Theta_{p^2}(pq^2)}
{\Theta_{p^2}(pq^2z^2) \; \Theta_{p^2}(q^2)} \;,
\label{eq:az}
\\
&& b(u) = \frac{\snh(u)}{\snh(v)} = q z^{-1} \;
\frac{\Theta_{p^2}(z^2) \; \Theta_{p^2}(pq^2)}
{\Theta_{p^2}(pz^2) \; \Theta_{p^2}(q^2)} \;, \\
&& c(u) = 1 \;, \\
&& d(u) = -k\,\snh(v-u)\snh(u) = -p^{1/2} q^{-1} z^{-2} \;
\frac{\Theta_{p^2}(z^2) \; \Theta_{p^2}(q^2z^2)}
{\Theta_{p^2}(pz^2) \; \Theta_{p^2}(pq^2z^2)} \;.
\label{eq:dz}\end{aligned}$$ The function $\snh(u)$ is defined by $\snh(u) = -i\sn(iu)$ where $\sn(u)$ is Jacobi’s elliptic function with modulus $k$. The variables $z,q,p$ are related to the variables $u,v$ by $$p = \exp\Big(-\frac{\pi K'}{K}\Big) \;, \qquad
q = - \exp\Big(-\frac{\pi v}{2K}\Big) \;, \qquad
z = \exp\Big(\frac{\pi u}{2K}\Big) \;,
\label{eq:pqz}$$ where the elliptic integrals $K,K'$ are given by (with ${k'}^2=1-k^2$): $$K = \int_0^1 \frac{dx}{\sqrt{(1-x^2)(1-k^2x^2)}} \qquad\mbox{and}\qquad
K' = \int_0^1 \frac{dx}{\sqrt{(1-x^2)(1-{k'}^2x^2)}} \;.$$ From now on, we shall consider $a$, $b$, $c$, $d$, as functions of $z$ given by (\[eq:pqz\]).\
The normalization factors are $$\begin{aligned}
&& \frac{1}{\mu(z)} = \frac{1}{\kappa(z^2)} \; \frac{(p^2;p^2)_\infty}
{(p;p)_\infty^2} \; \frac{\Theta_{p^2}(pz^2) \; \Theta_{p^2}(q^2)}
{\Theta_{p^2}(q^2z^2)} \;, \\
&& \frac{1}{\kappa(z^2)} = \frac{(q^4z^{-2};p,q^4)_\infty \;
(q^2z^2;p,q^4)_\infty \; (pz^{-2};p,q^4)_\infty \; (pq^2z^2;p,q^4)_\infty}
{(q^4z^2;p,q^4)_\infty \; (q^2z^{-2};p,q^4)_\infty \; (pz^2;p,q^4)_\infty \;
(pq^2z^{-2};p,q^4)_\infty} \;, \\
&& \tau(q^{1/2}z^{-1}) = q^{-1/2} z \;
\frac{\Theta_{q^4}(q^2z^2)}{\Theta_{q^4}(z^2)} \;,\end{aligned}$$ where the infinite multiple products are defined by: $$(z;p_1,\dots,p_m)_\infty = \prod_{n_i \ge 0} (1-z p_1^{n_1}
\dots p_m^{n_m}) \;.
\label{eq:prodinf}$$ $R$ satisfies the so-called quasi-periodicity property $$R_{12}(-z p^{\frac 12}) = (\sigma_1
\otimes {\mbox{1\hspace{-1mm}I}})^{-1} \, R_{21}(z^{-1})^{-1} \,
(\sigma_1 \otimes {\mbox{1\hspace{-1mm}I}}) \;.
\label{eq:quasiper}$$ It also obeys the crossing-symmetry property (\[eq:crossing\]), but not unitarity (\[eq:unitarity\]).\
This matrix defines the elliptic algebra ${{{\cal A}_{q,p}{{(\widehat{sl(2)}_{c})}}}}$ as $$R_{12}(z_1/z_2 ,q,p) \, L_1(z_1) \, L_2(z_2 ) =
L_2(z_2 ) \, L_1(z_1) \, R_{12}(z_1/z_2 ,q,p^*=pq^{-2c}) \;.
\label{eq:rll_elpa}$$
Non elliptic limit: quantum affine algebra ${{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}$
--------------------------------------------------------------------------------------
Starting from the above $R$-matrix of ${{{\cal A}_{q,p}{{(\widehat{sl(2)}_{c})}}}}$, and taking the limit $p\rightarrow 0$, one gets the ${{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}$ algebra, with its $R$-matrix given by $$R_{V}(z) = \rho(z^2) \left( \begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & \displaystyle \frac{q(1-z^2)}{1-q^2z^2} & \displaystyle
\frac{z(1-q^2)}{1-q^2z^2} & 0 \\
0 & \displaystyle \frac{z(1-q^2)}{1-q^2z^2} & \displaystyle
\frac{q(1-z^2)}{1-q^2z^2} & 0 \\
0 & 0 & 0 & 1 \\
\end{array} \right) \;.
\label{eq:ruqa}$$ The normalization factor is $$\rho(z^2) = q^{-1/2} \; \frac{(q^2z^2;q^4)_{\infty}^2}
{(z^2;q^4)_{\infty} \; (q^4z^2;q^4)_{\infty}} \;.$$ It is known [@FIJKMY] that the algebra ${{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}$ is only obtained after a suitable renormalization of the generators of ${{{\cal A}_{q,p}{{(\widehat{sl(2)}_{c})}}}}$ and a subsequent non-continuous limit $p \to 0$.
The algebra ${{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}$ is then defined by the relations $$\begin{aligned}
R_{12}(z_1/z_2) \, L^\pm_1(z_1) \, L^\pm_2(z_2)
&=&
L^\pm_2(z_2) \, L^\pm_1(z_1) \, R_{12}(z_1/z_2) \;,
\label{eq:rll_uq}
\\
R_{12}(q^{c/2} z_1/z_2) \, L^+_1(z_1) \, L^-_2(z_2)
&=&
L^-_2(z_2) \, L^+_1(z_1) \, R_{12}(q^{-c/2} z_1/z_2) \;.
\label{eq:rll_uq2}\end{aligned}$$
As indicated in the introduction, we do not discuss the problem of generator expansions here. The same caveat will hold throughout the whole paper, viz. we shall assume that suitable, consistent expansions of the Lax equations will exist to generate well-defined algebraic structures.
Scaling limit
-------------
The so-called scaling limit of an algebra will be understood as the algebra defined by the scaling limit of the $R$-matrix of the initial structure. It is obtained by setting in the $R$-matrix $p=q^{2r}$ (elliptic nome) and $z=q^{i\beta/\pi}$ (spectral parameter) with $q\rightarrow 1$, and $r$, $\beta$ being kept fixed. The spectral parameter in the Lax operator is now to be taken as $\beta$.
### Deformed double Yangian ${{{\cal D}Y_{r}^{V8}(sl(2))_{c}}}$
Taking the scaling limit of ${{{\cal A}_{q,p}{{(\widehat{sl(2)}_{c})}}}}$, one gets the ${{{\cal D}Y_{r}^{V8}(sl(2))_{c}}}$ algebra. Its $R$-matrix takes the form [@Ko2; @AAFR] (the superscript $V8$ is a token of the eight non vanishing entries of the vertex-type $R$-matrix): $$R_{V8}(\beta,r) = \rho_{V8}(\beta;r) \left( \begin{array}{cccc}
\displaystyle
\frac{\cos\frac{i\beta}{2r} \; \cos\frac{\pi}{2r}}
{\cos\frac{\pi+i\beta}{2r}}
& 0 & 0 & \displaystyle -\frac{\sin\frac{i\beta}{2r}\;\sin\frac{\pi}{2r}}
{\cos\frac{\pi+i\beta}{2r}} \\
0 & \displaystyle \frac{\sin\frac{i\beta}{2r}\;\cos\frac{\pi}{2r}}
{\sin\frac{\pi+i\beta}{2r}} & \displaystyle \frac{\cos\frac{i\beta}{2r} \;
\sin\frac{\pi}{2r}} {\sin\frac{\pi+i\beta}{2r}} & 0 \\
0 & \displaystyle \frac{\cos\frac{i\beta}{2r}\;\sin\frac{\pi}{2r}}
{\sin\frac{\pi+i\beta}{2r}} & \displaystyle \frac{\sin\frac{i\beta}{2r} \;
\cos\frac{\pi}{2r}} {\sin\frac{\pi+i\beta}{2r}} & 0 \\
\displaystyle -\frac{\sin\frac{i\beta}{2r} \; \sin\frac{\pi}{2r}}
{\cos\frac{\pi+i\beta}{2r}} & 0 & 0 & \displaystyle
\frac{\cos\frac{i\beta}{2r}\;\cos\frac{\pi}{2r}}
{\cos\frac{\pi+i\beta}{2r}} \\
\end{array} \right) \;.
\label{eq:dyra8}$$ The normalization factor is $$\rho_{V8}(\beta;r) = -\frac{S_{2}(-\frac{i\beta}{\pi} \;\vert\; r,2) \,
S_{2}(1+\frac{i\beta}{\pi} \;\vert\; r,2)}
{S_{2}(\frac{i\beta}{\pi} \;\vert\; r,2) \,
S_{2}(1-\frac{i\beta}{\pi} \;\vert\; r,2)}
\cotan \frac{i\beta}{2} \;.
\label{eq:rhoV8}$$ $S_{2}(x \vert \omega_{1},\omega_{2})$ is the Barnes’ double sine function of periods $\omega_{1}$ and $\omega_{2}$ defined by [@Barnes], quoted in [@JimMiw]: $$S_{2}(x \vert \omega_{1},\omega_{2}) =
\frac{\Gamma_{2}(\omega_{1}+\omega_{2}-x \;\vert\; \omega_{1},\omega_{2})}
{\Gamma_{2}(x \;\vert\; \omega_{1},\omega_{2})}$$ where $\Gamma_r$ is the multiple Gamma function of order $r$ given by $$\Gamma_{r}(x \vert \omega_{1},\dots,\omega_{r}) = \exp
\left(
\frac{\partial}{\partial s} \sum_{n_{1},\dots,n_{r} \ge 0}
(x+n_{1}\omega_{1}+\dots+n_{r}\omega_{r})^{-s}\Bigg\vert_{s=0}
\right) \;.$$ This ${R}$ matrix satisfies the quasi-periodicity property $${R}_{12}(\beta-i\pi r) = (\sigma_1 \otimes {\mbox{1\hspace{-1mm}I}})^{-1} \,
{R}_{21}(-\beta)^{-1} \, (\sigma_1 \otimes {\mbox{1\hspace{-1mm}I}}) \;,
\label{eq:quasiper2}$$ where $\sigma_1$ is the usual Pauli matrix.\
It also obeys the crossing-symmetry property (\[eq:crossing\]), but not (\[eq:unitarity\]).
The algebra ${{{\cal D}Y_{r}^{V8}(sl(2))_{c}}}$ is then defined by the relation $$R_{12}(\beta_{1}-\beta_{2},r) \, L_1(\beta_{1}) \, L_2(\beta_{2}) =
L_2(\beta_{2}) \, L_1(\beta_{1}) \,
R_{12}(\beta_{1}-\beta_{2},r-c) \;.
\label{eq:rll_dy8}$$
### Double Yangian ${{{\cal D}Y_{}^{}(sl(2))_{c}}}$
Starting now from the quantum affine algebra ${{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}$ and taking its scaling limit, one obtains the double Yangian algebra ${{{\cal D}Y_{}^{}(sl(2))_{c}}}$ [@KT]. Its $R$-matrix is given by
$$R(\beta) = \rho(\beta) \left( \begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & \displaystyle \frac{i\beta}{i\beta+\pi} & \displaystyle
\frac{\pi}{i\beta+\pi} & 0 \\[.3cm]
0 & \displaystyle \frac{\pi}{i\beta+\pi} & \displaystyle
\frac{i\beta}{i\beta+\pi} & 0 \\
0 & 0 & 0 & 1 \\
\end{array} \right) \;.
\label{eq:dy}$$
The normalization factor is $$\rho(\beta) = \frac{\Gamma_{1}(\frac{i\beta}{\pi} \;\vert\; 2) \;
\Gamma_{1}(2+\frac{i\beta}{\pi} \;\vert\; 2)}
{\Gamma_{1}(1+\frac{i\beta}{\pi} \;\vert\; 2)^2} \;.$$ Taking the limit $r\rightarrow \infty$ of the $R$-matrix of ${{{\cal D}Y_{r}^{V8}(sl(2))_{c}}}$ (corresponding to the previous $p\rightarrow 0$ limit), one also gets the double Yangian algebra.
Notice that in both previous cases, the limit procedure may be applied directly to the Lax matrices, leading to the explicit, continuous labelled algebras, respectively denoted ${{\cal
A}_{\hbar,\eta}{{(\widehat{sl(2)}_{c})}}}$ and ${{\cal A}_{\hbar,0}{{(\widehat{sl(2)}_{c})}}}$ [@KLP].
The different limit procedures in the vertex case are summarized in Figure \[fig:lmv\].
(80,60) (15,50)[(1,0)[50]{}]{} (15,0)[(1,0)[50]{}]{} (0,10)[(0,1)[30]{}]{} (80,10)[(0,1)[30]{}]{} (0,50)[(0,0)[${{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}$]{}]{} (0,0)[(0,0)[${{{\cal A}_{q,p}{{(\widehat{sl(2)}_{c})}}}}$]{}]{} (80,50)[(0,0)[${{{\cal D}Y_{}^{}(sl(2))_{c}}}$]{}]{} (80,0)[(0,0)[${{{\cal D}Y_{r}^{V8}(sl(2))_{c}}}$]{}]{} (40,53)[(0,0)[scaling $q \rightarrow 1$]{}]{} (40,47)[(0,0)[$z=q^{i\beta/\pi}$]{}]{} (40,3)[(0,0)[scaling $q \rightarrow 1$]{}]{} (40,-3)[(0,0)[$z=q^{i\beta/\pi},p=q^{2r}$]{}]{} (-10,25)[(0,0)[$p \rightarrow 0$]{}]{} (90,25)[(0,0)[$r \rightarrow \infty$]{}]{}
Finite algebras
----------------
Up to now, the various limits led to affine structures. We now consider another kind of limit where the algebra is “factorized”. The resulting structure is based on a finite $sl(2)$ algebra. This is interpreted as a highly degenerate consistent representation of the affine algebras at $c=0$, where all generators are expressed in terms of only four ones.
### Sklyanin algebra
The Sklyanin algebra [@Skl] is constructed from ${{{\cal A}_{q,p}{{(\widehat{sl(2)}_{c})}}}}$ taken at $c=0$. The $R$-matrix (\[eq:elpa\]) can be written as $$R(z) = {\mbox{1\hspace{-1mm}I}}\otimes {\mbox{1\hspace{-1mm}I}}+ \sum_{\alpha =1}^3 W_\alpha(z) \sigma_\alpha
\otimes \sigma_\alpha \;,
\label{eq:RSklyanin}$$ where $\sigma_\alpha$ are the Pauli matrices and $W_\alpha(z)$ are expressed in terms of the Jacobi elliptic functions. A particular $z$-dependence of the $L(z)$ operators is chosen, leading to a factorization of the $z$-dependence in the $RLL$ relations. Indeed, setting $$L(z)= S_0 + \sum_{\alpha=1}^3 W_\alpha(z) S_\alpha \sigma_\alpha \;,
\label{eq:LSklyanin}$$ one obtains an algebra with four generators $S^\alpha$ ($\alpha=0,...,3$) and commutation relations $$\begin{aligned}
[S_0, S_\alpha ] &=& -i J_{\beta \gamma } (S_\beta S_\gamma + S_\gamma
S_\beta ) \;, \nonumber \\
{}[S_\alpha ,S_\beta ] &=& i (S_0 S_\gamma + S_\gamma S_0) \;,
\label{eq:Sklyanin}\end{aligned}$$ where $J_{\alpha \beta}=\displaystyle\frac{W_\alpha^2 -
W_\beta^2}{W_\gamma^2 - 1}$ and $\alpha$, $\beta$, $\gamma$ are cyclic permutations of 1, 2, 3. The structure functions $J_{\alpha \beta}$ are actually independent of $z$. Hence we get an algebra where the $z$-dependence has been dropped out.
### ${\cal U}_{r}(sl(2))$
The same factorization procedure (\[eq:RSklyanin\]-\[eq:LSklyanin\]) applied to ${{{\cal D}Y_{r}^{V8}(sl(2))_{c}}}$ leads to a ${\cal U}_{r}(sl(2))$ algebra described by (\[eq:Sklyanin\]) with now $J_{12}=-J_{31}=\tan^2 \frac{\pi}{2r}$ and $J_{23}=0$. We recognize the algebra ${\cal U}_{q'}(sl(2))$ if we set $q'=e^{i\pi/r}$.
**Remark:** The scaling limit of the Sklyanin algebra (\[eq:Sklyanin\]) also leads to the algebra ${\cal U}_{r}(sl(2))$.
### Other factorizations
Applying the factorization procedure (\[eq:RSklyanin\]-\[eq:LSklyanin\]) to the quantum affine algebra ${{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}$, one simply gets the finite ${\cal U}_{q}(sl(2))$ algebra.\
Let us remark that this algebra is also the $p\rightarrow 0$ limit of the Sklyanin algebra.
If we finally apply the factorization procedure to the double Yangian ${{{\cal D}Y_{}^{}(sl(2))_{c}}}$, one gets $J_{\alpha \beta}=0$. Setting the central generator $S_0$ to 1, we recognize the classical ${\cal U}(sl(2))$ algebra.
Note that ${\cal U}(sl(2))$ can also be viewed as:
*i)* the $r\rightarrow \infty$ limit of ${\cal U}_{r}(sl(2))$;
*ii)* the $q\rightarrow 1$ limit (“scaling limit”) of ${\cal U}_{q}(sl(2))$.
The different limit procedures in the finite vertex case are summarized in figure \[fig:flmv\].
(80,60) (15,50)[(1,0)[50]{}]{} (15,0)[(1,0)[50]{}]{} (0,10)[(0,1)[30]{}]{} (80,10)[(0,1)[30]{}]{} (0,50)[(0,0)[$\cU_q(sl(2))$]{}]{} (0,0)[(0,0)[$\cS_{qp}(sl(2))$]{}]{} (80,50)[(0,0)[$\cU(sl(2))$]{}]{} (80,0)[(0,0)[$\cU_r(sl(2))$]{}]{} (40,53)[(0,0)[$q \rightarrow 1$]{}]{} (40,3)[(0,0)[scaling $q \rightarrow 1$]{}]{} (40,-3)[(0,0)[$p=q^{2r}$]{}]{} (-10,25)[(0,0)[$p \rightarrow 0$]{}]{} (90,25)[(0,0)[$r \rightarrow \infty$]{}]{}
Face type algebras
==================
Elliptic algebra ${{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}$
----------------------------------------------------------------------
The starting point in the face case is the ${{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}$ algebra. Let $\{h,c,d\}$ be a basis of the Cartan subalgebra of ${{(\widehat{sl(2)}_{c})}}$. If $r,s,s'$ are complex numbers, we set $\lambda = {{{{\textstyle{\frac{1}{2}}}}}}\;(s+1)h + s'c +
(r+2)d$. The elliptic parameter $p$ and the dynamical parameter $w$ are related to the deformation parameter $q$ by $p=q^{2r}$, $w=q^{2s}$.\
The $R$ matrix of ${{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}$ is [@Fe; @JKOS] $$R(z;p,w) = \rho(z;p)
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & b(z) & c(z) & 0 \\
0 & {{\bar c}}(z) & {{\bar b}}(z) & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right)
\label{eq:Relpb}$$ where $$\begin{aligned}
b(z) &=& q \frac{(pw^{-1}q^2;p)_{\infty} \; (pw^{-1}q^{-2};p)_{\infty}}
{(pw^{-1};p)_{\infty}^2} \; \frac{\Theta_{p}(z)}{\Theta_{p}(q^2z)}
\;, \\
{{\bar b}}(z) &=& q \frac{(wq^2;p)_{\infty} \; (wq^{-2};p)_{\infty}}
{(w;p)_{\infty}^2} \; \frac{\Theta_{p}(z)}{\Theta_{p}(q^2z)} \;, \\
c(z) &=& \frac{\Theta_{p}(q^2)}{\Theta_{p}(w)} \;
\frac{\Theta_{p}(wz)}{\Theta_{p}(q^2z)} \;, \\
{{\bar c}}(z) &=& z \frac{\Theta_{p}(q^2)}{\Theta_{p}(pw^{-1})} \;
\frac{\Theta_{p}(pw^{-1}z)}{\Theta_{p}(q^2z)} \;.\end{aligned}$$ The normalization factor is $$\rho(z;p) = q^{-1/2} \frac{(q^2z;p,q^4)_{\infty}^2}
{(z;p,q^4)_{\infty} \; (q^4z;p,q^4)_{\infty}} \;
\frac{(pz^{-1};p,q^4)_{\infty} \; (pq^4z^{-1};p,q^4)_{\infty}}
{(pq^2z^{-1};p,q^4)_{\infty}^2} \;.
\label{eq:rhoelpb}$$ The elliptic algebra ${{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}$ is then defined by [@Fe; @JKOS] $$R_{12}(z_1/z_2,\lambda+h) \, L_1(z_1,\lambda) \, L_2(z_2,\lambda+h^{(1)})
=
L_2(z_2,\lambda) \, L_1(z_1,\lambda+h^{(2)}) \, R_{12}(z_1/z_2,\lambda) \;.
\label{eq:rll_elpb}$$
Dynamical quantum affine algebras ${{{\cal U}_{q,\lambda}{{(\widehat{sl(2)}_{c})}}}}$
-------------------------------------------------------------------------------------
Starting from the ${{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}$ $R$-matrix, and taking the limit $p\rightarrow 0$, one gets the ${{{\cal U}_{q,\lambda}{{(\widehat{sl(2)}_{c})}}}}$ one.\
The $R$ matrix of ${{{\cal U}_{q,\lambda}{{(\widehat{sl(2)}_{c})}}}}$ is $$R(z;w) = \rho(z) \left( \begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & \displaystyle \frac{q(1-z)}{1-q^2z} & \displaystyle
\frac{(1-q^2)(1-wz)}{(1-q^2z)(1-w)} & 0 \\
0 & \displaystyle \frac{(1-q^2)(z-w)}{(1-q^2z)(1-w)} & \displaystyle
\frac{q(1-z)}{(1-q^2z)}\,\frac{(1-wq^2)(1-wq^{-2})}{(1-w)^2} & 0 \\
0 & 0 & 0 & 1 \\
\end{array} \right) \;.
\label{eq:DQA}$$ The normalization factor is $$\rho(z) = q^{-1/2} \; \frac{(q^2z;q^4)_{\infty}^2}{(z;q^4)_{\infty} \;
(q^4z;q^4)_{\infty}} \;.$$
Non dynamical limit
-------------------
Taking the limit $w \rightarrow 0$ in ${{{\cal U}_{q,\lambda}{{(\widehat{sl(2)}_{c})}}}}$, one gets the algebra ${{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}$ with $R$-matrix:
$$R_{F}(z) = \rho(z) \left( \begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & \displaystyle \frac{q(1-z)}{1-q^2z} & \displaystyle
\frac{1-q^2}{1-q^2z} & 0 \\
0 & \displaystyle \frac{z(1-q^2)}{1-q^2z} & \displaystyle
\frac{q(1-z)}{1-q^2z} & 0 \\
0 & 0 & 0 & 1 \\
\end{array} \right) \;.
\label{eq:ruqb}$$
The normalization factor is $$\rho(z) = q^{-1/2} \; \frac{(q^2z;q^4)_{\infty}^2}{(z;q^4)_{\infty} \;
(q^4z;q^4)_{\infty}} \;.
\label{eq:rhouqb}$$
**Remark 1:** The matrix (\[eq:ruqa\]) differs from the matrix (\[eq:ruqb\]) by rescaling $z \rightarrow z^2$ and symmetrization between the $e_{12} \otimes e_{21}$ and $e_{21} \otimes e_{12}$ terms. The corresponding algebraic structures will be denoted respectively ${{{\cal U}^F_{q}{{(\widehat{sl(2)}_{c})}}}}$ for (\[eq:ruqb\]) and ${{{\cal U}^V_{q}{{(\widehat{sl(2)}_{c})}}}}$ for (\[eq:ruqa\]).
Actually, the matrix $R(z)$ is computed from the universal ${\cal R}$ matrix of ${{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}$ by $R(z) = (\pi \otimes \pi) {\cal R}(z)$ where $\pi$ is a spin $1/2$ evaluation representation [@IIJMNT]. Implementation of the spectral parameter $z$ in the universal ${\cal R}$ matrix is obtained by $$\begin{aligned}
{\cal R}(z) &=& Ad(z^{\rho} \otimes 1) {\cal R} \qquad \mbox{in the
vertex case,} \\
{\cal R}(z) &=& Ad(z^d \otimes 1) {\cal R} \qquad \mbox{in the face case.}\end{aligned}$$ Hence, the $R$ matrix of ${{{\cal U}^V_{q}{{(\widehat{sl(2)}_{c})}}}}$ is associated to the principal gradation of the ${{(\widehat{sl(2)}_{c})}}$ algebra, whilst the $R$ matrix ${{{\cal U}^F_{q}{{(\widehat{sl(2)}_{c})}}}}$ is associated to the homogeneous gradation.
**Remark 2:** The scaling limit of the $R$-matrix (\[eq:ruqb\]) of ${{{\cal U}^F_{q}{{(\widehat{sl(2)}_{c})}}}}$ gives back the $R$-matrix (\[eq:dy\]) of ${{{\cal D}Y_{}^{}(sl(2))_{c}}}$.
Dynamical deformed double Yangian ${{{\cal D}Y_{r,s}^{}(sl(2))_{c}}}$
---------------------------------------------------------------------
Starting again from the ${{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}$ case, and taking now the scaling limit $p=q^{2r}$ (elliptic nome), $z=q^{2i\beta/\pi}$ (spectral parameter), $w=q^{2s}$ (dynamical parameter) with $q\rightarrow 1$, one gets a new structure, interpreted as a dynamical deformed centrally extended double Yangian ${{{\cal D}Y_{r,s}^{}(sl(2))_{c}}}$.\
The $R$ matrix of ${{{\cal D}Y_{r,s}^{}(sl(2))_{c}}}$ is $$R(\beta;r,s) = \rho(\beta;r) \left( \begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & b(\beta) & c(\beta) & 0 \\
0 & {{\bar c}}(\beta) & {{\bar b}}(\beta) & 0 \\
0 & 0 & 0 & 1 \\
\end{array} \right) \;,
\label{eq:dyrs}$$ where $$\begin{aligned}
b(\beta) &=& \frac{\Gamma_{1}(r-s \;\vert\; r)^2}
{\Gamma_{1}(r-s+1 \;\vert\; r) \, \Gamma_{1}(r-s-1 \;\vert\; r)} \;
\frac{\sin\frac{i\beta}{r}}{\sin\frac{\pi+i\beta}{r}} \;, \\
c(\beta) &=& \frac{\sin\frac{\pi s+i\beta}{r}}{\sin\frac{\pi s}{r}} \;
\frac{\sin\frac{\pi}{r}}{\sin\frac{\pi+i\beta}{r}} \;, \\
{{\bar b}}(\beta) &=& \frac{\Gamma_{1}(s \;\vert\; r)^2}
{\Gamma_{1}(s+1 \;\vert\; r) \, \Gamma_{1}(s-1 \;\vert\; r)} \;
\frac{\sin\frac{i\beta}{r}}{\sin\frac{\pi+i\beta}{r}} \;, \\
{{\bar c}}(\beta) &=& \frac{\sin\frac{\pi s-i\beta}{r}}{\sin\frac{\pi s}{r}} \;
\frac{\sin\frac{\pi}{r}}{\sin\frac{\pi+i\beta}{r}} \;.\end{aligned}$$ The normalization factor is the same as formula (\[eq:rhoV8\]), rewritten as $$\rho(\beta;r) = \frac{S_{2}^2(1+\frac{i\beta}{\pi} \;\vert\; r,2)}
{S_{2}(\frac{i\beta}{\pi} \;\vert\; r,2) \,
S_{2}(2+\frac{i\beta}{\pi} \;\vert\; r,2)} \;.
\label{eq:normdyrs}$$
The algebra ${{{\cal D}Y_{r,s}^{}(sl(2))_{c}}}$ is then defined by the relations $$R_{12}(\beta_1-\beta_2,\lambda+h) \,
L_1(\beta_1,\lambda) \, L_2(\beta_2,\lambda+h^{(1)})
=
L_2(\beta_2,\lambda) \, L_1(\beta_1,\lambda+h^{(2)}) \,
R_{12}(\beta_1-\beta_2,\lambda) \;.
\label{eq:rll_dyrs}$$
Dynamical double Yangian ${{{\cal D}Y_{s}^{}(sl(2))_{c}}}$
----------------------------------------------------------
Taking the limit $r\rightarrow \infty$ in ${{{\cal D}Y_{r,s}^{}(sl(2))_{c}}}$, one gets a new, dynamical, centrally extended double Yangian ${{{\cal D}Y_{s}^{}(sl(2))_{c}}}$.\
The $R$ matrix of ${{{\cal D}Y_{s}^{}(sl(2))_{c}}}$ is given by $$R(\beta) = \rho(\beta) \left( \begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & \displaystyle\frac{i\beta}{i\beta+\pi} &
\displaystyle\frac{\pi s+i\beta}{s(i\beta+\pi)} & 0 \\
0 & \displaystyle\frac{\pi s-i\beta}{s(i\beta+\pi)} &
\displaystyle\frac{s^2-1}{s^2}\,\frac{i\beta}{i\beta+\pi} & 0 \\
0 & 0 & 0 & 1 \\
\end{array} \right) \;.
\label{eq:dys}$$ The normalization factor is $$\rho(\beta) = \frac{\Gamma_{1}(\frac{i\beta}{\pi} \;\vert\; 2) \;
\Gamma_{1}(2+\frac{i\beta}{\pi} \;\vert\; 2)}
{\Gamma_{1}(1+\frac{i\beta}{\pi} \;\vert\; 2)^2} \;.$$
**Remark 1:** This $R$-matrix (\[eq:dys\]) is also obtained by taking the scaling limit of the $R$-matrix (\[eq:DQA\]) of ${{{\cal U}_{q,\lambda}{{(\widehat{sl(2)}_{c})}}}}$.
**Remark 2:** The $|s| \rightarrow \infty$ limit of the $R$-matrix (\[eq:dys\]) gives back the $R$-matrix (\[eq:dy\]) of ${{{\cal D}Y_{}^{}(sl(2))_{c}}}$.
Dynamical deformed double Yangian ${{{\cal D}Y_{r,s}^{-\infty}(sl(2))_{c}}}$ in the trigonometric limit \[subsect:dyrsi\]
-------------------------------------------------------------------------------------------------------------------------
Starting again from ${{{\cal D}Y_{r,s}^{}(sl(2))_{c}}}$ and taking $s\ll 0$, but retaining the oscillating behaviour in $s$, one gets ${{{\cal D}Y_{r,s}^{-\infty}(sl(2))_{c}}}$, another dynamical deformed centrally extended double Yangian structure.\
The $R$ matrix of ${{{\cal D}Y_{r,s}^{-\infty}(sl(2))_{c}}}$ reads $$R(\beta;r,s) = \rho(\beta;r)
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & \displaystyle
\frac{\sin\frac{i\beta}{r}}{\sin\frac{\pi+i\beta}{r}} &
\displaystyle \frac{\sin\frac{\pi s+i\beta}{r}}{\sin\frac{\pi
s}{r}} \;
\frac{\sin\frac{\pi}{r}}{\sin\frac{\pi+i\beta}{r}} & 0 \\
0 & \quad \displaystyle \frac{\sin\frac{\pi
s-i\beta}{r}}{\sin\frac{\pi s}{r}}
\; \frac{\sin\frac{\pi}{r}}{\sin\frac{\pi+i\beta}{r}} \quad &
\quad
\displaystyle \frac{\sin\pi\frac{s+1}{r}
\sin\pi\frac{s-1}{r}}{\sin^2\frac{\pi
s}{r}} \frac{\sin\frac{i\beta}{r}}{\sin\frac{\pi+i\beta}{r}}
\quad & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right) \;.
\label{eq:dyrsi}$$ The normalization factor is the same as for ${{{\cal D}Y_{r,s}^{}(sl(2))_{c}}}$, see (\[eq:normdyrs\]).
**Remark 1:** The limit $r\rightarrow \infty$ of the $R$-matrix (\[eq:dyrsi\]) gives again the $R$-matrix (\[eq:dys\]) of ${{{\cal D}Y_{s}^{}(sl(2))_{c}}}$.
**Remark 2: Correspondence with ${{{\cal U}_{q,\lambda}{{(\widehat{sl(2)}_{c})}}}}$**\
The previous $R$ matrix is homothetical to that of ${{{\cal U}_{q,\lambda}{{(\widehat{sl(2)}_{c})}}}}$ by the following identifications of parameters: $$z = e^{-2\beta/r} \;, \qquad
q = e^{i\pi/r} \;, \qquad
w = e^{2i\pi s/r} \;.
\label{eq:identif}$$ The same identification of parameters applied to the $R$-matrix (\[eq:dyrs\]) of ${{{\cal D}Y_{r,s}^{}(sl(2))_{c}}}$ leads to an $R$-matrix close to that of ${{{\cal U}_{q,\lambda}{{(\widehat{sl(2)}_{c})}}}}$, but with $\Gamma$-function dependence in the dynamical parameter. This would define a new dynamical algebraic structure ${\cal U}^{\Gamma}_{q,\lambda}{{(\widehat{sl(2)}_{c})}}$.
Deformed double Yangian ${{{\cal D}Y_{r}^{F}(sl(2))_{c}}}$
----------------------------------------------------------
Taking now the limit $s\rightarrow i\infty$ in ${{{\cal D}Y_{r,s}^{}(sl(2))_{c}}}$, one gets a non dynamical structure ${{{\cal D}Y_{r}^{F}(sl(2))_{c}}}$.\
The $R$ matrix of ${{{\cal D}Y_{r}^{F}(sl(2))_{c}}}$ is given by $$R(\beta;r) = \rho(\beta;r)
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 &
\displaystyle\frac{\sin\frac{i\beta}{r}}{\sin\frac{\pi+i\beta}{r}} &
e^{\beta/r}
\displaystyle\frac{\sin\frac{\pi}{r}}{\sin\frac{\pi+i\beta}{r}}
&
0 \\
0 & \quad e^{-\beta/r}
\displaystyle\frac{\sin\frac{\pi}{r}}{\sin\frac{\pi+i\beta}{r}}
\quad & \quad
\displaystyle\frac{\sin\frac{i\beta}{r}}{\sin\frac{\pi+i\beta}{r}}
\quad & 0
\\
0 & 0 & 0 & 1 \\
\end{array}
\right) \;.
\label{eq:dyrF}$$ The normalization factor is the same as for ${{{\cal D}Y_{r,s}^{}(sl(2))_{c}}}$.
**Remark 1:** The limit $r\rightarrow \infty$ of the $R$-matrix (\[eq:dyrF\]) gives again the $R$-matrix of ${{{\cal D}Y_{}^{}(sl(2))_{c}}}$.
**Remark 2: Correspondence with ${{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}$**\
This matrix is homothetical to that of ${{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}$ – eq. (\[eq:ruqb\]) – by the following identifications of parameters: $$z = e^{-2\beta/r} \;, \qquad q = e^{i\pi/r} \;.
\label{eq:identif2}$$
The different limit procedures in the face case are summarized in Figure \[fig:lmf\].
(120,130) (15,120)[(1,0)[50]{}]{} (15,50)[(1,0)[50]{}]{} (65,0)[(1,0)[50]{}]{} (0,110)[(0,-1)[50]{}]{} (5,45)[(1,-1)[40]{}]{} (80,110)[(0,-1)[50]{}]{} (85,45)[(1,-1)[40]{}]{} (130,60)[(0,-1)[50]{}]{} (98,114)[(1,-1)[13]{}]{} (121,89)[(1,-1)[11]{}]{} (128,10)[(-1,6)[13]{}]{} (10,115)[(1,-3)[35]{}]{} (90,115)[(1,-3)[35]{}]{} (108,90)[(-3,-4)[22]{}]{} (50,0)[(0,0)[${{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}$]{}]{} (130,0)[(0,0)[${{{\cal D}Y_{}^{}(sl(2))_{c}}}$]{}]{} (0,50)[(0,0)[${{{\cal U}_{q,\lambda}{{(\widehat{sl(2)}_{c})}}}}$]{}]{} (80,50)[(0,0)[${{{\cal D}Y_{s}^{}(sl(2))_{c}}}$]{}]{} (0,120)[(0,0)[${{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}$]{}]{} (80,120)[(0,0)[${{{\cal D}Y_{r,s}^{}(sl(2))_{c}}}$]{}]{} (115,95)[(0,0)[${{{\cal D}Y_{rs}^{-\infty}(sl(2))_{c}}}$]{}]{} (132,70)[(0,0)[${{{\cal D}Y_{r}^{}(sl(2))_{c}}}$]{}]{} (40,123)[(0,0)[scaling $q \rightarrow 1$]{}]{} (40,117)[(0,0)[$z=q^{2i\beta/\pi},p=q^{2r},w=q^{2s}$]{}]{} (40,53)[(0,0)[scaling $q \rightarrow 1$]{}]{} (40,47)[(0,0)[$z=q^{2i\beta/\pi},w=q^{2s}$]{}]{} (90,3)[(0,0)[scaling $q \rightarrow 1$]{}]{} (90,-3)[(0,0)[$z=q^{2i\beta/\pi}$]{}]{} (140,35)[(0,0)[$r \rightarrow \infty$]{}]{} (-8,85)[(0,0)[$p \rightarrow 0$]{}]{} (72,85)[(0,0)[$r \rightarrow \infty$]{}]{} (15,25)[(0,0)[$w \rightarrow 0$]{}]{} (95,25)[(0,0)[$|s| \rightarrow \infty$]{}]{} (92,67)[(0,0)[$r \rightarrow \infty$]{}]{} (112,110)[(0,0)[$s \ll 0$]{}]{} (137,85)[(0,0)[$s \rightarrow i\infty$]{}]{} (33,75)[(0,0)[$\begin{array}{c} p \rightarrow 0 \\ w
\rightarrow 0 \end{array}$]{}]{} (117,50)[(0,0)[$\begin{array}{c} r \rightarrow
\infty \\ s
\rightarrow i\infty \end{array}$]{}]{} (127.3,11.5)(128,10)(128.2,11.6)(127.3,11.5)
Finite dimensional algebras
---------------------------
By constrast with the vertex case, the finite face-type elliptic algebras have not yet been obtained from the affine algebras by a factorization procedure. The starting point of our description will therefore be the $R$-matrix representation of ${\cal B}_{q,\lambda}(sl(2))$ given in [@Bab].
### Elliptic algebra ${\cal B}_{q,\lambda}(sl(2))$
The $R$-matrix of ${\cal B}_{q,\lambda}(sl(2))$ is $$R(w) = q^{-1/2}
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & q & \displaystyle
\frac{1-q^2}{1-w} & 0 \\
0 & \displaystyle -\frac{w(1-q^2)}{1-w}\; & \displaystyle
\;\frac{q(1-wq^2)(1-wq^{-2})}{(1-w)^2} & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right) \;.$$
**Remark:** The limit $w\rightarrow 0 $ of this matrix gives the usual $R$-matrix of $\cU_q(sl(2))$ $$R = q^{-1/2}
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & q & 1-q^2 & 0 \\
0 & 0 & q & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right) \;.
\label{eq:uqfini}$$
### Dynamical algebra ${\cal U}_{s}(sl(2))$
Taking the scaling limit $w=q^{2s}$ with $q\rightarrow 1$, we obtain the dynamical algebra $\cU_s(sl(2))$ with the $R$-matrix $$R =
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & s^{-1} & 0 \\
0 & -s^{-1} & 1-s^{-2} & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right) \;.
\label{eq:us}$$
The limit $|s| \rightarrow \infty$ of (\[eq:us\]) gives ${\mbox{1\hspace{-1mm}I}}$, which is the evaluated $R$-matrix of $\cU(sl(2))$. It is not clear to us whether this particular matrix (\[eq:us\]) can be used for an $RLL$ formulation of the algebra. However, we will show in the second part that it is indeed obtained as evaluation of a *universal* twist action on the *universal* $R$-matrix $1 \otimes 1$ of $\cU(sl(2))$.
We now describe the twist connections between the various algebraic structures previously defined. We first discuss twist-like actions between vertex-type algebras; we then introduce TLAs between ${{{\cal U}^V_{q}{{(\widehat{sl(2)}_{c})}}}}$ and ${{{\cal U}^F_{q}{{(\widehat{sl(2)}_{c})}}}}$. We then give the TLA between face-like algebras. The TLAs are classified here according to the “arrival” algebraic structure, i.e. with the highest number of parameters. We end up with the homothetical TLAs.
Vertex type algebras
====================
Twist operator ${{{\cal U}^V_{q}{{(\widehat{sl(2)}_{c})}}}} \rightarrow {{{\cal A}_{q,p}{{(\widehat{sl(2)}_{c})}}}}$
--------------------------------------------------------------------------------------------------------------------
The existence of a twist operator between ${{{\cal U}^V_{q}{{(\widehat{sl(2)}_{c})}}}}$ and ${{{\cal A}_{q,p}{{(\widehat{sl(2)}_{c})}}}}$ was proved at the level of universal matrices in [@JKOS]. Once the operators are evaluated, one gets $${R}[{{{\cal A}_{q,p}{{(\widehat{sl(2)}_{c})}}}}] = E^{(1)}_{21}(z^{-1};p) \; R[{{{\cal U}^V_{q}{{(\widehat{sl(2)}_{c})}}}}] \;
{E^{(1)}_{12}}(z;p)^{-1} \;.$$ The twist operator $E^{(1)}(z;p)$ is given by [@Fro1; @Fro2] $$E^{(1)}(z;p) = \rho_{E}(z;p) \left( \begin{array}{cccc}
a_{E}(z) & 0 & 0 & d_{E}(z) \\
0 & b_{E}(z) & c_{E}(z) & 0 \\
0 & c_{E}(z) & b_{E}(z) & 0 \\
d_{E}(z) & 0 & 0 & a_{E}(z) \\
\end{array} \right) \;,
\label{eq:E1}$$ where $$\begin{aligned}
\label{eq:adE}
a_{E}(z) \pm d_{E}(z) &=& \frac{(\mp p^{1/2}qz;p)_{\infty}}
{(\mp p^{1/2}q^{-1}z;p)_{\infty}} \\
b_{E}(z) \pm c_{E}(z) &=& \frac{(\mp pqz;p)_{\infty}}
{(\mp pq^{-1}z;p)_{\infty}} \;.
\label{eq:bcE}\end{aligned}$$ The normalization factor is $$\rho_{E}(z;p) = \frac{(pz^2;p,q^4)_{\infty} \; (pq^4z^2;p,q^4)_{\infty}}
{(pq^2z^2;p,q^4)_{\infty}^2} \;.
\label{eq:normrhoE}$$
Deformed double Yangians ${{{\cal D}Y_{r}^{V}(sl(2))_{c}}}$
-----------------------------------------------------------
### Deformed double Yangian ${{{\cal D}Y_{r}^{V6}(sl(2))_{c}}}$
We need to define an algebraic structure not previously derived in this paper.\
The $R$ matrix (\[eq:dyra8\]) of the deformed double Yangian ${{{\cal D}Y_{r}^{V8}(sl(2))_{c}}}$ can be related to the two-body $S$ matrix of the Sine–Gordon theory $S_{SG}(\beta,r)$ by a rigid twist operator. The connection goes as follows. One defines the following $R$-matrix [@Ko2]: $$R_{V6}(\beta,r) = \cotan(\frac{i\beta}{2}) S_{SG}(\beta,r)
=
\rho_{V6}(\beta;r)
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\[1mm]
0 & \displaystyle
\frac{\sin\frac{i\beta}{r}}{\sin\frac{\pi+i\beta}{r}} &
\displaystyle \frac{\sin\frac{\pi}{r}}
{\sin\frac{\pi+i\beta}{r}} & 0 \\[5mm]
0 & \displaystyle \frac{\sin\frac{\pi}{r}}
{\sin\frac{\pi+i\beta}{r}} &
\displaystyle \frac{\sin\frac{i\beta}{r}}
{\sin\frac{\pi+i\beta}{r}} & 0 \\[5mm]
0 & 0 & 0 & 1 \\
\end{array}
\right) \;,
\label{eq:dyra6}$$ where $\rho_{V6}(\beta;r) = \rho_{V8}(\beta;r)$, see (\[eq:rhoV8\]). This $R$-matrix is assumed to define by the $RLL$ formalism an algebraic structure denoted ${{{\cal D}Y_{r}^{V6}(sl(2))_{c}}}$.\
One has now $$R[{{{\cal D}Y_{r}^{V8}(sl(2))_{c}}}] = K_{21} \; R[{{{\cal D}Y_{r}^{V6}(sl(2))_{c}}}] \; K_{12}^{-1} \;.$$ The rigid twist operator $K$ is given by $$K = \frac{1}{2} \left( \begin{array}{rrrr}
1 & -i & -i & -1 \\
-1 & -i & i & -1 \\
-1 & i & -i & -1 \\
1 & i & i & -1 \\
\end{array} \right) \;.
\label{eq:twistcst}$$
**Remark 1:** We note that $$K = V \otimes V \qquad \mbox{with} \qquad V = \frac{1}{\sqrt{2}}
\left(
\begin{array}{rr} 1 & -i \\ -1 & -i \\
\end{array}
\right) \;.$$ This implies an isomorphism between ${{{\cal D}Y_{r}^{V8}(sl(2))_{c}}}$ and ${{{\cal D}Y_{r}^{V6}(sl(2))_{c}}}$ where the Lax operators are connected by $L_{V8} = V L_{V6} V^{-1}$.
**Remark 2:** The rigid twist leaves invariant the $R$-matrix of the undeformed double Yangian, upon which $V$ induces an automorphism.
**Remark 3:** The $R$ matrix (\[eq:dyra6\]) is homothetical to that of ${{{\cal U}^V_{q}{{(\widehat{sl(2)}_{c})}}}}$ – eq. (\[eq:ruqa\]) – by the following identifications of parameters: $$z = e^{-\beta/r} \;, \qquad q = e^{i\pi/r} \;.
\label{eq:identiv}$$
**Remark 4:** By applying the twist $K$ to the $R$-matrix (\[eq:ruqa\]) we obtain an $R$-matrix $R[{\cal U}^{V8}_{q}{{(\widehat{sl(2)}_{c})}}]$ with eight non-vanishing entries which may equivalently describe ${{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}$ according to remark 1.\
Moreover, $R[{\cal U}^{V8}_{q}{{(\widehat{sl(2)}_{c})}}]$ appears to be homothetical to the $R$-matrix obtained by redefining the parameters of (\[eq:dyra8\]) according to (\[eq:identiv\]).
### Twist operator ${{{\cal D}Y_{}^{}(sl(2))_{c}}} \rightarrow {{{\cal D}Y_{r}^{V8}(sl(2))_{c}}}$
The $R$-matrix of ${{{\cal D}Y_{r}^{V8}(sl(2))_{c}}}$ can be obtained from the $R$-matrix of ${{{\cal D}Y_{}^{}(sl(2))_{c}}}$ by a twist-like action: $${R}[{{{\cal D}Y_{r}^{V8}(sl(2))_{c}}}] = E^{(2)}_{21}(-\beta;r) \; R[{{{\cal D}Y_{}^{}(sl(2))_{c}}}] \;
{E^{(2)}_{12}}(\beta;r)^{-1} \;.$$ The twist operator $E^{(2)}(\beta;r)$ is the scaling limit of the twist operator $E^{(1)}(z,p)$, see eq. (\[eq:E1\]). It is given by $$E^{(2)}(\beta;r) = \rho_{E}(\beta;r) \left( \begin{array}{cccc}
a_{E}(\beta) & 0 & 0 & d_{E}(\beta) \\
0 & b_{E}(\beta) & c_{E}(\beta) & 0 \\
0 & c_{E}(\beta) & b_{E}(\beta) & 0 \\
d_{E}(\beta) & 0 & 0 & a_{E}(\beta) \\
\end{array} \right) \;,$$ where $$\begin{aligned}
\label{eq:aEscal}
a_{E}(\beta) + d_{E}(\beta) &=& 1 \;, \\
a_{E}(\beta) - d_{E}(\beta) &=&
\frac{\Gamma_{1}(r-1+\frac{i\beta}{\pi}
\;\vert\; 2r)} {\Gamma_{1}(r+1+\frac{i\beta}{\pi} \;\vert\; 2r)}
\;, \\
b_{E}(\beta) + c_{E}(\beta) &=& 1 \;, \\
b_{E}(\beta) - c_{E}(\beta) &=&
\frac{\Gamma_{1}(2r-1+\frac{i\beta}{\pi}
\;\vert\; 2r)} {\Gamma_{1}(2r+1+\frac{i\beta}{\pi} \;\vert\; 2r)} \;.
\label{eq:dEscal}\end{aligned}$$ The normalization factor is $$\rho_{E}(\beta;r) =
\frac{\Gamma_{2}(r+1+\frac{i\beta}{\pi} \;\vert\; r,2)^2}
{\Gamma_{2}(r+\frac{i\beta}{\pi} \;\vert\; r,2) \;
\Gamma_{2}(r+2+\frac{i\beta}{\pi} \;\vert\; r,2)} \;.
\label{eq:normrhoEscal}$$
### Twist operator ${{{\cal D}Y_{}^{}(sl(2))_{c}}} \rightarrow {{{\cal D}Y_{r}^{V6}(sl(2))_{c}}}$
Combining the previous two twist-like actions, one gets $${R}[{{{\cal D}Y_{r}^{V6}(sl(2))_{c}}}] = E^{(3)}_{21}(-\beta;r) \; R[{{{\cal D}Y_{}^{}(sl(2))_{c}}}] \;
E^{(3)}_{12}(\beta;r)^{-1} \;.$$ The twist operator $E^{(3)}(\beta;r)$ is given by $E^{(3)}=K^{-1}E^{(2)}$, that is $$E^{(3)}(\beta;r) = {{{{\textstyle{\frac{1}{2}}}}}}\; \rho_{E}(\beta;r) \left(
\begin{array}{cccc}
1 & -1 & -1 & 1 \\
i(a_{E}-d_{E})(\beta) & i(b_{E}-c_{E})(\beta) & i(c_{E}-b_{E})(\beta) &
i(d_{E}-a_{E})(\beta) \\
i(a_{E}-d_{E})(\beta) & i(c_{E}-b_{E})(\beta) & i(b_{E}-c_{E})(\beta) &
i(d_{E}- a_{E})(\beta) \\
-1 & -1 & -1 & -1 \\
\end{array} \right) \;,
\label{eq:E6}$$ where $a_{E},b_{E},c_{E},d_{E}$ are given by the formulae (\[eq:aEscal\])–(\[eq:dEscal\]) and the normalization factor $\rho_{E}(\beta;r)$ by (\[eq:normrhoEscal\]).
The different twist procedures in the vertex case are summarized in Figure \[fig:twv\].
Vertex to face isomorphism
==========================
The two $R$ matrices (\[eq:ruqa\]) and (\[eq:ruqb\]) can be related by a twist operator: $$R[{{{\cal U}^F_{q}{{(\widehat{sl(2)}_{c})}}}}](z^2) = K^{(6)}_{21}(z^{-1}) \; R[{{{\cal U}^V_{q}{{(\widehat{sl(2)}_{c})}}}}](z) \;
{K^{(6)}_{12}(z)}^{-1} \;.$$ The twist operator $K^{(6)}$ is given by $$K^{(6)}(z) =
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & z^{-1/2} & 0& 0 \\
0 & 0 & z^{1/2} & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right)
=
\left(
\begin{array}{cc}
1&0 \\ 0&z^{1/2}
\end{array}
\right)
\otimes
\left(
\begin{array}{cc}
1&0 \\ 0&z^{-1/2}
\end{array}
\right)
\;.
\label{eq:K6}$$ which acts also as a *bona fide* conjugation since $K_{21}(z^{-1})=K_{12}(z)$. Moreover, a redefinition of the Lax operators in (\[eq:rll\_uq\],\[eq:rll\_uq2\]) as $$\begin{aligned}
L_F^-(z^2) &=&
\left(
\begin{array}{cc}
z^{-1/2} &0 \\ 0& z^{1/2}
\end{array}
\right)
L_V^-(z)
\left(
\begin{array}{cc}
z^{1/2} &0 \\ 0& z^{-1/2}
\end{array}
\right)
\\
L_F^+(z^2) &=&
\left(
\begin{array}{cc}
z^{-1/2}q^{-c/4} &0 \\ 0& z^{1/2}q^{c/4}
\end{array}
\right)
L_V^+(z)
\left(
\begin{array}{cc}
z^{1/2}q^{-c/4} &0 \\ 0& z^{-1/2}q^{c/4}
\end{array}
\right)
\label{eq:Ltransf}\end{aligned}$$ provides a genuine algebra isomorphism between ${{{\cal U}^F_{q}{{(\widehat{sl(2)}_{c})}}}}$ and ${{{\cal U}^V_{q}{{(\widehat{sl(2)}_{c})}}}}$.
Face type algebras
==================
Twist operator ${{{\cal U}^F_{q}{{(\widehat{sl(2)}_{c})}}}} \rightarrow {{{\cal U}_{q,\lambda}{{(\widehat{sl(2)}_{c})}}}}$
--------------------------------------------------------------------------------------------------------------------------
The two $R$ matrices of ${{{\cal U}^F_{q}{{(\widehat{sl(2)}_{c})}}}}$ and ${{{\cal U}_{q,\lambda}{{(\widehat{sl(2)}_{c})}}}}$ can be related by a twist operator: $$R[{{{\cal U}_{q,\lambda}{{(\widehat{sl(2)}_{c})}}}}] = F^{(3)}_{21}(w) \; R[{{{\cal U}^F_{q}{{(\widehat{sl(2)}_{c})}}}}] \; {F^{(3)}_{12}}(w)^{-1}
\label{eq:twF3} \;.$$ The twist operator $F^{(3)}(w)$ is given by $$F^{(3)}(w) =
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & \displaystyle\frac{w(q-q^{-1})}{1-w} & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right) \;.
\label{eq:F3}$$
Dynamical face elliptic affine algebra ${{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}$
--------------------------------------------------------------------------------------------
### Twist operator ${{{\cal U}^F_{q}{{(\widehat{sl(2)}_{c})}}}} \rightarrow {{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}$
The existence of a twist operator between ${{{\cal U}^F_{q}{{(\widehat{sl(2)}_{c})}}}}$ and ${{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}$ was proved at the level of universal matrices in [@JKOS]. Once the operators are evaluated, one gets $$R[{{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}] = F^{(1)}_{21}(z^{-1};p,w) \; R[{{{\cal U}^F_{q}{{(\widehat{sl(2)}_{c})}}}}] \;
{F^{(1)}_{12}}(z;p,w)^{-1} \;.$$ The twist operator $F^{(1)}(z;p,w)$ is given by $$F^{(1)}(z;p,w) = \rho_{F}(z;p)
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & X_{11}(z) & X_{12}(z) & 0 \\
0 & X_{21}(z) & X_{22}(z) & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right) \;,$$ where $$\begin{aligned}
\label{eq:Xij}
X_{11}(z) &=& {{_{2}\phi_{1}\left(\begin{array}{c}{wq^2}\quad{q^2}\\
{w}\end{array};{p,pq^{-2}z}\right)}} \;, \\
X_{12}(z) &=& \frac{w(q-q^{-1})}{1-w} \;
{{_{2}\phi_{1}\left(\begin{array}{c}{wq^2}\quad{pq^2}\\
{pw}\end{array};{p,pq^{-2}z}\right)}} \;, \\
X_{21}(z) &=& z \; \frac{pw^{-1}(q-q^{-1})}{1-pw^{-1}} \;
{{_{2}\phi_{1}\left(\begin{array}{c}{pw^{-1}q^2}\quad{pq^2}\\
{p^2w^{-1}}\end{array};{p,pq^{-2}z}\right)}} \;, \\
X_{22}(z) &=& {{_{2}\phi_{1}\left(\begin{array}{c}{pw^{-1}q^2}\quad{q^2}\\
{pw^{-1}}\end{array};{p,pq^{-2}z}\right)}} \;.\end{aligned}$$ The $q$-hypergeometric function ${{_{2}\phi_{1}\left(\begin{array}{c}{q^a}\quad{q^b}\\
{q^c}\end{array};{q,z}\right)}}$ is defined by $${{_{2}\phi_{1}\left(\begin{array}{c}{q^a}\quad{q^b}\\
{q^c}\end{array};{q,z}\right)}} = \sum_{n=0}^\infty
\frac{(q^a;q)_{n}(q^b;q)_{n}}{(q^c;q)_{n}(q;q)_{n}} \; z^n
\qquad \mathrm{where} \qquad
(x;q)_{n} = \prod_{k=0}^{n-1} (1-xq^k) \;.$$ The normalization factor is $$\rho_{F}(z;p) = \frac{(pz;p,q^4)_{\infty} \;
(pq^4z;p,q^4)_{\infty}}
{(pq^2z;p,q^4)_{\infty}^2}\;.
\label{eq:normfact}$$
### Twist operator ${{{\cal U}_{q,\lambda}{{(\widehat{sl(2)}_{c})}}}} \rightarrow {{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}$
Combining the last two twist-like actions, one obtains $$R[{{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}] = F^{(5)}_{21}(z^{-1};p,w) \; R[{{{\cal U}_{q,\lambda}{{(\widehat{sl(2)}_{c})}}}}] \;
{F^{(5)}_{12}(z;p,w)}^{-1} \;.$$ The twist operator $F^{(5)}(z;p,w)$ is given by $F^{(5)}=F^{(1)}
{F^{(3)}}^{-1}$, that is $$F^{(5)}(z;w) = \rho_{F}(z;p)
\left( \begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & X_{11}'(z) & X_{12}'(z) & 0 \\
0 & X_{21}'(z) & X_{22}'(z) & 0 \\
0 & 0 & 0 & 1 \\
\end{array} \right) \;,$$ where $$\begin{aligned}
X'_{11}(z) &=& X_{11}(z) \;, \\
X'_{12}(z) &=& X_{12}(z) - \frac{w(q-q^{-1})}{1-w} \; X_{11}(z) \;, \\
X'_{21}(z) &=& X_{21}(z) \;, \\
X'_{22}(z) &=& X_{22}(z) - \frac{w(q-q^{-1})}{1-w} \; X_{21}(z) \;.\end{aligned}$$ and $X_{ij}(z)$ are given in (\[eq:Xij\]).\
The normalization factor $\rho_{F}(z;p)$ is given by (\[eq:normfact\]).
Dynamical double Yangian ${{{\cal D}Y_{s}^{}(sl(2))_{c}}}$
----------------------------------------------------------
By taking the scaling limit of the connection (\[eq:twF3\]), one gets $$R[{{{\cal D}Y_{s}^{}(sl(2))_{c}}}] = F^{(4)}_{21}(s) \; R[{{{\cal D}Y_{}^{}(sl(2))_{c}}}] \;
{F^{(4)}_{12}}(s)^{-1} \;.$$ The twist operator $F^{(4)}(s)$ is the scaling limit of the twist operator $F^{(3)}$, see eq. (\[eq:F3\]). It is given by $$F^{(4)}(s) =
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & -s^{-1} & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right) \;.$$
Deformed double Yangian ${{{\cal D}Y_{r}^{F}(sl(2))_{c}}}$
----------------------------------------------------------
### Twist operator ${{{\cal D}Y_{r}^{F}(sl(2))_{c}}}$ $\rightarrow {{{\cal D}Y_{r}^{V}(sl(2))_{c}}}$
The two deformed double Yangians ${{{\cal D}Y_{r}^{V}(sl(2))_{c}}}$ and ${{{\cal D}Y_{r}^{F}(sl(2))_{c}}}$ obtained from the vertex type algebras on one hand, and from face type algebras on the other hand, are related by twist-like actions. One has: $$R[{{{\cal D}Y_{r}^{F}(sl(2))_{c}}}](\beta) = K^{(6)}_{21}(-\beta) \; R[{{{\cal D}Y_{r}^{V6}(sl(2))_{c}}}](\beta) \;
{K^{(6)}_{12}(\beta)}^{-1} \;.$$ The twist operator $K^{(6)}$ is actually equal to the twist operator (\[eq:K6\]) by setting $z=e^{-\beta/r}$: $$K^{(6)}(\beta) =
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & e^{\beta/2r} & 0 & 0 \\
0 & 0 & e^{-\beta/2r} & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right) \;.$$ Using the rigid twist operator (\[eq:twistcst\]), one gets also: $$R[{{{\cal D}Y_{r}^{F}(sl(2))_{c}}}](\beta) = K^{(8)}_{21}(-\beta) \; R[{{{\cal D}Y_{r}^{V8}(sl(2))_{c}}}](\beta) \;
{K^{(8)}_{12}(\beta)}^{-1} \;.$$ The twist operator $K^{(8)}$ is given by $K^{(8)}=K^{(6)} K^{-1}$, that is $$K^{(8)}(\beta) = \frac{1}{2} \left( \begin{array}{cccc}
1 & -1 & -1 & 1 \\
ie^{\beta/2r} & ie^{\beta/2r} & -ie^{\beta/2r} & -ie^{\beta/2r} \\
ie^{-\beta/2r} & -ie^{-\beta/2r} & ie^{-\beta/2r} & -ie^{-\beta/2r} \\
-1 & -1 & -1 & -1 \\
\end{array} \right) \;.$$ This twist provides a link between the face type and vertex type double Yangian structures.
### Twist operator ${{{\cal D}Y_{}^{}(sl(2))_{c}}} \rightarrow {{{\cal D}Y_{r}^{F}(sl(2))_{c}}}$
The connection between $R$-matrices of ${{{\cal D}Y_{}^{}(sl(2))_{c}}}$ and ${{{\cal D}Y_{r}^{F}(sl(2))_{c}}}$ can be established by three different combinations of previously constructed twist-like actions. These three combinations of course give by construction the same twist operator $F^{(7)} = K^{(8)} E^{(2)} =
K^{(6)} K^{-1} E^{(2)}$. One has therefore $$R[{{{\cal D}Y_{r}^{F}(sl(2))_{c}}}] = F^{(7)}_{21}(-\beta;r,) \; R[{{{\cal D}Y_{}^{}(sl(2))_{c}}}] \;
{F^{(7)}_{12}(\beta;r)}^{-1} \;.$$ The twist operator $F^{(7)}$ is given by $$F^{(7)}(\beta;r) = {{{{\textstyle{\frac{1}{2}}}}}}\; \rho_{E}(\beta;r)
\left(
\begin{array}{cccc}
1 & -1 & -1 & 1 \\
i(a_{E}-d_{E})e^{\beta/2r} & i(b_{E}-c_{E})e^{\beta/2r} &
i(c_{E}-b_{E})e^{\beta/2r} & i(d_{E}-a_{E})e^{\beta/2r} \\
i(a_{E}-d_{E})e^{-\beta/2r} & i(c_{E}-b_{E})e^{-\beta/2r} &
i(b_{E}-c_{E})e^{-\beta/2r} & i(d_{E}-a_{E})e^{-\beta/2r} \\
-1 & -1 & -1 & -1 \\
\end{array}
\right) \;,
\label{eq:F7}$$ where $a_{E},b_{E},c_{E},d_{E}$ are given by (\[eq:aEscal\])–(\[eq:dEscal\]) and the normalization factor $\rho_{E}(\beta;r)$ by (\[eq:normrhoEscal\]).
Trigonometric Dynamical deformed double Yangian ${{{\cal D}Y_{r,s}^{-\infty}(sl(2))_{c}}}$
------------------------------------------------------------------------------------------
### Twist operator ${{{\cal D}Y_{r}^{F}(sl(2))_{c}}} \rightarrow {{{\cal D}Y_{r,s}^{-\infty}(sl(2))_{c}}}$
The connection between ${{{\cal D}Y_{r}^{F}(sl(2))_{c}}}$ and ${{{\cal D}Y_{r,s}^{-\infty}(sl(2))_{c}}}$ is achieved by the twist operator $F^{(3)}$: $$R[{{{\cal D}Y_{r,s}^{-\infty}(sl(2))_{c}}}] = F^{(3)}_{21}(s) \; R[{{{\cal D}Y_{r}^{F}(sl(2))_{c}}}] \;
{F^{(3)}_{12}(s)}^{-1} \;.$$ The twist operator $F^{(3)}$ is actually equal to the twist operator (\[eq:F3\]) by setting $q=e^{i\pi/r}$ and $w=e^{2i\pi s/r}$: $$F^{(3)}(s) =
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & -e^{i\pi s/r} \displaystyle
\frac{\sin\frac{\pi}{r}}{\sin\frac{\pi s}{r}} & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right) \;.$$
### Twist operator ${{{\cal D}Y_{}^{}(sl(2))_{c}}} \rightarrow {{{\cal D}Y_{r,s}^{-\infty}(sl(2))_{c}}}$
Combination of two twist-like operations yields the connection between ${{{\cal D}Y_{}^{}(sl(2))_{c}}}$ and ${{{\cal D}Y_{r,s}^{-\infty}(sl(2))_{c}}}$: $$R[{{{\cal D}Y_{r,s}^{-\infty}(sl(2))_{c}}}] = F^{(10)}_{21}(-\beta;r,s) \; R[{{{\cal D}Y_{}^{}(sl(2))_{c}}}] \;
{F^{(10)}_{12}(\beta;r,s)}^{-1} \;.$$ The twist operator $F^{(10)}$ is given by $F^{(10)} = F^{(3)}
F^{(7)}$, that is $$\begin{aligned}
&&
F^{(10)}(\beta;r,s) = {{{{\textstyle{\frac{1}{2}}}}}}\; \rho_{E}(\beta;r)
\nonumber\\
&& \qquad
\left( \begin{array}{cccc}
1 & -1 & -1 & 1 \\
i(a_{E}-d_{E})e_{+} & i(b_{E}-c_{E})e_{-} & i(c_{E}-b_{E})e_{-} &
i(d_{E}-a_{E})e_{+} \\
i(a_{E}-d_{E})e^{-\beta/2r} & i(c_{E}-b_{E})e^{-\beta/2r} &
i(b_{E}-c_{E})e^{-\beta/2r} & i(d_{E}-a_{E})e^{-\beta/2r} \\
-1 & -1 & -1 & -1 \\
\end{array} \right) \nonumber\\\end{aligned}$$ where $e_{\pm} = e^{\beta/2r} \mp e^{i\pi s/r} e^{-\beta/2r} \frac{\sin(\pi
/r)}{\sin(\pi s/r)}$, the functions $a_{E},b_{E},c_{E},d_{E}$ are given by the formulae (\[eq:aEscal\])–(\[eq:dEscal\]) and the normalization factor $\rho_{E}(\beta;r)$ by (\[eq:normrhoEscal\]).
### Twist operator ${{{\cal D}Y_{s}^{}(sl(2))_{c}}} \rightarrow {{{\cal D}Y_{r,s}^{-\infty}(sl(2))_{c}}}$
Again, combination of two twist-like operations yields the connection between ${{{\cal D}Y_{s}^{}(sl(2))_{c}}}$ and ${{{\cal D}Y_{r,s}^{-\infty}(sl(2))_{c}}}$: $$R[{{{\cal D}Y_{r,s}^{-\infty}(sl(2))_{c}}}] = F^{(8)}_{21}(-\beta;r,s) \; R[{{{\cal D}Y_{s}^{}(sl(2))_{c}}}] \;
{F^{(8)}_{12}(\beta;r,s)}^{-1} \;.$$ The twist operator $F^{(8)}$ is given by $F^{(8)} = F^{(10)}
{F^{(4)}}^{-1}$, that is $$\begin{aligned}
&& F^{(8)}(\beta;r,s) = {{{{\textstyle{\frac{1}{2}}}}}}\; \rho_{E}(\beta;r) \nonumber \\
&& \qquad \left( \begin{array}{cccc}
1 & -1 & -1-s^{-1} & 1 \\
i(a_{E}-d_{E})e_{+} & i(b_{E}-c_{E})e_{-} &
i(c_{E}-b_{E})e_{-}(1-s^{-1}) &
i(d_{E}-a_{E})e_{+} \\
i(a_{E}-d_{E})e^{-\beta/2r} & i(c_{E}-b_{E})e^{-\beta/2r} &
i(b_{E}-c_{E})e^{-\beta/2r}(1-s^{-1}) &
i(d_{E}-a_{E})e^{-\beta/2r} \\
-1 & -1 & -1-s^{-1} & -1 \\
\end{array} \right) \nonumber \\
\label{eq:F8}\end{aligned}$$ where $e_{\pm}$, $a_{E},b_{E},c_{E},d_{E}$ and $\rho_{E}(\beta;r)$ have the same meaning as above.
Dynamical deformed double Yangian ${{{\cal D}Y_{r,s}^{}(sl(2))_{c}}}$
---------------------------------------------------------------------
### Twist operator ${{{\cal D}Y_{r,s}^{-\infty}(sl(2))_{c}}} \rightarrow {{{\cal D}Y_{r,s}^{}(sl(2))_{c}}}$
The $R$-matrices of ${{{\cal D}Y_{r,s}^{-\infty}(sl(2))_{c}}}$ and ${{{\cal D}Y_{r,s}^{}(sl(2))_{c}}}$ are connected by a diagonal TLA (not depending on the spectral parameter): $$R[{{{\cal D}Y_{r,s}^{}(sl(2))_{c}}}] = G_{21}(r,s) \; R[{{{\cal D}Y_{r,s}^{-\infty}(sl(2))_{c}}}] \;
{G_{12}(r,s)}^{-1} \;.$$ The twist operator $G$ is given by $$G(r,s) = \left( \begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & g^{-1} & 0 & 0 \\
0 & 0 & g & 0 \\
0 & 0 & 0 & 1 \\
\end{array} \right)
=
\left(
\begin{array}{cc}
1&0 \\ 0&g
\end{array}
\right)
\otimes
\left(
\begin{array}{cc}
1&0 \\ 0&g^{-1}
\end{array}
\right)
\;,
\label{eq:G}$$ where $$g(r,s) = \displaystyle\frac{\Gamma_{1}(r-s \; \vert \; r)}
{\Gamma_{1}(r-s+1 \; \vert \; r)^{1/2} \;
\Gamma_{1}(r-s-1 \; \vert \; r)^{1/2}}\;.
\label{eq:defg}$$
**Remark:** Equivalently, $G$ expressed in terms of the parameters $q=e^{i\pi/r}$ and $w=e^{2i\pi s/r}$ realizes a TLA between ${{{\cal U}_{q,\lambda}{{(\widehat{sl(2)}_{c})}}}}$ and ${\cal U}^{\Gamma}_{q,\lambda}{{(\widehat{sl(2)}_{c})}}$ defined in remark 2, Section \[subsect:dyrsi\].
### Twist operator ${{{\cal D}Y_{s}^{}(sl(2))_{c}}} \rightarrow {{{\cal D}Y_{r,s}^{}(sl(2))_{c}}}$
Combining the last two twists, one gets: $$R[{{{\cal D}Y_{r,s}^{}(sl(2))_{c}}}] = F^{(6)}_{21}(-\beta;r,s) \; R[{{{\cal D}Y_{s}^{}(sl(2))_{c}}}] \;
{F^{(6)}_{12}(\beta;r,s)}^{-1} \;.$$ The twist operator $F^{(6)}$ is given by $F^{(6)} = G F^{(8)}$, that is $$\begin{aligned}
&& F^{(6)}(\beta;r,s) = {{{{\textstyle{\frac{1}{2}}}}}}\; \rho_{E}(\beta;r) \nonumber \\
&&
\left(
\begin{array}{cccc}
1 & -1 & -1-s^{-1} & 1 \\
i(a_{E}-d_{E})e_{+}g^{-1} & i(b_{E}-c_{E})e_{-}g^{-1} &
i(c_{E}-b_{E})e_{-}(1-s^{-1})g^{-1} & i(d_{E}-a_{E})e_{+}g^{-1} \\
i(a_{E}-d_{E})e^{-\beta/2r}g & i(c_{E}-b_{E})e^{-\beta/2r}g &
i(b_{E}-c_{E})e^{-\beta/2r}(1-s^{-1})g &
i(d_{E}-a_{E})e^{-\beta/2r}g \\
-1 & -1 & -1-s^{-1} & -1 \\
\end{array}
\right) \nonumber \\\end{aligned}$$ where $e_{\pm}$, $a_{E},b_{E},c_{E},d_{E}$ and $\rho_{E}(\beta;r)$ have the same meaning as above and $g$ is given by (\[eq:defg\]).
### Twist operator ${{{\cal D}Y_{}^{}(sl(2))_{c}}} \rightarrow {{{\cal D}Y_{r,s}^{}(sl(2))_{c}}}$
Similarly, by a combination of previous twists, one gets: $$R[{{{\cal D}Y_{r,s}^{}(sl(2))_{c}}}] = F^{(2)}_{21}(-\beta;r,s) \; R[{{{\cal D}Y_{}^{}(sl(2))_{c}}}] \;
{F^{(2)}_{12}(\beta;r,s)}^{-1} \;.$$ The twist operator $F^{(2)}$ is given by $F^{(2)} = G F^{(10)}$, that is $$\begin{aligned}
&& F^{(2)}(\beta;r,s) = {{{{\textstyle{\frac{1}{2}}}}}}\; \rho_{E}(\beta;r) \nonumber \\
&& \qquad \left( \begin{array}{cccc}
1 & -1 & -1 & 1 \\
i(a_{E}-d_{E})e_{+}g^{-1} & i(b_{E}-c_{E})e_{-}g^{-1}
& i(c_{E}-b_{E})e_{-}g^{-1} &
i(d_{E}-a_{E})e_{+}g^{-1} \\
i(a_{E}-d_{E})e^{-\beta/2r}g & i(c_{E}-b_{E})e^{-\beta/2r}g &
i(b_{E}-c_{E})e^{-\beta/2r}g & i(d_{E}-a_{E})e^{-\beta/2r}g \\
-1 & -1 & -1 & -1 \\
\end{array} \right) \nonumber \\\end{aligned}$$ where $e_{\pm}$, $a_{E},b_{E},c_{E},d_{E}$ and $\rho_{E}(\beta;r)$ have the same meaning as above and $g$ is given by (\[eq:defg\]).
### Twist operator ${{{\cal D}Y_{r}^{F}(sl(2))_{c}}} \rightarrow {{{\cal D}Y_{r,s}^{}(sl(2))_{c}}}$
Finally, connection between ${{{\cal D}Y_{r}^{F}(sl(2))_{c}}}$ and ${{{\cal D}Y_{r,s}^{}(sl(2))_{c}}}$ is provided by $$R[{{{\cal D}Y_{r,s}^{}(sl(2))_{c}}}] = F^{(11)}_{21}(r,s) \; R[{{{\cal D}Y_{r}^{F}(sl(2))_{c}}}] \;
{F^{(11)}_{12}(r,s)}^{-1} \;.$$ The twist operator $F^{(11)}$ is given by $F^{(11)} = G F^{(3)}$, that is $$F^{(11)}(r,s) =
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & g^{-1} & -e^{i\pi s/r} \displaystyle
\frac{\sin\frac{\pi}{r}}{\sin\frac{\pi s}{r}} \, g^{-1} & 0 \\
0 & 0 & g & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right) \;,$$ where $g$ is given by (\[eq:defg\]).
The different twist procedures in the face case are summarized in Figure \[fig:twf\].
Connections with ${\cal A}_{q,p;\pi}{{(\widehat{sl(2)}_{c})}}$ and derived algebras
-----------------------------------------------------------------------------------
### Twist ${{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}\to {\cA}_{q,p;\pi}{{(\widehat{sl(2)}_{c})}}$
The $R$-matrix of ${\cal A}_{q,p;\pi}{{(\widehat{sl(2)}_{c})}}$ given in [@HouYang] (actually their $R^+$-matrix) is $$R = z^{1/2r}\rho(z) \left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\[1mm]
0 &
\displaystyle\frac{\Theta_{p}(z)\Theta_{p}(q^{-2}w)}
{\Theta_{p}(q^2z)\Theta_{p}(w)}
&
\displaystyle\frac{\Theta_{p}(zw)\Theta_{p}(q^2)}
{\Theta_{p}(q^2z)\Theta_{p}(w)}
& 0 \\[5mm]
0 &
\displaystyle\frac{\Theta_{p}(z^{-1}w)\Theta_{p}(q^2)}
{\Theta_{p}(q^2z)\Theta_{p}(w)}
&
\displaystyle\frac{\Theta_{p}(z)\Theta_{p}(q^2 w)}
{\Theta_{p}(q^2z)\Theta_{p}(w)}
& 0 \\[4mm]
0 & 0 & 0 & 1 \\
\end{array}
\right)
\label{eq:raqppi}$$ where $\rho(z)$ is the same as (\[eq:rhoelpb\]). This $R$-matrix is obtained from (\[eq:Relpb\]) by exchanging factors in $b$ and $\bar b$ so as to reconstruct a full $\Theta$-function dependence and correcting the $z^{1/2r}$ factor. All this can be achieved by a factorized diagonal twist of the form of $G$ (\[eq:G\]).
### Twist ${{{\cal D}Y_{r,s}^{-\infty}(sl(2))_{c}}} \to {\cA}_{\hbar,\eta;\pi}{{(\widehat{sl(2)}_{c})}}$
Again, the $R$-matrix of the scaling limit ${\cA}_{\hbar,\eta;\pi}{{(\widehat{sl(2)}_{c})}}$ [@HouYang] of ${\cA}_{q,p;\pi}{{(\widehat{sl(2)}_{c})}}$ can be obtained from that of ${{{\cal D}Y_{r,s}^{-\infty}(sl(2))_{c}}}$ (\[eq:dyrsi\]) by a factorized diagonal twist. It also has the form of $G$ (\[eq:G\]), with now $g^2 = \frac{\sin\pi(s-1)/r}{\sin\pi s/r}$.
Finite dimensional algebras
---------------------------
In both cases where TLA actions are known for non affine algebras, they are evaluations of universal twists.
### Elliptic algebra ${\cal B}_{q,\lambda}(sl(2))$
The twist operator that links ${\cal U}_{q}(sl(2))$ to ${\cal B}_{q,\lambda}(sl(2))$ is [@Bab]: $$F^{(i)}(w) = \left( \begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & \displaystyle\frac{(q-q^{-1})w}{1-w} & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array} \right)\;.$$ The universal form of the twist is [@Bab] $$\cF(w) = \sum_{n=0}^{\infty} \frac{(q^2w)^n(q-q^{-1})^n}
{(n)_{q^{-2}}! (q^{-2} w(t^2\otimes 1); q^{-2})_n}
(et)^n \otimes (tf)^n \;,
\label{eq:Funivw}$$ where $$(x;q)_n = \prod_{i = 0}^{n-1} (1-x q^{i}) \;.
\label{eq:prodfini}$$
### Dynamical algebra ${\cal U}_{s}(sl(2))$
Its $R$ matrix can be obtained by action of the twist $$F^{(ii)}(s) = \left( \begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & -s^{-1} & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array} \right)$$ on the $R$ matrix of ${\cal U}(sl(2))$: $R={\mbox{1\hspace{-1mm}I}}_{4\times 4}$.
We find the universal form of the twist to be $${\cF}= \sum_{n=0}^\infty \frac{1}{n!}\left( \prod_{k=0}^{n-1}
[(1+k-s) 1 - h]\otimes 1\right)^{-1}\ e^n\otimes f^n \;.$$ This twist is the scaling limit of the universal twist (\[eq:Funivw\]). We checked that it satisfies the shifted cocycle condition (\[eq:cocycle\_decale\]).
Homothetical twists \[sect:homothetical\]
=========================================
We recall that homothetical TLAs connect two $R$-matrices *up to a scalar factor*: $$\widetilde{R} = f(z,p,q) F_{21}(z^{-1}) R F_{12}(z)^{-1} \;.$$ From now on, we shall denote such a relation by: $$\widetilde{R} \;\sim\; F_{21}(z^{-1}) R F_{12}(z)^{-1} \;.$$ We now describe two sets of homothetical TLAs. The first one starts from the unit evaluated $R$-matrix of ${{{\cal U}_{}{{(\widehat{sl(2)}_{c})}}}}$ and leads to unitary $R$-matrices. The second one goes backwards along direction of the scaling limits.
It is important to notice at this point that the Lie algebraic structure of ${{{\cal U}_{}{{(\widehat{sl(2)}_{c})}}}}$ is *not* described by the $RLL$ formalism using its unit $R$-matrix (this was also the case for $\cU(sl(2))$). In fact, the Lie algebraic structure is described by the semi-classical $r$-matrix, i.e. the next-to-leading order of the evaluated universal $R$-matrix of ${{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}$.
Unitary matrices
----------------
Four homothetical TLAs can be defined between the unit matrix and the vertex quantum affine algebras. By construction, a TLA on the unit matrix will lead to unitary $R$-matrices, while vertex quantum affine algebras are defined by crossing-symmetry but *non*-unitary $R$-matrices.
### Twist operator ${{{\cal U}_{}{{(\widehat{sl(2)}_{c})}}}} \rightarrow {{{\cal U}^V_{q}{{(\widehat{sl(2)}_{c})}}}}$
$$R[{{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}] \;\sim\; H^{(1)}_{21}(z^{-1}) \; \mbox{1\hspace{-1mm}I}
\; {H^{(1)}_{12}}(z)^{-1} \;.$$
The twist operator $H^{(1)}(z)$ is given by $$H^{(1)}(z) = \left( \begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & \displaystyle
\frac{q^{1/2}z^{(-1-\epsilon)/2} -
q^{-1/2}z^{(1+\epsilon)/2}}{qz^{-1} - q^{-1}z}
&
\displaystyle
\frac{q^{1/2}z^{(-1+\epsilon)/2} -
q^{-1/2}z^{(1-\epsilon)/2}}{qz^{-1} - q^{-1}z}
& 0 \\[.5cm]
0 & \displaystyle
\frac{q^{1/2}z^{(-1+\epsilon)/2} -
q^{-1/2}z^{(1-\epsilon)/2}}{qz^{-1} - q^{-1}z}
&
\displaystyle
\frac{q^{1/2}z^{(-1-\epsilon)/2} -
q^{-1/2}z^{(1+\epsilon)/2}}{qz^{-1} - q^{-1}z}
& 0 \\
0 & 0 & 0 & 1 \\
\end{array} \right) \;,$$ where $\epsilon$ is an arbitrary non-vanishing parameter.
### Twist operator ${{{\cal U}_{}{{(\widehat{sl(2)}_{c})}}}} \rightarrow {{{\cal A}_{q,p}{{(\widehat{sl(2)}_{c})}}}}$
$$R[{{{\cal A}_{q,p}{{(\widehat{sl(2)}_{c})}}}}] \;\sim\; H^{(2)}_{21}(z^{-1}) \; \mbox{1\hspace{-1mm}I} \;
{H^{(2)}_{12}}(z)^{-1} \;.$$
The twist operator $H^{(2)}(z)$ is given by $$H^{(2)}(z) =
\left(
\begin{array}{cccc}
A & 0 & 0 & D \\
0 & B & C & 0 \\
0 & C & B & 0 \\
D & 0 & 0 & A \\
\end{array}
\right) \;,$$ such that $$\begin{aligned}
A(1/z) \pm D(1/z) &=& [ a(z)\pm d(z)][A(z)\pm D(z)] \;,\nonumber\\
B(1/z) \pm C(1/z) &=& [ b(z)\pm c(z)][B(z)\pm C(z)] \;,
\label{eq:ABCD}\end{aligned}$$ the functions $a$, $b$, $c$, $d$ being the entries of the $R$-matrix of ${{{\cal A}_{q,p}{{(\widehat{sl(2)}_{c})}}}}$ (\[eq:elpa\]). Solutions of (\[eq:ABCD\]), viewed as a system of functional equations for $A$, $B$, $C$, $D$, do exist since the functions $a \pm
d$, $b \pm c$ (\[eq:az\])-(\[eq:dz\]) all have precisely the form $f(z)/f(z^{-1})$. One can choose for instance $$\begin{aligned}
A(z) \pm D(z) &=& (\mp q^{-1}z^{-1}p^{1/2};p)_\infty
(\mp qzp^{1/2};p)_\infty
\;, \\
B(z) \pm C(z) &=& f(z)\; (\mp pq{-1}z^{-1};p)_\infty (\mp
pqz;p)_\infty \;,\end{aligned}$$ where $$f(z) = \frac{q^{1/2}z^{-1}-q^{-1/2}z \pm q^{1/2} \mp q^{-1/2}}
{qz^{-1}-q^{-1}z} \;.$$
### Twist operator ${{{\cal U}_{}{{(\widehat{sl(2)}_{c})}}}} \rightarrow {{{\cal D}Y_{}^{}(sl(2))_{c}}}$
$$R[{{{\cal D}Y_{}^{}(sl(2))_{c}}}] \;\sim\; H^{(3)}_{21}(-\beta) \; \mbox{1\hspace{-1mm}I}
\; {H^{(3)}_{12}}(\beta)^{-1} \;.$$
The twist operator $H^{(3)}(\beta)$ is given by $$H^{(3)}(\beta) =
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & \displaystyle
\frac{\pi - i\beta(1+\epsilon)}
{2(\pi - i\beta)}
&
\displaystyle \frac{\pi - i\beta(1-\epsilon)}
{2(\pi - i\beta)}
& 0 \\[.5cm]
0 & \displaystyle
\frac{\pi - i\beta(1-\epsilon)} {2(\pi - i\beta)}
&
\displaystyle
\frac{\pi - i\beta(1+\epsilon)}
{2(\pi - i\beta)}
& 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right) \;.$$
### Twist operator ${{{\cal U}_{}{{(\widehat{sl(2)}_{c})}}}} \rightarrow {{{\cal D}Y_{r}^{V6}(sl(2))_{c}}}$
$$R[{{{\cal D}Y_{r}^{V6}(sl(2))_{c}}}] \;\sim\; H^{(4)}_{21}(-\beta) \;
\mbox{1\hspace{-1mm}I} \; {H^{(4)}_{12}}(\beta)^{-1} \;.$$
The twist operator $H^{(4)}(\beta)$ is given by $$H^{(4)}(\beta) =
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & \displaystyle
\frac{\sin \frac{\pi - i\beta - \epsilon i\beta }{2r}}
{\sin \frac{\pi - i\beta}{r}}
&
\displaystyle
\frac{\sin \frac{\pi - i\beta + \epsilon i\beta }{2r}}
{\sin \frac{\pi - i\beta}{r}}
& 0 \\[.5cm]
0 & \displaystyle
\frac{\sin \frac{\pi - i\beta + \epsilon i\beta }{2r}}
{\sin \frac{\pi - i\beta}{r}}
&
\displaystyle
\frac{\sin \frac{\pi - i\beta - \epsilon i\beta }{2r}}
{\sin \frac{\pi - i\beta}{r}}
& 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right) \;.$$
Inverse scaling procedures
--------------------------
### Inverse scaling procedure ${{{\cal D}Y_{}^{}(sl(2))_{c}}}$ to $\cU_q^V{{(\widehat{sl(2)}_{c})}}$
Using the correspondence (\[eq:identiv\]), the formula (\[eq:E6\]) gives a homothetical twist between ${{{\cal D}Y_{}^{}(sl(2))_{c}}}$ and $\cU_q^V{{(\widehat{sl(2)}_{c})}}$, that is, the inverse of the scaling procedure $\cU_q^V{{(\widehat{sl(2)}_{c})}}\rightarrow {{{\cal D}Y_{}^{}(sl(2))_{c}}}$: $$R[{{{\cal U}^V_{q}{{(\widehat{sl(2)}_{c})}}}}](z=e^{-\beta/r},q=e^{i\pi/r}) \;\sim\;
E^{(3)}_{21}(-\beta;r) \; R[{{{\cal D}Y_{}^{}(sl(2))_{c}}}](\beta;r) \;
E^{(3)}_{12}(\beta;r)^{-1} \;.$$
### Inverse scaling procedure ${{{\cal D}Y_{}^{V8}(sl(2))_{c}}}$ to ${{{\cal A}_{q,p}{{(\widehat{sl(2)}_{c})}}}}$
The identification between the $R$ matrices of ${{{\cal U}^V_{q}{{(\widehat{sl(2)}_{c})}}}}$ and ${{{\cal D}Y_{r}^{V6}(sl(2))_{c}}}$ through the formulae (\[eq:identiv\]) allows us to get a homothetical twist operator between ${{{\cal D}Y_{r}^{V8}(sl(2))_{c}}}$ and ${{{\cal A}_{q,p}{{(\widehat{sl(2)}_{c})}}}}$, that is, the inverse of the scaling procedure ${{{\cal A}_{q,p}{{(\widehat{sl(2)}_{c})}}}}\rightarrow
{{{\cal D}Y_{r}^{V8}(sl(2))_{c}}}$. More precisely, one has: $$R[{{{\cal A}_{q,p}{{(\widehat{sl(2)}_{c})}}}}](z=e^{-\beta/r},q=e^{i\pi/r}) \;\sim\;
E^{(4)}_{21}(z^{-1};p) \; R[{{{\cal D}Y_{}^{V8}(sl(2))_{c}}}](\beta;r) \;
E^{(4)}_{12}(z;p)^{-1} \;.$$ The twist operator $E^{(4)}(z,p)$ is given by $E^{(4)}=E^{(1)}K^{-1}$, that is $$E^{(4)}(z,p) = {{{{\textstyle{\frac{1}{2}}}}}}\; \rho_{E}(z;p)
\left(
\begin{array}{rrrr}
(a_{E}-d_{E}) & -(a_{E}+d_{E}) & -(a_{E}+d_{E}) & (a_{E}-d_{E}) \\
i(b_{E}+c_{E}) & i(b_{E}-c_{E}) & i(c_{E}-b_{E}) & -i(b_{E}+c_{E}) \\
i(b_{E}+c_{E}) & i(c_{E}-b_{E}) & i(b_{E}-c_{E}) & -i(b_{E}+c_{E}) \\
(d_{E}-a_{E}) & -(a_{E}+d_{E}) & -(a_{E}+d_{E}) & (d_{E}-a_{E}) \\
\end{array}
\right) \;,
\label{eq:E8}$$ where $a_{E},b_{E},c_{E},d_{E}$ are given by the formulae (\[eq:adE\],\[eq:bcE\]) and the normalization factor $\rho_{E}(z;p)$ by (\[eq:normrhoE\]).
### Inverse scaling procedure ${{{\cal D}Y_{}^{}(sl(2))_{c}}}$ to $\cU_q^F{{(\widehat{sl(2)}_{c})}}$
Using the correspondence (\[eq:identif2\]), the formula (\[eq:F7\]) gives a homothetical twist between ${{{\cal D}Y_{}^{}(sl(2))_{c}}}$ and $\cU_q^F{{(\widehat{sl(2)}_{c})}}$, that is, the inverse of the scaling procedure $\cU_q^F{{(\widehat{sl(2)}_{c})}}\rightarrow {{{\cal D}Y_{}^{}(sl(2))_{c}}}$: $$R[{{{\cal U}^F_{q}{{(\widehat{sl(2)}_{c})}}}}](z=e^{-2\beta/r},q=e^{i\pi/r}) \;\sim\;
F^{(7)}_{21}(-\beta;r) \; R[{{{\cal D}Y_{}^{}(sl(2))_{c}}}](\beta;r) \;
F^{(7)}_{12}(\beta;r)^{-1} \;.$$
### Inverse scaling procedure ${{{\cal D}Y_{s}^{}(sl(2))_{c}}}$ to ${{{\cal U}_{q,\lambda}{{(\widehat{sl(2)}_{c})}}}}$
Using the correspondence (\[eq:identif\]), the formula (\[eq:F8\]) gives a homothetical twist between ${{{\cal D}Y_{s}^{}(sl(2))_{c}}}$ and ${{{\cal U}_{q,\lambda}{{(\widehat{sl(2)}_{c})}}}}$, that is, the inverse of the scaling procedure ${{{\cal U}_{q,\lambda}{{(\widehat{sl(2)}_{c})}}}} \rightarrow {{{\cal D}Y_{s}^{}(sl(2))_{c}}}$: $$R[{{{\cal U}_{q,\lambda}{{(\widehat{sl(2)}_{c})}}}}](z=e^{-2\beta/r},q=e^{i\pi/r},w=e^{2i\pi s/r})
\;\sim\; F^{(8)}_{21}(-\beta;r,s) \;
R[{{{\cal D}Y_{s}^{}(sl(2))_{c}}}](\beta;r) \; F^{(8)}_{12}(\beta;r,s)^{-1} \;.$$
### Inverse scaling procedure ${{{\cal D}Y_{r,s}^{}(sl(2))_{c}}}$ to ${{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}$
The identification between the $R$ matrices of ${{{\cal U}_{q,\lambda}{{(\widehat{sl(2)}_{c})}}}}$ and ${{{\cal D}Y_{r,s}^{-\infty}(sl(2))_{c}}}$ through the formulae (\[eq:identif\]) allows us to get a homothetical twist operator between ${{{\cal D}Y_{r,s}^{}(sl(2))_{c}}}$ and ${{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}$, that is, the inverse of the scaling procedure ${{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}\rightarrow
{{{\cal D}Y_{r,s}^{}(sl(2))_{c}}}$. One has: $$R[{{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}] \;\sim\; F^{(9)}_{21}(z;p,w) \; R[{{{\cal D}Y_{r,s}^{}(sl(2))_{c}}}] \;
{F^{(9)}_{12}(z;p,w)}^{-1} \;.$$ The twist operator $F^{(9)}$ is given by $F^{(9)} = F^{(5)} G^{-1} =
F^{(1)} {F^{(3)}}^{-1} G^{-1}$, that is $$F^{(9)}(z;p,w) =
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & g X_{11} & g^{-1} (X_{12} - \frac{w(q-q^{-1})}{1-w} X_{11}) & 0 \\
0 & g X_{21} & g^{-1} (X_{22} - \frac{w(q-q^{-1})}{1-w} X_{21}) & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right) \;,$$ where the $X_{ij}(z)$ are given by the formulae (\[eq:Xij\]), $g$ is given by (\[eq:defg\]).
(120,60) (65,50)[(-1,0)[50]{}]{} (65,0)[(-1,0)[50]{}]{} (0,40)[(0,-1)[30]{}]{} (0,40)[(0,-1)[29]{}]{} (0,40)[(0,-1)[28]{}]{} (80,40)[(0,-1)[30]{}]{} (80,40)[(0,-1)[29]{}]{} (0,50)[(0,0)[${{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}$]{}]{} (0,0)[(0,0)[${{{\cal A}_{q,p}{{(\widehat{sl(2)}_{c})}}}}$]{}]{} (80,50)[(0,0)[${{{\cal D}Y_{}^{}(sl(2))_{c}}}$]{}]{} (80,0)[(0,0)[${{{\cal D}Y_{r}^{V8}(sl(2))_{c}}}$]{}]{} (95,50)[(1,-1)[20]{}]{} (95,50)[(1,-1)[19]{}]{} (115,20)[(-1,-1)[20]{}]{} (115,20)[(-1,-1)[19]{}]{} (120,25)[(0,0)[${{{\cal D}Y_{r}^{V6}(sl(2))_{c}}}$]{}]{} (40,54)[(0,0)[$E^{(3)}(z;p)$]{}]{} (40,-4)[(0,0)[$E^{(4)}(z;p)$]{}]{} (-10,25)[(0,0)[$E^{(1)}(z;p)$]{}]{} (70,25)[(0,0)[$E^{(2)}(\beta;r)$]{}]{} (110,10)[(0,0)[$K$]{}]{} (115,43)[(0,0)[$E^{(3)}(\beta;r)$]{}]{}
(120,130) (65,120)[(-1,0)[50]{}]{} (65,50)[(-1,0)[50]{}]{} (115,0)[(-1,0)[50]{}]{} (0,60)[(0,1)[50]{}]{} (0,60)[(0,1)[49]{}]{} (45,5)[(-1,1)[40]{}]{} (45,5)[(-1,1)[39]{}]{} (80,60)[(0,1)[50]{}]{} (80,60)[(0,1)[49]{}]{} (125,5)[(-1,1)[40]{}]{} (125,5)[(-1,1)[39]{}]{} (134,10)[(0,1)[55]{}]{} (134,10)[(0,1)[54]{}]{} (111,101)[(-1,1)[13]{}]{} (111,101)[(-1,1)[12]{}]{} (132,78)[(-1,1)[11]{}]{} (132,78)[(-1,1)[10]{}]{} (128,10)[(-1,6)[13]{}]{} (114.7,86.8)(115,88)(115.7,87)(114.7,86.8) (114.9,85.6)(115.2,86.8)(115.9,85.8)(114.9,85.6) (45,10)[(-1,3)[35]{}]{} (45,10)[(-1,3)[34.5]{}]{} (45,10)[(-1,3)[34]{}]{} (125,10)[(-1,3)[35]{}]{} (125,10)[(-1,3)[34.5]{}]{} (86,60)[(3,4)[22]{}]{} (86,60)[(3,4)[21.1]{}]{} (50,0)[(0,0)[${{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}$]{}]{} (130,0)[(0,0)[${{{\cal D}Y_{}^{}(sl(2))_{c}}}$]{}]{} (0,50)[(0,0)[${{{\cal U}_{q,\lambda}{{(\widehat{sl(2)}_{c})}}}}$]{}]{} (80,50)[(0,0)[${{{\cal D}Y_{s}^{}(sl(2))_{c}}}$]{}]{} (0,120)[(0,0)[${{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}$]{}]{} (80,120)[(0,0)[${{{\cal D}Y_{r,s}^{}(sl(2))_{c}}}$]{}]{} (125,95)[(0,0)[${{{\cal D}Y_{rs}^{-\infty}(sl(2))_{c}}}$]{}]{} (138,71)[(0,0)[${{{\cal D}Y_{r}^{}(sl(2))_{c}}}$]{}]{} (31,70)[(0,0)[$F^{(1)}$]{}]{} (106,50)[(0,0)[$F^{(2)}$]{}]{} (16,25)[(0,0)[$F^{(3)}$]{}]{} (96,25)[(0,0)[$F^{(4)}$]{}]{} (-7,85)[(0,0)[$F^{(5)}$]{}]{} (73,85)[(0,0)[$F^{(6)}$]{}]{} (90,4)[(0,0)[$F^{(7)}$]{}]{} (50,54)[(0,0)[$F^{(8)}$]{}]{} (40,124)[(0,0)[$F^{(9)}$]{}]{} (126,55)[(0,0)[$F^{(10)}$]{}]{} (135,85)[(0,0)[$F^{(3)}$]{}]{} (139,35)[(0,0)[$F^{(7)}$]{}]{} (91,75)[(0,0)[$F^{(8)}$]{}]{} (109,110)[(0,0)[$G$]{}]{}
(120,60) (65,50)[(-1,0)[50]{}]{} (115,0)[(-1,0)[50]{}]{} (45,5)[(-1,1)[40]{}]{} (45,5)[(-1,1)[39]{}]{} (45,5)[(-1,1)[38]{}]{} (125,5)[(-1,1)[40]{}]{} (125,5)[(-1,1)[39]{}]{} (125,5)[(-1,1)[38]{}]{} (50,0)[(0,0)[$\cU_q(sl(2))$]{}]{} (130,0)[(0,0)[$\cU(sl(2))$]{}]{} (0,50)[(0,0)[$\cU_{q,\lambda}(sl(2))$]{}]{} (80,50)[(0,0)[$\cU_s(sl(2))$]{}]{} (16,25)[(0,0)[$F^{(i)}$]{}]{} (96,25)[(0,0)[$F^{(ii)}$]{}]{}
Conclusion
==========
We have now constructed several $R$-matrix representations for algebraic structures, deduced from vertex or face elliptic quantum $sl(2)$ algebras by suitable limit procedures. We have shown that these structures exhibited associativity properties characterized by (dynamical) Yang–Baxter equations for their evaluated $R$-matrices. Finally, we have constructed a reciprocal set of twist-like transformations, acting on the evaluated $R$-matrices canonically as $R^{T}_{12} = T_{21} R_{12} T_{12}^{-1}$.
The next step is now to try to get explicit universal formulae for these $R$-matrices and twist operators. This in turn requires to specify the exact form under which individual generators are encapsulated in the Lax matrices, and obtain thus the full description of the associative algebras which we wish to study.
Let us immediately indicate that we need in particular to separate (as is explained in [@KLP]) the two algebraic structures contained in the single $R$-matrix formulations labelled here as (deformed) (dynamical) double Yangians ${{{\cal D}Y_{...}^{}(sl(2))_{c}}}$. Expansion of the Lax matrix in terms of integer labelled generators will lead to the (deformed) (dynamical) versions of the genuine double Yangian [@KT; @Kh; @IK; @Io]; expansion in terms of Fourier modes by a contour integral will lead to the “scaled elliptic” algebras [@Ko2; @KLP] more correctly labelled $\cA_{\hbar,\eta}{{(\widehat{sl(2)}_{c})}}$. Once this is done, we can then start to investigate the following issues
- Representations, vertex operators.
- Hopf or quasi-Hopf algebra structure, leading to:
- Universal $R$-matrices and twists.
Concerning these last two points a number of already known explicit results lead us to draw reasonable conjectures on some of the newly discovered algebraic structures in our work.
Known universal $R$-matrices and twists
---------------------------------------
Universal $R$-matrices are known for ${{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}$ [@KT2]; ${{{\cal A}_{q,p}{{(\widehat{sl(2)}_{c})}}}}$ [@JKOS] and ${{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}$ [@JKOS; @Fe]; $\cB_{q,\lambda}(sl(2))$ [@Bab]. They are also known for the double Yangian $\cD Y(sl(N))_c$ [@Kh] (proved for $N=2$, conjectured for $N\ge 3$). Universal twists have been constructed in the finite-algebra case from $\cU_q(sl(N))$ to $\cB_{q,\lambda}(sl(N))$ [@BBB; @JKOS; @ABRR]; and in the affine case from ${{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}$ to ${{{\cal A}_{q,p}{{(\widehat{sl(2)}_{c})}}}}$ [@JKOS] and ${{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}$ [@Fro1; @Fro2].
Conjectures
-----------
We therefore expect that universal $R$-matrices and twist operators may be obtained for the complete set of algebraic structures represented by Figure \[fig:twv\] in the vertex case and Figure \[fig:twf\] in the face case. The structures ${{{\cal D}Y_{...}^{}(sl(2))_{c}}}$ are here to be interpreted as genuine, integer-labelled double Yangians. The explicit construction of universal objects in this frame seems achievable, along the lines followed in [@Kh] and [@KT]. The problem of constructing universal objects associated with the continuous-labelled algebras of $\cA_{\hbar,\eta}$-type is more delicate, since one needs in particular to contrive a direct universal connection between continuous-labelled generators in $\cA_{\hbar,\eta}$ and discrete-labelled generators in $\cA_{qp}$, or between $\cA_{\hbar,0}$ and ${{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}$.
The case of unitary matrices
----------------------------
We have described in Section \[sect:homothetical\] homothetical twist-like connections between ${\mbox{1\hspace{-1mm}I}}_{4\times 4}$, interpreted as the evaluated $R$-matrix ${\mbox{1\hspace{-1mm}I}}$ for the centrally extended algebra ${{{\cal U}_{}{{(\widehat{sl(2)}_{c})}}}}$, and unitary $R$-matrices realizing a $RLL$-structure “proportional” to ${{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}$. Interpretation of this $RLL$-structure, and its derived relations at elliptic level, remains obscure. The canonical construction of universal $R$-matrices for ${{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}$ [@KT2] and their subsequent evaluation [@IIJMNT] leaves open the possibility of an alternative construction of universal $\rightarrow$ evaluated $R$-matrix which lead to unitary (and crossing-symmetrical) $R$-matrices; it may arise either by dropping the triangularity requirement $\cR\in
\cB_+\otimes\cB_-\subset {{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}\otimes{{{\cal U}_{q}{{(\widehat{sl(2)}_{c})}}}}$, or by relaxing analyticity constraints on the evaluated $R$-matrix.\
Homothetical TLAs also appear between double Yangian-like structures and their antecedent structures through the scaling procedure. The same possibilities hold for the differently normalized $R$-matrix structures obtained by applications of these homothetical TLAs.
The notion of dynamical elliptic algebra
----------------------------------------
Finally let us briefly comment on the notion of “dynamical” algebraic structure. This notion was applied throughout this paper to algebras incorporating an extra parameter $\lambda$ belonging to the Cartan algebra, subsequently shifted along a general Cartan algebra direction. This shift is therefore retained in the Yang-Baxter equation for evaluated $R$-matrices of face type (but *not* of vertex type, for which the extra parameter is simply a $c$-number and the shift takes place along the central charge direction, set to zero in the evaluation representation [^2]). A particular illustration of this fact arises in the case of classical and quantum $R$-matrix for Calogero–Moser models [@ABB] where $\lambda$ is identified with the momentum of the Calogero–Moser particles, hence the denomination “dynamical” for the $R$-matrices. In the algebraic structures described here however, $\lambda$ is not yet promoted to the r[ô]{}le of generator, hence this denomination is slightly abusive. There exists however at least one example of algebraic structure, $\cU_{q,p}{{(\widehat{sl(2)}_{c})}}$ [@Ko1; @JKOS2], where $\lambda$ and its conjugate $\displaystyle\frac{\partial}{\partial\lambda}$ are “added” to the algebra ${{{\cal B}_{q,p,\lambda}{{(\widehat{sl(2)}_{c})}}}}$; however $\cU_{q,p}{{(\widehat{sl(2)}_{c})}}$ is not a Hopf, even quasi-Hopf, algebra. We expect therefore that similar genuinely dynamical algebraic structures may be associated in the same way to all “dynamical” algebras described here, and may play important r[ô]{}le in solving the models where such algebras arise.
**Acknowledgements**
This work was supported in part by CNRS and EC network contract number FMRX-CT96-0012. M.R. was supported by an EPSRC research grant no. GR/K 79437. We wish to thank O. Babelon for enlightening discussions, S.M. Khoroshkin for fruitful explanations and P. Sorba for his helpful and clarifying comments. J.A. wishes to thank the LAPTH for its kind hospitality.
[99]{}
M. Jimbo, R. Kedem, H. Konno, T. Miwa and R. Weston, *Difference equations in spin chains with a boundary,* Nucl. Phys. **B448**, 429 (1995) and `hep-th/9502060`.
G. Andrews, R. Baxter, P.J. Forrester, *Eight-vertex SOS model and generalized Rogers-Ramanujan type identities,* J. Stat. Phys. **35** (1984) 193.
H. Konno, *An elliptic algebra $U_{q,p}(\widehat{sl}(2))$ and the fusion RSOS model,* Commun. Math. Phys. **195** (1998) 373.
H. Konno, *Degeneration of the elliptic algebra ${\cal A}_{q,p}(\widehat{sl}(2))$ and form factors in the Sine-Gordon theory,* Proceedings of the Nankai-CRM joint meeting on “Extended and Quantum Algebras and their Applications to Physics”, Tianjin, China, 1996, to appear in the CRM series in mathematical physics, Springer Verlag, and `hep-th/9701034`.
S. Khoroshkin, A. LeClair, S. Pakuliak, *Angular quantization of the Sine–Gordon model at the free fermion point,* `hep-th/9904082`.
R.J. Baxter, *Partition function of the eight-vertex lattice model,* Ann. Phys. **70** (1972) 193.
A.A. Belavin, *Dynamical symmetry of integrable quantum systems,* Nucl. Phys. B **180** (1981) 189.
E. Date, M. Jimbo, T. Miwa, M. Okado, *Fusion of the eight-vertex SOS model,* Lett. Math. Phys. **12** (1986) 209.
M. Jimbo, T. Miwa, M. Okado, Lett. Math. Phys. **14** (1987) 123, and *Solvable lattice models related to the vector representation of classical simple Lie algebras,* Commun. Math. Phys. **116** (1988) 507.
E. K. Sklyanin, *Some algebraic structures connected with the Yang–Baxter equation,* Funct. Anal. Appl. **16** (1982) 263 and *Some algebraic structures connected with the Yang–Baxter equation. Representations of quantum algebras,* Funct. Anal. Appl. **17** (1983) 273.
O. Foda, K. Iohara, M. Jimbo, R. Kedem, T. Miwa, H. Yan, *An elliptic quantum algebra for $\widehat{sl}_2$,* Lett. Math. Phys. **32** (1994) 259 and `hep-th/9403094`.
M. Jimbo, H. Konno, S. Odake, J. Shiraishi, *Quasi-Hopf twistors for elliptic quantum groups,* to appear in “Transformation Groups” and `q-alg/9712029`.
H. Awata, H. Kubo, S. Odake, J. Shiraishi, *Quantum ${\cal W}_N$ algebras and Macdonald polynomials,* Commun. Math. Phys. **179** (1996) 401 and `q-alg/9508011` and J. Shiraishi, H. Kubo, H. Awata, S. Odake, *A Quantum Deformation of the Virasoro Algebra and the Macdonald Symmetric Functions,* Lett. Math. Phys. **38** (1996) 33 and `q-alg/9507034`.
E. Frenkel, N. Reshetikhin, *Quantum affine algebras and deformations of the Virasoro and $W$-algebras,* Commun. Math. Phys. **178** (1996) 237 and `q-alg/9505025`.
B. Feigin, E. Frenkel, *Quantum $W$ algebras and elliptic algebras,* Commun. Math. Phys. **178** (1996) 653 and `q-alg/9508009`.
J. Avan, L. Frappat, M. Rossi, P. Sorba, *Deformed ${\cal W}_{N}$ algebras from elliptic $sl(N)$ algebras,* Commun. Math. Phys. **199** (1999) 697 and `math.QA/9801105`.
J. Avan, L. Frappat, M. Rossi, P. Sorba, *Universal construction of $q$-deformed ${\cal W}$ algebras,* Commun. Math. Phys. **202** (1999) 445 and `math.QA/9807048`.
G. Felder, *Elliptic quantum groups,* Proc. ICMP Paris 1994, pp 211. See also J.L. Gervais, A. Neveu, *Novel triangle relation and absence of tachyons in Liouville string field theory,* Nucl. Phys. B **238** (1984) 125.
B. Enriquez, G. Felder, *Elliptic quantum groups $E_{\tau,\eta}(sl_2)$ and quasi-Hopf algebras,* `q-alg/9703018`.
M. Jimbo, H. Konno, S. Odake, J. Shiraishi, *Elliptic algebra $U_{q,p}(\widehat{sl}_2)$: Drinfeld currents and vertex operators,* to appear in Commun. Math. Phys, `math.QA/9802002`.
J. Avan, O. Babelon, E. Billey, *The Gervais–Neveu–Felder equation and quantum Calogero–Moser systems,* Commun. Math. Phys. **178** (1996) 281.
O. Babelon, D. Bernard, E. Billey, *A quasi-Hopf algebra interpretation of quantum $3-j$ and $6-j$ symbols and difference equations,* Phys. Lett. **B375** (1996) 89.
B. Jurčo, P. Schupp, *AKS scheme for face and Calogero-Moser-Sutherland type models,* `solv-int/9710006`.
B.Y. Hou, W.L. Yang, *Dynamically twisted algebra ${\cal
A}_{q,p;\pi}(\widehat{gl_{2}})$ as current algebra generalizing screening currents of $q$ deformed Virasoro algebra,* `q-alg/9709024`.
N.Yu. Reshetikhin, M.A. Semenov-Tian-Shansky, *Central extensions of quantum current groups,* Lett. Math. Phys. **19** (1990) 133.
J. Ding, I.B. Frenkel, *Isomorphism of two realizations of quantum affine algebra $U_{q}(\widehat{gl}(n))$,* Commun. Math. Phys. **156** (1993) 277.
M. Jimbo, H. Konno, T. Miwa, *Massless XXZ model and degeneration of the elliptic algebra ${\cal A}_{q,p}(\widehat{sl}(2))$,* in Ascona 1996, “Deformation theory and symplectic geometry”, 117-138 and `hep-th/9610079`.
S. Khoroshkin, D. Lebedev, S. Pakuliak, *Elliptic algebra ${\cal
A}_{q,p}(\hat{sl_2})$ in the scaling limit,* Commun. Math. Phys. **190** (1998) 597 and `q-alg/9702002`.
S.M. Khoroshkin, V.N. Tolstoy, *Yangian double and rational $R$ matrix,* Lett. Math. Phys. (1995) and `hep-th/9406194`.
S.M. Khoroshkin, *Central extension of the Yangian double,* Collection SMF, 7[è]{}me rencontre du contact franco-belge en alg[è]{}bre, Reims 1995, `q-alg/9602031`.
K. Iohara, M. Kohno, *A central extension of ${\cal
D}Y_{\hbar}(\widehat{gl}(2))$ and its vertex representations,* Lett. Math. Phys. **37** (1996) 319 and `q-alg/9603032`.
K. Iohara, *Bosonic representations of Yangian double ${\cal
D}Y_{\hbar} (\widehat{gl})$ with $\widehat{gl}=\widehat{gl}_N,\widehat{sl}_N$,* J. Phys. A (Math. Gen.) **29** (1996) 4593 and `q-alg/9603033`.
V.G. Drinfel’d, *Quasi-Hopf algebras,* Leningrad Math. Journ. **1** (1990) 1419.
C. Fr[ø]{}nsdal, *Quasi Hopf Deformations of Quantum Groups,* [Lett. Math. Phys. **40** (1997) 117,]{} `q-alg/9611028.`
C. Fr[ø]{}nsdal, *Generalization and exact deformations of quantum groups,* Publication RIMS Kyoto University **33** (1997) 91.
D. Arnaudon, E. Buffenoir, E. Ragoucy, Ph. Roche, *Universal solutions of quantum dynamical Yang-Baxter equations,* Lett. Math. Phys. **44** (1998) 201–214, and `q-alg/9712037`.
E. Buffenoir, Ph. Roche, *Harmonic analysis on the quantum Lorentz group,* Commun. Math. Phys. 207 (1999) 499-555, and `q-alg/9710022`.
L. D. Faddeev, N.Yu. Reshetikhin and L.A. Takhtajan, *Quantization of Lie groups and Lie algebras*, Leningrad Math. J. **1** (1990) 193.
V.G. Drinfeld, *Quantum Groups,* Proc. Int. Congress of Mathematicians, Berkeley, California, Vol. **1**, Academic Press, New York (1986), 798.
M. Idzumi, K. Iohara, M. Jimbo, T. Miwa, T. Nakashima, T. Tokihiro, *Quantum affine symmetry in vertex models,* Int. J. Mod. Phys. **A8** (1993) 1479, and `hep-th/9208066`.
O. Babelon, *Universal Exchange Algebra for Bloch Waves and Liouville Theory,* Comm. Math. Phys. **139** (1991) 619.
D. Arnaudon, J. Avan, L. Frappat, M. Rossi, *Deformed double [Y]{}angian structures,* `math.QA/9905100`, to appear in Rev. Math. Phys.
E.W. Barnes, *The theory of the double gamma function,* Philos. Trans. Roy. Soc. **A196** (1901) 265-388.
M. Jimbo, T. Miwa, *Quantum KZ equation with $\vert q \vert = 1$ and correlation functions of the XXZ model in the gapless regime,* J. Phys. A (Math. Gen.) **29** (1996) 2923 and `hep-th/9601135`.
S. Khoroshkin and V. Tolstoy, *Uniqueness theorem for universal $R$-matrix,* Lett. Math. Phys. **24** (1992), 231.
[^1]: Cladistics : a system of biological taxonomy that defines taxa uniquely by shared characteristics not found in ancestral groups and uses inferred evolutionary relationships to arrange taxa in a branching hierarchy such that all members of a given taxon have the same ancestors.
[^2]: This fact was clarified to us by O. Babelon.
|
---
author:
- Marco Chianese
- 'and Stephen F. King'
bibliography:
- 'DarkMatter.bib'
title: 'The Dark Side of the Littlest Seesaw: freeze-in, the two right-handed neutrino portal and leptogenesis-friendly fimpzillas'
---
Introduction
============
Neutrino oscillation experiments have provided the first evidence for new particle physics beyond the Standard Model (BSM) in the form of neutrino mass and mixing [@nobel]. Although the origin of neutrino mass and mixing remains unknown [@XZbook; @King:2013eh], there has been continuing experimental progress, for example on atmospheric mixing which is consistent with being maximal [@Abe:2017uxa; @Adamson:2017qqn; @NOvA_new; @Esteban:2016qun; @deSalas:2017kay; @Capozzi:2018ubv].
The leading candidate for a theoretical explanation of neutrino mass and mixing remains the seesaw mechanism [@Minkowski:1977sc; @Yanagida:1979as; @GellMann:1980vs; @Schechter:1980gr; @Mohapatra:1979ia; @Mohapatra:1980yp]. However the seesaw mechanism involves a large number of free parameters at high energy, and is therefore difficult to test. One approach to reducing the number of seesaw parameters is to consider the minimal version involving only two right-handed neutrinos (2RHN) [@King:1999mb; @King:2002nf]. In such a scheme the lightest neutrino is massless. An early simplification [@Frampton:2002qc] involved two texture zeros in the Dirac neutrino mass matrix consistent with cosmological leptogenesis [@Fukugita:1986hr; @Guo:2003cc; @Ibarra:2003up; @Mei:2003gn; @Guo:2006qa; @Antusch:2011nz; @Harigaya:2012bw; @Zhang:2015tea]. Although the normal hierarchy of neutrino masses, favoured by current data, is incompatible with the 2RHN model with two texture zeros [@Harigaya:2012bw; @Zhang:2015tea], the one texture zero case originally proposed [@King:1999mb; @King:2002nf] remains viable.
The Littlest Seesaw (LS) model is based on the 2RHN model with one texture zero, and in addition involves a well defined and constrained Yukawa involving just two independent Yukawa couplings [@King:2013iva; @Bjorkeroth:2014vha; @King:2015dvf; @Bjorkeroth:2015ora; @Bjorkeroth:2015tsa; @King:2016yvg; @Ballett:2016yod], leading to a highly predictive scheme. The LS model also provides a rather minimal explanation of cosmological leptogenesis, providing the lightest right-handed neutrino has a mass of order $10^{10}-10^{11}$ GeV [@King:2018fqh]. However, to date there has been no attempt in the literature to address the origin of Dark Matter (DM) in the LS model.
The existence of DM in the Universe provides cosmological evidence for new physics beyond the Standard Model. Although there are many possible candidates for DM particles, it is interesting to try to connect the new physics required for DM particles to that required for neutrino mass and mixing, and there are many works in this direction [@Caldwell:1993kn; @Mohapatra:2002ug; @Krauss:2002px; @Ma:2006km; @Asaka:2005an; @Boehm:2006mi; @Kubo:2006yx; @Ma:2006fn; @Hambye:2006zn; @Lattanzi:2007ux; @Ma:2007gq; @Allahverdi:2007wt; @Gu:2007ug; @Sahu:2008aw; @Arina:2008bb; @Aoki:2008av; @Ma:2008cu; @Gu:2008yj; @Aoki:2009vf; @Gu:2010yf; @Hirsch:2010ru; @Esteves:2010sh; @Kanemura:2011vm; @Lindner:2011it; @JosseMichaux:2011ba; @Schmidt:2012yg; @Borah:2012qr; @Farzan:2012sa; @Chao:2012mx; @Gustafsson:2012vj; @Blennow:2013pya; @Law:2013saa; @Hernandez:2013dta; @Restrepo:2013aga; @Chakraborty:2013gea; @Ahriche:2014cda; @Kanemura:2014rpa; @Huang:2014bva; @Varzielas:2015joa; @Sanchez-Vega:2015qva; @Fraser:2015mhb; @Adhikari:2015woo; @Ahriche:2016rgf; @Sierra:2016qfa; @Lu:2016ucn; @Batell:2016zod; @Ho:2016aye; @Escudero:2016ksa; @Bonilla:2016diq; @Borah:2016zbd; @Biswas:2016yan; @Hierro:2016nwm; @Bhattacharya:2016qsg; @Chakraborty:2017dfg; @Bhattacharya:2017sml; @Ho:2017fte; @Ghosh:2017fmr; @Nanda:2017bmi; @Narendra:2017uxl; @Bernal:2017xat; @Borah:2018gjk; @Batell:2017cmf; @Pospelov:2007mp; @Falkowski:2009yz; @Falkowski:2011xh; @Cherry:2014xra; @Bertoni:2014mva; @Allahverdi:2016fvl; @Karam:2015jta]. For example, a recent work considers the type-I seesaw mechanism together with a fermion singlet dark matter particle, stabilised by a discrete $Z_2$ symmetry arising from a broken $Z_4$ [@Bhattacharya:2018ljs]. The construction in Ref. [@Bhattacharya:2018ljs] suggests that there exists a scalar field mediator between the two sectors whose vacuum expectation value not only generates the mass of the dark matter, but also takes part in the neutrino mass generation. This example is representative of many such attempts to connect the seesaw mechanism to DM, namely by invoking a Higgs portal type coupling to constrain the seesaw scale.
In this paper we propose a minimal and realistic model to simultaneously account for the neutrino masses through a type-I seesaw and provide a viable dark matter candidate which exists in a dark sector consisting of a single dark fermion $\chi$ and a single dark complex scalar $\phi$, both having an odd dark parity $Z_2$. Unlike many works in the literature, we focus on the right-handed neutrino portal (rather than the Higgs portal [@Pospelov:2007mp; @Burgess:2000yq; @Davoudiasl:2004be; @Bird:2006jd; @Kim:2006af; @Finkbeiner:2007kk; @DEramo:2007anh; @Barger:2007im; @SungCheon:2007nw; @MarchRussell:2008yu; @McDonald:2008up; @Piazza:2010ye; @Pospelov:2011yp; @Batell:2012mj; @Kouvaris:2014uoa; @Kainulainen:2015sva; @Krnjaic:2015mbs; @Tenkanen:2016jic; @Heikinheimo:2016yds; @Heikinheimo:2017ofk]) and on the implications of this for neutrino physics and dark matter phenomenology. In other words we suppose that the production of dark sector particles is achieved dominantly via their couplings to the right-handed neutrinos $N_R$, which are in turn coupled to the thermal bath via their neutrino Yukawa couplings to left-handed neutrinos and Higgs scalar. Thus the production of DM particles depends crucially on the same neutrino Yukawa couplings which determine neutrino mass and mixing. Another unique feature of our approach here is that we constrain the parameter space by the requirement of achieving a realistic pattern of neutrino masses and mixing, while many works in the literature only consider a toy model with a single neutrino mass for example. In order to achieve this, we focus on the LS model above with two right-handed neutrinos $N_R$, which successfully describes the neutrino data with a very small number of parameters.
The requirement that the right-handed neutrino portal plays a dominant role in the production mechanism of dark sector particles, then provides a highly constrained and direct link between dark matter and neutrino mass and mixing. In order to achieve the correct relic density in this scenario we need to assume that the coupling of $N_R$ to the dark sector is very small, which puts us in the so called “freeze-in” scenario [@Hall:2009bx] which is not usually considered in the framework of the right-handed neutrino portal (see Ref. [@Bernal:2017kxu] for a recent detailed analysis on the freeze-in production mechanism). However in such a scenario, it becomes possible to obtain a successful relic density for almost arbitrarily large right-handed neutrino and dark sector masses. In particular, it is possible to achieve right-handed neutrino masses in the range $10^{10}-10^{11}$ GeV required for leptogenesis. This means that the LS model can simultaneously achieve both leptogenesis and (when extended by a dark sector as here) a successful DM relic density. In such a case, the DM particles are ultra-heavy frozen-in particles, which we refer to as “fimpzillas”.
The layout of the remainder of the paper is as follows. In Section 2 we discuss the Lagrangian of the model describing in detail the features of the LS model. In Section 3 we report the Boltzmann equations required to determine the DM relic abundance according to the freeze-in production mechanism. In Section 4 we discuss our numerical results, highlighting the regions of the parameter space that provide a link between DM and neutrino mass and mixing. Finally, in Section 5 we draw our conclusions.
The model
=========
We consider an extension of the Littlest Seesaw model where the two right-handed neutrinos $N_{Ri}$ are coupled to a dark sector consisting of a scalar $\phi$ and a fermion $\chi$. For the sake of simplicity, we assume that the former is a complex field while the latter has a vector-like mass. The lighter particle between $\phi$ and $\chi$ plays the role of dark matter. The Standard Model (SM) and the Dark Sector (DS) are distinguished by a $Z_2$ symmetry that also stabilises the DM particles. In particular, the DS particles, $\phi$ and $\chi$, are odd under $Z_2$, while the other fields are even. In Tab. \[tab:matter\] we report the matter content of the model providing how the new fields transform under the electroweak gauge group $SU(2)_L \otimes U(1)_Y$ and the discrete symmetry $Z_2$.
$N_R$ $\phi$ $\chi$
----------- ----------- ----------- -----------
$SU(2)_L$ [**1**]{} [**1**]{} [**1**]{}
$U(1)_Y$ 0 0 0
$Z_2$ + - -
: \[tab:matter\]New fields representations of the model, where $N_R$ are the two right-handed neutrinos, while $\phi$ and $\chi$ are a new DS complex scalar and fermion, respectively.
Hence, the full Lagrangian of the model can be divided in four parts $$\mathcal{L} = \mathcal{L}_{\rm SM} + \mathcal{L}_{\rm Seesaw} + \mathcal{L}_{\rm DS} + \mathcal{L}_{\rm portal}\,,
\label{eq:lag}$$ where the first term is the SM Lagrangian, the Seesaw term is responsible for neutrino masses, the DS part contains all the kinetic and mass terms of the dark particles $\phi$ and $\chi$, while the last term consists of the interactions that connect the visible and the dark sectors. In particular, the last three terms read $$\begin{aligned}
\mathcal{L}_{\rm Seesaw} & = & - Y_{\alpha\beta} \overline{L_L}_\alpha \tilde{H} N_{R\beta} - \frac12 M_{R}\overline{N^c_{R}} N_{R} +h.c.\,, \label{eq:lagNS} \\
\mathcal{L}_{\rm DS} & = & \overline{\chi}\left(i \slashed{\partial} - m_\chi \right)\chi + \left|\partial_\mu \phi\right| - m^2_\phi \left|\phi\right|^2 + V\left(\phi\right)\,, \label{eq:lagDS} \\
\mathcal{L}_{\rm portal} & = & y_{\rm DS} \phi \, \overline{\chi}N_{R} + h.c \,, \label{eq:lagPortal}\end{aligned}$$ where $L_{L\alpha}$ are the left-handed lepton doublets ($\alpha=e,\mu\tau$ and $\beta=1,2$) and $$H = \left(\begin{array}{c}G^+ \\ \frac{v_{\rm SM} + h^0 + i G^0}{\sqrt2}\end{array}\right)$$ is the SM Higgs doublet with $\tilde{H} = i \tau_2 H^*$. In Eq. , the quantities $m_\chi$ and $m_\phi$ are the masses of the fermion and the scalar, respectively, and $V\left(\phi\right)$ is a general potential for the scalar field allowed by the $Z_2$ symmetry. We assume that the discrete $Z_2$ symmetry is an exact symmetry of the model so that the scalar field does not acquire a v.e.v. (a detailed analysis of the scalar potential is beyond the scope of this paper). Finally, the visible and dark sectors are connected through the right-handed neutrino portal defined in Eq. , where for the sake of simplicity we have assumed the same real coupling $y_{\rm DS}$ between the two right-handed neutrinos and the DS particles. It is worth noticing that in this framework the Higgs portal defined by the coupling $y_{H\phi} \left|H\right|^2\left|\phi\right|^2$ is also allowed. However, the aim of the present analysis is to investigate the impact of the right-handed neutrino portal on the DM phenomenology and to highlight the interesting connection between neutrinos and DM particles. Hence, for this reason we consider the coupling $y_{H\phi}$ to be negligible.
Let us now discuss in detail the Littlest Seesaw model defined by the Lagrangian given in Eq. . The first term is a Yukawa-like coupling while the second one is the Majorana mass term for the right-handed neutrinos. After the electroweak symmetry breaking, the light effective left-handed Majorana neutrino mass matrix is obtained by type-I seesaw formula [@Minkowski:1977sc; @Yanagida:1979as; @GellMann:1980vs; @Schechter:1980gr; @Mohapatra:1979ia; @Mohapatra:1980yp] $$m_\nu = -m_D M^{-1}_R {m_D}^T \,,
\label{eq:ssformula}$$ where the neutrino Dirac mass matrix $m_D$ is defined as $$m_D = \frac{v_{\rm SM}}{\sqrt 2} Y\,,
\label{mD1}$$ being $v_{\rm SM}=246$ GeV the SM Higgs v.e.v. In the LS model [@King:2013iva; @Bjorkeroth:2014vha; @King:2015dvf; @Bjorkeroth:2015ora; @Bjorkeroth:2015tsa; @King:2016yvg; @Ballett:2016yod], in the basis where the right-handed neutrino Majorana mass matrix is diagonal, $$M_R = \left(\begin{array}{cc} M_{\rm R1} & 0 \\ 0 & M_{\rm R2} \end{array}\right)\,,$$ the neutrino Dirac mass matrix has the form in the LR convention $$m_D = \left(\begin{array}{lr} 0 & b \, e^{i \frac{\eta}{2}} \\ a & 3b \, e^{i \frac{\eta}{2}} \\ a & b \, e^{i \frac{\eta}{2}} \end{array}\right) \,,
\label{mD2}$$ while the neutrino Majorana mass matrix for the light neutrinos $\nu_L$ is given effectively in the seesaw approximation by $$m_\nu = m_a \left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 1 & 1 \end{array}\right) + m_b \, e^{i \eta} \left(\begin{array}{ccc} 1 & 3 & 1 \\ 3 & 9 & 3 \\ 1 & 3 & 1 \end{array}\right)\,.
\label{eq:numass}$$ In the above expressions, $a$ and $b$ are two real couplings while $\eta$ is their relative phase, and $$m_a = \frac{a^2}{M_{R1}} \qquad {\rm and} \qquad m_b = \frac{b^2}{M_{R2}} \,.
\label{mD3}$$ Hence, in the LS model the Yukawa matrix in Eq. reads $$Y = \sqrt{\frac{2 \,m_a\, M_{\rm R1}}{v^2_{SM}}} \left(\begin{array}{cc} 0 & 0 \\ 1 & 0 \\ 1 & 0\end{array}\right) + \sqrt{\frac{2 \,m_b\, M_{\rm R2}}{v^2_{SM}}} e^{i \frac{\eta}{2}} \left(\begin{array}{cc} 0 & 1 \\ 0 & 3 \\ 0 & 1\end{array}\right)\,.
\label{eq:yuk}$$ The result in Eq. follows from Eqs. , and . From a model building perspective, it may be achieved by the effective Yukawa coupling of the first right-handed neutrino having an alignment proportional to $\left(0,1,1\right)$, while that of the second right-handed neutrino having an alignment proportional to $\left(1,3,1\right)$. The theoretical origin of such alignments may be related to a spontaneously broken family symmetry under which the three families of electroweak lepton doublets transform as an irreducible triplet, as discussed further in Refs. [@King:2013iva; @Bjorkeroth:2014vha; @King:2015dvf; @Bjorkeroth:2015ora; @Bjorkeroth:2015tsa; @King:2016yvg; @Ballett:2016yod]. The phenomenological desirability of such a structure, from the point of view of both low energy neutrino data and leptogenesis, was recently discussed in Ref. [@King:2018fqh].
This minimal framework allows for a very good fit of neutrino mixing and mass parameters. In particular, according to Ref. [@Ballett:2016yod] we consider the following benchmark values for the three parameters that provide a nice agreement with the experimental neutrino measurements: $$m_a = 26.74~{\rm \,meV}\,, \qquad m_b = 2.682~{\rm \,meV}\,, \qquad {\rm and} \qquad \eta = \frac23\pi\,.
\label{eq:nu-data}$$ Hence in the model, there are five free parameters: the two right-handed neutrino masses, the two masses $m_\chi$ and $m_\phi$, and the right-handed neutrino portal coupling $y_{\rm DS}$. However, for the sake of simplicity, we consider the case where the two right-handed neutrinos have the same mass $M_{R1} = M_{R2} = M_{R}$, where $M_{R}$ is then identified with the energy scale of the seesaw. The other quantities are then fixed by the experimental neutrino measurements according to Eq. . In the following, we focus on the case $m_\chi \leq m_\phi$ implying that the fermions $\chi$ are stable and play the role of DM particles. The other case is a trivial modification and, therefore, is not discussed here. In the next Section, we report the Boltzmann equations important for the DM production in the early Universe.
The Boltzmann equations in freeze-in dark matter production
===========================================================
The Boltzmann equations encode how the yield $Y_i$ of particles of species $i$ evolves with the temperature $T$. The most general expression takes the form [@Kolb:1990vq][^1] $$\begin{aligned}
\mathcal{H}\,T\left(1+\frac{T}{3 g^\mathfrak{s}_*\left(T\right)}\frac{d g^\mathfrak{s}_*}{d T}\right)^{-1}\frac{d Y_i}{d T} &=& \sum_{kl} \left<\Gamma_{i\rightarrow kl}\right>Y_i^{\rm eq}\left(\frac{Y_i}{Y_i^{\rm eq}}-\frac{Y_k \, Y_l}{Y_k^{\rm eq}Y_l^{\rm eq}}\right) \nonumber \\
&& - \sum_{jk} \left<\Gamma_{j\rightarrow ik}\right>Y_j^{\rm eq}\left(\frac{Y_j}{Y_j^{\rm eq}}-\frac{Y_i \, Y_k}{Y_i^{\rm eq}Y_k^{\rm eq}}\right) \\
&&+ \mathfrak{s} \sum_{jkl} \left<\sigma_{ij\rightarrow kl}\, v_{ij}\right> Y_i^{\rm eq}Y_j^{\rm eq}\left(\frac{Y_i \, Y_j}{Y_i^{\rm eq}Y_j^{\rm eq}}-\frac{Y_k \, Y_l}{Y_k^{\rm eq}Y_l^{\rm eq}}\right) \,, \nonumber
\label{eq:Boltz}\end{aligned}$$ where $\mathcal{H}$ and $\mathfrak{s}$ are, respectively, the Hubble parameter and the entropy density of the thermal bath[^2] $$\mathcal{H} = 1.66 \sqrt{g_*\left(T\right)}\frac{T^2}{M_{\rm Planck}} \qquad{\rm and}\qquad \mathfrak{s} = \frac{2\pi^2}{45} g^\mathfrak{s}_*\left(T\right) T^3 \,,
\label{eq:hubble}$$ where $M_{\rm Planck}=1.22\times10^{19}$ GeV is the Planck mass, and $g_*$ and $g^\mathfrak{s}_*$ are the degrees of freedom of the relativistic species in the thermal bath. In the Boltzmann equation , the quantity $Y^{\rm eq}_i$ is the yield at the thermal equilibrium that takes the expression $$Y^{\rm eq}_{i} \equiv \frac{n^{\rm eq}_{i}}{\mathfrak{s}}\qquad{\rm with}\qquad n^{\rm eq}_{i} = \frac{g_i \, m_i^2 \, T}{2\pi^2} K_2\left(\frac{m_i}{T}\right)\,,
\label{eq:yeq}$$ where $m_i$ and $g_i$ are respectively the mass and the internal degrees of freedom of particles $i$, and $K_2$ denotes the modified Bessel function of the second kind of order 2. The Boltzmann equation takes into account all the decay processes $i\rightarrow kl$ and $j\rightarrow ik$ (first and second terms) and all the scattering ones $ij\rightarrow kl$ (third term). In particular, the thermally averaged decay width is given by $$\left< \Gamma_{i\rightarrow kl} \right> = \frac{K_1\left(m_i/T\right)}{K_2\left(m_i/T\right)}\Gamma_{i\rightarrow kl}\,,$$ with $K_1$ being the modified Bessel function of the second kind of order 1, while the thermally averaged cross section takes the form [@Edsjo:1997bg] $$\left<\sigma_{ij\rightarrow kl}\, v_{ij}\right> = \frac{1}{n^{\rm eq}_{i}\, n^{\rm eq}_{j}}\frac{g_i \, g_j}{S_{kl}}\frac{T}{512\pi^6} \int_{\left(m_i+m_j\right)^2}^\infty ds \, \frac{p_{ij} \, p_{kl} \,K_1\left(\sqrt{s}/T\right)}{\sqrt{s}} \int \overline{\left|\mathcal{M}\right|^2}_{ij\rightarrow kl} \, d\Omega\,.$$ where $s$ is the Mandelstam variable equal to the square of the centre-of-mass energy, $S_{kl}$ is a symmetry factor, $p_{ij}$ ($p_{kl}$) is the initial (final) centre-of-mass momentum and $\overline{\left|\mathcal{M}\right|^2}$ is the averaged squared amplitude evaluated in the centre-of-mass frame.
In order to compute the DM relic abundance, we need in principle to solve four coupled Boltzmann equations: two equations for the two right-handed neutrinos $n_1$ and $n_2$, one for the dark scalars $\phi$ and one for the DM particles $\chi$. However, in all the cases analysed, we consider the masses of the four new particles larger than the electroweak scale. In this limit, the neutrino Yukawa couplings given in Eq. are large enough to bring the two right-handed neutrino in thermal equilibrium with the thermal bath. Hence, their yield follows the thermal distribution defined in Eq. . This allows us to solve just the two coupled Boltzmann equations for the particles in the dark sector. Furthermore, we also have $g^\mathfrak{s}_* = g^\mathfrak{s}_{*n} + g^\mathfrak{s}_{*\rm SM}$ for $T > M_R$ and $g^\mathfrak{s}_* = g^\mathfrak{s}_{*\rm SM}$ for $T < M_R$, with $g^\mathfrak{s}_{*n} = 2\times2\times7/8$ and $g^\mathfrak{s}_{*\rm SM} = 106.75$ that corresponds to the total number of relativistic degrees of freedom in the SM at temperature above the electroweak scale. In both limits we have $g_* = g^\mathfrak{s}_*$.
In the present analysis, we consider the right-handed neutrino portal coupling $y_{\rm DS}$ to be very small. Only in this case, the neutrino sector defined by the Yukawa couplings in Eq. could play an important role in DM production. Indeed, due to the smallness of the neutrino Yukawa couplings fixed by the seesaw formula , the scatterings involving active neutrinos and leptons could dominate over the other processes in the DM production only when the right-handed neutrino portal is suppressed by a very small coupling as well. When this occurs, we would have an interesting relation between neutrino physics and dark matter. In the limit $y_{\rm DS} \ll 1$, DM particles are produced through the so-called [*freeze-in*]{} production mechanism [@Hall:2009bx]. Differently from the standard [*freeze-out*]{} production mechanism [@Lee:1977ua; @Bernstein:1985th; @Gondolo:1990dk], the DM particles are never in thermal equilibrium and are instead gradually produced from the thermal bath through very weak interactions, while the rate of the back-reactions that would destroy DM particles are further suppressed since $Y_{\chi,\phi} \ll Y_{\chi,\phi}^{\rm eq}$ at high temperatures. On the other hand, in the freeze-out regime the coupling $y_{\rm DS}$ has to be assumed large enough to bring the dark sector in thermal equilibrium, spoiling the neutrino-dark matter relation.
Moreover, in the freeze-in scenario, since the dark sector is almost decoupled with the SM one ($y_{\rm DS} \ll 1$), all the constraints coming from collider, direct and indirect DM searches are practically circumvented. As will be discussed later, in the range of masses considered the correct DM abundance is achieved for a right-handed neutrino portal coupling smaller than $10^{-4}$. Consequently, since the DM annihilation processes important for indirect DM searches are in general suppressed by $y_{\rm DS}^4$, the corresponding indirect signals are very small. On the other hand, due to the absence of a direct coupling to quarks, the processes important for collider and direct DM searches are even more suppressed.
Once the two Boltzmann equations are solved, the total DM relic abundance is then given by $$\Omega_{\rm DM}h^2 = \frac{\rho_{\rm DM,0}}{\rho_{\rm crit}/h^2}\,,
\label{eq:omegaPRE}$$ where $\rho_{\rm DM,0}$ is today’s energy density of DM particles and $\rho_{\rm crit}/h^2 = 1.054 \times10^{-5}~{\rm GeV \, cm^{-3}}$ is the critical density [@Patrignani:2016xqp]. In oder to provide a realistic model for viable DM candidates, this quantity has to be equal to its experimental value that has been measured by Planck Collaboration at 68% C.L. [@Ade:2015xua]: $$\left.\Omega_{\rm DM}h^2\right|_{\rm obs} = 0.1188 \pm 0.0010\,.
\label{eq:omegaOBS}$$
In this framework, by considering the simple case of degenerate right-handed neutrinos ($M_{R1} = M_{R2} = M_R$) one may classify two different types of orderings among the masses of new particles:
- [**Ordering type A:**]{} $m_\phi \geq M_R,\, m_\chi$;
- [**Ordering type B:**]{} $M_R \geq m_\phi,\, m_\chi$.
In all the cases, the fermion particles $\chi$ are the lightest dark particles and play the role of dark matter. As it will be clear later, the main difference in the DM production for the two hierarchies resides in the fact that the two-body decay of right-handed neutrinos into dark particles is allowed or not depending on the mass ordering. Therefore the results are insensitive to the detailed sub-orderings within the above classification. Before we discuss in detail the expressions of the two Boltzmann equations for the two different types of ordering, it is worth observing that other more involved orderings are allowed if one relaxes the assumption of having degenerate right-handed neutrinos. However, all the following considerations can be applied for just one right-handed neutrino at a time, according to its mass ordering with respect to the masses of dark particles. For this reason, we prefer to focus on the simplest case of equal right-handed neutrino masses.
Ordering type A: $m_\phi \geq M_R,\, m_\chi$
--------------------------------------------
In this case, the Boltzmann equations for the two dark species are given by $$\begin{aligned}
\mathcal{H}\,T\left(1+\frac{T}{3 g^\mathfrak{s}_*\left(T\right)}\frac{d g^\mathfrak{s}_*}{d T}\right)^{-1} \frac{d Y_\phi}{d T} & = & - \mathfrak{s} \left<\sigma\, v\right>_{\phi\phi}^{\rm DS} \left({Y_\phi^{\rm eq}}\right)^2 - \mathfrak{s} \left<\sigma\, v\right>^{\rm \nu-Yukawa}_{\chi\phi} Y_\phi^{\rm eq}Y_\chi^{\rm eq} \nonumber \\
& & + \left<\Gamma_{\phi}\right>\left(Y_\phi-\frac{Y_\phi^{\rm eq}}{Y_\chi^{\rm eq}}Y_\chi\right) \,, \label{eq:phi1} \\
\mathcal{H}\,T\left(1+\frac{T}{3 g^\mathfrak{s}_*\left(T\right)}\frac{d g^\mathfrak{s}_*}{d T}\right)^{-1} \frac{d Y_\chi}{d T} & = & - \mathfrak{s} \left<\sigma\, v\right>_{\chi\chi}^{\rm DS} \left({Y_\chi^{\rm eq}}\right)^2 - \mathfrak{s} \left<\sigma\, v\right>^{\rm \nu-Yukawa}_{\chi\phi} Y_\phi^{\rm eq}Y_\chi^{\rm eq} \nonumber \\
& & - \left<\Gamma_{\phi}\right>\left(Y_\phi-\frac{Y_\phi^{\rm eq}}{Y_\chi^{\rm eq}}Y_\chi\right) \,, \label{eq:chi1}\end{aligned}$$ where the right-handed neutrinos $N_R$ are taken to be in thermal equilibrium with photons. In the above expressions, we have neglected all the other subdominant terms suppressed by the condition $Y_{\chi,\phi} \ll Y_{\chi,\phi}^{\rm eq}$ according to the freeze-in production paradigm. Moreover, we do not take into account all the other scattering processes with right-handed neutrinos that are suppressed by the active-sterile neutrino mixing $\theta\equiv m_D M^{-1}_R$ (see Refs. [@Buchmuller:1990vh; @Pilaftsis:1991ug]). This is indeed a good approximation in the case where the right-handed neutrinos are heavier than the electroweak energy scale. Hence, we can mainly distinguish three different classes of processes (see Fig. \[fig:Feyn1\]), whose expressions are reported in the Appendix.
In particular, we have:
- [**Dark Sector scatterings:**]{} the first term in the two Boltzmann equations refers to the scattering processes $\phi\phi^*\rightarrow n_in_j$ and $\chi\overline{\chi}\rightarrow n_in_j$, respectively. In particular, we have $$\begin{aligned}
\left<\sigma\, v\right>_{\phi\phi}^{\rm DS} & = & \sum_{i,j=1,2} \left<\sigma_{\phi\phi^*\rightarrow n_in_j}\, v\right> \,, \\
\left<\sigma\, v\right>_{\chi\chi}^{\rm DS} & = & \sum_{i,j=1,2} \left<\sigma_{\chi\overline{\chi}\rightarrow n_in_j}\, v\right> \,.\end{aligned}$$ The amplitudes of such processes are obtained with different contractions of the right-handed neutrino portal in Eq. and, therefore, the corresponding thermal averaged cross sections are proportional to the coupling $y_{\rm DS}^4$;
- [**Neutrino Yukawa scatterings:**]{} the second term in both equations corresponds to the scattering processes that originate from the neutrino Yukawa interaction given in Eq. . According to the Goldstone Boson Equivalence Theorem, we consider also the processes involving the other degrees of freedom of the Higgs doublet, $G^0$ and $G^\pm$. Hence, we have $$\left<\sigma\, v\right>^{\rm \nu-Yukawa}_{\chi\phi} = \sum_{i=1}^3 \left[\left<\sigma_{\chi \phi\rightarrow \nu_i h^0}\, v\right> + \left<\sigma_{\chi \phi\rightarrow \nu_i G^0}\, v\right> + \left<\sigma_{\chi \phi\rightarrow \ell^\pm_i G^\mp}\, v\right>\right]\,.$$ The size of such $\nu-$Yukawa processes is set by the product $y_{\rm DS}^2\left|y_\nu\right|^2$, where $y_\nu$ is proportional to the Yukawa couplings in Eq. . In the neutrino mass basis, we have $y_\nu = \left(U^\dagger_\nu Y\right)_{ij}/\sqrt2$ or $y_\nu = Y_{ij}$ for neutrinos or leptons in the final states, respectively. The matrix $U_\nu$ is the Pontecorvo–Maki–Nakagawa–Sakata matrix describing neutrino oscillations. In the present analysis, we consider the best-fit values of neutrino oscillation parameters reported by the latest neutrino global analysis [@Esteban:2016qun] (see also Refs. [@Capozzi:2018ubv; @deSalas:2017kay]);
- [**Scalar decay:** ]{} the last terms are related to the two-body decays of $\phi$ particles into $\chi$ and right-handed neutrino, which are kinematically allowed if $m_\phi > m_\chi + M_R$. The corresponding partial decay widths are proportional to $y_{\rm DS}^2$ and take the expression $$\Gamma_{\phi \rightarrow \chi n_i}^{\rm 2-body} = \frac{y_{\rm DS}^2\,m_\phi}{16 \pi}\left(1-\frac{m_\chi^2}{m_\phi^2}-\frac{M_R^2}{m_\phi^2}\right)\lambda\left(1,\frac{m_\chi}{m_\phi},\frac{M_R}{m_\phi}\right) \,,$$ where $\lambda$ is the Kallen function. The total decay width is simply given by the sum of the two widths, i.e. $$\Gamma_{\phi}^{\rm 2-body} = \sum_{i=1}^2 \Gamma_{\phi \rightarrow \chi n_i}^{\rm 2-body} \,.$$
It is worth observing that the DS and $\nu-$Yukawa scattering processes scale in a different way with the right-handed neutrino portal coupling $y_{\rm DS}$. As will be discussed in detail, this implies that there are regions of the parameter space where one of the two different processes dominates in the DM production. While the two scattering processes are responsible for the production of particles in the dark sector, the decays allow to convert all the dark scalars $\phi$ into DM particles $\chi$. Indeed, if $m_\phi > m_\chi + M_R$, all the dark scalars particles are converted into the lighter fermions according to the discrete $Z_2$ symmetry. These scalar two-body decays are in general faster than the Universe expansion set by the Hubble parameter and occur at temperature higher than the electroweak energy scale. Hence, in case of $m_\phi > m_\chi + M_R$, the total DM relic abundance can be cast in terms of an effective DM yield defined as $$Y_{\rm DM} \left(T\right)= Y_\chi \left(T\right) + Y_\phi \left(T\right) \,,$$ which then obeys to the following effective Boltzmann equation $$\begin{aligned}
\mathcal{H}\,T\left(1+\frac{T}{3 g^\mathfrak{s}_*\left(T\right)}\frac{d g^\mathfrak{s}_*}{d T}\right)^{-1} \frac{d Y_{\rm DM}}{d T} & = & - \mathfrak{s} \left<\sigma\, v\right>_{\phi\phi}^{\rm DS} \left({Y_\phi^{\rm eq}}\right)^2 - \mathfrak{s} \left<\sigma\, v\right>_{\chi\chi}^{\rm DS} \left({Y_\chi^{\rm eq}}\right)^2 \nonumber \\
& & - 2 \, \mathfrak{s} \left<\sigma\, v\right>^{\rm \nu-Yukawa}_{\chi\phi} Y_\phi^{\rm eq}Y_\chi^{\rm eq}\,.\end{aligned}$$ This differential equation can be easily integrated providing the today’s DM yield at $T=0$, denoted as $Y_{\rm DM,0}$. Such a quantity is given by the sum of three different contributions related to DS and $\nu-$Yukawa processes: $$Y_{\rm DM,0} = Y^{\rm DS}_\phi + Y^{\rm DS}_\chi + 2\,Y^{\rm \nu-Yukawa}\,,
\label{eq:DMyield}$$ where $$\begin{aligned}
Y^{\rm DS}_\phi & = & \int_0^\infty dT \, \frac{\mathfrak{s}}{\mathcal{H}\,T} \left(1+\frac{T}{3 g^\mathfrak{s}_*\left(T\right)}\frac{d g^\mathfrak{s}_*}{d T}\right) \left<\sigma\, v\right>_{\phi\phi}^{\rm DS} \left({Y_\phi^{\rm eq}} \right)^2 \,, \label{eq:1}\\
Y^{\rm DS}_\chi & = & \int_0^\infty dT \, \frac{\mathfrak{s}}{\mathcal{H}\,T} \left(1+\frac{T}{3 g^\mathfrak{s}_*\left(T\right)}\frac{d g^\mathfrak{s}_*}{d T}\right) \left<\sigma\, v\right>_{\chi\chi}^{\rm DS} \left({Y_\chi^{\rm eq}}\right)^2 \,, \label{eq:2}\\
Y^{\rm \nu-Yukawa} & = & \int_0^\infty dT \, \frac{\mathfrak{s}}{\mathcal{H}\,T} \left(1+\frac{T}{3 g^\mathfrak{s}_*\left(T\right)}\frac{d g^\mathfrak{s}_*}{d T}\right) \left<\sigma\, v\right>^{\rm \nu-Yukawa}_{\chi\phi}Y_\phi^{\rm eq}Y_\chi^{\rm eq} \,. \label{eq:3}\end{aligned}$$ Here, we have assumed negligible yields of dark particles as initial condition. We note that the above expressions are dimensionless. However, we can rewrite Eq. in a way that one can explicitly see how it depends on the seesaw energy scale $M_R$, the right-handed neutrino portal coupling $y_{\rm DS}$ and the Yukawa couplings $y_\nu$ evaluated at the GeV energy scale. In particular, we have $$Y_{\rm DM,0} = y_{\rm DS}^4\left(\frac{1~{\rm GeV}}{M_R}\right) \left[ \tilde{Y}^{\rm DS}_\phi + \tilde{Y}^{\rm DS}_\chi \right] + y_{\rm DS}^2 \tilde{y}_\nu^2\left(1~{\rm GeV}\right) \left[ 2 \, \tilde{Y}^{\rm \nu-Yukawa} \right] \,,
\label{eq:DMyieldDimLess}$$ where the dimensionless quantities $\tilde{Y}$ are suitably defined by using Eqs. , and , and the effective squared Yukawa coupling $\tilde{y}_\nu^2$ is equal to (see the Appendix) $$\tilde{y}_\nu^2 \left(M_R\right) = 2.47 \times 10^{-14} \left(\frac{M_R}{\rm 1~GeV}\right)\,.
\label{eq:effyuk}$$ We note once again that, due to the different dependence on the coupling $y_{\rm DS}$ and the seesaw scale $M_R$ (see Eq. ), one of the two different classes of scattering processes could dominate over the other one. Hence, we expect that the neutrino sector defined by the Yukawa couplings in Eq. , whose structure could be inferred by low-energy neutrino experiments, would drive the DM production in some regions of the parameter space. This implies an intriguing relation between neutrino physics and dark matter.
On the other hand, in case of $m_\phi \leq m_\chi + M_R$ the scalar decays are not kinematically allowed and both dark particles are stable, providing a [*two-component*]{} DM scenario. By integrating the two Boltzmann equations, we have that the two today’s yields of $\chi$ and $\phi$ particles are given by $$Y_{\chi,0} = Y^{\rm DS}_\chi + Y^{\rm \nu-Yukawa}\,,\qquad{\rm and}\qquad Y_{\phi,0} = Y^{\rm DS}_\phi + Y^{\rm \nu-Yukawa}\,.$$ According to Eq. , the total DM relic abundance predicted by the model takes the form $$\Omega_{\rm DM}h^2 =\left\{ \begin{array}{lcl} \frac{2 \, \mathfrak{s}_0 \, m_\chi \,Y_{\rm DM,0}}{\rho_{\rm crit}/h^2} & \qquad & {\rm for} \,\,\,\, m_\phi > m_\chi + M_R \\
& \qquad & \\
\frac{2 \, \mathfrak{s}_0 \, \left(m_\chi \,Y_{\chi,0} + m_\phi \,Y_{\phi,0} \right)}{\rho_{\rm crit}/h^2} & \qquad & {\rm for} \,\,\,\, m_\phi \leq m_\chi + M_R \end{array}\right.\,,
\label{eq:omegaPREA}$$ where $\mathfrak{s}_0=2891.2\,{\rm cm^3}$ is today’s entropy density [@Patrignani:2016xqp], and the factor of 2 takes into account the contribution of DM anti-particles. Before concluding, we remind that all the above equations and expressions still hold in both the two possible hierarchies $M_R \leq m_\chi \leq m_\phi$ and $m_\chi \leq M_R \leq m_\phi$, once the different phase space provided by the different mass hierarchy is taken into account.
Ordering type B: $M_R \geq m_\phi,\, m_\chi$
--------------------------------------------
In the ordering B case, the dark particles are mainly produced through the two-body decays of the two right-handed neutrinos $n_i \rightarrow \chi \phi^*,\phi\overline{\chi}$. The decay widths take the expression $$\Gamma_{n_i\rightarrow \chi \phi^*} = \frac{y_{\rm DS}^2\,M_R}{32 \pi}\left(1+\frac{m_\chi^2}{M_R^2}-\frac{m_\phi^2}{M_R^2}\right)\lambda\left(1,\frac{m_\chi}{M_R},\frac{m_\phi}{M_R}\right)\,.$$ Since the corresponding term in the Boltzmann equation is just proportional to $y_{\rm DS}^2$, such two-body decays provide the dominant contribution to the DM production. On the other hand, the contributions coming from the DS and $\nu-$Yukawa processes are almost sub-dominant. Moreover, in this scenario the dark scalar particles $\phi$ are slowly converted into the fermion ones $\chi$ through three-body decays with virtual right-handed neutrinos. In the limit $m_\phi \gg m_\chi$, the scalar three-body decay width is given by $$\Gamma_\phi^{\rm 3-body} = \frac{y_{\rm DS}^2\tilde{y}_\nu^2}{1536 \pi^3}\frac{m_\phi^3}{M^2_R}\left(1+\frac{m_\phi^2}{2M_R^2}\right)\,,
\label{eq:dec3}$$ where $\tilde{y}_\nu^2$ is the effective squared Yukawa coupling defined in Eq. . Such decays, which are suppressed by the small Yukawa coupling and by the right-handed neutrino mass, only occur at very late times. Hence, the ordering B case would lead to a two-component dark matter scenario where both dark scalars and fermions contribute to the final DM relic abundance. In this case, the two Boltzmann equations further simplifies and the today’s yields of $\chi$ and $\phi$ are both equal to $$Y_{\rm \chi,0} = Y_{\rm \phi,0} = \int_0^\infty dT \, \frac{1}{\mathcal{H}\,T} \sum_{i=1,2} \left[\left<\Gamma_{n_i\rightarrow \chi \phi^*} \right> Y^{\rm eq}_{n_i} \right] = \left(\frac{135 \, M_{\rm Planck}}{1.66 \left(2 \pi^3\right) g^\mathfrak{s}_* \sqrt{g_*}}\right)\frac{\Gamma_{n_i\rightarrow \chi \phi^*}}{M^2_R} \,,
\label{eq:DMC}$$ and the final DM relic abundance is then given by $$\Omega_{\rm DM}h^2 = \frac{2 \, \mathfrak{s}_0 \left(m_\chi Y_{\rm \chi,0} + m_\phi Y_{\rm \phi,0} \right) }{\rho_{\rm crit}/h^2} \,.$$ By comparing the above expression with Eq. and using Eq. , we obtain the following analytical expression for the right-handed neutrino portal coupling $y_{\rm DS}$ required to account for the correct DM relic abundance, $$y_{\rm DS} = 1.22\times 10^{-12} \left(\frac{g^\mathfrak{s}_*}{106.75}\right)^{3/2} \sqrt{\frac{M_R}{m_\phi+m_\chi}} \left[\left(1+\frac{m_\chi^2}{M_R^2}-\frac{m_\phi^2}{M_R^2}\right)\lambda\left(1,\frac{m_\chi}{M_R},\frac{m_\phi}{M_R}\right)\right]^{-\frac12}\,. \label{eq:caseB}$$ By plugging this expression in Eq. , we obtain that the following prediction for the lifetime of dark scalars $$\tau_\phi^{\rm 3-body} \simeq 8.45\times 10^{11} \left(1+\frac{m_\phi^2}{2 M_R^2}\right)^{-1} \left(\frac{\rm 10^3~GeV}{m_\phi}\right)^2 \left(\frac{106.75}{g^\mathfrak{s}_*}\right)^{3/2}~{\rm sec}\,,$$ where, for the sake of simplicity, we have considered the limit $M_R \gg m_\phi \gg m_\chi$. Such values for the lifetime imply that the ordering B case is almost ruled out. Indeed, for a dark scalar mass $10^3~{\rm GeV} \leq m_\phi \leq 10^{12}~{\rm GeV}$ (compatible with the assumptions behind the reported Boltzmann equations), we accordingly have $8.45 \times 10^{11}~{\rm sec} \geq \tau_\phi^{\rm 3-body} \geq 8.45 \times 10^{-7}~{\rm sec}$. This means that the lifetime is too large so that the dark scalars decays into the dark fermions occur after the electroweak breaking or even after the Big Bang Nucleosynthesis. On the other hand, for small dark scalar masses, the lifetime is smaller than the age of the Universe ($\sim 4.35 \times 10^{17}$ sec), scenario that is strongly constrained by cosmological observations and indirect DM searches and in contrast with the two-component DM assumption. Hence, we find that the ordering type B case can provide an allowed two-component DM scenario only if the dark scalar mass is smaller than the sum of the masses of final particles involved in the three-body decay, i.e. $m_\phi \simeq m_\chi$. In this case, the decays are not kinematically allowed and the dark scalars are stable particles, so providing a viable two-component DM model.
Hence, since the ordering type B is almost ruled out unless $m_\phi \simeq m_\chi$ and the right-handed neutrino coupling is analytically provided by Eq. , in the next Section we focus only on the numerical results obtained in the ordering type A, i.e. $m_\phi \geq M_R,\,m_\chi$.
Numerical results
=================
The four free parameters of the model (the three masses of the new particles, $m_\chi$, $m_\phi$ and $M_R$, and the right-handed neutrino portal coupling $y_{\rm DS}$) are constrained by requiring the equality between the observed DM relic abundance and the predicted one . In particular, for a given choice of the seesaw energy scale $M_R$ and the two masses of the dark particles, one can obtain the value for the coupling $y_{\rm DM}$ that provides the correct DM relic abundance by solving Eq. .
![\[fig:benchmark1\]Example of DM production through freeze-in mechanism for the benchmark values: $M_R = 10^3$ GeV, $m_\chi = 10^5$ GeV, $m_\phi = 10^7$ GeV (ordering type A with $M_R \leq m_\chi \leq m_\phi$) and $y_{\rm DS} = 7.5\times 10^{-6}$. [*Left Panel*]{}: yields of DM particles and dark scalars as a function of the auxiliary variable $x = M_R/T$. [*Right Panel*]{}: interactions rates of the different processes involved in the Boltzmann equations and .](BenchmarkYield1.pdf "fig:"){width="49.00000%"} ![\[fig:benchmark1\]Example of DM production through freeze-in mechanism for the benchmark values: $M_R = 10^3$ GeV, $m_\chi = 10^5$ GeV, $m_\phi = 10^7$ GeV (ordering type A with $M_R \leq m_\chi \leq m_\phi$) and $y_{\rm DS} = 7.5\times 10^{-6}$. [*Left Panel*]{}: yields of DM particles and dark scalars as a function of the auxiliary variable $x = M_R/T$. [*Right Panel*]{}: interactions rates of the different processes involved in the Boltzmann equations and .](BenchmarkRate1.pdf "fig:"){width="49.00000%"}
In Fig. \[fig:benchmark1\], we report a benchmark case of ordering A with $M_R = 10^3$ GeV, $m_\chi = 10^5$ GeV, $m_\phi = 10^7$ GeV and $y_{\rm DS} = 7.5\times 10^{-6}$. As can been seen in the left panel, at high temperatures the DS and $\nu-$Yukawa scattering processes produce particles in the dark sector and the two yields increases as the Universe is cooling. However, such interactions are not strong enough to bring the dark sector in thermal equilibrium. Indeed, the corresponding interaction rates are always smaller than the Hubble parameter, as shown in the right panel of the same figure. At $T=m_\phi$, the yield $\phi$ would freeze-in, but at this temperature its decay rate becomes efficient since $\left<\Gamma_\phi\right> \geq \mathcal{H}$ for $T\lesssim m_\phi$. Hence, the scalar particles are converted into the lighter fermions through the two-body decays that then freeze-in providing the correct DM relic abundance. This conversion can be easily recognised in the left panel by looking to the kink of the $\chi$ yield occurring at $T \simeq m_\phi$. Such a feature suggests that the DM production is mainly driven by the decay of scalar particles that are produced by DS and $\nu-$Yukawa scatterings.
![\[fig:benchmark2\]Example of DM production through freeze-in mechanism for the benchmark values $M_R = 10^{10}$ GeV, $m_\chi = 10^{12}$ GeV, $m_\phi = 10^{14}$ GeV (ordering type A with $M_R \leq m_\chi \leq m_\phi$) and $y_{\rm DS} = 1.9\times 10^{-8}$. The description of the plots is the same of Fig. \[fig:benchmark1\].](BenchmarkYield2.pdf "fig:"){width="49.00000%"} ![\[fig:benchmark2\]Example of DM production through freeze-in mechanism for the benchmark values $M_R = 10^{10}$ GeV, $m_\chi = 10^{12}$ GeV, $m_\phi = 10^{14}$ GeV (ordering type A with $M_R \leq m_\chi \leq m_\phi$) and $y_{\rm DS} = 1.9\times 10^{-8}$. The description of the plots is the same of Fig. \[fig:benchmark1\].](BenchmarkRate2.pdf "fig:"){width="49.00000%"}
It is worth noticing that, for $M_R=10^3$ GeV, the rates of DS processes are larger than the $\nu-$Yukawa one. This means that the DS processes provide the dominant contribution to the DM production. On the other hand, according to Eq. , for larger values of the seesaw energy scale $M_R$ the DS processes become less efficient and the DM production starts to be instead driven by $\nu-$Yukawa scatterings. In Fig. \[fig:benchmark2\] we show a benchmark case of ordering type A where $\nu-$Yukawa scatterings dominate the DM production. In particular, we consider the values $M_R = 10^{10}$ GeV, $m_\chi = 10^{12}$ GeV, $m_\phi = 10^{14}$ GeV and $y_{\rm DS} = 1.9\times 10^{-8}$. In this case, both the yields of $\phi$ and $\chi$ particles increase and freeze-in at $T=m_\phi$. This is due to the fact that, as show in the right panel, the dominant contribution to the DM production is provided by the $\nu-$Yukawa processes that simultaneously produce $\phi$ and $\chi$ particles for $T \geq m_\phi$. Then, when $\left<\Gamma_\phi\right> \geq \mathcal{H}$, the scalar particles decay into the dark fermions whose yield just doubles.
![\[fig:RelativeContributions\][*Left Panel*]{}: right-handed neutrino coupling $y_{\rm DS}$ as a function of the seesaw energy scale $M_R$. [*Right Panel*]{}: relative contribution of DS (solid lines) and $\nu-$Yukawa (dashed lines) scattering processes to the DM relic abundance. In both panels, the different colours correspond to different values for the ratio $m_\phi/m_\chi$ for ordering type A.](Coupling.pdf "fig:"){width="49.00000%"} ![\[fig:RelativeContributions\][*Left Panel*]{}: right-handed neutrino coupling $y_{\rm DS}$ as a function of the seesaw energy scale $M_R$. [*Right Panel*]{}: relative contribution of DS (solid lines) and $\nu-$Yukawa (dashed lines) scattering processes to the DM relic abundance. In both panels, the different colours correspond to different values for the ratio $m_\phi/m_\chi$ for ordering type A.](RelativeContribution.pdf "fig:"){width="49.00000%"}
![\[fig:NuDMrelation\]Right-handed neutrino coupling $y_{\rm DM}$ in the plane $M_R$ – $m_\phi/m_\chi$ able to reproduce the correct DM relic abundance. The black line highlights the choices of parameters at which the equality of the DS and $\nu-$Yukawa contributions occurs for ordering type A.](MainPlot.png){width="65.00000%"}
In Fig.s \[fig:RelativeContributions\] and \[fig:NuDMrelation\] we report the main results of the present numerical analysis focusing on the ordering type A ($m_\phi \geq M_R,\,m_\chi$). In particular, in the left panel of Fig. \[fig:RelativeContributions\] it is depicted how the right-handed neutrino coupling $y_{\rm DS}$ depends on the right-handed neutrino mass $M_R$. The four different lines correspond to different values for the ratio $m_\phi / m_\chi$, with the blue line representing the case of a two-component DM scenario ($m_\phi = m_\chi$). Hence, we find that the value for the dark coupling required to achieve the correct DM abundance is almost independent on the ratio $m_\chi / M_R$. A slightly different coupling $y_{\rm DS}$ is instead required when $m_\phi \simeq M_R$. However, such a case does not provide a viable DM model since the dark scalars are converted too slowly into dark fermions through the three-body decays as discussed for the ordering type B. For this reason, we do not report here the corresponding results.
Moreover, we note that for small seesaw energy scale the coupling is almost constant, while for larger values of $M_R$ it decreases. Such a change of the behaviour of the coupling $y_{\rm DS}$ occurs at the energy where $\nu-$Yukawa scattering processes start to dominate over the DS ones in the DM production. This can be easily understood by looking to the right panel of Fig. \[fig:RelativeContributions\] that shows the relative contribution to the DM relic abundance of the DS (solid lines) and $\nu-$Yukawa (dashed lines) processes. In the plot, the different colours correspond to different values of the ratio $m_\phi / m_\chi$, according to the left panel. In Fig. \[fig:NuDMrelation\], instead, it is displayed the required value for the coupling $y_{\rm DS}$ in the plane $M_R$ – $m_\phi/m_\chi$. The solid line represents the equality between the contributions to the DM relic abundance of the two different classes of scattering processes. For all the points of the parameter space that provide the correct DM relic abundance with $m_\phi > M_R$, the lifetime of $\phi$ particles is found to be smaller than $10^{-15}$ sec, implying that the scalar decays occur at a temperature higher than the electroweak energy scale. This result confirms the validity of the assumptions behind all the expressions we have used in the present numerical analysis. Finally, we remark that the final DM abundance is found to depend on mass ratio $m_\phi/m_\chi$ only, while it is almost independent on $m_\chi/M_R$. Hence, the results reported in Fig.s \[fig:RelativeContributions\] and \[fig:NuDMrelation\] are valid for any choice of masses that satisfies the relations $m_\phi \geq M_R,\,m_\chi$.
Conclusions
============
We have proposed a minimal model to simultaneously account for a realistic neutrino spectrum through a type-I seesaw mechanism and a viable dark matter relic density. The model is an extension of the Littlest Seesaw model in which the two right-handed neutrinos of the model are coupled to a $Z_2$-odd dark sector via right-handed neutrino portal couplings. In other words we suppose that the production of dark sector particles is achieved dominantly via their couplings to the right-handed neutrinos $N_R$, which are in turn coupled to the thermal bath via their neutrino Yukawa couplings to left-handed neutrinos and Higgs scalar.
In this model, we have seen that a highly constrained and direct link between dark matter and neutrino physics may be achieved by considering the freeze-in production mechanism of dark matter. In such a framework we have shown that the same neutrino Yukawa couplings which describe neutrino mass and mixing may also play a dominant role in the dark matter production. We have investigated the allowed regions in the parameter space of the scheme that provide the correct neutrino masses and mixing and simultaneously give the correct dark matter relic abundance.
In the above model, we have seen that the results may be classified into two types of cases characterised by whether the right-handed neutrinos are heavier or lighter than the heavier of the two dark particles. We have seen that the ordering type B with $M_R \geq m_\phi,\, m_\chi$ is almost ruled out unless $m_\phi \simeq m_\chi$, which is the condition that prevents late decays of the dark scalars, and would lead to a viable two-component DM scenario. We have therefore mainly focused on the ordering type A for which $m_\phi \geq M_R, m_\chi$. This is also the most interesting case since then the neutrino Yukawa coupling may play an important role in the production of dark sector particles in the early Universe, and hence in providing the correct relic density.
For type A orderings, we have found that the allowed neutrino portal coupling depends on the right-handed neutrino mass $M_R$ and on the ratio $m_\phi/m_\chi$, while it is almost independent on the ratio $m_\chi/M_R$. Remarkably, for large right-handed neutrino mass, the neutrino Yukawa couplings are fixed via the seesaw mechanism to account neutrino mass and mixing and also by their dominant role in the DM production. The only free parameter is then the common right-handed neutrino mass, which determines the required dark sector right-handed neutrino portal coupling. Such a feature provide a direct link between neutrino physics and dark matter phenomenology, within the framework of the minimally extended Littlest Seesaw model. In certain cases the right-handed neutrino masses may be arbitrarily large, for example in the range $10^{10}-10^{11}$ GeV required for vanilla leptogenesis, with a successful relic density arising from frozen-in dark matter particles with masses around this scale, which we refer to as “fimpzillas”.
In conclusion, the present paper has made progress in connecting neutrino physics to dark matter in three different ways. Firstly we have proposed a realistic model not only of dark matter but also neutrino mass and mixing, via the Littlest Seesaw model with two right-handed neutrinos, as compared to many neutrino related dark matter models in the literature which only consider a single right-handed neutrino. Secondly, we have considered the freeze-in mechanism for dark matter which is the first time it has been considered in the literature in the connection with the right-handed neutrino portal.[^3] Thirdly, we have focused on cases where the same Yukawa couplings which control neutrino mass and mixing also control the relic abundance of dark matter. Within this framework, assuming ordering type A for which $m_\phi \geq M_R, m_\chi$, we have seen that the parameter space is very tightly constrained with the neutrino portal coupling uniquely given by the right-handed neutrino mass $M_R$ for a given ratio of dark sector masses $m_\phi/m_\chi$. If the right-handed neutrino masses are in the range $10^{10}-10^{11}$ GeV required for vanilla leptogenesis, this leads to the new idea of superheavy frozen-in dark matter or “fimpzillas”.
Acknowledgments {#acknowledgments .unnumbered}
===============
S.F.K. acknowledges the STFC Consolidated Grant ST/L000296/1 and the European Union’s Horizon 2020 Research and Innovation programme under Marie Skłodowska-Curie grant agreements Elusives ITN No. 674896 and InvisiblesPlus RISE No. 690575. M.C. acknowledges financial support from the STFC Consolidated Grant L000296/1.
Amplitudes {#A}
==========
In this Appendix, we report the squared matrix elements for all the scattering processes that are important for the DM production as a function of the corresponding Mandelstam variables $s$, $t$ and $u$. In the following computations, we have used the Feynman rules for Majorana fermions reported in Ref. [@Denner:1992me; @Denner:1992vza]. For the DS processes drawn in Fig. \[fig:Feyn1\] we have [$$\begin{aligned}
\overline{\left|\mathcal{M}\right|^2}_{\phi\phi^*\rightarrow n_in_j} & = & - \frac{\left|y_{{\rm DS}i}\, y_{{\rm DS}j}\right|^2}{\left(t-m_\chi^2\right)^2} \left[\left(t-m_\phi^2+M_{Ri}^2\right)\left(t-m_\phi^2+M_{Rj}^2\right)+t\left(s-M_{Ri}^2-M_{Rj}^2\right)\right] \nonumber \\
& & - \frac{\left|y_{{\rm DS}i}\, y_{{\rm DS}j}\right|^2}{\left(u-m_\chi^2\right)^2} \left[\left(u-m_\phi^2+M_{Ri}^2\right)\left(u-m_\phi^2+M_{Rj}^2\right)+u\left(s-M_{Ri}^2-M_{Rj}^2\right)\right] \\
& & \mp \frac{2 M_{Ri} M_{Rj} \,{\rm Re}\left(y_{{\rm DS}i}\, y_{{\rm DS}j}^*\right)}{\left(t-m_\chi^2\right)\left(u-m_\chi^2\right)} \left[\frac{s+t+u}{2}-2m_\phi^2 \right] \nonumber \,, \\
%%%%%%%
&& \nonumber \\
%%%%%%%
\overline{\left|\mathcal{M}\right|^2}_{\chi\overline{\chi}\rightarrow n_in_j} & = &\left|y_{{\rm DS}i}\, y_{{\rm DS}j}\right|^2 \left[\frac{\left(t-m_\chi^2-M_{Ri}^2\right)\left(t-m_\chi^2-M_{Rj}^2\right)}{4\left(t-m_\phi^2\right)^2} +\frac{\left(u-m_\chi^2-M_{Ri}^2\right)\left(u-m_\chi^2-M_{Rj}^2\right)}{4\left(u-m_\phi^2\right)^2} \right] \nonumber \\
& & \mp \frac{M_{Ri} M_{Rj} \,{\rm Re}\left(y_{{\rm DS}i}\, y_{{\rm DS}j}^*\right)\,\left(s-2m_\chi^2\right)}{4\left(t-m_\phi^2\right)\left(u-m_\phi^2\right)} \,,\end{aligned}$$]{} where the sign $-$ ($+$) in the interference term is in case of equal (different) right-handed neutrinos, i.e. $i=j$ ($i \neq j$). On the other hand, neglecting the neutrino and lepton masses, the squared matrix element for the $\nu-$Yukawa scattering processes in Fig. \[fig:Feyn1\] take the expressions [$$\begin{aligned}
\overline{\left|\mathcal{M}\right|^2}_{\phi\overline{\chi}\rightarrow h^0\nu_i} & = & \frac{\left|y_{{\rm DS}1} \left(U_\nu^\dagger Y\right)_{i1}\right|^2}{\left(s-M_{R1}^2\right)^2} \left[\left(s-m_\chi^2-m_\phi^2\right)\left(s-m_{h^0}^2\right)+2 \left(s-M_{R1}^2\right)\left(t-m_\chi^2\right)\right] \nonumber \\
& & + \frac{\left|y_{{\rm DS}2} \left(U_\nu^\dagger Y\right)_{i2}\right|^2}{\left(s-M_{R2}^2\right)^2}\left[\left(s-m_\chi^2-m_\phi^2\right)\left(s-m_{h^0}^2\right)+2 \left(s-M_{R2}^2\right)\left(t-m_\chi^2\right)\right] \\
& & + \frac{2\,{\rm Re}\left(y_{{\rm DS}1} \, y_{{\rm DS}2}^* \left(U_\nu^\dagger Y\right)_{i1} \left(U_\nu^\dagger Y\right)^{*}_{i2}\right)}{\left(s-M_{R1}^2\right)\left(s-M_{R2}^2\right)}\left[\left(s-m_\chi^2-m_\phi^2\right)\left(s-m_{h^0}^2\right)+ 2 \left(s-M_{R1}M_{R2}\right)\left(t-m_\chi^2\right)\right] \nonumber \,, \\
\nonumber\end{aligned}$$ $$\begin{aligned}
\overline{\left|\mathcal{M}\right|^2}_{\phi\overline{\chi}\rightarrow G^0\nu_i} & = & \frac{\left|y_{{\rm DS}1} \left(U_\nu^\dagger Y\right)_{i1}\right|^2}{\left(s-M_{R1}^2\right)^2} \left[\left(s-m_\chi^2-m_\phi^2\right)\left(s-m_{G^0}^2\right)+2 \left(s-M_{R1}^2\right)\left(t-m_\chi^2\right)\right] \nonumber \\
& & + \frac{\left|y_{{\rm DS}2} \left(U_\nu^\dagger Y\right)_{i2}\right|^2}{\left(s-M_{R2}^2\right)^2}\left[\left(s-m_\chi^2-m_\phi^2\right)\left(s-m_{G^0}^2\right)+2 \left(s-M_{R2}^2\right)\left(t-m_\chi^2\right)\right] \\
& & + \frac{2\,{\rm Re}\left(y_{{\rm DS}1} \, y_{{\rm DS}2}^* \left(U_\nu^\dagger Y\right)_{i1} \left(U_\nu^\dagger Y\right)^{*}_{i2}\right)}{\left(s-M_{R1}^2\right)\left(s-M_{R2}^2\right)}\left[\left(s-m_\chi^2-m_\phi^2\right)\left(s-m_{G^0}^2\right)+ 2 \left(s-M_{R1}M_{R2}\right)\left(t-m_\chi^2\right)\right] \nonumber \,, \\
&& \nonumber \\
\overline{\left|\mathcal{M}\right|^2}_{\phi\overline{\chi}\rightarrow G^+\ell^-_i} & = & \frac{\left|y_{{\rm DS}1} Y_{i1}\right|^2}{2 \left(s-M_{R1}^2\right)^2} \left[\left(s-m_\chi^2-m_\phi^2\right)\left(s-m_{G^+}^2\right)+2 \, s \left(t-m_\chi^2\right)\right] \nonumber \\
& & + \frac{\left|y_{{\rm DS}2} Y_{i2}\right|^2}{2\left(s-M_{R2}^2\right)^2}\left[\left(s-m_\chi^2-m_\phi^2\right)\left(s-m_{G^+}^2\right)+2 \, s \left(t-m_\chi^2\right)\right] \\
& & + \frac{{\rm Re}\left(y_{{\rm DS}1} \, y_{{\rm DS}2}^*Y_{i1} Y^{*}_{i2}\right)}{\left(s-M_{R1}^2\right)\left(s-M_{R2}^2\right)}\left[\left(s-m_\chi^2-m_\phi^2\right)\left(s-m_{G^+}^2\right)+ 2 \, s \left(t-m_\chi^2\right)\right] \nonumber \,, \\
&& \nonumber \\
\overline{\left|\mathcal{M}\right|^2}_{\phi\overline{\chi}\rightarrow G^-\ell^+_i} & = & \left[\frac{\left|y_{{\rm DS}1} Y_{i1}\right|^2 M_{R1}^2}{\left(s-M_{R1}^2\right)^2} +\frac{\left|y_{{\rm DS}2} Y_{i2}\right|^2 M_{R2}^2}{\left(s-M_{R2}^2\right)^2} + \frac{2{\rm Re}\left(y_{{\rm DS}1} \, y_{{\rm DS}2}^*Y_{i1} Y^{*}_{i2}\right) M_{R1}M_{R2}}{\left(s-M_{R1}^2\right)\left(s-M_{R2}^2\right)} \right] \left(t-m_\chi^2\right) \,.\end{aligned}$$]{} By taking the equalities $y_{{\rm DS}1} = y_{{\rm DS}2}$ and $M_{R1} = M_{R2}$, the sum of all the $\nu-$Yukawa scattering processes can be parametrized in terms of the effective squared Yukawa coupling reported in Eq. , whose analytical expression is given by $$\begin{aligned}
\tilde{y}^2_\nu & = & 4 \sum_{i=1}^3\left[\left|\left(U_\nu^\dagger Y\right)_{i1}\right|^2 + \left|\left(U_\nu^\dagger Y\right)_{i2}\right|^2 + 2\,{\rm Re}\left(\left(U_\nu^\dagger Y\right)_{i1} \left(U_\nu^\dagger Y\right)^{*}_{i2}\right) \right] \nonumber \\
& & + \sum_{i=1}^3\left[\left|Y_{i1}\right|^2 + \left|Y_{i2}\right|^2 + 2\,{\rm Re}\left(Y_{i1} Y^{*}_{i2}\right) \right]\,.\end{aligned}$$
[^1]: The Botlzmann equations are here written in terms of the temperature $T$ rather than the time or the auxiliary variable $x = M_R/T$. This implies different signs in the right-hand side of the Boltzmann equations because the photon temperature decreases during the evolution of the Universe.
[^2]: We use the symbol $ \mathfrak{s}$ to represent entropy density and reserve the symbol $s$ for the Mandelstam variable.
[^3]: Note added: As this paper was being completed, a neutrino portal dark matter model appeared, including regions of parameter space in which the freeze-in mechanism is responsible for dark matter production [@Becker:2018rve].
|
---
abstract: 'We analyze the notion of *guessing model*, a way to assign combinatorial properties to arbitrary regular cardinals. Guessing models can be used, in combination with inaccessibility, to characterize various large cardinals axioms, ranging from supercompactness to rank-to-rank embeddings. The majority of these large cardinals properties can be defined in terms of suitable elementary embeddings $j\colon V_\gamma \to V_\lambda$. One key observation is that such embeddings are uniquely determined by the image structures $j [ V_\gamma ]\prec V_\lambda$. These structures will be the prototypes guessing models. We shall show, using guessing models $M$, how to prove for the ordinal $\kappa_M=j_M (\operatorname{crit}(j_M))$ (where $\pi_M$ is the transitive collapse of $M$ and $j_M$ is its inverse) many of the combinatorial properties that we can prove for the cardinal $j(\operatorname{crit}(j))$ using the structure $j[V_\gamma]\prec V_{j(\gamma)}$. $\kappa_M$ will always be a regular cardinal, but consistently can be a successor and guessing models $M$ with $\kappa_M=\aleph_2$ exist assuming the proper forcing axiom. By means of these models we shall introduce a new structural property of models of ${\ensuremath{\text{{\sf PFA}}}}$: the existence of a “Laver function” $f \colon \aleph_2 \to H_{\aleph_2}$ sharing the same features of the usual Laver functions $f\colon\kappa\to H_\kappa$ provided by a supercompact cardinal $\kappa$. Further applications of our analysis will be proofs of the singular cardinal hypothesis and of the failure of the square principle assuming the existence of guessing models. In particular the failure of square shows that the existence of guessing models is a very strong assumption in terms of large cardinal strength.'
author:
- Matteo Viale
bibliography:
- 'APLMV\*\*.bib'
title: Guessing models and generalized Laver diamond
---
Introduction
============
The notation used is standard and follows [@jech] and [@Kan09]. The reader should look-up Section \[subsect.notation\] for all undefined notions.
A structure $\mathfrak{R}=\langle R,\in, A\rangle$ is a *suitable initial segment* if $R = V_ \alpha $ for some ordinal $ \alpha $ or $ R = H_ \theta $ for some regular cardinal $ \theta $ and $A\subseteq P_\omega ( R )$.
For any set $X$ let $$\kappa_X \coloneqq \min\{\alpha\in X:\alpha\text{ is an ordinal and }X\cap\alpha \neq\alpha\},$$ $ \kappa _X $ being undefined when $ X\cap{\ensuremath{\text{{\rm Ord}}}}$ is an ordinal.
Let $\mathfrak{R}$ be a suitable initial segment and $M\prec R$.
1. Given a cardinal $\delta \leq \kappa_M$, $X\in M$ and $d\in P(X)\cap R$ we say that:
- $d$ is $(\delta,M)$-*approximated* if $d\cap Z\in M$ for all $Z\in M\cap P_{\delta} ( R )$.
- $d$ is $M$-*guessed* if $d\cap M=e\cap M$ for some $e\in M\cap P(X)$.
2. $M\prec R$ is a $\delta$*-guessing model for $X$* if every $(\delta,M)$-approximated subset of $X$ is $M$-guessed.
3. $M\prec R$ is a $\delta$*-guessing model* if $M$ is a $\delta$-guessing model for $X$, for all $X\in M$.
4. $M\prec R$ is a *guessing model* if $M\prec R$ is a $\delta$-guessing model, for some $\delta \leq \kappa_M$.
We shall show in Section \[sect.largecard\], exploiting ideas of Magidor [@MAG71; @magidor.characterization_supercompact], that simple statements regarding the existence of appropriate $\aleph_0$-guessing models give a fine hierarchy of large cardinal hypothesis above supercompactness. For uncountable $\delta$, the notion of $\delta$-guessing model is motivated by the main results of [@VIAWEI10] and [@weiss]. For example [@weiss Theorem 5.4] can be rephrased as follows:
> It is relatively consistent with the existence of a supercompact cardinals that there is $W$ model of ${\ensuremath{\text{{\sf ZFC}}}}$ in which for eventually all regular $\theta$ there is an $\aleph_1$-guessing model $M\prec H_\theta^W$ with $\kappa_M$ successor of a regular cardinal.
On the other hand Proposition 3.2 and Theorem 4.8 from [@VIAWEI10] show that ${\ensuremath{\text{{\sf PFA}}}}$ implies that for every regular $\theta\geq\aleph_2$ there are $\aleph_1$-guessing models $M\prec H_\theta$ with $\kappa_M=\aleph_2$.
In these two papers, converse implications were proved. For example [@VIAWEI10 Corollary 6.6] can be stated as follows:
> Assume $V\subseteq W$ are a pair of transitive models of ${\ensuremath{\text{{\sf ZFC}}}}$ which have the $\kappa$-covering and $\kappa$-approximation property for some $\kappa$ inaccessible in $V$. Then the existence of an $\aleph_1$-guessing models $M\prec (V_\theta)^W$ with $\kappa_M=\kappa$ implies that $\kappa_M$ is $|\gamma|^V$-strongly compact cardinal in $V$ for all $\gamma<\theta$.
The first two results above show that $\delta$-guessing models for uncountable $\delta$ are a means to transfer large cardinal features of inaccessible cardinals to successor cardinals and the latter result above combined with the analysis we give in Section \[sect.largecard\] of the highest segment in the large cardinals hierarchy shows that this is a two way correspondance: the existence of a $\delta$-guessing model model $M$ in some transitive class model $W$ of ${\ensuremath{\text{{\sf ZFC}}}}$ will most often be a sufficient condition to show that $\kappa_M$ is a large cardinal with a high degree of strong compactness in some transitive inner model $V$ of $W$.
The paper is organized as follows. In Section \[sect.basicproperties\] we develop the basic features of guessing models, which are used in Section \[sect.largecard\] to define fine hierarchies of large cardinals ranging from supercompactness to rank to rank embeddings. In Sections \[sect.internalclosure\] and \[sect.isotypesGM\] we analyze the type of guessing models that exist in models of ${\ensuremath{\text{{\sf MM}}}}$ and ${\ensuremath{\text{{\sf PFA}}}}$. In particular in Section \[sect.internalclosure\] we show that assuming ${\ensuremath{\text{{\sf MA}}}}$ every $\aleph_1$-guessing model of size $\aleph_1$ is $ \aleph_1$-internally unbounded (Definition \[def.intclos\]) and that assuming ${\ensuremath{\text{{\sf PFA}}}}$ there are stationarily many $\aleph_1$-guessing models which are $\aleph_1$-internally club. The most interesting results of the paper are proved in Sections \[sect.isotypesGM\] and \[sect.laverdiamond\]. In Section \[sect.isotypesGM\] it is shown that the isomorphism type of a $\delta$-internally club $\delta$-guessing model $M\prec H_\theta$ is uniquely determined by the order-type of the set of cardinals in $M$ (Theorem \[thm.isotypealeph1guesmod\]). This generalize a classification result for $\aleph_0$-guessing models (Lemma \[lem.isotype0guesmod\]) due to Magidor. In Section \[sect.laverdiamond\] we define the notion of a strong $\mathcal{J}$-Laver function $f \colon \kappa\rightarrow H_\kappa$ with respect to a class $\mathcal{J}$ of elementary embeddings $j \colon V\rightarrow M$ all with critical point $\kappa$. We first show that any Laver function $f \colon \kappa\rightarrow H_\kappa$ produced by the “standard proof” of Laver diamond under the assumption that $\kappa$ is supercompact is a strong $\mathcal{J}_\kappa$-Laver function ($\mathcal{J}_\kappa$ is the class of elementary embeddings induced by generics for the stationary tower below some stationary set of guessing models $M$ with $\kappa_M=\kappa$). We next prove, using Theorem \[thm.isotypealeph1guesmod\], that under ${\ensuremath{\text{{\sf PFA}}}}$ there are strong $\mathcal{J}_{\aleph_2}$-Laver functions $f\colon\aleph_2\rightarrow H_{\aleph_2}$. This is a new property of models of ${\ensuremath{\text{{\sf PFA}}}}$ which may lead to further applications. On the other hand we expect that Theorem \[thm.isotypealeph1guesmod\] will be of help in outlining may other structural properties of models of forcing axioms. Finally in Section \[sect.applications\] we give new proofs that ${\ensuremath{\text{{\sf PFA}}}}$ implies failure of square principles and that ${\ensuremath{\text{{\sf PFA}}}}$ implies the singular cardinal hypothesis which factors through the use of guessing models.
Acknowledgements
----------------
I thank Boban Veličković for many useful comments and discussions on the themes of research explored in this article and Alessandro Andretta for his valuable advices on how to improve the presentation of the material exposed in this paper. I thank also Peter Holy for some useful remarks on the proofs of Theorems \[theorem.GM->non\_square\] and \[the.GMSCH\]. Finally I thank the referees for many of their criticism and comments, they greatly helped me to prepare the final version of this work.
Notation {#subsect.notation}
--------
${\ensuremath{\text{{\rm Ord}}}}$ denotes the class of all ordinals and ${\ensuremath{\text{{\rm Card}}}}$ the class of all cardinals. The class of all limit points of $ X \subseteq {\ensuremath{\text{{\rm Ord}}}}$ is denoted by $ \operatorname{Lim}X $, and $ \operatorname{Lim}$ is $ \operatorname{Lim}({\ensuremath{\text{{\rm Ord}}}}) $. If $a$ is a set of ordinals, $\operatorname{otp}a$ denotes the order type of $a$. For a regular cardinal $\delta$, $\operatorname{cof}( \delta )$ denotes the class of all ordinals of cofinality $\delta$, and $\operatorname{cof}({<} \delta)$ denotes those of cofinality less than $\delta$. For any $X$, let $P_\delta ( X ) = \{ z \in P ( X ) : | z | < \delta \}$ and $[X]^{\delta}=\{z\in P(X) : \operatorname{otp}( z \cap {\ensuremath{\text{{\rm Ord}}}})= \delta \}$.
Given two families of sets $\mathcal{F}$ and $\mathcal{G}$, $\mathcal{F}$ is covered by $\mathcal{G}$ — equivalently: $\mathcal{G}$ is cofinal in $\mathcal{F}$ — if every $x\in\mathcal{F}$ is contained in some $y\in\mathcal{G}$. A family $\mathcal{F}\subseteq P(P(X))$ is a filter on $X$ if it is upward closed with respect to inclusion and closed under finite intersections, $\mathcal{F}$ is fine if $\{Z\in P(X): x\in Z\}\in\mathcal{F}$ for all $x\in X$. A set $S$ is positive with respect to $\mathcal{F}$ if $S\cap T$ is non-empty for all $T\in \mathcal{F}$. A filter $\mathcal{F}$ is normal if for all choice functions $f\colon P(X)\setminus \{\emptyset\}\to X$ there is $x\in X$ such that $\{Z\in P(X): f(Z)=x\}$ is positive with respect to $\mathcal{F}$. A filter $\mathcal{F}$ is $\kappa$-complete if it is closed with respect to intersections of size less than $\kappa$. Finally $\mathcal{F}$ is an ultrafilter if it is a maximal filter.
For $f\colon P_\omega X \to X$ we let ${\ensuremath{\text{{\rm Cl}}}}_f \coloneqq \{ x \in P(X) : f [P_\omega x] \subset x \}$. The club filter on $X$ is the normal and $\omega_1$-complete filter contained in $P(P(X))$ generated by the sets ${\ensuremath{\text{{\rm Cl}}}}_f$. A subset of $P(X)$ is a club if it is in the club filter, i.e., contains ${\ensuremath{\text{{\rm Cl}}}}_f$ for some $f\colon P_\omega X \to X$. $S\subseteq P(X)$ is stationary if it is positive with respect to the club filter.
If $X \subset X'$, $R \subset P(X)$, $U \subset P(X')$, then the projection of $U$ to $X$ is $U \restriction X \coloneqq \{ u \cap X : u \in U \} \subset P(X)$ and the lift of $R$ to $X'$ is $R^{X'} \coloneqq \{ x' \in P(X') : x' \cap X \in R \} \subset P(X')$.
Given $\mathcal{F}$ subset of $P(P(X))$ and $Y\subseteq X\subseteq Z$ we let $\mathcal{F}\restriction Y=\{S\restriction Y: S\in\mathcal{F}\}$ and $\mathcal{F}^Z=\{S^Z: S\in\mathcal{F} \}$ be the projection and the lift of $\mathcal{F}$. If $\mathcal{F}$ is a (normal) filter then $\mathcal{F}\restriction Y$ and $\mathcal{F}^Z$ are (normal) filters.
Given a structure $\mathfrak{R}=\langle R,\in, A_i : i\in I\rangle$ we shall say that $M\prec \mathfrak{R}$ if $M\subseteq R$ and $\langle M,\in, A_i\cap M: i\in I\rangle$ is an elementary substructure of $\mathfrak{R}$. Often we shall write $M\prec R$ instead of $M\prec \mathfrak{R}$.
Given $R$ well founded binary relation on $X$ we let $\pi_{R}$ be the collapsing map of the structure $\langle X, R\rangle$. We denote $\pi_{\in\restriction X^2}$ by $\pi_X$.
For forcings, we write $p < q$ to mean $p$ is stronger than $q$. Names either carry a dot above them or are canonical names for elements of $V$, so that we can confuse sets in the ground model with their names. Given a filter $G$ on $\mathbb{P}$, $\sigma_G(\dot{A})=\{\sigma_G(\dot{x}) : \exists p\in G ( p\Vdash \dot{x}\in\dot{A}) \}$ is the standard interpretation of $\mathbb{P}$-names given by $G$.
If $W$ is a transitive model of ${\ensuremath{\text{{\sf ZFC}}}}$, $ \alpha $ is an ordinal, and $\theta$ a cardinal in $W$, we let $H_\theta^W$ and $V_\alpha^W$ be the relativizations of $ H_ \theta $ and $ V_ \alpha $ to $ W $. We shall need for reference and motivation of our results the following definitions:
Let $V\subseteq W$ be a pair of transitive models of ${\ensuremath{\text{{\sf ZFC}}}}$.
- $(V,W)$ satisfies the $\mu$-covering property if for every $x \in W$ with $x \subset {\ensuremath{\text{{\rm Ord}}}}$ and $\operatorname{otp}(x) < \mu$ there is $z \in V$ such that $x \subset z$ and $\operatorname{otp}(z)<\mu$.
- $(V,W)$ satisfies the $\mu$-approximation property if for all $x \in W$, $x \subset {\ensuremath{\text{{\rm Ord}}}}$, it holds that $x \cap z \in V$ for all $z \in V$ such that $\operatorname{otp}(z)<\mu$, then $x \in V$.
A forcing $\mathbb{P}$ is said to satisfy the $\mu$-covering property or the $\mu$-approximation property if for every $V$-generic $G \subset \mathbb{P}$ the pair $(V, V[G])$ satisfies the $\mu$-covering property or the $\mu$-approximation property respectively.
Given a class of forcing notions $\Gamma$, ${\ensuremath{\text{{\sf FA}}}}(\Gamma)$ holds if for any poset $\mathbb{P}\in\Gamma$ and every family $\mathcal{D}$ of $\aleph_1$-many dense subsets of $\mathbb{P}$ there is a $\mathcal{D}$-generic filter $G\subseteq\mathbb{P}$, i.e a filter $G$ which has non-empty intersection with every element in $\mathcal{D}$.
The proof of [@woodin Theorem 2.53] yields the following reformulation of forcing axioms:
\[lem.MMWoo\] Given a class of forcing notions $\Gamma$, ${\ensuremath{\text{{\sf FA}}}}(\Gamma)$ holds if and only if for any poset $\mathbb{P}\in\Gamma$ and all sufficiently large regular $\theta$, there are stationarily many structures $M\prec H(\theta)$ of size $\aleph_1$ which have an $M$-generic filter $G$ for $\mathbb{P}$.
If $\Gamma$ is the family of ccc-posets, we shall denote ${\ensuremath{\text{{\sf FA}}}}(\Gamma)$ by ${\ensuremath{\text{{\sf MA}}}}$. If $\Gamma$ is the family of proper posets, we shall denote ${\ensuremath{\text{{\sf FA}}}}(\Gamma)$ by ${\ensuremath{\text{{\sf PFA}}}}$. If $\Gamma$ is the family of stationary set preserving posets ${\ensuremath{\text{{\sf FA}}}}(\Gamma)$ is Martin’s maximum ${\ensuremath{\text{{\sf MM}}}}$. We refer the reader to [@jech] for the definition of the relevant $\Gamma$’s. We recall however that any ccc partial order is proper and any proper partial order is stationary set preserving.
Finally, the Singular Cardinal Hypothesis ($ {\ensuremath{\text{{\sf SCH}}}}$) says that $ \kappa ^{\operatorname{cf}( \kappa )} = \kappa ^+ $ for all singular cardinals $ \kappa > 2^{ \aleph_0 }$.
Basic properties of guessing models {#sect.basicproperties}
===================================
The following are basic properties of guessing models
\[prop.basiguessing\] Let $\mathfrak{R}$ be a suitable initial segment and $M\prec R$.
1. \[prop.basiguessing-1\] $\kappa_M$ is a regular cardinal.
2. \[prop.basiguessing-2\] $M$ is a $0$-guessing model iff it is an $\aleph_0$-guessing model.
3. \[prop.basiguessing-3\] If $M$ is a $\delta$-guessing model, then it is also a $\gamma$-guessing model for all cardinal $\gamma\geq\delta$.
4. \[prop.basiguessing-4\] If $M$ is a $\delta$-guessing model and $2^{<\delta}<\kappa_M$, $M$ is an $\aleph_0$-guessing model.
5. \[prop.basiguessing-5\] If $M$ is an $\aleph_0$-guessing model, $\kappa_M$ and $M\cap\kappa_M$ are strongly inaccessible cardinals.
6. \[prop.basiguessing-6\] If $M$ is a $\delta$-guessing model and for some regular cardinal $\gamma\leq\delta$, $P_\gamma\xi\subseteq M$ for all $\xi<\delta\cap M$, then $M\cap{\ensuremath{\text{{\rm Ord}}}}$ is closed under suprema of sets of order type $ \leq\gamma$. In particular a guessing model $M$ is always closed under countable suprema since $P_{\aleph_0} M\subseteq M$.
\[prop.basiguessing-1\]: This is a standard property of elementary substructures, we enclude a proof just for the sake of completeness. Assume towards a contradiction $\operatorname{cf}(\kappa_M)<\kappa_M$, and fix $E\in M$ cofinal in $\kappa_M$ of order type $\delta<\kappa_M$. Then since $\delta\in M\cap\kappa_M$, we have that $\delta\subseteq M$ and thus $E\subseteq M$. Now either $\kappa_M\subseteq M$ which contradicts the very definition of $\kappa_M$ or $\kappa_M$ is not the least ordinal in $M$ such that $M\cap\kappa_M$ is bounded below $\kappa_M$ which again contradicts the very definition of $\kappa_M$.
\[prop.basiguessing-2\] and \[prop.basiguessing-3\] are immediate.
\[prop.basiguessing-4\] Observe that if $Z\in M$ and $|Z|<\delta$, $R\models |P(Z)|\leq 2^{<\delta}$. So there is a bijection $\phi\in M$ from some ordinal $\alpha<\kappa_M$ and $P(Z)$. Then $P(Z)=\phi[\alpha]\subseteq M$: this follows since $\alpha\subseteq M$ because $\alpha<\kappa_M$ and $\alpha=\operatorname{dom}(\phi)\in M$. Thus if $d\in P(X)$ for some $X\in M$ and $Z\in M$ is any set of size less than $\delta$, $d\cap Z\in P(Z)\subseteq M$. Thus any $d\in P(X)$ is $(\delta,M)$-approximated for all $X\in M$. Since $M$ is $\delta$-guessing, any $d\in P(X)$ is $M$-guessed for any $X\in M$. Thus $M$ is $\aleph_0$-guessing.
\[prop.basiguessing-5\] We first show that $\kappa\cap M$ is a regular cardinal in $R$. Assume not and pick $C\subseteq \kappa_M\cap M$ in $R$ of order type $\operatorname{cf}(\kappa\cap M)<\kappa\cap M$. Since $M$ is $\aleph_0$-guessing, $C=E\cap M$ for some $E\in M$. Now it is not hard to check that: $$M\models E\text{ is an unbounded subset of }\kappa_M \text{ of order type less than }\kappa_M.$$ For this reason there is a unique order preserving bijection $\phi\in M$ from some ordinal $\xi$ less than $\kappa_M$ into $E$. By elementarity $\xi\in M$. Since $\xi<\kappa_M$, $\xi\subseteq M$. Thus $E=\phi[\xi]\subseteq M$. Thus $C=E$ which implies that $\sup(\kappa_M\cap M)=\kappa_M$, contradicting the very definition of $\kappa_M$.
Now assume $2^{\delta}\geq\kappa_M\cap M$ for some $\delta<\kappa_M\cap M$. By elementarity, since $\delta\in M$, we get that $2^{\delta}\geq\kappa_M$. Now let $\phi\colon 2^\delta\rightarrow P(\delta)$ be a bijection in $M$. Let $X=\phi(\kappa_M\cap M)$. Then $X\subseteq \delta\subseteq M$. Since $M$ is $\aleph_0$-guessing, $X=Y\cap M$ for some $Y\in P(\delta)\cap M$, since $Y\subseteq\delta\subseteq M$, $X=Y$, thus $\kappa_M>\kappa_M\cap M=\phi^{-1}(Y)\in M$ which contradicts the very definition of $\kappa_M$. This proves that $\kappa_M\cap M$ is strongly inaccessible. Now by elementarity $M$ models that $\kappa_M$ is strong limit. Thus $\kappa_M$ is strong limit and regular in $R$ i.e. strongly inaccessible.
\[prop.basiguessing-6\] To simplify the exposition we prove the proposition for $\gamma=\aleph_0$ and $\delta=\aleph_1$. We leave to the reader the proof of the general case. So assume that some $\aleph_1$-guessing model $M$ is not closed under countable suprema. Now let $\xi\in M$ have uncountable cofinality be such that $\sup ( M \cap \xi ) \not \in M$ has countable cofinality. This means that $M\cap [ \sup ( M \cap \xi ) , \xi )$ is empty.
Fix in $R$, $d^*=\{\alpha_n : n \in \omega \}\subseteq M\cap\xi$ increasing and cofinal sequence converging to $\xi$. Then for any $d\in M\cap P_{\omega_1} \xi$, $d$ is a bounded subset of $\xi\cap M$, since $\sup(d)\in M$ and $\sup(M\cap\xi)\not\in M$. Thus $d^*$ is an $(\aleph_1,M)$-approximated subset of $M$, since $d^*\cap d$ is a finite subset of $M$ for any countable $d\in M$. Since $M$ is an $\aleph_1$-guessing model, $d^*=d^*\cap M=e\cap M$ for some $e\in M\cap P(\xi)$. Now $$M\models e\text{ is an unbounded subset of $\xi$,}$$ thus $\operatorname{otp}(e)\geq\operatorname{cf}(\xi)\geq\omega_1$. In particular $$\operatorname{otp}(e\cap M)\geq\operatorname{otp}(\omega_1\cap M)=\omega_1>\omega=\operatorname{otp}(d^*).$$ Thus $e\cap M\neq d^*$ which is the desired contradiction.
Propositions \[prop.basiguessing\]\[prop.basiguessing-5\] and \[prop.basiguessing\]\[prop.basiguessing-6\] are reformulation in terms of guessing models of Proposition 3.6 and Theorem 3.5 from [@weiss]. Note also that Proposition \[prop.basiguessing\]\[prop.basiguessing-1\] holds for any $M\prec R$ and not just for guessing models.
From part \[prop.basiguessing-3\] of Proposition \[prop.basiguessing\] we obtain at once
Assume $M\prec V_\theta$ is a $\delta$-guessing model which is not an $\aleph_0$-guessing model. Then $2^{<\delta}\geq\kappa_M$.
Thus existence of guessing models has effects on the exponential function. We shall see in Section \[sect.applications\] that the existence of an $\aleph_1$-internally unbounded (Definition \[def.intclos\]) $\aleph_1$-guessing model $M$ is an assumption strong enough to imply the ${\ensuremath{\text{{\sf SCH}}}}$ for all cardinals in $[ \kappa_M , \sup ( M \cap {\ensuremath{\text{{\rm Card}}}}) )$.
Large cardinals and $\aleph_0$-guessing models. {#sect.largecard}
===============================================
In this section we show that many large cardinal axioms present in the literature can be formulated in terms of the existence of appropriate $\aleph_0$-guessing models. Our aim is to show that $\aleph_0$-guessing models allows uniform and simple definitions of fine hierarchies of large cardinal notions above supercompactness which in many cases give an equivalent formulation of well known large cardinals assumptions. As a sample of the type of results we can aim for, consider the formula $ \phi(\kappa,\lambda,\gamma) $ $$\exists M\prec V_\lambda \left ( M \text{ is $\aleph_0$-guessing and $\kappa_M=\kappa$
and $\operatorname{otp}(M\cap\lambda)\geq\gamma$} \right ) .$$ We shall show the following:
- $\kappa$ is supercompact iff $\forall\lambda\geq\kappa\exists\gamma\phi(\kappa,\lambda,\gamma)$
- $I_3$ holds iff $\exists\lambda>\kappa\phi(\kappa,\lambda,\lambda)$,
- $ I_1$ holds iff $\exists\lambda>\kappa\phi(\kappa,\lambda+1,\lambda+1)$,
- if $\kappa$ is a huge cardinal, then $\exists\lambda>\kappa\phi(\kappa,\lambda,\kappa)$,
- if there exists $\lambda$ such that $\phi(\kappa,\lambda,\kappa+1)$, $\kappa$ is a limit of huge cardinals,
- if there exists $\lambda$ such that $\phi(\kappa,\lambda,\kappa+ 2)$, $\kappa$ is huge and limit of huge cardinals,
- and so on …
Using statements slightly more involved than $\phi(\kappa,\lambda,\gamma)$ one might approximate many other large cardinals, for example $n$-huge cardinals.
Supercompactness
----------------
Recall that a cardinal $\kappa$ is supercompact if for all $\lambda\geq\kappa$ there is a fine, normal and $\kappa$-complete ultrafilter on $P_\kappa\lambda$.
Magidor ([@MAG71 Lemma 2, Lemma 3] or [@magidor.characterization_supercompact Theorem 1]) has characterized supercompactness as follows:
\[thm.magspct\] $\kappa$ is supercompact iff for every $\lambda\geq\kappa$ there is a non trivial elementary embedding $j\colon V_\gamma\rightarrow V_\lambda$ with $j( \operatorname{crit}(j))=\kappa$.
The main Lemma of section 2 in [@magidor.characterization_supercompact] can be rephrased in our setting as:
\[lem.isotype0guesmod\] $M\prec V_\lambda$ is an $\aleph_0$-guessing model if and only if its transitive collapse is $V_\gamma$ for some $ \gamma $.
We prove just one direction, the other one is proved by a similar argument. Recall that $M\prec V_\lambda$ is an $\aleph_0$-guessing model iff it is a $0$-guessing model. Now assume $M\prec V_\lambda$ is a $0$-guessing model. We proceed by induction on $\beta\in M\cap\lambda$ to show that $M\cap V_\beta$ collapses to some $V_{\gamma_\beta}$ via $\pi_M\restriction V_\beta$. This is clear if $\beta$ is a limit ordinal since $$\pi_M[V_\beta]=\bigcup_{\alpha<\beta}\pi_M[
V_\alpha] = \bigcup_{\alpha<\beta} V_{\gamma_\alpha}=V_{\gamma_\beta}.$$ If $\beta=\alpha+1$ then $$V_{\gamma_\beta}=P(V_{\gamma_\alpha})=P(\pi_M[V_\alpha]).$$ Thus for every $Y\in V_{\gamma_\beta}$, $Y=\pi_M [ X_Y ]$ for some $X_Y\in P ( M \cap V_\alpha )$. Now $M$ is a $0$-guessing model, $V_\alpha\in M$ and every $X\in P(V_\alpha\cap M)$ is $0$-approximated. Thus we have that for every $Y$, $X_Y$ is $M$-guessed i.e. $X_Y=M\cap E_Y$ for some $E_Y\in M$. Clearly such an $E_Y\in V_{\alpha+1}$. Therefore $$V_{\gamma_\beta}=\{\pi_M[E_Y]: E_Y\in V_\beta\cap M\}=\pi_M [ V_\beta ],$$ which is what we had to prove.
Note that if $M\prec V_\lambda$ and $\pi_M[M]=V_\gamma$ then $j=\pi_M^{-1}$ is an elementary embedding of $V_\gamma$ into $V_\lambda$. Thus Magidor’s Theorem \[thm.magspct\] can be reformulated as follows:
$\kappa$ is supercompact iff for every $\lambda\geq\kappa$ there is an $\aleph_0$-guessing model $M\prec V_\lambda$ with $\kappa_M = \kappa$, i.e. $\forall \lambda \geq \kappa \exists \gamma \phi ( \kappa , \lambda ,\gamma ) $.
Rank initial segment embeddings
-------------------------------
The following is an immediate consequence of Magidor’s observations:
$j\colon V_{\lambda+1}\rightarrow V_{\lambda+1}$ is elementary iff $j[V_{\lambda+1}]=M\prec V_{\lambda+1}$ is an $\aleph_0$-guessing model.
Thus the existence of an $\aleph_0$-guessing model $M\prec V_{\lambda+1}$ such that $\operatorname{otp}(M\cap\lambda)=\lambda$ is an equivalent formulation of the axiom $I_1$. In the same manner one can formulate $I_3$.
Hugeness
--------
Recall that a cardinal $\kappa$ is $n$-huge in $V$ if there is $j\colon V\rightarrow M$ such that $\operatorname{crit}(j)=\kappa$, $M^{j^n(\kappa)}\subseteq M$ and $M\subseteq V$. An equivalent formulation is given by the following result [@Kan09 Theorem 22.8]
\[thm.nhuge\] $\kappa$ is $n$-huge if and only if for some $\delta_n>\cdots>\delta_0=\kappa$ there is a normal fine ultrafilter on $\{X\subseteq \delta_n:\forall 0<i\leq n \operatorname{otp}(X\cap\delta_i)=\delta_{i-1}\}$.
We can use guessing models to get a tight approximation of $n$-hugeness: we show how to pin hugeness in terms of the hierarchy defined using the formula $\phi$.
\[lem.hugeGM\] The following holds:
1. \[lem.hugeGM-1\] If $\kappa$ is huge then $\exists\lambda\phi(\kappa,\lambda,\kappa)$.
2. \[lem.hugeGM-2\] If there exists $\lambda$ such that $\phi(\kappa,\lambda,\kappa+1)$, then $\kappa$ is a limit of huge cardinals.
3. \[lem.hugeGM-3\] If there exists $\lambda$ such that $\phi(\kappa,\lambda,\kappa+ 2)$, then $\kappa$ is huge and limit of huge cardinals.
\[lem.hugeGM-1\] Assume $\kappa$ is huge. Let $j\colon V\rightarrow M$ witness it. Then $j[V_{j(\kappa)}]\in M$ and $M$ models that $j [ V_{j ( \kappa ) } ] \prec j ( V_{ j ( \kappa ) } ) = ( V_{j^2 ( \kappa ) } )^M$. Since $V_{j(\kappa)}=(V_{j(\kappa)})^M$ is the transitive collapse of $j[V_{j(\kappa)}]$ we get that, in $M$, $j[V_{j(\kappa)}]$ is an $\aleph_0$-guessing model and $\operatorname{otp}(j[V_{j(\kappa)}]\cap j^2(\kappa))=j(\kappa)$. In particular $M$ models that there is an $\aleph_0$-guessing model $N\prec (V_{j^2(\kappa})^M$ such that $\operatorname{otp}(N\cap j^2(\kappa))=j(\kappa)$. By pulling down, we get that $V$ models that there is an $\aleph_0$-guessing model $N \prec V_{j ( \kappa )}$ such that $\operatorname{otp}( N \cap j( \kappa ) ) = \kappa$, i.e $\phi ( \kappa , j ( \kappa ) , \kappa )$ holds.
\[lem.hugeGM-2\] Let $M$ be a witness of $\phi(\kappa,\lambda,\kappa+1)$, i.e.:
- $M\prec V_\lambda$ is an $\aleph_0$-guessing model,
- $\kappa_M=\kappa$,
- $\operatorname{otp}(M\cap\lambda)=\gamma\geq\kappa+1$.
Let $\delta<\lambda$ in $M$ be such that $\operatorname{otp}(M\cap\delta)=\kappa_M$ and $\bar{\kappa}=\kappa\cap M$.
We first show that $\bar{\kappa}$ is huge: Let $j=\pi_M^{-1}$. Then $j\colon V_\gamma\rightarrow V_{\lambda}$ is elementary and belongs to $V$, $j(\bar{\kappa})=\kappa$ and $j(\kappa)=\delta$. Now, since $\kappa=\kappa_M$, $\kappa$ is regular by Proposition \[prop.basiguessing\]\[prop.basiguessing-1\] and, by elementarity of $j$, we also get that $\delta=j(\kappa)$ is regular. For this reason $$A=\{X\subseteq\kappa: \operatorname{otp}(X)=\bar{\kappa}\}\subseteq V_\kappa$$ and $$B=\{X\subseteq\delta: \operatorname{otp}(X)=\kappa\} \subseteq V_\delta.$$ Now observe that $\operatorname{otp}(M\cap\lambda)=\gamma\geq\kappa+1$, thus $A\in V_{\kappa+1}\subseteq V_\gamma$. Thus $j(A)=B\in M$ and $M\cap \delta\in j(A)$.
Now we can define in $V$ a normal fine ultrafilter $\mathcal{U}\subseteq V_{\kappa+1}\subseteq V_\gamma=\operatorname{dom}( j )$ by: $$E\in\mathcal{U} \iff M\cap \delta\in j(E).$$ Then $\mathcal{U}\subseteq V_{\kappa+1}$ concentrates on $A$ and thus, by the above theorem, witnesses that $\bar{\kappa}$ is huge in $V$.
Finally we show that $\kappa$ is a limit of huge cardinals: Since $\bar{\kappa}=M\cap\kappa$ is huge, we get that: $$\forall \alpha\in M\cap\kappa \left ( V_\lambda \models \exists
\bar{\kappa} ( \alpha < \bar{\kappa} < \kappa \wedge \bar{\kappa}\text{ is huge})\right ).$$ Since $M\prec V_\lambda$: $$\forall \alpha\in M\cap\kappa \left ( M \models
\exists \bar{\kappa} ( \alpha < \bar{\kappa} < \kappa \wedge \bar{\kappa}\text{ is huge}) \right )$$ Thus: $$M \models \forall \alpha<\kappa \exists \bar{\kappa}
\left ( \alpha < \bar{ \kappa } < \kappa \wedge \bar\kappa \text{ is huge}\right )$$ Since $M\prec V_\lambda$, we have that $V_\lambda$ models this assertion and we are done.
\[lem.hugeGM-3\] The proof is exactly as in \[lem.hugeGM-2\], just observe that $\gamma\geq\kappa+2$ and $\mathcal{U}\subseteq V_{\kappa+1}$, thus $\mathcal{U}\in V_\gamma$. Then $j(\mathcal{U})$ is defined and in $V_\lambda$ witnesses that $\kappa$ is huge.
Part \[lem.hugeGM-2\] of Lemma \[lem.hugeGM\] is not optimal. For example with some more work we could show that $\phi(\kappa,\lambda,\kappa+1)$ implies that there is a normal measure on $\kappa$ which concentrates on huge cardinals.
To pin $n$-hugeness using guessing models we need a refinement of $\phi$. Let $\psi_n(\kappa,\lambda,\vec{\delta})$ asserts the existence of an $\aleph_0$-guessing model $M\prec V_\lambda$ such that if $\vec{\delta}=\{\delta_0<\cdots<\delta_n\}$ then:
- $\delta_0=\kappa_M$,
- $\delta_n\leq\lambda$,
- $\delta_i\in M$ and $\operatorname{otp}(M\cap\delta_{i+1})=\delta_i$ for all $i<n$.
Then we can prove:
The following holds:
1. If $\kappa$ is $n$-huge then $\exists\vec{\delta}
\psi_n (\kappa,\max\vec{\delta},\vec{\delta})$.
2. If there exists $\vec{\delta}$ such that $\psi_n(\kappa,\max\vec{\delta}+1,
\vec{\delta})$, then $\kappa$ is a limit of $n$-huge cardinals.
3. If there exists $\vec{\delta}$ such that $\psi_n(\kappa,\max\vec{\delta}+2,
\vec{\delta})$, then $\kappa$ is $n$-huge and is a limit of $n$-huge cardinals.
We leave the proof of the lemma to the interested reader.
Internal closure of guessing models {#sect.internalclosure}
===================================
In this and in the next section, we come back to an analysis of the properties of guessing models and we also address some consistency issues regarding their existence.
If $M\prec V_\lambda$ is an $\aleph_0$-guessing model, $\kappa_M$ is inaccessible and $P_{\gamma}M\subseteq M$ for all $\gamma\in M\cap\kappa_M$. Such a degree of closure cannot be achieved for $\aleph_1$-guessing models which are not $\aleph_0$-guessing, however we can prove that such models have a reasonable degree of closure in most cases. To this aim we need to recall the following definitions:
\[def.intclos\] Let $\mathfrak{R}$ be a suitable initial segment. For a model $M\prec R$ and a cardinal $\delta \leq \kappa _M$, we say that:
- $ M $ is $\delta$-internally unbounded if $M \cap P_{\delta}( M ) $ is cofinal in $P_{\delta}( M )$,
- $ M $ is $\delta$-internally club if $M\cap P_{\delta} M$ is a club subset of $P_{\delta} M$,
- $ M $ is $\delta$-internally stationary if $M\cap P_{\delta} M$ is a stationary subset of $P_{\delta} M$.
We let $\mathrm{IC}^\delta (\mathfrak{R})$ be the set of $M\prec R$ which are $\delta$-internally club, $\mathrm{IS}^\delta ( \mathfrak{R})$ be the set of $M\prec R$ which are $\delta$-internally stationary and $\mathrm{IU}^\delta ( \mathfrak{R} )$ be the set of $M\prec R$ which are $\delta$-internally unbounded.
Recall that the pseudo-intersection number $\mathfrak{p}$ is the minimal size of a family $X\subseteq P(\omega)$ which is closed under finite intersections and for which there is no infinite $a\subseteq\omega$ such that $a\subseteq^* b$ (i.e. $a\setminus b$ is finite) for all $b\in X$. We show the following:
\[lem.pgeqkappaIUGM\] Assume $M\prec R$ for a suitable initial segment $\mathfrak{R}$ is an $\aleph_1$-guessing model such that $\mathfrak{p}>|M|$. Then $M$ is in $\mathrm {IU}^{\aleph_1} R$.
Assume not and pick $M\prec R$ guessing model witnessing it. The family $\{x\setminus z: z\in M\cap P_{\omega_1} M\}$ has the finite intersection property and has size at most $|M|<\mathfrak{p}$. Thus there is $y\subseteq x$ such that $y\cap z$ is finite for all countable $z\in M$. Thus $y$ is $M$-approximated. Let $d\in M$ be such that $d\cap M=y$. Then $d$ is countable, else, since $d\in M$ and $\omega_1\subseteq M$, $d\cap M$ is uncountable and thus different from $y$. This means that $d=d\cap M=y$. This is impossible since $d\cap y$ is finite by choice of $y$.
With more work we could first prove the same conclusion of the above lemma replacing the assumption “$|M|<\mathfrak{p}$” with “${\ensuremath{\text{{\sf SCH}}}}$ holds and $\kappa_M\leq\mathfrak{p}$”. By Theorem \[the.GMSCH\], ${\ensuremath{\text{{\sf SCH}}}}$ would be redundant if the set of $\aleph_1$-guessing model which are internally unbounded is stationary, so we could reformulate the Lemma as follows:
“Assume $\kappa_M\leq\mathfrak{p}$ and there are stationarily many $\aleph_1$-guessing model $M\prec H_\theta$ which are $\aleph_1$-internally unbounded. Then any guessing $M\prec H_\theta$ is $\aleph_1$-internally unbounded.”
It is open whether there can be an $\aleph_1$-guessing model $M$ in a universe of sets where $\mathfrak{p}<\kappa_M$ and also if there can be an $\aleph_1$-guessing model $M$ which is not $\aleph_1$-internally unbounded.
The following theorem will be used in Sections \[sect.isotypesGM\] and \[sect.laverdiamond\].
\[the.PFAIUGM\] Assume ${\ensuremath{\text{{\sf MM}}}}$. Then for evey regular $\theta\geq\aleph_2$ the following sets are stationary:
1. \[the.PFAIUGM-1\] the set of $\aleph_1$-guessing models $M\prec H_\theta$ of size $\aleph_1$ which are $\aleph_1$-internally club (for this result ${\ensuremath{\text{{\sf PFA}}}}$ suffices),
2. \[the.PFAIUGM-2\] the set of $\aleph_1$-guessing models $M\prec H_\theta$ of size $\aleph_1$ which are $\aleph_1$-internally unbounded but not $\aleph_1$-internally stationary,
3. \[the.PFAIUGM-3\] the set of $\aleph_1$-guessing models $M\prec H_\theta$ of size $\aleph_1$ which are $\aleph_1$-internally stationary but not $\aleph_1$-internally club.
We shall just sketch its proof since it is a straightforward consequence of the combination of results of [@VIAWEI10] with a series of results appearing in Krueger’s papers [@krueger.IA] and [@krueger.internally_club]. The interested reader will find all the missing details in the relevant papers. We shall in any case give at any stage of the proof a careful reference to the parts of these papers where the key arguments are presented.
We begin stating the following result [@VIAWEI10 Lemma 4.6]:
\[lem.ICGM\] Assume $\lambda\geq\aleph_2$ is regular and $\mathbb{P}$ is a poset with the $\omega_1$-approximation and $\omega_1$-covering properties which collapses $2^\lambda$ to $\aleph_1$. Then there is in $V^{\mathbb{P}}$ a ccc-poset $\dot{\mathbb{Q}}_{\mathbb{P}}$ with the following property:
> If there is an $M$-generic filter for $\mathbb{P}*\dot{\mathbb{Q}}_{\mathbb{P}}$, where $ M \in V $ and $M\prec (H_\theta)^V$, then $M\cap H_\lambda\prec H_\lambda$ is an $\aleph_1$-guessing model.
Krueger in [@krueger.IA] and [@krueger.internally_club] essentially showed the following:
\[prop.ICGM\] There are posets $\mathbb{P}_i=\mathbb{C}*\dot{\mathbb{R}}_i$ for $i<3$ satisfying the hypothesis of Lemma \[lem.ICGM\] such that:
1. \[prop.ICGM-1\] any model $M\prec H_\theta$ in $V$ of size $\aleph_1$ which has a $\mathbb{P}_0*\dot{\mathbb{Q}}_{\mathbb{P}_0}$-generic filter, is such that $M\cap H_\lambda$ is $\aleph_1$-internally club
2. \[prop.ICGM-2\] any model $M\prec H_\theta$ in $V$ of size $\aleph_1$ which has a $\mathbb{P}_1*\dot{\mathbb{Q}}_{\mathbb{P}_1}$-generic filter, is such that $M\cap H_\lambda$ is $\aleph_1$-internally unbounded but not $\aleph_1$-internally stationary,
3. \[prop.ICGM-3\] any model $M\prec H_\theta$ in $V$ of size $\aleph_1$ which have a $\mathbb{P}_2*\dot{\mathbb{Q}}_{\mathbb{P}_2}$-generic filter is such that $M\cap H_\lambda$ is $\aleph_1$-internally stationary but not $\aleph_1$-internally club.
Assume Theorem \[prop.ICGM\] is granted. Then we can use the formulation of ${\ensuremath{\text{{\sf MM}}}}$ given by Lemma \[lem.MMWoo\] to get models of size $\aleph_1$ which have generic filter for each $\mathbb{P}_i*\dot{\mathbb{Q}}_{\mathbb{P}_i}$. The combination of Lemma \[lem.ICGM\] with Theorem \[prop.ICGM\] will then prove Theorem \[the.PFAIUGM\].
We sketch a proof of Theorem \[prop.ICGM\].
Define each $\mathbb{P}_i=\mathbb{C}*\dot{\mathbb{R}}_i$ as a two steps iteration, where $\mathbb{C}$ is Cohen forcing and $\dot{\mathbb{R}_i}\in V^{\mathbb{C}}$ are $\mathbb{C}$-names for posets. To define $\dot{\mathbb{R}_i}$, fix $G$ a $V$-generic filter for $\mathbb{C}$, let $X=(H_\lambda)^V$, fix a partition of $\omega_1$ in two disjoint stationary sets $E_0,E_1\in V$. $\mathbb{R}_i=\sigma_G(\dot{\mathbb{R}_i})$ are the following posets in $V[G]$:
- $\mathbb{R}_0$ is the poset of continuous maps $f\colon \alpha+1\to (P_{\omega_1}X)^{V}$ with $\alpha$ a countable ordinal.
- $\mathbb{R}_1$ is the poset of continuous maps $f\colon \alpha+1\to (P_{\omega_1}X)^{V[G]}$ where $\alpha$ is a countable ordinal and $f(\xi)\in V$ iff $\xi\in E_0$ for all $\xi\leq\alpha$.
- $\mathbb{R}_2$ is the poset of continuous maps $f\colon \alpha+1\to (P_{\omega_1}X)^{V[G]}\setminus V$ with $\alpha$ a countable ordinal.
The order of each $R_i$ is end extension.
In [@krueger.IA] and [@krueger.internally_club] it is essentially shown:
$\mathbb{P}_i$ is stationary set preserving and has the $\omega_1$-covering and $\omega_1$-approximation properties for all $i<3$.
It is a standard argument that each $\mathbb{P}_i$ is stationary set preserving: see [@krueger.IA Proposition 3.7] for a proof that $\mathbb{R}_0$ and $\mathbb{R}_2$ are semiproper in $V[G]$, modify that proof to check that also $\mathbb{R}_1$ is semiproper in $V[G]$. The conclusion for each $\mathbb{P}_i$ follows from the fact that each $\mathbb{P}_i$ is a two step iteration of semiproper posets.
[@krueger.internally_club Proposition 2.4] proves the $\omega_1$-approximation property for the poset $\mathbb{P}_0$. The interested reader can supply the modifications needed to prove the same proposition for $\mathbb{P}_1$, $\mathbb{P}_2$.
To check the $\omega_1$-covering property for each $\mathbb{P}_i$ we use that each $\mathbb{R}_i$ is $\aleph_1$-distributive in $V[G]$ [@krueger.internally_club Lemma 2.2]. Thus all new countable sets of ordinals added by any $\mathbb{P}_i$ are appearing already in $V[G]$ where $G$ is a $V$-generic filter for $\mathbb{C}$. Since $\mathbb{C}$ satisfies the $\omega_1$-covering property, this shows that the $\omega_1$-covering property holds for each $\mathbb{P}_i$.
Still following Krueger’s cited papers we can show the following:
- any model $M\prec H_\theta$ in $V$ of size $\aleph_1$ which has a $\mathbb{P}_0*\dot{\mathbb{Q}}_{\mathbb{P}_0}$-generic filter, is such that $M\cap H_\lambda$ is $\aleph_1$-internally club (see the discussion on pages 5-6 of [@krueger.internally_club]),
- any model $M\prec H_\theta$ in $V$ of size $\aleph_1$ which has a $\mathbb{P}_1*\dot{\mathbb{Q}}_{\mathbb{P}_1}$-generic filter, is such that $M\cap H_\lambda$ is $\aleph_1$-internally unbounded but not $\aleph_1$-internally stationary (adapt with obvious modifications the proof of [@krueger.IA Theorem 3.9]),
- any model $M\prec H_\theta$ in $V$ of size $\aleph_1$ which have a $\mathbb{P}_2*\dot{\mathbb{Q}}_{\mathbb{P}_2}$-generic filter is such that $M\cap H_\lambda$ is $\aleph_1$-internally stationary but not $\aleph_1$-internally club (see the proof of [@krueger.IA Theorem 3.6] and modify it according to the definition of $\mathbb{P}_1$).
Actually $\mathbb{P}_1$ and $\mathbb{P}_2$ are just semiproper, while $\mathbb{P}_0$ is proper [@krueger.internally_club Proposition 2.3]. This completes the proof of the theorem.
Isomorphism types of guessing models {#sect.isotypesGM}
====================================
In this section we will show that for $ \delta $-guessing models $M$ which are $ \delta $-internally club, the isomorphism type is uniquely determined by the ordinal $M\cap\kappa_M$ and the order-type of the set of cardinals in $M$. In the case of $\aleph_0$-guessing models this is Magidor’s result that any $\aleph_0$-guessing model $M\prec V_\lambda$ is isomorphic to some $V_\gamma$, however when we want to extend this result to $\delta$-guessing models we must put some extra condition to constrain the variety of possible isomorphism types.
Given a set $M$, let ${\ensuremath{\text{{\rm Card}}}}_M$ be the set of cardinals in $M$ and $$\chi_M\colon {\ensuremath{\text{{\rm Card}}}}_M\rightarrow\sup ( M \cap {\ensuremath{\text{{\rm Ord}}}}) , \quad
\lambda \mapsto\sup(M\cap \lambda ) .$$ The next result generalizes Magidor’s Lemma \[lem.isotype0guesmod\] on the isomorphism type of $\aleph_0$-guessing models.
\[thm.isotypealeph1guesmod\] Let $\mathfrak{R}_i=\langle H_{\theta_i},\in\rangle$ with $\theta_i$ regular cardinals for $i=0,1$. Assume $M_i\prec H_{\theta_i}$ are $\delta$-guessing models which are $\delta$-internally club and:
1. \[thm.isotypealeph1guesmod-1\] $\kappa_{M_0}=\kappa_{M_1}=\kappa$,
2. \[thm.isotypealeph1guesmod-2\] $M_0\cap \kappa=M_1\cap \kappa$,
3. \[thm.isotypealeph1guesmod-3\] $P_\delta(\delta)\cap M_0=P_\delta(\delta)\cap M_1$
4. \[thm.isotypealeph1guesmod-4\] $\operatorname{otp}({\ensuremath{\text{{\rm Card}}}}_{M_0})=\operatorname{otp}({\ensuremath{\text{{\rm Card}}}}_{M_1})$.
Then $M_0$ and $M_1$ are isomorphic.
If $\delta^{<\delta}<\kappa$, then both $M_i$ are $\aleph_0$-guessing models by part \[prop.basiguessing-4\] of Proposition \[prop.basiguessing\] hence Magidor’s Lemma \[lem.isotype0guesmod\] applies. So for the rest of the proof we can assume $\kappa \leq\delta^{<\delta} $. We will show that $$\langle M_0\cap\theta_0,P(\theta_0)\cap M_0,\in\rangle$$ is isomorphic to $$\langle M_1\cap\theta_1,P(\theta_1)\cap M_1,\in\rangle.$$ This suffices by the following:
Assume $M_0\prec H_{\theta_0}$ and $M_1\prec H_{\theta_1}$ are such that $\langle M_i\cap\theta_i,M_i\cap P(\theta_i),\in\rangle$ are isomorphic structures. Then also $\langle M_i,\in\rangle$ are isomorphic structures for $i=0,1$.
Notice that the isomorphism $\pi$ of $\langle M_0\cap\theta_0,P(\theta_0)\cap M_0,\in\rangle$ with $\langle M_1\cap\theta_1,P(\theta_1)\cap M_1, \in\rangle$ is unique and factors through $\pi_{M_1}^{-1}\circ\pi_{M_0}$ where $\pi_{M_i}$ are the collapsing maps of $M_i$. In particular $\pi$ is uniquely determined by the order isomorphism on $M_i\cap\theta_i$ and maps $A\in M_0\cap P(\theta_0)$ to the unique $B\in M_1\cap P(\theta_1)$ such that $B\cap M_1=\pi[A]$.
For each $x\in M_0$ we want to find $z_x\in M_1$ so that the map $\pi^*( x ) \coloneqq z_x$ is an isomorphism of $\langle M_0,\in\rangle$ onto $\langle M_1,\in\rangle$ extending $\pi$. So, given $x\in M_0$, Pick $A_x\subseteq |\operatorname{trcl}(x) | = \lambda_x$ with $A_x,\lambda_x\in M_i$ such that $A_x$ codes (modulo the map $\phi_*\colon {\ensuremath{\text{{\rm Ord}}}}^2\to{\ensuremath{\text{{\rm Ord}}}}$ which linearly orders pairs of ordinals according to the square order on ${\ensuremath{\text{{\rm Ord}}}}^2$) an extensional and well founded relation $R_x\in M_i$ on $\lambda_x$ such that the transitive collapse of $\langle\lambda_x, R_x\rangle$ is $\langle \operatorname{trcl}(x),\in\rangle$. Let $\gamma=\pi(\lambda_X)$ and $B_x=\pi(A_x)$. Observe that $M_1$ models that $\phi_*^{-1}[B_x]$ is an extensional and well founded binary relation on $\gamma$ since $\langle\gamma\cap M_1,\phi_*^{-1}[B_x] \cap M_1\rangle$ is isomorphic to $\langle\lambda_x\cap M_0, R_x\cap M_0\rangle$ and the latter is a well-founded extensional relation. Let $y\in M_1$ be the transitive collapse of $\gamma$ with respect to the binary relation $\phi_*^{-1}[B_x]$ and $z_x\in M_1$ be the set of elements in $y$ of highest rank. We leave to the reader to check that the map $x\mapsto z_x$ is an isomorphism of $\langle M_0,\in\rangle$ onto $\langle M_1,\in\rangle$ extending $\pi$.
The proof now goes by induction on $ \mu \coloneqq \operatorname{otp}({\ensuremath{\text{{\rm Card}}}}_{M_i})\setminus\kappa_M$. Let $\{\alpha_i^\eta:\eta<\mu\}={\ensuremath{\text{{\rm Card}}}}_{M_i}\setminus\kappa$. We show by induction on $\eta<\mu$ that we can define unique isomorphisms $$\label{eq.iso}
\pi_\eta\colon\langle M_0\cap\alpha_0^\eta,P(\alpha_0^\eta)\cap M_0,\in\rangle\to
\langle M_1\cap\alpha_1^\eta,P(\alpha_1^\eta)\cap M_1,\in\rangle.$$ The map $\pi_\eta$ can be defined only if the following conditions are satisifed:
1. \[req.1\] $\operatorname{otp}(\alpha_0^\eta\cap M_0)=\operatorname{otp}(\alpha_1^\eta\cap M_1)$,
2. \[req.2\] $\pi_\eta[a]=\pi_\eta(a)$ for all sets of ordinals $a\in M_0\cap P_\delta(\alpha_0^\eta)$,
3. \[req.3\] $\pi_\eta(A)\cap M_1=\pi_\eta[A\cap M_0]$ for all sets of ordinals $A\in M_0\cap P(\alpha_0^\eta)$.
We have that \[req.3\] implies \[req.2\] which implies \[req.1\].
We shall proceed by induction to define a coherent sequence $$\{\pi_\eta : \eta < \operatorname{otp}(M_i\cap{\ensuremath{\text{{\rm Card}}}}\setminus\kappa)\}$$ of isomorphisms of the structures $\langle M_i\cap\alpha_i^\eta,P(\alpha_i^\eta)\cap M_i, \in \rangle$. Given $\{\pi_\eta:\eta<\gamma)\}$ in order to define $\pi_\gamma$ we shall check step by step that \[req.1\], \[req.2\], and \[req.3\] are satisfied by the unique possible extension of $\cup_{\eta<\gamma}\pi_\eta=\pi^*$ to an isomorphism of the structures $$\langle M_i\cap\alpha_0^\eta , P(\alpha_i^\eta) \cap M_i, \in \rangle.$$ We will need that the models $M_i$ are internally club to check that condition \[req.2\] is satisfied by $\pi^*$ and the guessing property of the models $M_i$ to check that $\pi^*$ can be extended to a $\pi_\gamma$ which satisfies condition \[req.3\].
The induction will split in three cases:
Case 0: $\alpha_i^0=\kappa=\kappa_{M_i}$.
: \[cas.0\]
Clearly the identity map defines an isomorphism of the ordinal $M_i\cap\kappa$ with itself. By our assumptions, $M_i\cap P_\delta\kappa$ are club subsets $C_i$ of $P_\delta(M_i\cap\kappa)$. Let $C=C_0\cap C_1$. Then $C\subseteq M_0\cap M_1$ and $Id:\kappa\cap M_0\to\kappa\cap M_i$ can be extended to $\pi^*: C\cup(\kappa\cap M_0)\to C\cup(\kappa\cap M_1)$ by letting $\pi^*(c)=\pi[c]$ for all $c\in C$. Now pick $d\in C$, since the collapsing map $\pi_d\in M_0\cap M_1$ and $\operatorname{otp}(d)<\delta$, if $e\subseteq d$: $$\begin{aligned}
\label{keycl.thm.isotypeGM}
e\in M_0 & \iff e\in P(d)\cap M_0
\\
& \iff \pi_d [ e ] \in M_0 \cap P ( \operatorname{otp}( d ) ) = M_1 \cap P( \operatorname{otp}( d ) )
\\
& \iff e\in P(d)\cap M_1\end{aligned}$$ where $\pi_d\in M_0\cap M_1$ is the transitive collapse of $d$ to its order type.
Since $C$ is cofinal in $P_\delta(M_i\cap\kappa)$ we have that $M_0\cap P_\delta(\kappa)=M_1\cap P_\delta(\kappa)$. Thus $\pi^*(d)=d$ is an isomorphism of the structures $$\langle M_i\cap\kappa,P_\delta(\kappa)\cap M_i,\in\rangle.$$
We extend $\pi^*$ to an isomorphism of the structures $\langle M_i\cap\kappa,P(\kappa)\cap M_i,\in\rangle$ using the guessing property of each $M_i$ as follows: $$\begin{aligned}
d\in M_0\cap P(\kappa) & \iff
d\cap M_0 \text{ is $(\delta,M_0)$-approximated}
\\
& \iff \text{$d\cap M_1$ is $(\delta,M_1)$-approximated}
\\
& \iff \text{$d\cap M_1=e(d)\cap M_1$ for some $e(d)\in M_1\cap P(\kappa)$.}\end{aligned}$$ The mapping $\pi_0$ which is the identity on $M_0\cap\kappa$ and sends $d\mapsto e(d)$ is an isomorphism of $\langle M_0\cap\kappa,P(\kappa)\cap M_0,\in\rangle$ with $\langle M_1\cap\kappa,P(\kappa)\cap M_1,\in\rangle$.
Now assume the induction has been carried up to some ordinal $\eta<\xi$ by defining a sequence of coherent and unique isomorphisms $$\pi_\beta\colon\langle M_0\cap\alpha_0^\beta,P(\alpha_0^\beta)\cap M_0,\in\rangle\to
\langle M_1\cap\alpha_1^\beta,P(\alpha_1^\beta)\cap M_1,\in\rangle$$ for all $\beta<\eta$.
Case 1:
: $\alpha_0^\eta$ is a limit cardinal.
First of all observe that $$\pi^*=\cup_{\xi<\eta}\pi_\xi\restriction M_0\cap\alpha_0^\xi$$ is the order isomorphism between $M_0\cap\alpha_0^\eta$ and $M_1\cap\alpha_1^\eta$.
Our aim is to show
\[keycl.thm.isotypeGM2\] $\pi^*[e]\in M_1\cap P_{\omega_1} (\alpha_1^\eta)$ iff $e\in M_0\cap P_{\omega_1} (\alpha_0^\eta)$.
First we choose clubs $C^*_i\subseteq M_i\cap P_\delta(\alpha_i^\eta)$. Next we observe that if $\xi=\operatorname{otp}(M_i\cap\alpha_i^\eta)$, $\pi_{M_i}[C^*_i]=C_i'$ are club subsets of $P_\delta\xi$. We let $C=C_0'\cap C_1'$ and $C_i$ be the club subsets of $M_i\cap P_\delta (M_i\cap\alpha_i^\eta)$ which collapse to $C$.
Then:
1. \[subcas.cas2-1\] $\pi^*[d]\in C_1$ iff $d\in C_0$ for all $d\in P_\delta (M_0\cap\alpha_i^\eta)$,
2. \[subcas.cas2-2\] $C_i$ are cofinal subsets of $P_\delta (M_i\cap\alpha_i^\eta)$,
3. \[subcas.cas2-3\] $\pi^*[e]=\pi_{\pi^*(d)}^{-1}\circ\pi_d[e]$ where $\pi_d\in M_0$ and $\pi_{\pi^*(d)}\in M_1$ are the maps which collapse $d$ and $\pi^*(d)$ to their common order type.
Now the Claim is easily proved as follows: given $e\in M_0$, by \[subcas.cas2-2\] we can find $d\in C_0$ such that $e\subseteq d$, by \[subcas.cas2-1\] $\pi^*[d]\in M_1$, and by \[subcas.cas2-3\] $\pi^*[e]=\pi_{\pi^*(d)}^{-1}\circ\pi_d[e]$. Since $\pi_d[e]\in P(\operatorname{otp}(d))\cap M_0=P(\operatorname{otp}(d))\cap M_1$ and $\pi_{\pi^*(d)}^{-1}\in M_1$ we get that $\pi^*[e]\in M_1$. With a simmetric argument we can prove that $\pi^{*-1}[e]\in M_0$ if $e\in M_1\cap P_\delta (M_0\cap\alpha_0^\eta)$.
Finally we can extend $\pi^*$ to $\pi_\eta$ by the usual trick employed in the previous cases.
Case 2:
: \[cas.2\] $\alpha_0^\eta$ is a successor cardinal.
We are given $\pi_\beta$ isomorphism of $$\langle\alpha_0^\beta\cap M_0, P(\alpha_0^\beta)\cap M_0, \in\rangle$$ with $$\langle\alpha_1^\beta\cap M_1, P(\alpha_1^\beta)\cap M_1, \in\rangle.$$ Any ordinal $\delta$ in $\alpha_i^{\beta+1}$ is coded by a binary relation on $\alpha_i^\beta$ whose transitive collapse is $\delta$. Now let $\phi_i\in M_i$ be functions from $\alpha_i^{\beta+1}$ to $P(\alpha_i^\beta)$ such that for each $\gamma<\alpha_i^{\beta+1}$, $\phi_*^{-1}[\phi_i(\gamma)]$ is a binary relation that codes $\gamma$ (where $\phi_*\in M_0\cap M_1$ is some recursive bijection of ${\ensuremath{\text{{\rm Ord}}}}^2$ with ${\ensuremath{\text{{\rm Ord}}}}$).
Then we can extend $\pi_\beta$ to $\pi^*$ on $M_0\cap\alpha_0^{\beta+1}$ as follows, $\pi^*(\gamma)=\delta$ iff $\phi_1(\delta)=\pi_\beta(\phi_0(\gamma))$. We leave to the reader to check that $\pi^*$ is the order isomorphism of the sets $M_i\cap\alpha_i^{\beta+1}$ by the following steps:
- $\operatorname{dom}(\pi^*)=M_0\cap\alpha_0^{\beta+1}$ and $\operatorname{rng}(\pi^*)=M_1\cap\alpha_1^{\beta+1}$,
- $\alpha<\gamma$ iff $\pi^*(\alpha)<\pi^*(\gamma)$,
- $\alpha=\gamma$ iff $\pi^*(\alpha)=\pi^*(\gamma)$.
Arguing as in Claim \[keycl.thm.isotypeGM2\] $$\pi^*[a]\in M_1\cap P_{\omega_1}(\alpha_1^\eta) \iff a\in M_0\cap P_{\omega_1}(\alpha_0^\eta).$$ Now we proceed with the usual trick to define $\pi_\eta$ using the guessing property of each $M_i$ and the isomorphism $\pi^*$ between $P_{\omega_1}(\alpha_i^\eta)\cap M_i$.
This completes the proof of the theorem.
The proof actually show that what matters is not that the models $M_i$ are internally club but that each $M_i$ “sits” inside $P_\delta(M_i\cap\theta_i)$ in a similar way. To make this a precise assertion assume the case of the theorem in which $\delta=\aleph_1 = | M_i |$, let each $M_i=\bigcup\{M_i^\alpha:\alpha<\omega_1\}$ be a continuous increasing union of the countable sets $M_i^\alpha$. Then we can relax the requirement on each $M_i$ of being internally club to the requirement $$\{\alpha<\omega_1: M_i^\alpha\in M_i\text{ for }i=0,1\}$$ is unbounded in $\omega_1$.
It is not clear under which conditions on $A_i$ the theorem can hold with respect to guessing models $M_i\prec\mathfrak{R}_i=\langle H_{\theta_i}, \in, A_i\rangle$ where both $A_i$ are proper class in $H_{\theta_i}$.
\[rem.PFAIsotype\] In models of ${\ensuremath{\text{{\sf PFA}}}}$ there is a well-order of the reals in type $\omega_2$ definable in $H_{\aleph_2}$ using as parameter a ladder system on $\omega_1$. (A ladder system on $ \omega _1$ is a $\mathcal{C}=\{C_\alpha:\alpha<\omega_1\}$ where $C_\alpha \subseteq \alpha $, $ \bigcup_{} C_ \alpha = \alpha $ and $ \operatorname{otp}( C_ \alpha ) = \omega $ for all $\alpha$ limit.) Thus, assuming ${\ensuremath{\text{{\sf PFA}}}}$, if there is a ladder system in $M_0\cap M_1$, assumption \[thm.isotypealeph1guesmod-3\] in Theorem \[thm.isotypealeph1guesmod\] can be removed — this will be crucial in the proof of Theorem \[thm.PFASLD\].
Assuming Woodin’s $(*)$-axiom (a strenghtening of $\mathsf{BMM}$), there is always a ladder sequence in $M_0\cap M_1$ provided that $\omega_1\subseteq M_i$. However it is not known whether $(*)$ is compatible with the full ${\ensuremath{\text{{\sf PFA}}}}$.
Faithful models {#subsect.faithful}
---------------
In this section assume $\theta$ is inaccessible in $V$. The above characterization of isomorphism types for $\delta$-guessing, $\delta$-internally club models is not completely satisfactory since it could be the case that two such models $M_0,M_1\prec H_\theta$ have the same isomorphism type, are such that $\kappa_{M_0}=\kappa_{M_1}=\kappa$, $M_0\cap\kappa=M_1\cap\kappa$, but for some cardinal $\lambda\in M_0\cap M_1\setminus \kappa$, $\chi_{M_0}(\lambda)=\chi_{M_1}(\lambda)$ and $\chi_{M_0}\restriction \lambda\neq\chi_{M_1}\restriction \lambda$. We shall show that for $\aleph_0$-guessing models this cannot be the case, thus we would like that this rigidity property of $\aleph_0$-guessing models holds also for arbitrary guessing models. We shall see that in models of ${\ensuremath{\text{{\sf MM}}}}$ there is a stationary set of $\aleph_1$-guessing models which have this rigidity property. Let for a suitable initial segment $\mathfrak{R}=\langle R,A,\in\rangle$: $$\mathrm {G}_\kappa^\delta(\mathfrak{R})=\{M\prec R: M\text{ is a $\delta$-guessing model and }\kappa_M=\kappa\}$$ For $S$ stationary subset of $P(V_\theta)$, let $T(S)=\{\chi_M\restriction \gamma\colon M\in S, \gamma\in {\ensuremath{\text{{\rm Card}}}}\cap M\}$.
The following holds:
1. $T(\mathrm {G}_\kappa^{\aleph_0}(V_\theta))$ is a tree of functions ordered by end extension.
2. \[thm.faithfulmodels2\] Assume ${\ensuremath{\text{{\sf MM}}}}$. Then there is $S$ stationary subset of $\mathrm {G}_{\aleph_2}^{\aleph_1}(V_\theta)\cap \mathrm {IC}^{\aleph_1}(V_\theta)$ such that $T(S)$ is a tree of functions ordered by end extension.
We need the following definition. Given a set of ordinals $S$ such that $S$ is a stationary subset of $\sup(S)$ let: $$P^*(S)=\{T\subseteq S: T\text{ is stationary in }\sup(S)\}.$$
\[def.faithfulmodel\] $M\prec V_\theta$ is an *$S$-faithful model* if for all $T\in P(S)\cap M$, $T$ reflects on $\sup(M\cap S))$ iff $T\in P^*(S)$.
$M\prec V_\theta$ is a *$\lambda$-faithful model* if $M$ is $E^\lambda_{\aleph_0}$-faithful.
$M\prec V_\theta$ is a *faithful model* if $M$ is $E^\lambda_{\aleph_0}$-faithful for all regular $\lambda\in M$.
The following lemma motivates the definition of faithful models:
Assume $M_0,M_1\prec \langle V_\theta,\in,<_*\rangle$ where $<_*$ is a well-order of $V_\theta$ are $\lambda$-faithful models for some regular $\lambda\in M_0\cap M_1$ and $\chi_{M_0}(\lambda)=\chi_{M_1}(\lambda)$. Then $\chi_{M_0}\restriction \lambda=\chi_{M_1}\restriction \lambda$.
Let $\mathrm{S}_\lambda=\{S_\alpha:\alpha<\lambda\}\in M_0\cap M_1$ be the least partition under $<_*$ of $E^\lambda_{\aleph_0}$ in stationary sets, then $\mathrm{S}_\lambda\in M_0\cap M_1$ and: $$\alpha\in M_i\iff M_i\models S_\alpha\text{ is stationary } \iff S_\alpha \text{ reflects on }\chi_{M_i}(\lambda)$$ Thus $$M_i\cap\lambda=\{\alpha: S_\alpha \text{ reflects on }\chi_{M_i}(\lambda)\}$$ and we are done.
If $M\prec V_\theta$ is an $\aleph_0$-guessing model then $M$ is a faithful model.
$M$ is isomorphic to $V_\gamma$ for some $\gamma$ by Lemma \[lem.isotype0guesmod\]. Thus for any regular cardinal $\lambda\in M$ and $S\in M\cap P(\lambda)$: $$S \text{ reflects on }\chi_M(\lambda) \iff \pi_M(S) \text{ is a stationary subset of } \pi_M(\lambda)$$ and $\lambda$ is regular iff $\pi_M(\lambda)$ is regular.
By the two lemmas the first part of the theorem is proved. To prove the second part of the theorem we proceed as follows:
Let in $V$ $$X=\bigcup\{P^*(E^\lambda_{\aleph_0}):\lambda<\theta\text{ is regular}\}$$ Fix also in $V$ a family $\{S_\alpha:\alpha<\omega_1\}$ of disjoint stationary subsets of $\omega_1$ such that $\min S_\alpha\geq\alpha$ for all $\alpha$ and $\{S_\alpha:\alpha<\omega_1\}$ is a maximal antichain on $P(\omega_1)/ \mathsf{NS}_{\omega_1}$.
Let $\mathbb{C}$ be Cohen forcing. In $V[G]$ where $G$ is $V$-generic for $\mathbb{C}$ we define the poset $\mathbb{P}$ as follows.
A condition $p\in\mathbb{P}$ is a pair $(f_p,\phi_p)$ such that:
- $f_p\colon \alpha+1\rightarrow V\cap (P_{\omega_1}V_\theta)^{V[G]}$ is a continuous map.
- $\phi_p\colon \alpha+1\rightarrow X$ is such that for all $\eta<\xi\leq\alpha$: $$\xi\in S_\eta \text{ iff } \sup(f_p(\xi)\cap\sup\phi_p(\eta))\in\phi_p(\eta) .$$
$p\leq q$ if $f_p$ extends $f_q$ and $\phi_p$ extends $\phi_q$. We omit the proof of the following lemma:
The poset $\mathbb{R}=\mathbb{C}*\dot{\mathbb{P}}$ is stationary set preserving and has the $\omega_1$-covering and $\omega_1$-approximation properties.
By ${\ensuremath{\text{{\sf MM}}}}$ in $V$, there are stationarily many $N\prec H_{(2^\theta)^+}$ of size $\aleph_1$ which have a generic filter for the poset $\mathbb{R}*\mathbb{Q}_{\mathbb{R}}$, where $\mathbb{Q}_{\mathbb{R}}$ is the ccc-poset in $V^{\mathbb{R}}$ used in the proof of Theorem \[the.PFAIUGM\]. For any such $N$ we can check the following properties of $M=N\cap V_\theta$: $$M\prec V_\theta \text{ is an $\aleph_1$-guessing faithful model which is internally club.}$$ This sketches the proof of the second part of the theorem.
The existence of faithful models as in Definition \[def.faithfulmodel\] is a principle of diagonal reflection. This type of reflection properties for successor cardinals already appeared in several works of Foreman and others to a great extent also in [@foreman_magidor_shelah]. Cox in [@cox10 Definition 7] has introduced a maximal principle of diagonal reflection which follows from a strengthening of ${\ensuremath{\text{{\sf PFA}}}}$ and implies the existence of faithful models as in Definition \[def.faithfulmodel\].
Laver functions in models of ${\ensuremath{\text{{\sf PFA}}}}$. {#sect.laverdiamond}
===============================================================
Let $\mathcal{J}$ be a family of elementary embeddings all with the same critical point $\kappa$.
- $f\colon \kappa\rightarrow H_\kappa$ is a weak $\mathcal{J}$-Laver function if for all $x\in V$ and for all $\lambda>\operatorname{rank}(x)$ there is $j\colon V\rightarrow M$ in $\mathcal{J}$ such that $j(f)(\kappa)=x$ and $V_\lambda\subseteq M$.
- $f\colon \kappa\rightarrow H_\kappa$ is a strong $\mathcal{J}$-Laver function if for all $x$ and for all $\lambda>\operatorname{rank}(x)$ there is $j\colon V\rightarrow M$ such that $j(f)(\kappa)=x$ and $j[V_\lambda]\in M$.
Weak (strong) $\mathcal{J}$-Laver diamond holds at $\kappa$ holds if there is a weak (strong) $\mathcal{J}$-Laver function. We shall denote weak (strong) $\mathcal{J}$-Laver diamond by $\mathrm{WLD}(\mathcal{J})$ ($\mathrm{SLD}(\mathcal{J})$).
We shall show in Section \[subsec.LDeqJJ\^\] that the usual proof of Laver diamond from a supercompact cardinal $\kappa$ actually produce a witness for strong $\mathcal{J}_\kappa$-Laver diamond, where $\mathcal{J}_\kappa$ is the class of elementary embeddings induced by the stationary tower forcing below some $\mathrm{G}_\kappa(H_\theta)$ (see Definition \[def.JkappaLD\] and Theorem \[secLD.thmeqLD\]). In Section \[subsec.PFASLD\] we show that assuming ${\ensuremath{\text{{\sf PFA}}}}$ there is a strong $\mathcal{J}_{\aleph_2}$-Laver diamond on $\aleph_2$ (Theorem \[thm.PFASLD\]).
Properties of elementary embeddings {#subsec.elemb}
-----------------------------------
In this section we shall briefly recall some basic properties of elementary embeddings, a reference text for this material is [@larson Chapter 2]. The reader must have familiarity with the basic properties of ultrapower embeddings and of the stationary tower forcing.
Let $V, M$ be transitive class model of ${\ensuremath{\text{{\sf ZFC}}}}$ and $j\colon V\rightarrow M$ be an elementary embedding with $\operatorname{crit}(j)=\kappa$. Then:
1. If $j[X]\in M$ for some set $X\supseteq\kappa$. $$\mathcal{U}_X=\{A\in V\cap P(P_\kappa(X)):j[X]\in j(A)\}$$ is a filter (possibly not in $V$) on $P_\kappa X\cap V$ which measures all sets in $V$, i.e if $S\in P(P_\kappa X)\cap V$, $S$ in $\mathcal{U}_X$ or $P_\kappa(X)\setminus S\in\mathcal{U}_X$. We call a filter with this property a $V$-ultrafilter on $P_\kappa(X)$.
2. Let $$V^{P_\kappa X}/\mathcal{U}_X =M_X=\{[f]_X: f\colon P_\kappa X\to V\text{ is in }V\}$$ where $$[f]_X = \{g: j(g)(j[X])=j(f)(j[X]) \text{ and $g$ is of minimal rank}\}$$ and if $R$ is either $\in$ or $=$, $$[f]_X \mathrel{R}_X [g]_X \iff j(g)(j[X]) \mathrel{R} j(f)(j[X]).$$ Then $\langle M_X,\in_X,=_X\rangle$ can be identified with its transitive collapse and $j=i_X\circ j_X$ where $$j_X \colon V \to M_X, \qquad x \mapsto \langle x: M \in P_\kappa X\rangle]_X$$ is elementary and $$i_X \colon M_X \to M, \qquad [f]_X \mapsto j(f)(j[X])$$ is also elementary.
3. Assume $X$ is transitive and $\kappa\subseteq X$. Let $\lambda_X=\sup X\cap {\ensuremath{\text{{\rm Ord}}}}$. Then every $x\in X$ is represented in the ultrapower $M_X$ by the equivalence class $[ \langle \pi_M[x\cap M]: M\in P_\kappa X \rangle ]_X$. In particular $$\alpha = [ \langle \operatorname{otp}(\alpha\cap M) : M\in P_\kappa X \rangle ]_X$$ for all $\alpha\leq\lambda_X$ and $$\kappa = [ \langle M \cap \kappa : M \in P_\kappa X \rangle ]_X.$$ For these reasons it is possible to check that $\operatorname{crit}(i_X)>\lambda_X$.
4. If $X\supseteq Y\supseteq\kappa$ are both transitive, we also get a natural elementary embedding $$i_{XY} \colon M_Y\to M_X$$ which maps $[f]_Y$ to $$[\langle f(M\cap Y): M\in P_\kappa X\rangle]_X$$ and is such that $i_Y=i_X\circ i_{XY}$. One can also check that that for all $x\in Y$, $$i_{XY}(x)=i_Y(x)=i_X(x)=x.$$
These properties lead to the following
\[secLD.keyremark0\] Let $V,M$ be transitive models of ${\ensuremath{\text{{\sf ZFC}}}}$ and $j:V\rightarrow M$ be an elementary embedding with critical point $\kappa$ such that $j[X]\in M$ for some transitive set $X\in V$. Assume $f\colon \kappa\rightarrow H_\kappa$ and $j(f)(\kappa)=x$. Then for all transitive set $Y\subseteq X$ such that $x\in Y$, $$j_Y(f)(\kappa)=j_X(f)(\kappa)=j(f)(\kappa)=x.$$
Notice that all of the above observation do not subsume any relation between $V$ and $M$, i.e. possibly $M \nsubseteq V$ and $V\nsubseteq M$ and so everythig said so far applies to embeddings induced by normal measures in $V$ as well as to embeddings induced by the stationary tower forcing over $V$.
A brief account of the stationary tower {#subsec.WOOST}
---------------------------------------
Given a strongly inaccessible cardinal $\delta$, $\mathbb{P}_\delta$ is the poset whose conditions are $S\in H_\delta$ such that $S\subseteq P(\cup S)$ is stationary in $P(\cup S)$.
$S\leq T$ iff $\cup T\subseteq \cup S$ and $S\restriction\cup T$ is a subset of $T$.
Let $G$ be $V$-generic for $\mathbb{P}_\delta$. Let $R$ be $\in$ or $=$, $\operatorname{dom}(f)=P(X)$ and $\operatorname{dom}(g)=P(Y)$ with $X,Y\in H_\delta$, then $$f \mathrel{R_G} g \iff
\{t\subseteq X\cup Y: f(t\cap X) \mathrel{R} g (t\cap Y)\}\in G.$$ Define $M_G=\{[f]_G: f\in V,\, f:P(X)\to V,\, X\in H_\delta\}$ where $$[f]_G=\{ g\in V: f \mathrel{=_G} g\text{ and $g$ has minimal rank}\} .$$ With abuse of notation we shall say that $$[f]_G \mathrel{R_G} [g]_G \text{ if } f \mathrel{R_G} g.$$ With these definition it can be seen that:
1. $M_G$ is the direct limit of the ultrapowers $M_X=V^{P_\kappa X}/ ( G\restriction X )$ under the embeddings $j_{XY}$ defined for $Y\subseteq X$ by $$j_{XY}([f]_Y)=[\langle f(M\cap Y):M\in P(X)\rangle ]_X,$$
2. \[eqn.WOOjG\] $j_X[X]=j_G[X]=[{\ensuremath{\text{{\rm id}}}}_{P ( X )}]_G\in M_G$ for all $X\in H_\delta$.
3. \[eqn.WOOjG2\] $M_G\models\phi([f_1]_G,\dots,[f_k]_G)$ iff for some $H_\gamma\supseteq \cup\bigcup\operatorname{dom}(f_1)\cup\dots\cup\bigcup\operatorname{dom}(f_k)$ $$\{X\prec H_\gamma: V\models\phi(f_1(X\cap\bigcup\operatorname{dom}(f_1)),\dots,f_k(X\cap\bigcup\operatorname{dom}(f_k))\}\in G$$
Woodin has proved the following fundamental result [@larson Theorem 2.5.8]:
Assume $\delta$ is a Woodin cardinal and $G$ is $V$-generic for $\mathbb{P}_\delta$. Then $M_G\subseteq V[G]$ is well founded and $V[G]\models M_G^{<\delta}\subseteq M_G$.
We let $j\colon V\rightarrow M$ be a generic ultrapower embedding if $j=j_G$ for some $V$-generic filter $G$ for $\mathbb{P}_\delta$ where $\delta$ is a Woodin cardinal in $V$.
Equivalent formulations of strong Laver diamond for a supercompact cardinal $\kappa$ {#subsec.LDeqJJ^}
------------------------------------------------------------------------------------
In this section we shall see that if $\kappa$ is supercompact then $\mathrm{SLD}(\mathcal{J})$ holds for the class $\mathcal{J}$ of ultrapower embeddings induced by normal measures on $P_\kappa X$ and that this principle is equivalent to $\mathrm{SLD}(\mathcal{J}_\kappa)$ where $\mathcal{J}_\kappa$ is a class of embeddings induced by stationary towers below a stationary set of guessing models.
For the sake of completeness we begin with the proof of the existence of a strong Laver diamond at a supercompact cardinal $\kappa$. This is what the ordinary proof of Laver diamond gives but it is not spelled out in the usual argument.
Assume $\kappa$ is supercompact. Let $\mathcal{J}$ be the class of elementary embeddings $j\colon V\rightarrow M$ with critical point $\kappa$ such that $M\subseteq V$. Then $\mathrm{SLD}(\mathcal{J})$ holds.
Assume not and for each $f\colon \kappa\rightarrow H_\kappa$, let $\lambda_f$ be least such that for some $x_f\in V_{\lambda_f}$, $j_\mathcal{U}(f)(\kappa)\neq x_f$ for every $\theta\geq\lambda_f$ and every normal measure $\mathcal{U}$ on $P_\kappa V_{\theta}$.
We first notice that for any $f$ we might restrict our attention to normal measures on $V_{\lambda_f}$: to see this assume that $j_{\mathcal{U}}(f)(\kappa)=x_f$ for some normal measure $\mathcal{U}$ on $P_\kappa V_{\theta}$. Then $j_{\mathcal{V}}(f)(\kappa)=x_f$ by Fact \[secLD.keyremark0\], where $ \mathcal{V} =\mathcal{U}\restriction V_{\lambda_f}$.
Let $\phi ( g , \delta )$ hold if $g\colon \alpha\rightarrow H_\alpha$ and $\delta$ is the least $\gamma$ such that for some $x\in V_\gamma$, $j_\mathcal{E}( g ) ( \alpha ) \neq x$ for every normal measure $\mathcal{E}$ on $P_\alpha\gamma$.
Let $f:\kappa\rightarrow H_\kappa$ be defined as follows: $f ( \alpha ) = 0$ if for no $\delta$, $\phi ( f \restriction \alpha , \delta ) $ holds, else $f ( \alpha )$ is some $x_\alpha\in V_\delta$ that witnesses that $\phi(f\restriction\alpha,\delta)$ holds for some $\delta$.
Let $\theta$ be large enough so that $\lambda_f\in V_\theta$ for all $f\colon \kappa\rightarrow H_\kappa$. Now let $\mathcal{E}$ be a normal measure on $P_\kappa V_\theta$. Notice that $j[V_\theta]\in M_{\mathcal{E}}$, $V_\theta\subseteq M_{\mathcal{E}}$ and $\phi ( g , \lambda_g)$ holds in $V_\theta$ for all $g \colon \kappa\rightarrow V_\kappa$. Thus we get that $\phi(g,\lambda_g)$ holds in $M$ for all such $g$.
By definition of $f$, we get that in $M_{\mathcal{E}}$, $j_\mathcal{E}( f ) (\kappa)$ is some $x$ of least rank such that for any measure $\mathcal{U}\in M$ on $(P_\kappa V_{\lambda_f})^M$, $j_\mathcal{U}(f)( \kappa ) \neq x$.
Notice that $M$ is closed under $| V_\theta |$-sequences and that for every $\gamma<\theta$, $V_\gamma\subseteq M$. Let $\mathcal{E}^*$ be the projection of $\mathcal{E}$ on $V_{\lambda_f}$. Then $\mathcal{E}^*\in V_{\lambda_f+2}\subseteq M$ and thus it is a normal measure in $M$. Notice that $j_{\mathcal{E}}=k\circ j_{\mathcal{E}^*}$ with $\operatorname{crit}(k)>\lambda_f$. Since $x\in V_{\lambda_f}$, this means that $$j_{\mathcal{E}^*}(f)(\kappa)=j_{\mathcal{E}}(f)(\kappa)=x$$ contradicting the choice of $x$ as a witness of $\phi(f,\lambda_f)$ in $M$.
Let: $$\begin{aligned}
\mathrm {G}_\kappa^\delta(\mathfrak{R}) &= \{M\prec R: M\text{ is a $\delta$-guessing model and }\kappa_M=\kappa\}
\\
\mathrm {G}_\kappa(\mathfrak{R}) &= \{M\prec R: M\text{ is a guessing model and }\kappa_M=\kappa\} \end{aligned}$$
\[def.JkappaLD\] Assume $\mathrm{G}_\kappa(H_\lambda)$ is stationary for all $\lambda\geq\kappa$, $\mathcal{J}_\kappa$ is the family of generic ultrapower embeddings $j \colon V\rightarrow{M}$ defined as follows:
$j\in\mathcal{J}_\kappa$ if there is $G$ such that:
- $G$ is a $V$-generic filter for the full stationary tower on some Woodin cardinal $\delta>\kappa$,
- $\mathrm {G}_\kappa(H_\theta)\in G$ for some regular $\theta\in (\kappa,\delta)$
- $j = j_{H_\theta}$ where $$j_{H_\theta}\colon V\rightarrow V^{P(P(H_\theta))}/ ( G\restriction H_\theta)$$ is the canonical embedding induced by $G\restriction H_\theta$.
We have all the elements to state the main result of this section:
\[secLD.thmeqLD\] Assume $\kappa$ is supercompact and there are class many Woodin cardinals. Let $\mathcal{J}$ be the family of ultrapower embeddings induced by normal measures on $P_\kappa X$ for some set $X$. Then the following are equivalent:
1. \[secLD.thmeqLD-1\] $f$ is a strong $\mathcal{J}$-Laver function.
2. \[secLD.thmeqLD-2\] $f$ is a strong $\mathcal{J}_\kappa$-Laver function.
3. \[secLD.thmeqLD-3\] $\{M\in\mathrm {G}_\kappa^{H_\theta}: \pi_M [ x ] = f ( M \cap \kappa ) \}$ is stationary for all $\theta \geq \kappa$.
We start with the proof that \[secLD.thmeqLD-1\] implies \[secLD.thmeqLD-2\].
Recall the following result of Burke [@Bur97 Lemma 3.1 ]:
Assume $\mathcal{I}\subseteq P(P(X))$ is a normal ideal. Let $\breve{\mathcal{I}}$ denote the dual filter. There is $S _{\breve{\mathcal{I}}}$ stationary subset of $P(2^{2^{|X|}})^+)$ such that $\mathcal{I}$ is the projection of the non-stationary ideal restricted to $\mathrm {S}_{\breve{\mathcal{I}}}$.
In particular if $\theta$ is regular and such that $\mathcal{I}\in H_\theta$, an $S _{\breve{\mathcal{I}}}$ which witnesses the lemma for $\mathcal{I}$ is the set of $M\prec H_\theta$ such that $M\cap X\not\in T$ for all $T\in M\cap\mathcal{I}$.
In the sequel we shall assume that $S _{\breve{\mathcal{I}}}$ is the above stationary set where $\theta=\theta_{\breve{\mathcal{I}}}$ is chosen least possible.
\[secLD.keyremark1\] Assume:
1. $\mathcal{U}\in V$ is a normal measure on $P_\kappa H_\lambda$,
2. $\delta>\lambda$ is a Woodin cardinal in $V$,
3. $j_G\colon V\rightarrow M_G$ is the generic embedding induced by a $V$-generic filter for the full stationary tower up to $\delta$ such that $S _{\mathcal{U}}\in G$.
Then $\mathcal{U}=G\restriction H_\lambda$, $j_G=k\circ j_\mathcal{U}$ where $j_\mathcal{U}\colon V\rightarrow M_{\mathcal{U}}$ is the ultrapower embedding induced by $\mathcal{U}$ and $\operatorname{crit}( k ) > \lambda$.
$S=S _{\mathcal{U}}\in G$ iff $S\cap C\in G$ for all $C$ clubsets of $P(\cup S)$. Moreover if $T\in G$, then $T\restriction H_\lambda\in G$ for all stationary sets $T$ such that $H_\lambda\subseteq \cup T$. Since the club filter restricted to $S$ projects to $\mathcal{U}$, we can conclude that $\mathcal{U}= G\restriction H_\lambda$ and we are done.
Now let $f:\kappa\to H_\kappa$ be a strong $\mathcal{J}$-Laver function. Given $x\in H_\lambda$, find a normal measure $\mathcal{U}\in V$ on some $P_\kappa H_\lambda$ such that $j_\mathcal{U}(f)(\kappa)=x$.
Let $\delta>\lambda$ be a Woodin cardinal and $G$ be generic for $\mathbb{P}_\delta$ such that $S=S _{\mathcal{U}}\in G$. Then $j_{G\restriction H_\lambda}\in \mathcal{J}_\kappa$ and $$j_{G\restriction H_\kappa}(f)(\kappa)=j_{\mathcal{U}}(f)(\kappa)=x$$ by Fact \[secLD.keyremark0\]. This concludes the proof that \[secLD.thmeqLD-1\] implies \[secLD.thmeqLD-2\].
We now show that \[secLD.thmeqLD-2\] implies \[secLD.thmeqLD-1\].
We shall need the following result:
\[prop.GMtoNorm\] Assume that for some regular $\theta$, $M\prec H_\theta$ is an $\aleph_0$-guessing model. Then for all regular $\lambda\in M$ such that $|P(P(H_\lambda))|<\theta$, there is $\mathcal{U}\in M$ normal measure on $P_\kappa H_\lambda$ such that $M \cap H_\lambda \in \bigcap ( \mathcal{U} \cap M )$.
Let $\pi_M$ be the transitive collapse of $M$ and $j_M$ be the inverse map. Since $\kappa_M=\kappa$ is inaccessible by Proposition \[prop.basiguessing\]\[prop.basiguessing-5\], $\pi_M[M]=H_{\bar{\theta}}$ for some regular cardinal $\bar{\theta}$ and, if $\bar{\kappa}=\pi_M[\kappa]$, $j_M\colon H_{\bar{\theta}}\rightarrow H_\theta$ is an elementary embedding with critical point $\bar{\kappa}$. If $\lambda\in M$ is regular, $|P(P(H_\lambda))|< \theta$ and $\bar{\lambda}=\pi_M[\lambda]$, we get that $\bar{\lambda}$ is regular and $|P(P(H_{\bar{\lambda}}))|<\bar{\theta}$. Define $\mathcal{\bar{U}}$ on $P_{\bar{\kappa}}H_{\bar{\lambda}}$ by $A\in\mathcal{\bar{U}}$ iff $j_M[H_{\bar{\lambda}}]=M\cap H_\lambda\in j_M(A)$. Then $\mathcal{\bar{U}}\subseteq P(P(H_{\bar{\lambda}}))$ and thus $\mathcal{\bar{U}}\in H_{\bar{\theta}}$. Then $j_M(\mathcal{\bar{U}})=\mathcal{U}\in M$ is an ultrafilter on $P_\kappa H_\lambda$ and $M\cap H_\lambda\in j(A)$ for all $A\in\mathcal{\bar{U}}$ i.e. $M\cap\lambda\in\bigcap j_M[\mathcal{\bar{U}}]=\bigcap(\mathcal{U}\cap M)$, the desired conclusion.
Assume $f$ is a strong $\mathcal{J}_\kappa$-Laver function, we want to show that it is also a strong $\mathcal{J}$-Laver function.
Given $x\in V_\lambda$, pick $\bar{j}\in\mathcal{J}_\kappa$, a large enough regular $\theta>\lambda$ and a $V$-generic filter $G$ for the full stationary tower on some Woodin cardinal $\delta>\theta$ such that:
- $x\in H_\theta$,
- $\mathrm {G}_\kappa^{H_\theta}\in G$.
- $\bar{j}$ is induced by the projection of $G$ to $H_\theta$,
- $\bar{j}(f)(\kappa)=x$.
By Proposition \[prop.GMtoNorm\], for each $M\in \mathrm {G}_\kappa^{H_\theta}$ there is a normal measure $\mathcal{U}_M\in M$ on $P_\kappa\lambda$ such that $M\cap\lambda\in\bigcap (\mathcal{U}_M\cap M)$.
By a standard density argument for the full stationary tower, since $\mathrm {G}_\kappa^{H_\theta}\in G$ there will be some normal measure $\mathcal{U}\in V$ such that $$\{M\in\mathrm {G}_\kappa^{H_\theta}:\mathcal{U}_M=\mathcal{U}\}\in G.$$ This means that $\mathcal{U}\subseteq G$ and thus that $\mathcal{U}$ is the projection of $G$ to $H_\lambda$.
Since $H_\lambda\subseteq H_\theta$ are both transitive, contain $\kappa$, and $j_G[H_\kappa]\in M_G$, Fact \[secLD.keyremark0\] applied to $j_G$, $V_\lambda$ and $H_\theta$ brings that $$j_\mathcal{U}(f)(\kappa)=j_G(f)(\kappa)=\bar{j}(f)(\kappa)=x.$$ However $\mathcal{U}\in\mathcal{J}$. Thus $f$ is also a $\mathcal{J}$-Laver function.
The equivalence of \[secLD.thmeqLD-2\] and \[secLD.thmeqLD-3\] is a standard property of the stationary tower and follows by the following
\[keyremark1\] For a given $f\colon \kappa\rightarrow H_\kappa$ and $x\in H_\theta$, there is $\bar{j}\colon V\rightarrow \bar{M}$ in $\mathcal{J}_\kappa$ such that $\bar{j}[H_\theta]\in \bar{M}$ and $\bar{j}(f)(\kappa)=x$ iff $\{X\in \mathrm {G}_\kappa(H_\theta): f(X\cap\kappa)=\pi_X[x\cap X]\}$ is stationary.
Let $\bar{j}$ be induced by some $j_G$ such that $\mathrm {G}_\kappa(H_\theta)\in G$ and $G$ is a $V$ generic filter for a full stationary tower up to some Woodin cardinal $\delta$. By Fact \[secLD.keyremark0\] for any $f\colon \kappa\rightarrow H_\kappa$ and any $x\in H_\theta$ $\bar{j}(f)(\kappa)=x$ iff $j_G(f)(\kappa)=x$.
If we unfold the definition of $j_G$, we get that for any $f\colon \kappa\rightarrow H_\kappa$ and any $x\in H_\theta$: $$j_G(f)(\kappa)=x \iff \{X\prec H_\theta: f(X\cap\kappa)=\pi_X[x\cap X]\}\in G.$$ In particular we have that $S=\{X\in\mathrm {G}_\kappa^{H_\theta}: f(X\cap\kappa)=\pi_X[x\cap X]\}\in G$ is stationary and we are done. The converse implication is proved by a similar argument.
This concludes the proof of the theorem.
Laver functions in models of ${\ensuremath{\text{{\sf PFA}}}}$ {#subsec.PFASLD}
--------------------------------------------------------------
We can prove the main theorem of the whole Section \[sect.laverdiamond\]:
\[thm.PFASLD\] Assume ${\ensuremath{\text{{\sf PFA}}}}$ holds and there are class many Woodin cardinals. Then there is a strong $\mathcal{J}_{\aleph_2}$-Laver function $f \colon \aleph_2\to H_{\aleph_2}$ i.e. a function $f$ such that: $$\{M\in\mathrm {G}_{\aleph_2}(H_\theta): \pi_M(x)=f(M\cap\kappa)\}$$ is stationary for all $\theta\geq\kappa$ and for all $x$.
We shall prove the following strengthening of the conclusion:
Let $$\mathrm{H}^*_\theta=\{M\in\mathrm {G}_{\aleph_2}^{\aleph_1}(H_\theta):
M\text{ is $\aleph_1$-internally club and } \text{ and } |M|=\aleph_1\}$$ Then for some function $f:\aleph_2\to H_{\aleph_2}$ we have that: $$\{M\in \mathrm{H}^*_\theta:\pi_M(x)=f(M\cap\kappa)\}$$ is stationary for all $\theta\geq\kappa$ and for all $x\in H_\theta$.
By Theorem \[the.PFAIUGM\]\[the.PFAIUGM-1\], $\mathrm{H}^*_\lambda$ is stationary for every $\lambda\geq\omega_2$ in models of ${\ensuremath{\text{{\sf PFA}}}}$.
The rest of the section is devoted to the proof of the claim. Assume towards a contradiction that for each $g\colon \aleph_2\rightarrow H_{\aleph_2}$, there is $x_g\in H_\gamma$ such that $$\{M\in \mathrm{H}^*_\gamma: g(M\cap\kappa)=\pi_M(x_g)\}$$ is non-stationary.
Let $\theta > \lambda$ be regular and such that $x_g\in H_\theta$ for all such $g$. Then the following statement $\psi(\theta)$ holds in $V$ as well as in $H_{\theta^+}$:
> For every $g\colon \aleph_2\rightarrow H_{\aleph_2}$ there is $x_g\in H_\theta$ such that $$\{X\in\mathrm{H}^*_\theta: g(X\cap\kappa)=\pi_X[x_g\cap X]\}$$ is non-stationary.
Now we proceed to define $f\colon \aleph_2\rightarrow H_{\aleph_2}$ as follows.
First of all by the isomorphism type Theorem \[thm.isotypealeph1guesmod\] in combination with Remark \[rem.PFAIsotype\] we have the following:
> Assume ${\ensuremath{\text{{\sf PFA}}}}$ holds. Let $M_i\prec H_{\theta_i}$ be models in $\mathrm {H}^*_{\theta_i}$ for $i<2$ such that $M_0\cap\aleph_2=M_1\cap\aleph_2$ and there is a ladder system $\mathcal{C}=\{C_\alpha:\alpha<\omega_1\}$ in $M_0\cap M_1$. Then for some $\lambda\in M_0\cap M_1$, $M_0$ is isomorphic to $M_1\cap H_\lambda$ or conversely.
(See Remark \[rem.PFAIsotype\] for the definition of a ladder system on $\omega_1$.) The above shows:
Assume ${\ensuremath{\text{{\sf PFA}}}}$ holds. Then, given a ladder system $\mathcal{C}$ on $\omega_1$, for a stationary set of $\alpha<\aleph_2$ we can define a unique transitive structure $N_\alpha$ of size $\aleph_1$ and $\theta_\alpha\in N_\alpha$ such that:
- $\mathcal{C}\in N_\alpha$,
- $\alpha=(\omega_2)^{N_\alpha}$,
- if $M\in\mathrm{H}^*_{\theta^+}$ is such that $\mathcal{C}\in M$ and $M\cap\aleph_2=\alpha$, then the transitive collapse of $M$ is $N_\alpha$,
- $\theta_\alpha=\pi_M(\theta)$,
- $\psi(\theta_\alpha)$ holds in $N_\alpha$, i.e., for all $g:(\omega_2)^{N_\alpha}\to (H_{\omega_2})^{N_\alpha}$ in $N_\alpha$, there is some $x_g\in (H_{\theta_\alpha})^{N_\alpha}$ such that $$\{M\in(\mathrm{H}_{\theta_\alpha}^*)^{N_\alpha}:\pi_M(x_g)=g(M\cap\alpha)\}$$ is non-stationary in $N_\alpha$.
Let $A$ be the set of $\alpha$ such that $N_\alpha$ exists and let $$\phi \colon A \to H_{\aleph_2}, \qquad \phi ( \alpha ) = N_\alpha $$ be the enumerating function. Notice that $\phi\upharpoonright\alpha\in H_{\aleph_2}$ for all $\alpha\in A$, since $|N_\alpha|=\aleph_1$. Now we redo the proof of Laver’s theorem using the structures $N_\alpha$ as follows.
If the following does not hold for $\alpha$ we set $f(\alpha)=0$. Else we require
1. $\alpha\in A$, so that:
- $N_\alpha$ models that $\mathrm{H}_{\alpha}^{\aleph_1}(H_{\theta_\alpha})$ is stationary,
- $\psi(\theta_\alpha)$ holds in $N_\alpha$,
2. $f\restriction \alpha\in N_\alpha$.
In this case we set $f(\alpha)$ to be some $x\in (H_{\theta_\alpha})^{N_\alpha}$ such that $N_\alpha$ models that $$\{X\in\mathrm {H}_{\theta_\alpha}^*: f(X\cap\alpha)=\pi_X(x)\}$$ is non-stationary.
The set of $\alpha\in A$ such that $f\restriction \alpha\in N_\alpha$ is a club subset of $A$.
Let $M\in \mathrm {H}^*_{\theta^+}$ such that $\phi=\{N_\alpha:\alpha\in A\}\in M$. Let $\alpha=M\cap\aleph_2$. Then the transitive collapse of $M$ is $N_\alpha$ and we get that:
1. $\{N_\gamma:\gamma\in \alpha\cap A\}=\pi_M[\phi]\in N_\alpha$, and for all $\gamma\in A\cap M$, $N_\gamma$ has size $\aleph_1$ and thus is contained in $$M\cap H_{\aleph_2}=\pi_M[H_{\aleph_2}\cap M] .$$
2. $\psi(\theta_\alpha)$ holds in $N_\alpha$.
For all $\xi\in A$, to define $f\restriction \xi$ we just need to know the sequence $\langle N_\gamma:\gamma\in A\cap\xi\rangle$. Since $\phi\in M$, this means that $f\restriction \gamma\in M$ for all $\gamma\in M\cap\aleph_2$. Since $M$ is an $\aleph_1$-guessing model we get that $f\restriction \alpha=g\restriction \alpha$ for some $g\in M$. Thus $f\restriction \alpha=\pi_M(g)\in N_\alpha$.
Now let $j_G\colon V\rightarrow M_G$ be an elementary embedding induced by a $V$-generic filter $G$ for the full stationary tower on some Woodin cardinal $\delta>\theta$ such that $\mathrm {H}^*_{\theta^+}\in G$. Then, appealing repeatedly to items \[eqn.WOOjG\] and \[eqn.WOOjG2\] of Section \[subsec.WOOST\], we can see that the following holds in $M_G$:
1. $\operatorname{crit}(j_G)=\omega_2$,
2. $\omega_2\in j_G(A)$ since $j_G[H_{\theta^+}]=M\in (\mathrm{H}^*_{j(\theta)^+})^{M_G}$ (by items \[eqn.WOOjG\] and \[eqn.WOOjG2\] of Section \[subsec.WOOST\]) and thus in particular
- $(H_{\theta^+})^V=\pi_M[M]=(N_{(\omega_2)^V})^{M_G}=j_G(\phi)((\omega_2)^V)$,
- $\theta_{(\omega_2)^V}=\theta$,
- $(\mathrm{H}^*_{\theta})^{(N_{(\omega_2)^V})}=(\mathrm{H}^*_{\theta})^V$.
3. $f=j_G(f)\restriction (\omega_2)^V\in H_{\theta^+}= (N_{(\omega_2)^V})^{M_G}$
4. $(N_{(\omega_2)^V})^{M_G}$ models that $\psi(\theta)$ holds.
In particular in $M_G$, $j_G(f)((\omega_2)^V)$ is defined to be some $x\in N_{(\omega_2)^V}$ such that $$N_{(\omega_2)^V}\models
\{X\in\mathrm {H}^*_{\theta}:
f(X\cap\alpha)=\pi_X(x)\}\text{ is non-stationary.}$$ but $N_{(\omega_2)^V}=(H_{\theta^+})^V$, so we get that: $$(H_{\theta^+})^V\models
\{X\in\mathrm{H}^*_{\theta}:
f(X\cap\alpha)=\pi_X(x)\}\text{ is non-stationary.}$$ which occurs only if $$\{X\in\mathrm{H}^*_{\theta}:
f(X\cap\alpha)=\pi_X(x)\}\text{ is non-stationary in $V$.}$$ This is impossible since $j_G(f)((\omega_2)^V)=x$ iff $$\{X\in\mathrm{H}^*_{\theta}: f(X\cap\alpha)=\pi_X(x)\}\in G$$ which can occur only if the latter set is stationary in $V$. The proof of the claim and of the theorem is completed.
Further applications of guessing models {#sect.applications}
=======================================
We show that the failure of the weakest forms of square principle and the singular cardinal hypothesis are simple byproduct of the existence of guessing models. In particular the first application yields that the existence of a guessing models subsumes strong large cardinal hypotheses.
The failure of square principles {#subsect.square}
--------------------------------
Recall the following definitions:
\[def.square\] A sequence $\langle \mathcal{C}_\alpha : \alpha \in {\ensuremath{\text{{\rm Lim}}}}\cap E \cap \lambda \rangle$ is called a *$\square_E(\kappa, \lambda)$-sequence* if it satisfies the following properties.
1. $0 < |\mathcal{C}_\alpha| < \kappa$ for all $\alpha \in {\ensuremath{\text{{\rm Lim}}}}\cap E \cap \lambda$,
2. $C \subset \alpha$ is club for all $\alpha \in {\ensuremath{\text{{\rm Lim}}}}\cap E \cap \lambda$ and $C \in \mathcal{C}_\alpha$,
3. $C \cap \beta \in \mathcal{C}_\beta$ for all $\alpha \in {\ensuremath{\text{{\rm Lim}}}}\cap E \cap \lambda$, $C \in \mathcal{C}_\alpha$ and $\beta \in \operatorname{Lim}C$,
4. there is no club $D \subset \lambda$ such that $D \cap \delta \in \mathcal{C}_\delta$ for all $\delta \in \operatorname{Lim}( D ) \cap E \cap \lambda$.
We say that $\square_E(\kappa, \lambda)$ holds if there exists a $\square_E(\kappa, \lambda)$-sequence, and $\square(\kappa, \lambda)$ stands for $\square_\lambda(\kappa, \lambda)$.
Note that $\square_{\tau, <\kappa}$ implies $\square(\kappa, \tau^+)$ and that $\square(\lambda)$ is $\square(2, \lambda)$.
The theorem below is just a rephrasing using the notion of guessing models of the results on the failure of square principles Weiß obtained assuming his ineffability property or thin lists (see [@weiss]).
\[theorem.GM->non\_square\] Suppose there is a $\delta$-guessing model $M\prec H_\theta$ for some $\delta<\kappa_M$ and some regular $\theta>\kappa_M$. Then for every regular $\lambda\geq \kappa_M$ in $M\cap\theta$ such that $\sup(M\cap\lambda)<\lambda$, $\square_{\operatorname{cof}(<\kappa_M)}(\kappa_M, \lambda)$ fails.
The assumption that $M\cap\lambda$ is bounded in $\lambda$ for any regular $\lambda\in M$ above $\kappa_M$ might seem redundant and I conjecture that for any guessing model $M\prec H_\theta$ and any regular cardinal $\lambda\in M$ above $\kappa_M$, $\sup(M\cap\lambda)<\lambda$. We show below in Proposition \[prop.kunen\] (rephrasing Kunen’s proof that there cannot be an elementary $j\colon V_{\gamma+2}\rightarrow V_{\gamma+2}$) that this is the case for $\aleph_0$-guessing model.
We now turn to the proof of Theorem \[theorem.GM->non\_square\].
Assume not. Since $M$ is a $\delta$-guessing model, $M$ is closed under countable suprema, thus $\gamma=\sup(M\cap\lambda)<\lambda$ has uncountable cofinality. Pick a sequence $\langle \mathcal{C}_\alpha : \alpha\in \lambda, \operatorname{cf}(\alpha)<\kappa_M \rangle\in M$ witnessing $\square_{\operatorname{cof}(<\kappa_M)}(\kappa_M, \lambda)$. Since $|\mathcal{C}_\xi|<\kappa_M$ for all $\xi<\lambda$, $\mathcal{C}_\xi\subseteq M$ for all $\xi\in M$. Pick $C\in \mathcal{C}_\gamma$. Then $C\cap\xi\in\mathcal{C}_\xi\subseteq M$ for all $\xi\in M$ which are limit points of $C$. Since $M$ is closed under countable suprema, there are club many such $\xi$ of countable cofinality in $M$. Now given $z\in M\cap P_{\delta}\lambda$, find $\xi\in C\cap M$ above $\sup(z)$ and $D\in \mathcal{C_\xi}$ such that $C\cap\xi=D$. Then $C\cap z=D\cap z\in M$ since $z,D\in M$. Thus $C$ is $(\delta,M)$-approximated. Since $M$ is a $\delta$-guessing model, there is $E\in M$ be such that $C\cap M=E$. Then $$M\models E \text{ is a club subset of }\lambda$$ and for all $\xi\in M$ limit points of $E$, $$E\cap\xi\cap M=C\cap\xi\cap M=D\cap M$$ for some $D\in\mathcal{C}_\xi$. This shows that $M$ models that $E$ is a counterexample to $\langle \mathcal{C}_\alpha : \alpha \in \lambda \rangle$ being a $\square_{\operatorname{cof}(<\kappa_M)}(\kappa_M, \lambda)$-sequence.
\[prop.kunen\] Assume $M\prec H_\theta$ is an $\aleph_0$-guessing model. Then $M\cap\lambda$ is bounded in $\lambda$ for all regular $\lambda\in M$ above $\kappa_M$.
If $M\prec H_\theta$ is a $0$-guessing model such that $M\cap\lambda$ is unbounded in $\lambda$ for some regular $\lambda\in M$, we have that the transitive collapse of $M\cap H_\lambda$ is $H_\lambda$. Thus for any $A\in M\cap P(H_{\lambda})$ we get an elementary embedding $$j\colon \langle H_\lambda,\in, \pi_M(A)\rangle\to\langle H_\lambda,\in, A\rangle$$ with critical point $\pi_M(\kappa_M)<\lambda$ given by $j=\pi_M^{-1}$. We want to reach a contradiction mimicking by a Kunen’s celebrated result that there is no elementary $j:V\to V$.
Suppose first that $\lambda$ is inaccessible. Then there is a club subset of $\gamma<\lambda$ such that $j(\gamma)=\gamma$. For any such $\gamma$ we would get that $M\cap V_{\gamma+2}$ collapses to $V_{\gamma+2}$. This would give that $$\pi_{M\restriction V_{\eta+2}}^{-1}\colon V_{\eta+2}\rightarrow V_{\eta+2}$$ is elementary. This is impossible by [@Kan09 Corollary 21.24].
Suppose now $\lambda$ is a successor. In this case we will argue mimicking the proof of Kunen’s inconsistency given by Zapletal [@Zap96]. We recall the following notions of pcf-theory (the reader can find in Abraham and Magidor’s chapter in [@HST] the basic development of pcf-theory): A scale on $(\prod_n\gamma_n,<^*)$ is a family of functions well-ordered under $<^*$ and cofinal with respect to $<^*$, where $f<^*g$ if $f(n)<g(n)$ for eventually all $n$. The $<^*$-exact upper bound of a family of functions $\mathcal{F}\subseteq{\ensuremath{\text{{\rm Ord}}}}^\omega$ is a least upper bound of $\mathcal{F}$ under $<^*$ when such least upper bound can be defined. If $\mathcal{F}$ has a least upper bound, then it is unique modulo finite changes. Shelah showed that for any $\eta$ singular cardinal of countable cofinality there is a $<^*$-well-ordered family $\mathcal{F}\subseteq \eta^\omega$ of order type $\eta^+$ which has a least upper bound $f\in\eta^\omega$ such that $\sup_n f(n)=\eta$ and $f(n)$ is a regular cardinal for all $n$. A $\mathcal{F}$ with these properties is called a scale on $\prod_n f(n)$.
Now we proceed as follows: We can assume that $\lambda$ is the least regular such that $j(\lambda)=\lambda=\gamma^+=j(\gamma)^+$. Notice that for any cardinal $\eta$, $j(\eta)=\eta$ if $j(\eta^+)=\eta^+$. So the minimality of $\lambda$ entails that $\gamma$ is the least fixed point of $j$. This means that $\gamma=\sup_n j^n( \operatorname{crit}( j ) )$. Now by Shelah’s result a scale $\mathcal{G}\subseteq \gamma^\omega$ of order type $\lambda$ with exact upper bound $g$ exists in $H_\delta$ for any $\delta>\lambda$. Since $\theta>\lambda$ and $M\prec H_\theta$, such a $\mathcal{G}$ with exact upper bound $g$ can be found in $M$. Then $\pi_M(\mathcal{G})=\mathcal{F}$ by elementarity is again a scale on $\prod_n f(n)$ where $f=\pi_M(g)$. Now let us consider $j=\pi_{M\restriction H_\lambda\cup\{\mathcal{G}\}}^{-1}$. Then $$j\colon \langle H_\lambda,\in, \mathcal{F}\rangle\to\langle H_\lambda,\in, \mathcal{G}\rangle$$ is elementary and $ j [ \mathcal{F} ] = \{ j ( f_\xi ) : \xi < \lambda\}$ will have as an upper bound the function $$h ( n )=\sup j[f(n)]=j[\gamma_n]<j(\gamma_n)=j(f)(n)=g(n).$$ Moreover $j[\mathcal{F}]\subseteq j(\mathcal{F})=\mathcal{G}=\{g_\alpha:\alpha<j(\lambda)=\lambda\}$ is a cofinal subset so the two families of increasing functions will have the same exact upper bound $j(f)=g$. This is impossible since any $f\in j[\mathcal{F}]$ is everywhere dominated by $h$ which is dominated by $g$ modulo finite, so $g$ cannot be an exact upper bound for $j [ \mathcal{F}]$.
If the proof of the proposition is recast using $\delta$-guessing model, one can rule out the case $\lambda$ successor by essentially the same argument. However if $M\prec H_\theta$ is a $\delta$-guessing model and $\lambda\in M$ is inaccessible such that $\operatorname{otp}(M\cap\lambda)=\lambda$, then $\pi_M[M\cap H_\lambda]$ could be different from $H_\lambda$. For this reason we cannot reproduce for an arbitrary $\delta$-guessing models the proof of the proposition in this case.
A proof of ${\ensuremath{\text{{\sf SCH}}}}$ {#subsect.SCH}
--------------------------------------------
We give a proof of ${\ensuremath{\text{{\sf SCH}}}}$ assuming there are $\aleph_1$-internally unbounded, $\aleph_1$-guessing models $M$ of size less than $\kappa_M$. This assumption follows for example from ${\ensuremath{\text{{\sf PFA}}}}$ and also from the existence of a supercompact cardinal. We recall the following definition from [@covering_properties]:
\[def.covering\_matrix\] Suppose $\lambda$ is a cardinal with $\operatorname{cf}\lambda = \omega$. $$\mathcal{D} = \langle D(n, \alpha) : n < \omega,\ \alpha < \lambda^+ \rangle$$ is called a *strong covering matrix on $\lambda^+$* if
1. $\bigcup_{n < \omega} D(n, \alpha) = \alpha$ for all $\alpha < \lambda^+$,
2. $D(m, \alpha) \subset D(n, \alpha)$ for all $\alpha < \lambda^+$ and $m < n < \omega$,
3. for all $\alpha < \alpha' < \lambda^+$ there is $n < \omega$ such that $D(m, \alpha) \subset D(m, \alpha')$ for all $m \geq n$,
4. for all $x \in P_{\omega_1} \lambda^+$ there is $\gamma_x < \lambda^+$ such that for all $\alpha \geq \gamma_x$ there is $n < \omega$ such that $D(m, \alpha) \cap x = D(m, \gamma_x) \cap x$ for all $m \geq n$,\[property.covering\_matrix\]
5. $|D(n, \alpha)| < \lambda$ for all $\alpha < \lambda^+$ and $n < \omega$.
The following simple facts are proved in [@covering_properties]:
\[fact.coveringmatrix\] Assume $\lambda>2^{\aleph_0}$ has countable cofinality. Then there is a strong covering matrix $\mathcal{D}$ on $\lambda^+$.
\[fact.coveringmatrix1\] Assume that for all $\lambda>2^{\aleph_0}$ of countable cofinality, there is a strong covering matrix $\mathcal{D}$ on $\lambda^+$ and an unbounded subset $A$ of $\lambda^+$ such that $P_{\omega_1}A$ is covered by $\mathcal{D}$. Then ${\ensuremath{\text{{\sf SCH}}}}$ holds.
\[lemma.covering\_matrix\_slender\] Suppose $\lambda$ is a cardinal with $\operatorname{cf}\lambda = \omega$ and $\mathcal{D}$ is a strong covering matrix on $\lambda^+$. Let $\theta$ be sufficiently large. Suppose $M \in P_{\omega_2} H_\theta$ is an $\aleph_1$-internally unbounded model and $\mathcal{D} \in M$. Then there is $n < \omega$ such that $D(m, \sup (M \cap \lambda^+) ) \cap x \in M$ for all $x \in P_{\omega_1} \lambda^+ \cap M$ and $m \geq n$.
Assume not and for each $n$ pick $x_n\in M\cap P_{\omega_1}\lambda^+$ such that $D(n, \sup (M \cap \lambda^+) ) \cap x_n \not\in M$. By $\aleph_1$-internally unboundedness of $ M $, there is a countable $x\in M$ containing all the $x_n$. Now $\gamma_x\in M$ by elementarity and thus there is $n_0$ such that $D(n, \sup (M \cap \lambda^+) )\cap x= D(n, \gamma_x )\cap x\in M$ for all $n\geq n_0$. This means that $$D(n, \sup (M \cap \lambda^+) )\cap x_n =
D ( n, \sup (M \cap \lambda^+) ) \cap x \cap x_n\in M$$ since $D(n, \sup (M \cap \lambda^+) )\cap x\in M$ and $x_n\in M$. This is the desired contradiction.
\[the.GMSCH\] Suppose that for all regular $\theta \geq \kappa$, there are stationarily many $\aleph_1$-guessing models $M\in\mathrm {G}_\kappa^{H_\theta}$ of size $\aleph_1$ which are $\aleph_1$-internally unbounded. Then [$\text{{\sf SCH}}$]{} holds.
Let $\lambda$ be a cardinal with $\operatorname{cf}\lambda = \omega$. By Fact \[fact.coveringmatrix\] there exists a strong covering matrix on $\lambda^+$ and by Fact \[fact.coveringmatrix1\] it suffices to show there is an unbounded $A \subset \lambda^+$ such that $P_{\omega_1} A$ is covered by $\mathcal{D}$, that is, for all $x \in P_{\omega_1} A$ there is $\alpha < \lambda^+$ and $n < \omega$ such that $x \subset D(n, \alpha)$.
Let $\theta$ be sufficiently large. Pick an $\aleph_1$-guessing model $M\prec H_\theta$ which is $\aleph_1$-internally unbounded and is also such that $\mathcal{D}\in M$. By Proposition \[prop.basiguessing\] we may assume $\operatorname{cf}\sup (M \cap \lambda^+) \geq \omega_1$. By Lemma \[lemma.covering\_matrix\_slender\] there is $n' < \omega$ such that $D ( m, \sup ( M \cap \lambda^+ ) ) \cap x \in M$ for all $x \in P_{\omega_1} \lambda^+ \cap M$ and $m \geq n'$. As $M$ is an $\aleph_1$-guessing model, this means that for all $m \geq n'$ there is $A_m \in M$ such that $D ( m , \sup (M \cap \lambda^+) ) = A_m \cap M$.
Since $\operatorname{cf}\sup (M \cap \lambda^+) \geq \omega_1$ and ${\ensuremath{{\textstyle\bigcup}}}\{ D(m, \sup (M \cap \lambda^+)) : m < \omega \} = \sup (M \cap \lambda^+)$ there is an $n' \leq n < \omega$ such that $A_n$ is unbounded in $\sup (M \cap \lambda^+)$. As $A_n \in M$, this implies $A_n$ is unbounded in $\lambda^+$.
Let $x \in M\cap P_{\omega_1} A_n $. Then $$x=A_n \cap x = D(n, \sup (M \cap \lambda^+))\cap x \subseteq D(n, \sup (M \cap \lambda^+)) .$$ Thus $H_\theta$ models that $x$ is covered by some $D(n,\alpha)$. Since $x\in M$, also $M$ models it. Since this occurs for an arbitrary $x\in M\cap P_{\omega_1} X$, $M$ models $P_{\omega_1} A_n$ is covered by $\mathcal{D}$, whence it really holds.
Conclusions and open problems
=============================
We close this paper with a list of open problems and some guesses on their possible solutions:
1. Assuming ${\ensuremath{\text{{\sf PFA}}}}$ in $W$, $\mathrm {G}^{\aleph_1}_{\aleph_2} W_\theta$ is stationary for all inaccessible $\theta$. Is it possible to build a transitive inner model $V$ of $W$ such that $\aleph_2$ is supercompact in $V$? Note that this would be the case if in $V$, $\mathrm {G}^{\aleph_0}_{\aleph_2} V_\theta$ is stationary for all inaccessible $\theta$. In [@VIAWEI10] and [@weiss] there are several positive partial answers when we assume that $W$ is a forcing extension of $V$. A possible attempt to overcome this latter assumption would be to isolate in models of ${\ensuremath{\text{{\sf MM}}}}$ some stationary subset $T$ of $\mathrm {G}^{\aleph_1}_{\aleph_2} W_\theta$, and then try to argue that $\aleph_2$ is $\theta$-supercompact in $L(\{M\cap\theta: M\in T\})$ or in some simple transitive class model of ${\ensuremath{\text{{\sf ZFC}}}}$ defined using $\{M\cap\theta: M\in T\}$ as a parameter to define it.
2. Is it at all consistent that there are $\delta$-guessing models which are not $\aleph_1$-guessing for some $\delta > \aleph_1$? It seems reasonable to expect this is the case and could be achieved using Krueger’s mixed support iterations techniques as developed in [@krueger].
3. Is it consistent that for a guessing model $M$, $\kappa_M$ is the successor of a singular cardinal? Or that for a singular $\delta$, there are $\delta$-guessing models $M$ which are not $\xi$-guessing for any $\xi<\delta$? I think that this shouldn’t be possible.
4. Can the isomorphism of types theorem be proved also for $\delta$-guessing models which are not $\delta$-internally club?
5. Given $f\colon \kappa\rightarrow H_\kappa$, what is the class of functions $g\colon P(H_\theta)\rightarrow H_\theta$ such that $\{M\in\mathrm {G}_\kappa^{H_\theta}: f(M\cap\kappa)=g(M)\}$ is stationary? Can the supercompactness of $\kappa$ be characterized by the existence of an $f\colon \kappa\rightarrow H_\kappa$ such that $\{M\prec H_\theta: f(M\cap\kappa)=g(M)\}$ is stationary for a large family of $g\colon P_\kappa H_\theta\rightarrow H_\theta$?
|
---
abstract: 'Image aesthetic quality assessment has got much attention in recent years, but not many works have been done on a specific genre of photos: Group photograph. In this work, we designed a set of high-level features based on the experience and principles of group photography: Opened-eye, Gaze, Smile, Occluded faces, Face Orientation, Facial blur, Character center. Then we combined them and 83 generic aesthetic features to build two aesthetic assessment models. We also constructed a large dataset of group photographs - GPD- annotated with the aesthetic score. The experimental result shows that our features perform well for categorizing professional photos and snapshots and predicting the distinction of multiple group photographs of diverse human states under the same scene.'
address:
- 'School of computer science and technology, Tiangong University, Tianjin, 300387, China'
- 'Tianjin Key Laboratory of Autonomous Intelligence Technology and Systems, Tianjin,300387, China'
- 'Business School, Nankai University, Tianjin, 300072, China'
author:
- Yaoting Wang
- Yongzhen Ke
- Kai Wang
- Cuijiao Zhang
- Fan Qin
bibliography:
- 'mybibfile.bib'
title: Aesthetic Quality Assessment for Group photograph
---
Image aesthetic quality assessment,group photograph,machine learning,feature design,dataset
Introduction
============
With the rapid growth of image applications, the traditional image quality evaluation no longer satisfies the practical need. Thus the image aesthetic quality assessment (IAQA) was born. IAQA uses the computer simulation of human perception and cognition of beauty to automatically assess the “beauty” of images (i.e., Computer evaluation of image aesthetic quality) [@1]. It mainly responds to the aesthetic stimuli formed by the image under the influence of aesthetic elements such as composition, color, luminance, and depth of field.
In daily life, we often encounter a situation where we need to take group photographs for souvenirs. So how to estimate the aesthetic values of a group photograph and further provide a guidance system for real-time group photo shooting will become meaningful. The current methods of IAQA mainly focus on the effects of composition, color, light and shadow, depth of field, and other components on the aesthetics of the entire image, which can classify the professional photographs and snapshots, as shown in Figure1(a). Nevertheless, when evaluating the aesthetics of group photographs, people are not only concerned with the above factors but also focus on the state of the person in the image, such as whether someone’s eyes closed, does not look at the camera, the face is blocked, does not smile and other factors. If these factors are not taken into account in the aesthetic quality assessment of group photography, the evaluation will not be accurate. An example is shown in Figure 1(b), the two images assessed by the general method have similar ratings. However, when considering the criteria for group photography, it is clear that we are more satisfied with the first one.
Therefore, we propose a method to assess the aesthetic quality of the group photograph. Firstly, we extracted the texture, brightness, low depth of field, color, and other features commonly used in IAQA. Moreover, we designed seven specific features that conform to group photography experience and principles, such as whether the face in the photo is occluded, whether someone closes their eyes, whether they smile or not. We then constructed a dataset (GPD) specifically for group photography aesthetic quality assessment. It contains a total of 1000 pictures, which are selected from the network, the existing IAQA dataset, and photos taken by ourselves. Finally, with the extracted features, we trained a classifier and a regression model on the GPD-dataset, which are used to classify photos into good and bad categories and predict aesthetic scores, respectively.
To summarize, our main contributions in this paper are as follows:
- We design 7 new features related to group photography as the standard for aesthetic assessment.
- We build GPD dataset annotated with the aesthetic score, and develop an online annotation system to collect users’ aesthetic evaluation of photographs.
- We propose a classifier and a regression model trained by our features, which outperform existing methods in in group photo aesthetic evaluation.
Related works
=============
Early research on IAQA focused on low-level visual features and then training classifiers or regressors to evaluate image aesthetics [@2; @3; @4; @5]. In 2004, Microsoft Asia Research Institute and Tsinghua University [@5] jointly proposed a method that can automatically distinguish the photographs taken by professional photographers from those taken by customers. This work is considered as the earliest research on IAQA. They used a 21-class, a total of 846-dimensional low-level global features to learn the classification model to classify the test images aesthetically. In 2006, [@3] began to use local features for aesthetic assessment, combining the low-level features such as color, texture, shape, picture size and high-level features such as depth of field, tripartite rule, regional contrast and so on which are usually used for image retrieval, and then trained the SVM classifier for the binary classification of image aesthetic quality. [@4] proposed using global edge distribution, color distribution, hue counting, contrast, and brightness to represent the image. Based on these features, the naive Bayesian classifier is trained. All the above works are aesthetic evaluations that are unrelated to content. Since 2010, some aesthetic assessment research related to content has appeared [@6; @7; @8]. In 2014, with the emergence of AVA [@9], a large-scale aesthetic analysis dataset, significant progress had been made in the automatic aesthetic analysis by using deep learning technology [@10; @11; @12; @13; @14]. The classification accuracy rate of ILGNet-Inc.V4 proposed by [@10; @11; @12] ranked first in the world on the ava dataset. In recent years, researchers have mainly studied the problem of IAQA from different tasks. In order to solve the problem of the need for subjective labeling when the image database is established, Ning Ma [@15] proposed a deep attractiveness rank net (DARN) model to learn aesthetic scores. [@16] proposed a query-based aesthetic assessment deep learning model that makes different aesthetic evaluations based on different styles of images. [@17] considers not only objective factors but also subjective factors of user reviews for aesthetic assessment. [@18] extended the one-dimensional score to a multi-dimensional aesthetic space score. [@19] proposed an A-Lamp CNN architecture to learn the fine-grained and the overall layout aesthetic assessment simultaneously. In addition, in terms of the prediction of image aesthetic distribution, jinxing et al. [@20] proposed the method that predicts image aesthetic distribution, opening the direction of aesthetic prediction in the era of deep learning. There are also some research results in the aesthetic assessment of faces and portraits [@21; @22; @23]. In the study of group photo images, [@24] proposed the spring-electric model, which recommended the appropriate station and the proportion of the characters to the photograph. However, they did not evaluate the aesthetic quality from the perspective of the subject. To our best knowledge, there are currently no research on the aesthetic assessment of group photography.
Aesthetic Factors for Group photography
=======================================
In this section, we will discuss the extraction of features for representing the aesthetic quality of a group photograph. We extract two major groups of features: group photographic features conformed to the group photographic rules and low-level generic aesthetic features proposed in [@3; @27; @28]. These features are combined to obtain better estimates of the aesthetic scores. The following subsections explain each group of features.
Group Photography Features
--------------------------
When people assess group photographs, they usually pay more attention to the facial information and position of the person in the image. Therefore, we utilize proven face recognition tools [@face] and [@baidu] to extract face-related information and perform further feature design based on this information.
Assuming $N$ faces are detected in a group photograph, the detected face sequence $F$ is represented as follows: $$F=\{face^1,face^2,\dots,face^i\} \quad i\in\{1,2,\dots,N\},$$ where $face^i$ represents the facial information of the ith person, which includes: The coordinates of the top-left point of the facial box ($x,y$); The height and width of the facial box ($h,w$); Confidence ($c_i$) of different eye states ($S$ broken into 6 states, detailed in section 3.1.1); The gaze direction of the left and right eyes ($D_l,D_r$); The value of the smile ($m$); Rotation angle of the head ($\gamma$); The occlusion degree of seven regions of the face ($o_i$,detailed in section 3.1.2); The position coordinates of the person in the photograph ($P$); The degree of blur of the face ($b$).
### Open-eyed
Eyes are the windows to one’s soul. When we shoot portraits, we tend to focus on people’s eyes, and so do group photos. If someone’s eyes are closed or obscured in a group picture, the beauty of that will be greatly decreased. Therefore, we consider the eyes states of each person in the photo, which including: Eye opening without glasses $S_1$; Eye opening with ordinary glasses $S_2$ ; Wearing sunglasses $S_3$; Covered eye $S_4$; Eye closing without glasses $S_5$ and eye closing with ordinary glasses $S_6$. We use [@face] to predict the confidence of each status of the left and right eye, respectively, which are {$c_1,c_2,c_3,c_4,c_5,c_6$}. The $\sum_{i=1}^{6} c_i$ equals 100. We select the state of the maximum of six confidences as the condition of the eye. If the condition of eyes belongs to one of these three states ($S_1,S_2,S_3$), we judge the person is open-eyed and thus formulate that as: $$S_r=f(\max_{1\leq j\leq 6}c_j^{(r)}),$$ $$S_l=f(\max_{1\leq j\leq 6}c_j^{(l)}),$$ $$E_i=\begin{cases}
1 & \text{if} \; S_l \in \{S_1,S_2,S_3\} \text\;{and} \;S_r \in \{S_1,S_2,S_3\} \\
0 & \text{otherwise}
\end{cases}
\quad i\in\{1,2,\dots,N\},$$ where $S_l$ and $S_r$ represent the final state prediction of the left and right eyes, respectively. $c_j^{(l)}$ and $c_j^{(r)}$ are the confidence of 6 states corresponding to the left and right eyes. $f$ is the mapping between confidence and corresponding status. We further calculate the proportion of people whose open-eyed. Based on the experience for group photograph assessment, we found the proportion has a non-linearity relationship with the evaluation result. i.e., when all the people open their eyes, the assessment is high. Once someone closes their eyes, the evaluation of the image falls into the bad category, and then gradually decrease with the number of close-eyed people. Thus we fit the formula (5), where $\frac{1}{N}\sum_{i=1}^{N} E_i$ is the proportion and $f_1$ is the final feature of open-eyed. $$f_1=\begin{cases}
1 & \text{if} \;\frac{1}{N}\sum_{i=1}^{N} E_i=1 \\
1-2^{-\frac{1}{N}\sum_{i=1}^{N} E_i} & \text{otherwise}
\end{cases}
\quad i\in\{1,2,\dots,N\},$$
### Occluded Faces
In group photography, the most basic requirement is that everyone’s face is not occluded. If someone is masked, no matter how splendid the color, composition, light and shadow of the photo is, the photo will be discarded without hesitation. Hence, whether the face is occluded or not is another crucial criterion for judging the quality of the group photograph.
We use the method provided by [@baidu] to obtain the information about occluded faces in the photograph. The face is segmented into seven regions (see Figure 2. (a)): the left and right eye, the left and right cheek, mouth, jaw, and nose. An occlusion degree will be calculated of each region, which is a floating-point number in the range \[0, 1\], where 1 means that the region is completely occluded. When the occlusion degree of any region exceeds the recommendation threshold provided in BaiduAI, we judge that the person’s face is occluded. We define the occluded face as: $$O_i=\begin{cases}
1 & \exists\; o_i \geq \theta_j \\
0 & \text{otherwise}
\end{cases}
\quad i\in\{1,2,\dots,N\},j\in\{1,2,\dots,7\},$$ where $O_i$ indicates whether the face of the ith person is occluded, and 1 denotes occlusion. The $o_j$ and $\theta_j$ are the value of the occlusion degree and threshold of each region. Then we calculate the proportion of the number of un-occluded people defined as ($1-\frac{1}{N}\sum_{i=1}^{N} O_i$). Same as $f_1$, the proportion and evaluation also satisfies the nonlinearity relationship. The $f_2$ is the occluded faces feature, and the formula is described as follows. $$f_2=\begin{cases}
1 & \text{if} \;\sum_{i=1}^{N} O_i=0 \\
1-2^{-(1-\frac{1}{N}\sum_{i=1}^{N} O_i)} & \text{otherwise}
\end{cases}
\quad i\in\{1,2,\dots,N\}.$$
### Face Orientation
A word that photographers often say during photography is “Looking at the camera”. If a person in viewfinder looks at the camera, but the head tilted, the photo is not a high-quality image. Therefore, we get the yaw angle of head as $\gamma$, where $\gamma$ $\in$ \[-180,180\]. When $\gamma$ $\in$ \[-30,30\], it is considered that facing the camera. We record whether the character is facing the camera as $H_i$, which equals means yes. Same as above, the proportion of the number of people without head-tilted to the total number of people $N$ is calculated. This proportion also fits a non-linear relationship. $f_3$ is the face orientation feature. The formula is as follows. $$H_i=\begin{cases}
1 & \gamma_i \in[-30,30]\\
0 & \text{otherwise}
\end{cases}
\quad i\in\{1,2,\dots,N\},$$ $$f_3=\begin{cases}
1 & \text{if} \;\frac{1}{N}\sum_{i=1}^{N} H_i=1 \\
1-2^{-\frac{1}{N}\sum_{i=1}^{N} H_i} & \text{otherwise}
\end{cases}
\quad i\in\{1,2,\dots,N\},$$
### Gaze
For formal group photography, everyone’s focus on the lens is an important criterion. So we designed a feature to represent the proportion of people looking at the camera. There are three prerequisites before estimating gaze: eyes open, facing the camera, eyes are not occlusion. We utilize the information detected by face++ to calculate the direction of the gaze. The gaze estimation process is as follows:
1. Determine the center of the circle: $O=(C_1+C_2)/2$.
2. Determine the radius: $R=max(w,h)$.
3. Calculating the average gaze: $D=(D_l+D_r)/2$.
4. Calculating the gaze junction point coordinates: $p=O+R*D$.
Where $C_1$ and $C_2$ are the landmarks of the left and right eyeball center, $w$ and $h$ are the width and height of the rectangle of face, respectively. The gaze direction vector of the left and right eye are recorded as $D_r=(x_r,y_r)$ and $D_l=(x_l,y_l)$. We use the face landmarks to define a rectangular range (see Figure 2.(b)), If the gaze junction point falls within the range, it is judged that the people is looking at the lens. It is defined as follow, $$G_i=\begin{cases}
1 & p_i \in Range_i\\
0 & \text{otherwise}
\end{cases}
\quad i\in\{1,2,\dots,N\},$$ where $P_i$ represents the coordinates of the gaze junction of the ith person in the frame, and $Range_i$ is the rectangular range of the ith person, $G_i=1$ means looking at the camera. Then we take the ratio of the people looking at the lens, and the ratio also meets the nonlinear relationship with the assessment. $f_4$ is the gaze feature. $$f_4=\begin{cases}
1 & \text{if} \;\frac{1}{N}\sum_{i=1}^{N} G_i=1 \\
1-2^{-\frac{1}{N}\sum_{i=1}^{N} G_i} & \text{otherwise}
\end{cases}
\quad i\in\{1,2,\dots,N\},$$
### Facial blur
Whether the face is clear or not is essential to the quality of a photograph. Therefore, we obtain the blur degree $b_i$ of face by Face++. We employ the recommended threshold $v$ (generally $v$ is 50) as the threshold. If the blur degree greater than the threshold, we consider the person’s face was not captured clearly. It can be formalization as: $$B_i=\begin{cases}
1 & b_i > v \\
0 & b_i \leq v
\end{cases}
\quad i\in\{1,2,\dots,N\},$$ where $B_i$ indicates whether the face of the ith person in the photograph is blurred or not. Then we calculated the percentage of the number of people whose facial blur degree exceeded the threshold as $\frac{1}{N}\sum_{i=1}^{N} B_i$. The higher the percentage, the higher the quality. $f_5$ is facial blur feature. $$f_5=\begin{cases}
1 & \text{if} \;\sum_{i=1}^{N} B_i=0 \\
1-\frac{1}{N}\sum_{i=1}^{N} B_i & \text{otherwise}
\end{cases}
\quad i\in\{1,2,\dots,N\}.$$
### Smile
Smile plays a vital role in the emotional expression of group photography. Through observation, we found that a large proportion of people smiling in group photograph is often attractive and easier to remember than no smile in group photograph. We use $m$ to represent the degree of smile. There is a threshold $w$ for the degree of smile provided by Face++. We count the number of people with a smile formulated as $\sum_{i=1}^{N} M_i$, which the degree of smile greater than the threshold. Then we take the ratio of the number of people with smile as the smile feature $f_6$, which is defined as: $$M_i=\begin{cases}
1 & m_i > w \\
0 & m_i \leq w
\end{cases}
\quad i\in\{1,2,\dots,N\},$$ $$f_6=\begin{cases}
1 & \text{if} \;\sum_{i=1}^{N} M_i=0 \\
\frac{1}{N}\sum_{i=1}^{N} M_i & \text{otherwise}
\end{cases}
\quad i\in\{1,2,\dots,N\}.$$
### Character center
Through observation and experience, we found that in a good group photo, the positions of the people are usually horizontally centered and uniformly arranged, particularly the formal group photos. Therefore, the horizontal position of people is also positively correlated to the quality of group photograph.We sequentially detect the horizontal x-axis coordinate of the center of each person’s face, represented as $x_i$, and then average the x-axis coordinates, which is represented by $P_x$ define as: $$P_x=\frac{\sum_{i=1}^{N}x_i}{N}\quad i\in\{1,2,\dots,N\}.$$ Next, we compute the relative position $R$ of the character center and the picture. We formulate that as $R=P_x/W$, where $W$ is the width of the frame. We divided the photograph evenly into five parts. If $R$ is in the range of 0.4 to 0.6, it means that $R$ is located in the center of the photograph, i.e., the people position is horizontally centered. We call $f_7$ the character center feature. $$f_7=\begin{cases}
1 & 0.4\leq R\leq 0.6 \\
0 & \text{otherwise}
\end{cases}
\quad i\in\{1,2,\dots,N\}.$$
Generic Aesthetic Features
--------------------------
In addition to group photographic features, we selected 83 features from the generic aesthetic features mentioned in [@3; @27; @28], such as exposure, saturation and texture based on wavelet transform, as aesthetic features to evaluate group photography aesthetics. These features can be divided into four categories: color, local, texture, and composition. The above features are not the focus of this paper, so briefly described in Table 1.
**Category** **Short Name** **\#** **Description**
-------------- ----------------------------- --------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------
Brightness, Hue, Saturation f8-f13 mean brightness, saturation, and hue of the image and the center of the picture [@3].
Emotion f52-f54 emotional measure based on brightness and saturation[@28].
Colorfulness f55 colorfulness measured, using the Earth Mover’s Distance (EMD) between the histogram of an image and the histogram having a uniform color distribution [@28].
Color f56-f71 amount of black, silver, gray, white, maroon, red, purple, fuchsia, green, lime, olive, yellow, navy, water blue [@27].
Disconnected Region f28 image segmentation, based on K-means, number of disconnected regions in the image [@3].
Local HSV f29-f43 average H, S and V values for each of the top 5 connected regions [@3].
Ratio f44-f48 the size ratio of the top 5 connected regions with respect to the image. [@3]
Wavelet textures f14-f25 after three-level wavelet transform, wavelet textures for each channel (Hue, Saturation, Brightness) and each level (1-3), sum of all levels for each channel [@3].
GLCM f72-f83 features based on the GLCM: contrast, correlation, homogeneity, energy for Hue, Saturation and Brightness channel [@28].
Image size f26-f27 Image size, sum of the length and width; image proportion, ratio of the length and width. [@3]
Low Depth of Field (DOF) f49-f51 low depth of field indicator; ratio of wavelet coefficients of inner rectangle vs. whole image (for Hue, Saturation and Brightness channel) [@3].
Dynamics f84-f89 absolute angles, relative angles, and lengths of static (horizontal, vertical) and dynamic (oblique) lines [@28].
Level of Detail f90 number of segments after waterfall segmentation [@28].
: 83 generic aesthetic features
Group photography Dataset
=========================
The datasets related to photography aesthetics include AVA [@9], AADB [@13]. AVA included 250,000 images, each with the corresponding aesthetic classification and rating labels. AADB contains 10,000 images, which has more balanced distribution of professional photograph and snapshot. Each image is annotated with score and eleven attributes. Nevertheless, there is no dataset for aesthetic evaluation of group photography at present. To this end, we collected a group photography dataset GPD by ourselves, which consists of three parts: group photographs shot by ourselves, selected from the existing aesthetic photography dataset, and obtained through internet. GPD contains a total of 1000 group photographs, and each image has been scored. Samples of GPD is shown in Figure 3.
#### (a)
Shooting by ourselves: Our research team used mobile phones and SLR cameras to take some group photographs. During the photography, the subjects constantly changed position and their expression. Most of the time, the photographer was in the status of continuous shooting, and deliberately took some photographs under the condition of out of focus, overexposure, not following the composition, and blurring caused by shaking the hands. The photographs taken by ourselves are mostly image pairs, i.e., multiple photos of different states are taken in the same scene. This is for better explain the inaccuracy of traditional method evaluating the group photographs. This section contains a total of 600 images.
#### (b)
Selected from the existing dataset: We selected part of the group photographs from the AVA and AADB datasets. The sources of these images are mostly social networking sites such as Flickr and DPChallenge. Most of the photographs were shot and uploaded by amateur photographers. We selected the group photographs among them, but the aesthetic quality of these is not high, and there are photographic problems such as blurring and overexposure. So this partition balances the distribution of quality in GPD, making GPD more robust. This section contains a total of 224 images.
#### (c)
Download from the Internet: We selected group photos from image sites such as Baidu Pictures, Petal.com. This partition includes 74 images, all of which are formal group photographs. They are taken by professional photographers and have high aesthetic quality. The photographs in this partition are more attractive than the previous two.
To obtain the aesthetic annotations of the group photograph, we designed an online annotation tool, which can rate the group photograph that appears randomly - made the assessment based on the first impression - on the website by users. The scores range from 1 to10. We give tips on the website for scoring, “please pay attention to the following factors when scoring: face occlusion, eyes closed, gaze, smile, and general aesthetic factors such as lighting, composition, color, and picture clarity.” In the end, each photograph is assessed by 5 to 20 people, and the average score of each image is taken as its ground truth label. In addition, the website has an image upload model, so users can voluntarily upload their own group photographs for the GPD. Figure 4 shows the probability distribution of GPD.
In GPD dataset, there are two kinds of annotation for each image, one is that the binary value represents the quality of the image, which is used for classifier training, and the other is the score, which is used for regression training. The binary label is obtained by binarizing the score label with 6 (the median value of the aesthetics label in the dataset) as the dividing line
Aesthetic Quality Assessment for Group Photograph
=================================================
In order to verify the effectiveness of our proposed group photo aesthetic features, we proposed a method whose system flow as in Figure 5. We first construct a group photo dataset, including the image and the label (ground truth), and then perform image preprocessing on all images. On the processed image, we extract the group photograph features and generic aesthetic features, and store them into a vector. After feature extraction, the dataset is divided into a training and test set to training a classifier and a regression. The classifier classifies the photo into two categories: good or bad. The regressor evaluates the image aesthetics with a score of 1 to 10, and finally uses the trained classifier and regression to predict the photo in the test set. We compared the results with the test set label to estimate the accuracy of the classifier and regressor.
Before generic aesthetic features extraction, we preprocess all the images. The processes include: Adjusting the image size to 128 \*128 pixels, which can not only retain enough image information but also meet the efficiency of calculation; Converting RGB color space into HSV and LUV color space, some features need to be extracted from these two color spaces; The K-means is used to segment the image according to chromaticity in the LUV color space; The Waterfall segmentation [@29] is used to segment the image into continuous regions in the HSV color space. On the basis of these image preprocessing, the features are extracted according to the description in Table 1.
Before group photograph feature extraction, we utilized Face++ to detect and save the information of facial recognition, the state of the person’s eyes, the smile degree, the rotation angle of face, the degree of facial blur and the landmarks of face in each image from the GPD. We applied Baidu AI’s face detection tool to detect and save the face occlusion of the person. Based on these information, the group photograph features are calculated according to Section 3.1.
Experiments
===========
This section shows the effectiveness of our proposed features, and comparison of the performances of our method with other methods, in the specific genre: group photography. Firstly, we used the random forest to obtain the importance of each feature to analyze their impact on assessment. Secondly, we applied k-fold cross validation (k = 10) to split GPD into train set and test set, then trained a classifier using support vector machine (SVM) and a regression model using random forest regression (RF). Finally, we report the performance of this method compared with other methods based on deep learning.
Importance of features
----------------------
Before evaluating the importance of features, all 90 features were normalized by the Z-score standardization method, i.e., using conversion function : $(X-mean)/std$. We used the Gini-based Random Forest [@30] to analyze the respective importance ranking of all features for the model. The top 33 features which importance greater than 0.011 are shown in Figure 6.
It can be seen that there are 5 group photograph features in the ranks. Among them, the importance of gaze features and opened-eyes features is much higher than other, indicating that the eyes state is important in the group photo evaluation. The importance of the central position of the character, face occlusion, and smile also exceeded the average value, which also played a positive role in the model. The two features of facial blur and face orientation do not appear in top33, because the feature extraction of facial blur depends on the image resolution. If the image itself is low-resolution, the face also blur. The estimation of face orientation is challenging which affected by the light direction, shooting angle, etc., so it is not accurate. The length of the static line is the third important feature, which demonstrates that the feature of horizontal line composition are positive for group photo aesthetic assessment. The three features of brightness, saturation and hue in the center of the image are as same as our hypothesis that the group photography should satisfy the central composition rule. We also found that emotional features (PAD), Pleasure and Arousal has some influence, Pleasure reflects the degree of the people’s love for images, The Arousal reflects the level of neurophysiological activation, dominance reflects people’s anger and fear, and there is no direct relationship with the evaluation of group photos, which is basically consistent with our hypothesis.
Classifier
----------
Through the feature importance analysis based on random forest, it can be concluded that not all features are effective for group photo assessment. So, as same as [@28], we used two feature selection methods (filter-based and wrapper-based) to filter out the useless features: one is based on the accuracy of single feature classification and the other is recursive feature elimination (RFE) - a feature selection method based on wrapper. We used the sklearn-svm package [@sklearn] to train the classification model using the standard RBF kernel ($\gamma = 2.0 .C=1.0$), and use 10-fold cross-validation to ensure the fairness of the experiment. The average AUC of 10-fold cross-validation was adopted as the quality measure of the classifier. AUC is defined as Area under the ROC Curve.
Because the average score of GPD is 6.05, we employed 6 as the boundary to divide the group image into two categories: good and bad. The ROC curve of the model trained by each group photo feature is shown in Figure 7(a), which performance is similar to the importance ranking. The AUC of the gaze feature is 0.73 and the AUC of the opened-eye feature is 0.68. It also shows that the two features are effective for group photo assessment, and the effect of facial blur feature is not ideal, which is due to the challenge of the face recognition in low-quality images.
Figure 7 (b) shows the ROC and AUC of three models built on the combination of group photograph features (GPF) and generic aesthetic features (GAF). Among them, the features applied in the GAF&GPF model are 20 features selected from all features by the above two feature selection methods, the features used in the GAF model are selected from the generic aesthetic features, and the features used in the GPF model only contain group photo features. The selected feature set is shown in Table 2. It can be seen that the GAF&GPF model completely wraps the GAF model, and the AUC value reached 0.81. The AUC of the GPF model is larger than the GAF model, but smaller than the GAF&GPF model, which indicates that GAF combined with GPF can make the evaluation performance to the best. Table 3 compares the three models from four measurements: accuracy, precision, recall and F1. We found that the model trained by the combination of the generic aesthetic features and group photo features is better than the other two models in each measurement.
[|l|l|l|l|l|l|]{} \# & GPF Model &
------------
GAF&GPF
Classifier
------------
: Feature set used by each model, ’\*’ indicates the selected
&
------------
GAF
Classifier
------------
: Feature set used by each model, ’\*’ indicates the selected
&
-----------
GAF&GPF
Regressor
-----------
: Feature set used by each model, ’\*’ indicates the selected
&
-----------
GAF
Regressor
-----------
: Feature set used by each model, ’\*’ indicates the selected
\
f1-f7 & \* & \* & & \* &\
f8-f13 & & \* & \* & \* & \*\
f55 & & & & \* & \*\
f56-f71 & & & & \* & \*\
f28 & & \* & & &\
f29-f43 & & \* & \* & \* & \*\
f44-f48 & & & \* & & \*\
f14-f25 & & \* & \* & &\
f72-f83 & & \* & \* & \* & \*\
f26-f27 & & \* & \* & \* & \*\
f49-f51 & & & & & \*\
f84-f89 & & & & & \*\
Models Accuracy Precision Recall F1
--------------------- ---------- ----------- -------- --------
GAF& GPF classifier 0.7097 0.7968 0.7543 0.7285
GAF classifier 0.6573 0.5721 0.5969 0.5612
GPF classifier 0.6889 0.7878 0.6771 0.7025
: Comparison: performance of three models
regression model
----------------
We adopted random forest regression algorithm to train the regression model, through 10-fold cross verification to determine the parameters: the maximum depth is 5, and the number of basic learners is 130. Firstly, the random forest algorithm is used for feature selection. Like the training classifier, three different feature subsets are selected from the feature set which is shown in Table 2. Using these three feature sets to train three models on the GPD (20% randomly selected as the test set and 80% as the training set) for 100 times. We use the $R^2$ (coefficient of determination) to measure the regression. $R^2$ is always between 0 and 1, best score is 1, which is defined as: $$R^2=1-\frac{\sum_{i}^{N_{test}}(\hat{Y}^{(i)}-Y^{(i)})^2}{\sum_{i}^{N_{test}}(\bar{Y}^{(i)}-Y^{(i)})^2},$$ where $\hat{Y}$ is the prediction score, $Y$ is label (ground truth), $\bar{Y}$ is the average value of the test image label, and $N_test$ is the number of test images. Finally, Averaging the $R^2$ of 100 times to avoid the coincidence caused by random sampling from dataset, the comparison of the experimental results is shown in Table 4.
Models Maximum $R^2$ Average $R^2$
--------------------- --------------- ---------------
GAF & GPF Regressor 0.563 0.415
GPF Regressor 0.529 0.372
GAF Regressor 0.379 0.241
: Comparison of performance of the three regression models
The results show that the average $R^2$ of the GAF&GPF model reaches 0.415 and the Maximum $R^2$ reaches 0.563 in these 100 times trainings, which is not particularly high, but the best performance of the three models. It also shows that the group photo features and general aesthetic features are effective for group photo evaluation. The $R^2$ of GPF model is also higher than that of the model trained by generic aesthetic features, which proved that people pay more attention to the rules we proposed for group photography. General aesthetic features have relatively little impact on the assessment of the group photos.
Comparison
----------
In order to verify that the generic aesthetic features cannot fit the image aesthetic assessment for group photos, and the other method cannot distinguish the photos of different people‘s status under the same scene, we preformed the following comparison. We have taken four groups of photos, where each group contains a standard group photo and three photos that do not conform to the group photography rules. They are divided into three categories: “Looking away”, “Occlusion” and “Not in the center ”. Then we utilized four methods: NIMA-res [@11], NIMA-mobile [@11], Kong [@13] and our regression model to evaluate them. The discrimination of the standard image and the other types is defined as $\delta=s_{aes}(I_{standard})-s_{aes}(I_{other})$, where $s_{aes}(I_{standard})$ and $s_{aes}(I_{other})$ are the score of standard group photo and other types. We calculate the difference between other types and standard photos in each group to measure the discrimination of each model. Figure 8 shows the comparative experimental results.
![Compared with the results of the deep learning method, the score at the bottom of the picture represents $\delta_{NIMA-RES}$/ $\delta_{NIMA-mobile}$ / $\delta_{Kong}$ / $\delta_{our}$[]{data-label="8"}](images/figure8.pdf)
Looking at Figure 8, taking the first row (a) as an example, the degree of differentiation using the deep learning method are very small or even negative which are 0.231, -0.097, -0.453, respectively. It shows that these methods only from the perspective of the general image to assess the photo, do not consider the people’s state. The $\delta$ of our regression model can reach 1.793, which makes a good distinction between standard group photo and “Looking away”. This is mainly because our assessment method is based on the constraint of people’s state, then combined with general features to assessment group photos. It can be found from the observation of column (c) that the face in first group, the third group and the fourth group have serious occlusion. Using our method to evaluate, the discrimination is close to 1. In the second group, the rightmost character is slightly obscured by objects, and the discrimination is 0.301. However, the discrimination of the deep learning method in the evaluation of such photos is small, all of which are floating up and down 0, and there is Irregular, which proved that the occlusion feature is also effective in group photo evaluation. From Figure 8, it can be seen that the discrimination (in the range of \[1.4-2.3\]) of column (a) is generally higher than that of column (b) and (c) (in the range of \[0.3-1.7\]). This fully corresponds our expectations, as well as the importance ranking of photo features, the impact of eye‘s state is greater than the face occlusion and the position of the person on the photo assessment. We also observed that when assess column (d), there’s a good chance that $\delta$ being negative , which indicates that the deep learning methods consider that the object on the side has a higher aesthetic score than the object on the center. The rule of thirds may be effective when assessment landscape photos, but it is not applicable in the group photos. It also demonstrated that the method based on deep learning relies on a large number of aesthetic photos, without professional knowledge, so it only learns some generic shooting rules and aesthetic features, and it is difficult to make a correct assessment of images in a specific field. On the whole, the discrimination of the assessment method based on deep learning is between -0.5 and 1. The assessment of group photos does not take into account the state of people, and can’t distinguish between good and bad photos when assess multiple group photos in the same scene, but the $\delta$ of our model is between 0.3 and 2.3, which can make a good discrimination of such photos.
Conclusion and future work
==========================
In this work, by analyzing the aesthetic features of group photography, we address the problem that the general method cannot accurately evaluate the group photograph, and introduce group photography features to facilitate investigation of this problem. Furthermore, we construct a group photography dataset (GPD), and built an online annotation tool for collecting the label of GPD. In the experiments, we validated that the proposed method can better evaluate group photography than previous methods that only considered generic features. However, our group photography scene is relatively single. Moreover, there is still a lot of space for improvement in the extraction of group photography features and generic aesthetic features in the future. To further improve the accuracy of the aesthetic evaluation of group photography, much work remains to be done.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research was partially supported by National Natural Science Foundation of China \[Grant No. 61771340,61602344\] and Natural Science Foundation of Tianjin, China \[Grant No. 18JCYBJC15300\].
|
---
abstract: 'Let $\Omega$ be an open set in Euclidean space ${\mathbb{R}}^m,\, m=2,3,...$, and let $v_{\Omega}$ denote the torsion function for $\Omega$. It is known that $v_{\Omega}$ is bounded if and only if the bottom of the spectrum of the Dirichlet Laplacian acting in $\Leb^2(\Omega)$, denoted by $\lambda(\Omega)$, is bounded away from $0$. It is shown that the previously obtained bound $\|v_{\Omega}\|_{\Leb^{\infty}(\Omega)}\lambda(\Omega)\ge 1$ is sharp: for $m\in\{2,3,...\}$, and any $\epsilon>0$ we construct an open, bounded and connected set $\Omega_{\epsilon}\subset {\mathbb{R}}^m$ such that $\|v_{\Omega_{\epsilon}}\|_{\Leb^{\infty}(\Omega_{\epsilon})} \lambda(\Omega_{\epsilon})<1+\epsilon$. An upper bound for $v_{\Omega}$ is obtained for planar, convex sets in Euclidean space $M={\mathbb{R}}^2$, which is sharp in the limit of elongation. For a complete, non-compact, $m$-dimensional Riemannian manifold $M$ with non-negative Ricci curvature, and without boundary it is shown that $v_{\Omega}$ is bounded if and only if the bottom of the spectrum of the Dirichlet-Laplace-Beltrami operator acting in $\Leb^2(\Omega)$ is bounded away from $0$.'
author:
- |
[M. van den Berg]{}\
School of Mathematics, University of Bristol\
University Walk, Bristol BS8 1TW\
United Kingdom\
`mamvdb@bristol.ac.uk`\
date: 30 March 2017
title: Spectral bounds for the torsion function
---
3truecm
1truecm **Keywords**: Torsion function; Dirichlet Laplacian; Riemannian manifold; non-negative Ricci curvature.\
[*AMS*]{} 2000 [*subject classifications.*]{} 58J32; 58J35; 35K20.\
[*Acknowledgement.*]{} MvdB acknowledges support by The Leverhulme Trust through International Network Grant *Laplacians, Random Walks, Bose Gas, Quantum Spin Systems*.
Introduction\[sec1\]
====================
Let $\Omega$ be an open set in ${\mathbb{R}}^m,$ and let $\Delta$ be the Laplace operator acting in $L^2({\mathbb{R}}^m)$. Let $(B(s),s\ge 0, {\mathbb{P}}_x,x\in {\mathbb{R}}^m)$ be Brownian motion on ${\mathbb{R}}^m$ with generator $\Delta$. For $x\in \Omega$ we denote the first exit time, and expected lifetime of Brownian motion by
$$T_{\Omega}=\inf\{s\ge 0: B(s)\notin \Omega\},$$
and $$\label{e2}
v_{\Omega}(x)=\mathbb{E}_x[T_{\Omega}],\, x\in \Omega,$$ respectively, where $\mathbb{E}_x$ denotes the expectation associated with $\mathbb{P}_x$. Then $v_{\Omega}$ is the torsion function for $\Omega$, i.e. the unique solution of $$\label{e3}
-\Delta v=1,\, v\in H_0^1(\Omega).$$ The bottom of the spectrum of the Dirichlet Laplacian acting in $\Leb^2(\Omega)$ is denoted by $$\label{e4}
\lambda(\Omega)=\inf_{\varphi\in H_0^1(\Omega)\setminus\{0\}}\frac{\displaystyle\int_\Omega|D\varphi|^2}{\displaystyle\int_\Omega \varphi^2}.$$ It was shown in [@vdB], [@vdBC] that $\|v_{\Omega}\|_{\Leb^{\infty}(\Omega)}$ is finite if and only if $\lambda(\Omega)>0$. Moreover, if $\lambda(\Omega)>0$, then $$\label{e5}
\lambda(\Omega)^{-1}\le \|v_{\Omega}\|_{\Leb^{\infty}(\Omega)}\le (4+3m\log 2)\lambda(\Omega)^{-1}.$$ The upper bound in was subsequently improved (see [@HV]) to $$\|v_{\Omega}\|_{\Leb^{\infty}(\Omega)}\le \frac18(m+cm^{1/2}+8)\lambda(\Omega)^{-1},$$ where $$c=(5(4+\log 2))^{1/2}.$$
In Theorem \[the3\] below we show that the coefficient $1$ of $\lambda(\Omega)^{-1}$ in the left-hand side of is sharp.
\[the3\] For $m\in\{2,3,\dots\}$, and any $\epsilon>0$ there exists an open, bounded, and connected set $\Omega_{\epsilon}\subset{\mathbb{R}}^m$ such that $$\label{e11}
\|v_{\Omega_{\epsilon}}\|_{\Leb^{\infty}(\Omega_{\epsilon})} \lambda(\Omega_{\epsilon})<1+\epsilon.$$
The set $\Omega_{\epsilon}$ is constructed explicitly in the proof of Theorem \[the3\].
It has been shown by L. E. Payne (see (3.12) in [@P]) that for any convex, open $\Omega\subset{\mathbb{R}}^m$ for which $\lambda(\Omega)>0$, $$\label{e9}
\|v_{\Omega}\|_{\Leb^{\infty}(\Omega)}\lambda(\Omega)\ge \frac{\pi^2}{8},$$ with equality if $\Omega$ is a slab, i.e. the connected, open set, bounded by two parallel $(m-1)$-dimensional hyperplanes. Theorem \[the2\] below shows that for any sufficiently elongated, convex, planar set (not just an elongated rectangle) $\|v_{\Omega}\|_{\Leb^{\infty}(\Omega)}\lambda(\Omega)$ is approximately equal to $\frac{\pi^2}{8}$. We denote the width and the diameter of a bounded open set $\Omega$ by $w(\Omega)$ (i.e. the minimal distance of two parallel lines supporting $\Omega$), and $\textup{diam}(\Omega)=\sup\{|x-y|:x\in \Omega,\,y\in \Omega\}$ respectively.
\[the2\]If $\Omega$ is a bounded, planar, open, convex set with width $w(\Omega)$, and diameter $\textup{diam}(\Omega)$, then $$\|v_{\Omega}\|_{\Leb^{\infty}(\Omega)}\lambda(\Omega)\le \frac{\pi^2}{8}\left(1+7\cdot3^{2/3}\left(\frac{w(\Omega)}{\textup{diam}(\Omega)}\right)^{2/3}\right).$$
In the Riemannian manifold setting we denote the bottom of the spectrum of the Dirichlet-Laplace-Beltrami operator by . We have the following.
\[the1\]Let $M$ be a complete, non-compact, $m$-dimensional Riemannian manifold, without boundary, and with non-negative Ricci curvature. There exists $K<\infty,$ depending on $M$ only, such that if $\Omega\subset M$ is open, and $\lambda(\Omega)>0,$ then $$\label{e8}
\lambda(\Omega)^{-1}\le \|v_{\Omega}\|_{\Leb^{\infty}(\Omega)}\le 2^{(3m+8)/4}\cdot3^{m/2}K^2 \lambda(\Omega)^{-1},$$ where $K$ is the constant in the Li-Yau inequality in below.
The proofs of Theorems \[the3\], \[the2\], and \[the1\] will be given in Sections \[sec4\], \[sec3\] and \[sec2\] respectively.
Below we recall some basic facts on the connection between torsion function and heat kernel. It is well known (see [@EBD3], [@GB], [@GB1]) that the heat equation $$\Delta u(x;t)=\frac{\partial u(x;t)}{\partial t},\quad x\in M,\quad t>0, $$ has a unique, minimal, positive fundamental solution $p_M(x,y;t),$ where $x\in M$, $y\in M$, $t>0$. This solution, the heat kernel for $M$, is symmetric in $x,y$, strictly positive, jointly smooth in $x,y\in M$ and $t>0$, and it satisfies the semigroup property $$p_M(x,y;s+t)=\int_{M}dz\ p_M(x,z;s)p_M(z,y;t),$$ for all $x,y\in M$ and $t,s>0$, where $dz $ is the Riemannian measure on $M$. See, for example, [@RS] for details. If $\Omega$ is an open subset of $M,$ then we denote the unique, minimal, positive fundamental solution of the heat equation on $\Omega$ by $p_{\Omega}(x,y;t)$, where $x\in \Omega,y\in \Omega,t>0$. This Dirichlet heat kernel satisfies, $$p_{\Omega}(x,y;t)\le p_M(x,y;t),\, x\in \Omega, y\in \Omega,t>0.$$ Define $u_{\Omega}:\Omega \times (0,\infty)\mapsto {\mathbb{R}}$ by $$ u_{\Omega}(x;t)=\int_{\Omega}dy\, p_{\Omega}(x,y;t).$$ Then, $$ u_{\Omega}(x;t)=\int_{\Omega}dy\, p_{\Omega}(x,y;t)={\mathbb{P}}_x[T_{\Omega}>t],$$ and by $$\label{e17}
v_{\Omega}(x)=\int_0^{\infty}dt\,{\mathbb{P}}_x[T_{\Omega}>t]=\int_0^{\infty}dt\,\int_{\Omega}dy\, p_{\Omega}(x,y;t).$$ It is straightforward to verify that $v_{\Omega}$ as in satisfies .
Proof of Theorem \[the3\]\[sec4\]
=================================
We introduce the following notation. Let $C_L=(-\frac{L}{2},\frac{L}{2})^{m/2}$ be the open cube with measure $L^m$, and delete from $C_L$, $N^m$ closed balls with radii $\delta$, where each ball $B(c_i;\delta)$ is positioned at the centre of an open cube $Q_i$ with measure $(L/N)^m$. These open cubes are pairwise disjoint, and contained in $C_L$. Let $0<\delta<\frac{L}{2N}$, and put $$\label{e41}
\Omega_{\delta,N,L}=C_L-\cup_{i=1}^{N^m}B(c_i;\delta).$$ Below we will show that for any $\epsilon>0$ we can choose $\delta, N$ such that $$\|v_{\Omega_{\delta,N,L}}\|_{\Leb^{\infty}(\Omega_{\delta,N,L})} \lambda(\Omega_{\delta,N,L})<1+\epsilon.$$
![$\Omega_{\delta,N,L}$ with $m=2,N=10,\delta=\frac{L}{8N}.$[]{data-label="fig1"}](figurev2-1_0.pdf)
In Lemma \[lem2\] below we show that $\lambda(\Omega_{\delta,N,L})$ is approximately equal to the first eigenvalue, $\mu_{1,B(0;\delta),L/N},$ of the Laplacian with Neumann boundary conditions on $\partial C_{L/N}$, and with Dirichlet boundary conditions on $\partial B(0;\delta)$. The requirement $\mu_{1,B(0;\delta),L/N}$ not being too small stems from the fact that the approximation of replacing the Neumann boundary conditions on $C_L$ is a surface effect which should not dominate the leading term $\mu_{1,B(0;\delta),L/N}$.
\[lem2\] If $\delta\le \frac{L}{4N},\ N\ge 10$, and $\frac{N}{L^2}\le \mu_{1,B(0;\delta),L/N}$, then $$\lambda(\Omega_{\delta,N,L})\le \mu_{1,B(0;\delta),L/N}+32m\bigg(\frac{5}{4}\bigg)^m\bigg(\frac{N}{L^2}+\frac{1}{N^{1/2}}\mu_{1,B(0;\delta),L/N}\bigg)$$
Let $\varphi_{1,B(0;\delta),L/N}$ be the first eigenfunction (positive) corresponding to $\mu_{1,B(0;\delta),L/N}$, and normalised in $\Leb^2(C_{L/N}-B(0;\delta))$. In order to prove the lemma we construct a test function by periodically extending $\varphi_{1,B(0;\delta),L/N}$ to all cubes $Q_1,\dots Q_{N^m}$ of $\Omega_{\delta,N,L}$. We denote this periodic extension by $f$. We define $$C_{L,N}=C_{L(1-\frac{2}{N})}.$$ So $C_{L,N}$ is the sub-cube of $C_L$ with the outer layer of cubes of size $L/N$ removed. Let $$\tilde{f}=\bigg(1-\frac{\textup{dist}(x,C_{L,N})}{L/(4N)}\bigg)_+f.$$ Then $\tilde{f}\in H_0^1(\Omega_{\delta,N,L}),$ and $$\label{e46}
\|\tilde{f}\|_{\Leb^2(\Omega_{\delta,N,L})}\ge \int_{C_{L,N}}f^2=(N-2)^m,$$ since $f$ restricted to any of the cubes $Q_i$ in $\Omega_{\delta,N,L}$ is normalised. Furthermore $$\begin{aligned}
|D\tilde{f}|^2&\le \bigg(1-\frac{\textup{dist}(x,C_{L,N})}{L/(4N)}\bigg)^2|Df|^2+1_{C_L-C_{L,N}}\bigg(\bigg(\frac{4N}{L}\bigg)^2f^2+\frac{8N}{L}f|Df|\bigg)\nonumber \\ &
\le |Df|^2+\bigg(\frac{4N}{L}\bigg)^21_{C_L-C_{L,N}}f^2+\frac{8N}{L}1_{C_L-C_{L,N}}f|Df|.\end{aligned}$$ Hence $$\begin{aligned}
\label{e48}
&\int_{\Omega_{\delta,N,L}}|D\tilde{f}|^2\le \int_{\Omega_{\delta,N,L}}|Df|^2+\bigg(\frac{4N}{L}\bigg)^2\int_{C_L-C_{L,N}}f^2\nonumber \\ & \hspace{4cm} +\frac{8N}{L}\bigg(\int_{C_L-C_{L,N}}|Df|^2\bigg)^{1/2}\bigg(\int_{C_L-C_{L,N}}f^2\bigg)^{1/2}\nonumber \\ &
=N^m\mu_{1,B(0;\delta),L/N}+\big(N^m-(N-2)^m\big)\bigg(\bigg(\frac{4N}{L}\bigg)^2+\frac{8N}{L}\big(\mu_{1,B(0;\delta),L/N}\big)^{1/2}\bigg)\nonumber \\ &
\le N^m\mu_{1,B(0;\delta),L/N}+\big(N^m-(N-2)^m\big)\bigg(\bigg(\frac{4N}{L}\bigg)^2+8N^{1/2}\mu_{1,B(0;\delta),L/N}\bigg),\end{aligned}$$ where we have used the last hypothesis in the lemma. By , , the Rayleigh-Ritz variational formula, and the hypothesis $N\ge 10$, $$\begin{aligned}
\label{e49}
\lambda(\Omega_{\delta,N,L})&\le \mu_{1,B(0;\delta),L/N}\nonumber \\ & \ \ \ +\frac{N^m-(N-2)^m}{(N-2)^m}\bigg(\bigg(\frac{4N}{L}\bigg)^2+\big(8N^{1/2}+1\big)\mu_{1,B(0;\delta),L/N}\bigg)\nonumber \\ &\le
\mu_{1,B(0;\delta),L/N}+32m\bigg(\frac{5}{4}\bigg)^m\bigg(\frac{N}{L^2}+\frac{1}{N^{1/2}}\mu_{1,B(0;\delta),L/N}\bigg).\end{aligned}$$
To obtain an upper bound for $\|v_{\Omega_{\delta,N,L}}\|_{\Leb^{\infty}(\Omega_{\delta,N,L})}$, we change the Dirichlet boundary conditions on $\partial C_L$ to Neumann boundary conditions. This increases the corresponding heat kernel, torsion function, and $\Leb^{\infty}$ norm respectively. By periodicity, we have that $$\label{e50}
\|v_{\Omega_{\delta,N,L}}\|_{\Leb^{\infty}(\Omega_{\delta,N,L})}\le \|\tilde{v}_{C_{L/N}-B(0;\delta)}\|_{\Leb^{\infty}(C_{L/N}-B(0;\delta))},$$ where $\tilde{v}_{C_{L/N}-B(0;\delta)}$ is the torsion function with Neumann boundary conditions on $\partial C_{L/N}$, and Dirichlet boundary conditions on $\partial B(0;\delta)$. Denote the spectrum of the corresponding Laplacian by $\{\mu_j:=\mu_{j,B(0;\delta),L/N},j=1,2,\dots\}$, and let $\{\varphi_j:=\varphi_{1,B(0;\delta),L/N},j=1,2,\dots\}$ denote a corresponding orthonormal basis of eigenfunctions. We denote by $\pi_{\delta,N/L}(x,y;t), x\in C_{L/N}-B(0;\delta), y \in C_{L/N}-B(0;\delta), t>0$ the corresponding heat kernel. Then $$\label{e51}
\pi_{\delta,N/L}(x,y;t)=\sum_{j=1}^{\infty}e^{-t\mu_j}\varphi_j(x)\varphi_j(y),$$ and $$\begin{aligned}
\label{e52}
\tilde{v}&_{C_{L/N}-B(0;\delta)}(x)\nonumber \\ &
=\int_0^{\infty}dt\ \int_{C_{L/N}-B(0;\delta)}dy\ \pi_{\delta,N/L}(x,y;t)\bigg(\frac{\varphi_1(y)}{\|\varphi_1\|}+1-\frac{\varphi_1(y)}{\|\varphi_1\|}\bigg)\nonumber \\ &
=\frac{1}{\mu_1}\frac{\varphi_1(x)}{\|\varphi_1\|}+\int_0^{\infty}dt\ \int_{C_{L/N}-B(0;\delta)}dy\ \pi_{\delta,N/L}(x,y;t)\bigg(1-\frac{\varphi_1(y)}{\|\varphi_1\|}\bigg)\nonumber \\ & \le\frac{1}{\mu_1}+ \int_0^Tdt\ \int_{C_{L/N}-B(0;\delta)}dy\ \pi_{\delta,N/L}(x,y;t)\nonumber \\ &\hspace{1cm}+\int_T^{\infty}dt\ \int_{C_{L/N}-B(0;\delta)}dy\ \pi_{\delta,N/L}(x,y;t)\bigg(1-\frac{\varphi_1(y)}{\|\varphi_1\|}\bigg)\nonumber \\ & \le\frac{1}{\mu_1}+T+\int_T^{\infty}dt\ \int_{C_{L/N}-B(0;\delta)}dy\ \pi_{\delta,N/L}(x,y;t)\bigg(1-\frac{\varphi_1(y)}{\|\varphi_1\|}\bigg),\end{aligned}$$ where $\|\varphi_1\|=\|\varphi_1\|_{\Leb^{\infty}(C_{L/N}-B(0;\delta))}$. By , we have that the third term in the right-hand side of equals $$\label{e53}
\sum_{j=1}^{\infty}\mu_j^{-1}e^{-T\mu_j}\varphi_j(x)\int_{C_{L/N}-B(0;\delta)}dy\ \varphi_j(y)\bigg(1-\frac{\varphi_1(y)}{\|\varphi_1\|}\bigg).$$ The term with $j=1$ in is bounded from above by $$\begin{aligned}
\mu_1^{-1}\|\varphi_1\|\int_{C_{L/N}-B(0;\delta)}& \|\varphi_1\|\bigg(1-\frac{\varphi_1}{\|\varphi_1\|}\bigg)\nonumber \\ &
=\mu_1^{-1}\|\varphi_1\|\int_{C_{L/N}-B(0;\delta)}\big(\|\varphi_1\|-\varphi_1\big)\nonumber \\ &
\le \mu_1^{-1}\bigg(\|\varphi_1\|^2\bigg(\frac{L}{N}\bigg)^m-1\bigg),\end{aligned}$$ where we used the fact that $1=\int_{C_{L/N}-B(0;\delta)}\varphi_1^2\le \|\varphi_1\|\int_{C_{L/N}-B(0;\delta)}\varphi_1$. It was shown on p.586, lines -3,-4, in [@vdBFNT] (with appropriate adjustment in notation) that $$\|\varphi_1\|^2\le \bigg(\frac{N}{L}\bigg)^m\bigg(1-s\mu_1-\frac{mL^2}{3esN^2}\bigg)^{-1},s\ge 0,$$ provided the last term in the round brackets is non-negative. The optimal choice for $s$ gives that $$\|\varphi_1\|^2\le \bigg(\frac{N}{L}\bigg)^m\bigg(1-\frac{(4m\mu_1)^{1/2}L}{(3e)^{1/2}N}\bigg)^{-1},\ \mu_1<\frac{3eN^2}{4mL^2}.$$ By further restricting the range for $\mu_1,$ we have that the first term with $j=1$ in is then bounded from above by $$\label{e57}
\mu_1^{-1}\frac{2L(m\mu_1/(3eN^2))^{1/2}}{1-2L(m\mu_1/(3eN^2))^{1/2}}\le \frac{(2m)^{1/2}L}{\mu_1^{1/2}N}, \ \mu_1\le\frac{3eN^2}{16mL^2}.$$ The terms with $j\ge 2$ in give, by Cauchy-Schwarz for both the series in $j$, and the integral over $C_{L/N}-B(0;\delta)$, a contribution $$\begin{aligned}
\label{e58}
&\biggr\rvert\sum_{j=2}^{\infty}\mu_j^{-1}e^{-T\mu_j}\varphi_j(x)\int_{C_{L/N}-B(0;\delta)}\varphi_j\bigg(1-\frac{\varphi_1}{\|\varphi_1\|}\bigg)\biggr\rvert\nonumber \\ &
\le \mu_2^{-1}\sum_{j=2}^{\infty}e^{-T\mu_j}|\varphi_j(x)|\int_{C_{L/N}-B(0;\delta)}|\varphi_j|\nonumber \\ &
\le \mu_2^{-1}\bigg(\frac{L}{N}\bigg)^{m/2}\bigg(\sum_{j=2}^{\infty}e^{-T\mu_j}\bigg)^{1/2}\bigg(\sum_{j=2}^{\infty}e^{-T\mu_j}|\varphi_j(x)|^2\bigg)^{1/2}\nonumber \\ &
\le \mu_2^{-1}\bigg(\frac{L}{N}\bigg)^{m/2}\bigg(\sum_{j=2}^{\infty}e^{-T\mu_j}\bigg)^{1/2}\big(\pi_{\delta,N/L}(x,x;T)\big)^{1/2}.\end{aligned}$$ To bound the first series in , we note that the $\mu_j$’s are bounded from below by the Neumann eigenvalues of the cube $C_{L/N}$. So choosing $T=(L/N)^2$ we get that $$\bigg(\sum_{j=2}^{\infty}e^{-L^2\mu_j/N^2}\bigg)^{1/2}\le \bigg(1+\sum_{j=1}^{\infty}e^{-\pi^2j^2}\bigg)^{m/2}\le \bigg(\frac43\bigg)^{m/2}.$$ Similarly to the proof of Lemma 3.1 in [@vdBFNT], we have that $$\begin{aligned}
\label{e60}
\big(\pi_{\delta,N/L}(x,x;L^2/N^2)\big)^{1/2}&\le \big(\pi_{0,N/L}(x,x;L^2/N^2)\big)^{1/2}\nonumber \\ &\le \bigg(\frac{N}{L}\bigg)^{m/2}\bigg(1+2\sum_{j=1}^{\infty}e^{-\pi^2j^2}\bigg)^{m/2}\nonumber \\ &\le
\bigg(\frac43\bigg)^{m/2}\bigg(\frac{N}{L}\bigg)^{m/2}.\end{aligned}$$ Finally, $\mu_2\ge \frac{\pi^2N^2}{L^2}$, together with , , , , , and the choice $T=(L/N)^2$ gives that $$\begin{aligned}
\label{e61}
\|v_{\Omega_{\delta,N,L}}\|_{\Leb^{\infty}(\Omega_{\delta,N,L})}\le \mu_1^{-1}+\frac{(2m)^{1/2}L}{\mu_1^{1/2}N}+\bigg(\frac43\bigg)^{m}\frac{L^2}{N^2}, \, \mu_1\le\frac{3eN^2}{16mL^2}.\end{aligned}$$ [*Proof of Theorem \[the3\].*]{} Let $1<\alpha<2$. By and , we have that $$\begin{aligned}
\label{e62}
\lambda(\Omega_{\delta,N,L})\|v_{\Omega_{\delta,N,L}}\|_{\Leb^{\infty}(\Omega_{\delta,N,L})}\le & \bigg(\mu_1+32m\bigg(\frac54\bigg)^m\bigg(\frac{N}{L^2}+\frac{1}{N^{1/2}}\mu_1\bigg)\bigg) \nonumber \\ & \times \bigg(\mu_1^{-1}+\frac{(2m)^{1/2}L}{\mu_1^{1/2}N}+\bigg(\frac43\bigg)^{m}\frac{L^2}{N^2}\bigg),\end{aligned}$$ provided $$\frac{N}{L^2}\le \mu_1\le\frac{3eN^2}{16mL^2}.$$
First consider the planar case $m=2$. Recall Lemma 3.1 in [@vdBFNT]: for $\delta<L/(6N)$, $$\label{e64}
\frac{N^2}{100L^2}\bigg(\log \frac{L}{2\delta N}\bigg)^{-1}\le \mu_{1,B(0;\delta),L/N}\le \frac{8\pi N^2}{(4-\pi)L^2}\bigg(\log \frac{L}{2\delta N}\bigg)^{-1}.$$ Let $$\label{e65}
\delta^*:=\delta^*(\alpha,N,L)=\frac{L}{2N}e^{-N^{2-\alpha}},$$ where $1<\alpha<2$. Let $N_1\in {\mathbb{N}}$ be such that for all $N\ge N_1$, $\delta^*<L/(6N)$. We now use to see that there exists $C>1$ such that $$\label{e66}
C^{-1}\frac{N^{\alpha}}{L^2}\le \mu_{1,B(0;\delta^*),L/N}\le C\frac{N^{\alpha}}{L^2}.$$ (In fact $C=\max\{100,8\pi/(4-\pi)\}$). We subsequently let $N_2\in {\mathbb{N}}$ be such that for all $N\ge N_2$, $$\frac{N}{L^2}\le C^{-1}\frac{N^{\alpha}}{L^2}\le C\frac{N^{\alpha}}{L^2}\le \frac{3eN^2}{16mL^2}.$$ By , , and all $N\ge \max\{N_1,N_2\}$ we have that $$\label{e68}
\lambda(\Omega_{\delta^*,N,L})\|v_{\Omega_{\delta^*,N,L}}\|_{\Leb^{\infty}(\Omega_{\delta^*,N,L})}\le 1+\mathcal{C}\big(N^{1-\alpha}+N^{(\alpha-2)/2}\big),$$ where $\mathcal{C}$ depends on $C$ and on $m$ only. Finally, we let $N_3\in {\mathbb{N}}$ be such that for all $N\ge N_3$, $$\mathcal{C}\big(N^{1-\alpha}+N^{(\alpha-2)/2}\big)<\epsilon.$$ We conclude that holds with $\Omega_{\epsilon}=\Omega_{\delta^*,N,L}$ with $\delta^*$ given by , and $N\ge \max\{N_1,N_2,N_3\}.$
Next consider the case $m=3,4,\dots$. We apply Lemma 3.2 in [@vdBFNT] to the case $K=B(0;\delta)$, and denote the Newtonian capacity of $K$ by $\textup{cap}(K)$. Then $\textup{cap}(B(0;\delta))=\kappa_m\delta^{m-2}$, where $\kappa_m$ is the Newtonian capacity of the ball with radius $1$ in ${\mathbb{R}}^m$. Then Lemma 3.2 gives that there exists $C\ge 1$ such that $$\label{e70}
C^{-1}\bigg(\frac{N}{L}\bigg)^m\delta^{m-2}\le\mu_{1,B(0;\delta,L/N)}\le C\bigg(\frac{N}{L}\bigg)^m\delta^{m-2},$$ provided $$\label{e71}
\kappa_m\delta^{m-2}\le \frac{1}{16}(L/N)^{m-2}.$$ We choose $$\label{e72}
\delta^*:=\delta^*(\alpha,N,L)=LN^{(\alpha-m)/(m-2)}.$$ This gives inequality by . The requirement holds for all $N\ge N_1$, where $N_1$ is the smallest natural number such that $N_1^{2-\alpha}\ge 16\kappa_m$. The remainder of the proof follows the lines below with the appropriate adjustment of constants, and the choice of $\delta^*$ as in . $\square $
We note that the choice $\alpha=\frac43$ in either or in gives, by , the decay rate $$\label{e73}
\lambda(\Omega_{\delta^*,N,L})\|v_{\Omega_{\delta^*,N,L}}\|_{\Leb^{\infty}(\Omega_{\delta^*,N,L})}\le 1+2\mathcal{C}N^{-1/3}.$$
Proof of Theorem \[the2\]\[sec3\]
=================================
In view of Payne’s inequality it suffices to obtain an upper bound for $\|v_{\Omega}\|_{\Leb^{\infty}(\Omega)}\lambda(\Omega)$. We first observe, that by domain monotonicity of the torsion function, $v_{\Omega}$ is bounded by the torsion function for the (connected) set bounded by the two parallel lines tangent to $\Omega$ at distance $w(\Omega)$. Hence $$\label{e33}
\|v_{\Omega}\|_{\Leb^{\infty}(\Omega)}\le \frac{w(\Omega)^2}{8}.$$ In order to obtain an upper bound for $\lambda(\Omega)$, we introduce the following notation. For a planar, open, convex set, with finite measure, we let $z_1,z_2$ be two points on the boundary of $\Omega$ which realise the width. That is there are two parallel lines tangent to $\partial \Omega$, at $z_1$ and $z_2$ respectively, and at distance $w(\Omega)$. Let the $x$-axis be perpendicular to the vector $z_1z_2$, containing the point $\frac12(z_1+z_2)$. We consider the family of line segments parallel to the $x$-axis, obtained by intersection with $\Omega$, and let $l_1,l_2$ be two points on the boundary of $\Omega$ which realise the maximum length $L$ of this family. The quadrilateral with vertices, $z_1,z_2,l_1,l_2$ is contained in $\Omega$. This quadrilateral in turn contains a rectangle with side-lengths $h$, and $\big(1-\frac{h}{w(\Omega)}\big)L$ respectively, where $h\in [0,w(\Omega))
$ is arbitrary. Hence, by domain monotonicity of the Dirichlet eigenvalues, we have that $$\lambda(\Omega)\le \pi^2h^{-2}+\pi^2\bigg(1-\frac{h}{w(\Omega)}\bigg)^{-2}L^{-2}.$$ Minimising the right-hand side above with respect to $h$ gives that $$h= \frac{(w(\Omega)L^2)^{1/3}}{1+\big(\frac{L}{w(\Omega)}\big)^{2/3}}.$$ It follows that $$\label{e36}
\lambda(\Omega)\le\frac{\pi^2}{w(\Omega)^2}\bigg(1+\bigg(\frac{w(\Omega)}{L}\bigg)^{2/3}\bigg)^3.$$ As $w(\Omega)\le L$ we obtain by that $$\begin{aligned}
\label{e37}
\lambda(\Omega)\le \frac{\pi^2}{w(\Omega)^2}\left(1+7\left(\frac{w(\Omega)}{L}\right)^{2/3}\right).\end{aligned}$$ In order to complete the proof we need the following.
\[lem1\] If $\Omega$ is an open, bounded, convex set in ${\mathbb{R}}^2$, and if $L$ is the length of the longest line segment in the closure of $\Omega$, perpendicular to $z_1z_2$, then $$\label{e38}
\textup{diam}(\Omega)\le 3L.$$
Let $d_1,d_2\in \partial\Omega$ such that $|d_1-d_2|=\textup{diam}(\Omega)$. We denote the projections of $d_1,d_2$ onto the line through $z_1,z_2$ by $e_1,e_2$ respectively. Let $z$ be the intersection of the lines through $z_1,z_2$ and $d_1,d_2$ respectively. Then, by the maximality of $L$, we have that $|d_1-e_1|\le L, |d_2-e_2|\le L.$ Furthermore, by convexity, $|e_1-z|+|e_2-z|\le w(\Omega)$. Hence, $$\begin{aligned}
|d_1-d_2|\le |d_1-e_1|+|e_1-z|+|d_2-e_2|+|e_2-z|\le 2L+w(\Omega)\le 3L.\end{aligned}$$
By , we have that $$\lambda(\Omega)\le\frac{\pi^2}{w(\Omega)^2}\left(1+7\cdot 3^{2/3}\left(\frac{w(\Omega)}{\textup{diam}(\Omega)}\right)^{2/3}\right).$$ This implies Theorem \[the2\] by . $\square $
Proof of Theorem \[the1\]\[sec2\]
=================================
We denote by $d:M\times M\mapsto {\mathbb{R}}^+$ the geodesic distance associated to $(M,g)$. For $x\in M,\,R>0,$ $B(x;R)=\{y\in M:d(x,y)<R\}$. For a measurable set $A\subset M$ we denote by $|A|$ its Lebesgue measure. The Bishop-Gromov Theorem (see [@BC]) states that if $M$ is a complete, non-compact, $m$-dimensional, Riemannian manifold with non-negative Ricci curvature, then for $p\in M$, the map $r\mapsto\frac{\vert B(p;r)\vert}{r^m}$ is monotone decreasing. In particular $$\label{e18}
\frac{\vert B(p;r_2)\vert}{\vert B(p;r_1)\vert}\le
\left(\frac{r_2}{r_1}\right)^m,\ 0<r_1\le r_2.$$ Corollary 3.1 and Theorem 4.1 in [@LY], imply that if $M$ is complete with non-negative Ricci curvature, then for any $D_2>2$ and $0<D_1<2$ there exist constants $0<C_1\le C_2<\infty$ such that for all $x\in
M,\ y\in M,\ t>0$, $$\label{e19}
C_1\frac{e^{-d(x,y)^2/(2D_1t)}}{ (\vert B(x;t^{1/2})\vert \vert
B(y;t^{1/2})\vert )^{1/2}}\le p_M(x,y;t)\le
C_2\frac{e^{-d(x,y)^2/(2D_2t)}}{ (\vert B(x;t^{1/2})\vert \vert
B(y;t^{1/2})\vert )^{1/2}}.$$ Finally, since by the measure of any geodesic ball with radius $r$ is bounded polynomially in $r$, the theorems of Grigor’yan in [@GB] imply stochastic completeness. That is, for all $x\in M,$ and all $t>0$, $$\int_Mdy\, p_M(x,y;t)=1.$$
[*Proof of Theorem \[the1\].*]{} We choose $D_1=1,\, D_2=3$ in , and define the corresponding number $K=\max\{C_2,C_1^{-1}\}$. Then $$\label{e21}
K^{-1}e^{-d(x,y)^2/(2t)}\le (\vert B(x;t^{1/2})\vert \vert
B(y;t^{1/2})\vert )^{1/2}p_M(x,y;t)\le
Ke^{-d(x,y)^2/(6t)}.$$
Let $q\in M$ be arbitrary, and let $R>0$ be such that $\Omega(q;R):=B(q;R)\cap \Omega\ne \emptyset$. The spectrum of the Dirichlet Laplacian acting in $L^2(\Omega(q;R))$ is discrete. Denote the bottom of this spectrum by $\lambda(\Omega(q;R))$. Then $\lambda(\Omega(q;R))\ge \lambda(\Omega)$. By the spectral theorem, monotonicity of Dirichlet heat kernels, and the Li-Yau bound , we have that $$\begin{aligned}
\label{e22}
p_{\Omega(q;R)}(x,x;t)&\le e^{-t\lambda(\Omega(q;R))/2}p_{\Omega(q;R)}(x,x;t/2)\nonumber \\ & \le e^{-t\lambda(\Omega(q;R))/2}p_M(x,x;t/2)\nonumber \\ &
\le Ke^{-t\lambda(\Omega(q;R))/2}|B(x;(t/2)^{1/2})|^{-1}.\end{aligned}$$ By the semigroup property and the Cauchy-Schwarz inequality, for any open set $\Omega\subset M$, we have that $$\begin{aligned}
\label{e23}
p_{\Omega}(x,y;t)&=\int_{\Omega}dz\, p_{\Omega}(x,z;t/2)\,p_{\Omega}(z,y;t/2)\nonumber \\ &\le \left(\int_{\Omega}dz\, p_{\Omega}^2(x,z;t/2)\right)^{1/2}\left(\int_{\Omega}dz\, p_{\Omega}^2(z,y;t/2)\right)^{1/2}\nonumber \\ &=
\big(p_{\Omega}(x,x;t)\,p_{\Omega}(y,y;t)\big)^{1/2}.\end{aligned}$$ We obtain by , (for $\Omega=\Omega(q;R)$), and $p_{\Omega(q;R)}(x,y;t)\le p_M(x,y;t),$ that $$\begin{aligned}
\label{e24}
p_{\Omega(q;R)}&(x,y;t)\le \big(p_{\Omega(q;R)}(x,x;t)\,p_{\Omega(q;R)}(y,y;t)\big)^{1/4}p_M(x,y;t)^{1/2}\nonumber \\ &\le K^{1/2}e^{-t\lambda(\Omega(q;R))/4}\big(|B(x;(t/2)^{1/2})||B(y;(t/2)^{1/2})|\big)^{-1/4}p_M^{1/2}(x,y;t).\end{aligned}$$ By and , we have that $$\begin{aligned}
\label{e25}
p_{\Omega(q;R)}(x,y;t)\le & Ke^{-t\lambda(\Omega(q;R))/4}\big(|B(x;(t/2)^{1/2})||B(y;(t/2)^{1/2})|\big)^{-1/4} \nonumber \\ &\times\big(|B(x;t^{1/2})||
B(y;t^{1/2})|\big)^{-1/4}e^{-d(x,y)^2/(12t)}.\end{aligned}$$ By the Li-Yau lower bound in , we can rewrite the right-hand side of to yield, $$\begin{aligned}
\label{e26}
p_{\Omega(q;R)}(x,y;t)&\le K^2e^{-t\lambda(\Omega(q;R))/4}p_M(x,y;6t)\nonumber \\ & \times \frac{\big(|B(x;(6t)^{1/2})||B(y;(6t)^{1/2})|\big)^{1/2}}{\big(|B(x;(t/2)^{1/2})||B(y;(t/2)^{1/2})||B(x;t^{1/2})||
B(y;t^{1/2})|\big)^{1/4}}.\end{aligned}$$ By Bishop-Gromov , we have that the volume quotients in the right-hand side of are bounded by $2^{3m/4}\cdot 3^{m/2}$ uniformly in $x$ and $y$. Hence $$p_{\Omega(q;R)}(x,y;t)\le 2^{3m/4}\cdot 3^{m/2}K^2e^{-t\lambda(\Omega(q;R))/4}p_M(x,y;6t).$$ Since manifolds with non-negative Ricci curvature are stochastically complete, we have that $$\begin{aligned}
\int_{\Omega(q;R)}dy\,p_{\Omega(q;R)}(x,y;t)&\le 2^{3m/4}\cdot 3^{m/2}K^2e^{-t\lambda(\Omega(q;R))/4}\int_Mdy\,p_M(x,y;6t)\nonumber \\ &=2^{3m/4}\cdot 3^{m/2}K^2e^{-t\lambda(\Omega(q;R))/4}.\end{aligned}$$ Integrating the inequality above with respect to $t$ over $[0,\infty)$ yields, $$v_{\Omega(q;R)}(x)\le 2^{(3m+8)/4}\cdot3^{m/2}K^2\lambda(\Omega(q;R))^{-1}\le 2^{(3m+8)/4}\cdot3^{m/2}K^2\lambda(\Omega)^{-1}.$$ Finally letting $R\rightarrow \infty$ in the left-hand side above yields the right-hand side of .
The proof of the left-hand side of is similar to the one in Theorem 5.3 in [@vdB] for Euclidean space. We have that $$\label{e30}
v_{\Omega(q;R)}(x)=\int_0^{\infty}dt\,\int_{\Omega(q;R)}dy\,p_{\Omega(q;R)}(x,y;t).$$ We first observe that $|\Omega(q;R)|<\infty$, and so the spectrum of the Dirichlet Laplacian acting in $L^2(\Omega(q;R))$ is discrete and is denoted by $\{\lambda_j(\Omega(q;R)), j\in {\mathbb{N}}\}$, with a corresponding orthonormal basis of eigenfunctions $\{\varphi_{j,\Omega(q;R)}, j\in {\mathbb{N}}\}$. These eigenfunctions are in $\Leb^{\infty}(\Omega(q;R))$. Then, by and the eigenfunction expansion of the Dirichlet heat kernel for $\Omega(q;R)$, we have that $$\begin{aligned}
\label{e31}
v_{\Omega(q;R)}(x)&\ge\int_0^{\infty}dt\,\int_{\Omega(q;R)}dy\,p_{\Omega(q;R)}(x,y;t)\frac{\varphi_{1,\Omega(q;R)}(y)}{\|\varphi_{1,\Omega(q;R)}\|_{\Leb^{\infty}(\Omega(q;R))}}\nonumber \\ &
=\int_0^{\infty}dt\,e^{-t\lambda_1(\Omega(q;R))}\frac{\varphi_{1,\Omega(q;R)}(x)}{\|\varphi_{1,\Omega(q;R)}\|_{\Leb^{\infty}(\Omega(q;R))}}\nonumber \\ &=\lambda_1(\Omega(q;R))^{-1}\frac{\varphi_{1,\Omega(q;R)}(x)}{\|\varphi_{1,\Omega(q;R)}\|_{\Leb^{\infty}(\Omega(q;R))}}.\end{aligned}$$ First taking the supremum over all $x\in \Omega(q;R)$ in the left-hand side of , and subsequently taking the supremum over all such $x$ in the right-hand side of gives $$\label{e32}
\|v_{\Omega(q;R)}\|_{\Leb^{\infty}(\Omega(q;R))}\ge \lambda(\Omega(q;R))^{-1}.$$ Observe that the torsion function is monotone increasing in $R$. Taking the limit $R\rightarrow \infty$ in the left-hand side of , and subsequently in the right-hand side of completes the proof. $\square $
[99]{}
<span style="font-variant:small-caps;">M. van den Berg</span>, Estimates for the torsion function and Sobolev constants. Potential Anal. 36 (2012), 607–616.
<span style="font-variant:small-caps;">M. van den Berg, T. Carroll</span>, Hardy inequality and $L^p$ estimates for the torsion function. Bull. Lond. Math. Soc. 41 (2009), 980–986.
M. van den Berg, V. Ferone, C. Nitsch, C. Trombetti, On Pólya’s inequality for torsional rigidity and first Dirichlet eigenvalue, Integral Equations and Operator Theory 86 (2016), 579–600.
<span style="font-variant:small-caps;">R. L. Bishop, R. J. Crittenden</span>, *Geometry of manifolds*, AMS Chelsea Publishing, Providence, RI (2001).
<span style="font-variant:small-caps;">E. B. Davies</span>, *Heat kernels and spectral theory*, Cambridge University Press, Cambridge (1989).
<span style="font-variant:small-caps;">A. Grigor’yan</span>, *Analytic and geometric backgroud of recurrence and non-explosion of the Brownian motion on Riemannian manifolds*, Bulletin (New Series) of the American Mathematical Society 36 (1999), 135–249.
<span style="font-variant:small-caps;">A. Grigor’yan</span>, *Heat kernel and Analysis on manifolds*, AMS-IP Studies in Advanced Mathematics, **47**, American Mathematical Society, Providence, RI; International Press, Boston, MA (2009).
<span style="font-variant:small-caps;">P. Li, S. T. Yau</span>, *On the parabolic kernel of the Schrödinger operator*, Acta Math. 156 (1986), 153–201.
<span style="font-variant:small-caps;">L. E. Payne</span>, *Bounds for solutions of a class of quasilinear elliptic boundary value problems in terms of the torsion function*, Proc. Royal Soc. Edinburgh 88A (1981), 251–265.
<span style="font-variant:small-caps;">R. S. Strichartz</span>, *Analysis of the Laplacian on the complete Riemannian manifold*, J. Funct. Anal. 52 (1983), 48–79.
<span style="font-variant:small-caps;">H. Vogt</span>, *$L_{\infty}$ estimates for the torsion function and $L_{\infty}$ growth of semigroups satisfying Gaussian bounds*, arXiv:1611.0376v1.
|
---
abstract: 'The Orthogonal Matching Pursuit (OMP) for compressed sensing iterates over a scheme of support augmentation and signal estimation. We present two novel matching pursuit algorithms with intrinsic regularization of the signal estimation step that do not rely on *a priori* knowledge of the signal’s sparsity. An iterative approach allows for a hardware efficient implementation of our algorithm, and enables real-world applications of compressed sensing. We provide a series of numerical examples that demonstrate a good performance, especially when the number of measurements is relatively small.'
author:
- 'Robert Seidel[^1]'
bibliography:
- 'literature.bib'
title: |
Orthogonal Matching Pursuit\
with Tikhonov and Landweber Regularization
---
*Keywords: Matching pursuit, regularization, compressed sensing, restricted isometry property*
Introduction
============
A well-known theorem in signal processing is the Nyquist-Shannon theorem: It states that any band-limited signal can be exactly recovered by sampling with a rate of no more than two times its highest frequency. In many applications, however, signals only carry little information compared to the space where the signal is acquired in. Compressed sensing—we refer to [@CW08] for a review—relies on the assumption that the gap between the dimension of the signal and its information content is expressed through the signal’s sparsity, i.e. the number of its non-zero entries. Compressed sensing has demonstrated its capabilities in many applications such as medical imaging [@LDP07], wireless communications [@LT10], and the Internet of Things [@LXW13], see also [@QBIN13] for a review.
Compressed sensing and the restricted isometry property
-------------------------------------------------------
The sampling process itself is described by the application of a linear functional to the signal. In the discrete setting, the acquisition of $m$ samples of an unknown signal $x \in {\mathbb{R}}^N$ can be written as $$Ax = y$$ with a *sampling matrix* $A \in {\mathbb{R}}^{m \times N}$. We say that $x \in {\mathbb{R}}^N$ is $k$-sparse if $\|x\|_0 := |\operatorname{supp}(x)| = k$. Given a set $S \subset \{1,...,N\}$ with $|S| = k$, we denote by $A_S \in {\mathbb{R}}^{m \times k}$ the matrix that consists of the columns of $A$ indexed by $S$. Similarly, we denote by $x_S \in {\mathbb{R}}^k$ the vector that consists of the elements of $x$ indexed by $S$.
It is easy to see that every $k$-sparse signal $x \in {\mathbb{R}}^N$ cannot be reconstructed by $m < 2k$ measurements. Indeed, given a sampling matrix $A \in {\mathbb{R}}^{m \times N}$ with $m < 2k$, a basic result of linear algebra yields $$\operatorname{rank}(A) \leq m < 2k,$$ and one can find a $2k$-sparse vector $w \in {\mathbb{R}}^N$ with $Aw = 0$. Decomposing $w$ into two $k$-sparse vectors $x,y \in {\mathbb{R}}^N$ with $w=x-y$, yields $$Ax - Ay = A(x-y) = Aw = 0,$$ i.e. $Ax = Ay$. We have shown that $A$ does not map all pairs of different $k$-sparse signals to pairs of different samples, and exact signal recovery is impossible.
The previous example shows that $A$ should be injective, at least on the set of sparse signals. This idea motivates the following definition due to @CT05 [@CT05].
Let $A \in {\mathbb{R}}^{m \times N}$ and $1 \leq k \leq N$. Then, the *restricted isometry constant (RIC)* $\delta_k$ of $A$ is defined as the smallest $\delta \geq 0$ such that $$\label{eqn:ric}
(1-\delta_k) \|x\|_2^2 \leq \|Ax\|_2^2 \leq (1+\delta_k) \|x\|_2^2$$ for all $k$-sparse $x \in {\mathbb{R}}^N$.
The restricted isometry constant of order $k$ is hard to verify for a given sampling matrix $A \in {\mathbb{R}}^{m \times N}$ as it requires the computation of $N! / (N-k)!$ submatrices consisting of $k$ columns taken from $A$. Surprisingly, many random matrices have a small RIC with a high probability. If, for example, the entries of $A$ are sampled independently from a standard Gaussian distribution, the matrix $\sqrt{m} A$ has a small RIC $\delta_k$ of order $k$ with high probability, if $$\label{eqn:ric-m}
m \geq C k \log(N/k).$$ We refer to [@CT06] for more details.
Signal reconstruction without noise
-----------------------------------
Naturally, conditions on the sensing matrix and reconstruction algorithms come in pairs. One famous example due to @CT05 is the basis pursuit (BP) [@CT05; @C08]: If $A \in {\mathbb{R}}^{m \times N}$, $1 \leq k \leq N$, and $\delta_{2k} < \sqrt{2}-1$, then every $k$-sparse $x \in {\mathbb{R}}^N$ is the unique solution of $$\min_{z \in R^N} \|z\|_1 \text{ s.t. } Az=y$$ with input $y=Ax$. The bound on $\delta_{2k}$ has been further improved in [@F10]. The BP can be understood as the convex relaxation of an $\ell_0$-pseudonorm functional [@T06].
Another class of reconstruction algorithms are greedy pursuit algorithms that can be motivated by the following: If the sensing matrix $A$ was an isometry, then $A^{-1} = A^*$, and the signal $x$ can be recovered from $A^*y = A^*Ax = x$. In this ideal scenario, the support of $x$ can be recovered by $\operatorname{supp}(x) = \operatorname{supp}(A^*y)$. If $A$ is not an isometry, we call $u = A^*y$ the *observation vector* of $y$. The orthogonal mating pursuit (OMP) due to @TG07 [@TG07] iteratively adds the coordinate of the biggest value in magnitude of $u$ to the recovered support set $S \subset \{1,...,N\}$ of $x$. The new signal estimate $x'$ is then computed by projecting $y$ onto the column space of $A_S$, and the next observation is given by $A^*r$, where $r = y - Ax'$ is the residual. The OMP is summarized in Algorithm \[alg:omp\].
Input: sensing matrix $A \in {\mathbb{R}}^{m \times N}$, samples $y \in {\mathbb{R}}^m$, sparsity level $k$.
Initialize: $r \leftarrow y$, $S \leftarrow \{\}$, $x \leftarrow 0 \in {\mathbb{R}}^N$.
For $k$ iterations:
1. *Part 1: Support augmentation*
2. Observe $A^*y$ and find the index $s$ of the largest element in magnitude, i.e. $$s \leftarrow \operatorname*{arg~max}_{j=1,...,N} |\langle A_j, r\rangle|.$$
3. Add the element to the support, i.e. $S \leftarrow S \cup \{s\}$.
4. *Part 2: Project measured signal*
5. Obtain new signal estimate by $$\begin{aligned}
x_S &\leftarrow \operatorname*{arg~min}_{z \in {\mathbb{R}}^{|S|}} \|y - A_S z\|_2, \\
x_{S^C} &\leftarrow 0.
\end{aligned}$$
6. Update the residual by $r \leftarrow y - A_S x_S$.
Note that Step 3 of OMP is equivalent to computing the pseudoinverse $A_S^\dagger$, provided that $A_S$ has a trivial kernel. In particular, the new signal estimate is obtained by $$\begin{aligned}
x_S &\leftarrow A_S^\dagger y, \\
x_{S^C} &\leftarrow 0.\end{aligned}$$
@TG07 proved the following non-uniform recovery result for the OMP algorithm [@TG07]: Given a Gaussian sensing matrix $A \in {\mathbb{R}}^{m \times N}$, the support of every $k$-sparse signal $x \in {\mathbb{R}}^N$ is recovered by OMP with input $y = Ax$ with high probability, if $$\label{eqn:omp-m}
m \geq C k \log N$$ holds. Using the restricted isometry condition, @WZWT17 [@WZWT17] derived a uniform sharp condition for exact support recovery with OMP: If $A \in {\mathbb{R}}^{m \times N}$ satisfies $$\label{eqn:omp-ric}
\delta_{k+1} < \frac{1}{\sqrt{k+1}},$$ then the support of every $k$-sparse signal $x \in {\mathbb{R}}^N$ is exactly recovered by OMP with input $y = Ax$ within $k$ iterations. Conversely, for every sparsity level $k$, there exist a $k$-sparse signal $x \in {\mathbb{R}}^N$, and a sensing matrix $A \in {\mathbb{R}}^{m \times N}$ with $$\delta_{k+1} = \frac{1}{\sqrt{k+1}},$$ such that OMP cannot recover the support of $x$ within $k$ iterations.
Compared to solving the minimization problem of BP, matching pursuit algorithms are known to have a low computational complexity [@NV09; @TG07]: They add exactly one coordinate per iteration to the support estimate and solve the projection problem for the new signal estimate. They are therefore good candidates for efficient hardware implementations.
Signal reconstruction in the presence of noise
----------------------------------------------
A naturally arising question is the robustness of signal recovery in the presence of noise. In this scenario, the signal acquisition reads as $$y = Ax + v,$$ where $v \in {\mathbb{R}}^m$ is an unknown noise term. There are several robustness results for BP [@CRT06; @CDS98; @CT06] and OMP [@WZWT17; @CW11; @W15]. In particular, for the OMP algorithm, a second assumptions besides (\[eqn:omp-ric\]) needs to be made to guarantee successful signal recovery. Namely, if $A \in {\mathbb{R}}^{m \times N}$ satisfies (\[eqn:omp-ric\]) and $$\min_{i \in \operatorname{supp}(x)} |x_i| > \frac{2 \varepsilon}{1 - \sqrt{K+1} \delta_{k+1}},$$ then the support of every $k$-sparse signal $x \in {\mathbb{R}}^N$ is recovered by OMP with input $y = Ax$ with the stopping rule $\|r\|_2 \leq \varepsilon$ within $k$ iterations, where $r$ is the OMP residual (see Algorithm \[alg:omp\]), and $\varepsilon \geq \|v\|_2$ is the noise energy. A similar *sufficient* condition for exact recovery can be found, see [@WZWT17] for more details.
Similar to the basis pursuit, regularization has been introduced to the orthogonal matching pursuit by various means. We have identified two major strands in the available literature: The refinement of the support set augmentation and the regularization of the projection step. Most of the literature focuses on regularization of the support augmentation, i.e. Part 1 in Algorithm \[alg:omp\]:
- The regularized OMP (ROMP) due to @NV09 [@NV09; @NV10] computes the observation vector $u = A^*r$ of the residual $r$, and selects up to $k$ support indices from a trusted interval of coefficient magnitude. Namely, ROMP will select a set $J \subset \{1,...,N\}$ of the $k$ largest coefficients in magnitude, and seek a subset $J_0 \subseteq J$ such that the smallest coefficient is not bigger than twice the largest coefficient selected, i.e. $$|u_i| \leq 2 |u_j| \text{ for all } i,j \in J_0.$$ If there are multiple such sets $J_0$, ROMP will choose the one with the maximal energy $\|u_{J_0}\|_2$. The set $J_0$ is then added to the set $S$, which completes the support augmentation. The remainder of the algorithm is similar to the OMP.
- The compressive sampling matching pursuit (CoSaMP) due to @NT09 [@NT09] uses the coordinates $T \subset \{1,...,N\}$ of the $2k$ largest coefficients in magnitude of the observation vector $u = A^*r$ as estimate for $\operatorname{supp}(x)$. A $2k$-sparse signal $\xi \in {\mathbb{R}}^N$ is then estimated by the least squares problem $$\xi_T = \operatorname*{arg~min}_{z \in {\mathbb{R}}^{2k}} \|A_T z - y\|_2,$$ and a $k$-sparse approximation for $x$ is given by the $k$ largest entries of $\xi$ in magnitude. The remainder of the algorithm is similar to the OMP.
- Finally, the hard thresholding pursuit (HTP) due to @F11 [@F11] selects the coordinates of the $k$ largest coefficients in $x + A^*(y - A x)$ as support estimate $S$, and then projects $y$ onto $\operatorname{Im}(A_S)$. These steps are iterated with an arbitrary $k$-sparse initialization for $x \in {\mathbb{R}}^N$ until a halting criterion is met.
For a fixed sparsity level $k$, the reconstruction error of these algorithms is linear in $\|v\|_2$, see [@NV10; @NT09; @F11]. The same holds true for the BP [@CRT06].
To the author’s best knowledge, there is only one algorithm that applies regularization to the signal estimation in OMP (Step 3 in Algorithm \[alg:omp\]): The stochastic gradient pursuit (SGP) algorithm due to @LCHW17 [@LCHW17] replaces the computation of the pseudoinverse in Step 3 with the least mean squares (LMS) estimate of $A_S^\dagger y$. This approach is motivated by the LMS adaptive filter due to @WMLJ76 [@WMLJ76], whereby the rows $u_1, ..., u_m \in {\mathbb{R}}^{1 \times |S|}$ of $A_S$ are considered to be the stochastic input to a digital filter with desired outputs $y \in {\mathbb{R}}$, and $x_S$ plays the role of that filter’s unknown weight vector. The objective of the LMS algorithm is to minimize the mean-squared error $${\mathbb{E}}\left[(y_\ell - u_\ell x_S)^2\right] = {\mathbb{E}}\left[y_\ell^2\right] - 2 {\mathbb{E}}\left[y_\ell u_\ell\right] x_S + x_S^T {\mathbb{E}}\left[u_\ell^T u_\ell\right] x_S$$ by stochastic gradient descent, where the expectation is taken over all rows of $A_S$ with equal probability. The unknown expectations in the gradient $$\nabla_{x_S} {\mathbb{E}}\left[(y_\ell - u_\ell x_S)^2\right] = - 2 {\mathbb{E}}\left[y_\ell u_\ell\right] + 2 {\mathbb{E}}\left[u_\ell^T u_\ell\right] x_S$$ are approximated by gradients of single samples. We refer to [@WMLJ76] for more details. The SGP is summarized in Algorithm \[alg:sgp\].
Input: sensing matrix $A \in {\mathbb{R}}^{m \times N}$, samples $y \in {\mathbb{R}}^m$, residual threshold $\tau$, sparsity estimate $k_\text{max}$.
Initialize: $r \leftarrow y$, $S \leftarrow \{\}$, $x \leftarrow 0 \in {\mathbb{R}}^N$, $\mu = \frac{2 m}{3 k_\text{kmax}}$.
Until $\|r\|_2 \leq \tau$:
1. Observe $A^*y$ and find the index $s$ of the largest element in magnitude, i.e. $$s \leftarrow \operatorname*{arg~max}_{j=1,...,N} |\langle A_j, r\rangle|.$$
2. Add the element to the support, i.e. $S \leftarrow S \cup \{s\}$.
3. Obtain new signal estimate by LMS iteration:
1. Initialize $z_0 \leftarrow x_S \in {\mathbb{R}}^{|S|}$.
2. For $\ell = 1,...,M$: $$\begin{aligned}
a_\ell &\leftarrow A_S[\ell,:] \in {\mathbb{R}}^{1 \times |S|} \quad \text{(the $\ell$-th row of $A_S$)} \\
d_\ell &\leftarrow y_\ell \in {\mathbb{R}}\\
e_\ell &\leftarrow d_\ell - a_\ell \cdot z_{\ell-1} \in {\mathbb{R}}\\
z_\ell &\leftarrow z_{\ell-1} + \mu \cdot e_\ell \cdot a^T \in {\mathbb{R}}^{|S|}
\end{aligned}$$
3. Then, update $x_S$ by $$\begin{aligned}
x_S &\leftarrow z_M, \\
x_{S^C} &\leftarrow 0.
\end{aligned}$$
4. Update the residual by $r \leftarrow y - A_S x_S$.
Note that the regularization of the support augmentation relies on knowledge about the sparsity level $k$ of the unknown signal $x$. This information is required in every iteration when the next support set estimate of appropriate size has to be determined. In contrast, both OMP and SGP add no more than one coordinate per iteration to the support estimate, and the SGP algorithm uses the sparsity information only to find an upper bound for the LMS step size $\mu$, see Algorithm \[alg:sgp\].
Contributions
-------------
The contributions of this work are as follows:
1. Introduce two other means of regularization in the signal estimation step: We could identify only one publication [@LCHW17] where the signal estimation step of OMP is regularized. We broaden this picture and introduce two other well-established regularization methods in the signal estimation step of the OMP, namely Tikhonov regularization and Landweber iteration. This approach does not rely on *a priori* knowledge of the signal’s sparsity level.
2. Hardware feasibility: In many application areas, limitations of sensor size and battery lifetime create the need for efficient hardware implementations [@SM12]. The SGP algorithm’s LMS estimate is tailored to be a hardware efficient reconstruction method [@LCHW17]. This work broadens the class of iterative hardware efficient algorithms where regularization an intrinsically built-in feature of the proposed algorithm.
3. Provide uniform comparison: This work provides a series of numerical experiments that serve as a proof of concept for the previously made claims. We thereby employ the same stopping criterion for all algorithms, making a fair comparison is possible. As a by-product, this stopping criterion improves the stopping criterion for SGP as proposed in [@LCHW17].
Regularization of the pseudoinverse
===================================
Most of the reviewed literature employs regularization in the support augmentation step of OMP, i.e. Part 1 of Algorithm \[alg:omp\]. This work aims to regularize the computation of the signal estimate, i.e. Part 2 of Algorithm \[alg:omp\]. We assume throughout this whole section that $|S| \leq m$, and $\ker A_S = \{0\}$. The latter holds true with high probability for many matrices with random entries, see e.g. [@BG09].
We begin with the noise-free scenario. Given the exact measurements $y$ and a support set estimate $S$, the signal estimate $x$ is given by $$\begin{aligned}
x_S &= \operatorname*{arg~min}_{z \in {\mathbb{R}}^{|S|}} \|y - A_S z\|_2 = A_S^\dagger y, \\
x_{S^C} &= 0.
\end{aligned}$$ If $S = \operatorname{supp}x$, it is easy to see that the signal estimate corresponds to the true signal. If only an approximation $y^\varepsilon$ of $y$ with $\|y^\varepsilon - y\|_2 \leq \varepsilon$ for $\varepsilon > 0$ is available, the OMP signal estimate reads as $$\label{xS}
\begin{aligned}
x_S^\varepsilon &= \operatorname*{arg~min}_{z \in {\mathbb{R}}^{|S|}} \|y^\varepsilon - A_S z\|_2 = A_S^\dagger y^\varepsilon \\
x_{S^C}^\varepsilon &= 0.
\end{aligned}$$ It is a standard result from numerical linear algebra that, depending on the condition number $\kappa$ of the matrix $A_S$, the measurement error $y^\varepsilon - y$ amplifies by the action of the pseudoinverse. The condition number is defined as $$\kappa = \frac{\sigma_\text{max}(A_S)}{\sigma_\text{min}(A_S)},$$ where $\sigma_\text{max}(A_S)$ and $\sigma_\text{min}(A_S)$ are the biggest and smallest non-zero singular values of $A_S$, respectively. If $A$ has the restricted isometry constant $\delta_k$ of order $k \geq |S|$, equation (\[eqn:ric\]) implies that the singular values of $A_S$ lie between $\sqrt{1-\delta_k}$ and $\sqrt{1+\delta_k}$, and therefore $$\kappa \leq \frac{\sqrt{1+\delta_k}}{\sqrt{1-\delta_k}}.$$
In the light of (\[eqn:omp-m\]), a favorable condition number is reached, when the number of measurements is large, compared to the ambient dimension $N$ and the RIC-order $k$. The effect of regularizing the signal estimate in the OMP is therefore stronger in the regime where $m < C k \log (N/k)$.
There are two standard approaches due to Tikhonov and Landweber that deal with unstable solutions under data perturbations. One famous regularization method is Tikhonov regularization, where the estimate $x^\varepsilon$ is replaced by $$\label{xS-tikhonov}
\begin{aligned}
x_S^{\alpha, \varepsilon} &= (A_S^*A_S + \alpha I)^{-1} A_S^*y^\varepsilon, \\
x_{S^C}^{\alpha, \varepsilon} &= 0,
\end{aligned}$$ for some $\alpha > 0$.
The regularized signal estimate is an approximation of the pseudoinverse in the following sense [@EHN00]:
Let $x^{\alpha, \varepsilon}$ be defined as in (\[xS-tikhonov\]) with $S = \operatorname{supp}x$. Let $y = Ax$ and $\|y^\varepsilon - y\|_2 \leq \varepsilon$. For $\alpha > 0$, the noise amplification is controlled by the regularization parameter as $$\|x^{\alpha,0} - x^{\alpha,\varepsilon}\| \leq \frac{\varepsilon}{\sqrt \alpha}.$$ If $\alpha = \alpha(\varepsilon)$ is such that $$\lim_{\varepsilon \to 0} \alpha(\varepsilon) = 0 \quad \text{and} \quad \lim_{\varepsilon \to 0} \frac{\varepsilon^2}{\alpha(\varepsilon)} = 0,$$ then $$\lim_{\varepsilon \to 0} x^{\alpha(\varepsilon), \varepsilon} = A^\dagger y.$$
A straightforward extension of the signal estimation of the OMP algorithm is to replace the direct signal estimate (\[xS\]) by its regularized counterpart (\[xS-tikhonov\]). The Tikhonov regularized OMP (T-OMP) is summarized in Algorithm \[alg:t-omp\]. Note that the T-OMP algorithm relies on the computation of a matrix inverse, which may prevent a hardware efficient implementation.
Input: sensing matrix $A \in {\mathbb{R}}^{m \times N}$, samples $y \in {\mathbb{R}}^m$, noise level $\varepsilon$, regularization parameter $\alpha > 0$.
Initialize: $r \leftarrow y$, $S \leftarrow \{\}$, $x \leftarrow 0 \in {\mathbb{R}}^N$.
Until $\|r\|_2 \leq \varepsilon$:
1. Observe $A^*y$ and find the index $s$ of the largest element in magnitude, i.e. $$s \leftarrow \operatorname*{arg~max}_{j=1,...,N} |\langle A_j, r\rangle|.$$
2. Add the element to the support, i.e. $S \leftarrow S \cup \{s\}$.
3. Obtain new signal estimate by $$\begin{aligned}
x_S &\leftarrow (A_S^*A_S + \alpha I)^{-1} A_S^*y, \\
x_{S^C} &\leftarrow 0.
\end{aligned}$$
4. Update the residual by $r \leftarrow y - A_S x_S$.
Another well-known regularization method is the Landweber iteration. The new signal estimate reads as $$\label{xS-landweber}
\begin{aligned}
x_S^{\ell, \varepsilon} &= x_S^{\ell-1, \varepsilon} + \omega A_S^*(y^\varepsilon - A_S x_S^{\ell-1, \varepsilon}), \\
x_{S^C}^{\ell, \varepsilon} &= 0,
\end{aligned}$$ $$$$ where $0 < \omega \leq \|A_S\|^{-2}_\text{op}$, and $x^{0,\varepsilon} = 0$. The Landweber iteration is an approximation of the pseudoinverse in the following sense [@EHN00]:
If $y = Ax$, $\|y^\varepsilon - y\|_2 \leq \varepsilon$ and $(x^{\ell,0})_\ell$, $(x^{\ell, \varepsilon})_\ell$ are two iteration sequences given by (\[xS-landweber\]), then $$x^{\ell,0} \to A^\dagger y \text{ as } \ell \to \infty, \quad \text{and} \quad \|x^{\ell, 0} - x^{\ell, \varepsilon}\| \leq \sqrt{\ell\,} \varepsilon.$$
This iterative regularization does not rely matrix inversion and therefore allows for a hardware efficient implementation. Note that the constraint $0 < \omega \leq \|A_S\|^{-2}_\text{op}$ can be satisfied by choosing $\omega = \|A\|_F^{-2} \leq \|A_S\|_F^{-2} \leq \|A_S\|^{-2}_\text{op},$ which in turn is easy to compute. The Landweber regularized OMP (L-OMP) is summarized in Algorithm \[alg:l-omp\]. Note that the inner loop (Step 3b) is initialized with $x_S = 0$ in the first iteration of the outer loop. In the subsequent iterations, the inner loop takes the previous signal estimate $x_{S}$ as initialization for the Landweber iteration.
Input: sensing matrix $A \in {\mathbb{R}}^{m \times N}$, samples $y \in {\mathbb{R}}^m$, noise level $\varepsilon$, regularization parameter $\lambda \in {\mathbb{N}}^+$.
Initialize: $r \leftarrow y$, $S \leftarrow \{\}$, $x \leftarrow 0 \in {\mathbb{R}}^N$.
Until $\|r\|_2 \leq \varepsilon$:
1. Observe $A^*y$ and find the index $s$ of the largest element in magnitude, i.e. $$s \leftarrow \operatorname*{arg~max}_{j=1,...,N} |\langle A_j, r\rangle|.$$
2. Add the element to the support, i.e. $S \leftarrow S \cup \{s\}$.
3. Obtain new signal estimate by the iteration:
1. Initialize $0 < \omega \leq \|A_S\|_{\text{op}}^{-2}$.
2. For $\ell = 1,...,\lambda$: $$\begin{aligned}
x_S &\leftarrow x_S + \omega A_S^* (y - A_S x_S), \\
x_{S^C} &\leftarrow 0.
\end{aligned}$$
4. Update the residual by $r \leftarrow y - A_S x_S$.
**Remark:** A similar approach can be found in [@F11], where the same iterative scheme is used in the Fast Hard Thresholding Pursuit (FHTP). Similar ideas are discussed in [@NT09] for the CoSaMP. However, note that both algorithms rely on *a priori* information on the sparsity level $k$ for the support augmentation step.
Numerical results
=================
In this section, we evaluate the performance of the proposed T-OMP and L-OMP algorithms by a numerical experiment. We generate a random sensing matrix $A \in {\mathbb{R}}^{m \times N}$ where each entry is taken independently from a standard Gaussian distribution. Following [@LCHW17], the columns of $A$ are then normalized. For the ambient dimension, we set $N=256$, and we consider numbers of measurement $m \in \{16, 32, 64\}$. The sparsity level is fixed at $k=8$, and the non-zeros entries of the signal vector $x \in {\mathbb{R}}^N$ are taken independently from a uniform distribution supported on $[-1, 1]$. The noise level is measured via the signal-to-noise ratio (SNR), given by $$\text{SNR} = 10 \log_{10} \left( \frac{\|Ax\|_2^2}{\|v\|_2^2} \right).$$ Each component of the noise vector $v \in {\mathbb{R}}^m$ is sampled independently from a standard Gaussian distribution. The noise vector is scaled thereafter such that the desired SNR is attained. The reconstruction performance is measured by normalized root-mean-square error (NRMSE) defined by $$\text{NRMSE} = \frac{1}{\sqrt{N}} \frac{\|x - \hat x\|_2}{\Delta},$$ where $x$ is the true signal, $\hat x$ is the reconstructed signal, and $\Delta$ is the spread between the largest and smallest entry of $x$.
![Reconstruction performance of the T-OMP algorithm with different regularization parameters $\alpha$ for different signal-to-noise ratios. The number of measurements is $\bm{m = 16}$. The curves for OMP, SGP, and CoSaMP serve as benchmark.[]{data-label="fig:t-omp16"}](fig1_m16_metric3.pdf "fig:"){width="\figwidth"} ![Reconstruction performance of the T-OMP algorithm with different regularization parameters $\alpha$ for different signal-to-noise ratios. The number of measurements is $\bm{m = 16}$. The curves for OMP, SGP, and CoSaMP serve as benchmark.[]{data-label="fig:t-omp16"}](fig1_m16_metric5.pdf "fig:"){width="\figwidth"}
In order to make the presented algorithms’ performance comparable, we employ the same stopping rule in all our experiments: The iteration is terminated as soon as $\|r\|_2 \leq \varepsilon$, where $r$ is the matching pursuit’s residual, and $\varepsilon = \|v\|_2$ is the true energy of the generated noise. This stopping rule is somewhat of theoretical nature, since the true noise energy is unknown in applications. However, only a uniform halting criterion will allow for a fair competition between the algorithms assessed. Note that @LCHW17 propose a fixed halting criterion $\|r\|_2 \leq \sqrt{0.0164}$ which is independent from the magnitude of the noise. All implemented algorithms use the number of measurements $m$ as maximum number of iterations. The presented values are the average of 1,000 trials for each combination of method and noise level.
Figure \[fig:t-omp16\] displays the reconstruction performance of the proposed T-OMP algorithm for $m = 16$. The presented algorithms SGP, CoSaMP and T-OMP outperform OMP, with the exception of CoSaMP in the high noise regime. Looking at the SGP algorithm, the performance gap between parameter choice $\tau = \sqrt{0.0164}$ (as proposed in [@LCHW17]) and $\tau = \varepsilon$ highlights the importance of a uniform stopping criterion for a fair algorithm comparison. Interestingly, T-OMP with $\alpha = 1$ and $\alpha = 10$ reaches a stable estimated support size as the noise level decreases. The same holds true for SGP with a fixed, noise-independent $\tau$.
![Reconstruction performance of the L-OMP algorithm with different regularization parameters $\lambda$ for different signal-to-noise ratios. The number of measurements is $\bm{m = 16}$. The curves for OMP, SGP, and CoSaMP serve as benchmark. Note that the curves for OMP (blue) and L-OMP with $\lambda = 100$ (pink) are often overlapping.[]{data-label="fig:l-omp16"}](fig2_m16_metric3.pdf "fig:"){width="\figwidth"} ![Reconstruction performance of the L-OMP algorithm with different regularization parameters $\lambda$ for different signal-to-noise ratios. The number of measurements is $\bm{m = 16}$. The curves for OMP, SGP, and CoSaMP serve as benchmark. Note that the curves for OMP (blue) and L-OMP with $\lambda = 100$ (pink) are often overlapping.[]{data-label="fig:l-omp16"}](fig2_m16_metric5.pdf "fig:"){width="\figwidth"}
Figure \[fig:l-omp16\] displays the reconstruction performance of the proposed L-OMP algorithm in the same setting. For $\lambda = 100$, we observe that L-OMP produces results almost identical to the OMP algorithm as the pseudo-inverse is very closely approximated. Surprisingly, the Landweber method with only one iteration in the inner loop, i.e. $\lambda= 1$, finds a good reconstruction of the true signal in terms of NRMSE while it tends to misestimate the support size. For $\lambda = 10$, we observe a good approximation of the OMP solution which cannot outperform CoSaMP and SGP. However, note that SGP requires $m = 16$ iterations in its inner loop.
![Reconstruction performance of the T-OMP algorithm with different regularization parameters for different signal-to-noise ratios. The number of measurements is $\bm{m = 64}$. The curves for OMP, SGP, and CoSaMP serve as benchmark.[]{data-label="fig:t-omp64"}](fig1_m64_metric3.pdf "fig:"){width="\figwidth"} ![Reconstruction performance of the T-OMP algorithm with different regularization parameters for different signal-to-noise ratios. The number of measurements is $\bm{m = 64}$. The curves for OMP, SGP, and CoSaMP serve as benchmark.[]{data-label="fig:t-omp64"}](fig1_m64_metric5.pdf "fig:"){width="\figwidth"}
![Reconstruction performance of the L-OMP algorithm with different regularization parameters for different signal-to-noise ratios. The number of measurements is $\bm{m = 64}$. The curves for OMP, SGP, and CoSaMP serve as benchmark. Note that the curves for OMP (blue) and L-OMP with $\lambda = 100$ (pink) are overlapping.[]{data-label="fig:l-omp64"}](fig2_m64_metric3.pdf "fig:"){width="\figwidth"} ![Reconstruction performance of the L-OMP algorithm with different regularization parameters for different signal-to-noise ratios. The number of measurements is $\bm{m = 64}$. The curves for OMP, SGP, and CoSaMP serve as benchmark. Note that the curves for OMP (blue) and L-OMP with $\lambda = 100$ (pink) are overlapping.[]{data-label="fig:l-omp64"}](fig2_m64_metric5.pdf "fig:"){width="\figwidth"}
We now repeat the numerical experiments with a higher number of measurements, $m = 64$. In the light of (\[eqn:ric-m\]), the random matrix $A_S$ is now more likely to have a good condition number. Furthermore, in the light of (\[eqn:omp-m\]), the OMP algorithm has now a higher success probability. Indeed, Figures \[fig:t-omp64\]–\[fig:l-omp64\] confirm that the OMP algorithm cannot be outperformed on any noise level and regularization does not improve the reconstruction performance. Note that for an increasing SNR, the OMP now captures the correct support size. Looking at the iterative methods (Figure \[fig:l-omp64\]), the SGP tends to overestimate the support size. In this case, the Landweber method can be a good alternative as it approximates the OMP solution very well after only 10 iterations in the inner loop, while the SGP requires $m = 64$ iterations in its LMS step. Note that this setup reproduces the results of [@LCHW17]. Clearly, the proposed choice of $\tau = \sqrt{0.0164}$ in the SGP does not demonstrate the algorithm’s full reconstruction capabilities.
![Reconstruction performance of the T-OMP algorithm with different regularization parameters for different signal-to-noise ratios. The number of measurements is $\bm{m = 32}$. The curves for OMP, SGP, and CoSaMP serve as benchmark.[]{data-label="fig:t-omp32"}](fig1_m32_metric3.pdf "fig:"){width="\figwidth"} ![Reconstruction performance of the T-OMP algorithm with different regularization parameters for different signal-to-noise ratios. The number of measurements is $\bm{m = 32}$. The curves for OMP, SGP, and CoSaMP serve as benchmark.[]{data-label="fig:t-omp32"}](fig1_m32_metric5.pdf "fig:"){width="\figwidth"}
![Reconstruction performance of the L-OMP algorithm with different regularization parameters for different signal-to-noise ratios. The number of measurements is $\bm{m = 32}$. The curves for OMP, SGP, and CoSaMP serve as benchmark. Note that the curves for OMP (blue) and L-OMP with $\lambda = 100$ (pink) are overlapping.[]{data-label="fig:l-omp32"}](fig2_m32_metric3.pdf "fig:"){width="\figwidth"} ![Reconstruction performance of the L-OMP algorithm with different regularization parameters for different signal-to-noise ratios. The number of measurements is $\bm{m = 32}$. The curves for OMP, SGP, and CoSaMP serve as benchmark. Note that the curves for OMP (blue) and L-OMP with $\lambda = 100$ (pink) are overlapping.[]{data-label="fig:l-omp32"}](fig2_m32_metric5.pdf "fig:"){width="\figwidth"}
Finally, Figures \[fig:t-omp32\]–\[fig:l-omp32\] show the same simulation with $m = 32$ measurements. Both T-OMP and L-OMP outperform OMP in the high and medium noise regime. One can easily see that the choice $\alpha = 10$ leads to a poor overall performance due to a too strong regularization. T-OMP with $\alpha = 0.1$ has the best average success rates—at the price of an overestimated support. The L-OMP reaches the best overall performance with only one iteration ($\lambda = 1$), which demonstrates the advantages of regularized methods over the direct computation of the pseudoinverse.
Conclusion
==========
In this work, we have derived two extensions of the OMP algorithm for compressed sensing based on Tikhonov regularization and Landweber iteration. A series of numerical experiments confirms the positive effect of regularization, especially in situations where the sampling matrix does not act like an almost-isometry on the set of sparse vectors. In particular, in situations where the number of measurements is comparably small, T-OMP and L-OMP outperform OMP and CoSaMP with a unified stopping criterion. Unlike CoSaMP, the proposed algorithms no not rely on *a priori* information of the signal’s sparsity. The L-OMP algorithm is an alternative iterative method for signal reconstruction that allows for a hardware-friendly implementation in the spirit of [@LCHW17], as it renounces the computation of a pseudoinverse or the solution of a linear system, and shows good results after only few steps.
An important open question for applications are good parameter choice rules or heuristics for the regularization parameters in T-OMP and L-OMP, and halting criteria that lead to provable recovery guarantees. Parameter choice rules and halting criteria usually rely on *a priori* information on the signal such as the sparsity or noise energy, such that the fine-tuning of T-OMP and L-OMP remains a highly application-specific problem.
The OMP algorithm and its robust modifications iterate over a scheme of *support augmentation* and *signal estimation*. With this modularity in mind, our regularization approach can be employed within the framework of other matching pursuits as well. For example, a study of CoSaMP or ROMP can be of future interest, where both the support augmentation (by the algorithms’ corresponding techniques) and the signal estimation (by our proposed approach) are simultaneously regularized. By this fused approach, even higher reconstruction rates than proposed in this work can possibly be achieved.
[^1]: R. Seidel is with the Institut für Mathematik, Technische Universität Berlin. He was supported by the German Academic Exchange Service and the Taiwanese Ministry of Science and Technology through the *Taiwan-Germany Summer Institute Program*. The author wishes to thank his supervisor An-Yeu (Andy) Wu, and all reviewers for their comments on this publication.
|
---
abstract: 'We numerically study thermodynamic and structural properties of the one-component Gaussian core model (GCM) at very high densities. The solid-fluid phase boundary is carefully determined. We find that the density dependence of both the freezing and melting temperatures obey the asymptotic relation, $\log T_f$, $\log T_m \propto -\rho^{2/3}$, where $\rho$ is the number density, which is consistent with Stillinger’s conjecture. Thermodynamic quantities such as the energy and pressure and the structural functions such as the static structure factor are also investigated in the fluid phase for a wide range of temperature above the phase boundary. We compare the numerical results with the prediction of the liquid theory with the random phase approximation (RPA). At high temperatures, the results are in almost perfect agreement with RPA for a wide range of density, as it has been already shown in the previous studies. In the low temperature regime close to the phase boundary line, although RPA fails to describe the structure factors and the radial distribution functions at the length scales of the interparticle distance, it successfully predicts their behaviors at shorter length scales. RPA also predicts thermodynamic quantities such as the energy, pressure, and the temperature at which the thermal expansion coefficient becomes negative, almost perfectly. Striking ability of RPA to predict thermodynamic quantities even at high densities and low temperatures is understood in terms of the decoupling of the length scales which dictate thermodynamic quantities from the interparticle distance which dominates the peak structures of the static structure factor due to the softness of the Gaussian core potential.'
author:
- Atsushi Ikeda
- Kunimasa Miyazaki
title: Thermodynamic and Structural Properties of the High Density Gaussian Core Model
---
Introduction
============
Complex fluids such as colloidal suspensions and emulsions are often regarded as macroscopic models of atomic or molecular systems. They are ideal benches to test liquid theories developed to describe thermodynamic, dynamic, and structural properties of atomic and molecular liquids [@Hansen2006]. It is not only because the size of constituent unit of complex fluids are much larger than atomic counterpart but also because their interparticle interactions can be tailor-made and tuned relatively easily. While the pair interactions of atomic systems are exclusively characterized by short-ranged and strong repulsions with weak and longer-ranged attractions, leading to the typical phase diagram demarcating gas, liquid, and crystalline phases [@Hansen2006], those for the complex fluids are far more diverse. This diversity leads to very rich and often counter-intuitive macroscopic behaviors [@Likos2001a]. Amongst those interactions, the [*ultra-soft*]{} potential have attracted particular attention recently in the soft-condensed matter community [@Likos2006b; @Stillinger; @Stillinger1997; @Lang2000; @Louis2000a; @Prestipino2005; @Mladek2006b; @Mausbach2006; @Zachary2008; @Krekelberg2009c; @Pond2009; @Krekelberg2009d; @Shall2010; @Pond2011; @Marquest1989; @Likos1998; @Likos2001b; @Mladek2006a; @Zhang2010; @Watzlawek1999; @Foffi2003b; @Zaccarelli2005; @Mayer2008; @Pamies2009; @Berthier2009f; @Berthier2010e]. The ultra-soft potentials are isotropic repulsive potential characterized by weak and bounded repulsion at short distance and the mild repulsive tails whose steepness is much smaller than the typical atomic potential. These potentials are realized in many complex fluids such as star-polymers [@Watzlawek1999; @Foffi2003b; @Zaccarelli2005; @Mayer2008], dendrimers [@Likos2006b; @Gotze2005; @Mladek2008], and the polymers in good solvent [@Louis2000a; @Kruger1989; @Dautenhahn1994; @Louis2000b; @Bolhuis2001]. Thermodynamic phase diagrams of the ultra-soft particle systems have very distinct and exotic properties such as the re-melting from solid to fluid phase at high densities, the re-entrant peak at the intermediate densities [@Stillinger; @Lang2000; @Prestipino2005; @Zachary2008; @Zhang2010; @Watzlawek1999; @Pamies2009], negative thermal expansion coefficient [@Stillinger1997; @Mausbach2006], and the cascades of the various crystalline phases at very high densities [@Watzlawek1999; @Pamies2009]. The Gaussian core model (GCM) is one of the simplest examples of the ultra-soft potential systems. GCM consists of the point particles interacting with a Gaussian shaped repulsive potential; $$\begin{aligned}
v(r) = \epsilon \exp[-(r/\sigma)^2], \end{aligned}$$ where $r$ is the interparticle separation, $\epsilon$ and $\sigma$ are the parameters which characterize the energy and length scales, respectively. GCM was first introduced by Stillinger [@Stillinger] and has been studied by many groups [@Stillinger1997; @Lang2000; @Louis2000a; @Prestipino2005; @Mladek2006b; @Mausbach2006; @Zachary2008; @Krekelberg2009c; @Pond2009; @Krekelberg2009d; @Shall2010; @Pond2011]. Despite of its simple form of the pair potential, GCM exhibits many typical thermodynamic behaviors of the ultra-soft particles. According to thermodynamic phase diagram obtained by numerical simulations [@Stillinger; @Prestipino2005], GCM basically behaves like hard spheres at very low densities and temperatures; the crystalline structure in the solid phase is fcc and the freezing/melting temperatures sharply increase with density. However, as the density increases further, the freezing temperature reaches a maximal value and beyond this point it changes to a decreasing function of the density. This re-entrance takes place at $\rho\sigma^{3}\approx 0.25$, where $\rho$ is the number density. Concomitantly, the crystalline structure changes from fcc to bcc. Recently, thermodynamic and transport anomalies of the fluid phase in the vicinity of the reentrant peaks are investigated [@Mausbach2006; @Krekelberg2009c; @Pond2009; @Krekelberg2009d; @Shall2010; @Pond2011]. Microscopic and structural properties such as the static structure factor in the fluid phase are also reported and documented [@Lang2000; @Louis2000a; @Mladek2006b; @Zachary2008]. These studies revealed that, as the density increases beyond the reentrant peak but at a fixed temperature, thermodynamic and structural properties of GCM becomes more ideal-gas-like, signaled by the lowering of the peak of the structure factors and the better agreement with simple approximations such as the random phase approximation (RPA). Most of studies in the past, however, have focused on the densities not far from the reentrant peak or the relatively high temperatures. Less attention has been paid for the high density and low temperature regimes, especially in the vicinity of the solid-fluid phase boundary. Near the phase boundary line, the thermodynamic and structural properties are expected to be highly non-trivial even at the high density limit. Based on the duality argument of the ground state of GCM in the reciprocal space, Stillinger has conjectured that the freezing and melting temperatures, $T_f$ and $T_m$, are given by an asymptotic form; $$\begin{aligned}
\log T_f, ~\log T_m \propto -\rho^{2/3}
\label{st}\end{aligned}$$ in the high density limit [@Stillinger]. However, this conjecture has not been confirmed numerically. Recently, we studied the nucleation and glassy dynamics of the one-component GCM in the supercooled state at the unprecedentedly high densities, $0.5 \leq \rho\sigma^3 \leq 2$ [@Ikeda2011]. It was found that the crystal nucleation rate decreases drastically as the density increases and concomitantly dynamics of the constituent particles becomes very sluggish. The density time correlation function exhibits typical behavior of the supercooled liquids near the glass transition point, such as the two-step and non-exponential structural relaxation. The relaxation time steeply increases as the temperature is lowered at a fixed density. Surprisingly, these are well described by the mode-coupling theory, implying that the high density and one-component GCM is more amenable to the mean-field picture of the glass transition than other typical glass formers. These observations call for more detailed analysis of the high density GCM at the low temperature regime. Especially, it is tempting to consider GCM in the high density limit as the ideal and clean model system to study the glass transition. Thermodynamic and structural characterization are prerequisites for dynamical study [@Ikeda2011; @Ikeda_II] but the detailed study is lacking.
In this work, we numerically investigate thermodynamic and structural properties of the one-component GCM up to the density $\rho\sigma^3 = 2.4$. We determine the solid-fluid phase boundary and show that the Stillinger’s scaling, Eq. (\[st\]), holds at $\rho\sigma^3 \gtrsim 1.2$. Thermodynamic and microscopic structural properties of GCM are also analyzed carefully over a wide range of temperature and density. The potential energy, pressure, thermal expansion coefficient, and the static structure factors are evaluated and compared with the prediction of the liquid state theory. Surprisingly good agreement with the random phase approximation (RPA) is found for thermodynamic quantities for a wide range of temperature, including the low temperature regimes where the same approximation poorly describes the static structure factor and radial distribution function. This counterintuitive observation can be attributed to the ultra-soft nature of GCM for which the microscopic structure near the first shell of the system decouples with the macroscopic properties.
This paper is organized as follows. In Section II, technical details of simulations and the method to compute the phase boundary are discussed. In Section III, we present simulation results for the phase diagram, thermodynamic quantities, and structural functions. We compare the simulation results in the fluid phase with RPA predictions in Section IV. We summarize and discuss the results in Sec. V.
Simulation Method
=================
MD and MC simulation
--------------------
Thermodynamic state of GCM is fully characterized by the density $\rho$ and temperature $T$. In this work, we focus on the density and temperature range of $0.3 < \rho^{\ast} < 2.4$ and $10^{-6} < T^{\ast} < 1$, where $\rho^{\ast}\equiv \rho\sigma^{3}$ and $T^{\ast} \equiv {k_{\mbox{\scriptsize B}}}T/\epsilon$. In order to analyze thermodynamic properties and determine the solid-fluid phase boundary, molecular dynamics (MD) simulations with Nos[é]{} thermostat are carried out under the periodic boundary conditions. To integrate the equations of motion, we use a reversible algorithm similar to the Velocity-Verlet method [@Frenkel2001] with time steps of $0.1\tau$, which is sufficiently short to preserve the Nos[é]{} Hamiltonian. Here $\tau = \sqrt{m\sigma^2/\epsilon}$ is the time unit, where $m$ is the mass of a particle. For evaluation of the free energy of the reference state (see below), Monte Carlo (MC) simulation is used. In a trial MC move, the maximum displacement of a particle is adjusted to keep the acceptance ratio about 50 %. In both simulations, the total number of particles is $N=3456$. This is twice the cube of an integer (in this case 12), a natural choice for the bcc crystal in a cubic simulation box. The cutoff length of the potential is taken as $5\sigma$. The pair potential at the cutoff length is $1.4 \times 10^{-11} \epsilon$ which is much smaller than the typical kinetic energy at the lowest temperature studied in this work.
Evaluation of the free energy
-----------------------------
The chemical potential as a function of the temperature and pressure, $\mu(T,P)$, is required in order to determine the phase boundary. We evaluate it using the free energy $f(T,\rho)$ and pressure $P(T,\rho)$.
We calculate the free energy using the thermodynamic integration method combined with the particle insertion method [@Widom1963] for the fluid phase and the Frenkel-Ladd method [@Frenkel1984] for the crystalline phase. This procedure is the same as the one employed by Prestipino [*et al.*]{} [@Prestipino2005] for lower densities. The free energy of the system is the sum of the ideal $f_{id}$ and excess part $f_{ex}$. $f_{id}$ is given by $f_{id}(T,\rho) = {k_{\mbox{\scriptsize B}}}T (\log \Lambda^3 \rho \sigma^3 - 1)$, where $\Lambda = \sqrt{2\pi \hbar^2/m{k_{\mbox{\scriptsize B}}}T}$ is the de Broglie thermal wave length. According to the thermodynamic integration scheme, $f_{ex}(T,\rho)$ can be evaluated by integrating over the energy and pressure from the reference state point $(T_0,\rho_0)$ to the target state point $(T,\rho)$ using the following equations. $$\begin{aligned}
\frac{f_{ex}(T,\rho_0)}{T}
&= \frac{f_{ex}(T_0,\rho_0)}{T_0} + \int^{T}_{T_0}\!\!\! dT' \ \frac{u(T',\rho_0)}{T'^2},
\\
\frac{f_{ex}(T,\rho)}{T}
&= \frac{f_{ex}(T,\rho_0)}{T} + \int^{\rho}_{\rho_0}\!\!\! d\rho' \!\!\
\left\{
\frac{P(T,\rho')}{\rho'^2 T} - \frac{1}{\rho'}
\right\},
\end{aligned}
\label{ti2}$$ where $u$ is the potential energy per particle. For the fluid phase, the reference free energy $f_{ex}(T_0,\rho_0)$ is calculated by the particle insertion method [@Frenkel2001; @Widom1963] and the pressure is evaluated from the virial equation. In this method, the free energy is expressed in terms of the energy cost to insert one particle into the system as $$\begin{aligned}
\frac{f_{ex}(T_0,\rho_0)}{T_0} = - {k_{\mbox{\scriptsize B}}}\mbox{log} {\left\langle {\exp
\left(-\frac{E_{in}}{{k_{\mbox{\scriptsize B}}}T_0}\right)} \right\rangle} -
\frac{P_0}{\rho_0 T_0} + 1,
\label{particleinsertion}\end{aligned}$$ where $E_{in}$ is the interaction energy of an inserted particle with other particles in the system. The average should be taken over the ensemble of randomly inserted particles. For the crystalline phase, on the other hand, the reference free energy is computed using the Frenkel-Ladd method [@Frenkel2001; @Frenkel1984] which is a different kind of thermodynamic integration technique. In this method, we consider a hybrid Hamiltonian $\tilde{V}(\lambda)$, which interpolates between the Hamiltonian of the original system $V$ and that of the Einstein crystal $V_{ein}$ as $\tilde{V}(\lambda) = V + (1-\lambda) V_{ein}$, where $\lambda$ is the switching parameter. The free energy of the original system can be computed by the following equation, which is the integral over $\lambda$ of the Hamiltonian of the hybrid system evaluated from the simulation, $$\begin{aligned}
f_{ex} = f_{ex,ein} + \frac{1}{N}\int^{1}_0 d\lambda \ {\left\langle {V - V_{ein}} \right\rangle}_{\lambda},
\label{fl}\end{aligned}$$ where ${\left\langle {\cdots} \right\rangle}_{\lambda}$ is the ensemble average under a hybrid Hamiltonian $\tilde{V}(\lambda)$ and $f_{ex,ein}$ is the excess part of the free energy of the Einstein crystal. We choose $(T_0^{\ast}, \rho_0^{\ast})$ $=$(0.1, 0.01) and ($T_0^{\ast}$, $\rho_0^{\ast}$) $=$ (0.0794, 0.28) as the reference states for the fluid and crystalline phases, respectively. The ensemble averages in Eqs. (\[particleinsertion\]) and (\[fl\]) are evaluated using the MC simulations. The integration over $\lambda$ in Eq. (\[fl\]) is calculated by slicing $\lambda$ to the grids of the width of 0.05 and evaluating the value of the integrand at each grid point from independent MC simulations. Likewise, the integral over the isothermal and isochoric pathways in Eq. (\[ti2\]) is computed by slicing the pathways into many grid points. The energy and pressure at each grid point are computed using the MD simulations. The free energy for both the fluid and crystalline phases are obtained by combining these data points and the reference free energy. In order to determine the solid-fluid phase boundary over the density range of $0.3<\rho^{\ast}<2.4$ with satisfactory accuracies, more than 800 grid points were necessary.
Simulation results
==================
Phase diagram
-------------
In his pioneering work, Stillinger has conjectured that the solid-fluid phase boundary is asymptotically given by Eq. (\[st\]) in the high density limit [@Stillinger]. This conjecture is based on analysis of the density dependence of the potential energy of various crystalline structures (bcc, fcc, etc) at $T=0$. According to his analysis, the potential energy is expressed as $$\begin{aligned}
u = - \frac{\epsilon}{2} + \frac{\pi^{3/2} \rho \sigma^3 \epsilon}{2} \left\{ 1 +
A \exp(- K \rho^{2/3}) + \cdots \right\},
\label{st1} \end{aligned}$$ where $A$ and $K$ are constants which depend on the crystalline structure. Stillinger argued that this density dependence remains qualitatively unchanged at finite temperatures. Since the ordered structure of the crystalline phase is responsible for the term $\exp(- K \rho^{2/3})$ in Eq. (\[st1\]), the energy difference between the crystalline and fluid phase at the phase boundary should be also proportional to this term. This argument leads us to a conjecture that the melting/freezing temperature is proportional to this factor, which is Eq. (\[st\]). Here we verify this argument numerically.
![ The freezing ($T_f^{\ast}$) and melting ($T_m^{\ast}$) temperatures of GCM as a function of density (open up/down triangles). The result of Prestipino [*et al.*]{} is also plotted (filled circles with short-dashed line) [@Prestipino2005]. The long-dashed line is the threshold temperatures, $T_{th}$, above which RPA gives a reasonable description of the system (see Sec. IV). Open squares and dotted line indicate the temperature below which the thermal expansion coefficient becomes negative, $T_{\alpha}$, obtained from simulation and RPA, respectively. The short-dashed line at $\rho^{\ast}\approx
0.15
$ demarcates the fcc (left) and bcc (right) crystalline phases. ](fig1_spd.eps){width="0.95\columnwidth"}
\[spd\]
{width="0.7\columnwidth"}
\[stillinger\]
Figure \[spd\] presents the phase diagram of GCM obtained from our simulation. Both the freezing and melting temperature $T_f^{\ast}$ and $T_m^{\ast}$ are shown in this figure, but their values are very close to each other and indistinguishable in the scale of the figure. As expected, the melting and freezing temperatures dramatically decrease as the density increases, down to $T^{\ast} \approx 10^{-6}$ at the highest density we studied. The phase boundary for $\rho^{\ast} \lesssim 0.7$ obtained by Prestipino [*et al.*]{} [@Prestipino2005] is also plotted, in order to confirm that the present result perfectly matches with theirs for the density window where both results are available. The crystalline structure at high densities is bcc, as has been verified by the direct MD simulation of the nucleation [@Ikeda2011; @Ikeda_II]. In order to verify the scaling relation, Eq. (\[st\]), we plot the logarithm of the melting temperature as a function of $\rho^{2/3}$ in Fig. \[stillinger\]. One observes that the result rides on the scaling function at $\rho^{\ast} \gtrsim 1.2$.
Potential energy and pressure
-----------------------------
{width="0.7\columnwidth"}
\[du\]
The first order transition from the crystalline to fluid phase is accompanied with the discontinuous change in the structural order and the potential energy. We show the temperature dependence of the potential energy difference between the two phases $u^{\ast}_{fluid}(T_f,\rho) -u^{\ast}_{crystal}(T_f,\rho)$ as a function of $T_m$ in Fig. \[du\], where $u^{\ast}= u/\epsilon$ is the dimensionless potential energy. This figure shows that the energy difference is proportional to the melting temperature at the low temperatures/high densities, verifying the assumption which Stillinger has employed to conjecture Eq. (\[st\]). The entropy difference between the two phases can be estimated from this energy difference by $s_{fluid}(T_f,\rho) - s_{crystal}(T_f,\rho) = \left\{u_{fluid}(T_f,\rho) - u_{crystal}(T_f,\rho)\right\}/T_f$. From the result of Fig. \[du\], the entropy difference at the low temperature/high density limit can be estimated as $$\begin{aligned}
s_{fluid}(T_f,\rho) - s_{crystal}(T_f,\rho) \sim 0.45 {k_{\mbox{\scriptsize B}}}. \end{aligned}$$ This value should be compared with the results at lower densities in the earlier work; $0.81{k_{\mbox{\scriptsize B}}}$ at $\rho^{\ast} = 0.4$ and $0.54{k_{\mbox{\scriptsize B}}}$ at $\rho^{\ast} = 1.0$ [@Stillinger].
{width="0.95\columnwidth"}
\[prhot\]
Next we focus on the equation of state (EOS) of the high density GCM, [*i.e.*]{}, the pressure as a function of the density and temperature. The dimensionless pressure $P^{\ast} \equiv P\sigma^3/\epsilon$ is plotted in Figure \[prhot\]. In Figures \[prhot\] (a), (c), and (e), we plot the isothermal cut of EOS and the isochoric cut in Figures \[prhot\] (b), (d), and (f). The parameters $T^{\ast}$ and $\rho^{\ast}$ for each adjacent figures have been chosen so as for them to share the common freezing points; Fig. \[prhot\] (a) and (b) share the freezing point ($T_{f}^{\ast}, \rho_{f}^{\ast})=(7.94\times 10^{-3}, 0.33$), (c) and (d) share ($T_{f}^{\ast}, \rho_{f}^{\ast})=(5.0\times 10^{-4}, 1.00$), and (e) and (f) share ($T_{f}^{\ast}, \rho_{f}^{\ast})=(7.9\times 10^{-6}, 2.01$). Figure \[prhot\] (a) shows that the melting of the bcc crystalline phase (white circles) to the fluid phase (filled circles) takes place at at $\rho^{\ast}\approx 0.33$. The inset shows the narrow coexistence region around the transition density, at which the pressure becomes constant. Similar behaviors of the first order transition are also observed in Figs. \[prhot\] (c) and (e) but at much higher densities.
Figure \[prhot\] (b) shows the equation of state at $\rho^{\ast}=0.33$ over the temperature range of $1.0\times 10^{-3}<T^{\ast}<1.0$. At this relatively low density, the pressure is a monotonically increasing function of the temperature, which is usual behavior of ordinary fluids. However, at higher densities, as shown in Fig. \[prhot\] (d) and (f), there exists the temperature regime in which the pressure becomes a decreasing function of temperature. In this regime, the thermal expansion coefficient $\alpha=V^{-1}(\partial V/ \partial T)_P$ becomes negative. We determine the threshold temperature $T_{\alpha}$ at which $\alpha$ changes its sign by fitting the pressure by a smooth polynomial function and plot $T_{\alpha}$ in Fig. \[spd\] (open squares). At low densities, $T_{\alpha}$ is located in the vicinity of the phase boundary. With increasing density, however, the difference between $T_{\alpha}$ and $T_m$ increases monotonically. Although the existence of the anomalous negative thermal expansion coefficient of GCM has been reported in the literatures [@Stillinger; @Stillinger1997], its high density behavior has not been explored. We shall discuss this result and its asymptotic behavior of $T_{\alpha}$ at high densities in the following section.
Static structure factor
-----------------------
{width="0.95\columnwidth"}
\[sk\]
In Figure \[sk\], we show the static structure factors $S(k)$ in the fluid phase just above the freezing temperatures. $S(k)$ obtained from the Fourier transformation of the radial distribution function $g(r)$ is also shown. At all densities, $S(k)$’s exhibit sharp peaks similar to those of the ordinary simple fluids, such as the hard sphere and Lennard-Jones fluid, near the freezing temperatures. The peak position $k_{\max}$ shifts to high $k$’s as the density increases, $k_{\max} \propto \rho^{1/3}$, as one should expects at high densities. Note that, however, the height of the first peak is about 3.1 at all densities, which is slightly higher than the universal value 2.85 for the ordinary fluids which is known as Hansen-Verlet criterion [@Hansen2006; @Hansen1969].
Random phase approximation analysis
===================================
Random phase approximation
--------------------------
It is known that the thermodynamics and microscopic structure of the GCM fluid in high densities and temperatures are described by the random phase approximation (RPA) remarkably well [@Lang2000; @Louis2000a]. RPA is one of the approximation schemes of the liquid theory and a kind of the mean-field theory for the thermodynamics and structure of liquids [@Hansen2006]. In this section, we discuss how this approximation works at much lower temperatures.
It is convenient to divide thermodynamic quantities into uniform and fluctuation parts as follows. The potential energy can be represented as $$\begin{aligned}
u &=& - \frac{\epsilon}{2} + \frac{\pi^{3/2} \rho\sigma^3 \epsilon}{2} +
\Delta u,
\label{ueq} \end{aligned}$$ where the first two terms are the uniform part and $\Delta u$ is the fluctuation part. $\Delta u$ can be expressed as $$\begin{aligned}
\Delta u &=& \frac{1}{4\pi^2} \int^{\infty}_0 k^2 \tilde{v}(k) S(k) \ dk,
\label{du1}\end{aligned}$$ where $$\tilde{v}(k) = \pi^{3/2} \epsilon \sigma^3 \exp(-k^2 \sigma^2/4)$$ is the reciprocal expression of $v(r)$. Likewise, the pressure can be written, using the virial equation, as $$\begin{aligned}
P &=& {k_{\mbox{\scriptsize B}}}T \rho + \frac{\pi^{3/2} \rho^2 \sigma^3 \epsilon}{2} + \Delta P,\end{aligned}$$ where the first two terms are the uniform part and the third is the fluctuation part which can be written as $$\begin{aligned}
\Delta P &=& \frac{\rho}{4\pi^2} \int^{\infty}_0 \bigl( k^2 - k^4\sigma^2/6 \bigr) \tilde{v}(k) S(k) \ dk. \label{dz}\end{aligned}$$ In RPA, the direct correlation function of the system is approximated as $c_{{{\mbox{\scriptsize RPA}}}}(r) = -\beta v(r)$, where $\beta = 1/{k_{\mbox{\scriptsize B}}}T$. This approximation makes it possible to express various static quantities in simple and analytic forms. The static structure factor can be expressed as $$\begin{aligned}
S_{{{\mbox{\scriptsize RPA}}}}(k) = \frac{1}{1+\rho \beta \tilde{v}(k)}. \label{mf}\end{aligned}$$ The fluctuation parts of the potential energy and pressure are expressed as [@Louis2000a]: $$\begin{aligned}
\Delta u_{{{\mbox{\scriptsize RPA}}}}
&= - \frac{\epsilon}{2\gamma} {\mbox{Li}}_{3/2} (- \gamma ), \\
\Delta P_{{{\mbox{\scriptsize RPA}}}}
&= - \frac{\rho \epsilon}{2\gamma} \left\{ {\mbox{Li}}_{3/2} (- \gamma ) - {\mbox{Li}}_{5/2}(- \gamma ) \right\},
\label{upRPA}
\end{aligned}$$ where $\gamma = \pi^{3/2}\rho \sigma^3 \epsilon/{k_{\mbox{\scriptsize B}}}T$ is a dimensionless coupling parameter and ${\mbox{Li}}_{\nu}(x)$ is the $\nu$-th polylogarithm function [@Louis2000a; @Mladek2006b]. Furthermore, the radial distribution function at $r=0$ can be expressed analytically as $$\begin{aligned}
g_{{{\mbox{\scriptsize RPA}}}}(r=0) = 1 + \frac{1}{\pi^{3/2}\rho\sigma^3} {\mbox{Li}}_{3/2} (-\gamma).
\label{gRPA}\end{aligned}$$ The second term on the right hand side of this expression is negative for arbitrary densities and temperatures. At a very low temperature, the modulus of this term becomes larger than the first, leading to an unphysical negative $g(r=0)$. We refer to this temperature as the threshold temperature $T_{th}$. We plot $T_{th}$ in Fig. \[spd\] (long-dashed line). At high densities, $T_{th}$ can be expressed analytically as $$\begin{aligned}
\frac{{k_{\mbox{\scriptsize B}}}T_{th}}{\epsilon} = \pi^{3/2} \rho \sigma^3
\exp\left[- \Bigl( \frac{3\pi^2}{4} \rho\sigma^3 \Bigr)^{2/3} \right],
\label{th}\end{aligned}$$ which is obtained by the asymptotic expansion of polylogarithm function of Eq. (\[gRPA\]) (see Appendix). Interestingly, the threshold temperature follows the same asymptotic scaling law $\log T_{th} \propto - \rho^{2/3}$ as the melting and freezing temperatures, Eq. (\[st\]). Note that this asymptotic expression is very accurate down to moderate densities $\rho^{\ast} \sim 1$.
High temperature regime
-----------------------
--------------------------------------------------------- ---------------- ---------------- ---------------- --------------------------
$\rho$ 0.10 0.33 1.00 3.00
${T_{th}}$ [0.823]{} [0.535]{} [0.159]{} ${0.689 \times 10^{-2}}$
$\Delta u_{{{\mbox{\scriptsize RPA}}}}/\Delta u$ 0.97 0.94 0.94 0.98
$\Delta P_{{{\mbox{\scriptsize RPA}}}}/\Delta P$ 1.07 0.96 0.92 0.94
--------------------------------------------------------- ---------------- ---------------- ---------------- --------------------------
: RPA results compared with simulation results for the fluctuation parts of the potential energy and pressure at the threshold temperatures for various densities. []{data-label="table1"}
\[hight\_uz\]
We first assess the validity of RPA at temperatures above $T_{th}$. Table \[hight\_uz\] shows the ratio of RPA to simulation results of the fluctuation parts of the potential energy and pressure at $T=T_{th}$. The deviations are smaller than 10% and they monotonically become smaller as the temperature increases, implying the thermodynamic quantities are well-described by RPA even at $T_{th}$.
{width="0.95\columnwidth"}
\[hight\_gofr\]
We also computed the radial distribution function $g(r)$. Fig. [\[hight\_gofr\]]{} shows $g(r)$ obtained from simulation (open circles) and RPA (solid lines) at $T=T_{th}$ and at much higher temperatures at which $g_{{{\mbox{\scriptsize RPA}}}}(r=0) = 0.83$, for several densities. At high temperatures, agreement of simulation results with RPA is excellent in all densities and for all $r$’s. At $T=T_{th}$, however, RPA works poorly around $r=0$, as expected from Eq. (\[gRPA\]). On the other hand, RPA’s results perfectly match with the simulation results at larger $r$’s including the first shell peaks. The reason why thermodynamic quantities are well described by RPA even at $T_{th}$ whereas agreement of $g(r)$ near $r=0$ is poor can be attributed to the fact that the short-range part of $g(r)$ does not contribute to both the potential energy and pressure, as we shall discuss in the following subsection.
Low temperature regime
----------------------
We move to temperatures below $T_{th}$ and discuss the validity of RPA in describing thermodynamic properties of GCM at high densities. In Fig. \[sk\], the static structure factors obtained from RPA, $S_{{{\mbox{\scriptsize RPA}}}}(k)$, are shown in the solid lines. It is obvious that RPA cannot capture even the qualitative behaviors of $S(k)$; $S_{{{\mbox{\scriptsize RPA}}}}(k)$ remains flat at higher $k$’s and does not possess any prominent peak. However, as the main panel and inset of Fig. \[sk\] shows, RPA correctly predicts the low $k$’s behavior up to just below the wavevector at which the first peaks are located. It is in stark contrast with ordinary simple atomic fluids for which RPA works poorly for the whole range of wavevectors [^1]. The excellent agreement implies that the mean-field character of the dense GCM still survives in the length scales slightly longer than the typical interparticle distance.
Next, we compare RPA with simulation results for thermodynamic quantities. We first look at the equation of state. In Fig. \[prhot\], the pressure obtained from RPA are plotted in solid lines. As shown in Fig. \[prhot\] (a) and (b), the deviation of the values of RPA from those of simulation is very large at $\rho^{\ast}=0.33$ and the discrepancies increase with decreasing temperature. RPA also predicts a fictitious negative thermal expansion coefficient at this density as shown in Fig. \[prhot\] (b). In high densities, however, agreement of RPA with simulation is excellent for all temperatures down to the freezing temperatures as shown in Fig. \[prhot\] (c)–(f). This is very surprising because the temperatures in these figures are far below $T_{th}$, where RPA fails to describe overall shapes of $S(k)$ as discussed in the previous subsection.
We also calculated $T_{\alpha,{{\mbox{\scriptsize RPA}}}}$ by solving $\partial
P_{{{\mbox{\scriptsize RPA}}}}/\partial T = 0$ and plotted in Fig. \[spd\] (dotted line). Agreement of $T_{\alpha,{{\mbox{\scriptsize RPA}}}}$ with the simulation result is perfect except for the vicinity of the re-entrant melting region. The asymptotic expression of $T_{\alpha,{{\mbox{\scriptsize RPA}}}}$ which is valid at high densities can be written as (see Appendix) $$\begin{aligned}
\frac{{k_{\mbox{\scriptsize B}}}T_{\alpha,{{\mbox{\scriptsize RPA}}}}}{\epsilon} = \pi^{3/2} \rho\sigma^3 \exp\ \left[- \left(
\frac{ {15} \pi^2}{4} \rho\sigma^3 \right)^{2/5} -2 \right].
\label{alpha}\end{aligned}$$ This asymptotic expression works well down to $\rho^{\ast} \sim 1$ as in the case of $T_{th}$. Interestingly, the density exponent $2/5$ in this expression is smaller than $2/3$ for $T_{th}$ and $T_m$ (see Eqs. (\[st\]) and (\[th\])). Due to this difference, $T_{\alpha}$ monotonically deviates from $T_m$ as the density increases and it eventually becomes larger than $T_{th}$ in the high density limit (not shown).
{width="0.95\columnwidth"}
\[ut\]
We made similar comparison for the potential energy in Fig. \[ut\]. Fig. \[ut\] (a) and (b) show the temperature dependence of the potential energy in the fluid phase (filed circles) and crystalline phase (open circles) at two densities. The uniform part of $u$ is temperature independent (see Eq. (\[ueq\])), although it dominates the net values of $u$. In order to see the temperature dependence of $u$ more clearly, the fluctuation part $\Delta u$ are shown in Fig. \[ut\] (c) and (d). One observes that agreement of the simulation results with RPA is far better at high densities; At $\rho^{\ast}=0.33$, RPA underestimates $\Delta u$ and discrepancy from the simulation data are larger than the energy gap between the fluid and crystalline phases. At the higher density $\rho^{\ast}=2.01$, the discrepancy is less 10% even at the freezing temperature.
{width="0.95\columnwidth"}
\[lowt\_uz\]
In order to quantify the accuracy of RPA for both the potential energy and pressure, we plot the ratios of $\Delta u$ and $\Delta P$ of simulation to those of RPA in Fig. \[lowt\_uz\] against the inverse temperatures for several densities. The ratios decrease rapidly as the temperature decreases at the low density $\rho^{\ast}=0.33$, whereas, in the high density $\rho^{\ast}=2.01$, the ratios for both the potential energy and pressure remain to be more than 80% for the whole range of temperatures down to the melting temperature.
![ The integrand of Eq. (\[du1\]) (shaded areas), the integrand where $S(k)$ is replaced with $S_{{{\mbox{\scriptsize RPA}}}}(k)$ (solid lines) and $S(k)$ (broken lines) for (a) $T^{\ast} = 7.9 \times 10^{-3}$ and $\rho^{\ast}=0.33$, (b) $T^{\ast} = 5.0 \times 10^{-4}$ and $\rho^{\ast}=1.00$, (c) $T^{\ast} = 3.2 \times 10^{-5}$ and $\rho^{\ast}=1.65$ and (d) $T^{\ast} = 7.9 \times 10^{-6}$ and $\rho^{\ast}=2.01$. ](fig9_lowt_int.eps){width="0.95\columnwidth"}
\[lowt\_int\]
Given that RPA poorly describes even the qualitative behaviors of $S(k)$ and $g(r)$ at the high density and low temperature regimes, it is very surprising and counterintuitive that RPA is excellent at predicting quantitatively thermodynamic quantities $u$ and $P$. In order to rationalize this puzzling facts, we look again the integral expressions of the thermodynamic functions, Eq. (\[du1\]) and (\[dz\]). These expressions show that both $\Delta u$ and $\Delta P$ are expressed in terms of the integral of $S(k)$ multiplied with the pair potential over the wavevectors. In order to see which length scales dominate the integrand, we show the integrand of $\Delta u$ in Eq. (\[du1\]) with simulated $S(k)$ in Fig. \[lowt\_int\] (shaded area), along with those obtained using $S_{{{\mbox{\scriptsize RPA}}}}(k)$ (solid line). At the low density $\rho^{\ast}=0.33$, the integrand with simulated $S(k)$ is peaked at the peak position of $S(k)$ (broken line). This implies that $\Delta u$ at this density is dominated by the contribution at the interparticle distance, just like ordinary atomic fluids. The integrand obtained using RPA fails to account for this peak structure (solid line). However, with increasing density, the peak position of the integrand shifts to smaller wavevectors than the first peak position of $S(k)$. Concomitantly, agreement of the integrand obtained from simulation and RPA becomes better and better. This agreement originates from the fact that RPA can account excellently for the low wavevector behavior of the static structure factor as shown in Fig. \[sk\]. At very high densities, the particles start overlapping and the characteristic interparticle distance decouples with the length scales which dominate thermodynamic quantities of GCM. This is the reason why RPA remains the excellent approximation to predict thermodynamic quantities even far below the threshold temperatures.
Conclusions
===========
In this paper, we have presented a detailed analysis of thermodynamic and structural properties of the high density one-component GCM. Special emphasis has been put for static properties of the fluid phase. First, the solid-fluid phase boundary of the system is carefully evaluated up to the unprecedentedly high density $\rho^{\ast} =2.4$. Our result confirmed the scaling conjectured by Stillinger for the freezing and melting temperatures, $\log T_f, ~\log T_m \propto -\rho^{2/3}$, at $\rho^\ast \gtrsim 1.2$. The potential energy difference between the crystalline and fluid phases was shown to be linear in the freezing temperature and the entropy difference is almost constant at high densities, which verifies the assumption which Stillinger’s argument is based upon. The thermodynamic and structural properties of GCM in the fluid phase are analyzed in detail for a wide range of temperature and density. The potential energy $u$, the equation of state $P$, the static structure factor $S(k)$, and the radial distribution function $g(r)$ were evaluated by simulation. We compare the simulation results with the RPA results. In the high temperature regime, RPA provides almost perfect description for both thermodynamic quantities and the structural factor $S(k)$. RPA is rather poor at predicting $g(r)$ at $r \approx 0$. Threshold temperature $T_{th}$ below which RPA fails to describe $g(r=0)$ is relatively high. In the high density and low temperature regime, RPA fails to capture the peak structure of $S(k)$ even qualitatively, whereas it predicts correctly the low $k$’s behavior up to just below the wavevector at which the first peaks are located. Despite of poor performance of RPA at describing the structural properties, RPA successfully describes thermodynamic quantities such as the potential energy and pressure at high densities. Agreement of RPA with simulation results systematically improves as the density increases even near the phase boundary. The temperature below which the thermal expansion coefficient become negative is also accurately calculated from RPA. By scrutinizing the role of the microscopic structure of particles in the potential energy and pressure, we concluded that the surprising success of RPA is originated from the decoupling of the length scales which dictate the thermodynamic quantities and the interparticle distance. This decoupling is attributed to the mild and long-ranged repulsive tails of the pair potential of GCM. The fact that RPA is an excellent approximation even at the vicinity of the phase boundary, or even at the supercooled regime, at high densities, hints that the mean-field description is valid for the high density GCM and may play a crucial role to understand (glassy) dynamics let alone thermodynamic properties [@Ikeda2011; @Ikeda_II].
This work is partially supported by Grant-in-Aid for JSPS Fellows (AI), KAKENHI; \# 21540416, (KM), and Priority Areas “Soft Matter Physics” (KM).
Derivation of Eqs. (\[th\]) and (\[alpha\])
===========================================
In this appendix, we derive the asymptotic expressions of $T_{th}$ and $T_{\alpha,{{\mbox{\scriptsize RPA}}}}$ at high densities by using the asymptotic expansion of the polylogarithm [@Wood1992] $$\begin{aligned}
{\mbox{Li}}_{\nu}(-x) = - \frac{(\log x)^{\nu}}{\Gamma(\nu + 1)} + \mathcal{O}((-\log x)^{\nu - 2}), \label{polylog}\end{aligned}$$ where $\Gamma(x)$ is the gamma function.
![ The validity of the asymptotic expressions for $T_{th}$ and $T_{\alpha,{{\mbox{\scriptsize RPA}}}}$. Open circles and solid line are RPA values and its asymptotic expression, Eq. (\[th\]), of $T_{th}$, respectively. Open squares, dotted line, and dashed line are RPA values and its two asymptotic expressions, Eqs. (\[eq0\]) and (\[alpha\]), of $T_{\alpha,{{\mbox{\scriptsize RPA}}}}$, respectively. ](fig10_asymptotic.eps){width="0.95\columnwidth"}
\[asymptotic\]
At $T=T_{th}$, Eqs. (\[gRPA\]) and (\[polylog\]) lead to $$\begin{aligned}
&
\rho^\ast - \frac{4}{3 \pi^2}
\left(\log x_{th} \right)^{3/2}
+ \mathcal{O}\left(\left(\log x_{th}\right)^{-1/2}\right) = 0,
\end{aligned}$$ where $x_{th}= \pi^{3/2} \rho^\ast/T_{th}^{\ast}$. When $\rho^\ast$ is sufficiently large, we can neglect the third term on the left hand side, leading to the asymptotic expression of $T_{th}$ of Eq. (\[th\]). Fig. \[asymptotic\] shows that this asymptotic expression is very accurate down to $\rho^{\ast} \sim 1$. Likewise, at $T=T_{\alpha,{{\mbox{\scriptsize RPA}}}}$, Eqs. (\[upRPA\]) and (\[polylog\]) lead to $$\begin{aligned}
&
\rho^\ast
- \frac{4}{15 \pi^2} \left(\log x_{{{\mbox{\scriptsize RPA}}}}\right)^{5/2}
+ \frac{4}{3 \pi^2} \left(\log x_{{{\mbox{\scriptsize RPA}}}}\right)^{3/2}
\\
&
+ \mathcal{O}\left(\left(\log x_{{{\mbox{\scriptsize RPA}}}}\right)^{1/2}\right) = 0,
\end{aligned}
\label{ap1}$$ where $x_{{{\mbox{\scriptsize RPA}}}} = \pi^{3/2} \rho^\ast/T_{\alpha,{{\mbox{\scriptsize RPA}}}}^\ast$. If we neglect the third and fourth terms on the left hand side, we obtain $$\begin{aligned}
T_{\alpha,{{\mbox{\scriptsize RPA}}}}^{\ast}
= \pi^{3/2} \rho^\ast \exp\left[-\left(\frac{15\pi^2\rho^\ast}{4}\right)^{2/5}\right]. \label{eq0}\end{aligned}$$ However, Fig. \[asymptotic\] shows that this expression is not accurate even at very high densities. For sufficient accuracy, we have to keep the third term on the left hand side of Eq. (\[ap1\]). If we neglect only the forth term on the left hand side, Eq. (\[ap1\]) becomes $$\begin{aligned}
\frac{4}{15 \pi^2} s^{5/2} - \rho^{\ast -2/5} \frac{4}{3 \pi^2} s^{3/2} - 1 = 0, \end{aligned}$$ where $s = \rho^{\ast -2/5} \log (\pi^{3/2} \rho^\ast /T_{\alpha}^\ast)$. Since the second term on the left hand side of this equation is small due to the factor $\rho^{\ast -2/5}$, we can expand the solution as $s = s_0 + \rho^{\ast -2/5} s_1 +
\cdots$ and can solve the equation in each order of density. The first order solution including $s_0$ and $s_1$ is $$\begin{aligned}
s = \Bigl( \frac{15 \pi^2}{4} \Bigr)^{2/5} + 2 \rho^{\ast -2/5}, \end{aligned}$$ which leads to the asymptotic expression of $T_{\alpha,{{\mbox{\scriptsize RPA}}}}$ of Eq. (\[alpha\]). Fig. \[asymptotic\] shows that this expression is very accurate down to $\rho^{\ast} \sim 1$.
[10]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, ** (, ).
, ****, ().
, ****, ().
, ****, (); , ****, (); , ****, ().
, ****, ().
, , , , ****, ().
, , , ****, ().
, , , ****, (); , , , ****, ().
, , ****, ().
, , ****, ().
, , ****, ().
, , , , , ****, ().
, , , , , ****, ().
, , , , , , ****, ().
, ****, ().
, , , [arXiv:1101.1982]{}.
, ****, ().
, , , ****, ().
, , , ****, ().
, , , , ****, ().
, , , ****, ().
, , , ****, ().
, , , , , , , , ****, ().
, , , , , , , , , , , ****, ().
, , , , , , , , , , , , ****, ().
, , , ****, ().
, ****, ().
, , , ****, ().
, ****, ().
, , ****, ().
, , ****, ().
, ****, ().
, , , ****, ().
, , , ****, ().
, ****, ().
, **** .
, ** (, ).
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ** (unpublished).
[^1]: Even for the one component plasma, a typical long-range interacting system for which RPA is often employed to describe its static properties, RPA does not capture the structures at low $k$’s. See Ref.[@Ichimaru1982].
|
---
abstract: 'The next generation of space-based galaxy surveys are expected to measure the growth rate of structure to about a percent level over a range of redshifts. The rate of growth of structure as a function of redshift depends on the behavior of dark energy and so can be used to constrain parameters of dark energy models. In this work we investigate how well these future data will be able to constrain the time dependence of the dark energy density. We consider parameterizations of the dark energy equation of state, such as XCDM and $\omega$CDM, as well as a consistent physical model of time-evolving scalar field dark energy, $\phi$CDM. We show that if the standard, specially-flat cosmological model is taken as a fiducial model of the Universe, these near-future measurements of structure growth will be able to constrain the time-dependence of scalar field dark energy density to a precision of about 10%, which is almost an order of magnitude better than what can be achieved from a compilation of currently available data sets.'
author:
- 'Anatoly Pavlov, Lado Samushia, and Bharat Ratra'
title: 'Forecasting cosmological parameter constraints from near-future space-based galaxy surveys'
---
Introduction
============
Recent measurements of the apparent magnitude of Type Ia supernovae (SNeIa) continue to indicate, quite convincingly, that the cosmological expansion is currently accelerating [see, e.g., @Conley:2011ab; @Suzuki:2012ab; @Li:2011ab; @Barreira:2011ab].
If we assume that general relativity provides an adequate description of gravitational interactions on these cosmological length scales, then the kinematic properties of the Universe can be derived by solving the Einstein equations $$R_{\mu \nu} - \frac{1}{2}g_{\mu \nu}R = 8G\pi T_{\mu \nu}.
\label{equ:en}$$ Here $g_{\mu\nu}$ is the metric tensor, $R_{\mu\nu}$ and $R$ are the Ricci tensor and (curvature) scalar respectively, $T_{\mu\nu}$ is the stress-energy tensor of the Universe’s constituents, and $G$ is the Newtonian gravitational constant.
There is good observational evidence that the large-scale radiation and matter distributions are statistically spatially isotropic. The (Copernican) cosmological principle, which is also consistent with current observations, then indicates that the Friedmann-Lema[î]{}tre-Robertson-Walker (FLRW) models provide an adequate description of the spatially homogeneous background cosmological model.
In the FLRW models, the current accelerating cosmological expansion is a consequence of dark energy, the dominant, by far, term in the current cosmological energy budget. The dark energy density could be constant in time (and hence uniform in space) — Einstein’s cosmological constant $\Lambda$ [@Peebles:1984ab] — or gradually decreasing in time and thus slowly varying in space .
The “standard" model of cosmology is the spatially-flat $\Lambda$CDM model in which the cosmological constant contributes around 75% of the current energy budget. Non-relativistic cold dark matter (CDM) is the next largest contributor, at around 20%, with non-relativistic baryons in third place with about 5%. For a review of the standard model see @Ratra:2008ab and references therein.
Recent measurements of the anisotropies of the cosmic microwave background (CMB) radiation [e.g., @Komatsu:2011ab; @Reichardt:2012ab], in conjunction with significant observational support for a low density of non-relativistic matter [CDM and baryons together, e.g., @Chen:2003ab], as well as measurements of the position of the baryon acoustic oscillation (BAO) peak in the matter power spectrum [e.g., @Percival:2010ab; @Dantas:2011ab; @Carnero:2012ab; @Anderson:2012ab], provide significant observational support to the spatially-flat $\Lambda$CDM model. Other data are also not inconsistent with the standard $\Lambda$CDM model. These include strong gravitational lensing measurements [e.g., @Chae:2004ab; @Lee:2007ab; @Biesiada:2010ab], measurement of Hubble parameter as a function of redshift [e.g., @Samushia:2006ab; @Sen:2008ab; @Pan:2010ab; @Chen:2011ab], large-scale structure data [e.g., @Baldi:2011ab; @DeBoni:2011ab; @Brouzakis:2011ab; @Campanelli:2011ab], and galaxy cluster gas mass fraction measurements [e.g., @Allen:2008ab; @Samushia:2008ab; @Tong:2011ab]. For recent reviews of the situation see, e.g., @Blanchard:2010ab, @Sapone:2010ab, and @Jimenez:2011ab.
While the predictions of the $\Lambda$CDM model are in reasonable accord with current observations, it is important to bear in mind that dark energy has not been directly detected (and neither has dark matter). Perhaps as a result of this, some feel that it is more reasonable to assume that the left hand side of Einstein’s Eq. (\[equ:en\]) needs to be modified (instead of postulating a new, dark energy, component of the stress-energy tensor on the right hand side). While such modified gravity models are under active investigation, at present there is no compelling observational reason to prefer any of these over the standard $\Lambda$CDM cosmological model.
The $\Lambda$CDM model assumes that dark energy is a cosmological constant with equation of state $$p_\Lambda = -\rho_\Lambda,
\label{equ:eos}$$ where $p_\Lambda$ and $\rho_\Lambda$ are the pressure and energy density of the cosmological constant (fluid). This minimalistic model, despite being in good agreement with most observations available today, has some potential conceptual shortcomings that have prompted research into alternative explanations of the dark energy phenomenon.[^1]
To describe possible time-dependence of the dark energy density, it has become popular to consider a more general equation of state parametrization $$p_\omega = \omega(z)\rho_\omega.
\label{equ:wrho}$$ Here $p_\omega$ and $\rho_\omega$ are the pressure and energy density of the dark energy fluid with redshift $z$ dependent equation of state parameter $\omega(z)$. The simplest such parametrization is the XCDM one for which the equation of state parameter is constant and results in accelerated expansion if $\omega(z) = \omega_X < -1/3$. In this case the dark energy density decreases with time and this allows for the possibility that the fundamental energy density scale for dark energy is set at high energy in the early Universe and the slow decrease of the energy density over the long age of the Universe ensures that the characteristic dark energy density scale now is small (a few meV). This also ensures that the dark energy density remains comparable to the matter energy density over a longer period of time (compared to that for the $\Lambda$CDM model).
When $\omega_X = -1$ the XCDM parametrization reduces to the consistent (and complete) $\Lambda$CDM model. For any other value of $\omega_X$ the XCDM parametrization cannot consistently describe spatial inhomogeneities without further assumptions and extension [see, e.g., @Ratra:1991ab; @Podariu:2000ab]. Models in which $\omega(z)$ varies in time, $\omega$CDM models, are also unable to consistently describe spatial inhomogeneities without further assumptions and extension.
A physically and observationally viable alternative to the $\Lambda$CDM model, that consistently describes a slowly decreasing in time dark energy density, is the $\phi$CDM model . This model, in which a dark energy scalar field, $\phi$, slowly roles down its potential, resulting in a slowly decreasing dark energy density, alleviates some of the conceptual problems, mentioned above, associated with the $\Lambda$CDM model. The slowly rolling scalar field, at a given instant of time, can be approximated by a dark energy fluid with an appropriately negative equation of state parameter.
More specifically, a $\phi$CDM model with an inverse-power-law scalar field potential energy density $V(\phi)\propto \phi^{-\alpha}$, $\alpha > 0$, is a prototypical example that has been extensively studied. This model has a non-linear attractor or “tracker" scalar field solution that forces the initially sub-dominant dark energy density to come to dominate over the matter energy density, thus dominating the energy budget of the current Universe, and so resulting in the current accelerated cosmological expansion. In addition to therefore partially alleviating the “coincidence" problem of the $\Lambda$CDM model, the $\phi$CDM model generates the current tiny dark energy scale of order an meV, measured by the SNeIa, through decrease, via cosmological expansion over the long age of the Universe, of a much larger energy scale.
The $\alpha$ parameter controls the steepness of the scalar field potential, with larger values resulting in a stronger time dependence of the approximate equation of state parameter and $\alpha=0$ corresponds to the $\Lambda$CDM model limit. $\alpha$ has been constrained using currently available data [see e.g., @Chen:2004ab; @Wilson:2006ab; @Chen:2012ab; @Mania:2012ab and references therein]. The strongest current limits are that $\alpha$ has to be less than $\sim 0.7$ at 2$\sigma$ confidence [@SamushiaThesis].
In the $\phi$CDM model, or in the XCDM or $\omega$CDM parameterizations, the background evolution of the (spatially homogeneous) Universe differs from that in the $\Lambda$CDM case. This affects both the distance-redshift relation as well as the growth rate of large-scale structure. With precise measurements of distance and growth rate over a range of redshifts it will be possible to discriminate between cosmological models.[^2]
The BAO signature in the observed large-scale structure of the Universe allows for the measurements of radial and angular distances as functions of redshift [see, e.g., @Percival:2010ab; @Blake2011b; @Beutler2011; @Anderson:2012ab]. In addition, the redshift-space distortion signal allows for inferences about the strength of gravitational interactions on very large scales [see, e.g., @Percival2004; @Angulo2008; @Guzzo2008; @Blake2011a; @Samushia2012; @Reid:2012ab]. Currently available data sets have been used to measure distances and growth history up to a redshift $z \sim 0.8$ and the next generation of planned space-based galaxy redshift surveys of the whole extragalactic sky are expected to extend these measurements to a redshift $z \sim 2$. Possible candidates for such surveys include the Euclid satellite mission that has been approved by the European Space Agency [@Laureijs2011] and the WFIRST satellite that was ranked high by the recent Decadal Survey [@Green2011]. These surveys have been shown to have the potential to measure angular distances, Hubble parameter $H(z)$, and growth rate as functions of redshift to a few percent precision over a wide range of redshifts [@Wang2011; @samushia2010; @Majerotto2012; @Basse:2012].[^3]
As mentioned above, an alternative potential explanation of the observed accelerated expansion of the Universe is to replace general relativity by a modified theory of gravity. For example, in the $f(R)$-gravity models the Einstein-Hilbert gravitational action is modified to $$S = \frac{1}{16\pi G}\int d^{4}x\sqrt{-g}f(R),
\label{equ:Act_fR}$$ where the function $f(R)$ of the Ricci curvature $R$ can in general be of any form. In the special case when $f(R) = R$ one recovers the Einstein-Hilbert action which yields the Einstein equations of general relativity, Eq. (\[equ:en\]). For every dark energy model it is possible to find a function $f(R)$ that will result in exactly the same expansion history [see, e.g. @Sotiriou:2010ab; @Tsujikawa:2010ab; @Capazziello2011ab] thus potentially eliminating the need for dark energy. However, nothing prevents the coexistence of modified gravity and dark energy, with both contributing to powering the current accelerated cosmological expansion. It is of significant importance to be able to determine which scenario best describes what is taking place in our Universe.
In this paper we investigate how well anticipated data from the galaxy surveys mentioned above can constrain the time dependence of the dark energy. We will use the Fisher matrix formalism to obtain predictions for the $\phi$CDM model and compare these with those made using the (model-dependent) XCDM and $\omega$CDM parameterizations of dark energy. We will mostly assume that gravity is well described by general relativity, but will also look at some simple modified gravity cases. We find that the anticipated constraints on the parameter $\alpha$ of the $\phi$CDM model are almost an order of magnitude better than the ones that are currently available.
Compared to the recent analysis of @samushia2010, here we use an updated characterization of planned next-generation space-based galaxy surveys, so our forecasts are a little more realistic. We also consider an additional dark energy parametrization, XCDM, a special case of $\omega$CDM that was considered by @samushia2010, as well as the $\phi$CDM model, forecasting for which has not previously been done.
The paper is organized as follows. In Sec. \[sec:measurement\] we briefly describe the observables and their relationship to basic cosmological parameters. In Sec. \[sec:model\] we describe the models of dark energy that we study. Section \[sec:fisher\] outlines the method we use for predicting parameter constraints, with some details given in the Appendix. We present our results in Sec. \[sec:result\] and conclude in Sec. \[sec:conclusion\].
Measured power spectrum of galaxies {#sec:measurement}
===================================
The large-scale structure of the Universe, which most likely originated as quantum-mechanical fluctuations of the scalar field that drove an early epoch of inflation [see, e.g., @Fischler:1985ab], became (electromagnetically) observable at $z \sim 10^{3}$ after the recombination epoch. Dark energy did not play a significant role at this early recombination epoch because of its low mass-density relative to the densities of ordinary and dark matter as well as that of radiation (neutrinos and photons). At $z \sim$ 5 galaxy clusters began to form. Initially, in regions where the matter density was a bit higher than the average, space expanded a bit slower than average. Eventually the dark and ordinary matter reached a minimum density and the regions contracted. If an over-dense region was sufficiently large its baryonic matter collapsed into its dark-matter halo. The baryonic matter continued to contract even more due to its ability to lose thermal energy through the emission of electromagnetic radiation. This can not happen with dark matter since it does not emit significant electromagnetic radiation nor does it interact significantly (non-gravitationally) with baryonic matter. As a result the dark matter remained in the form of a spherical halo around the rest of the baryonic part of a galaxy. At $z \sim$ 2 the rich clusters of galaxies were formed by gravity, which gathered near-by galaxies together. Also by this time the dark energy’s energy density had become relatively large enough to affect the growth of large-scale structure.
Different cosmological models with different sets of parameters can result in the same expansion history and so it impossible to distinguish between such models by using only expansion history measurements. This is one place where measurements of the growth history of the large-scale structure of the Universe plays an important role. It is not possible to fix free parameters of two different cosmological models to give exactly the same expansion and growth histories simultaneously. It is therefore vital to observe both histories in order to obtain better constraints on parameters of a cosmological model.
In a cosmological model described by the FLRW metric, and to lowest order in dark matter over-density perturbations, the power-spectrum of observed galaxies is given by [@kaiser87] $$P_{g}(k,\mu) = P_{m}(k)(b\sigma_{8} + f\sigma_{8}\mu^{2})^{2}.
\label{equ:psigma}$$ Here subscript $g$ denotes galaxies, $P_m$ is the underlying matter power spectrum, $b$ is the bias of galaxies, $f$ is the growth rate, $\mu$ is the cosine of the angle between wave-vector $k$ and the line-of-sight direction, and $\sigma_8$ is the overall normalization of the power spectrum ($\sigma_8$ is the rms energy density perturbation smoothed over spheres of radius $8 h^{-1}$ Mpc, where $h = H_0/(100 {\rm km} {\rm s}^{-1}
{\rm Mpc}^{-1}$) and $H_0$ is the Hubble constant). Since, for a measured power spectrum of galaxies on a single redshift slice, the bias and growth rate are perfectly degenerate with the overall amplitude, in the equations below we will refer to $b\sigma_8$ and $f\sigma_8$ simply as $b$ and $f$.
The angular dependence of the power spectrum in Eq. (\[equ:psigma\]) can be used to infer the growth rate factor $f(z)$ which is defined as the logarithmic derivative of the linear growth factor $$f(z) = \frac{d \ln G}{d \ln a},
\label{equ:fz_dlg}$$ where $a$ is the cosmological scale factor, and the linear growth factor $G(t) = \delta(t)/\delta(t_{\rm in})$ shows by how much the perturbations have grown since some initial time $t_{\rm in}$.[^4]
The numerical value of the $f(z)$ function depends both on the theory of gravity and on the expansion rate of the Universe. Since the growth rate depends very sensitively on the total amount of non-relativistic matter, it is often parametrized as [see, e.g., @linder05 and references therein] $$f(z) \approx \Omega_{m}^{\gamma}(z),
\label{equ:fzOm}$$ where $$\Omega_{m}(z) = \frac{\Omega_{m}(1 + z)^{3}}{E^2(z)},
\label{equ:Omz}$$ and $$E(z) = H(z)/H_0 = \sqrt{\Omega_{m}(1 + z)^{3} + \Omega_{k}(1 + z)^{2} +
\Omega_{DE}(z)}.
\label{equ:Ez}$$\
Here $H(z)$ is the Hubble parameter and $H_0$ is its value at the present epoch (the Hubble constant), $\Omega_{m}$ is the value of the energy density parameter of non-relativistic matter at the present epoch ($z = 0$), $\Omega_k$ that of spatial curvature, and $\Omega_{DE}(z)$ is the energy density parameter which describes the evolution of the dark energy density and is different in different dark energy models.
The growth index, $\gamma$, depends on both a model of dark energy as well as a theory of gravity. When general relativity is assumed and the equation of state of dark energy is taken to be of the general form in Eq. (\[equ:wrho\]) then [see, e.g., @linder05 and references therein] $$\gamma \approx 0.55 + 0.05[1 + \omega(z=1)]
\label{equ:gamma}$$ to a few percent accuracy. In the $\Lambda$CDM cosmological model $\gamma \approx$ 0.55. An observed significant deviation from this value of $\gamma$ will present a serious challenge for the standard cosmological model.
The power spectrum is measured under the assumption of a fiducial cosmological model. If the angular and radial distances in the fiducial model differ from those in the real cosmology, the power spectrum will acquire an additional angular dependence via the @alcock79 [AP] effect, as discussed in @samushia2010, $$P_{g}(k, \mu) = \frac{1}{f_{\parallel}f_{\perp}^{2}}
P_{m}\left(\frac{k}{f_{\perp} F}\sqrt{F^2 +
\mu^{2}\left(1 - F^2\right)} \right)
\times\left\{b + \frac{\mu^{2}f}{F^{2} +
\mu^{2}(1 - F^{2})}\right\}^{2},
\label{equ:Pw}$$ where $$f_{\parallel}(z) = R_{r}(z)/\hat{R}_{r}(z),
\label{equ:f_par}$$ $$f_{\perp}(z) = D_{A}(z)/\hat{D}_{A}(z),
\label{equ:f_per}$$ $$F = f_{\parallel}/f_{\perp}.$$ Here $R_{r} = dr/dz$ is the derivative of the radial distance, $D_{A}$ is the angular diameter distance (both defined below), a hat indicates a quantity evaluated in the fiducial cosmological model, and a quantity without a hat is evaluated using the alternative cosmological model. The AP effect is an additional source of anisotropy in the measured power spectrum and allows for the derivation of stronger constraints on cosmological parameters.
Cosmological models {#sec:model}
===================
In an FLRW model with only non-relativistic matter and dark energy the distances $D_{A}(z)$ and $R_{r}(z)$ are $$D_{A}(z) = \frac{1}{h\sqrt{\Omega_{k}}(1 + z)}
\sinh\left( \sqrt{\Omega_{k}}\int_{0}^{z}\frac{dz'}{E(z')} \right),
\label{equ:Dz}$$ $$R_{r}(z) = \frac{1}{h(1 + z)E(z)}.
\label{equ:Rr}$$ Here $E(z)$ is defined in Eq. (\[equ:Ez\]). The functional form of $E(z)$ depends on the model of dark energy.
$\Lambda$CDM, XCDM and $\omega$CDM parameterizations
----------------------------------------------------
Here we describe the relevant features of the $\Lambda$CDM model and the dark energy parameterizations we consider.
If the dark energy is taken to be a fluid its equation of state can be written as $p = \omega(z)\rho$. For the $\Lambda$CDM model the equation of state parameter $\omega(z) = -1$ and the dark energy density is time independent.
In the XCDM parametrization $\omega (z) = \omega_X (< -1/3)$ is allowed to take any time-independent value, resulting in a time-dependent dark energy density.
In the $\omega$CDM parametrization the time dependence of $\omega(z)$ is parametrized by introducing an additional parameter $\omega_a$ through [@Chevallier:2001ab; @Linder:2003ab] $$w(z) = w_{0} + w_{a}\frac{z}{1 + z}.
\label{equ:wz}$$ The XCDM parametrization is the limit of the $\omega$CDM parametrization with $\omega_a = 0$. In the $\omega$CDM parametrization the function $\Omega_{DE}(z)$ that describes the time evolution of the dark energy density is $$\Omega_{DE}(z) = (1 - \Omega_{m} - \Omega_{k})
(1 + z)^{3(1 + w_{0} + w_{a})}\exp\left(-3w_{a}\frac{z}{1 + z}\right),
\label{equ:Fz}$$ and the corresponding expression for the XCDM case can be derived by setting $\omega_a = 0$ here.
$\phi$CDM model
---------------
In the $\phi$CDM model the energy density of the background, spatially homogeneous, scalar field $\phi$ can be found by solving the set of simultaneous ordinary differential equations of motion, $$\ddot{\phi} + 3\frac{\dot{a}}{a}\dot{\phi} + V'(\phi) = 0
\label{equ:phi_field},$$ $$\left(\frac{\dot{a}}{a} \right)^{2} = \frac{8\pi G}{3}(\rho + \rho_{\phi}) - \frac{k}{a^{2}},
\label{equ:phi_hubble}$$ $$\rho_{\phi} = \frac{1}{16\pi G}\left(\frac{1}{2}\dot{\phi}^{2}
+ V(\phi) \right).
\label{equ:phi_rho}$$ Here an over-dot denotes a derivative with respect to time, a prime denotes one with respect to $\phi$, $V(\phi)$ is the potential energy density of the scalar field, $\rho_\phi$ is the energy density of the scalar field, and $\rho$ that of the other constituents of the Universe.
Following we consider a scalar field with inverse-power-law potential energy density $$V(\phi) = \frac{\kappa}{2G}\phi ^{-\alpha}.
\label{equ:phi_V}$$ Here $\alpha$ is a positive parameter of the model to be determined experimentally and $\kappa$ is a positive constant. This choice of potential has the interesting property that the scalar field solution is an attractor with an energy density that slowly comes to dominate over the energy density of the non-relativistic matter (in the matter dominated epoch) and causes the cosmological expansion to accelerate. The function $\Omega_{DE}(z)$ in the case of $\phi$CDM is $$\Omega_{DE}(z) = \frac{1}{12}\left(\dot{\phi}^{2}
+ \frac{\kappa}{G}\phi ^{-\alpha} \right).
\label{equ:phi_Ode}$$
Fisher matrix formalism {#sec:fisher}
=======================
The precision of the galaxy power spectrum measured in redshift bins depends on the cosmological model, the volume of the survey, and the distribution of galaxies within the observed volume. See App. A for a summary of how to estimate the precision of measurements from survey parameters.
We assume that the power spectrum $P(k_i)^{\rm meas}$ has been measured in $N$ wave-number $k_i$ bins ($i = 1\ldots N$) and each measurement has a Gaussian uncertainty $\sigma_{i}$. From these measurements a likelihood function $$\mathcal{L} \propto \exp\left(-\frac{1}{2}\chi^{2} \right)
\label{equ:likelihood}$$ can be constructed where $$\chi^{2} = \sum_{i = 1}^{N}\frac{(P_{i}^{\rm meas} -
P_{i}(\vec{p}))^{2}}{\sigma_{i}^{2}} .
\label{equ:chi}$$ Here $\vec{p}$ are the set of cosmological parameters on which the power spectrum depends.
The likelihood function in Eq. (\[equ:likelihood\]) can be transformed into the likelihood of theoretical parameters $\vec{p}$ by Taylor expanding it around the maximum and keeping terms of only second order in $\delta \vec{p}$ as $\chi^{2}(\delta p)$ = $ F_{jk} \delta p^{j}
\delta p^{k}$, where $F_{jk}$ is the Fisher matrix[^5] of the parameter set $\vec{p}$ given by second derivatives of the likelihood function through $$F_{jk} = -\left\langle \frac{\partial^{2}\ln \mathcal{L}}{\partial p^{j}\partial p^{k}}
\right\rangle .
\label{equ:F_jk}$$
The Fisher matrix predictions are exact in the limit where initial measurements as well as derived parameters are realizations of a Gaussian random variable. This would be the case if the $P_{i}^{\rm meas}$ were perfectly Gaussian and the $P_{i}(\vec{p})$ were linear functions of $\vec{p}$, which would make the second order Taylor expansion of the likelihood around its best fit value exact. In reality, because of initial non-Gaussian contributions and nonlinear effects, the predictions of Fisher matrix analysis will be different (more optimistic) from what is achievable in practice. These differences are larger for strongly non-linear models and for the phase spaces in which the likelihood is non-negligible at some physical boundary ($\alpha=0$ in case of $\phi$CDM). A more realistic approach, that requires significantly more computational time and power, is to generate a large amount of mock data and perform a full Monte-Carlo Markov Chain (MCMC) analysis [see, e.g., @Perotto2006; @Martinelli2011 where the authors find significant differences compared to the results of the Fisher matrix analysis].
We assume that the full-sky space-based survey will observe H$\alpha$-emitter galaxies over 15000 $\rm{deg}^2$ of the sky. For the density and bias of observed galaxies we use predictions from @Orsi:2010ab and @Geach:2010ab respectively. We further assume that about half of the galaxies will be detected with a reliable redshift. These numbers roughly mirror what proposed space missions, such as the ESA Euclid satellite and the NASA WFIRST mission, are anticipated to achieve. For the fiducial cosmology we use a spatially-flat $\Lambda$CDM model with $\Omega_{m}$ = 0.25, the baryonic matter density parameter $\Omega_{b}$ = 0.05, $\sigma_{8}$ = 0.8, and the primordial density perturbation power spectral index $n_{s}$ = 1.0, for convenience we summarize all the parameters of the fiducial model in Table 1.
$\Omega_{m}$ $\Omega_{b}$ $\Omega_{k}$ h $\sigma_{8}$ $n_{s}$ Efficiency Redshift span Covered sky area in $\rm{deg}^2$
-------------- -------------- -------------- ----- -------------- --------- ------------ ------------------------- ----------------------------------
0.25 0.05 0.0 0.7 0.8 1.0 0.45 0.55 $\leq z \leq$ 2.05 15000
: Values of the parameters of the fiducial $\Lambda$CDM model and the survey.
\[tab: fiducial\]
We further assume that the shape of the power spectrum is known perfectly (for example from the results of the [*Planck*]{} satellite) and ignore derivatives of the real-space power spectrum with respect to cosmological parameters.
We predict the precision of the measured galaxy power spectrum and then transform it into correlated error bars on the derived cosmological parameters. At first we make predictions for the basic quantities $b$ and $f$ in the XCDM and $\omega$CDM parameterizations and in the $\phi$CDM model. Then it allows us to predict constraints on deviations from general relativity and see how these results change with changing assumptions about dark energy. Finally, we forecast constraints on the basic cosmological parameters of dark energy models.
For the XCDM parametrization these basic cosmological parameters are $p_{\rm XCDM}=(f, b, h, \Omega_m, \Omega_k, w_X)$. The $\omega$CDM parametrization has one extra parameter describing the time evolution of the dark energy equation of state parameter, $p_{\omega {\rm CDM}}
= (f, b, h, \Omega_m, \Omega_k, w_0, w_a)$. For the $\phi$CDM model the time dependence of the dark energy density depends only on one parameter $\alpha$ so we have $p_{\phi{\rm CDM}}=(f, b, h, \Omega_m, \Omega_k, \alpha)$. In order to derive constraints on the parameters of the considered cosmological models while altering assumptions about the correctness of general relativity, we transform Fisher matrices of each model from the parameter set described above to the following parameter set (that now includes $\gamma$ that parametrizes the growth rate) $p_{model} =
(\gamma, model)$, where by $model$ we mean all the parameters of a particular model, for example, for $\omega$CDM $model
= p_{\omega {\rm CDM}} = (f, b, h, \Omega_m, \Omega_k, w_0, w_a)$.
Results {#sec:result}
=======
Constraints on growth rate
--------------------------
Figure \[fig: 1\] shows predictions for the measurement of growth rate assuming different dark energy models. We find that in the most general case, when no assumption is made about the nature of dark energy, the growth rate can be constrained to a precision of better then 2% over a wide range of redshifts. This is in good agreement with previous similar studies [see, e.g., Fig. 1 of @samushia2010]. When we specify a dark energy model the constraints on growth rate improve by about a factor of two. There is very little difference between the results derived for different dark energy models: the precision is almost insensitive to the assumed model. Also, one can notice that the curves for the XCDM parametrization and for the $\phi$CDM model are almost identical. The likely explanation of this effect is that for a fixed redshift bin the $\phi$CDM model is well described by the XCDM parametrization with the value of the parameter $\omega_{X}$ = $p_{\phi}/\rho_{\phi}$, where the values of the scalar field pressure $p_{\phi}$ and energy density $\rho_{\phi}$ are evaluated at that redshift bin.
The measurements of growth rate can be remapped into constraints on parameters describing the deviation from general relativity. Figure \[fig: 2\] shows correlated constraints between the current re-normalized Hubble constant $h$ and the $\gamma$ parameter that describes the growth of structure. The $\phi$CDM model constraints on both $h$ and $\gamma$ are tighter than those for the XCDM or $\omega$CDM parameterizations. As expected, the most restrictive $\Lambda$CDM model results in the tightest constraints.
Constraints on dark energy model parameters
-------------------------------------------
We use measurements of growth and distance to constrain parameters of the dark energy models.
Figure \[fig: 3\] shows constraints on parameters of the $\omega$CDM parametrization [these should be compared to Figs. 4a and 5a of @samushia2010] . When no assumptions are made about the nature of gravity the constraints on $\omega_0$ and $\omega_a$ are very weak and degenerate. When we assume general relativity the constraints tighten significantly, resulting in $\sim 10$% accuracy in the measurement of $\omega_0$ and $\sim 25$% accuracy in the measurement of $\omega_a$.
The upper panel of Fig. \[fig: 4\] shows constraints on the parameters $\omega_X$ and $\Omega_m$ of the XCDM parametrization. Similar to the previous case, the constraints tighten significantly when we assume general relativity as the model of gravity. About a 2% measurement of $\omega_X$ and a 5% measurement of $\Omega_m$ are possible in this case. The lower panel of Fig. \[fig: 4\] show the related constraints on $\Omega_k$ and $\Omega_m$ for the XCDM parametrization. The constraints are similar to, but somewhat tighter than, those for the $\omega$CDM parametrization. This is because the XCDM parametrization has one less parameter than the $\omega$CDM parametrization. Spatial curvature can be constrained to about 15% precision in this case.
Figures \[fig: 5\] and \[fig: 6\] show constraints on parameters of the $\phi$CDM model. In the most general case, when no assumption is made about the nature of gravity, the constraints are weak and the parameters $\alpha$ and $\Omega_m$ are strongly correlated, with larger values of $\alpha$ requiring larger values of $\Omega_m$. When general relativity is assumed, the constraints become much stronger and parameter $\alpha$ can be constrained to be less than 0.1 at the 1-$\sigma$ confidence level. This is significantly better than any constraint available at the moment.
Figure \[fig: 7\] shows constraints on the parameters of the $\Lambda$CDM model. From the clustering data alone the spatial curvature can be constrained with almost 1% precision, largely because this model has the least number of free parameters.
The exact numerical values for the forecast error bars and likelihood contours should be taken with caution and not be interpreted as predictions for the performance of any specific survey (such as [*Euclid*]{} or [*WFIRST*]{}). Our main objective in this work was first to investigate how the modified gravity constraints change with different models of dark energy and second to demonstrate the improvement in $\phi$CDM model constraints achievable with future galaxy surveys. Because of this we were able to simplify our method by adopting a Fisher matrix formalism instead of a full MCMC approach and also use a simplified description of the survey baseline. For more realistic predictions of [*Euclid*]{} performance, see, e.g., @Laureijs2011 [@samushia2010; @Majerotto2012].
Conclusion {#sec:conclusion}
==========
We have forecast the precision at which planned near-future space-based spectroscopic galaxy surveys should be able to constrain the time dependence of dark energy density. For the first time, we have used a consistent physical model of time-evolving dark energy, $\phi$CDM, in which a minimally-coupled scalar field slowly rolls down its self-interaction potential energy density. We have shown that if general relativity is assumed, the deviation of the parameter $\alpha$ of the $\phi$CDM model can be constrained to better than $0.05$; this is almost an order of magnitude better than the best currently available result.
The constraints on basic cosmological parameters, such as the relative energy densities of non-relativistic matter and spatial curvature, depend on the adopted dark energy model. We have shown that in the $\phi$CDM model the expected constraints are more restrictive than those derived using the XCDM or $\omega$CDM parameterizations. This is due to the fact that the $\phi$CDM model has fewer parameters. Also, the XCDM and $\omega$CDM parameterizations assign equal weight to all possible values of $\omega$, while in the $\phi$CDM model there is an implicit theoretical prior on which equation of state parameter values are more likely, based on how easy it is to produce such a value within the model.
Since the observational consequences of dark energy and modified gravity are partially degenerate, constraints on modified gravity parameters will depend on the assumptions made about dark energy. In Table \[tab: deviations\] we show how the constraint on the $\gamma$ parameter depends on the adopted dark energy model. The constraints on $\gamma$ are most restrictive in the $\Lambda$CDM model. For the $\phi$CDM model the constraints on $\gamma$ are about a third tighter than those for the $\omega$CDM and XCDM parameterizations.
These results are very encouraging: data from an experiment of the type we have modeled will be able to provide very good, and probably revolutionary, constraints on the time evolution of dark energy.
Acknowledgments
===============
This work was supported by DOE grant DEFG030-99EP41093 and NSF grant AST-1109275. LS is grateful for support from European Research Council, SNSF SCOPES grant \# 128040, and GNSF grant ST08/4-442.
Appendix
========
In this Appendix we summarize how to estimate the precision of measurements from the survey parameters.
The Fisher matrix coefficients are given by $$F_{ij} = \frac{1}{2}\int_{k_{\rm min}}^{k_{\rm max}}\left(\frac{\partial\ln P}{\partial p^{i}} \right)\left(\frac{\partial\ln P}{\partial p^{j}} \right)V_{\rm eff}(k,\mu)\frac{d^{3} k}{(2\pi)^{3}} ,
\label{equ:F_ij_apx}$$ where the effective volume is $$V_{\rm eff} = V_{0}\frac{nP(k,\mu)}{1 + nP(k,\mu)} ,
\label{equ:V_eff}$$ and $V_{0}$ is the total survey volume and $n$ is the number density. Also, following @Tegmark:1997ab, we multiply the integrand in Eq.(\[equ:F\_ij\_apx\]) by a Gaussian factor $\exp\left(
-k^{2}\sigma_{z}\frac{dr(z)}{dz}\right)$, where $r(z)$ is the co-moving distance, in order to account for the errors in distance induced by the errors of redshift measurements, $\sigma_z = 0.001$. We model the theoretical power spectrum using an analytic approximation of @eisenstein98. We integrate in $k$ from $k_{\rm min} = 0$ to $k_{\rm max}$, where the $k_{\rm max}$ values depend on redshift and are chosen in such a way that the small scales that are dominated by non-linear effects are excluded. The range of scales that will be fitted to the future surveys will depend on how well the theoretical templates are able to describe small-scale clustering and is difficult to predict. The $k_{\rm max}$ values along with the expected bias and number density of galaxies are listed in Table \[tab: FisherData\].
In order to derive the Fisher matrix of a specific cosmological model we have to go from our initial parameter space to the parameter space of the cosmological model whose Fisher matrix we want. The transformation formula for the Fisher matrix is given by $$F'_{lm} = \frac{\partial p_{i}}{\partial p'_{l}}\frac{\partial p_{j}}{\partial p'_{m}}F_{ij} ,
\label{equ:F_trans}$$ where the primes denote the “new" Fisher matrix and parameters.
We now list the derivatives of the transformation coefficients of the $\phi$CDM model in the limit $\alpha \longrightarrow$ 0 and $\Omega_{k} \longrightarrow$ 0 (which corresponds to the fiducial spatially-flat $\Lambda$CDM model). The transformation coefficients relating $f_{\parallel}(z)$ and the parameters ($h, \Omega_{m},
\Omega_{k}, \alpha$) are $$\frac{\partial f_{\parallel}(z)}{\partial h} = -\frac{1}{h} ,
\label{equ:fpar_h}$$ $$\frac{\partial f_{\parallel}(z)}{\partial \Omega_{m}} = \frac{1}{2E(z)^{2}}[1 - (1 + z)^{3}] ,
\label{equ:fpar_Om}$$ $$\frac{\partial f_{\parallel}(z)}{\partial \Omega_{k}} = \frac{1}{2E(z)^{2}}[1 - (1 + z)^{2}] ,
\label{equ:fpar_Ok}$$ $$\frac{\partial f_{\parallel}(z)}{\partial \alpha} = -\frac{(1 - \Omega_{m})}{8E(z)^{2}} .
\label{equ:fpar_alpha}$$ For the other transformation coefficients, it is convenient to introduce the integral $$D(z) = \int_{0}^{z}\frac{dz'}{E(z')} .
\label{equ:int_Ez}$$ Then the transformation coefficients between $f_{\perp}(z)$ and the parameters ($h, \Omega_{m}, \Omega_{k}$, $\alpha$) are $$\frac{\partial f_{\perp}(z)}{\partial h} = -\frac{1}{h} ,
\label{equ:fper_h}$$ $$\frac{\partial f_{\perp}(z)}{\partial \Omega_{m}} = \frac{1}{2D(z)}\int_{0}^{z}\frac{dz'}{E(z')^{3}}[1 - (1 + z')^{3}],
\label{equ:fper_Om}$$ $$\frac{\partial f_{\perp}(z)}{\partial \Omega_{k}} = \frac{D(z)^{2}}{6} + \frac{1}{2D(z)}\int_{0}^{z}\frac{dz'}{E(z')^{3}}[1 - (1 + z')^{2}],
\label{equ:fper_Ok}$$ $$\frac{\partial f_{\perp}(z)}{\partial \alpha} = -\frac{(1 - \Omega_{m})}{8D(z)}\int_{0}^{z}\frac{dz'}{E(z')^{3}}.
\label{equ:fper_alpha}$$ Finally, the transformation coefficients between the growth factor $f(z)$ and the parameters ($\gamma, h, \Omega_{m}, \Omega_{k}, \alpha$) are $$\frac{\partial f(z)}{\partial \gamma} = \frac{f(z)}{\gamma}\ln f(z) ,
\label{equ:f_gamma}$$ $$\frac{\partial f(z)}{\partial \Omega_{m}} = \frac{\gamma f(z)}{\Omega_{m}E(z)^{2}}\left\lbrace E(z)^{2} - \Omega_{m}[(1 + z)^{3} - 1]\right\rbrace ,
\label{equ:f_Om}$$ $$\frac{\partial f(z)}{\partial \Omega_{k}} = -\frac{\gamma f(z)}{E(z)^{2}}[(1 + z)^{2} - 1] ,
\label{equ:f_Ok}$$ $$\frac{\partial f(z)}{\partial \alpha} = -\frac{\gamma f(z)}{4E(z)^{2}}[1 - \Omega_{m}] .
\label{equ:f_alpha}$$
$z$ $k_{max}$ $b(z)$ $n(z)$
------ ----------- -------- --------
0.55 0.144 1.0423 3220
0.65 0.153 1.0668 3821
0.75 0.163 1.1084 4364
0.85 0.174 1.1145 4835
0.95 0.185 1.1107 5255
1.05 0.197 1.1652 5631
1.15 0.2 1.2262 5972
1.25 0.2 1.2769 6290
1.35 0.2 1.2960 6054
1.45 0.2 1.3159 4985
1.55 0.2 1.4416 4119
1.65 0.2 1.4915 3343
1.75 0.2 1.4873 2666
1.85 0.2 1.5332 2090
1.95 0.2 1.5705 1613
2.05 0.2 1.6277 1224
: Values of the $k_{max}$, bias $b(z)$ from @Orsi:2010ab, and the number densities $n(z)$ taken from @Geach:2010ab.
\[tab: FisherData\]
[80]{} natexlab\#1[\#1]{}
Albrecht, A., et al. 2009, arXiv:0901.0721 \[astro-ph.IM\]
Alcock, C., & Paczyński, B. 1979, Nat., 281, 358
Allen, S. W. et al., 2008, , 383, 879
Anderson, L., et al. 2012, arXiv:1203.6594 \[astro-ph.CO\]
Angulo, R. E., Baugh, C. M., & Lacey, C. G. 2007, MNRAS, 387, 921
Baldi, M., & Pettrino, V. 2011, , 412, L1
Barreira, A., & Avelino, P. P. 2011, Phys. Rev. D, 84, 083521
Basse, T., et al. 2012, arXiv:1205.0548 \[astro-ph.CO\]
Beutler, F., et al. 2011, , 416, 3017
Biesiada, M., Pi[ó]{}rkowska, A., & Malec, B. 2010, , 406, 1055
Blake, C., et al. 2011a, MNRAS, 415, 2876
Blake, C., et al. 2011b, , 418, 1707
Blanchard, A. 2010, A[&]{}A Rev., 18, 595
Brouzakis, N., et al. 2011, , 1103, 049
Campanelli, L., et al. 2011, arXiv:1110.2310 \[astro-ph.CO\]
Capazziello, S., & De Laurentis, M. 2011, Phys. Rept., 509, 167
Carnero, A. et al. 2012, , 419, 1689
Chae, K.-H., et al. 2004, , 607, L71
Chen, G., & Ratra, B. 2003, , 115, 1143
Chen, G., & Ratra, B. 2004, , 612, L1
Chen, Y., & Ratra, B. 2011, Phys. Lett. B, 703, 406
Chen, Y., & Ratra, B. 2012, A[&]{}A, 543, A104
Chevallier, M., & Polarski, D. 2001, Int. J. Mod. Phys. D, 10, 213
Conley, A., et al. 2011, , 192, 1
Dantas, M. A., et al. 2011, Phys Lett. B, 699, 239
De Boni, C., et al. 2011, , 415, 2758
Eisenstein, D. J., & Hu, W. 1998, ApJ, 496, 605
Fischler, W., Ratra, B., & Susskind, L. 1985, Nucl. Phys. B, 259, 730
Frolov, A. V., & Guo, J.-Q. 2011, arXiv:1101.4995 \[astro-ph.CO\]
Geach, J. E., et al. 2010, MNRAS, 402, 1330
Grande, J., et al. 2011, , 1108, 007
Green, J, et al. 2011, arXiv:1108.1374 \[astro-ph.IM\]
Guzzo, L., et al. 2009, Nat., 541, 2008
Jimenez, R. 2011, Fortschr. Phys., 59, 602
Kaiser, N. 1987, MNRAS, 227, 1
Kamenshchik, A. Y., Tronconi, A., & Venturi, G. 2011, Phys. Lett. B, 702, 191
Komatsu, E., et al. 2011, , 192, 18
Laureijs, R., et al. 2011, arXiv:1110.3193 \[astro-ph.CO\])
Lee, S., & Ng, K.-W. 2007, , 76, 043518
Li, X.-D., et al. 2011, , 1107, 011
Linder, E. V. 2003, Phys. Rev. Lett., 90, 091301
Linder, E. V. 2005, Phys. Rev. D, 72, 043529
Maggiore, M., et al. 2011, Phys. Lett. B, 704, 102
Majerotto, E., et al. 2012, MNRAS, in press, arXiv:1205.6215 \[astro-ph.CO\]
Mania, D., & Ratra, B. 2012, Phys. Lett. B, 715, 9
Martinelli, M., et al. 2011, Phys. Rev. D, 83, 023012
Nunes, N. J., Schrempp, L., & Wetterich, C. 2011, , 83, 083523
Orsi, A., et al. 2010, , 402, 1330
Pan, N., et al. 2010, Class. Quantum Grav., 27, 155015
Peebles, P. J. E. 1984, , 284, 439
Peebles, P. J. E., & Ratra, B. 1988, , 325, L17
Peebles, P. J. E., & Ratra, B. 2003, Rev. Mod. Phys., 75, 559
Percival, W. J., et al. 2004, MNRAS, 385, L78
Percival, W. J., et al. 2010, , 401, 2148
Perivolaropoulos, L. 2010, J. Phys. Conf. Ser., 222, 012024
Perotto, L., et al. 2006, J. Cosmology Astropart. Phys., 0610, 013
Podariu, S., Nugent, P., & Ratra, B. 2001, , 553, 39
Podariu, S., & Ratra, B. 2000, , 532, 109
Ratra, B. 1991, Phys. Rev. D, 43, 3802
Ratra, B., & Peebles, P. J. E. 1988, Phys. Rev. D, 37, 3406
Ratra, B., & Vogeley, M. S. 2008, , 120, 235
Reichardt, C. L., et al. 2012, , 749, L9
Reid, B. A., et al. 2012, arXiv:1203.6641 \[astro-ph.CO\]
Saitou, R., & Nojiri, S. 2011, Eur. Phys. J. C, 71, 1712
Samushia, L. 2009, PhD thesis, arXiv:0908.4597 \[astro-ph.CO\]
Samushia, L. et al. 2011, , 410, 1993
Samushia, L., Percival, W. J., & Raccanelli, A. 2012, , 420, 2102
Samushia, L., & Ratra, B. 2006, , 650, L5
Samushia, L., & Ratra, B. 2008, , 680, L1
Sapone, D. 2010, Int. J. Mod. Phys. A, 25, 5253
Sen, A. A., & Scherrer, R. J. 2008, Phys. Lett. B, 659, 457
Silva, R., et al. 2012, A[&]{}A, 537, A11
Sotiriou, T. P., & Faraoni, V. 2010, Rev. Mod. Phys., 82, 451
Suzuki, N., et al. 2012, , 746, 85
Tegmark, M. 1997, Phys. Rev. Lett., 79, 3806
Tong, M., & Noh, H. 2011, Eur. Phys. J. C, 71, 1586
Tsujikawa, S. 2010, Lect. Notes Phys., 800, 99
Wang, Y., et al. 2010, MNRAS, 409, 737
Wilson, K. M., Chen, G., & Ratra, B. 2006, Mod. Phys. Lett. A, 21, 2197
Yang, R.-J., Zhu, Z.-H., & Wu, F. 2011, Int. J. Mod. Phys. A, 26, 317
![Predicted relative error on the measurements of growth rate as a function of redshift $z$ in redshift bins of $\Delta z = 0.1$ for different models of dark energy. The upper solid black line shows predictions for the case when no assumption is made about the nature of dark energy.[]{data-label="fig: 1"}](fig1.EPS){width="100.00000%"}
![Predicted one standard deviation confidence level contour constraints on the current renormalized Hubble constant $h$ and the parameter $\gamma$ that describes deviations from general relativity for different dark energy models.[]{data-label="fig: 2"}](fig2.EPS){width="100.00000%"}
![Upper panel shows one standard deviation confidence level contours constraints on parameters $\omega_{a}$ and $\omega_{0}$ of the $\omega$CDM parametrization, while lower panel shows these for parameters $\Omega_{k}$ and $\Omega_{m}$.[]{data-label="fig: 3"}](fig3.EPS "fig:"){width="75.00000%"} ![Upper panel shows one standard deviation confidence level contours constraints on parameters $\omega_{a}$ and $\omega_{0}$ of the $\omega$CDM parametrization, while lower panel shows these for parameters $\Omega_{k}$ and $\Omega_{m}$.[]{data-label="fig: 3"}](fig4.EPS "fig:"){width="75.00000%"}
![One standard deviation confidence level contour constraints on parameters of the XCDM parametrization.[]{data-label="fig: 4"}](fig5.EPS "fig:"){width="75.00000%"} ![One standard deviation confidence level contour constraints on parameters of the XCDM parametrization.[]{data-label="fig: 4"}](fig6.EPS "fig:"){width="75.00000%"}
![One standard deviation confidence level contour constraints on parameters $\alpha$ and $\Omega_{m}$ of the $\phi$CDM model. Lower panel shows a magnification of the tightest two contours in the lower left corner of the upper panel.[]{data-label="fig: 5"}](fig7.EPS "fig:"){width="75.00000%"} ![One standard deviation confidence level contour constraints on parameters $\alpha$ and $\Omega_{m}$ of the $\phi$CDM model. Lower panel shows a magnification of the tightest two contours in the lower left corner of the upper panel.[]{data-label="fig: 5"}](fig8.EPS "fig:"){width="75.00000%"}
![One standard deviation confidence level contour constraints on parameters $\Omega_{k}$ and $\Omega_{m}$ of the $\phi$CDM model. The lower panel shows a magnification of the two tightest contours in the center of the upper panel.[]{data-label="fig: 6"}](fig9.EPS "fig:"){width="75.00000%"} ![One standard deviation confidence level contour constraints on parameters $\Omega_{k}$ and $\Omega_{m}$ of the $\phi$CDM model. The lower panel shows a magnification of the two tightest contours in the center of the upper panel.[]{data-label="fig: 6"}](fig10.EPS "fig:"){width="75.00000%"}
![One standard deviation confidence level contour constraints on parameters $\Omega_{k}$ and $\Omega_{m}$ of the $\Lambda$CDM model.[]{data-label="fig: 7"}](fig11.EPS){width="80.00000%"}
DE model Fiducial $\gamma$ deviation
-------------- ------------------- -----------
$\omega$CDM 0.55 0.035
$\phi$CDM 0.55 0.023
XCDM 0.55 0.035
$\Lambda$CDM 0.55 0.016
: Predicted deviations of parameter $\gamma$ from its fiducial value, at one standard deviation confidence level, for different assumptions about dark energy.
\[tab: deviations\]
[^1]: Structure formation in the $\Lambda$CDM model is governed by the “standard" CDM structure formation model, which might be in some observational difficulty [see, e.g., @Peebles:2003ab; @Perivolaropoulos:2010ab].
[^2]: There are many other models under current discussion, besides the $\Lambda$CDM and $\phi$CDM models and XCDM and $\omega$CDM parameterizations we consider here for illustrative purposes. For a sample of the available options see, e.g., @Yang:2011ab, @Frolov:2011ab, @Nunes:2011ab, @Grande:2011ab, @Saitou:2011ab, @Silva:2012ab, @Kamenshchik:2011ab, and @Maggiore:2011ab.
[^3]: For constraints on cosmological parameters from data from space missions proposed earlier, see @Podariu:2001ab and references therein.
[^4]: Here we have expanded the energy density $\rho (t,{\vec x})$ in terms of a small spatially inhomogeneous fractional perturbation $\delta (t, {\vec x})$ about a spatially-homogeneous background $\rho_b(t)$: $\rho (t,{\vec x}) =
\rho_b(t) [ 1 + \delta (t, {\vec x})]$.
[^5]: For a review of the Fisher matrix formalism as applied to cosmological forecasting, see @albrecht09.
|
---
abstract: |
We study the large-$N_{c}$ behavior of the critical temperature $T_{c}$ for chiral symmetry restoration in the framework of the Nambu–Jona-Lasinio (NJL) model and the linear $\sigma$-model. While in the NJL case $T_{c}$ scales as $N_{c}^{0}$ and is, as expected, of the same order as $\Lambda_{QCD}$ (just as the deconfinement phase transition), in the $\sigma$-model the scaling behavior reads $T_{c}\propto N_{c}^{1/2}$. We investigate the origin of the different scaling behavior and present two improvements of the $\sigma$-model: (i) a simple, phenomenologically motivated temperature dependence of the parameters and (ii) the coupling to the Polyakov loop. Both approaches lead to the scaling $T_{c}\propto
N_{c}^{0}$.
author:
- 'Achim Heinz$^{\text{(a)}}$, Francesco Giacosa$^{\text{(a)}}$, and Dirk H.Rischke$^{\text{(a,b)}}$'
title: 'Restoration of chiral symmetry in the large-$N_{c}$ limit'
---
Although quantum chromodynamics (QCD) is a theory for quarks and gluons, in the vacuum they are confined inside hadrons. It is, however, expected that at sufficiently high temperature and/or density a phase transition to a deconfined gas of interacting quarks and gluons takes place [@Cabibbo:1975ig; @Rischke:2003mt]. Moreover, it is also expected that this deconfinement phase transition is related to the so-called chiral phase transition: chiral symmetry is broken in the vacuum and restored in a hot and/or dense medium, see Ref. [@Casher:1979vw] and the lattice simulations of Refs. [@Cheng:2006qk; @Aoki:2006br].
The precise connection between the deconfinement and the chiral phase transition is not yet clear. This is also due to the fact that both transitions can only be precisely defined in limiting situations which are not realized in nature. Namely, in the context of pure Yang-Mills theory (QCD with infinitely heavy quarks) the Polyakov loop is the order parameter for the deconfinement phase transition [@Polyakov:1978vu]: the expectation value of the Polyakov loop vanishes for low $T$ and $\mu$ (confined matter) and approaches unity in the deconfined phase. On the other hand, in the limit of zero quark masses the QCD Lagrangian is invariant under chiral transformations. The chiral condensate $\left\langle \bar{q}q\right\rangle
$ is the order parameter for the chiral phase transition: it is nonzero in the vacuum, decreases for increasing $T$ and $\mu$, and vanishes in the chirally restored phase. Nature is somewhere in between: the quark masses are neither infinite nor zero. The chiral condensate and the Polyakov loop are therefore only approximate order parameters.
Since QCD cannot be directly solved, various methods are used to perform explicit calculations. Besides the already mentioned lattice simulations, effective models containing quark degrees of freedom only, such as the NJL model [@Nambu:1961tp; @Klevansky:1992qe; @Vogl:1991qt; @Hatsuda:1994pi], and purely hadronic models, such as the linear $\sigma$-model [@Gell-Mann:1960ls; @Parganlija:2010fz], have been used to study the thermodynamics of QCD. Both approaches cannot describe the deconfinement phase transition: the degrees of freedom of NJL models are deconfined quarks at all temperatures and densities, while linear $\sigma$-models feature hadronic degrees of freedom in which quarks are always confined. However, in both approaches the critical temperature for the chiral phase transition is in agreement with recent lattice studies.
In order to amend these problems, generalizations of the NJL-model have been developed recently, in which the Polyakov loop has been coupled to the quarks [@Ratti:2005jh; @Fukushima:2008wg] (see also Ref. [@Schaefer:2007pw], in which –besides quarks– also mesons are present). Although confinement is not realized in a strict sense because mesons can still decay into quark-antiquark pairs [@Hansen:2006ee], an effective description of confinement is achieved through the behavior of the Polyakov loop at nonzero $T$ and $\mu$. Thus, both deconfinement and chiral phase transition can be studied in a unified framework.
Another widely used approach to study QCD both in the vacuum and in the medium is the so-called large-$N_{c}$ limit [@tHooft:1973jz; @Witten:1979kh] in which the number of colors $N_{c}$ is not fixed to $3$, but is sent to infinity. When enlarging the gauge symmetry of QCD from $SU(3)$ to $SU(N_{c})$ with $N_c\gg3$ one obtains a theory which still contains mesons and baryons, but is –although not solvable– substantially simpler [@Coleman:1980mx]. Relevant quantities can be expressed as a series in $N_{c}^{-n}$, so that it is possible to separate large-$N_{c}$ dominant and large-$N_{c}$ suppressed terms. This is the only known approach to understand some phenomenologically well-established properties of QCD such as the Zweig rule. In order for the large-$N_{c}$ limit to be consistent, the following scaling of the QCD coupling $g_{QCD}$ must be implemented: $$N_{c}\rightarrow\infty~~,~~g_{QCD}^{2}~N_{c}\rightarrow\text{finite .}%$$ In this way quark-antiquark meson masses scale with $N_{c}^{0}$ and the amplitude for a $k$-leg quark-antiquark interaction vertex scales as $N_{c}^{-(k-2)/2}$ and thus goes to zero for $N_{c}\rightarrow\infty.$ In particular, decay amplitudes are suppressed as $1/\sqrt{N_{c}}$ and therefore quark-antiquark mesons are stable and non-interacting in the large-$N_{c}$ limit. Thus, at nonzero $T$ a non-interacting gas of mesons is realized for $N_{c}$ $\gg3$.
It is expected that the deconfinement and the chiral critical temperatures are independent of the number of colors, see Refs.[@Thorn:1980iv; @McLerran:2007qj] and refs. therein. In fact, they should be proportional to the only existing QCD scale, $\Lambda_{QCD}\propto N_{c}^{0}$. In this work we aim to study this issue in detail: namely, we investigate the large-$N_{c}$ behavior of the chiral phase transition in the previously mentioned NJL and linear $\sigma$-model. Quite remarkably, we find that these models behave very differently: while in the NJL model the chiral phase transition $T_{c}$ scales as $N_{c}^{0},$ as expected, in the linear $\sigma
$-model $T_{c}$ scales as $\sqrt{N_{c}}$ and is thus not consistent with the above expectations and with the NJL result. Note that these results, although proven within the simplest possible versions for both the NJL and the linear $\sigma$-model ($N_{f}=2,$ no vector mesons, etc.), are based only on general large-$N_{c}$ properties and therefore hold also when considering more complicated and realistic generalizations.
The reason for the (inconsistent) behavior of the linear $\sigma$-models at large-$N_{c}$ is investigated and two improvements are discussed: (i) a phenomenologically based modification of the model, in which (at least) one coupling constant becomes explicitly temperature-dependent; (ii) the coupling of the $\sigma$-model to the Polyakov-loop degree of freedom. In both ways the correct limit $T_{c}\propto N_{c}^{0}$ is recovered. Thus, it is still possible to use chiral hadronic models for studying the chiral phase transition, although the present study shows that some modifications are needed in order to be consistent with the large-$N_{c}$ limit.
The NJL-model is a quark-based chiral model [@Nambu:1961tp; @Klevansky:1992qe; @Vogl:1991qt; @Hatsuda:1994pi], which has been widely used to study the chiral phase transition in the medium. It is based on a chirally symmetric four-quark point-like interaction. The Lagrangian in the case $N_{f}=2$ reads as function of $N_{c}$: $$\mathcal{L}_{NJL}(N_{c})=\bar{\psi}(\imath\gamma^{\mu}\partial_{\mu}%
-m_{q})\psi+\frac{3G}{N_{c}}\left[ (\bar{\psi}\psi)^{2}+(\bar{\psi}%
\imath\gamma_{5}\psi)\right] ^{2}\text{ ,}%$$ where $\psi^{t}=(u,d)$ is the quark spinor, $m_{q}$ is the bare quark mass and $G$ is the coupling constant with dimension energy$^{-2}$, whose $N_c$-scaling $G\rightarrow3G/N_{c}$ (following from the relation $G\propto g_{QCD}^{2}$) has been made explicit. The quark develops a constituent mass $m^{\ast}$ which is proportional to the chiral condensate $\left\langle \bar
{q}q\right\rangle $: $m^{\ast}=-2G\left\langle \bar{q}q\right\rangle
.$ In mean-field approximation the effective mass $m^{\ast}$ as a function of $T$ and $N_{c}$ reads [@Klevansky:1992qe] $$\label{mast}
1=\frac{m_{q}}{m^{\ast}}+\frac{3G}{N_{c}}\left( 2N_{c}+\frac{1}{2}\right)
\int_{0}^{\Lambda}\frac{dk~k^{2}}{\pi^{2}}\frac{2 \tanh\left( \frac
{\sqrt{k^{2}+m^{\ast2}}}{2T}\right) }{\sqrt{k^{2}+m^{\ast2}}}\text{ ,}%$$ where a cutoff $\Lambda$ has been introduced in order to regularize the loop integral. Note that the number $N_c$ of quarks running in the loop cancels with the factor $1/N_c$ from the coupling constant, such that the dominant term in Eq. (\[mast\]) is independent of $N_c$. In the chiral limit $m_{q}\rightarrow0$ the critical temperature for chiral symmetry restoration $T_{c}$ is obtained as the temperature at which the effective mass $m^{\ast}$, and therefore also the chiral condensate $\left\langle
\bar{q}q\right\rangle $, vanish. To leading order in $N_{c}$ it reads $$T_{c}(N_{c})\simeq\Lambda\sqrt{\frac{3}{\pi^{2}}}\sqrt{1-\frac{ \pi^{2}%
}{6\Lambda^{2}G}}\text{ }\propto N_{c}^{0}\text{ .}\label{tcnjl}%$$ When $m_{q}>0,$ a crossover is realized and the corresponding (pseudo-)critical temperature, defined as the point at which the first derivative $\left\vert dm^{\ast}/dT\right\vert $ is maximal, is also $N_{c}$-independent. This result, based on general scaling arguments, does not change when including the $s$-quark and (axial-)vector degrees of freedom. Moreover, it is also unaffected by the ’t Hooft terms describing the $U_{A}(1)$ anomaly which is suppressed in the large-$N_{c}$ limit. We thus conclude that in all versions of the NJL model the critical temperature for chiral symmetry restoration (second order or crossover) is independent of the number of colors. It has therefore, as expected, the same scaling as the deconfinement phase transition.
The linear $\sigma$-model is a purely hadronic theory constructed from the requirements of chiral symmetry and its spontaneous breaking [@Gell-Mann:1960ls; @Giacosa:2006tf; @Parganlija:2010fz], out of which the pions emerge as Goldstone bosons in the chiral limit.
In order to study the large-$N_{c}$ behavior of the chiral phase transition we consider, as in the NJL model, the case $N_{f}=2$ in the chiral limit, in which the Lagrangian as function of $N_{c}$ reads: $$\mathcal{L}_{\sigma}(N_{c})=\frac{1}{2}(\partial_{\mu}\Phi)^{2}+\frac{1}{2}%
\mu^{2}\Phi^{2}-\frac{\lambda}{4}\frac{3}{N_{c}}\Phi^{4}\text{ ,}\label{ls}%$$ where $\Phi^{t}=(\sigma,\vec{\pi})$ describes the scalar field $\sigma$ and the pseudoscalar pion triplet $\vec{\pi}$ and where the standard scaling behaviors $\lambda\rightarrow\frac{3}{N_{c}}\lambda$ (suppression of the interaction) and $\mu^{2}\rightarrow\mu^{2}$ (constancy of meson masses) have been implemented. For $\mu^{2}>0$ a nonzero chiral condensate $\varphi_{0}%
=\varphi(T=0)=\mu\sqrt{N_{c}/3\lambda}=\sqrt{N_{c}/3}f_{\pi}$ emerges ($f_{\pi}$ is the pion decay constant). The tree-level masses for the sigma and the pions are $m_{\sigma}^{2}=3\lambda f_{\pi}^{2}-\mu^{2}$ , $m_{\pi}^{2}=0$.
Many investigations of the linear $\sigma$-model at nonzero $T$ have been performed in the past, see e.g. Refs.[@Bochkarev:1995gi; @Lenaghan:1999si; @Petropoulos:2004bt] and refs. therein. Using the Cornwall-Jackiw-Tomboulis (CJT) formalism [@Cornwall:1974vz] in double-bubble approximation and excluding the trivial solution, we obtain the gap equation [@Lenaghan:1999si] $$\label{gap}
0= \varphi(T)^{2}-\frac{N_c}{3 \lambda}\, \mu^2 +
3 \int \left( G_{\sigma}+ G_{\pi}\right) \;.$$ The tadpole integrals over the full propagators $G_{\sigma}$ and $G_{\pi}$ of $\sigma$ and $\pi$ meson read (in the so-called trivial renormalization scheme where vacuum fluctuations are neglected) $$\int G_{i}=\int_{0}^{\infty}\frac{dk~k^{2}}{2\pi^{2}\sqrt{k^{2}+M_{i}^{2}}}
\left[ \exp\left( \frac{\sqrt{k^{2}+M_{i}^{2}}}%
{T}\right) -1\right]^{-1}\!\!\!,$$ where $M_i$ is the effective $T$-dependent mass of either $\sigma$ meson or pion. At the critical temperature $T_{c}$ the condensate $\varphi(T)$ and the masses of $\sigma$ and pion vanish. The gap equation (\[gap\]) leads to the expression $$T_{c}(N_{c})=\sqrt{2}f_{\pi}\sqrt{\frac{N_{c}}{3}}\propto N_{c}^{1/2}\text{
.}\label{tcs}%$$ For $N_{c}=3$ the critical temperature is, as known, $T_{c}=\sqrt{2}f_{\pi}$ [@Bochkarev:1995gi], but for an infinite number of colors the phase transition does not take place: for $N_{c}$ $\gg3$ a gas of non-interacting mesons is realized and the meson loops, which are responsible for chiral restoration, are suppressed. Note that the scaling behavior of Eq.(\[tcs\]) is not a prerogative of the simple Lagrangian (\[ls\]), but holds also in more general hadronic models as those of Ref. [@Parganlija:2010fz].
However, Eq. (\[tcs\]) contradicts Eq. (\[tcnjl\]). This mismatch, already noticed in Ref. [@Megias:2004hj], is puzzling because both approaches contain the same symmetries. Moreover, the linear $\sigma$-model can be obtained as the hadronized version of the NJL model. However, the hadronization procedure should be performed for each temperature $T$ and, as a consequence, the coupling constants in the linear $\sigma$-model should be functions of $T.$ In particular, the chiral condensate $\varphi(T)$ of the $\sigma$-model should not be larger than in the corresponding NJL model. In the following, we discuss two improvements of the linear $\sigma$-model which repair the mismatch in the $N_c$-scaling of $T_c$.
We first present a simple and phenomenologically motivated modification of the linear $\sigma$-model which leads to the correct large-$N_{c}$ results. This consists of replacing the parameter $\mu^{2}$ with a $T$-dependent function, $$\mu^{2}\longrightarrow\mu(T)^{2}=\mu^{2}\left( 1-\frac{T^{2}}{T_{0}^{2}}\right)
\text{ ,}\label{musc}%$$ where the parameter $T_{0}\simeq\Lambda_{QCD}\propto N_{c}^{0}$ introduces a new temperature scale. We have implemented the quadratic $T$ dependence suggested in Ref. [@Gasser:1986vb]. Inserting Eq. (\[musc\]) into Eq. (\[gap\]) and following the same steps that led to Eq. (\[tcs\]), the critical temperature now reads $$\label{tcsmuT}
T_{c}(N_{c})=T_{0}\left( 1+\frac{1}{2}\frac{T_{0}^{2}}{f_{\pi}^{2}}\frac
{3}{N_{c}}\right) ^{-1/2}\text{ ,}%$$ and is independent of $N_{c}$ in the limit $N_{c}\rightarrow\infty$: $\lim_{N_{c}\rightarrow\infty}T_{c}(N_{c})=T_{0}$. It is also clear how the meson tadpoles affect the chiral phase transition. Without their contribution, the transition temperature is simply $T_c = T_0$, like in the large-$N_c$ limit, and independent of $N_c$. The meson tadpoles are responsible for the term $\propto 3/(N_c
f_\pi^2)$ in Eq. (\[tcsmuT\]) and thus lead to a reduction of $T_c$, $T_c < T_0$, for any finite value of $N_c$. In the case $N_{c}=3$, using the numerical value $f_{\pi}=92.4~\text{MeV}$ and setting the temperature scale $T_{0}%
=\Lambda_{QCD}\simeq225~\text{MeV}$, the critical temperature $T_{c}$ is lowered to $T_{c}\simeq113~\text{MeV}$. Interestingly, in the framework of sigma models with (axial-)vector mesons, one has to make the replacement $f_{\pi}\rightarrow Zf_{\pi}$ with $Z\simeq 1.67$ [@Parganlija:2010fz]. This leads to a critical temperature $T_{c}\simeq157~$MeV, which is remarkably close to lattice results [@Cheng:2006qk; @Aoki:2006br]. With the help of the described modification the linear $\sigma$-model respects the large-$N_{c}$ limit and is compatible with the NJL model.
Note that we could have also introduced a $T$-dependent coupling constant $\lambda(T)$ instead of a $T$-dependent mass parameter $\mu(T)^2$. As long as $\lambda(T)$ does not depend on $N_c$, our conclusions remain unchanged. Note also that the tadpoles in Eq. (\[gap\]) are natural candidates to induce the quadratic $T$-dependence of $\mu(T)^2$ in Eq. (\[musc\]). However, they are proportional to $1/N_c$, while in our case we have to require that the loop contributions reponsible for the $T$-dependence in Eq. (\[musc\]) are independent of $N_c$. Natural candidates are, for instance, quark loops, like in the NJL model. We therefore expect that the quark-meson loop model of Ref. [@Schaefer:2007pw] shows the correct large-$N_c$ scaling of $T_c$.
Our second suggestion for improving the linear $\sigma$-model is to incorporate the coupling to the Polyakov loop, defined as $$l(x)=N_{c}^{-1}\mathrm{Tr}\left[
\mathcal{P}~\text{exp}\left( \imath g_{QCD}\int_{0}^{1/T}A_{0}(\tau
,x)d\tau\right) \right]\; ,$$ where the trace runs over all color degrees of freedom, $\mathcal{P}$ stands for path ordering, and $A_{0}(\tau,x)$ is the zero component of the gluon field $A_{\mu}$ [@Polyakov:1978vu]. In pure gauge theory the expectation of the Polyakov loop $l(T)=\left\langle
l(x)\right\rangle $ is an order parameter for the deconfinement phase transition: $l=0$ in the confined phase and $l=1$ in the deconfined phase, see the review in Ref. [@Fukushima:2010bq] and refs. therein.
Following Ref. [@Dumitru:2000in] (for a similar approach see also Ref. [@Mocsy:2003qw]) we couple the $\sigma$-model to the Polyakov loop $$\mathcal{L}_{\sigma\text{-Pol}}(N_{c})=\mathcal{L}_{\sigma}(N_{c}%
)+\frac{\alpha N_{c}}{4\pi}|\partial_{\mu}l|^{2}T^{2}-\mathcal{V}%
(l)-\frac{h^{2}}{2}\,\Phi^{2}|l|^{2}T^{2}\text{ .}%$$ where $\mathcal{L}_{\sigma}(N_{c})$ is taken from Eq. (\[ls\]) and the Polyakov loop is coupled to the meson fields. Moreover, a kinetic term and a potential $\mathcal{V}(l)$ for the Polyakov field $l$ have been introduced. (Since we are only interested in the large-$N_{c}$ behavior, the precise form of $\mathcal{V}(l)$ is irrelevant in the following. Terms of the kind $\sim l T \Phi^2$ could also be included [@Mocsy:2003qw] but would not affect the overall $N_c$-scaling, although they might change the order of the phase transition.) Applying the CJT formalism in double-bubble approximation the gap equation for the condensate $\varphi(T)$ now reads $$0= \varphi(T)^{2}-\frac{N_c}{3 \lambda}\, \left( \mu^{2}
- h^2 |l|^{2}T^{2} \right)
+3 \int\left( G_{\sigma}+G_{\pi}\right) \;,$$ from which the following expression for the critical temperature $T_{c}$ is derived: $$T_{c}=\frac{\mu}{\sqrt{h^{2}|l(T_{c})|^{2}+\frac{6\lambda}{N_{c}}}}\text{ .}%$$ Assuming that $l(T_c)$ is a constant independent of $N_c$, we again obtain $T_c \propto N_c^0$ in the limit $N_{c}\rightarrow\infty$. Detailed numerical results represent a task for the future. They depend on the form of the Polyakov-loop potential and on other parameters of the model. However, the important point here is that it is natural to recover the desired large-$N_{c}$ limit when the hadronic model is coupled to the Polyakov loop. The reason for this is that the chiral phase transition is triggered by the Polyakov loop [@Meisinger:1995ih].
In conclusion, we have shown that models of NJL-type and linear $\sigma$-type predict a different behavior of $T_{c}$ as function of $N_{c}$. In the quark-based NJL model $T_{c}$ is independent of $N_{c}$: this result agrees with the expectation that $T_{c}$ scales as $\Lambda_{QCD}\propto N_{c}^{0},$ just as the deconfinement phase transition. On the contrary, in linear $\sigma$-type models a scaling $T_{c}\propto\sqrt{N_{c}}$ is obtained. The different scaling originates from the particular mechanism which restores chiral symmetry in the two models. In the NJL model, quark loops are responsible for chiral symmetry restoration, which survive in the large-$N_{c}$ limit, while in the linear $\sigma$-model, meson loops induce the chiral phase transition, which disappear in the large-$N_{c}$ limit.
This mismatch is, at first sight, even more striking because the linear $\sigma$-model can be derived from the NJL model through a hadronization procedure. However, since one should perform this hadronization at each $T,$ the parameters of the $\sigma$-model are actually functions of $T.$ Indeed, when making (at least) one parameter of the $\sigma$-model $T$-dependent, the expected large-$N_{c}$ limit can be easily recovered. This result, although interesting, is based on an [*ad hoc*]{} modification of the $\sigma$-model: for this reason we have also studied a different approach, in which –inspired by Ref. [@Dumitru:2000in]– we have coupled the linear $\sigma$-model to the Polyakov loop. In this way the chiral phase transition is induced by the deconfinement phase transition and, as a consequence, $T_{c}$ scales as $N_{c}^{0}.$
We therefore conclude that the study of the chiral phase transition within purely hadronic models is possible, although care is needed in order to be in agreement with the large-$N_{c}$ limit. Detailed numerical studies with more realistic models including (pseudo)scalar and (axial-)vector mesons will be presented in the future.
**Acknowledgment:** The authors thank E. Seel, M. Grahl and R. Pisarski for useful discussions. A.H. thanks H-QM and HGS-HIRe for financial support.
[99]{}
N. Cabibbo and G. Parisi, Phys. Lett. **B59**, 67-69 (1975).
D. H. Rischke, Prog. Part. Nucl. Phys. **52** (2004) 197-296. \[nucl-th/0305030\].
A. Casher, Phys. Lett. B **83**, 395 (1979).
M. Cheng, N. H. Christ, S. Datta, J. van der Heide, C. Jung, F. Karsch, O. Kaczmarek, E. Laermann *et al.*, Phys. Rev. **D74** (2006) 054507. \[hep-lat/0608013\].
Y. Aoki, Z. Fodor, S. D. Katz, K. K. Szabo, Phys. Lett. **B643** (2006) 46-54. \[hep-lat/0609068\].
A. M. Polyakov, Phys. Lett. B **72** (1978) 477.
Y. Nambu and G. Jona-Lasinio, Phys. Rev. **122**, 345-358 (1961); Y. Nambu and G. Jona-Lasinio, Phys. Rev. **124**, 246-254 (1961).
U. Vogl and W. Weise, Prog. Part. Nucl. Phys. **27**, 195-272 (1991).
T. Hatsuda and T. Kunihiro, Phys. Rept. **247**, 221-367 (1994). \[hep-ph/9401310\].
S. P. Klevansky, Rev. Mod. Phys. **64**, 649-708 (1992).
M. Gell-Mann and M. Levy, Nuovo Cimento, **16**, 705 (1960). P. Ko and S. Rudaz, Phys. Rev. D **50** (1994) 6877. S. Gasiorowicz and D. A. Geffen, Rev. Mod. Phys. **41**, 531-573 (1969). M. Urban, M. Buballa, J. Wambach, Nucl. Phys. **A697**, 338-371 (2002). \[hep-ph/0102260\].
D. Parganlija, F. Giacosa, D. H. Rischke, Phys. Rev. **D82**, 054024 (2010). \[arXiv:1003.4934 \[hep-ph\]\]. S. Janowski, D. Parganlija, F. Giacosa, D. H. Rischke, \[arXiv:1103.3238 \[hep-ph\]\]. S. Gallas, F. Giacosa, D. H. Rischke, Phys. Rev. **D82** (2010) 014004. \[arXiv:0907.5084 \[hep-ph\]\].
C. Ratti, M. A. Thaler, W. Weise, Phys. Rev. **D73**, 014019 (2006). \[hep-ph/0506234\].
K. Fukushima, Phys. Rev. **D77** (2008) 114028. \[arXiv:0803.3318 \[hep-ph\]\].
B. -J. Schaefer, J. M. Pawlowski, J. Wambach, Phys. Rev. **D76**, 074023 (2007). \[arXiv:0704.3234 \[hep-ph\]\].
H. Hansen, W. M. Alberico, A. Beraudo, A. Molinari, M. Nardi, C. Ratti, Phys. Rev. **D75**, 065004 (2007). \[hep-ph/0609116\].
G. ’t Hooft, Nucl. Phys. **B72**, 461 (1974).
E. Witten, Nucl. Phys. **B160**, 57 (1979).
Spontaneous symmetry breaking occurs also in the large-$N_{c}$ limit, see S. R. Coleman and E. Witten, Phys. Rev. Lett. **45** (1980) 100. However, nuclear matter seems to be a property of our $N_{c}=3$ world only: L. Bonanno and F. Giacosa, Nucl. Phys. [**A859**]{}, 49-62 (2011). \[arXiv:1102.3367 \[hep-ph\]\].
C. B. Thorn, Phys. Lett. **B99**, 458 (1981).
L. McLerran and R. D. Pisarski, Nucl. Phys. **A796** (2007) 83-100. \[arXiv:0706.2191 \[hep-ph\]\].
F. Giacosa, Phys. Rev. **D75** (2007) 054007. \[hep-ph/0611388\]. A. Heinz, S. Struber, F. Giacosa and D. H. Rischke, Phys. Rev. D **79** (2009) 037502 \[arXiv:0805.1134 \[hep-ph\]\]. A. H. Fariborz, R. Jora, J. Schechter, Phys. Rev. D **77** (2008) 034006 \[arXiv:0707.0843 \[hep-ph\]\]; Phys. Rev. D **72** (2005) 034001 \[arXiv:hep-ph/0506170\].
A. Bochkarev and J. I. Kapusta, Phys. Rev. **D54** (1996) 4066-4079. \[hep-ph/9602405\].
J. T. Lenaghan and D. H. Rischke, J. Phys. G **G26**, 431-450 (2000). \[nucl-th/9901049\]
N. Petropoulos, arXiv:hep-ph/0402136.
J. M. Cornwall, R. Jackiw, E. Tomboulis, Phys. Rev. **D10**, 2428-2445 (1974).
E. Megias, E. Ruiz Arriola, L. L. Salcedo, Phys. Rev. **D74**, 065005 (2006). \[hep-ph/0412308\].
J. Gasser and H. Leutwyler, Phys. Lett. **B184**, 83 (1987). S. Leupold and U. Mosel, AIP Conf. Proc. **1322**, 64-72 (2010).
K. Fukushima and T. Hatsuda, Rept. Prog. Phys. **74** (2011) 014001 \[arXiv:1005.4814 \[hep-ph\]\].
A. Dumitru and R. D. Pisarski, Phys. Lett. **B504** (2001) 282-290. \[hep-ph/0010083\]. O. Scavenius, A. Dumitru, J. T. Lenaghan, Phys. Rev. [**C66**]{}, 034903 (2002). \[hep-ph/0201079\].
A. Mocsy, F. Sannino, K. Tuominen, Phys. Rev. Lett. [**92**]{} (2004) 182302 \[arXiv:hep-ph/0308135\].
P. N. Meisinger and M. C. Ogilvie, Phys. Lett. **B379** (1996) 163-168. \[hep-lat/9512011\].
|
---
abstract: 'In this paper we investigate the power suppressed contributions from two-particle and three-particle twist-4 light-cone distribution amplitudes (LCDAs) of photon within the framework of light-cone sum rules. Compared with leading twist LCDA result, the contribution from three-particle twist-4 LCDAs is not suppressed in the expansion by $1/Q^2$, so that the power corrections considered in this work can give rise to a sizable contribution, especially at low $Q^2$ region. According to our result, the power suppressed contributions should be included in the determination of the Gegenbauer moments of pion LCDAs with the pion transition form factor.'
author:
- 'Yue-Long Shen[^1]'
- 'Jing Gao[^2]'
- 'Cai-Dian Lü[^3]'
- 'Yan Miao[^4]'
title: 'Power corrections to the pion transition form factor from higher-twist distribution amplitudes of photon '
---
Introduction {#sect:Intro}
============
As one of the simplest hard exclusive processes, the pion transition form factor $F_{\gamma^{\ast} \gamma \to
\pi^{0}}(Q^2)$ at large momentum transfer is of great importance in exploring the strong interaction dynamics of hadronic reactions in the framework of QCD, and to determine the parameters in the LCDAs of pion. It is defined via the matrix element $$\begin{aligned}
\langle \pi(p) | j_{\mu}^{\rm em} | \gamma (p^{\prime}) \rangle =
g_{\rm em}^2 \, \epsilon_{\mu \nu \alpha \beta} \, q^{\alpha} \,
p^{\beta} \, \epsilon^{\nu}(p^{\prime}) F_{\gamma^{\ast} \gamma
\to \pi^0} (Q^2) \,,\, \qquad \epsilon_{0123}= -1\end{aligned}$$ where $q=p-p^{\prime}$, $p$ and $p'$ refer to the four-momentum of the pion and the on-shell photon respectively, the electro-magnetic current $$\begin{aligned}
j_{\mu}^{\rm em} = \sum_q \, g_{\rm em} \, Q_q \, \bar q \,
\gamma_{\mu} \, q \,. \label{em current definition}\end{aligned}$$In collinear factorization theorem, pion transition form factor can be factorized into the convolution of the hard kernel and the leading twist pion LCDA at leading power of $1/Q^2$ [@Lepage:1980fj; @Efremov:1979qk; @Duncan:1979ny; @Rothstein:2003wh], and the hard kernel has been calculated up to two-loop level [@delAguila:1981nk; @Braaten:1982yp; @Kadantseva:1985kb; @Melic:2002ij]. At one-loop level, the factorization formula is written by $$\begin{aligned}
F_{\gamma^{\ast} \gamma \to \pi^0}^{\rm LP} (Q^2)= \frac{\sqrt{2}
\, (Q_u^2-Q_d^2) \, f_{\pi}}{Q^2} \, \int_0^1 \, d x \, \left[
T^{(0)}_{2}(x) + T^{(1), \, \Delta}_{2}(x, \mu) \right ] \,
\phi_{\pi}^{\Delta} (x, \mu) + {\cal O}(\alpha_s^2) \,,
\label{one-loop factorization formula at LP}\end{aligned}$$ where the leading twist pion LCDA is defined as $$\begin{aligned}
\langle \pi(p) |\bar \xi (y) \, [y, 0] \, \gamma_{\mu} \,
\gamma_5 \, \xi(0) | 0 \rangle = -i \, f_{\pi} \, p_{\mu} \,
\int_0^1 \, d u \, e^{i\, u \, p \cdot y} \, \phi_{\pi} (u, \mu)
+ {\cal O}(y^2) \,,\end{aligned}$$ and the superscript “$\Delta$" indicates the scheme to deal with $\gamma_5$ in dimensional regularization which is a subtle problem in QCD loop diagrams [@Bonneau:1990xu; @Collins:1984xc; @Larin:1993tq; @Martin:1999cc; @Jegerlehner:2000dz; @Moch:2015usa; @Gutierrez-Reyes:2017glx]. Employing the trace technique, the $\gamma_5$ ambiguity of dimensional regularization was resolved by adjusting the way of manipulating $\gamma_5$ in each diagram to preserve the axial-vector Ward identity [@Braaten:1982yp]. In a recent paper [@Wang:2017ijn], the one loop calculation is revisited by applying the standard OPE technique [@Beneke:2004rc; @Beneke:2005gs; @Beneke:2005vv] with the evanescent operator(s) [@Dugan:1990df; @Herrlich:1994kh], in both the NDR and HV schemes for $\gamma_5$ in the $D$-dimensional space. At one-loop level it has been shown explicitly that the scheme dependence of the hard kernel and the twist-two pion LCDA is cancelled out precisely, which guarantees the form factor $F_{\gamma^{\ast} \gamma \to \pi^0} (Q^2)$ to be free from $\gamma_5$ ambiguity.
At leading power the pion transition form factor has also been studied with transverse momentum dependent (TMD) factorization approach at one-loop level [@Nandi:2007qx; @Musatov:1997pu; @Wu:2010zc], where the joint resummation of the large logarithms $\ln^2{{
k_{\perp}^2} /Q^2}$ and $\ln^2 x$ was performed in moment and impact-parameter space [@Li:2013xna]. The prediction of joint resummation improved TMD factorization approach can accommodate the anomalous BaBar measurements [@Aubert:2009mc] of $F_{\gamma^{\ast} \gamma \to \pi^0} (Q^2)$, which have stimulated intensive theoretical investigations with various phenomenological approaches as well as lattice QCD simulations (see for instance [@Masjuan:2012wy; @Hoferichter:2014vra; @Gerardin:2016cqj]). In Ref. [@Radyushkin:2009zg; @Polyakov:2009je], a leading twist pion LCDA with the non-vanishing end-point behavior was proposed to explain the anomalous BaBar data at high $Q^2$. Later it was found that this method is able to be achieved by introducing a sizable nonperturbative soft correction from the TMD pion wave function[@Agaev:2010aq].
To achieve more precise theoretical predictions, power corrections need to be taken into account especially at low $Q^2$. In [@Agaev:2010aq; @Agaev:2012tm], the soft correction to the leading twist contribution is evaluated with the dispersion approach and found to be crucial to suppress the contributions from higher Gegenbauer moments of the twist-2 pion LCDAs [@Kroll:2010bf; @Li:2013xna]. Furthermore, the subleading power “hadronic" photon correction can also be taken into account effectively with dispersion approach. Within this method the theoretical accuracy for predicting the pion-photon form factor is improved by including the next-to-next-to-leading order (NNLO) QCD correction to the twist-2 contribution and the finite-width effect of the unstable vector mesons in the hadronic dispersion relation [@Bakulev:2002uc; @Stefanis:2012yw; @Bakulev:2011rp; @Bakulev:2012nh; @Mikhailov:2016klg]. Another approach to accommodate the contribution from the “hadronic photon" is to introduce the LCDAs of photon. In [@Wang:2017ijn], the QCD factorization of the correlation function for the construction of the LCSRs for the hadronic photon contribution to the pion-photon form factor is established. Both the hard matching coefficient and the leading twist photon LCDAs are independent of the $\gamma_5$ prescription in dimensional regularization, and the next-to-leading logarithmic(NLL) resummation of the large logarithms was also perform by solving the renormalization group equations(RGE) in momentum space. The contribution from the twist-4 pion LCDA is also calculated at tree level in [@Khodjamirian:1997tk; @Wang:2017ijn]. There is strong cancellation between this contribution and the contribution from hadronic structure of photon, which makes the overall power correction not significant. The LCDAs of photon, including both two-particle and three-particle Fock state, have been studied up the twist-4 level [@Ball:2002ps]. The higher-twist LCDAs are not suppressed in many processes such as radiative leptonic $B$ meson decay $B \to \gamma\ell\nu $ [@Wang:2018wfj; @Shen:2018abs]. In this paper we will investigate the contribution from the full set the LCDAs of photon up to twist-4 to the pion transition form factor using LCSRs approach.
The outline of this paper is as follow: in Section \[sect:higher-twist-photon\] we present the analytic calculation of the pion transition form factor from the higher twist photon LCDAs within LCSRs framework. The numerical results and discussions are given in section \[numerical\]. The last section is closing remark.
Power corrections from the hadronic structure of photon {#sect:higher-twist-photon}
=======================================================
All two-particle and three-particle LCDAs of photon have been defined and classified up to twist-4, and the expressions of the LCDAs have also been obtained through the conformal expansion in the presence of the background field [@Ball:2002ps]. To evaluate the power suppressed contribution to the pion-photon form factor due to the hadronic photon effect, the following correlation function is employed $$\begin{aligned}
G_{\mu}(p^{\prime}, q) &=& \int d^4 z \, e^{-i \, q \cdot z} \,
\langle 0 | {\rm T} \left \{ j_{\mu, \perp}^{\rm em}(z),
j_{\pi}(0) \right \} | \gamma(p^{\prime}) \rangle \,
\nonumber \\
&=& - g_{\rm em}^2 \, \epsilon_{\mu \nu \alpha \beta}^{\perp} \,
q^{\alpha} \, p^{\prime \beta} \, \epsilon^{\nu}(p^{\prime}) \,
G(p^2, Q^2) \,, \label{vacuum to photon correlator}\end{aligned}$$ where pion interpolating current $j_{\pi}$ is defined by $$\begin{aligned}
j_{\pi} = {1 \over \sqrt{2}} \, \left ( \bar u \, \gamma_5 \, u -
\bar d \, \gamma_5 \, d \right ) \,.\end{aligned}$$ The power counting rule for the external momenta $$\begin{aligned}
| n \cdot p | \sim \bar n \cdot p \sim n \cdot p^{\prime} \sim
{\cal O}(\sqrt{Q^2}) \,,\end{aligned}$$ will be adopted to determine the perturbative matching coefficient entering the factorization formula of $G_{\mu}(p^{\prime}, q) $. Applying the standard definition for the pion decay constant $$\begin{aligned}
\langle 0 | j_{\pi}| \pi(p) \rangle = -i \, f_{\pi} \,
\mu_{\pi}(\mu) \,, \qquad \mu_{\pi}(\mu) \equiv {m_{\pi}^2 \over
m_u(\mu) + m_d(\mu) } \,,\end{aligned}$$ we can write down the hadronic dispersion relation of $G(p^2,
Q^2)$ $$\begin{aligned}
G(p^2, Q^2) = {f_{\pi} \, \mu_{\pi}(\mu) \over m_{\pi}^2 - p^2
- i 0} \, F^{\rm NLP}_{\gamma^{\ast} \gamma \to \pi^0} (Q^2) +
\int_{s_0}^{\infty} \,ds \, {\rho^{h}(s, Q^2) \over s-p^2-i 0} \,.\end{aligned}$$ The form factor $F^{\rm NLP}_{\gamma^{\ast} \gamma \to \pi^0}
(Q^2)$ will be extracted after the correlation function being calculated by OPE in deep Euclidean region. Employing dispersion relation, subtracting the continuum state contribution with the help of quark hadron duality assumption, and performing Borel transformation, the LCSRs for the subleading power contribution to the $\pi^0 \gamma^{\ast} \gamma$ form factor are derived as $$\begin{aligned}
F^{\rm 2PLT}_{\gamma^{\ast} \gamma \to \pi^0} (Q^2) &=& - {
\sqrt{2} \, \left ( Q_u^2 - Q_d^2 \right ) \over f_{\pi} \,\,
\mu_{\pi}(\mu) \,\, Q^2} \, \chi(\mu) \, \langle \bar q q \rangle
(\mu) \,
\int_0^{s_0} \, ds \, {\rm exp} \left[ - {s - m_{\pi}^2 \over M^2} \right ] \nonumber \\
&& \times \left [ \rho^{(0)}(s, Q^2) + {\alpha_s \, C_F \over 4\,
\pi} \, \rho^{(1)}(s, Q^2) \right ] + {\cal O}(\alpha_s^2) \,.
\label{NLL resummation for the NLP effect}\end{aligned}$$ where the magnetic susceptibility of the quark condensate $\chi(\mu)$ contains the dynamical information of the QCD vacuum, and the spectral functions $\rho^{(0,1)}(s, Q^2)$ can be found in [@Wang:2017ijn].
Now we will proceed to investigate the contribution from higher twist LCDAs of photon. Up to twist-4, the two-particle LCDAs of photon are defined as $$\begin{aligned}
&& \langle 0 |\bar q(x) [x, 0] \sigma_{\alpha \beta} \,\, q(0)|
\gamma(p) \rangle = i \, g_{\rm em} \, Q_q \, \langle \bar q q
\rangle(\mu) \, (p_{\beta} \, \epsilon_{\alpha} - p_{\alpha} \,
\epsilon_{\beta}) \, \int_0^1 \, d z \, e^{i \, z \, p \cdot x} \,
\bigg [ \chi(\mu) \, \phi_{\gamma}(z, \mu)
\nonumber \\
&& \,\,\,\,\,\,\,\,\,\,+ {x^2 \over 16} \, \mathbb{A}(z, \mu)
\bigg ] +{i \over 2} \, g_{\rm em} \, Q_q \, {\langle \bar q q
\rangle(\mu) \over p \cdot x} \, (x_{\beta} \, \epsilon_{\alpha} -
x_{\alpha} \, \epsilon_{\beta}) \,\nonumber
\int_0^1 \, d z \, e^{i \, z \, p \cdot x} \, h_{\gamma}(z, \mu) \,. \\
\nonumber && \langle 0 |\bar q(x) [x, 0] \gamma_{\alpha} \,\,
q(0)| \gamma(p) \rangle = g_{\rm em} \, Q_q \, f_{3 \gamma}(\mu)
\, \epsilon_{\alpha} \, \int_0^1 \, d z \, e^{i \, z \, p \cdot x}
\, \psi_\gamma^{(v)}(z, \mu) \,\\ && \langle 0 |\bar q(x) [x,
0]\gamma_{\alpha} \, \gamma_5 \,\, q(0)| \gamma(p) \rangle
={g_{\rm em}Q_q \, f_{3 \gamma}(\mu) \, \over 4} \,
\varepsilon_{\alpha \beta \rho \tau} \, p^{\rho} \, x^{\tau}
\,\epsilon^{\, \beta} \, \int_0^1 \, d z \, e^{i \, z \, p \cdot
x} \, \, \psi_\gamma^{(a)}(z, \mu)\,,\label{two
particle}\end{aligned}$$ where $\psi_\gamma^{(v)}(z,
\mu),\psi_\gamma^{(a)}(z, \mu)$ are twist-3 and $\mathbb{A}(z,
\mu), h_{\gamma}(z, \mu)$ are twist-4. Employing the light-cone expansion of the $u,d$-quark propagator and keeping the subleading-power contributions to the correlation function (\[vacuum to photon correlator\]) leads to $$\begin{aligned}
G_{\mu}(p', q) & \supset & {1\over \sqrt{2}}\int {d^4 k \over (2
\pi)^4} \, \int d^4 x \, e^{i \, (k-q) \cdot x} \, {k^{\nu} \over
k^2 } \, \sum_{q=u,d}{\delta_q}Q_qg_{em}\langle0| \bar q(x) \,
\sigma_{\mu \nu} \, \gamma_5 \, q(0)| \gamma(p') \rangle
\,-\left(q\leftrightarrow -p\right)
\nonumber \\
&=&{i\over 2\sqrt 2}\epsilon_{\mu\nu\rho\sigma}\int {d^4 k \over
(2 \pi)^4} {k^{\nu} \over k^2 }\, \int d^4 x \, e^{i \, (k-q)
\cdot x} \, \,\sum_{q=u,d}{\delta_q}Q_qg_{em} \langle 0| \bar q(x)
\, \sigma^{\rho\sigma} \, q(0)| \gamma(p') \rangle \,\nonumber
\\&-&\left(q\leftrightarrow -p\right),\end{aligned}$$ where $\delta_u=1,\delta_d=-1$. The above equation indicates that only twist-2 and twist-4 two-particle LCDAs can contribute to pion transition form factor in the LCSRs approach, which is different from the method based on TMD factorization[@Shen2019]. Making use of the definitions in Eq.(\[two particle\]), it is straightforward to write down $$\begin{aligned}
G_{\mu}^{\rm 2PHT}(p, q) &=& -{g_{\rm em}^2 \over 4 }\,
\epsilon_{\mu \nu \alpha \beta}^{\perp} \varepsilon^{\nu}\,
q^{\alpha} \, p^{\prime \beta} \, {Q_u^2-Q_d^2\over \sqrt 2
Q^4}\langle\bar qq\rangle(\mu)\int_0^1du\left[{\mathbb{A}(u,
\mu)\over (\bar u+ur)^2}+{\mathbb{A}(u, \mu)\over ( u+r\bar
u)^2}\right],\end{aligned}$$ where the contribution from $h_\gamma(z,\mu)$ vanishes due to the anti-symmetric structure. The resulting LCSRs for the two-particle higher-twist hadronic photon corrections to the pion transition form factors can be further derived as follows $$\begin{aligned}
F^{\rm 2PHT}_{\gamma^{\ast} \gamma \to \pi^0} (Q^2) &=& - {
\sqrt{2} \, \left ( Q_u^2 - Q_d^2 \right ) \over 4f_{\pi} \,\,
\mu_{\pi}(\mu) \,\,} \, \langle \bar q q \rangle (\mu) \,\bigg\{
{1\over Q^2} \mathbb{A}(u_0)e^{-{s_0-m_\pi^2\over M^2}}\nonumber
\\ &+&\int_{u_0}^{1} \, {du\over u^2 }\, {1\over M^2}{\rm exp}
\left[ - {\bar uQ^2 - um_{\pi}^2 \over uM^2} \right ] \mathbb{A}(u,
\mu),\bigg\}
\label{2pht}\end{aligned}$$ where $u_0=Q^2/(s_0+Q^2)$.
![Diagrammatical representation of the tree-level contribution to the QCD amplitude $\widetilde{G}_{\mu}$ with the contribution from two-particle photon LCDAs. []{data-label="fig:tree-level-pion-FF-LP"}](Fig1.eps "fig:"){width="0.45\columnwidth"}\
(a) (b)
\
(a) (b)
To compute higher-twist three-particle hadronic photon corrections to the pion transition form factors, the definition of three-particle photon LCDA is required. In the appendix we collect the definition of three-particle twist-4 photon LCDAs for an incoming photon state. Keeping the one-gluon/photon part for the light-cone expansion of the quark propagator in the background gluon/photon field $$\begin{aligned}
\langle 0|T\{{q}(x),\bar{q}(0)\}|0\rangle_{G}&\supset& i\int\frac{d^{4}k}{(2\pi)^{4}}
e^{-ik\cdot x}\int_{0}^{1}du[\frac{ux_{\mu}\gamma_{\nu}}{k^{2}}
-\frac{\not \! k\sigma_{\mu\nu}}{2k^4}]G^{\mu\nu}(ux)\nonumber
\\ &+&
ig_{em}Q_q\int^{\infty}_{0}\frac{d^{4}k}{(2\pi)^{4}}
e^{-ik\cdot x}\int_{0}^{1}du[\frac{ux_{\mu}\gamma_{\nu}}{k^{2}}
-\frac{\not \! k\sigma_{\mu\nu}}{2k^4}]F^{\mu\nu}(ux)\end{aligned}$$where $ G^{\mu\nu}=i[D_{\mu},D_{\nu}]$. By evaluating Fig. \[fig:tree-level-pion-FF-NLP\], we obtain $$\begin{aligned}
\Pi_{\mu}(p, q) & \supset &{1\over
2\sqrt{2}}g_{em}^2\sum_q\delta_qQ_q^2\epsilon_{\mu\alpha\rho\lambda}q^\alpha
\varepsilon^\rho p'^\lambda\langle \bar q q \rangle(\mu)\int_0^1 du\int
[{\cal D} \alpha_i]{1\over[ q-(\alpha_q + \, \bar u \,
\alpha_g-1) \, p']^4}\nonumber \\
&\times& \rho^{\rm 3PHT}(\alpha_i, u, \mu)-\left(q\leftrightarrow
-p\right)\end{aligned}$$ where $$\begin{aligned}
\rho^{\rm 3PHT}(\alpha_i, u, \mu)&=&
2\{(2u-1)[T_1(\alpha_i)-T_2(\alpha_i)+T_3(\alpha_i)+T_4(\alpha_i)-\widetilde{S}(\alpha_i)
+T_{4\gamma}(\alpha_i)]\nonumber \\
&+&S(\alpha_i, \mu)+S_\gamma(\alpha_i, \mu)+T_2(\alpha_i,
\mu)-T_1(\alpha_i, \mu)\}\end{aligned}$$ and the integration measure is defined as $$\begin{aligned}
\int [{\cal D} \alpha_i] \equiv \int_0^1 d \alpha_q \, \int_0^1 d
\alpha_{\bar q} \, \int_0^1 d \alpha_g \, \delta \left (1-\alpha_q
- \alpha_{\bar q} -\alpha_g \right )\,.\end{aligned}$$ Taking advantage of quark-hadron duality, we arrive at the LCSRs of the contribution from three-particle photon LCDAs $$\begin{aligned}
F^{\rm 3PHT}_{\gamma^{\ast} \gamma \to \pi^0} (Q^2) &=& - {
\sqrt{2} \, \left ( Q_u^2 - Q_d^2 \right ) \over 2f_{\pi} \,\,
\mu_{\pi}(\mu) \,\,} \, \langle \bar q q \rangle (\mu){1\over
Q^2} \,
\bigg\{\int_0^{s_0/(s_0+Q^2)}d\alpha_q\int_{s_0/(s_0+Q^2)-\alpha_q}^{1-\alpha_q}{d\alpha_g\over
\alpha_g}
\nonumber \\
&\times& \rho^{\rm 3PHT}(\alpha_q,\alpha_g,\alpha_{\bar
q}=1-\alpha_q-\alpha_g,u_{s_0},\mu)e^{-{s_0-m_\pi^2\over M^2}}\nonumber \\
&+&{1\over M^2}\int_0^{s_0}dse^{-{s-m_\pi^2\over
M^2}}\int_0^{s/(s+Q^2)}d\alpha_q\int_{s/(s+Q^2)-\alpha_q}^{1-\alpha_q}{d\alpha_g\over
\alpha_g}
\nonumber \\
&\times& \rho^{3PHT}(\alpha_q,\alpha_g,\alpha_{\bar
q}=1-\alpha_q-\alpha_g,u_s,\mu)\bigg\} \label{3pht}\end{aligned}$$ where $u_s=[s/(s+Q^2)-\alpha_q]/\alpha_g$. The overall higher-twist photon LCDAs contribution is written by $$\begin{aligned}
F^{\rm HT}_{\gamma^{\ast} \gamma \to \pi^0} (Q^2)=F^{\rm
2PHT}_{\gamma^{\ast} \gamma \to \pi^0} (Q^2)+F^{\rm
3PHT}_{\gamma^{\ast} \gamma \to \pi^0} (Q^2).\end{aligned}$$ Now we discuss the power behavior of our results. The power counting scheme for the sum rule parameters are given below: $$\begin{aligned}
s_0 \sim M^2 \sim {\cal O} (\Lambda^2) \,, \qquad \bar u_0 \sim
{\cal O} (\Lambda^2/Q^2). \label{scaling}\end{aligned}$$ Employing Eq.(\[scaling\]), one can obtain that the contribution from leading twist LCDA of photon is suppressed by a factor $\Lambda^2/Q^2$ [@Wang:2017ijn] compared with LP contribution. The higher twist contributions are conjectured to be also suppressed by [only]{} one power of $\Lambda^2/Q^2$ due to the absent correspondence between the twist counting and the large-momentum expansion [@Agaev:2010aq]. For the contribution from two-particle twist-4 LCDAs of photon, the result in Eq.(\[2pht\]) is suppressed by $\Lambda^4/Q^4$ compared with LP contribution as the power of twist-4 photon LCDAs is suppressed with respect to leading twist one. While for the contribution from three-particle twist-4 LCDAs in Eq.(\[3pht\]), the scaling of $\alpha_q$ is ${\cal O}(\Lambda^2/Q^2)$, and $\alpha_g$ is ${\cal
O}(1)$. Although there is an overall factor $1/Q^2$, the result is only suppressed by $\Lambda^2/Q^2$ for the spectral function $\rho^{\rm 3PHT}$ is not suppressed at endpoint region. This result confirms the conjecture in [@Agaev:2010aq].
Numerical analysis {#numerical}
==================
In the following we explore the phenomenological consequences of the hadronic photon correction to the pion-photon form factor, and the most important input is the LCDAs of photon. The models of twist-4 LCDAs of photon used in this paper are written by $$\begin{aligned}
\mathbb{A}(z, \mu) &=& 40 \, z^2 \, \bar z^2 \left [3 \,
\kappa(\mu) - \kappa^{+}(\mu) + 1 \right ] + 8 \, \left [
\zeta_2^{+} (\mu) - 3 \, \zeta_2(\mu) \right ] \,
\big [ z \, \bar z \, (2 + 13 \,z \, \bar z ) \nonumber \\
&& + \, 2\, z^3 \, (10 - 15\, z + 6 \, z^2) \, \ln z
+ 2 \, \bar z^3 \, (10 - 15 \,\bar z + 6 \, \bar z^2) \, \ln \bar z \big ] \,, \nonumber \\
h_{\gamma}(z, \mu) &=& -10 \, \left (1 + 2 \, \kappa^{+}(\mu)
\right) \,
C_{2}^{1/2}(2 \, z -1) \,, \nonumber \\
S(\alpha_i, \mu) &=& 30 \, \alpha_g^2 \, \bigg \{ \left
(\kappa(\mu) + \kappa^{+}(\mu) \right ) \, (1-\alpha_g)
+ (\zeta_1 + \zeta_1^{+}) (1 -\alpha_g) (1 -2 \, \alpha_g) \nonumber \\
&& + \, \zeta_2(\mu) \,\left [ 3 \, (\alpha_{\bar q} - \alpha_q)^2
- \alpha_g \, (1- \alpha_g) \right ] \bigg \} \,, \nonumber \\
\widetilde{S}(\alpha_i, \mu) &=& - 30 \, \alpha_g^2 \, \bigg \{
\left (\kappa(\mu) - \kappa^{+}(\mu) \right ) \, (1-\alpha_g)
+ (\zeta_1 - \zeta_1^{+}) (1 -\alpha_g) (1 -2 \, \alpha_g) \nonumber \\
&& + \, \zeta_2(\mu) \,\left [ 3 \, (\alpha_{\bar q} - \alpha_q)^2
- \alpha_g \, (1- \alpha_g) \right ] \bigg \} \,, \nonumber \\
S_{\gamma}(\alpha_i, \mu) &=& 60 \, \alpha_g^2 \, (\alpha_q +
\alpha_{\bar q} ) \,
\left [ 4 - 7 \, (\alpha_{\bar q} + \alpha_q ) \right ] \,, \nonumber \\
T_{1}(\alpha_i, \mu) &=& - 120 \, \left (3 \, \zeta_2(\mu) +
\zeta_2^{+}(\mu) \right ) \,
\left ( \alpha_{\bar q} - \alpha_q \right ) \, \alpha_{\bar q} \, \alpha_q \, \alpha_g \,, \nonumber \\
T_{2}(\alpha_i, \mu) &=& 30 \, \alpha_g^2 \, (\alpha_{\bar q} -
\alpha_q )
\left [ \left (\kappa(\mu) - \kappa^{+}(\mu) \right )
+ \left (\zeta_1(\mu) - \zeta_1^{+}(\mu) \right )(1-2 \, \alpha_g)
+ \zeta_2(\mu) \, (3 -4 \, \alpha_g)\right ] \,, \nonumber \\
T_{3}(\alpha_i, \mu) &=& -120 \, \left (3 \, \zeta_2(\mu) -
\zeta_2^{+}(\mu) \right ) \,
(\alpha_{\bar q} - \alpha_q ) \,\alpha_{\bar q} \, \alpha_q \, \alpha_g \,, \nonumber \\
T_{4}(\alpha_i, \mu) &=& 30 \, \alpha_g^2 \, (\alpha_{\bar q}
-\alpha_q)\, \left [ \left (\kappa(\mu) + \kappa^{+}(\mu) \right
) + \left (\zeta_1(\mu) + \zeta_1^{+}(\mu) \right )(1-2 \,
\alpha_g)
+ \zeta_2(\mu) \, (3 - 4 \, \alpha_g)\right ] \,, \nonumber \\
T_{4}^{\gamma}(\alpha_i, \mu) &=& 60 \, \alpha_g^2 \, (\alpha_q -
\alpha_{\bar q} ) \, \left [ 4 - 7 \, (\alpha_{\bar q} + \alpha_q
) \right ] \,.\end{aligned}$$ In the above equations, the conformal expansion of the photon LCDAs have been truncated up to the next-to-leading conformal spin. Due to the Ferrara-Grillo-Parisi-Gatto theorem [@Ferrara:1972xq], these parameters satisfy the following relations $$\begin{aligned}
\zeta_1(\mu) + 11 \,\zeta_2(\mu) - 2 \, \zeta_2^{+}(\mu) = {7
\over 2} \,.\end{aligned}$$ The scale evolution of the nonperturbative parameters is given by $$\begin{aligned}
\kappa^{+}(\mu)= \left ( {\alpha_s (\mu) \over \alpha_s (\mu_0)}
\right ) ^{\left ( \gamma^{+} - \gamma_{q \bar q} \right ) /
\beta_0} \, \kappa^{+}(\mu_0)\,, & \qquad & \kappa(\mu)= \left (
{\alpha_s (\mu) \over \alpha_s (\mu_0)} \right ) ^{\left (
\gamma^{-} - \gamma_{q \bar q} \right ) / \beta_0} \,
\kappa(\mu_0) \,, \nonumber \\
\zeta_1(\mu)= \left ( {\alpha_s (\mu) \over \alpha_s (\mu_0)}
\right ) ^{\left ( \gamma_{Q^{(1)}} - \gamma_{q \bar q} \right )
/ \beta_0} \, \zeta_1(\mu_0)\,, & \qquad & \zeta_1^{+}(\mu)=
\left ( {\alpha_s (\mu) \over \alpha_s (\mu_0)} \right ) ^{\left
( \gamma_{Q^{(5)}} - \gamma_{q \bar q} \right ) / \beta_0} \,
\zeta_1^{+}(\mu_0)\,, \nonumber \\
\zeta_2^{+}(\mu)= \left ( {\alpha_s (\mu) \over \alpha_s (\mu_0)}
\right ) ^{\left ( \gamma_{Q^{(3)}} - \gamma_{q \bar q} \right )
/ \beta_0} \, \zeta_2^{+}(\mu_0)\,, \label{evolution of twist-four
parameters}\end{aligned}$$ where the anomalous dimensions at one loop read [@Ball:2002ps] $$\begin{aligned}
\gamma^{+}= 3 \, C_A - {5 \over 3} \, C_F \,, & \qquad &
\gamma^{-}= 4 \, C_A - 3 \, C_F \,,\,\,\,\,\,\,\,\,\gamma_{q \bar q} =-3 \, C_F \,, \nonumber \\
\gamma_{Q^{(1)}} = {11
\over 2} \, C_A - 3 \, C_F \,,& \qquad & \gamma_{Q^{(3)}} = {13
\over 3} \, C_F \,, \,\,\,\,\,\,\,\,\gamma_{Q^{(5)}} = 5 \, C_A -
{8 \over 3} \, C_F \,.\end{aligned}$$ Numerical values of the input parameters entering the photon LCDAs up to twist-4 are collected in Table \[tab of parameters for photon DAs\], where for the estimates of the twist-4 parameters from QCD sum rules [@Balitsky:1989ry] 100 % uncertainties are assigned.
--------------------------------------------- ------------------------------------- -------------- ----------------- --------------------- -------------------- ------------------------ ------------------------
\[-3.5mm\] $\chi(\mu_0) \, $ $\langle \bar q q \rangle (\mu_0) $ $b_2(\mu_0)$ $\kappa(\mu_0)$ $\kappa^{+}(\mu_0)$ $\zeta_{1}(\mu_0)$ $\zeta_{1}^{+}(\mu_0)$ $\zeta_{2}^{+}(\mu_0)$
\[-1mm\]
\[-1mm\] $(3.15 \pm 0.3) \, {\rm GeV}^{-2}$ $-(246^{+28}_{-19} \, {\rm MeV})^3$ $0.07 $0.2 \pm 0.2$ $0$ $0.4 \pm 0.4$ $0$ $0$
\pm 0.07$
--------------------------------------------- ------------------------------------- -------------- ----------------- --------------------- -------------------- ------------------------ ------------------------
: Numerical values of the nonperturbative parameters entering the photon LCDAs at the scale $\mu_0= 1.0 \, {\rm GeV}$ [@Ball:2002ps; @Duplancic:2008ix].[]{data-label="tab of parameters for photon DAs"}
---------------------- ----- ------------------------- --------------- ------------- -------------
Models CZ BMS KMOW Holographic Platykurtic
\[-1mm\] $a_2$(1GeV) 0.5 $0.20^{+0.07}_{-0.08}$ $0.17\pm0.08$ 0.15 0.08
\[-1mm\] $a_4$(1GeV) 0 $-0.15^{+0.10}_{-0.09}$ $0.06\pm0.10$ 0.06 -0.02
---------------------- ----- ------------------------- --------------- ------------- -------------
: The numerical values of Gegenbauer momemts $a_2$ and $a_4$ in leading twist pion LCDA.[]{data-label="tab Gegenbauer moments"}
Now we are in the position to investigate the phenomenological significance of the contribution from higher twist photon LCDAs. For the factorization scale in the evaluation of the contribution of higher-twist photon LCDAs, we will take the value $\mu^2=
\langle x \rangle \, M^2 + \langle \bar x \rangle \, Q^2$ as widely employed in the sum rule calculations [@Agaev:2010aq]. The Borel mass $M^2$ and the threshold parameter $s_0$ can be determined by applying the standard strategies described in [@Wang:2015vgv; @Wang:2017jow], $$\begin{aligned}
M^2= (1.25 \pm 0.50) \, {\rm GeV^2} \,, \qquad s_0= (0.70 \pm
0.20) \, {\rm GeV^2} \,,\label{borel}\end{aligned}$$ where the variation ranges of these parameters are set to be large to allow sufficient theoretical uncertainty. It has been checked that the Borel mass and threshold parameter dependence of the contribution of higher-twist photon LCDAs is mild in the intervals in Eq.(\[borel\]). In Fig. \[fig:comparison\] the $Q^2$ dependence of the relevant power suppressed contributions is presented. Compared with the contribution from leading-twist photon LCDA, the two-particle twist-4 contribution is obviously suppressed as the curve declines more quickly and approaches zero at large $Q^2$. While for the contribution from three-particle twist-4 LCDAs of photon, the result is comparable with that from leading twist photon LCDA, as they are at the same power. As mentioned in [@Wang:2017ijn], there exists strong cancellation effect between the contribution from leading twist photon LCDA and the twist-4 pion LCDA , thus the overall power correction is mainly from the contribution from twist-4 LCDAs of photon.
To obtain the total result of the photon-pion form factor, we will need to specify the non-perturbative models for the twist-2 pion LCDA. In general it is expanded in terms of Gegenbauer polynomials $$\begin{aligned}
\phi_{\pi}(x ,\mu) = 6 \, x \, \bar x \, \sum_{n=0}^{\infty} \,
a_n(\mu) \, C_n^{3/2}(2 x-1)\,, \label{Gegenbauer expansion of
pion DA}\end{aligned}$$ where the Gegenbauer moments $a_n$ can be determined by the calculation with QCD sum rules or lattice simulation, or by fitting the experimental data. Following [@Wang:2017ijn], we take advantage of the the Chernyak- Zhitnitsky (CZ) model[@Chernyak:1981zz], the Bakulev-Mikhailov-Stefanis (BMS) model[@Bakulev:2001pa], the platykurtic model (PK)[@Stefanis:2014nla], the KMOW model[@Khodjamirian:2011ub], and the holographic model[@Brodsky:2007hb] for comparison. The Gegenbauer coefficients in the BMS model and the PK model are computed from the QCD sum rules with non-local condensates, the first and second nontrivial Gegenbauer moments of the KMOW model are determined by comparing the LCSR predictions for the pion electromagnetic form factor with the experimental data at intermediate-$Q^2$, and the holographic model of the twist-2 pion LCDA is motivated by the AdS/QCD correspondence. We collect the values of the Gegenbauer moments in different models in Table. \[tab Gegenbauer moments\]. The total results including power suppressed contributions are shown in Fig. \[fig:total\], where the BMS model is employed. It can be seen that the higher power photon LCDAs manifestly modify the LP result especially at “small" $Q^2$ region. We note that the photon-LCSRs employed in this paper is valid when $Q^2\gg2GeV^2$, thus the prediction of $F_{\gamma^{\ast} \gamma \to \pi^0} (Q^2)$ should not be taken serious below 2 GeV$^2$.
![Comparison of the power suppressed contribution to pion-photon form factor $Q^2F_{\gamma^{\ast}
\gamma \to \pi^{0}}(Q^2)$ from different sources . []{data-label="fig:comparison"}](Fig3.eps "fig:"){width="0.45\columnwidth"}\
\
\
The model dependence of pion-photon form factor on the leading twist pion LCDA is displayed in Fig. \[fig:models\]. As the contribution from higher twist photon LCDA enhances the form factors significantly, the prediction from every models cannot match the experimental data at $Q^2<10GeV^2$. This result is inconsistent with the predictions from dispersion approach[@Bakulev:2002uc; @Stefanis:2012yw; @Bakulev:2011rp; @Bakulev:2012nh; @Mikhailov:2016klg], where the BMS and PK models of pion LCDA work well. This discrepancy is not a surprise because the power suppressed contributions considered in both approaches are not from a systematic study based on the effective theory, and what is omitted is not clear. Our result indicates that there exist significant power suppressed contributions, and they should not be neglected in the phenomenological studies. Meanwhile, we cannot draw the conclusion that the models mentioned in this paper should be ruled out, because in our study the QCD corrections are not included, and contributions from the pion and photon LCDA with twist higher than 4 are not considered, let alone the unknown power suppressed contributions. Thus in the present paper we aim at sheding light on the importance of the power corrections, and more efforts must be devoted to the study on power suppressed contributions to obtain more accurate prediction.
We present our final predictions for $Q^2F_{\gamma^{\ast} \gamma
\to \pi^{0}}(Q^2)$ with both LP contribution and power corrections included in Fig. \[fig:exp\], where the combined theory uncertainties are due to the variations of the input parameters $a_2,a_4$ of pion LCDA, $\xi, \langle \bar qq\rangle, b_2$ in twist-2 photon LCDAs, $\kappa, \zeta_1,\zeta_2$ in twist-4 photon LCDAs, quark mass, and factorization scale, etc. Diagram (a), (b) and (c) in Fig. \[fig:exp\] are corresponding to the BMS model, holographic model and KMOW model of pion LCDA respectively. Among all the parameters, the most important uncertainty comes from the shape parameters $a_2,a_4$ of leading twist pion LCDA, which means the pion transition form factor is still sensitive to the Gegenbauer moments of leading twist pion LCDA after the power suppressed contributions considered. Thus the photon-pion transition process provides a good platform to determine the parameters in the LCDAs of pion, which can also be compared with the future lattice simulation with the help of quasi parton distribution amplitude [@Liu:2018tox; @Wang:2017qyg].
(a)\
(b) (c)\
Closing remark
==============
In this paper we performed a study on the power suppressed contributions from higher-twist LCDAs of photon within the LCSRs. The twist-3 LCDAs cannot contribute for their Lorentz structures, thus the contributions from two-particle and three-particle twist-4 LCDAs of photon are considered in this work. According to the power analysis, the three-particle twist-4 contribution is not suppressed compared with the leading twist photon LCDA result, so that the power corrections considered in this work can give rise to sizable contribution, especially at “low" $Q^2$ region. In addition, there exists strong cancellation between the contribution from leading twist photon LCDA and the twist-4 pion LCDA, and the importance of the twist-4 photon LCDAs is further highlighted. The numerical result also confirms that after including power corrections, the predicted $Q^2F_{\gamma^{\ast}
\gamma \to \pi^{0}}(Q^2)$ is significantly enhanced especially at at “low" $Q^2$ region, thus the power suppressed contributions should be included in the determination of the Gegenbauer moments of pion LCDAs. Note that for the higher-twist photon LCDAs contribution, we only presented a tree level calculation, the NLO QCD corrections which might modify the current result to some extent and stablize the factorization scale dependence are not considered. Furthermore, the other power suppressed contributions are also absent in the present study, a more systematic study based on effective theory is necessary for a thorough understanding of the NLP corrections to the pion transition form factor, which can be checked by the (potentially) more accurate experimental measurements at the BEPCII collider and the SuperKEKB accelerator.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We thank S. V. Mikhailov and N. G. Stefanis for valuable comments. This work is supported in part by the National Natural Science Foundation of China (NSFC) with Grant No. 11521505 and 11621131001. CDL would like to express a special thanks to the Mainz Institute for Theoretical Physics (MITP) for its hospitality and Support.
Definition of three-particle twist-4 LCDAs of photon
====================================================
In the following, we present the definition of the three-particle photon LCDAs up to twist-4. $$\begin{aligned}
&& \langle 0 |\bar q(x) g_s \, G_{\alpha \beta}(u \, x) \, \, q(0)|\gamma(p) \rangle \nonumber \\
&& = i \, g_{\rm em} \, Q_q \, \langle \bar q q \rangle(\mu) \,
(p_{\beta} \, \epsilon_{\alpha} - p_{\alpha} \, \epsilon_{\beta})
\, \int [{\cal D} \alpha_i] \, e^{i \, (\alpha_q + \, \bar u
\, \alpha_g-1) \, p \cdot x}\, S(\alpha_i, \mu)
\\
&& \langle 0 |\bar q(x) g_s \, \widetilde{G}_{\alpha \beta}( u \,
x) \,
i \, \gamma_5 \,\, q(0)|\gamma(p)\rangle \nonumber \\
&& = i \, g_{\rm em} \, Q_q \, \langle \bar q q \rangle(\mu) \,
(p_{\beta} \, \epsilon_{\alpha} - p_{\alpha} \, \epsilon_{\beta})
\, \int [{ D} \alpha_i] \, e^{i \, (\alpha_q + \, \bar u \,
\alpha_g-1) \, p \cdot x}\, \widetilde{S}(\alpha_i, \mu)
\\
&& \langle 0|\bar q(x) g_s \, \widetilde{G}_{\alpha \beta}(u \,
x) \,
\gamma_{\rho} \, \gamma_5 \,\, q(0)| \gamma(p) \rangle \nonumber \\
&& = - g_{\rm em} \, Q_q \, f_{3 \gamma}(\mu) \, p_{\rho} \,
(p_{\beta} \, \epsilon_{\alpha} - p_{\alpha} \, \epsilon_{\beta})
\, \int [{\cal D} \alpha_i] \, e^{i \, (\alpha_q + \, \bar u
\, \alpha_g-1) \, p \cdot x} \, A(\alpha_i, \mu)
\\
&& \langle 0 |\bar q(x) g_s \, G_{\alpha \beta}(u \, x) \,
i \, \gamma_{\rho} \,\, q(0)|\gamma(p) \rangle \nonumber \\
&& = g_{\rm em} \, Q_q \, f_{3 \gamma}(\mu) \, p_{\rho} \,
(p_{\beta} \, \epsilon_{\alpha} - p_{\alpha} \, \epsilon_{\beta})
\, \int [{\cal D} \alpha_i] \, e^{i \, (\alpha_q + \, \bar u
\, \alpha_g-1) \, p \cdot x}\, V(\alpha_i, \mu)
\\
&& \langle 0 |\bar q(x) g_{\rm em} \, Q_q \, F_{\alpha \beta}(u \, x) \, \, q(0)| \gamma(p) \rangle \nonumber \\
&& = i \, g_{\rm em} \, Q_q \, \langle \bar q q \rangle(\mu) \,
(p_{\beta} \, \epsilon_{\alpha} - p_{\alpha} \, \epsilon_{\beta})
\, \int [{\cal D} \alpha_i] \, e^{i \, (\alpha_q + \, \bar u
\, \alpha_g-1) \, p \cdot x} \, S_{\gamma}(\alpha_i, \mu) \,. \\
\nonumber \\
&& \langle 0 |\bar q(x) \,\, \sigma_{\rho \tau} \,\, g_s \,
G_{\alpha \beta}(u \, x)
\,\, q(0)| \gamma(p)\rangle \nonumber \\
&& = - \, g_{\rm em} \, Q_q \,\langle \bar q q \rangle(\mu) \,
\left [p_{\rho} \, \epsilon_{\alpha} \, g_{\tau \beta}^{\perp} -
p_{\tau} \, \epsilon_{\alpha} \, g_{\rho \beta}^{\perp} - (\alpha
\leftrightarrow \beta) \right ] \, \int [{\cal D} \alpha_i]
\,e^{i \, (\alpha_q + \, \bar u
\, \alpha_g-1) \, p \cdot x} \, T_{1}(\alpha_i, \mu) \nonumber \\
&& \hspace{0.4 cm} - \, g_{\rm em} \, Q_q \,\langle \bar q q
\rangle(\mu) \, \left [p_{\alpha} \, \epsilon_{\rho} \, g_{\tau
\beta}^{\perp} - p_{\beta} \, \epsilon_{\rho} \, g_{\tau
\alpha}^{\perp} - (\rho \leftrightarrow \tau) \right ] \, \int
[{\cal D} \alpha_i] \, e^{i \, (\alpha_q + \, \bar u
\, \alpha_g-1) \, p \cdot x}\, T_{2}(\alpha_i, \mu) \nonumber \\
&& \hspace{0.4 cm} - \, g_{\rm em} \, Q_q \,\langle \bar q q
\rangle(\mu) \, \frac{(p_{\alpha} \, x_{\beta} - p_{\beta} \,
x_{\alpha} ) (p_{\rho} \, \epsilon_{\tau} - p_{\tau}
\,\epsilon_{\rho})} {p \cdot x} \, \int [{\cal D} \alpha_i] \,
e^{i \, (\alpha_q + \, \bar u \, \alpha_g-1) \, p \cdot x} \,
T_{3}(\alpha_i, \mu) \nonumber \\
&& \hspace{0.4 cm} - \, g_{\rm em} \, Q_q \,\langle \bar q q
\rangle(\mu) \, \frac{(p_{\rho} \, x_{\tau} - p_{\tau} \, x_{\rho}
) (p_{\alpha} \, \epsilon_{\beta} - p_{\beta}
\,\epsilon_{\alpha})} {p \cdot x} \, \int [{\cal D} \alpha_i] \,
e^{i \, (\alpha_q + \, \bar u \, \alpha_g-1) \, p \cdot x} \,
T_{4}(\alpha_i, \mu) \,. \\
\nonumber \\
&& \langle 0 |\bar q(x) \sigma_{\rho \tau} \,\, g_{\rm em} \, Q_q
\, F_{\alpha \beta}(u \, x)
\,\, q(0)| \gamma(p) \rangle \nonumber \\
&& = - \, g_{\rm em} \, Q_q \,\langle \bar q q \rangle(\mu) \,
\frac{(p_{\rho} \, x_{\tau} - p_{\tau} \, x_{\rho} ) (p_{\alpha}
\, \epsilon_{\beta} - p_{\beta} \,\epsilon_{\alpha})} {p \cdot x}
\, \int [{\cal D} \alpha_i] \, e^{i \, (\alpha_q + \, \bar u
\, \alpha_g-1) \, p \cdot x}\,
T_{4}^{\gamma}(\alpha_i, \mu) + ... \,\end{aligned}$$ Note that we have employed the following notations for the dual field strength tensor and the integration measure $$\begin{aligned}
\widetilde{G}_{\alpha \beta}&=& {1 \over 2} \, \varepsilon_{\alpha
\beta \rho \tau } \, G^{\rho \tau} \,, \qquad \int [{\cal D}
\alpha_i] \equiv \int_0^1 d \alpha_q \, \int_0^1 d \alpha_{\bar q}
\, \int_0^1 d \alpha_g \, \delta \left (1-\alpha_q - \alpha_{\bar
q} -\alpha_g \right )\end{aligned}$$
[99]{}
G. P. Lepage and S. J. Brodsky, Phys. Rev. D [**22**]{} (1980) 2157.
A. V. Efremov and A. V. Radyushkin, Phys. Lett. [**94B**]{} (1980) 245.
A. Duncan and A. H. Müeller, Phys. Lett. [**90B**]{} (1980) 159.
I. Z. Rothstein, Phys. Rev. D [**70**]{} (2004) 054024 \[hep-ph/0301240\].
F. del Aguila and M. K. Chase, Nucl. Phys. B [**193**]{} (1981) 517.
E. Braaten, Phys. Rev. D [**28**]{} (1983) 524.
E. P. Kadantseva, S. V. Mikhailov and A. V. Radyushkin, Yad. Fiz. [**44**]{} (1986) 507 \[Sov. J. Nucl. Phys. [**44**]{} (1986) 326\].
B. Melic, D. Müeller and K. Passek-Kumericki, Phys. Rev. D [**68**]{} (2003) 014013 \[hep-ph/0212346\].
G. Bonneau, Int. J. Mod. Phys. A [**5**]{} (1990) 3831.
J. C. Collins, “[*Renormalization : An Introduction to Renormalization, The Renormalization Group, and the Operator Product Expansion*]{},” Cambridge University Press, 1984.
S. A. Larin, Phys. Lett. B [**303**]{} (1993) 113 \[hep-ph/9302240\].
C. P. Martin and D. Sanchez-Ruiz, Nucl. Phys. B [**572**]{} (2000) 387 \[hep-th/9905076\].
F. Jegerlehner, Eur. Phys. J. C [**18**]{} (2001) 673 \[hep-th/0005255\].
S. Moch, J. A. M. Vermaseren and A. Vogt, Phys. Lett. B [**748**]{} (2015) 432 \[arXiv:1506.04517 \[hep-ph\]\].
D. Guti¨¦rrez-Reyes, I. Scimemi and A. A. Vladimirov, Phys. Lett. B [**769**]{} (2017) 84 \[arXiv:1702.06558 \[hep-ph\]\].
Y. M. Wang and Y. L. Shen, JHEP [**1712**]{}, 037 (2017) doi:10.1007/JHEP12(2017)037 \[arXiv:1706.05680 \[hep-ph\]\].
M. Beneke, Y. Kiyo and D. S. Yang, Nucl. Phys. B [**692**]{} (2004) 232 \[hep-ph/0402241\].
M. Beneke and D. Yang, Nucl. Phys. B [**736**]{} (2006) 34 \[hep-ph/0508250\].
M. Beneke and S. Jager, Nucl. Phys. B [**751**]{} (2006) 160 \[hep-ph/0512351\].
M. J. Dugan and B. Grinstein, Phys. Lett. B [**256**]{} (1991) 239.
S. Herrlich and U. Nierste, Nucl. Phys. B [**455**]{} (1995) 39 \[hep-ph/9412375\]. S. Nandi and H. n. Li, Phys. Rev. D [**76**]{} (2007) 034008 \[arXiv:0704.3790 \[hep-ph\]\].
I. V. Musatov and A. V. Radyushkin, Phys. Rev. D [**56**]{} (1997) 2713 \[hep-ph/9702443\].
X. G. Wu and T. Huang, Phys. Rev. D [**82**]{} (2010) 034024 \[arXiv:1005.3359 \[hep-ph\]\].
H. N. Li, Y. L. Shen and Y. M. Wang, JHEP [**1401**]{} (2014) 004 \[arXiv:1310.3672 \[hep-ph\]\].
B. Aubert [*et al.*]{} \[BaBar Collaboration\], Phys. Rev. D [**80**]{} (2009) 052002 \[arXiv:0905.4778 \[hep-ex\]\].
P. Masjuan, Phys. Rev. D [**86**]{} (2012) 094021 \[arXiv:1206.2549 \[hep-ph\]\].
M. Hoferichter, B. Kubis, S. Leupold, F. Niecknig and S. P. Schneider, Eur. Phys. J. C [**74**]{} (2014) 3180 \[arXiv:1410.4691 \[hep-ph\]\].
A. G¨¦rardin, H. B. Meyer and A. Nyffeler, Phys. Rev. D [**94**]{} (2016) 074507 \[arXiv:1607.08174 \[hep-lat\]\].
A. V. Radyushkin, Phys. Rev. D [**80**]{} (2009) 094009 \[arXiv:0906.0323 \[hep-ph\]\].
M. V. Polyakov, JETP Lett. [**90**]{} (2009) 228 \[arXiv:0906.0538 \[hep-ph\]\].
S. S. Agaev, V. M. Braun, N. Offen and F. A. Porkert, Phys. Rev. D [**83**]{} (2011) 054020 \[arXiv:1012.4671 \[hep-ph\]\].
S. S. Agaev, V. M. Braun, N. Offen and F. A. Porkert, Phys. Rev. D [**86**]{} (2012) 077504 \[arXiv:1206.3968 \[hep-ph\]\].
P. Kroll, Eur. Phys. J. C [**71**]{} (2011) 1623 \[arXiv:1012.3542 \[hep-ph\]\].
A. P. Bakulev, S. V. Mikhailov and N. G. Stefanis, Phys. Rev. D [**67**]{}, 074012 (2003) doi:10.1103/PhysRevD.67.074012 \[hep-ph/0212250\]. N. G. Stefanis, A. P. Bakulev, S. V. Mikhailov and A. V. Pimikov, Phys. Rev. D [**87**]{} (2013) 094025 \[arXiv:1202.1781 \[hep-ph\]\].
A. P. Bakulev, S. V. Mikhailov, A. V. Pimikov and N. G. Stefanis, Phys. Rev. D [**84**]{} (2011) 034014 \[arXiv:1105.2753 \[hep-ph\]\].
A. P. Bakulev, S. V. Mikhailov, A. V. Pimikov and N. G. Stefanis, Phys. Rev. D [**86**]{} (2012) 031501 \[arXiv:1205.3770 \[hep-ph\]\].
S. V. Mikhailov, A. V. Pimikov and N. G. Stefanis, Phys. Rev. D [**93**]{} (2016) 114018 \[arXiv:1604.06391 \[hep-ph\]\].
A. Khodjamirian, Eur. Phys. J. C [**6**]{} (1999) 477 \[hep-ph/9712451\]. P. Ball, V. M. Braun and N. Kivel, Nucl. Phys. B [**649**]{} (2003) 263 \[hep-ph/0207307\].
Y. M. Wang and Y. L. Shen, JHEP [**1805**]{}, 184 (2018) doi:10.1007/JHEP05(2018)184 \[arXiv:1803.06667 \[hep-ph\]\]. Y. L. Shen, Z. T. Zou and Y. B. Wei, Phys. Rev. D [**99**]{}, no. 1, 016004 (2019) doi:10.1103/PhysRevD.99.016004 \[arXiv:1811.08250 \[hep-ph\]\]. Y. L. Shen, Z. T. Zou and Y. Li, arXiv:1901.05244 \[hep-ph\]\].
S. Ferrara, A. F. Grillo, G. Parisi and R. Gatto, Phys. Lett. [**38B**]{} (1972) 333.
G. Duplancic, A. Khodjamirian, T. Mannel, B. Melic and N. Offen, JHEP [**0804**]{} (2008) 014 \[arXiv:0801.1796 \[hep-ph\]\]. I. I. Balitsky, V. M. Braun and A. V. Kolesnichenko, Nucl. Phys. B [**312**]{} (1989) 509. Y. M. Wang and Y. L. Shen, Nucl. Phys. B [**898**]{} (2015) 563 \[arXiv:1506.00667 \[hep-ph\]\].
Y. M. Wang, Y. B. Wei, Y. L. Shen and C. D. L¨¹, JHEP [**1706**]{}, 062 (2017) doi:10.1007/JHEP06(2017)062 \[arXiv:1701.06810 \[hep-ph\]\].
V. L. Chernyak and A. R. Zhitnitsky, Nucl. Phys. B [**201**]{} (1982) 492 Erratum: \[Nucl. Phys. B [**214**]{} (1983) 547\].
A. P. Bakulev, S. V. Mikhailov and N. G. Stefanis, Phys. Lett. B [**508**]{} (2001) 279 Erratum: \[Phys. Lett. B [**590**]{} (2004) 309\] \[hep-ph/0103119\]. N. G. Stefanis, Phys. Lett. B [**738**]{}, 483 (2014) doi:10.1016/j.physletb.2014.10.018 \[arXiv:1405.0959 \[hep-ph\]\].
A. Khodjamirian, T. Mannel, N. Offen and Y.-M. Wang, Phys. Rev. D [**83**]{} (2011) 094031 \[arXiv:1103.2655 \[hep-ph\]\].
S. J. Brodsky and G. F. de Teramond, Phys. Rev. D [**77**]{} (2008) 056007 \[arXiv:0707.3859 \[hep-ph\]\].
I. C. Cloët, L. Chang, C. D. Roberts, S. M. Schmidt and P. C. Tandy, Phys. Rev. Lett. [**111**]{} (2013) 092001 \[arXiv:1306.2645 \[nucl-th\]\]. J. Gronberg [*et al.*]{} \[CLEO Collaboration\], Phys. Rev. D [**57**]{} (1998) 33 \[hep-ex/9707031\]. S. Uehara [*et al.*]{} \[Belle Collaboration\], Phys. Rev. D [**86**]{} (2012) 092007 \[arXiv:1205.3249 \[hep-ex\]\]. Y. S. Liu, W. Wang, J. Xu, Q. A. Zhang, S. Zhao and Y. Zhao, \[arXiv:1810.10879 \[hep-ph\]\]. W. Wang, S. Zhao and R. Zhu, Eur. Phys. J. C [**78**]{}, no. 2, 147 (2018) \[arXiv:1708.02458 \[hep-ph\]\].
[^1]: shenylmeteor@ouc.edu.cn
[^2]: gaojing@ihep.ac.cn
[^3]: lucd@ihep.ac.cn
[^4]: 782771338@qq.com
|
---
abstract: 'In atomic force microscopy (AFM), the angle relative to the vertical ($\theta_{i}$) that the tip apex of a cantilever moves is determined by the tilt of the probe holder, and the geometries of the cantilever [[beam]{}]{} and actuated eigenmode $i$. Even though the effects of $\theta_{i}$ on static and single-frequency AFM are known (increased effective spring constant, sensitivity to sample anisotropy, etc.), the higher eigenmodes used in multifrequency force microscopy lead to additional effects that have not been fully explored. Here we use Kelvin probe force microscopy (KPFM) to investigate how $\theta_{i}$ affects not only the signal amplitude and phase, but can also lead to behaviors such as destabilization of the KPFM voltage feedback loop. We find that longer cantilever [[beam]{}]{}s and modified sample orientations improve voltage feedback loop stability, even though variations to scanning parameters such as shake amplitude and lift height do not.'
address:
- ' Department of Physics, University of Maryland, College Park, MD 20742, USA'
- ' Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, MD 20742, USA'
- ' Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, MD 20742, USA'
- ' Department of Electrical and Computer Engineering, University of Maryland, College Park, MD 20742, USA'
- ' Department of Physics, University of Maryland, College Park, MD 20742, USA'
- ' Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, MD 20742, USA'
- ' Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, MD 20742, USA'
- ' Department of Electrical and Computer Engineering, University of Maryland, College Park, MD 20742, USA'
author:
- 'Joseph L. Garrett'
- 'Lisa J. Krayer'
- 'Kevin J. Palm'
- 'Jeremy N. Munday'
bibliography:
- 'AFM\_geometry.bib'
title: Effect of lateral tip motion on multifrequency atomic force microscopy
---
The development of specialized cantilever probes enabled atomic force microscopy (AFM)[@Binnig1986]. Later, it was realized that the holder tilts the cantilever and the trajectory of the tip apex which both increases the effective static spring constant and causes the phase of Amplitude Modulation (AM) AFM to be sensitive to both the anisotropy and slope of samples[@Marcus2002; @DAmato2004; @Heim2004; @Hutter2005a]. For higher eigenmodes $i$, the angle between the tip apex trajectory and the vertical axis ($\theta_{i}$) also depends the geometries of the cantilever and eigenmode, so that recent experiments were able to use eigenmodes with different $\theta_{i}$ to probe forces in several directions[@Kawai2010a; @Sigdel2013; @Reiche2015; @Meier2016; @Huang2017; @Naitoh2017]. Bimodal AFM, in which two eigenmodes are driven by excitation of the cantilever base, was used for most of these experiments, but it is only one of many multifrequency techniques[@Nonnenmacher1991; @Jesse2007; @Platz2008; @Tetard2010; @Garcia2012a; @Ebeling2013b; @Zerweck2005; @Rajapaksa2011; @Sugawara2012; @Arima2015; @Garrett2016; @Tumkur2016; @Jahng2016; @Ambrosio2017], and the effects of $\theta_{i}$ have not yet been explored for the general multifrequency case.
[[Sideband multifrequency AFM methods are promising ways to investigate optoelectronic materials and devices at the nanoscale]{}]{}[@Zerweck2005; @Rajapaksa2011; @Sugawara2012; @Arima2015; @Garrett2016; @Tumkur2016; @Jahng2016; @Ambrosio2017]. [[In order to eliminate long-range artifacts and improve spatial resolution, they drive a signal by]{}]{} mixing a modulated tip-sample force with piezo-driven cantilever oscillations. [[A prominent sideband method is photo-induced force microscopy (PIFM), which has been used for nanoscale imaging of Raman spectra[@Rajapaksa2011], nanoparticle resonances[@Tumkur2016], and refractive index changes[@Ambrosio2017]. However, there is considerable debate about how to extract quantitative data from PIFM scans[@Jahng2016; @Tumkur2016; @Ambrosio2017] because it is unclear how the force couples into the probe and optical forces themselves are difficult to characterize [*a priori*]{}.]{}]{}
[[Because the electrostatic force is well-characterized and controllable compared to optical forces, it offers an opportunity to test the sideband actuation technique.]{}]{} Frequency Modulation (FM) and Heterodyne (H) Kelvin probe force microscopy (KPFM) are sideband [[methods]{}]{} that use [[the electrostatic force]{}]{} to drive cantilever oscillations, which are in turn input into a feedback loop that measures the tip-sample potential difference. In a recent experiment, height variation of around 10 nm destabilized the H-KPFM voltage feedback loop, but FM-KPFM scans were stable for variations of over 100 nm[@Garrett2017a]. Because FM- and H-KPFM are primarily distinguished by the eigenmode used to amplify the KPFM signal, the cause of their qualitatively different behavior likely originates from the geometry of the eigenmodes. Moreover, the details of cantilever dynamics have been shown to be critical to understanding AM-KPFM[@Elias2011; @Satzinger2012], a much simpler technique that drives and detects its signal at a single frequency[[, and which can be used for comparison]{}]{}.
![ The tip apex moves at an angle relative to the vertical for each eigenmode $i$ ($\theta_{i}$), which depends on the angle of the probe holder ($\theta_{\text{holder}}$), the geometry of the cantilever, and the geometry of the eigenmode ($\Phi_{i}$). The inset shows the tip apex with the first eigenmode excited ($i$=1), in which the amplitude of the eigenmode ($Y_{1}$), the tip apex displacement ($\vec{r}_{1}$), and $\theta_{1}$ are labeled. []{data-label="fig:CantileverResponse"}](Figure1.pdf){width=".43\textwidth"}
In this letter, we use KPFM measurements to answer the questions: (a) how does the $\theta_{i}$ of each eigenmode affect the signals of KFPM, (b) why does the KPFM feedback instability differ between H- and FM-KPFM, and (c) how [[do]{}]{} the effects of $\theta_{i}$ [[appear in sideband]{}]{} multifrequency force microscopy methods?
The motion of a cantilever [[beam]{}]{} can be expressed as a sum of eigenmodes, each a solution to the Euler-Bernoulli beam equation[@Butt1995; @Lozano2009]: $$\begin{aligned}
z_{\text{cant}}(x,t) = \sum_{i=1}^{\infty} Y_{i}(t)\Phi_{i}(x),
\label{eq:beam_motion}\end{aligned}$$ where $Y_{i}(t)$ contains the time-dependence, $\Phi_{i}$ is the shape of the $i$th cantilever [[beam]{}]{} eigenmode (normalized so that $\Phi_{i}(L)=1$, where $L$ is the length of the cantilever [[beam]{}]{}), and $z_{\text{cant}}$ is the displacement of the cantilever [[beam]{}]{} (see figure \[fig:CantileverResponse\]). To maintain generality, the exact form of $\Phi_{i}$ is not specified until the numerical evaluation of $\theta_{i}$, at which point the solution for a rectangular cantilever [[beam]{}]{} is used[@Butt1995; @Lozano2009]. Thus the following analysis holds even for non-rectangular [[cantilever beams]{}]{} and probes with large tip cones, both which may have atypical $\Phi_{i}$[@Tung2010; @Labuda2016].
[[To calculate the trajectory of the tip apex,]{}]{} the [[probe]{}]{} is characterized by its tip cone height $h$, contact angle $\delta$, and contact position $x_{t}$ (figure \[fig:Multiple\_cantilevers\]). [[The position of the tip apex is the location of base of the tip cone {$x_{t},Y_{i}\Phi_{i}(x_{t})$} plus the position of the tip apex relative to the base of tip cone, {$h\cos(\xi(Y_{i})-\delta),h\sin(\xi(Y_{i})-\delta)$},]{}]{} where $\xi(Y_{i}) = \tan^{-1}(Y_{i}\partial_{x}\Phi_{i}(x_{t}))$ is the angle of the vector normal to the cantilever at $x_{t}$. [[Because the probe is held at an angle $\theta_{\text{holder}}$]{}]{} (here, [[0.2 radians]{}]{}), the displacement of the tip apex from equilibrium becomes, in the small oscillation limit ($Y_{i}\ll L$): $$\begin{aligned}
\vec{r}_{i} &= \mathbf{R}\bigg[ \begin{matrix} h(\cos(\xi(Y_{i})-\delta)-\cos(\delta)) \\ Y_{i}\Phi_{i} (x_{t}) +h(\sin(\xi(Y_{i})-\delta)+\sin(\delta)) \end{matrix} \bigg],
\label{eq:full_r}\end{aligned}$$ where $\mathbf{R}=\left[ \begin{smallmatrix} \cos(\theta_{\text{holder}}) & \sin(\theta_{\text{holder}}) \\ -\sin(\theta_{\text{holder}}) & \cos(\theta_{\text{holder}}) \end{smallmatrix} \right]$ is a 2D rotation matrix around the base of the cantilever [[beam]{}]{}. For a single eigenmode in the $Y_{i}\ll L$ limit, the tip apex moves in a straight line at an angle with respect to the vertical: $$\begin{aligned}
\theta_{i}= \lim_{Y_{i}/L\rightarrow 0}\cos^{-1}\big(\vec{r}_{i}\cdot(Y_{i}\hat{z})\big).
\label{eq:linear_r_approx}\end{aligned}$$ [[Note that equations \[eq:full\_r\] and \[eq:linear\_r\_approx\] imply that much of the trajectory of the tip apex is in the $\hat{x}$ direction, even for very small excitations. For example, a 10 nm amplitude excitation of the first eigenmode of the cantilever beam in figure \[fig:Multiple\_cantilevers\]b causes the tip apex to move $\approx3.9$ nm in the $\hat{x}$ direction and 8.6 nm in the $\hat{z}$ direction.]{}]{} Because the potential energy of an eigenmode must be the same whether the motion of the end of cantilever [[beam]{}]{} ($\Phi_{i}(L)$) or the tip [[apex]{}]{} ($\vec{r}_{i}$) is considered, an effective spring constant ($k_{i}^{\text{eff}}$) for forces acting on the tip apex parallel to $\vec{r}_{i}$ (perpendicular forces excite only eigenmodes $\neq i$) can be defined[@Reiche2015]: $$\begin{aligned}
\label{eq:sensitivity}
k_{i}^{\text{eff}} &= \lim_{Y_{i}/L\rightarrow 0}\frac{Y_{i}^{2}}{|\vec{r}_{i}(Y_{i})|^{2}}k_{i}, \end{aligned}$$ where $k_{i}$ is the spring constant for an upward force acting at $x=L$[@Melcher2007].
![ Cantilever geometry determines the direction of the tip apex motion. (a,b) SEM images show cantilevers of length 350 $\mu$m and 90 $\mu$m, respectively ($\mu$masch, CSC37/Pt-B and NSC35/Pt-B). (c) Each cantilever is characterized by its tip cone height $h$, contact position $x_{t}$, contact angle $\delta$, and length $L$. (d,e) The full calculation of $\vec{r}_{i}$ (solid line, equation \[eq:full\_r\]) and linear approximation (dashed line, equation \[eq:linear\_r\_approx\]) show agreement. For each eigenmode, $\theta_{i}$ is greater for the short cantilever than for the long cantilever. (f) The slope of the sample is characterized by its normal vector ($\hat{n}$) and the angle it makes with the vertical ($\theta_{n}$).[]{data-label="fig:Multiple_cantilevers"}](Figure2.pdf){width=".48\textwidth"}
![ [[An AC voltage, $V_{AC}$, is applied to the [[cantilever]{}]{} at frequency $\omega_{A}$, while tip-sample separation is controlled by piezo-driven oscillation at frequency $\omega_{T}$ and the sample is grounded. The oscillations at $\omega_{T}$ mix with the electrostatic force driven by $V_{AC}$ at frequency $\omega_{A}$ to drive the tip apex at the detection frequency, $\omega_{D}$, which is amplified by one of the cantilever’s resonance frequencies and detected by a lock-in amplifier. When KPFM feedback is used, the grey signal paths are added to the circuit, and $\omega_{A}=(\omega_{D}-\omega_{T})/2$ is changed to $\omega_{A}=\omega_{D}-\omega_{T}$.]{}]{} []{data-label="fig:BoxDiagram"}](Figure3.pdf){width=".43\textwidth"}
{width=".90\textwidth"}
L ($\mu$m) $\frac{\omega_{1}}{2\pi}$ (MHz) $\frac{\omega_{2}}{2\pi}$ $\frac{\omega_{3}}{2\pi}$ $\frac{\omega_{4}}{2\pi}$ $\frac{\omega_{5}}{2\pi}$ $\frac{\omega_{6}}{2\pi}$ $\frac{\omega_{7}}{2\pi}$
------------ --------------------------------- --------------------------- --------------------------- --------------------------- --------------------------- --------------------------- ---------------------------
90 0.25 1.62 4.58 - - - -
350 0.02 0.13 0.37 0.72 1.20 1.79 2.50
: Cantilever resonance frequencies
\[table:cantilever\_resonances\]
The tip [[apex trajectory]{}]{} affects [[[AFM]{}]{}]{} techniques [[that]{}]{} use a modulated tip-sample force $\vec{F}_{\text{dir}}$ to actuate the cantilever either directly or through sideband coupling while relying on [[piezo-driven oscillation with amplitude $A_{T}$]{}]{} at frequency $\omega_{T}$ for topography control [[(here, $\omega_{T}=\omega_{1}$ in table \[table:cantilever\_resonances\] is used)]{}]{}.
Sideband techniques generate a signal by modulating a separation-dependent force $\vec{F}_{\text{dir}}$ at frequency $\omega_{M}$, which is then mixed with the piezo-driven oscillations, typically $A_{T}$. Here, the resonance frequency used for detection determines the modulation frequency $\omega_{M} = \omega_{i}-\omega_{T}$ (table \[table:cantilever\_resonances\]). By using the force gradient, sideband methods exclude the non-local effects of the cantilever [[beam]{}]{} that are present when $\vec{F}_{\text{dir}}$ is used for direct actuation, such as in AM-KPFM[@Zerweck2005; @Sugawara2012; @Jahng2016].
To confirm that the cantilever beam’s contribution to the total force is small even when higher eigenmodes are used, the force on the beam is computed for both direct actuation ($-\partial U/ \partial Y_{i}$) and sideband actuation ($-\partial^{2} U/ \partial Y_{i}^{2}$), where $U$ is the electrostatic potential energy between the probe and the surface evaluated using the proximity force approximation and the geometry of the longer probe. The contribution from the tip apex is calculated by modeling it as a 30 nm radius sphere 10 nm above the surface. The percent of the signal originating from the cantilever beam using direct actuation is found to be 17-53% for the first seven eigenmodes, while with sideband actuation 0.1-0.2% of the signal originates from the beam. The small contribution from the beam validates the approximation that the electrostatic force acts on the tip apex for sideband actuation of higher eigenmodes.
In the small-oscillation approximation[@Jahng2016; @Garrett2016], the force driving sideband oscillation [[is $\vec{F}_{\text{side}}\cos(\omega_{D}t)$, where]{}]{} $$\begin{aligned}
\label{eq:SidebandActuation}
\vec{F}_{\text{side}} = \partial_{d}\vec{F}_{\text{dir}}\frac{A_{T}}{2}\cos(\theta_{i}-\theta_{n}),\end{aligned}$$ [[in which]{}]{} $d$ is the tip-sample separation, [[$\omega_{D}$ is the detection frequency]{}]{}, and the $\cos(\theta_{i}-\theta_{n})$ factor originates from the angle between the trajectory [[of the tip apex]{}]{} and the force vector (parallel to $\hat{n}$). The [[displacement of the tip apex at $\omega_{D}$ is then $\vec{r}_{j}(t)=A_{D}\cos(\omega_{D}t)\hat{r}_{j}$, where eigenmode $j$ is driven and the signal detected by the lock-in amplifier is]{}]{} $$\begin{aligned}
A_{D} &= \frac{Q_{j}}{k^{\text{eff}}_{j}}\vec{F}\cdot\hat{r}_{j},
\label{eq:direct}\end{aligned}$$ for both the sideband and direct driving forces [[(figure \[fig:BoxDiagram\]). A change in the sign of $A_{D}$ corresponds to a phase shift by $\pi$ radians.]{}]{}
The interplay of $\theta_{j}$ and sample slope can then be observed in the [[signal $A_{D}$]{}]{} normalized by the [[its value]{}]{} on a flat surface ($\tilde{A}_{D} \equiv \frac{A_{D}}{A_{D}(\theta_{n}=0)}$):
$$\begin{aligned}
\label{eq:single_freq_A}
\tilde{A}_{D}^{\text{dir}} &= \frac{\cos(\theta_{j}-\theta_{n})}{\cos(\theta_{j})},\\
\tilde{A}_{D}^{\text{side}} &=\frac{\cos(\theta_{n}-\theta_{j})\cos(\theta_{n}-\theta_{i})}{\cos(\theta_{j})\cos(\theta_{i})},
\label{eq:mult_freq_A}\end{aligned}$$
where it is assumed that $\hat{n}$ is in the x-z plane and $\theta_{i},\theta_{j}\neq\pm\pi/2$. Note that if $|\theta_{i}-\theta_{n}|>\frac{\pi}{2}>\theta_{i}$, $\tilde{A}_{D}$ changes sign.
Equations \[eq:single\_freq\_A\] and \[eq:mult\_freq\_A\] predict how the geometry of tip apex motion causes scanning probe methods to be sensitive to sample slope. To test the equations, a silicon trench is fabricated using e-beam lithography to pattern a 2 $\mu$m $\times$ 100 $\mu$m line on a silicon wafer which is then etched using reactive ion etching (RIE) and coated with 5 nm of chromium for conductivity. The edges of the trench are imaged, in attractive mode [@Paulo2002], (Cypher, Asylum Research), trace and retrace images are averaged, and each column of pixels is summed and averaged (figure \[fig:Signals\_and\_geometry\]a,b).
[[In the static limit,]{}]{} when an AC voltage is applied to a probe [[at frequency $\omega_{A}$]{}]{}, the tip-sample electrostatic force has components at three frequencies[@Nonnenmacher1991; @Zerweck2005]: $\vec{F}_{\text{es}} = \vec{F}_{DC} + \vec{F}_{\omega}\cos(\omega_{A} t) + \vec{F}_{2\omega}\cos(2\omega_{A} t).$ [[Either $\vec{F}_{\omega}$ or $\vec{F}_{2\omega}$ can be used in equation \[eq:SidebandActuation\] to drive the sideband signal by choosing $\omega_{M} = \omega_{A}$ or $2\omega_{A}$, respectively. The signal then depends on the gradient of the original modulation force[@Zerweck2005; @Sugawara2012; @Miyahara2017]. For FM-KPFM, $\omega_{A}\ll\omega_{1}$[@Zerweck2005].]{}]{} Closed loop KPFM measures the contact potential difference between the probe and sample using a feedback loop to nullify a signal driven by the force $\vec{F}_{\omega}$. Alternatively, open loop KPFM uses oscillation driven by $\vec{F}_{2\omega}$ combined with the $\vec{F}_{\omega}$ signal to estimate the potential difference [[$\Delta V$ from the relationship between the forces $\vec{F}_{2\omega} = \vec{F}_{\omega}V_{AC}/(4\Delta V)$]{}]{}[@Takeuchi2007; @Collins2015]. The relationship between $\vec{F}_{2\omega}$ [[(which drives $A_{2\omega}$ according to equation \[eq:direct\])]{}]{} and KPFM feedback loop [[itself]{}]{} can be seen in figure \[fig:Signals\_and\_geometry\]c: the feedback becomes unstable at locations where $A_{2\omega}$ changes sign. Moreover, any change in $A_{2\omega}$ makes KPFM susceptible to topographic cross-talk[@Barbet2014]. The [[signal]{}]{} is driven by $\vec{F}_{2\omega}$ because it reveals the behavior of the KPFM feedback loop, without requiring feedback to be used and is not susceptible to patch potentials or tip change.
The effect of slope is revealed by observing how the normalized signal ($\tilde{A}_{2\omega}$) changes as the [[tip apex]{}]{} approaches an edge of the trench at different orientations, for AM-, FM- and H-KPFM with the first three eigenmodes of each cantilever[[, and $V_{AC}$ = 3 V]{}]{}. In figure \[fig:Signals\_and\_geometry\] the trench edge is crossed with three different orientations: (i) the vector from the base of the cantilever [[beam]{}]{} to its tip [[apex]{}]{} points down the slope ($\theta_{n}>0$, from the higher to the lower level) (ii) parallel to the slope ($\hat{n}$ out of plane) and (iii) up the slope ($\theta_{n}<0$). One trend predicted by equation \[eq:mult\_freq\_A\] is observed: $\tilde{A}_{2\omega}$ tends to increase as $\theta_{n}$ increases. However, the decrease of $\tilde{A}_{2\omega}$ is greater for the short cantilever [[beam]{}]{} than for the long cantilever [[beam]{}]{}. For the short cantilever [[beam]{}]{}, the $\theta_{n}<0$ edge leads to $\tilde{A}_{2\omega} < 0$ for every technique except FM-KPFM.
Other scan parameters affect $\tilde{A}_{2\omega}$ much less. $A_{T}$, used for topography control, is varied from 10 to 40 nm, but the shape of $\tilde{A}_{2\omega}$ retains a negative portion as the $\theta_{n}<0$ edge is crossed. Similarly, using a two-pass method and varying the lift height from 2 nm to 16 nm does not prevent $\tilde{A}_{2\omega}<0$ at the $\theta_{n}<0$ edge. Thus, if KPFM feedback is unstable for geometric reasons, adjustments to the scan settings do not typically stablize it.
![ (a) A cantilever scans, using H-KPFM, across a 2 $\mu$m wide chromium-coated silicon trench (64$\times$256 pixels, 800 nm/s). (b) On the downward slope (left), the normalized signal $\tilde{A}_{2\omega}$ becomes larger, but on the upward slope (right), the signal decreases. (c) Measured and (d) predicted values of $\tilde{A}_{2\omega}$ are plotted against the local slope of the trench. Higher eigenmodes tend to show a greater change with slope, as predicted from their larger $\theta_{i}$. Bimodal AFM is also used to scan across the surface, [[while biased to 3 V]{}]{}. (e) The change in phase shows peaks at the edges, but unlike the H-KPFM case, the relative amplitude of the phase change decreases for higher eigenmodes, because the of increased $k^{\text{eff}}_{i}$. (f) The amplitude decreases in the middle of the trench, but not at the edges, and changes by $<0.5\%$. []{data-label="fig:Multiple_eigenmodes"}](Figure5.pdf){width=".44\textwidth"}
To test the predictions with a wider range of $\theta_{i}$, the trenches are scanned again with the long probe in H-KPFM mode using the first eigenmode for topography control and amplifying the $\vec{F}_{2\omega}$ signal with eigenmodes 2-7 [[(ie. $\omega_{A}=\omega_{M}/2 = (\omega_{i}-\omega_{1})/2$, so that $\omega_{D}=\omega_{i}$ for 2$\leq i\leq$7, table \[table:cantilever\_resonances\])]{}]{}. Because each eigenmode has a slightly greater $\theta_{i}$ than the one before it (ie. $\theta_{i+1}>\theta_{i}$), equation \[eq:mult\_freq\_A\] predicts that the effect of sample slope is greater for the higher eigenmodes than the lower ones, and the experiment confirms this trend, although the seventh eigenmode changes less than the sixth (figure \[fig:Multiple\_eigenmodes\]b-d). The experimental data do not all fall on a single line (figure \[fig:Multiple\_eigenmodes\]c), perhaps because the region on the sample from which the $\vec{F}_{2\omega}$ force originates deviates from the single-slope assumption. [[For eigenmodes 3-7, the data agree better with equation \[eq:mult\_freq\_A\], which has no free parameters, than with the null hypothesis that the signal does not depend on slope, thus confirming that the direction of the force affects how it drives the tip apex. However, equation \[eq:mult\_freq\_A\] tends to underestimate $\tilde{A}_{2\omega}$, particularly for slopes $<-0.5$, which suggests that other factors, such as the tip cone and changes to the piezo-driven oscillation, $A_{T}$, may also matter. An initial test of effect of slope on piezo-driven oscillation with bimodal AFM shows a change in the phase at the edges of the trench (figure \[fig:Multiple\_eigenmodes\]e,f). Because the sideband excitation technique is similar for different forces, the results here indicate that $\theta_{i}$ affects the whole class of methods.]{}]{}
The direction of the tip [[apex]{}]{} trajectory depends on cantilever geometry and the eigenmodes used, and influences [[sideband]{}]{} multifrequency force microscopy methods. It can even change the sign of the signal, which leads to feedback instability in KPFM. The results here show that considerable topographic restrictions exist for multifrequency methods when short cantilevers are used. Because short cantilevers enable faster scanning than long cantilevers[@Walters1996a], the restriction amounts to a speed limitation for any given roughness. Because the equations above separate the calculation of $\theta_{i}$ (\[eq:beam\_motion\]-\[eq:sensitivity\]) from the analysis of [[the sideband signal]{}]{} (\[eq:SidebandActuation\]-\[eq:mult\_freq\_A\]), either portion can be combined with numerical methods to account for non-rectangular cantilever [[beams]{}]{}, or non-analytic forces. Knowledge of the effect of geometry will assist in the development of [[additional]{}]{} multifrequency methods and will make the interpretation of current methods more accurate. In particular, the improved stability of KPFM will enable high resolution voltage mapping of rough or textured surfaces, which will allow for improved nanoscale characterization of optoelectronic structures such as solar cells and for the study of light induced charging effects resulting from hot carrier generation or plasmoelectric excitation of nanostructured metals[@Sheldon2014; @Tumkur2016; @Garrett2017a].
The authors acknowledge funding support from the Office of Naval Research Young Investigator Program (YIP) under Grant No. N00014-16-1-2540, and the support of the Maryland NanoCenter and its FabLab. LK acknowledges that this material is based upon work supported by the National Science Foundation Graduate Research Fellowship under DGE 1322106.
|
---
abstract: 'Controlling segregation is both a practical and a theoretical challenge. In this Letter we demonstrate a manner in which rotation-induced segregation may be controlled by altering the geometry of the rotating containers in which granular systems are housed. Using a novel drum design comprising concave and convex geometry, we explore a means by which radial size-segregation may be used to drive axial segregation, resulting in an order of magnitude increase in the axial segregation rate. This finding, and the explanations provided of its underlying mechanisms, could lead to radical new designs for a broad range of particle processing applications.'
author:
- 'S. González'
- 'C. R. K Windows-Yule'
- 'S. Luding'
- 'D. J.Parker'
- 'A.R. Thornton$^{1,}$'
bibliography:
- 'new200409.bib'
- 'granulates.bib'
- 'biblio.bib'
- 'paper.bib'
title: 'Shaping Segregation: Convexity vs. concavity'
---
[^1]
Granular flows in rotating drums are widely used to study mixing, segregation, and pattern formation [@seiden11]. While the most studies focus on a circular cylinder geometry, several recent works have explored different drum geometries [@Hill2001; @Meier2006; @MeierLueptowOttino2007; @Naji2009; @christov2010; @PrasadKhakhar2010; @Pohlman2012]. [Non-circular drums are important from both an application perspective, as they are used in various industries, and from a theoretical perspective, to further validate theoretical approaches developed primarily from simpler cylindrical configurations.]{} Segregation is known to occur in rotated granular materials. Radial segregation occurs after a few rotations [@Hill2001] and, if rotation continues for an adequately long duration, axial segregation may also appear [@zik1994]. The question arises: is it possible to control this axial segregation?
The role of geometry in segregation is well known [@Metcalfe1995], with experiments looking at a variety of convex [@KawaguchiTsutsumiTsuji2006; @Naji2009] and concave [@Cleary2003; @Morton2004; @GuptaKatterfeldSoetemanLuding2010; @PrasadKhakhar2010] drums. However, to the best of our knowledge this is the first study utilising mixed convex-concave systems. In geometry concave polygons are defined by two numbers $\{n/m\}$; $n$ is the number of sides (points) and the polygon is formed by connecting every $m^{{th}}$ point with straight lines. We chose the simplest regular concave polygon the $\{5/2\}$-star polygon, or pentagram. In this Letter, we first focus on the dynamics of mono-sized particles in simple concave drums and later explain how the novel combination of concave and convex shapes can be used to control segregation.
[*Experimental set-up.*]{} Drums of length $a = 119mm$ and width $\Delta z \in (10,24)$ mm are partially filled with glass beads of diameter $d = 3.5 \pm 0.3$ and rotated at a constant rate $\Omega = \pi/2$ rad/s. [Data is acquired from the experimental system using both optical techniques and positron emission particle tracking (PEPT), enabling the bed’s exterior and interior to be explored. PEPT is a non-intrusive technique which records the motion of a single ‘tracer’ particle in order to extrapolate of a variety of time-averaged quantities pertaining to the system as a whole. Although not necessary to the understanding of this Letter, for the interested reader, a comprehensive overview of the PEPT technique may be found in our references [@parker2002positron; @wildmanSingle].]{}
In simulations, experimental system dimensions and particle properties are used, with particles’ contact forces represented using a standard spring dash-pot model [@cundall79; @Luding2008a; @Luding2008b] as implemented in [@thornton2012]. The drum rotation is achieved by changing the direction of gravity at fixed $\Omega = \pi/2$ rad/s in order to consistently provide a continuous free-surface avalanche [@Mellmann2001]. [Simulations are conducted with both periodic boundary conditions in the axial direction, as well as with solid side-walls allowing both direct comparison with experimental results in the case of the convex-concave drum, and investigation of the effect of drum geometry on flow for purely convex or concave systems using periodic walls. ]{}
[*The effect of the filling fraction.*]{} We observe four differing flow regimes depending on $F$. If grains occupy a volume smaller than one leg of the pentagram, they flow intermittently from leg to leg. If the $F$ increases such that there are always grains in at least two legs, flow is constant but its angle changes continuously. When two to four legs are filled, flow is continuous but with two qualitatively different flow profiles depending on the drum’s angle. Once grains occupy more than four legs, flow becomes intermittent and grain displacement is strongly limited, decreasing transport in the bulk, with dynamics mostly due to geometrical rearrangements. We focus on the regime $40\% \leq F \leq 60\%$, where unsteady flow is produced by a concave drum rotating at a constant rate. The geometric shape naturally causes periodic changes in the flowing layer as a function of the instantaneous orientation of the pentagram, as recently reported in other geometries [@Pohlman2012].
[*Comparison of Pentagram and Pentagon.*]{} In cylindrical and general convex drums, steady flow has a roughly constant kinetic energy, $E$, independent of the angle of rotation. For the pentagram, however, this oscillates strongly: when a pentagram points up, flow is slow while when pointing down, flow is much faster. Fig. \[fig2\] shows velocity fields for both pentagram (a, b) and pentagon (c, d). The pentagram shows great variation in the magnitude of $v$ between the up (b) and down configuration (a): when the pentagram points down, the avalanche occurs in a thick layer. As the drum rotates, more space becomes available, producing a saltating flow. This creates a fast avalanche in the down part of the flow, and the consequent movement of all the flowing layer. Thus, $E$ shows five maxima during a cycle (Fig. \[fig3\]). By allowing particles more space to flow, a large, fast, avalanche is produced. This avalanche is not symmetric along the free-surface. Most of the kinetic energy is on the downside, where the free volume makes it easier to flow. Eventually, the leg is filled with particles and the avalanche recovers its slow flow, before the process repeats. We now focus on how this feature can be used to control segregation. To do this, one must introduce the pentagram’s convex counterpart, the pentagon. As the pentagon rotates, the total length of the flowing layer changes, creating an oscillation in $E$ with the same period as for the pentagram (see Fig. \[fig3\]). However, this flow, and its velocity, are much more consistent in the pentagon, with a smaller variation between minimum and maximum.
The periodic structure of the $E$ can be understood by simple arguments. If one considers the speed of the flowing layer and its depth constant as much smaller than the filling height, $H$, then $E$ is proportional to the length, $L$, of the flowing layer [@Pohlman2012]. Disregarding the angle of the walls, and assuming a straight free-surface, $L$ scales approximately as $L \propto 1/\cos(\theta)$, with $\theta \in [0,2\pi/5]$ the angle of rotation modulo the shape’s symmetry, in this case $2\pi/5$. Hence, $E\propto 1/\cos(\theta)^2$. The agreement of simulations with this simple model is remarkable (see Fig. \[fig3\]), [although deviations from this simple sinusoidal form arise for the concave drum. These deviations are not surprising, as while in a pentagon the flowing layer is at the edge of the geometric region of constant volume and this region is always connected, giving a relatively consistent filling fraction, $F$, the same does not hold for pentagrams and other concave shapes. Consequently, both of the above assumptions are likely to be broken for such geometries. Specifically, one observes the presence of local maxima preceding each of the main peaks in kinetic energy. It is also notable that the initial maximum in $E$ is markedly higher than the following peaks. The former of these deviations can be explained by the fact that particles in the lower region of the surface flow avalanche first over the lowermost leg, before being followed by grains in the middle and upper regions, thus leading to the observed ‘two-part’ increase in $E$ [^2]. The latter, meanwhile, can be explained by the initial presence of localised jamming within the system, whereby a collection of particles in a jammed state are able to reach a higher point in the system before avalanching, naturally resulting in a higher-than-average kinetic energy. It is finally worth noting that the existence of side-walls acts to frustrate the observed local maxima, while the general sinusoidal evolution of $E$ is found to persist - i.e. the concave system approaches more closely the theoretical form.]{}
[*Shape-induced axial segregation.*]{} Although the influence of container shape on segregation has already been reported [@Mao-Bin2005; @hu2005] this is the first time that it is used in a rotating drum. It is also known that modifying the geometry (e.g. adding obstacles or mixing blades) can *reduce* segregation [@Shi2007] but it has not been shown how to *augment* and *control* it – a matter of obvious practical importance. For bi-disperse granulates in any rotating container, small particles will migrate towards the drum’s centre [@Mounty2007; @seiden11] (see Fig. \[fig4\] (a) and (b)). For adequately wide systems, upon continuous rotation the system will segregate axially [@zik1994], a process orders of magnitude slower than radial segregation. However, if two different geometries are used along the axial direction, e.g. a half-pentagram, half-pentagon drum, the usually slow segregation along this axis can be enhanced and its direction controlled (see Fig. \[fig4\] (c) and (d)). In both experiment and simulation, two sections of equal width are combined. We use particles of $d = 4.0, 2.5$ mm, in an equal volume distribution. The rapid axial segregation happens only with a convex-concave combination, as for convex shapes there exists little difference in the level of the flow, just the length of avalanching layer [@ArntzOttherBeeftinkBoomBriels2013]. We performed several experiments, putting together circular and square sections, pentagonal and square, and differently oriented square sections. None of these configurations presented axial segregation on the time scale of observation ($\sim 20$ revolutions). [A clearer representation of the segregation – both axial and radial – achieved in the convex-concave system described above may be seen in Fig. \[peptSeg\]. It should be noted that, due to the somewhat constrained nature of the system under investigation, the degree of axial segregation observed is likely to be *lower* than it would be in comparatively longer drums, due to the restricted motion of particles in the axial direction. Thus, the significant segregation observed even in these unfavourable conditions is a highly pleasing result, as one may well expect larger systems to provide still greater separation]{}.
Grains tend to minimise their potential energy, i.e. move towards the concave side, which also possesses more free-volume. Note that the few large grains in the run-out leg of Fig. \[fig4\] (c) will eventually fall to the pentagonal side. Radial segregation occurs in each side, so large particles go to the surface and small to the centre. Since for this packing fraction the avalanche in the pentagonal side of the drum ($z<0$) is slower than in the pentagram-shaped section ($z>0$), large particles can move to the empty side since they are faster and there is space available for them. Once the two avalanches reach the same angle there is no more flux of particles. This process is repeated five times per revolution (see Fig. \[fig:CMz\] (a)). When the big particles drop from the run-out leg of the pentagram to the pentagon, the centre of mass of the large particles shifts towards the pentagon. However, the process is not completely irreversible; some, but fewer, large particles go again to the pentagram side as the drum rotates. In this way, an oscillating movement of the centre of mass of each species is observed: big particles fall to the pentagonal side when the run-out leg is empty; once the flow covers the run-out leg some large particles return to the pentagram side. By this mechanism, there is a net transport of large particles to the convex side of the drum, while the concave side becomes dominated by small particles. Since this mechanism relies on the fast radial segregation, it is orders of magnitude faster than the axial segregation previously reported for axially homogeneous drums [@zik1994].
[Experimental evidence of the mechanism proposed above may be seen in Fig. \[peptV\]. From these images it is clear that, as expected, there exists a difference in the level and angle of inclination of the bed’s surface between the convex and concave sides of the drum. Moreover, the velocity fields in panels (a) and (b) demonstrate the avalanching region on the bed’s concave side to be both faster and deeper than for the convex half, again in agreement with our hypothesis. Panels (c) and (d) show two-dimensional, depth averaged velocity fields for the $z$-$y$ plane, where $y$ denotes a vertical axis perpendicular to the axial ($z$) axis. In image (d), which corresponds to the time-averaged motion of the large particles in the system, we see evidence of the recirculatory transport discussed above, and whose effect on the system’s mass centre is shown in Fig. \[fig:CMz\] (a). It is interesting to note that such motion is seemingly absent for the small particles in the same system (panel (c)).]{}
Finally, it must be noted that the final degree of segregation is not the equal every $F$. Fig. \[fig:CMz\] (b) shows the change in the number of large particles in the pentagonal side of the drum for different filling fractions. If $F$ too low, the avalanche on the pentagram side of the drum arrives concurrently with the one in the pentagon and axial segregation is slower. One could argue, that excluded volume effects make the small particles go preferably to the pentagram side since the big particles do not fit into the legs so easily, as reported in [@hu2005]. However, this mechanism alone does not explain the maximum in segregation at $\sim 50\%$ filling fraction. This can only be due to the differential flows along the axial direction and the consequent conversion of radial to axial segregation previously discussed.
[*Conclusion.*]{} In this letter we have studied granular flows inside the simplest possible regular concave drum, that is, the pentagram-drum. Different regimes are found for a fixed angular velocity depending on the filling fraction. From intermittent avalanching (low filling fraction) to geometrical rearrangements (high filling) passing by continuous flow (intermediate filling). These flow patterns differ qualitatively from those observed in convex drums. We have used this insight to control the segregation of a binary granular system, achieving fast geometrically-induced size separation along the axial direction, by re-directing radial segregation into axial segregation. The possible applications of this mechanism are potentially revolutionary. Firstly, one may produce axial segregation orders of magnitude faster than previously reported. Secondly, the direction and rate of segregation can be controlled. Finally, it makes us reconsider the role of boundary conditions when dealing with granular materials; this is the first step in shaping segregation at our will.
The practical importance of this discovery can be far-reaching in industries ranging from pharmaceuticals to mining. We foresee several applications for our discovery including, but by no means limited to, rotating kilns – allowing differential residence times depending on the size of the particles – and milling devices – whereby creating a sandwich of concave sections with a convex shape in the middle, large particles can be conducted into the middle of the mill, thus increasing efficiency by keeping the grinders and larger particles in the mill while moving the fines to the ends, where they could be removed. [ This study also provides great scope for future work in the extension, refinement and practical application of the findings presented here.]{}
We thank W. Zweers for the construction of the experimental set-up; V. Ogarko for generating the bi-disperse packings; W. den Otter and N. Rivas for the critical reading of the manuscript. The simulations performed for this paper are undertaken with [MercuryDPM.org](MercuryDPM.org); primarily developed by T. Weinhart, A. R. Thornton and D. Krijgsman at the University of Twente. This study was supported by the Stichting voor Fundamenteel Onderzoek der Materie (FOM), financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO), through the FOM project 07PGM27.
[^1]:
[^2]: See the supplementary video available at
|
---
abstract: |
We have performed 3-D numerical magnetohydrodynamic (MHD) jet experiments to study the instabilities associated with strongly toroidal magnetic fields. In contemporary jet theories, a highly wound up magnetic field is a crucial ingredient for collimation of the flow. If such magnetic configurations are as unstable as found in the laboratory and by analytical estimates, our understanding of MHD jet driving and collimation has to be revised.
A perfectly conducting Keplerian disc with fixed density, rotational velocity and pressure is used as a lower boundary for the jet. Initially, the corona above the disc is at rest, permeated by a uniform magnetic field, and is in hydrostatic equilibrium in a softened gravitational field from a point mass. The mass ejection from the disc is subsequently allowed to evolve according to deviations from the initial pressure equilibrium between disc and corona.
The energy equation is solved, with the inclusion of self-consistently computed heating by viscous and magnetic dissipation. We find that magnetic dissipation may have profound effects on the jet flow as: 1) it turns on in highly wound up magnetic field regions and helps to prevent critical kink situations; 2) it influences jet dynamics by re-organizing the magnetic field structure and increasing thermal pressure in the jet; and 3) it influences mass loading by increasing temperature and pressure at the base of the jet.
The resulting jets evolve into time-dependent, non-axisymmetric configurations, but we find only minor disruption of the jets by e.g. the kink instability.
author:
- 'F. Thomsen'
- '[Å]{}. Nordlund'
bibliography:
- 'aamnem99.bib'
- 'aa.bib'
date: 'Received date, accepted date'
nocite: '[@Michel; @Kudoh]'
title: Stability and heating of magnetically driven jets from Keplerian accretion discs
---
Introduction
============
Though astrophysical jets have been observed in a rather wide range of accreting systems, it is generally assumed that the mechanism for acceleration and collimation is generic [@Livio; @spruit00 for recent reviews]. Contemporary jet theories rely on magnetic forces as the jet producing mechanism, either with the flow emanating from the disc surface and associated with an open magnetic field structure [@BP82] or with the flow emanating from the disc-star interface and associated with a closed field structure connecting disc and central star [@pringle]. In the former so-called disc-wind scenario, which is currently receiving most attention, the driving force, acting to overcome the gravitational pull of the central object, is typically regarded as a component of the centrifugal force along the magnetic field. However, adopting an inertial frame of reference the driving mechanism may equally be interpreted as magnetic [@spruit96]. The inertia of the outflowing gas eventually becomes dynamically important and the gas stops corotating with the underlying disc. At this point the magnetic field, which cannot easily slip through the highly ionized gas, is wound up in a cork screw manner between the disc and the vertically accelerated and less quickly rotating outflow. The tension (“hoop stress”) of the wound-up magnetic field configuration is generally believed to be the collimating force, but collimation by a poloidal field has been proposed as well [@Spruit] motivated by instability arguments against the wound-up field scenario.
The assumed large scale magnetic field in the disc-wind scenario may either be provided by dynamo processes in the disc [@Brandenburg2000; @Brandenburg+2000] or captured from the environment and advected inwards by the accreting matter [@CaoSpruit]. Numerical work has been carried out to investigate situations in which a significant fraction of the field lines loop back into the disc [@Romanovab; @TBR]. These experiments have focused on the ejection and acceleration mechanisms as well as the evolution of the magnetosphere close to the disc, and have not followed the flow further out for extended times.
The numerical work done in relation to the “classical” large scale open field scenario may be divided into experiments attempting to include (parts of) the accretion disc in the computational domain [@US85; @BL95] and more recent work [@Ustyugova; @OPS; @Meier] which has used the disc only as a fixed lower boundary to avoid problems with radial collapse of the disc and thereby relatively short time evolution of the experiment. Both types of experiments have been axisymmetric and as such have not questioned the potentially (kink) unstable wound up magnetic field configuration on which the collimation process relies. However, Spruit et al. [-@spruit+00] have recently proposed fast reconnection processes in more disordered non-axisymmetric large scale magnetic fields in the jet like gamma ray burst (GRB) scenario. The magnetic dissipation is proposed for the GRB fireballs e.g. to produce the observed radiation with better efficiency. We find that such magnetic processes may have severe impact on jet dynamics and stability in particular.
The main purpose of the work presented here is to establish whether or not the jets in the disc-wind scenario are prone to catastrophic MHD instabilities. This calls for an implementation of a setup in three dimensions which will be described in Sect. \[sec:model\]. For simplicity, we assume a large-scale open magnetic field structure and use the disc as a fixed boundary. The details of the initial conditions and boundary conditions are found in Sect. \[sec:ic\] and \[sec:bc\] respectively. Results are presented in Sect. \[sec:results\], with Sect. \[sec:dynamics\] concentrating on the jet dynamics and Sect. \[sec:stability\] on the observed 3-D jet stability in the experiments. In Sect. \[sec:discussion\] we discuss the results with special emphasis on thermal properties and magnetic field structure. Finally, the paper is summarized and conclusions are presented in Sect. \[sec:conclusion\].
Model {#sec:model}
=====
The jet flow is described numerically by solving the MHD equations in the following form; $$\frac{\partial \rho}{\partial t} = - \nabla \cdot \rho \mathbf{u},$$ $$\frac{\partial(\rho \mathbf{u})}{\partial t} =
- \nabla \cdot \left( \rho \mathbf{uu} - \tau^\prime \right) + \nabla P +
\mathbf{J \times B} - \rho \nabla \Phi,$$ $$\mu \mathbf{J} = \nabla \times \mathbf{B},$$ $$\mathbf{E} = \eta \mathbf{J} - \mathbf{u \times B},$$ $$\frac{\partial \mathbf{B}}{\partial t} = - \nabla \times \mathbf{E},$$ $$\frac{\partial e}{\partial t} =
- \nabla \cdot e \mathbf{u} - P \nabla \cdot \mathbf{u} + Q,$$ where $\rho$ is the mass density, $\mathbf{u}$ the velocity vector, $\tau^\prime$ the viscous stress tensor, $P$ the thermal gas pressure, $\Phi$ the gravitational potential, $\mathbf{B}$ the magnetic flux density vector, $\mathbf{J}$ the electric current density vector, $\mathbf{E}$ the intensity of the electric field, $\eta$ the electric resistivity, $e$ the internal energy per unit volume and $Q$ the sum of viscous and Joule dissipation.
Dimensions and numerics
-----------------------
-------------- ----------------------------------- ----------------------------------------- -----------------------------------------
Velocity $\sqrt{ GM/\lambda }$ $138\,\mathrm{km/s}$ $0.2\, c$
Time $\sqrt{\lambda^3/(GM)}$ $0.58\, \mathrm{days}$ $0.51\, \mathrm{days}$
Mass density $\mathcal{B}^2\lambda/(\mu_0 GM)$ $4.2 \times 10^{-14}\, \mathrm{g/cm^3}$ $1.8 \times 10^{-17}\, \mathrm{g/cm^3}$
-------------- ----------------------------------- ----------------------------------------- -----------------------------------------
Both for numerical reasons and to be able to adapt the calculations to stellar as well as galactic scales, the physical quantities are given in units of characteristic quantities of the system. The form of the equations are not changed by converting to this system of units. The appropriate units may be constructed from the mass of the central object, $M$, the magnetic flux density, $\mathcal{B}$, and the length scale, $\lambda$. The characteristic length scale $\lambda$ is assumed to correspond to the inner disc radius. If not otherwise stated, the dimensionless quantities listed in Table \[tab:units\] are used in the following sections.
The code uses a sixth order accurate method for partial derivatives and a fifth order accurate interpolation method with the variables represented in non-uniform (Eulerian) staggered meshes [@code; @Rognvaldsson99]. In the discrete representation of quantities, the scalar variables ($e$ and $\rho$) are zone centered and components of the vector variables ($\mathbf{p}
= \rho \mathbf{u}$ and $\mathbf{B}$) are face centered in a unit mesh volume. The solution for the eight variables, $\rho$, $e$, $p_x$, $p_y$, $p_z$, $B_x$, $B_y$, $B_z$ is advanced in time by a third order predictor-corrector procedure [@hyman], modified to accommodate variable time steps. The code has previously been verified by a number of standard test problems and henceforth been used for experiments involving 3-D turbulence [@Nordlund+94a], investigations of problems related to coronal heating [@Nordlund+Galsgaard96spm], buoyant rise of magnetic flux tubes [@Dorch+Nordlund97buoy], dynamo experiments [@Dorch+99a], stellar convection [@Nordlund+Stein99budapest] and magnetized cooling flows [@Rognvaldsson+00a].
Cartesian coordinates ($x, y, z,$) are used to describe the disc-corona system in the computational box. To calculate the derivative or the interpolated value at a given point, the six nearest grid points are involved. As exactly the same operators are used in the whole grid, three layers of *ghost zones* must be added to prevent periodic wrapping of the computational domain. For (obsolete) reasons of computational efficiency, we use $x$ (first index) as the vertical direction and place ghost zones in the index range $i \in [1:3] \wedge [m_x-3:m_x]$. The $y$- and $z$-directions are taken to be periodic. The ($y,z$) cross section of the computational domain is quadratic, both in terms of number of grid points ($m_y = m_z$) and physical size ($L_y =
L_z$), and centered so that $y \in [-L_y/2,L_y/2]$ and $z \in [-L_z/2,L_z/2]$. Only odd numbers of $m_y$ and $m_z$ have been chosen, making $(x,y,z)=(0,0,0)$ at zone center $(i,j,k)=(4,(m_y + 1)/2, (m_z+1)/2)$.
Initial conditions {#sec:ic}
------------------
To investigate the magnetic driving and collimation of outflows it is desirable to choose an initial state in which all other dynamical effects are small. Therefore, to obtain such a state, the corona is assumed at rest and in hydrostatic equilibrium in the gravitational field of a point mass [@OP97b]. Self-gravity of the gas is neglected. A current free configuration is chosen so that the Lorentz force vanishes. In this way, the momentum equation for the corona reduces to $$\nabla P = - \rho \nabla \Phi.$$ By the use of a polytropic equation of state, $P = K \rho^\gamma$, the reduced momentum equation may be integrated, giving the density distribution for the corona, $$\rho = \left[ (\Phi_0 - \Phi) \frac{\gamma-1}{K \gamma} \right]^{\frac
{1}{\gamma - 1}}, \quad \gamma \neq 1.
\label{eq:density}$$ A positive value of the integration constant, $\Phi_0 \geq 0$, may be introduced to keep finite densities at large distances. The value of the polytropic exponent is chosen to be $\gamma = 5/3$, corresponding to an adiabatic mono-atomic ideal gas.
For numerical reasons, the gravitational potential is smoothed by introducing a so-called softening parameter, $l_\mathrm{s}$, $$\Phi = - \frac{1}{\sqrt{x^2 + y^2 + z^2 + l_\mathrm{s}^2}}.
\label{eq:Phi}$$ Due to the location of the gravitational potential in the staggered meshes, a non-zero softening parameter is necessary to make the potential well-defined at $x = y = z = 0$. Furthermore, a softened potential makes the potential gradient and the initial density gradient less steep close to the central object and therefore easier to handle by the numerical code. The softening parameter was chosen to be $l_\mathrm{s} =
\sqrt{1/2}$, for the experiments actually reported herein.
The disc surface (lower boundary) is assumed to have a density distribution described by Eq. \[eq:density\], and the corona is in pressure balance with the underlying disc. In the case of negligible thermal pressure, the softened potential gives rise to an equilibrium Keplerian like velocity structure in the disc given by $$\begin{aligned}
u_y &=& \frac{z}{(y^2 + z^2 + l_\mathrm{s}^2)^{3/4}}, \nonumber \\
u_z &=& - \frac{y}{(y^2 + z^2 + l_\mathrm{s}^2)^{3/4}}.\end{aligned}$$ Numerical experiments relying on a smoothed or softened potential produce generally weaker jets [@BL95]. Due to the reduced rotation velocity obtained when softening is introduced a longer time scale for the build up of the azimuthal field (and the jet flow) is anticipated. Devising identical experiments and only varying softening, we find no significant effect in jet features but only a scaling of dynamical quantities. The main effect of softening is an increase of the rotational period which scales the characteristic time and the specific kinetic energy (cf. Fig. \[fig:discvel\]).
In the chosen numerical setup, a density jump at the disc-corona interface must be applied with some caution, as the interpolation of the zone centered values could artificially bring mass into the corona. A density jump may however be implemented to have effect on the energy flux as is demonstrated in Sect. \[sec:bc\] below. The pressure balance between disc and corona together with the assumed disc density structure expressed by Eq. \[eq:density\] implies that a density jump may equivalently be regarded as a jump in internal energy per unit mass, $T \sim P/\rho$. Hence, the jump $\xi \equiv (T_\mathrm{corona}/T_\mathrm{disc})_0$ at the disc-corona interface expresses a jump in (isothermal) sound speed as well; $c_\mathrm{s,disc}^2 = c_\mathrm{s,corona}^2 / \xi$. For $c^2_\mathrm{s,disc} \ll (u^2_y + u^2_z)$, the radial pressure gradient in the disc, $\partial p / \partial r \sim (\rho c_\mathrm{s}^2)_\mathrm{disc}
/ R$, becomes negligible compared to the centrifugal force and the pressure gradient does not influence the radial balance of the disc. Anyway, since the disc is not simulated, the disc structure will not be of a numerical concern.
A constant vertical magnetic field, $B_x = B_0$, is chosen to penetrate both disc and corona initially. In a more realistic case, one may expect the field to be inclined with respect to the disc surface and the field strength is likely to increase towards the center due to advection by the accreting matter [@CaoSpruit]. Because of the periodic boundaries, such a magnetic field configuration would not be divergence free in the present setup. Furthermore, a field configuration facilitating wind ejection is expected to be generated automatically by the rotation of the disc [@OP97b]. Therefore, only the simple constant vertical magnetic field configuration has been implemented as initial condition.
The magnetic flux density is specified by the parameter $\beta_\mathrm{i}
\equiv P_\mathrm{gas,i}/P_\mathrm{mag,i}$, at $r_\mathrm{i}=1$. In the numerical experiment, the density distribution is determined from Eq. \[eq:density\] with $K = (\gamma -1)/\gamma$. Using $P_\mathrm{gas} = K \rho^\gamma$ and $P_\mathrm{mag} = B_0^2/2$ it follows that $$B_0 = \sqrt{ \frac{2K}{\beta_\mathrm{i}}
\left( \Phi_0 + \frac{1}{\sqrt{1+l_\mathrm{s}^2}}
\right)^{\frac{\gamma}{\gamma - 1}}}.
\label{eq:B0}$$
Boundary conditions {#sec:bc}
-------------------
The disc boundary is located in the $x=0$ plane and is assumed to be a fixed base for the jet. As such, the rotational velocity, mass density and internal energy density is kept constant at their initially prescribed values. Jet fluxes of mass, energy and momentum are assumed to have negligible impact on the disc surface layer. For instance, the disc is assumed to supply mass to the surface at the same rate as the jet carries mass away. This is implemented as a *steady* condition where the mass flux $p_x$ is symmetric at the lower boundary.[^1] The energy flux at the disc surface has been implemented as either a zero x-derivative condition or a cold condition. The cold condition is applied by assuming a jump in internal energy per unit mass (temperature) between disc and corona. The internal energy density in the disc ghost zones are found by antisymmetrizing the $x$-face centered values of $T \equiv e/\rho$ around $(e_0/\rho_0)/\xi$; $$T[b-i+1/2] = 2 (e_0/\rho_0)/\xi - T[b+i-1/2],$$ with $b=4$ and $i=1,2,3,4$ so that the values in square brackets are the grid locations. The fixed internal energy density and mass density at the disc surface is denoted $e_0$ and $\rho_0$ respectively. The temperature jump is specified by the ratio $\xi=(T_\mathrm{corona}/T_\mathrm{disc})_0$ evaluated at the disc-corona interface. In effect, a cold inflow condition is specified by the internal energy density flux, $h_x \equiv p_x T$.
The inflow velocity is not specified, but allowed to evolve freely according to the forces acting on the disc surface. However, the $x$-flux of $x$-momentum (the $xx$-component of the Reynolds stress tensor) is not allowed to change the outflow velocity on the surface. Instead, thermal and magnetic forces are taken to be the dominant mechanisms for injecting gas into the corona. The pressure in the disc (ghost zones) and on the boundary is kept fixed at the values prescribing the initial pressure balance. Any pressure gradient at the disc surface is caused solely by deviations from the initial hydrostatic equilibrium occurring in the corona.
For the evolution of the magnetic field the disc is taken to be perfectly conducting. In general, the magnetic field is only specified initially and is subsequently evolving according to the electric field. Keeping this in mind when imposing boundary conditions, violation of the divergence free condition of the magnetic field may be prevented in a natural way. Particularly, one finds that the disc velocities determine the electric field and thereby the evolution of the magnetic field at the lower boundary. The electric field on the disc surface is specified as $$\begin{aligned}
E_y & = & - u_z B_0, \nonumber \\ E_z & = & u_y B_0,\end{aligned}$$ where $B_0$ is the constant vertical field specified initially by Eq. \[eq:B0\]. The issues of boundary driving is discussed further in Sect. \[sec:bd\].
At the upper boundary, an extrapolation of the electric field would not be stable and another approach has to be applied which is discussed in Sect. \[sec:qtb\]. For the mass, momentum and energy density fluxes, a simple extrapolation into the ghost zones may be applied to allow the densities to evolve on the boundary.
### Boundary driving {#sec:bd}
At the lower boundary the disc is kept rotating with a fixed angular velocity. This should in principle be sufficient to determine the evolution of the $y$- and $z$-components of the magnetic field. It turns out, though, that the velocity driving has to be specified in a somewhat non-local manner. This is because of the non-local nature of the numerical differential operators used in the code. The numerical scheme involves the three nearest points on each side along the direction of differentiation. Therefore, to avoid ripples when driving, the driving has to be passed smoothly to the active grid. To be specific, the velocity field just inside the active grid is determined by passing a third order polynomial through the boundary and the two neighboring zones further inside the grid, $$u_i[5] = \frac{4}{9} u_i[4] +
\frac{6}{9} u_i[6] - \frac{1}{9} u_i[7], \quad i=y,z,$$ where the lower boundary is located at zone index 4. The above chosen polynomial representation has been found in other but similar experiments to give the smoothest driving [@code]. It has not been subjected to tests in the present work.
Care has to be taken also to avoid ripples caused by the periodicity of the $y$- and $z$-boundaries and by the shape of the box. To avoid shear at the box sides and reversed vorticity at the box corners, the disc velocity profile has to be terminated inside the periodic boundaries. As above, the non-local nature of the differential operators demands the decrease of disc rotation to span a few grid points. For the same reason, the velocity cutoff has to take place at some distance from the boundaries to ensure a region of practically zero rotation. To satisfy these conditions, a hyperbolic tangent function is used to cut off the velocity; $$f_i(r) = \frac{1}{2}\left[ \tanh \left( \frac{L_i}{2}-s_i-r \right) +1 \right],
\quad i=y,z.
\label{eq:velcut}$$ Here $L_i$ is the size of the box in the $i$ direction and $s_i$ is the distance that the midpoint of the hyperbolic cutoff is shifted from the boundary. The resulting velocity profile, $u_{i,\mathrm{cut}}=u_i f_i$, is shown in Fig. \[fig:discvel\] for $L_y = L_z = 30$ and $s_y = s_z =
2.5$ in a slice along $y=0$ and $z \ge 0$.
### The quasi-transmitting boundary {#sec:qtb}
A symmetric condition on the electric field does not allow tangential components of the magnetic field to evolve on the boundary. In particular, when the initial winding of the vertical magnetic field reaches the upper boundary as a toroidal Alfvén wave, it would be reflected. To minimize reflection (reversal of the toroidal magnetic field) and possible disruptive and/or decelerating effects on the flow, an upstream sensing of changes in the magnetic field is used to guide the evolution of the boundary field. This may appear somewhat artificial, but it corresponds roughly to apply values along an outgoing characteristic of a linear torsional Alfvén wave. Such a wave propagates with the Alfvén speed $u_\mathrm{A} = B_x/\sqrt{\rho}$. As a fair estimate, changes in the tangential field are taken to propagate with the (fixed) Alfvén speed determined by $$u_{\mathrm{A},x} = \frac{B_0}{\sqrt{(\Phi_0 - \Phi)^{\frac{1}{\gamma - 1}}}},
\label{eq:uadv}$$ evaluated at the upper boundary.
The tangential magnetic field on the boundary is assumed to evolve towards some upstream value at a distance $\Delta x_\mathrm{up}$, below the boundary. The travel time is estimated from the Alfvén speed; $t_\mathrm{up} =
\Delta x_\mathrm{up} / u_{\mathrm{A},x}$. The condition implemented on the electric field specifies the evolution of the tangential magnetic field as $$\frac {\partial B_i}{\partial t} =
\frac {\Delta B_{i,\mathrm{up}}}{t_\mathrm{up}}, \quad
i=y,z.$$ The work done on the field may be controlled by the parameter $\chi$, which specifies the convergence values for the magnetic field at the upper boundary; $$\Delta B_{i,\mathrm{up}} = B_i[m_x-4] - \chi B_i[m_x-4-n_x], \quad i = y, z.
\label{eq:Brise}$$ Here, $n_x$ specifies the index width of the sensing distance. Choosing $\chi \lesssim 1$ mimics in a simple way the work done on the magnetic field by the part of the jet outside the computational box.
It must be emphasized, that the $\chi$ parameter is to be regarded as a real physical parameter expressing to what extent the exterior is “braking” the field rotation. If, for example, the field is anchored to massive regions in the ambient medium outside the computational domain, the braking will cause the field to become inclined with respect to the outer boundary. Hence, the choice of $\chi$ is not only a question of numerics, but reflects a real physical degree of freedom.
Note on approach
----------------
The model philosophy adopted here is different from previous work, particularly in the applied boundary conditions. Instead of using large efforts on designing (artificially) open outer boundary conditions, periodic boundaries have been applied. The argument being, that it is inconceivable that real jets would not be perturbed by motions in the surrounding medium anyway. In the present work, the outer boundaries as well as numerical noise provide the perturbations triggering the possible instabilities we wish to investigate. Obviously, this approach cannot provide precise determinations of e.g. growth rates, but it is used to allow instabilities to develop in a natural way and provide indications on whether our understanding of jet collimation needs to be revised or not.
Results {#sec:results}
=======
The numerical experiments all show the same overall evolutionary sequence which may be summarized as follows:
1. Winding and opening of the magnetic field lines.
2. Relaxation of initial magnetic acceleration and spurious magnetic reflections.
3. Build up of collimated (but unsteady) outflow.
Initially, the winding of the magnetic field propagates through the computational box as a torsional Alfvén wave. The winding propagates with velocities expected to vary with distance approximately as Eq. \[eq:uadv\]. Accordingly the winding propagates faster at large radii as noted also by Ouyed and Pudritz [-@OP97b].
The upper boundary condition on the magnetic field allows (partially) for the Alfvén pulse to be transmitted. Some reflections are unavoidable, but eventually the reflections are transmitted through the boundaries and the system settles down. The relaxation of the minor Alfvén pulse reflections concludes the two initial phases of the numerical experiments. At which evolutionary stage a (significant) collimated outflow is initiated in a real accreting system can only be speculated upon. In the present type of jet experiment, the initial phases do not reveal much relevant physics and the initial conditions as well as boundary conditions are probably not well suited for such investigations. However, the setup eventually provides a collimated outflow with the expected qualities that we want to investigate.
A set of parameters have been introduced to truncate the physical problem and adopt it into a numerical setup. To probe the effect of these parameters, such as softening, box size and velocity cutoff, a range of experiments were carried out. The truncation of the problem and the effect of the related parameters have been presented above. In what follows, we will concentrate on a small number of experiments which have been conducted particularly to investigate stability issues. These experiments are listed in Table \[tab:experiments\].
run $ m_x \times m_y \times m_z$ $ L_x \times L_y \times L_z$ $\beta_\mathrm{i}$ $\xi$ $t_\mathrm{max}$
-------------------------- ------------------------------ ------------------------------ -------------------- ------- ------------------
\[run:largejet1\] [**]{} $98\times71\times71$ $40\times30\times30$ 5 1 500
\[run:largejet2\] [**]{} $98\times71\times71$ $40\times30\times30$ 10 1 500
\[run:largejet4\] [**]{} $98\times71\times71$ $40\times30\times30$ 5 100 500
\[run:hugejet1\] [**]{} $128\times85\times85$ $60\times40\times40$ 10 1 400
\[run:hugejet2\] [**]{} $200\times101\times101$ $200\times25\times25$ 10 1 260
: []{data-label="tab:experiments"}
Jet dynamics {#sec:dynamics}
------------
By examining the rates of change of energies, the action of various forces and issues of stationarity may be analyzed. The rate of change of total energy in a volume of gas is controlled by the rate of energy transport in and out of the volume, i.e. the net flux, $\Delta F=F(x_\mathrm{lower})-F(x_\mathrm{upper})$, the rate of energy conversion through work, $W$, and dissipation, $Q$. Consequently, the rate of change of magnetic, $\mathcal{M}$, kinetic, $\mathcal{K}$, thermal, $\mathcal{T}$, and gravitational energy, $\mathcal{G}$, in a volume may be expressed as follows, $$\frac{\partial \mathcal{M}}{\partial t} =
\Delta F_\mathrm{mag} + W_\mathrm{Lorentz} - Q_\mathrm{Joule},$$ $$\frac{\partial \mathcal{K}}{\partial t} =
\Delta F_\mathrm{kin} + \Delta F_\mathrm{visc} +
W_\mathrm{grav} - W_\mathrm{gas} - W_\mathrm{Lorentz}
- Q_\mathrm{visc},$$ $$\frac{\partial \mathcal{T}}{\partial t} =
\Delta F_\mathrm{enth} + W_\mathrm{gas} + Q_\mathrm{visc} + Q_\mathrm{Joule},$$ $$\frac{\partial \mathcal{G}}{\partial t} =
\Delta F_\mathrm{grav} - W_\mathrm{grav}.$$ The rate of total work done by gravity in the box, $W_\mathrm{grav}$, the rate of work done against gas pressure gradients, $W_\mathrm{gas}$, and the rate of work done against the Lorentz force, $W_\mathrm{Lorentz}$, are given by $$W_\mathrm{grav} = - \int_V \mathbf{u} \cdot \rho \nabla \Phi \, dV,$$ $$W_\mathrm{gas} = \int_V \mathbf{u} \cdot \nabla P \, dV,$$ $$W_\mathrm{Lorentz} = - \int_V \mathbf{u} \cdot \left(
\mathbf{J \times B} \right) \, dV.$$ Here $dV \equiv dx\,dy\,dz$ is an element of the volume, $V = [x_\mathrm{lower},
x_\mathrm{upper}] \times [-L_y/2,L_y/2] \times [-L_z/2,L_z/2]$ . The energy fluxes through a cross section, $x$, are given by $$F_\mathrm{mag}(x) =
\int_S (E_yB_z - E_zB_y) \, dS_x, \label{eq:flux1}$$ $$F_\mathrm{kin}(x) =
\int_S u_x \left( \frac{1}{2} \rho u^2 \right) \, dS_x,$$ $$F_\mathrm{enth}(x) =
\int_S \rho u_x \gamma e \, dS_x,$$ $$F_\mathrm{grav}(x) =
\int_S \rho u_x \Phi \, dS_x, \label{eq:flux2}$$ where $dS_x \equiv dy \, dz$ represents the element of a surface normal to the $x$-direction and the surface, $S = [-L_y/2,L_y/2] \times [-L_z/2,L_z/2]$, is an entire cross section of the box.
To demonstrate the quasi-stationary state of the flow and the general energy conversion mechanisms, the time averaged energy fluxes (Eqs. \[eq:flux1\]–\[eq:flux2\]) are plotted in Fig. \[fig:totalflux\] as functions of height above the disc. The fluxes are obtained as differences between the curves as indicated by the arrows in the plots.
For both experiments it is seen that more magnetic energy is transported into the volume than is carried out. The magnetic energy does not increase accordingly, since the difference between magnetic energy flux in and out of the volume is balanced on average by Lorentz work and Joule dissipation. The conversion of magnetic energy flux into kinetic energy flux is dominant close to the disc, whereas magnetic energy flux is primarily converted into thermal energy flux in the upper two thirds of the box for both the experiments. Just above the disc surface, a (hydrostatic) balance is seen to be maintained between gravitational energy flux and the sum of enthalpy and kinetic energy flux. This corresponds to a situation where gravity is balanced by gas pressure and $x$-advection of momentum. The two experiments differ significantly in the amount of magnetic energy flux converted into thermal energy flux as a consequence of the difference in the magnetic energy reserve at hand.
\
\
\
Details of the energy conversion processes as a function of time are shown in Fig. \[fig:largejet1-dedt\]. After a sharp increase of kinetic energy, caused by the initial Alfvén pulse, the rate of change of kinetic energy oscillates as does the rate of work done by the Lorentz force. Both the efflux of kinetic energy and the work done by gravity are fairly constant, but large variations in the work done by gas pressure are seen for run \[run:largejet1\]. The characteristics of the events at the “pressure work peaks” ($t \approx 270, 400, 450$) for run \[run:largejet1\] are all indications of major, dynamically important magnetic reconnection events. Such events relax the wound up magnetic field configuration, whereby Lorentz work is reduced as there is no longer the same amount of azimuthal magnetic field for driving (and pinching) the flow. Consequently, the rate of change of kinetic energy drops. Before this happens, the thermal energy has been building up for some time (positive rate of change of thermal energy) and much of this energy is now released by pressure work. This promptly makes the rate of change of kinetic energy positive. As the magnetic pinching is significantly reduced, one possible effect of the pressure work “explosions” is to generate filaments and disruptions of the jet into the ambient medium. The magnetic pressure of the surrounding vertical magnetic field eventually halts this (irregular) radial expansion (cf.Fig. \[fig:collforces\]). The events at $t \approx 400$ and $t\approx 450$ are followed by an increase in the rate of Lorentz work which supports this picture. As a consequence of the work done by gas pressure, and the increased enthalpy flux out of the volume following the reconnection events, the rate of change of thermal energy becomes negative. The slight decrease in net Poynting flux is in line with a constant Poynting flux at the lower boundary but a decreased Poynting flux out of the volume. The sudden deficit in Poynting flux through the upper boundary is a consequence of azimuthal field relaxation in the box, causing the tangential field components to decrease at exit.
Comparing the two experiments run \[run:largejet1\] and run \[run:largejet2\] the weak field experiment (run \[run:largejet2\]) appears much less violent with respect to magnetic reconnection events. The most prominent events to be identified occur relatively late, at $t \approx 320$ and $t \approx 400$. Apparently these events do not cause significant changes in the jet dynamics, as no abrupt peaks in the pressure work are detected. Instead, another prominent feature of the jet dynamics may be identified, namely the oscillatory pattern in the rate of Lorentz work and kinetic energy flux. Such harmonic oscillations (in the radial flow) between an inner ram pressure barrier (centrifugal barrier) and an outer magnetic pressure barrier have been predicted analytically [@SautyTsinganos] and shown in axisymmetric experiments to steepen into fast magneto-sonic shocks [@OP97b]. The oscillations are most clearly present until approximately the time where the first prominent reconnection event takes place and complicates the flow pattern.
### Forces and flow features {#sec:features}
A look at the forces reveals that close to the central object the gas is launched from the disc surface by a thermal pressure gradient between disc and corona. Just above the disc the acceleration process may be regarded as either magnetic and centrifugal. Close to the rotation axis the magnetic point of view may conveniently be adopted as the magnetic field is here highly wound up and a vertical gradient of the field is the main acceleration mechanism. At larger radial distances, the magnetic field is not too wound up for the gas to be flung more radially outwards. Here the acceleration mechanism bares the same characteristics as in the centrifugally driven wind scenario. The central jet region appears very well collimated from the start, as the magnetic acceleration process for this region is inherently vertical and causes little radial flow. The fraction of the jet flung more radially outwards is eventually collimated by the magnetic force. Fig. \[fig:collforces\] shows the collimating magnetic force decomposed into toroidal and radial magnetic pressure. The ambient vertical magnetic field has generally a small collimating effect, but when magnetic reconnection events relaxes the wound up field structure the vertical field takes over and prevents locally and for a period of time the flow from expanding into the ambient medium.
The mean vertical velocity increases approximately linearly with height and the terminal velocities obtained in the jet at the upper boundary are of the order of the Kepler velocity at the inner disc radius. However, there is some dependence of the mean terminal velocities on the strength of the initial field as expected from the predictions by Michel (Michel, 1969; see also Kudoh et al., 1997). The time averaged terminal velocities are consistent with a power law dependency, $u_\infty \propto B_0^{2/3}$, in the relatively early quiescent stages of the jet evolution. At later stages, disruptive events become dynamically important and the assumptions for the predicted terminal velocity dependency, e.g. that the gas dynamics is governed purely by magnetic effects, do not hold. The more violent disruptive events in the strong field experiment redirect a relatively larger fraction of the vertical flow into transverse gas motions. Eventually, the mean vertical velocity of the strong field experiment becomes less than the mean vertical velocity of the weak field experiment.
To illustrate the complexity of the resulting flow motions, Fig. \[fig:flow\] shows velocity vectors in a cross section of the flow. Contours indicating the Alfvén surface, super-sonic regions and regions of back-flow are drawn. The super-sonic and super-Alfvénic jet beam is surrounded by a supersonic shell at larger radii. Furthermore, a prominent back-flow region is noted just outside the jet beam in the upper right quadrant of the plot.
Jet stability {#sec:stability}
-------------
Though potentially disruptive events may be identified in the dynamics, these events are found only to generate filaments and cause non-destructive distortions of the jet beam. To monitor the onset and evolution of instabilities further, the magnetic field topology is investigated. Phenomenological investigations of the evolution of the field topology are carried out by visualizing isosurfaces of magnetic flux density. Fig. \[fig:kink1\] shows the evolution of an isosurface of the magnetic flux density. The magnetic field is seen to develop a complex and highly non-axisymmetric structure, which relaxes periodically allowing the field to return to a less complex, more nearly cylindrical topology. In each such build-up/relaxation cycle several modes may be identified qualitatively such as the sausage instability and the elliptical ($|m|=2$) modes.
The total viscous and magnetic dissipation as a function of time are shown in Fig. \[fig:jet1+2-dissip\] for a test volume. In the initial phase the wound up field structure is building up and the magnetic dissipation is seen to increase sharply until $t \approx 100$. From the magnetic dissipation, several minor and a couple of major magnetic reconnection events are identified to occur in this region for run \[run:largejet1\] (upper panel). The evolution of the magnetic field shown in Fig. \[fig:kink1\] is seen to be connected to a significant increase in the magnetic dissipation. A slight increase is seen also in the viscous dissipation, indicating that a small fraction of the energy released by field re-organization is transferred into kinetic energy and subsequently dissipated.
The dissipation of the magnetic field does not match the Poynting flux into the test volume in general. The surplus of magnetic energy is partly used for driving the jet flow, as demonstrated in Sect. \[sec:dynamics\], and partly to build up the winding between relaxation events. In addition, there is a continuum of “background” magnetic dissipation between the relaxation events which may have important consequences for the jet stability. The significance of the magnetic diffusion varies in correspondence to e.g. the characteristic flow velocity as. Specifically, one expects in the $\beta_\mathrm{i} = 10$ case (run \[run:largejet2\]) the characteristic (vertical) jet velocity to be less than in the $\beta_\mathrm{i} = 5$ case (run \[run:largejet1\]) as noted in Sect. \[sec:features\]. Accordingly, the magnetic Reynolds number, $Re_\mathrm{M} = UL/\eta$, will be smaller in the weak field experiment and magnetic diffusion relatively more important. The magnetic diffusion is better capable of counter balancing the continual field winding and the build-up phase of the azimuthal field is prolonged. For the same reason the relaxation events themselves appear less violent, as seen by comparing the panels of Fig. \[fig:jet1+2-dissip\], and will cause less destructive, more localized distortions of the flow.
The damping of the spiraling jet motion at heights where the jet is in general super-Alfvénic is related to what appears as field “unwinding” just outside of the super-Alfvénic central beam. Fig. \[fig:unwind\] shows the orientation of the magnetic field in a cross section at $x \approx 20$ and illustrates the unwinding. The jet beam swings clockwise in the plot, and the change in field orientation is seen particularly on the front of the Alfvén surface in the direction of the (clockwise) spiral motion.
The unwinding of the field comes about as the spiral motion of the jet beam forces the wound up field into the ambient almost vertical field. The wound up field is oriented in a direction practically perpendicular to the ambient vertical field and reconnection occurs when these oppositely directed field lines are forced into a small region. This is seen as sheets of Joule dissipation marking the regions of magnetic diffusion at the jet flanks in the left panel of Fig. \[fig:zoomins\]. The right panel of Fig. \[fig:zoomins\] shows such a sheet in more detail and the orientation of the field lines reconnecting. These magnetic reconnections, occurring in a “cocoon” surrounding the central jet beam, are the reason for the observed back-flow in Fig. \[fig:flow\] and the field unwinding in Fig. \[fig:unwind\]. More specifically, the upper right region of back-flow in Fig. \[fig:flow\] corresponds to where the reconnection events are expected to catapult gas backwards in this scenario. The gas at the “nose” of the Alfvén surface (at $y \approx 2$, $z \approx -1$) is accelerated oppositely, i.e. forward in the rotation direction and upwards. Also seen in left panel of Fig. \[fig:zoomins\] is a central region of significant magnetic dissipation. This is the region where reconnection occurs as a consequence of field interlocking resembling the scenario observed in experiments concerning coronal heating [@xc]. The right panel of Fig. \[fig:zoomins\] confirms this interpretation.
Discussion {#sec:discussion}
==========
Obviously, parameter space has not been probed in all detail in the work presented here. Issues which clearly need more attention in future work include the magnetic configuration, thermal conditions and mass loading. Probing other magnetic configurations would involve a significant change of the numerical code, whereas the inclusion of detailed cooling expressions would be straightforward [@Rognvaldsson99; @Rognvaldsson+00a]. Magnetic field configurations which fan away from the jet axis in a more dipole like fashion may provide less stabilization high above the disc and far from the rotation axis. However, close to the central parts of the disc the field lines will, in a potential configuration as suggested by Cao and Spruit [-@CaoSpruit], bend towards the jet axis as a function of height and not appear much different from the vertical configuration used in this work. The stabilization and collimation provided by the poloidal field may be reduced if the magnetic field is produced in a dynamo active disc. In such a scenario, Brandenburg et al. [-@Brandenburg+2000] have found that significant amounts of toroidal magnetic field are transported into the corona from the outer parts of the disc. This could seriously influence the stability.
Theoretically, the stability of twisted magnetic flux tubes is expected to depend on the diameter to length ratio (aspect ratio), such that relatively long tubes are most unstable. This is strictly valid only in the ideal magnetohydrodynamics case. In practice, the rate of magnetic dissipation tends to increase for more narrow tubes, which reduces the importance of the aspect ratio [@kink]. In the work presented here, no systematic change was found in the appearance of the jet when the aspect ratio of the experiment was changed (e.g. comparing runs \[run:largejet2\], \[run:hugejet1\] and \[run:hugejet2\]).
As a consequence of magnetic dissipation occurring primarily in two regions (cf. Sect. \[sec:stability\]) the jet flow consists of a hot central beam, with temperature of the order of the virial temperature at the inner disc radius, and a hot cocoon. Hot jet flanks has been observed in YSO’s and are proposed to be generated by shocks associated with time variability and entrainment in the flow (e.g. Hartigan et al., 1993). The results presented in Sect. \[sec:stability\] suggest that magnetic dissipation may be a major additional mechanism for heating in this region. Preliminary results (run \[run:largejet4\]) indicate that relatively cold disc outflow results in less distorted, more well-defined flows and relatively more dense jets. A similar result has been reported by Hardee et al. [-@HCR] who found dense jets to maintain a high-speed spine which prevents disruption of the internal jet structure though large helical and elliptical distortions were present. The experiment, as it is, is already suitable for addressing further which consequences the temperature of the disc outflow may have on the jet dynamics. However, further investigations with special emphasis on the transsonic region close to the disc surface are desirable to investigate the issues of mass loading and disc-jet interactions in general.
Summary and conclusions {#sec:conclusion}
=======================
A high order numerical scheme has successfully been adopted and a suitable mesh refinement specified. Initial conditions resembling previous axisymmetric numerical experiments have been chosen to ease comparison. The polytropic equation of state is only used initially, to prescribe the initial density distribution of the corona. The most important features that differ from previous jet experiments are:
- The model is three dimensional rather than axisymmetric. Due to the periodic boundaries, a cutoff of the disc at large radii is applied.
- The thermal energy equation is solved, with self-consistently computed heating by viscous and Joule dissipation.
- “Free” mass outflow from the disc, i.e. the mass flux is allowed to adjust self-consistently to the forces near the disc surface.
- Parameterized Poynting flux through the upper boundary, representing external work.
- The experiments have been evolved far beyond the initial transient, and display a quasistationary behavior, as evidenced for example by the nearly constant total energy flux.
The jet dynamics has been investigated by analyzing the mechanisms of energy conversion. The rotational energy of the disc is carried by the magnetic field into the corona and is first predominantly converted into kinetic energy. In the upper two thirds of the computational domain the magnetic energy is predominantly converted into thermal energy. General features predicted by steady state theory and axisymmetric numerical experiments, such as knot generation and terminal velocity dependency on the magnetic field strength have been confirmed in the relatively early and quiescent stages of the experiments. At later stages the flow becomes quite unsteady as instabilities set in, but no serious disruption of the flow occurs. The jet stability is found to be influenced by the magnetic dissipation — this has not previously been investigated in the context of jet flows. The main findings concerning magnetic dissipation are the following:
- The heating by magnetic dissipation is significant, and leads to jet temperatures of the order of the virial temperature of the innermost Kepler orbit.
- Magnetic reconnection occurs primarily in two regions: in a central region of the jet due to field interlocking and in a jet cocoon due to the spiraling motion of the jet beam forcing the wound up field into the ambient vertical field.
- Magnetic dissipation helps to prevent critical kinking, and the jet is able to sustain a quasistationary flow, with only limited excursions.
- Reconnection events are seen to result in mass ejection into the ambient medium and cause filament structures in the jet beam.
- Reconnection events are seen to significantly influence the dynamics at the jet flanks where deceleration and even back-flow can be found quite close to the central super-Alfvénic beam.
- Heating in the region just above the disc is likely to have a significant effect on the mass loading.
[Å]{}N acknowledges partial support by the Danish Research Foundation through its establishment of the Theoretical Astrophysics Center, Copenhagen. FT acknowledges the support received from the Theoretical Astrophysics Center, in granting access to supercomputer facilities both locally and at UNI-C, [Å]{}rhus.
[^1]: In the 1-D case, a symmetric or zero $x$-derivative condition on the mass flux $p_x$ results in a steady state where $\partial \rho / \partial t = - \partial p_x / \partial x = 0$ at the boundary.
|
---
abstract: 'We report on $25$ families of projective Calabi-Yau threefolds that do not have a point of maximal unipotent monodromy in their moduli space. The construction is based on an analysis of certain pencils of octic arrangements that were found by [C. Meyer]{} [@Mey]. There are seven cases where the Picard-Fuchs operator is of order two and $18$ cases where it is of order four. The birational nature of the Picard-Fuchs equation can be used effectively to distinguish between families whose members have the same Hodge numbers.'
author:
- Slawomir Cynk and Duco van Straten
title: |
Picard-Fuchs operators\
for octic arrangements I\
(*The case of orphans)\
*
---
[^1]
Introduction
============
The phenomenon of mirror symmetry among Calabi-Yau threefolds has attracted a lot of attention and has led to major developments in mathematics and physics, see e.g. [@Horics], [@GHJ]. Especially the marvelous discovery by [P. Candelas]{}, [X. de la Ossa]{} and coworkers [@COGP] of the relation between enumeration of rational curves on a Calabi-Yau threefold and period integrals on another mirror manifold has been an inspiration to many researchers. Both the determination of instanton numbers in [@COGP] and the construction of mirror pairs by [V. Batyrev]{} [@Bat1] as generalised by [M. Gross]{} and [B. Siebert]{} [@GS], depend on [*families*]{} of Calabi-Yau manifolds that degenerate at the boundary of their moduli space at a point of [*maximal unipotent monodromy*]{} (MUM), [@Mor]. In many natural families of Calabi-Yau threefolds, like hypersurfaces in toric varieties defined by reflexive polytopes, there do exist such MUM-points in their moduli space.\
However, it has been known for some time that there are quite simple examples of one-parameter families $$f: \mathcal{Y} {\longrightarrow}S$$ of Calabi-Yau threefolds for which there are no such MUM points. First examples of this kind were described by [J.-C. Rohde]{} [@Roh1] and later by [A. Garbagnati]{} and [B. van Geemen]{} [@GG]. However, these examples are somewhat atypical in the sense that the cohomological local system ${\mathbb{H}}=R^3f_*({\mathbb{C}})$ decomposes as a tensor product $${\mathbb{H}}= {\mathbb{F}}\otimes {\mathbb{V}},$$ where ${\mathbb{F}}$ is a constant VHS with Hodge numbers $(1,0,1)$ and ${\mathbb{V}}$ a variable $(1,1)$-VHS. As a consequence, although the cohomology space is four dimensional, the Picard-Fuchs operator is of order two, and [ Rohde]{} [@Roh2] asked the question if there exist examples where the Picard-Fuchs operator had order four.
In [@Str] we described an example of such a family, but the members of that family had the defect of not being projective. [W. Zudilin]{} [@Zud] suggested to name [*orphan*]{} to describe such families, as they do not have a MUM. We decided to follow his suggestion, and in this paper we will describe a series of new orphans which have the virtue of being projective as well. The first such example was announced in [@CvS1] and has the full symplectic group $Sp_4({\mathbb{C}})$ as differential Galois group.\
**Main Result**
\
For a precise definition of the notion of [*related families*]{} we refer to the discussion in section $7$.\
We also want to point out two interesting phenomena that we discovered during the analysis of the examples.
- We found one new (conjectural) instance of a Calabi-Yau threefold with a Hilbert modular form of weight $(4,2)$ and level $6\sqrt{2}$, much like the famous example of [Consani]{} and [Scholten]{}, [@CS]. (Hilbert modularity for that example was shown in [@DPS].)
- We found an example of cohomology change in a family of Calabi-Yau threefolds at a point with monodromy of finite order. As a consequence, the central fibre of any semi-stable model is reducible, contrary to what happens in families of K3-surfaces, according to [Kulikov]{}s theorem.
Double octics
=============
By a [*double octic*]{} we understand a double cover $Y$ of ${\mathbb{P}}^3$ ramified over a surface of degree $8$. It can be given by an equation of the form $$u^2=f_8(x,y,z,w)$$ and thus can be seen as a hypersurface in weighted projective space ${\mathbb{P}}(1,1,1,1,4)$. For a general choice of the degree eight polynomial $f_8$ the variety $Y$ is a smooth Calabi-Yau space with Hodge numbers $h^{11}=1, h^{12}=149$. When the octic $f_8$ is a product of eight linear factors, we speak of an [*octic arrangement*]{}. These form a nine dimensional sub-family and for these, the double cover $Y$ has singularities at the intersections of the planes. In the generic such situation $Y$ is singular along $8.7/2=28$ lines, and by blowing up these lines (in any order) we obtain a smooth Calabi-Yau manifold $\widetilde{Y}$ with $h^{11}=29, h^{12}=9$.
By taking the eight planes in special positions, the double cover $Y$ acquires further singularities. As explained in [@Mey], if the arrangement does not have double planes, fourfold lines or sixfold points, there exist a diagram $$\begin{diagram}
\node{\widehat{Y}} \arrow{e,t}{\hat{\pi}} \arrow{s,l}{2:1} \node{Y}\arrow{s,r}{2:1}\\
\node{\widehat{{\mathbb{P}}}^3} \arrow{e,t}{\pi} \node{{\mathbb{P}}^3}
\end{diagram}$$ where $\pi$ is a sequence of blow ups, $\hat{\pi}$ a crepant resolution, and the vertical maps are two-fold ramified covers.\
In this way a myriad of different Calabi-Yau threefolds $\widehat{Y}$ can be constructed. One of the nice things is that one can read off the Hodge number $h^{12}$ as the dimension of the space of deformations of the arrangements that do not change the combinatorial type, [@CvS]. In his doctoral thesis [@Mey], [C. Meyer]{} found $450$ combinatorially different octic arrangements, determined their Hodge numbers and started the study of their arithmetical properties.\
Among these $450$ arrangements there were $11$ arrangements with $h^{12}(\widehat{Y})=0$, so lead to rigid Calabi-Yau threefolds and $63$ one-parameter families of arrangements with $h^{12}(\widehat{Y}_t)$ leading to one-parameter families (all defined over ${\mathbb{Q}}$) of Calabi-Yau threefolds, parametrised by ${\mathbb{P}}^1$: for general $t \in {\mathbb{P}}^1$, the crepant resolutions $$\begin{diagram}
\node{\widehat{Y}_t} \arrow{e,t}{\hat{\pi}}\node{Y_t}
\end{diagram}$$
of the double octics $Y_t$ can be put together into a family $\overline{\mathcal{Y}}$ over $S:={\mathbb{P}}^1\setminus \Sigma$, where $\Sigma \subset {\mathbb{P}}^1$ is a finite set of special values, where the combinatorial type of the arrangement changes. At these points, the configuration becomes rigid, or ceases to be of Calabi-Yau type: the arrangement contains a double plane, a fourfold line or a sixfold point. When we use the sequence of blow-ups to resolve the generic fibre and apply this to all members of the family, we arrive at a diagram of the form
$$\begin{diagram}
\node{\widehat{\mathcal{Y}}} \arrow{e,J} \arrow{s,l}{f}\node{\widehat{\overline{\mathcal{Y}}}}\arrow{s,r}{\overline{f}}\\
\node{S} \arrow{e,J} \node{{\mathbb{P}}^1}
\end{diagram}$$
where $f$ is a smooth map, with fibre $\widehat{Y}_t$, a smooth Calabi-Yau threefold with $h^{12}=1$. In general, the fourfold $\hat{\overline{\mathcal{Y}}}$ will have singularities sitting over the special points $s \in \Sigma $.\
Recently, in [@CKC] the analysis of [Meyer]{} was found to be complete and only three further examples with $h^{12}=0$ exist over number fields and there is one further example of a family with $h^{12}=1$, defined over ${\mathbb{Q}}(\sqrt{-3})$. Furthermore, in that paper various symmetries and birational maps between different arrangements were found.\
We will take a closer look at these $63$ [Meyer]{}-families. In order to facilitate comparison with literature, we will keep the numbering from [@Mey]. In four cases (arrangements ${\bf 33}$, ${\bf 155}$, ${\bf 275}$, ${\bf 276}$) small adjustments in the parametrisation of the family were made.\
Although arithmetical information on varieties in these families is readily available via the counting of points in finite fields, we found it extremely hard to understand details of the resolution and topology from the combinatorics of the arrangements. For example, the Jordan type of the local monodromy around the special points turned out to be very delicate. In particular, we failed to find a clear combinatorial way to recognise the appearance of a MUM-point.\
As an example, we look at configuration ${\bf 69}$ of [Meyer]{}. It consists of six planes making up a cube, with two additional planes that pass through a face-diagonal and opposite vertices of the cube, as in the following picture.
{height="5cm"}
[***Arrangement 69.***]{}
The configuration is rigid and its resolution is a rigid Calabi-Yau with $h^{11}=50, h^{12}=0$.
By sliding the intersection point at the corner of one of the two planes, we arrive at configuration ${\bf 70}$, with $h^{11}=49, h^{12}=1$. Clearly, this pencil contains another rigid configuration, namely arrangement ${\bf 3}$.
{height="5cm"}\
{height="5cm"}
[***Family of arrangements 70 and arrangement 3*** ]{}
But there are also two degenerations involving two double planes and it is not so clear what the corresponding fibres of the semi-stable reduction look like, nor were we be able to determine topologically the monodromy around these points. It turns out this family is one of the simplest orphans we know of and in this paper we will be dealing with the $25$ cases without such a MUM-point and which thus do not make it onto the list of Calabi-Yau operators [@AZ], [@AESZ], [@CYDB]. In the sequel [@CvS2] to this botanical paper, we will report on Picard-Fuchs operators for the remaining $38$ MUM-cases.
$(1,1,1,1)$-variations
======================
Generalities
------------
Let us consider more generally a fibre square diagram $$\begin{diagram}
\node{\mathcal{Y}} \arrow{e,J} \arrow{s,l}{f} \node{\overline{\mathcal{Y}}} \arrow{s,r}{\overline{f}}\\
\node{S} \arrow{e,J} \node{{\mathbb{P}}^1}
\end{diagram}$$
where $f: \mathcal{Y} {\longrightarrow}S:= {\mathbb{P}}^1 \setminus \Sigma$ is a smooth proper map of Calabi-Yau threefolds. The datum of the local system $${\mathbb{H}}_{{\mathbb{C}}}:=R^3f_* {\mathbb{C}}_{\mathcal{Y}}$$ is equivalent, after choice of a base point $s \in S$, to that of its monodromy representation $$\rho: \pi_1({\mathbb{P}}^1\setminus \Sigma, s) {\longrightarrow}Aut(H^3(Y_s,{\mathbb{C}}))$$ There is an underlying lattice bundle ${\mathbb{H}}_{{\mathbb{Z}}}$, coming from the integral cohomology and a non-degenerate skew-symmetric intersection pairing, causing this representation to land in $$Aut(H^3(Y_s,{\mathbb{Z}})/torsion)=Sp_m({\mathbb{Z}}),\;\;\; m= \dim H^3(Y_s).$$
Furthermore, ${\mathbb{H}}$ carries the structure of a [*polarised variation of Hodge structures (VHS)*]{}, meaning basically that the fibres ${\mathbb{H}}_t=H^3(Y_t,{\mathbb{C}})$ of the local system carry a pure (polarised) Hodge structure with Hodge numbers $(1,h^{12},h^{12},1)$. It is a fundamental fact, proven by [Schmid]{} [@Schm1], that one may complement this VHS defined on ${\mathbb{P}}^1 \setminus \Sigma$ by adding for each $s \in \Sigma$ a so called [*Mixed Hodge Structure*]{} (MHS) $(H_s,W_{\bullet}, F^{\bullet})$. Here $H_s$ is a ${\mathbb{Q}}$-vector space that can be identified with the sections of the ${\mathbb{Q}}$-local system ${\mathbb{H}}_{{\mathbb{Q}}}$ over an arbitrary small slit disc centered at $s$. The local monodromy transformation $T:=T_s:H_s \to H_s$ at $s$ can be written as $$T=U S$$ where $U$ is unipotent and $S$ is semi-simple. The [*monodromy logarithm*]{} $$N= -\log U=(1-U)+\frac{1}{2}(1-U)^2+\frac{1}{3}(1-U)^3+\ldots$$ is nilpotent and determines a weight filtration $W_{\bullet}$ on $H_s$ which is characterised by the property that $N:W_k \to W_{k-2}$ and $$N^k: Gr^W_{d+k} H_s \stackrel{\simeq}{{\longrightarrow}} Gr^W_{d-k} H_s.$$ The Hodge filtration $F^{\bullet}$ in $H_s \otimes {\mathbb{C}}$ arises as limit from the Hodge filtration on the spaces ${\mathbb{H}}_t$, when $t \mapsto s$, and it is a fundamental fact that for each $s\in \Sigma$ the $F^{\bullet}$ defines a pure Hodge structure of weight $k$ on the graded pieces $Gr^W_k H_s$.\
In the geometrical case [Steenbrink]{} [@Ste] has constructed this mixed Hodge structure on $H_s$ using a semi-stable model
$$\begin{diagram}
\node{D} \arrow{e,J} \arrow{s} \node{{\mathcal{Z}}} \arrow{e} \arrow{s} \node{\overline{{\mathcal{Y}}}}\arrow{s,r}{\overline{f}}\\
\node{\{s\}}\arrow{e,J} \node{\Delta} \arrow{e} \node{{\mathbb{P}}^1}
\end{diagram}$$
Here $\Delta$ is a small disc, $\Delta \to {\mathbb{P}}^1$ is a finite covering map, ramified over one of the $s \in \Sigma$, ${\mathcal{Z}}$ is smooth and the fibre $D$ over $s$ is a (reduced) normal crossing divisor with components $D_i$ inside ${\mathcal{Z}}$. The complex of relative logarithmic differential forms $$\Omega_{{\mathcal{Z}}/\Delta}^{\bullet}(\log D)$$ can be used to describe the cohomology of the fibres and its extension to $\Delta$. The complex comes with two filtrations $F^{\bullet}$, $W_{\bullet}$, which induces filtrations on the hypercohomology groups $${\mathbb{H}}^d(\Omega_{{\mathcal{Z}}/\Delta}^{\bullet}(\log D)\otimes {\mathcal{O}}_D),$$ which then leads to the limiting mixed Hodge structure on $H_s$. We refer to [@PetSte] for a detailed account.\
If the family $f:\mathcal{Y} \to S$ is defined over ${\mathbb{Q}}$, there is also a treasure of arithmetical information associated to the situation. We obtain for each rational point of ${\mathbb{P}}^1 \setminus \Sigma$ a Galois-representation on the $l$-adic cohomology $H^3_{\textup{{\'e}t}}({Y}_t\otimes_{{\mathbb{Q}}}\overline{Q},{\mathbb{Q}}_l)$, and these together make up an $l$-adic sheaf.\
We will be mainly interested in the case where $$h^{12}=1,$$ so the local system ${\mathbb{H}}$ on ${\mathbb{P}}^1\setminus \Sigma$ is a so-called $(1,1,1,1)$-variations and its representations lands in $Sp_4({\mathbb{Z}})$. In particular, for each of the $63$ families of [ Meyer]{}, we obtain a family of double octics $$0=u^2-f_8(x,y,z,w;t).$$ By crepant resolution of the general fibre a family (dropping the earlier $\hat{}\;$) we obtain such families $$f: \mathcal{Y} \to S:={\mathbb{P}}^1 \setminus \Sigma$$ and from it an associated $(1,1,1,1)$-variation over ${\mathbb{P}}^1\setminus \Sigma$.\
Strictly speaking, an arrangement defined by $f_8=0$ does not define a single family. If we multiply $f_8$ with a $t$-dependent function $\varphi(t)$, the family defined by $$0=u^2-\varphi(t)f_8(x,y,z,w;t)$$ is said to be a [*twist*]{} of the family $$0=u^2-f_8(x,y,z,w;t)$$ Although twisting can be crucially important, its effect is usually easy to analyse, and we will consider families differing by a twist as essentially the same.\
Degenerations of (1,1,1,1)-variations
-------------------------------------
There are four possibilities for the mixed Hodge diamond of the limiting mixed Hodge structures appearing for $(1,1,1,1)$-VHS. The $k$-th row (counted from the bottom) of the diamond gives the Hodge numbers of $Gr^W_k$; the monodromy logarithm operator $N$ acts in the vertical direction, shifting downwards by two rows. The definition of the weight filtration makes the diagram symmetric with respect to reflection in the central horizontal line, whereas complex conjugation is a symmetry of the Hodge-diamond along the central vertical axis. The numbers in each slope $=1$ (so SW-NE-direction) row of the diagram have to add up to the corresponding Hodge number of the variation, so are all equal to $1$ in our case. The cases that arise are:\
[***F-point***]{} $$\begin{array}{ccccccc}
&&&0&&&\\
&&0&&0&&\\
&0&&0&&0&\\
1&&1&&1&&1\\
&0&&0&&0&\\
&&0&&0&&\\
&&&0&&&\\
\end{array}$$ In this case $N=0$, so this case occurs if and only if the monodromy is of [*finite order*]{}. The limiting mixed Hodge structure is in fact pure of weight three. Often an automorphism of finite order will split the Hodge structure $Gr^W_3$:\
$$(1\;\; 1\;\; 1\;\; 1) \mapsto (1\;\; 0\;\; 0\;\; 1)\; +( 0\;\;1\;\;1\;\;0)$$
On the arithmetic side, one expect that when this happens over ${\mathbb{Q}}$, the characteristic polynomial of Frobenius will factor as $$(1-a_pT+pT^2)(1-c_pT+p^3T^2)$$ where the $a_p$ and $c_p$ are Fourier coefficients of resp. a weight $2$ and a weight $4$ cusp form for some congruence subgroup $\Gamma_0(N)$ of $Sl_2({\mathbb{Z}})$.\
There are also cases where no splitting occurs, but the Euler-factors are determined by a Hilbert modular form of weight $(4,2)$ for some real quadratic extension of ${\mathbb{Q}}$.\
[***C-point***]{} $$\begin{array}{ccccccc}
&&&0&&&\\
&&0&&0&&\\
&0&&1&&0&\\
1&&0&&0&&1\\
&0&&1&&0&\\
&&0&&0&&\\
&&&0&&&\\
\end{array}$$ In this case $N \neq 0$, $N^2=0$ and there is a single Jordan block. The pure part $Gr^W_3$ is a rigid Hodge structure with Hodge numbers $(1,0,0,1)$. Furthermore, $Gr^W_4$ and $Gr^W_2$ are one-dimensional and are identified via $N$. This type appears when a Calabi–Yau threefold acquires one or more ordinary double points, nowadays often called [*conifold points*]{}, which explains our name $C$-type point for it. However, one should be aware that $C$-point do occur not only where ordinary nodes appear, but also for many other kinds of singularities.\
On the arithmetical side, one expects to get a $2$-dimensional Galois representation $Gr^W_3$ with characteristic polynomial of Frobenius of the form $$1-a_p T+p^3T^2,$$ where the Frobenius traces are Fourier coefficient of a weight $4$ cusp form for a congruence sub-group $\Gamma_0(N)$ of $Sl_2({\mathbb{Z}})$, for some level $N$.\
[***K-point***]{} $$\begin{array}{ccccccc}
&&&0&&&\\
&&0&&0&&\\
&1&&0&&1&\\
0&&0&&0&&0\\
&1&&0&&1&\\
&&0&&0&&\\
&&&0&&&\\
\end{array}$$
In this case we also have $N \neq 0$, $N^2=0$ but there are two Jordan blocks. In this case the pure part $Gr^W_3 =0$ and $Gr^W_4$, $Gr^W_2$ are Hodge structures with Hodge numbers $(1,0,1)$, which are identified via $N$. The Hodge structure looks like that of the transcendental part of a K3-surface with maximal Picard number, which explains our name $K$-point for it.\
On the arithmetical side, one expects to get a $2$-dimensional Galois representation $Gr^W_2$ with characteristic polynomial of Frobenius of the form $$1-a_p T+p^2T^2,$$ where the Frobenius traces are Fourier coefficient of a weight 3 cusp form for a congruence sub-group $\Gamma_0(N)$ of $Sl_2({\mathbb{Z}})$, for some level $N$ and character. Such forms always have complex multiplication (CM). For a nice overview see [@Schu1] and [@Schu2].\
[***MUM-point***]{} $$\begin{array}{ccccccc}
&&&1&&&\\
&&0&&0&&\\
&0&&1&&0&\\
0&&0&&0&&0\\
&0&&1&&0&\\
&&0&&0&&\\
&&&1&&&\\
\end{array}$$
Here $N^3 \neq 0$ and there is a single Jordan block of maximal size. The Hodge structures $Gr^W_{2k}$ ($k=0,1,2,3$) are one-dimensional and necessarily of Tate type. This happens for the quintic mirror at $t=0$ and is one of the main defining properties of Calabi–Yau operators.\
So at a MUM-point, the resulting mixed Hodge structure is an iterated extension of Tate–Hodge structures. [Deligne]{} [@Del] has shown that the instanton numbers $n_1, n_2, n_3, \ldots$ can be seen to encode precisely certain [*extension data*]{} attached to the variation of Hodge structures near the MUM-point.\
Picard-Fuchs operators
======================
Generalities
------------
By a [*choice of volume forms on the fibres of*]{} $\overline{f}: \overline{{\mathcal{Y}}} \to {\mathbb{P}}^1$ we mean a rational section $\omega$ of the relative dualising sheaf $$\overline{f}_* \omega_{\overline{\mathcal{Y}}/{\mathbb{P}}^1} .$$ It restricts to a holomorphic $3$-form $\omega(t)$ on each regular fibre $Y_t$ outside the divisor of poles and zero’s of $\omega$. If $\gamma(t) \in H_3(Y_t,{\mathbb{Z}})$ is a family of horizontal cycles, defined in a contractible neighborhood $U$ of $t \in S$, the function $$\Phi_\gamma: U \to {\mathbb{C}},t \mapsto \int_{\gamma(t)} \omega(t)$$ is called a [*period integral*]{} of $f:{\mathcal{Y}}\to {\mathbb{P}}^1$. It follows from the finiteness of the deRham cohomology group $H^3(Y_t,{\mathbb{C}})$ by [ *differentiation under the integral sign*]{} that all period functions $\Phi_{\gamma}(t)$ satisfies the same linear ordinary differential equation, called the [*Picard-Fuchs equation*]{}. The corresponding differential operator is called [*Picard-Fuchs operator*]{} and we will write $${\mathcal{P}}={\mathcal{P}}(\overline{{\mathcal{Y}}},\omega) \in {\mathbb{Q}}\langle t,\frac{d}{dt}\rangle .$$ The order of the operator is clearly at most $\dim H^3(Y_t)$.\
In the case of double octics given by an affine equation $$u^2-f_8(x,y,z,t)$$ we will always take the volume form $$\omega:=\frac{dxdydz}{u}$$ as three-form and thus the period integrals we are dealing with are $$\Phi_{\gamma}(t):=\int_{\gamma(t)} \frac{dxdydz}{u}=\int_{\gamma(t)} \frac{dxdydz}{\sqrt{f_8}}$$ over cycles $\gamma(t) \in H_3(Y_t, {\mathbb{Z}})$.
Determination of Picard-Fuchs operators
---------------------------------------
We have been using two fundamentally different methods to find Picard-Fuchs operators for concrete examples.\
[*Conifold expansion method.*]{} If in a family of varieties we can locate a vanishing cycle, then the power series expansion of the period integral can always be computed algebraically, [@CvS1]. The operator is then found from the recursion of the coefficients.
Especially for the case of double octics, in many of the $63$ families one can identify a [*vanishing tetrahedron*]{}: for a special value of the parameter one of the eight planes passes through a triple point of intersection, defined by three other planes. In appropriate coordinates we can write our affine equation as $$u^2=xyz(t-x-y-z)P(x,y,z,t)$$ where $P$ is the product of the other five planes and depend on a the parameter $t$. We assume $P(0,0,0,0) \neq 0$. One can now identify a nice cycle $\gamma(t)$ in the double octic, which consists of two parts $\gamma(t)_+,\;(u \ge 0)$ and $\gamma(t)_{-},\;(u\le 0)$ which project onto the real tetrahedron $T_t$ bounded by the plane $x=0$, $y=0$, $z=0$, $x+y+z=t$. $\gamma(t)_{+}$ and $\gamma_{-}$ are glued together at the boundary of $T_t$, thus making up a three sphere in the double octic. For $t=0$ the tetrahedron and thus the sphere $\Gamma(t)$ shrink to a point. We can write $$\Phi_{\gamma}(t)=\int_{\gamma(t)} \omega=2F(t)$$ where $$F(t)=\int_{T_t} \frac{dxdydz}{\sqrt{(xyz(t-x-y-z)P(x,y,z,t)}}=t \int_{T} \frac{dxdydz}{\sqrt{(xyz(1-x-y-z)P(tx,ty,tz,t)}},$$ where we used the substitution $(x,y,z) \mapsto (tx,ty,tz)$. By expanding the integrand in a series and perform termwise integration over the simplex $T:=T_1$, the period expands in a series of the form $$\Phi_{\gamma}(t) =\pi^2 t (A_0+A_1t+A_2t^2+\ldots)$$ with the coefficients $A_i \in {\mathbb{Q}}$ if $P_t(x,y,z) \in {\mathbb{Q}}[x,y,z,t]$ and can be computed explicitly, see [@CvS1]. By computing sufficiently many terms in the expansion, one may find the Picard-Fuchs operator by looking for the recursion on the coefficients $A_i$.\
[*Cohomology method.*]{} There are many variants for this method, but let us take for sake of simplicity the case of a smooth hypersurface $X \subset {\mathbb{P}}^{n}$. The middle dimensional (primitive) cohomology of $X$ can be identified with $H^n({\mathbb{P}}^n\setminus X)$ and it elements can be represented by residues of $n$-forms on ${\mathbb{P}}^n$ with poles along $X$ of the form $$\frac{P\Omega}{F^k}$$ where $F$ is the defining polynomial for $X$, $\Omega=\iota_E(dVol)$ the fundamental form for ${\mathbb{P}}^n$, and $P$ is a polynomial such that the above expression is homogeneous of degree $0$. If $P$ runs over the appropriate graded pieces of the Jacobi-ring of $F$, one obtains a basis $$\omega_1,\omega_2,\ldots,\omega_N$$ for $H^n({\mathbb{P}}^n\setminus X)$. If $F$ depends on an additional parameter $t$, one can differentiate these basis-elements with respect to $t$ and express the result in the given basis. One obtains thus a differential system $$\frac{d}{dt}
\left (\begin{array}{c}
\omega_1\\
\omega_2\\
\ldots\\
\omega_N\\
\end{array} \right)=A(t)\left(\begin{array}{c}
\omega_1\\
\omega_2\\
\ldots\\
\omega_N\\
\end{array}
\right)$$ from which one can obtain Picard-Fuchs equations for each $\omega_i$. This so-called [*Griffiths-Dwork*]{} method ([@Gri1], [@Dw1]) depends on the assumption that the hypersurface is smooth, so that the partial derivatives $\partial_iF$ form a regular sequence in the polynomial ring. If $X$ has singularities, one no longer obtains a basis for the cohomology. Rather one has to determine the Koszul-homology between the partial derivatives and enter into a spectral sequence and things become more complicated. In [@CvS1] we wrote:\
[*Due to the singularities of $f_8$, a Griffiths-Dwork approach is cumbersome, if not impossible.*]{}\
It was [P. Lairez]{} who proved us very wrong in this respect. His computer program, described in [@Lai], does not aim at finding a complete cohomology space, rather it looks for the smallest space stable under differentiation that contains the given rational differential form. It does so by going through the spectral sequence given by the pole order filtration, where at each step Gröbner basis calculations are done to increase the set of basis forms.\
For our computations we initially used the method of conifold expansion, but we discovered soon that\
[*Due to the singularities of $f_8$ the conifold expansion approach is cumbersome, if not impossible.*]{}\
Reading the Riemann-symbol
--------------------------
The amount of information that is contained in the Picard-Fuchs operator ${\mathcal{P}}$ can not be underestimated. The local system ${\mathbb{H}}_{{\mathbb{C}}}$ is isomorphic to local system of solutions $Sol({\mathbb{P}})$. Already the local monodromies $T_s$ around the special parameter values $s \in \Sigma$ are hard to obtain from topology or a semi-stable reduction. But this information can easily be read off from the operator, by studying the local solutions in series of the form $$t^{\alpha} \sum_{k=0}^N \sum_{n=0}^{\infty}A_{n,k}t^{n}\log(t)^k .$$
There is a delicate interaction between ${\mathcal{P}}$ and the Frobenius-operator (see [@Dw2]), so that arithmetical properties of the varieties are tightly linked to ${\mathcal{P}}$. It appears that the Picard-Fuchs operator just abstracts away sufficiently many details of the geometry and retains just the right amount of motivic information.\
We recall that the [*Riemann-symbol*]{} of a differential operator ${\mathcal{P}}\in {\mathbb{C}}\langle t,\frac{d}{dt}\rangle$ is a table recording for each singular point of the differential operator the corresponding [*exponents*]{}, i.e. solutions to the [*indicial equation*]{} [@Ince]. (In order to have a non-zero series solution of the type described above, one needs that $\alpha$ is an exponent at $0$.)\
We found it convenient to express the operators in terms of the logarithmic differentiation $$\Theta:=t \frac{d}{dt}$$ and write the operators in [*$\Theta$-form*]{} $${\mathcal{P}}:=P_0(\Theta)+tP_1(\Theta)+t^2P_2(\Theta)+\ldots t^rP_r(\Theta), \;\;\;P_r \neq 0$$ where the $P_i$ are polynomials, in our case of degree four. The exponents at $0$ are then just the roots of $P_0$, those of $\infty$ the roots of $P_r$, with a minus sign. To determine the exponents at other points, one just translate this point to the origin, and re-express the operator in $\Theta$-form.\
The exponents capture the semi-simple part of the monodromy at the corresponding singular point. The logarithmic terms appearing in the solutions encode the Jordan structure of the unipotent part. In general logarithmic terms may appear between solutions with integer difference in exponents, but in the geometrical context, as a rule, a logarithm appears [*always*]{} precisely when two exponents become [*equal*]{}.\
A [**C-point**]{} can be expected when the exponent are of the form $$\alpha-\epsilon\;\;\; \alpha\;\;\; \alpha\;\;\; \alpha+\epsilon$$ The archetypical case is $0\;\; 1\;\; 1\;\; 2$, indicating the presence of local solutions of the form $$\phi_0=1+a_1t+\ldots, \phi_1(t)=t+b_1t+\ldots, \phi_2(t)=log(t)\phi_1(t)+c_1 t+\ldots, \phi_3=t^3+d_1t^4+\ldots$$ as appear in a pure conifold smoothing.\
A [**$K$-point**]{} can be expected to appear when the exponents are of the form $$\alpha\;\;\; \alpha\;\;\; \beta\;\;\; \beta,\;\;\;(\alpha <\beta) .$$ The archetypical case is $0\;\; 0\;\; 1\;\; 1$, indicating the presence of local solutions of the form $$\phi_0=1+a t+\ldots, \phi_1(t)=\log(t)\phi_0(t)+b t+\ldots,
\phi_2(t)=t+c t^2+\ldots, \phi_3=\log(t)\phi_1(t)+d t^2+\ldots$$
A [**MUM-point**]{} can be expected to appear, when all exponents are equal $$\alpha\;\;\; \alpha\;\;\; \alpha\;\;\; \alpha\;\;\;$$ The archetypical case is $0\;\;0\;\;0\;\;0$, with the famous Frobenius basis of solutions of the form $$\phi_0=1+at+\ldots,\phi_1(t)=\log(t)\phi_0(t)+\ldots,\phi_2(t)=\log(t)^2\phi_0(t)+\ldots, \Phi_3(t)=\log^3(t)\phi_0(t)+\ldots$$
An [**$F$-point**]{} can be expected in all other cases. If all exponents are integral (and no logarithms appear) we have trivial monodromy and we speak of an [*apparent singularity*]{}, if the exponents are [*not*]{} $0\;\;1\;\;2\;\;3$, which would be the exponents at a regular point. The archetypical apparent singularity is signaled by the exponents $$0 \;\;\;1 \;\;\;3\;\;4 .$$ In general, we will call any $F$-point [*with non-equally spaced exponents*]{} an $A$-point.\
When we write the operator ${\mathcal{P}}$ in the form $$\frac{d^4}{dt^4} +a_1(t) \frac{d^3}{dt^3}+a_2(t) \frac{d^2}{dt^2}+a_3(t) \frac{d}{dt}+a_4(t)$$ where $a_i(t) \in {\mathbb{C}}(t)$, then the function $$Y(t):=e^{-\frac{1}{2} \int a_1(t) dt}$$ is called the [*Yukawa coupling*]{} and its (simple) zero’s typically are apparent singularities with exponents $0\;\; 1\;\; 3\;\; 4$.\
[***Basic transformation theory***]{}
In what follows, we will not distinguish between an operator ${\mathcal{P}}$ and the operator ${\mathcal{P}}' =\varphi(t) {\mathcal{P}}$ obtained from multiplying ${\mathcal{P}}$ by a rational function $\varphi(t)$, as they determine the same local system of solutions on ${\mathbb{P}}^1 \setminus \Sigma$. (Of course, in the finer theory of $\mathcal{D}$-modules one has to distinguish very well between ${\mathcal{P}}$ and ${\mathcal{P}}'$).\
Often one has to make simple transformations on differential operators.\
1) The simplest are those induced by Möbius transformations, coming from fractional linear transformations $$t \mapsto \frac{a t+b}{ct+d}$$ of the coordinate in ${\mathbb{P}}^1$. Of course, this just changes the position of the singular points, the corresponding exponents are preserved. We call two operators related in this way [*similar operators*]{}.\
2) If $\omega(t)$ and $\omega'(t)$ are two different choices of volume form on the fibres, then $$\omega'(t) =\varphi(t)\omega(t)$$ where $\varphi(t)$ is a rational function. The corresponding Picard-Fuchs operators ${\mathcal{P}}({\mathcal{Y}},\omega)$ and ${\mathcal{P}}({\mathcal{Y}},\omega')$ will be related in a certain way.\
More generally, if $y(t)$ satisfies ${\mathcal{P}}y(t)=0$, and $\phi(t)$ is a rational function, then the function $Y(t):=\varphi(t)y(t)$ will satisfy another differential equation ${\mathcal{Q}}Y(t)=0$ that is rather easy to determine. We will say that ${\mathcal{Q}}$ is [*strictly equivalent*]{} to ${\mathcal{P}}$. Its effect on the Riemann symbol will be a shift of all exponents by an amount given be the order of $\varphi(t)$ at the point in question. For example, the effect of multiplication by $t$ shifts the exponents at $0$ one up, those at $\infty$ one down.\
3) As already mentioned above, if $\phi(t)$ is a rational function of $t$, then the families of double octics $$u^2=f_8,\;\;\;\textup{and}\;\;\; u^2=\phi(t)f_8$$ are said to differ by a [*twist*]{}. Replacing $\phi(t)$ by $\phi(t)\varphi(t)^2$ does not change the fibration birationally, as can be seen by replacing $u$ by $\varphi(t)u$. The volume form $$\omega:=\frac{dxdydz}{\sqrt{f_8}}$$ for $u^2-f_8$ and $$\omega':=\frac{dxdydz}{\sqrt{\phi(t)f_8}}$$ for $u^2-\phi(t) f_8$ differ by the square root of a rational function $$\omega=\sqrt{\phi(t)}\omega'$$
More generally, if $y(t)$ satisfies a differential equation ${\mathcal{P}}y(t)=0$, then $Y(t):=\phi(t)y(t)$, where $\phi(t)$ is an algebraic function, will satisfy another differential equation ${\mathcal{Q}}$ that is easy to determine, knowing only ${\mathcal{P}}$. We will then call ${\mathcal{P}}$ and ${\mathcal{Q}}$ [*equivalent*]{}.\
Its effect on the Riemann symbol is also rather easy to understand. If $\phi(t)$ has near $a$ the character of $(t-a)^{\epsilon}$, then the exponents at $a$ get all shifted by the amount $\epsilon$:
$$\alpha,\beta,\gamma,\delta \mapsto \alpha+\epsilon,\beta+\epsilon,\gamma+\epsilon,\delta+\epsilon$$
4\) If we replace $t$ by $t=\psi(s)$ for some function $\psi(s)$ we can rewrite the operator the operator ${\mathcal{P}}$ in terms of the variable $s$ and obtain an operator $\psi^*{\mathcal{P}}$ in $s, \frac{d}{ds}$ that we call the [*pull-back*]{} of ${\mathcal{P}}$ along the map $\psi$. Very common are pull-backs by the map $t=s^n$, which geometrically is an $n$-fold covering map of ${\mathbb{P}}^1$, with total ramification at $0$ and $\infty$. This operation leads to a division of the exponents at $0$ and $\infty$: $$\alpha,\beta,\gamma,\delta \mapsto \alpha/n,\beta/n,\gamma/n,\delta/n$$
The most general transformations one has to allow are those given by algebraic coordinate transformations, which are multi-valued maps from ${\mathbb{P}}^1$ to itself, which properly understood are given by [*correspondences*]{} via a smooth curve $C$: $$\begin{diagram}
\node{C} \arrow{e,t}{p}\arrow{s,l}{q} \node{{\mathbb{P}}^1}\\
\node{{\mathbb{P}}^1}
\end{diagram}$$ and $\psi^*$ ’is’ $q_*p^*$, which just means that $$p^*{\mathcal{P}}= q^*{\mathcal{Q}}$$ In such a case we will say that ${\mathcal{P}}$ and ${\mathcal{Q}}$ are [*related operators*]{}. The effect of these transformations on the Riemann-symbol can be traced back the local ramification behaviour of $p$ and $q$; we will not spell out the details.\
It is easy to see that under pull-back one can not get rid of a $MUM$, $K$, $C$ or $A$-point. Only an $F$-point with equidistant exponents may turn into the non-singularity with exponents $0,1,2,3$. Note in particular that an operator with a MUM-point can not be related to an operator without a MUM-points, etc.
Orphans of order $2$
====================
It turns out that there are seven arrangements that lead to a second order operator. These are the arrangements $${\bf 4},\;\; {\bf 13},\;\; {\bf 34},\;\; {\bf 72},\;\; {\bf 261},\;\; {\bf 264},\;\; {\bf 270} .$$ The differential equations for each of these cases was computed; the results are recorded in Appendix B. All operators turn out to be of a very simple type, directly related to the Legendre differential equation, which is the hypergeometric equation $$\Theta^2-16 t(\Theta+1/2)^2$$ and Riemann symbol $$\left\{
\begin{array}{ccc}
0&1/16&\infty\\
\hline
0&0&1/2\\
0&0&1/2\\
\end{array}
\right\}.$$ This also is the Picard-Fuchs operator of the elliptic surface with Kodaira fibres $I_2,I_2,I_2^*$. The differential equations in the cases ${\bf 72}$ and ${\bf 270}$ are a bit different.
At first sight it is very surprising to find a second order equation for such octic triple integrals. As explained in [@Roh1], the appearance of a certain [*maximal automorphism*]{} causes the Picard-Fuchs operator to be of order two. In [@CKC] such maximal automorphism were identified in five of the seven cases. For the remaining two cases [**264**]{} and [**270**]{} we do not have such a simple explanation for the appearence of a second order Picard-Fuchs equation.\
A priori, there seem to be two different scenario’s in which the Picard-Fuchs operator of a family can reduce to an operator of order two. It could happen that the $(1,1,1,1)$-VHS splits as sum into a (rigid) $(1,0,0,1)$ Hodge structure and a variable $(0,1,1,0)$, coming from an elliptic curve. Or it could be that the $(1,1,1,1)$-VHS is a tensor product of a constant $(1,0,1)$-Hodge structure with a variable $(1,1)$-VHS coming from a family of elliptic curves. This $1,0,1$ should be the transcendental part of $H^2$ of a K3-surface with Picard number $20$.\
In our situation it is always the second alternative that has to occur, as we are looking at the Picard-Fuchs operator for the period integrals of the holomorphic volume form $\omega$, which in the first case would be constant.\
It is of interest to identify the K3-surface in the geometry of the arrangement. For example, for the octic corresponding to arrangement [**13**]{} $$Y_t: 0=u^2 - x y z (x+y) (y+z) w (x-z-w) (x-z-tw)$$ one can understand its relation to the K3-surface in the following way. Replacing $u$ by $u/(x-z)$ we see the double octic is equal to the normalisation of the [*double dectic*]{} $$u^2=xyz(x+y)(y+z)(x-z)w(x-z-w)(x-z-tw)(x-z) .$$ The first $6$ factors now only depend on the variables $x,y,z$ and determine a double sextic K3-surface $S$ with equation $$p^2=xyz(x+y)(y+z)(x-z)$$
It is the famous [*most algebraic K3-surface*]{} [@Vin], which comes with the weight $3$ modular form named $16$ in appendix A.
{height="5cm"}
The last four factors only depend on $w$ and $\xi:=x-z$, with $t$ as parameter and determine a double quartic family of elliptic curves $E_t$ given by the equation $$q^2=w(\xi-w) (\xi-tw) \xi$$ By dividing out the involution $\iota$ induced by $p\mapsto -p$, $q \to -q$ acting on $S \times E_t$ we get back our double dectic, $u=pq$. Hence we see that our original double octic is birational to $$S \times E_t/\iota {\longrightarrow}Y_t$$ so that we see that $Y_t$ is a simple instance of the [*Borcea-Voisin construction*]{}, see [@CM].\
It turns out that, unexpectedly, in all cases except ${\bf 270}$ we end up with the modular form $16$ as constant factor.
In case ${\bf 270}$ we get modular form $8$, attached to the double sextic K3-surface
{height="5cm"}
These two K3-surfaces appear in a nice pencil; in [@GT] one finds a very detailed account of their geometry and arithmetic.\
Orphans of order $4$
====================
Of the $63$ families it tuns out that $18$ are fourth order orphans. These are much more interesting and are collected in Appendix C. We sort them according to types of singularities that appear.
The two KKCC-operators
----------------------
It turns out that there are two different but similar operators with two points of type $K$ and two points of type $C$. In each case there is a pair of arrangements related to them.\
[***The arrangements ${\bf 33}$ and ${\bf 70}$ with $h^{11}=49$.***]{}\
These two arrangements are birational, via the map $$(x,y,z,v)\longmapsto (tyv, xy-yz, tzv-xz+{z}^{2}, {x}^{2}-xz-txv).$$
(Here and below we will always refer to the equations of the arrangements given in Appendix C.)
The Picard-Fuchs operator for ${\bf 33}$ has Riemann symbol $$\left\{
\begin{array}{cccc}
0&1&2&\infty\\
\hline
0&0&0&1/2\\
0&1/2&1&1/2\\
1&1/2&1&3/2\\
1&2&2&3/2\\
\end{array}
\right\}$$ from which we see that $0$ and $\infty$ are $K$-points, and $1$ and $2$ are $C$-points. The operator for $\bf{70}$ differs from it by $t \mapsto -t$. This operator was obtained in [@CvS1] by conifold expansion.
As explained above, to each rational $K$-point there is attached a weight three modular form and to each rational $C$-form a weight four modular form. It is well-known that these forms can be determined by counting points over finite fields and we will omit all details on their calculation.
We found it convenient to write the names of these forms [*above*]{} the corresponding points of the Riemann-symbol, as to form a [*decorated Riemann-symbol.*]{}
$$\left\{
\begin{array}{cccc}
8&32/2&8/1&16\\
\hline
0&1&2&\infty\\
\hline
0&0&0&1/2\\
0&1/2&1&1/2\\
1&1/2&1&3/2\\
1&2&2&3/2\\
\end{array}
\right\}$$ (For the naming of the modular forms we are using the reader is referred to Appendix A.) We see that two different types of $K3$-surfaces, corresponding to the forms $8$ and $16$ should appear in the semi-stable model of the singular fibres. This is in accordance with the fact that there is no symmetry that fixes the $C$-points and interchanges the $K$-points.\
[***The arrangements ${\bf 97}$ and ${\bf 98}$ with $h^{11}=45$.***]{}\
This is the other pair of arrangements that also lead to an operator with two $K$ and two $C$-points. The arrangements ${\bf 97}$ and ${\bf 98}$ are also birational, via the map\
$$(x,y,z,v)\longmapsto\left( (tv-z-v) \left( x+y+z+v \right) ,txz,( -tv+z+v) y, ( tv-z-v) ( x+y) \right)$$
The decorated Riemann symbol of the operator for ${\bf 97}$ is
$$\left\{
\begin{array}{cccc}
32/1&8&8/1&8\\
\hline
0&-1&-2&\infty\\
\hline
0&0&0&1/2\\
1/2&0&1&1/2\\
1/2&1&1&3/2\\
1&1&2&3/2\\
\end{array}
\right\}$$ By shifting the exponents by $1/2$ in $0$ and $-1/2$ in $\infty$ followed by a simple Möbius transformation one can transform it to operator ${\bf 98}$.\
We again see two $K$ points, but this time we find that at both of them the modular form is $8$, which suggests that the operator of [**98**]{} has a symmetry that interchanges the two $K$-points. Indeed, by shifting the finite $C$-point to the origin, one finds that the operator is symmetric under $t \mapsto -t$ and thus can be pulled-back by the squaring map from the nice operator $$\mathcal{A}:=\Theta^{2}( \Theta-1)^{2}+t\, \Theta^{2}(32\,\Theta^{2}+3)+4\,t^2 \left( 4\,\Theta+1 \right) \left( 2\,\Theta+1 \right) ^{2} \left( 4\,\Theta+3 \right)$$ which has extended Riemann symbol $$\left\{
\begin{array}{cccc}
8&8/1&32/1\\
\hline
0&-1/16&\infty\\
\hline
0&0&1/4\\
0&1/2&1/2\\
1&1/2&1/2\\
1&1&3/4\\
\end{array}
\right\}$$
The symmetry of the operator is also visible as a symmetry of the arrangements.\
The two KCCC operators
----------------------
It turns out that there are also two different operators with a single $K$ point, but with three additional $C$-points. The first of these operators is related to two essentially different pairs of double octic arrangements, namely\
[***The arrangements $\bf 35$ and $\bf 71$ with $h^{11}=49$.***]{}\
and\
[***The arrangements $\bf 247$ and $\bf 252$ with $h^{11}=37$.***]{}\
As the Riemann-symbol suggests, the Picard-Fuchs operators for these four cases are related by a simple Möbius transformation and multiplication with an algebraic function and we will only analyse one of the cases.
We know that ${\bf 247}$ and ${\bf 252}$ are birational arrangements, but we were unable to find a birational transformation between ${\bf 35}$ and ${\bf 71}$. The coincidence of the Picard-Fuchs operators (up to transformation) strongly suggest that there exist a [*correspondence*]{} between ${\bf 35}$ and ${\bf 247}$, but again we were unable to find it. This is an illustration of the power of Picard-Fuchs operators to make geometrical predictions.\
The decorated Riemann symbol of ${\bf 35}$ is $$\left\{
\begin{array}{cccc}
8/1&8/1&8&8/1\\
\hline
-1&0&1&\infty\\
\hline
0& 0&0&1/2\\
1&1/2&0&1\\
1&1/2&1&1\\
2& 1&1&3/2\\
\end{array}
\right\}$$ so at all $C$-points we find the modular form $8/1$.
Indeed, the operator of $\bf 35$ has symmetries interchanging the $C$-points. If we shift the exponents at $-1$ by $1/2$, at $\infty$ by $-1/2$, and then bringing the $K$-point to $0$ and the C-point at $-1$ to $\infty$, we obtain an operator symmetric under $t \mapsto -t$, so it is seen to be pull-back by the squaring map of $$\mathcal{B}:=
{\Theta}^{2} \left( 2\,\Theta-1 \right) ^{2}+t \left( 4\,{\Theta}^{2}+2\,\Theta+1 \right)
\left( 4\,\Theta+1 \right) ^{2}+ t^2 \left( 4\,\Theta+1 \right) \left( 4\,\Theta+3
\right) ^{2} \left( 4\,\Theta+5 \right)$$ with decorated Riemann-Symbol $$\left\{
\begin{array}{ccc}
8&8/1&8/1\\
\hline
0&-1/8&\infty\\
\hline
0&0&1/4\\
0&1/2&3/4\\
1/2&1/2&3/4\\
1/2&1&5/4
\end{array}
\right\}$$ At both conifold points we have modular form $8/1$, and it turns out that there is a further symmetry in the operator that exchanges these and thus can be obtained as pull-back from yet another operator with three singular points: $${\Theta}^{2} \left(4\,\Theta - 1 \right) ^{2}+ 2\,t \left(8\,\Theta+1 \right)
\left(32\,{\Theta}^{3}+28\,{\Theta}^{2}+19\,\Theta+4 \right) + t^2 \left(8\,\Theta+1
\right) \left( 8\,\Theta+9 \right) \left( 8\,\Theta+5 \right) ^{2}$$ Its decorated Riemann symbol is $$\left\{
\begin{array}{ccc}
8&?&8/1\\
\hline
0&-1/8&\infty\\
\hline
0&0&1/8\\
0&1/2&5/8\\
1/4&1&5/8\\
1/4&3/2&9/8\\
\end{array}
\right\}$$ Note that now there appears an singularity at $-1/8$ with monodromy of order $2$. So the Calabi-Yau threefold appearing at this point is special, but the $?$ indicates that we were not able yet to identify any modular forms.\
[***The arrangements $\bf 152$ and $\bf 198$ with $h^{11}=41$***]{}.\
These two arrangement give rise to another operator with one $K$ and three $C$-points. The coincidence of Hodge numbers and Picard-Fuchs operator (up to equivalence) suggest, that the two arrangements are birational. Again, we were unable to find the map.\
The decorated Riemann symbol of $\bf 152$ is $$\left\{
\begin{array}{cccc}
8&8/1&32/1&8/1\\
\hline
-1&0&1&\infty\\
\hline
0&0&0&1/2\\
1/2&1/2&0&1\\
1/2&1/2&2&1\\
1&1&2&3/2\\
\end{array}
\right\}$$ The operator of ${\bf 198}$ is equivalent to it: by shifting the exponents at $-1$ by $1/2$ and at $\infty$ by $-1/2$.
The appearance of the forms $8/1$ at both $0$ and $\infty$ suggests that there is a symmetry interchanging these points. And indeed, it turns out that the operator for ${\bf 152}$ is also a pull-back from the operator $\mathcal{A}$!
The ACCC-operator
-----------------
\
The arrangements $\bf 153$ and $\bf 197$ are birational, via the map\
$$(x,y,z,v)\longmapsto \left(( xt+vt-y ) tv,
( xt+vt -y-z ) y, ( x+v ) tz,
( xt+vt-y ) tx\right)$$
The decorated Riemann symbol for $\bf 153$ is: $$\left\{
\begin {array}{cccc}
32/1&32/2&8/1&32/2\\
\hline
0&-1&-2&\infty\\
\hline
0&0&0&1/2\\
1/2&1/2&1/2&1\\
1/2&1/2&3/2&1\\
1&1&2&3/2\\
\end {array} \right\}$$ The point $t=-2$ is very remarkable. By expanding the solutions around the singular point $-2$ one sees that the monodromy is of order two; no logarithmic terms arise. As a result, the corresponding limiting MHS remains pure of weight three. On the other hand, the arrangement at $t=-2$ specialises to the rigid arrangement $93$ and the corresponding double octic has a rigid Calabi-Yau (with modular form $8/1$) as resolution. This implies that any semi-stable fibre at $t=-2$ needs to have further components to account for the change in cohomology between special fibre at $-2$ and general fibre. We note that the theorem of [Kulikov]{} implies that a similar phenomenon can not happen for K3-surfaces. This and other examples will be studied in more detail in future paper.\
As suggested by the modular forms, there could be a symmetry interchanging the two $32/2$ points. By first shifting exponents by $1/2$ at $-2$ and $-1/2$ at $\infty$ and then bringing $-2$ to $\infty$, the operator is pulled back from the nice operator with three singular points:
$$\mathcal{C}:=
\Theta(4\Theta-1)(2\Theta-1)-t(4\Theta+1)^2(4\Theta^2+2\Theta+1)+t^2(4\Theta+1)
(2\Theta+1)(4\Theta+5)(\Theta+1)$$
with extended Riemann symbol
$$\left\{ \begin {array}{ccc}
32/1&32/2&8/1\\
\hline
0&1&\infty\\
\hline
0& 0&1/4\\
1/4&1/2&3/4\\
1/4&1/2&3/4\\
1/2& 1&5/4\\
\end {array} \right\} .$$
The KCCCC-operator
------------------
[*The arrangement $\bf 243$*]{} ($h^{11}=39$) leads to the rather complicated operator $$\Theta\, \left( \Theta-2 \right) \left( \Theta-1 \right) ^{2}
-\frac16\,t\Theta\, \left( \Theta-1 \right) \left( 19\,{\Theta}^{2}-19\,\Theta+9 \right)
+\frac13\,{t}^{2}{\Theta}^{2} \left( 11\,{\Theta}^{2}+4 \right)$$ $$-\frac1{24}\,{t}^{3} \left( 11\,{\Theta}^{2}+11\,\Theta+5 \right)
\left( 2\,\Theta+1 \right) ^{2}
+\frac1{48}\,{t}^{4} \left( 2\,\Theta+3 \right) ^{2} \left( 2\,\Theta+1 \right) ^{2}$$
Its decorated Riemann symbol is
$$\left\{
\begin {array}{ccccc}
12/1&32/2&6/1&8/1&8\\
\hline
0&1&\frac32&2&\infty\\
\hline
0& 0&0&0&1/2\\
1&1/2&1&1&1/2\\
1&1/2&1&1&3/2\\
2&1 &2&2&3/2\\
\end {array} \right\}$$
We claim it can not be simplified further, as the modular forms at the cusps are all different.
The ACCCCK-operator
-------------------
[***Arrangement ${\bf 250}$ and ${\bf 258}$***]{} define birational families of Calabi-Yau threefolds with $h^{11}=37$. The decorated Riemann-symbol of ${\bf 250}$ is $$\left\{
\begin {array}{cccccc}
6/1&8/1&h&8/1&6/1&8\\
\hline
-2 &-1&-1/2&0&1&\infty\\
\hline
0& 0&0& 0&0&1/2\\
1&1/2&1&1/2&1&1/2\\
1&1/2&3&1/2&1&3/2\\
2&1 &4& 1 &2&3/2\\
\end {array} \right\}$$ Very remarkably, at the apparent singularity at the point $-1/2$ there appears the Hilbert-modular form $h$ of level $6\sqrt{2}$ and weight $(4,2)$. When we shift this apparent singularity to the origin, we obtain an operator that is symmetric with respect to the involution $t \mapsto -t$. But there is no obvious corresponding symmetry in the family. Counting points, it appears that the number of points of the fibres at $t$ and $-t$ are equal or opposite mod $p$, according to the quadratic character $\left( \frac{2}{p} \right)$. In fact the transformation $$\scriptsize
\left( \begin{array}{c}
x\\y\\z\\t
\end{array}\right)
\longmapsto
\left(\begin{array}{l}
2\, \left( -y-z+v \right) x \left( x+v+y+z \right) \left( \left(
t+1/2 \right) y-v-x-z \right) \\
4\, \left( x+v+y+z \right) \left( -1/2\,{z}^{2}+ \left( v/2-x/2-y \right) z-1/2
\,{y}^{2}+ \left( v/2-x/2 \right) y+vx \right) z\\
\left( {y}^{2}+ \left( -v+x+2z \right) y + {z}^{2}+ \left( x-v \right) z-2vx \right) \left( x+v+y+z \right)
\left( \left( t+1/2 \right) y-v-x+z \right) \\
\left( y+z \right) \left( v-x-y-z \right) ^{2} \left( ty-v-x+y/2-z \right)
\end{array}\right)$$ gives a correspondance between a fiber of the family and the quadratic twist by 2 of the opposite fiber. In particular, the modular forms labelled $8/1$ actually occuring differ by this character. From this state of affairs it seems natural that at the symmetry point the Hilbert modular form for $\sqrt{2}$ appears. We plan study this example more carefully in a future paper.\
We remark that although the modular forms at corresponding fibres $0$ and $-1$ are in both cases $8/1$, they correspond to two non-birational rigid Calabi-Yau configuration ${\bf 69}$ ($h^{11}=50$) and ${\bf 93}$ ($h^{11}=46$).\
Similarly, at $1$ and $-2$ we have modular form $6/1$, but rigid Calabi-Yau configurations ${\bf 245}$ ($h^{11}=38$) resp. ${\bf 240}$ ($h^{11}=40$). So it seems improbable that there is a birational map relating the fibre at $t$ to the fibre at $-1/2-t$. Geometrically, the symmetry of the Picard-Fuchs operator is surprising.\
Using the symmetry we see that the operator is pulled back from a simpler operator with the following Riemann symbol: $$\left\{
\begin {array}{cccc}
h&8/1&6/1&\infty\\
\hline
0&1&9&\infty\\
\hline
0 & 0& 0& 1/4\\
1/2&1/2& 1& 1/4\\
3/2&1/2& 1& 3/4\\
2 & 1& 2& 3/4\\
\end {array} \right\}$$
The KCCCCC-operator
-------------------
[***Arrangement ${\bf 248}$ with $h^{12}=37$.***]{}
Decorated Riemann symbol
$$\left\{
\begin{array}{cccccc}
12/1&6/1&16&6/1&12/1&32/1\\
\hline
-2&-3/2&-1&-1/2&0&\infty\\
\hline
0&0&0&0&0&1/2\\
1&1&0&1&1&1\\
1&1&2&1&1&1\\
2&2&2&2&2&3/2\\
\end{array}
\right\}$$
There is a symmetry pairing the points with same modular form and fixing the points belonging to the modular forms $16$ and $32/1$. First shift the $K$-point to $0$, then the operator is seen the be pull-back via quadratic map from an operator with Riemann symbol $$\left\{
\begin{array}{cccc}
16&12/1&6/1&32/1\\
\hline
0&1&1/4&\infty\\
\hline
0&0&0&1/4\\
0&1&1&1/2\\
1&1&1&1/2\\
1&2&2&3/4\\
\end{array}
\right\} .$$
This symmetry, after having discovered it from the operator, can be seen in the arrangement.\
The ACCCCCC-operator
--------------------
[***Arrangements $\bf 266$ and ${\bf 273}$*** ]{} have both $h^{11}=37$ and have rather complicated but [*identical*]{} Picard-Fuchs equations, with Riemann symbol
$$\left\{
\begin{array}{cccccccc}
6/1&32/1&6/1&32/1&6/1&?&?&32/1\\
\hline
-1&-1/2&-1/4&0&1/2&(-1+\sqrt{-3})/4&(-1-\sqrt{-3})/4&\infty\\
\hline
0&0 &0&0 &0&0&0&1/2\\
1&1/2&1&1/2&1&1&1&1\\
1&1/2&1&1/2&1&3&3&1\\
2&1 &2&1 &2&4&4&3/2\\
\end{array}
\right\}$$
Arrangement ${\bf 266}$ and ${\bf 273}$ both have $8$ quadruple points and no five-fold point or triple lines. The difference between the two is rather subtle; ${\bf 266}$ contains six planes which are in general position, which is not the case for ${\bf 273}$. In ${\bf 266}$ there are six of the quadruple points in a plane, which is not the case for ${\bf 273}$, etc. So clearly the two configurations are projectively very different. On the other hand, they have the same Hodge number $h^{11}=37$, and the equality of their Picard-Fuchs operators clearly suggest the varieties belonging to the two arrangements are birational, but were unable to find any transformation.\
The operator can be reduced to one with three singular points via the following steps
- Make the exponents at all $32/1$-points equal to , by exponent-shift.
- Translate the A-points to $0$ and $\infty$. Of course, the resulting operator has coefficients in ${\mathbb{Q}}(\sqrt{-3})$.
- Make a pull-back of order three, i.e. we use $t^3$ as new coordinate.
- The operator has now four singular points, where the points at $0$ and $\infty$ have exponents $0,1/3,1,4/3$. Bring the two other points to $0$ and $\infty$.
- The result is an operator invariant under $t \mapsto -t$. Make a quadratic pull-back and bring the $A$ point to the origin.
- The result is, after a scaling of the coordinate, the following operator
$$4 \Theta (3 \Theta - 1)(\Theta-1) (3 \Theta - 4)
+6 t \Theta (3 \Theta - 1) (288 \Theta^2 - 96 \Theta + 35)
+144 t^2 (12 \Theta +1)(3 \Theta+1)^2 (12\Theta+7)$$
with decorated Riemann symbol $$\left\{
\begin{array}{ccc}
?&6/1&32/1\\
\hline
0&-1/36&\infty\\
\hline
0& 0&1/12\\
1/3&1/2&1/3\\
1&1/2&1/3\\
4/3& 1&7/12\\
\hline
F&C&C\\
\end{array}
\right\}$$
Birational nature of the Picard-Fuchs operator
==============================================
Birational maps and strict equivalence
--------------------------------------
We would like to formulate a statement expressing the idea that the Picard-Fuchs operator is in some sense a birational invariant of a family. We formulate three theorems to this effect.
[**Theorem 1:**]{} Consider $\mathcal{X} {\longrightarrow}S$ and $\mathcal{X'} {\longrightarrow}S$ two proper smooth families of Calabi-Yau varieties over $S={\mathbb{P}}^1 \setminus \Sigma$. Let $\omega$ and $\omega'$ be holomorphic volume forms on $\mathcal{X}$ and $\mathcal{X'}$. If there is a birational map $$\phi:\mathcal{X} {\longrightarrow}\mathcal{X'},$$ then the Picard-Fuchs operators ${\mathcal{P}}(\mathcal{X},\omega)$ and ${\mathcal{P}}(\mathcal{X'},\omega')$ are strictly equivalent.\
[**proof:**]{}\
Let $X$ and $X'$ be fibres over the same general point. Recall that V. Batyrev [@Bat2] proved that if $$\phi:X {\longrightarrow}X'$$ is a birational map between smooth Calabi-Yau manifolds, then it induces isomorphisms $$\phi^*: H^{p,q}(X') \to H^{p,q}(X)$$ of Hodge groups. As a consequence, $\phi$ induces isomorphisms $$\phi_*: H_n(X,{\mathbb{Q}}) {\longrightarrow}H_n(X',{\mathbb{Q}}), \;\;\;\phi^*: H^n(X',{\mathbb{Q}}) {\longrightarrow}H^n(X,{\mathbb{Z}})$$ If $\omega \in H^{n,0}(X)=H^0(X,\Omega_X^n)$ and $\omega' \in H^{n,0}(X')=H^0(X',\Omega_X'^n)$ are holomorphic volume forms on $X$ and $X'$, then $$\phi^*(\omega') =\phi \omega$$ for some $\phi \in {\mathbb{C}}^*$. Furthermore, one has $$\int_{\gamma} \phi^*(\omega') =\int_{\phi_*(\gamma)} \omega$$
In the relative situation we consider a birational map that induce fibrewise birational isomorphism $$\phi: \mathcal{X} \dashrightarrow \mathcal{X'}$$ If $\omega(t)$ and $\omega'(t)$ are volume forms on $\mathcal{X}$ resp. $\mathcal{X'}$, then $$\phi^*(\omega'(t)) =\varphi(t) \omega(t)$$ for some rational function $\varphi(t)$ So we have $$\varphi(t)\int_{\gamma(t)}\omega(t)=\int_{\gamma(t)} \phi^*(\omega'(t)) = \int_{\phi_*(\gamma(t))} \omega'(t)$$ which shows that the period integrals for $\mathcal{X}$ and $\mathcal{X'}$ differ by multiplication by a rational function. So the Picard-Fuchs operators $${\mathcal{P}}(\mathcal{X},\omega)\;\;\textup{and}\;\;\; {\mathcal{P}}(\mathcal{X'},\omega')$$ are strictly equivalent.$\Diamond$.
Moduli spaces
-------------
We consider a smooth Calabi-Yau threefold $X$ with $h^{12}=1$. By the famous theorem of [Bogomolov]{}, [Tian]{} and [Todorov]{}, the local deformation theory of any Calabi-Yau manifold is unobstructed, and so by the classical deformation theory of [Kodaira]{}, [ Spencer]{} [@KS] and [Kuranishi]{} [@Kur] $X$ posses a versal deformation over a smooth $1$-dimensional disc $\Delta$: $$\begin{diagram}
\node{X} \arrow{e,J} \arrow{s} \node{\mathcal{X}} \arrow{s,r}{f}\\
\node{ \{0\} } \arrow{e,J} \node{\Delta}
\end{diagram}$$ Even if $X$ is projective there will not exist a well-defined moduli space for $X$, but rather one has a moduli stack. However, when we choose an ample line bundle $L$ on $X$, we can form families over corresponding quasi projective moduli spaces,[@Vie]. As a result, we may produce many apriori different projective families $$f: \mathcal{X}_L {\longrightarrow}S_L$$ which all have $X$ as fibre. Now versality of the Kuranishi family $\mathcal{X} {\longrightarrow}\Delta$ implies that if $X$ appears as fibre $f^{-1}(s)$ of any projective family $f: \mathcal{X'} {\longrightarrow}S$ over a curve $S$, and having a non-zero Kodaira-Spencer map at $s$, then this family is locally analytic isomorphic to the above model family $\mathcal{X} {\longrightarrow}\Delta$.\
Let us call two projective families $f: \mathcal{X} {\longrightarrow}S$ and $f': \mathcal{X}' {\longrightarrow}S'$ [*related*]{}, if there exists curve $D$ and finite maps $g:D {\longrightarrow}S$, $g':D {\longrightarrow}S'$ and an isomorphism $$\varphi: g^*\mathcal{X} {\longrightarrow}g'^*\mathcal{X}'$$
Clearly, if the families $\mathcal{X} {\longrightarrow}S$ and $\mathcal{X} {\longrightarrow}S'$ are both obtained as pull-back from a single family over $T$, then the families are related.\
[**Theorem 3:**]{} If $\mathcal{X} {\longrightarrow}S$ and $\mathcal{X}' {\longrightarrow}S'$ are families of Calabi-Yau manifolds, $\omega$ and $\omega'$ volume forms on $\mathcal{X}$ and $\mathcal{X}'$, then the Picard-Fuchs operators ${\mathcal{P}}(\mathcal{X},\omega)$ and ${\mathcal{P}}(\mathcal{X}',\omega')$ are related.\
[**Theorem 2:**]{} Let $f_1:\mathcal{X}_1 {\longrightarrow}S_1$ and $f_2: \mathcal{X}_2 {\longrightarrow}S_2$ be two projective families and $$\hat{\varphi}: \hat{X}_1 {\longrightarrow}\hat{X}_2$$ an isomorphism between between the formal neighbourhoods $\hat{X}_i$ of a fibre $X_i=f_i^{-1}(s_i) \subset \mathcal{X}_i$, then there exists an isomorphisms $\phi$, $\psi$ $$\begin{diagram}
\node{U_1} \arrow{e,t}{\phi} \arrow{s,l}{f_1} \node{U_2} \arrow{s,r}{f_2}\\
\node{ V_1 } \arrow{e,t}{\psi} \node{V_2}
\end{diagram}$$
of étale neighbourhoods $U_i$ of $X_i \subset \mathcal{X}_i$, $s_i
\in S_i$ ($i=1,2$) and thus the families $\mathcal{X}_1 {\longrightarrow}S_1$ and $\mathcal{X}_2 {\longrightarrow}S_2$ are related.\
[**proof:**]{} This is a particular case of a very general Theorem (1.7) (Uniqueness) proven by [Artin]{} in [@Art]. It states that if $F$ is a functor of locally of finite presentation and $\overline{\xi} \in F(\overline{A})$ an effective versal family, then the triple $(X,x,\xi)$ is unique up to local isomorphism for the étale topology, meaning that if $(X',x',\xi')$ is another algebraisation, then there is a third one, dominating both.\
A more down to earth proof can be given by an application of the [*nested approximation theorem*]{},[@KPPRM], [@Ron], theorem 5.2.1. In the above situation it can be applied as follows: assume we have two algebraisations $\mathcal{X} {\longrightarrow}S$, $\mathcal{X'} {\longrightarrow}S'$, given by equations of the form $F(x,t)=0, G(x',t')=0$. We are looking for a algebraic maps $$\phi(x,t),\;\;\;\psi(t)$$ that map $\mathcal{X}$ to $\mathcal{X}'$ and $S$ to $S'$. Hence we look for solutions to the system of equations $$F(x,t)=0,\;\;\;G(\phi(x,t),\psi(t))=0$$ Now as $\mathcal{X}$ and $\mathcal{X}'$ are analytically equivalent, we know that there exist formal solutions (or even analytic) $\phi, \psi$ to the above equations and hence by the nested approximation theorem we obtain a solution in the ring of algebraic power series. $\Diamond$.\
[**Theorem 3:**]{} If $\mathcal{X} {\longrightarrow}S$ and $\mathcal{X}' {\longrightarrow}S'$ are related families of Calabi-Yau manifolds, $\omega$ and $\omega'$ volume forms on $\mathcal{X}$ and $\mathcal{X}'$, then the Picard-Fuchs operators ${\mathcal{P}}(\mathcal{X},\omega)$ and ${\mathcal{P}}(\mathcal{X}',\omega')$ are related.\
[**Corollary:**]{} If $X$ is Calabi-Yau threefold with $h^{12}=1$, which appears as fibre in any two families $\mathcal{X}_i {\longrightarrow}S_i$ ($i=1,2$). Assume that the Kodaira-Spencer map of both families is non-zero at $X$. Then the Picard-Fuchs operator $\mathcal{P}_1$ for $\mathcal{X}_1$ and $\mathcal{P}_2$ for $\mathcal{X}_2$ are related.\
We claim that the following theorem holds, but the details will be given elsewhere.\
[**Theorem 4:**]{}\
If $\mathcal{X}_1 {\longrightarrow}S_1$ and $\mathcal{X}_2 {\longrightarrow}S_2$ are two smooth families of Calabi-Yau threefolds. Let $X_1 \subset \mathcal{X}_1$ and $X_2 \subset \mathcal{X}_2$ be two fibres. Assume that\
1) the Kodaira-Spencer map at $X_1$ and $X_2$ is non-zero.\
2) $X_1$ and $X_2$ are birational.\
Then the two families are related.\
Theorem ${\bf 4}$ would be very useful. For example, the Calabi-Yau threefolds of arrangements ${\bf 152}$ and ${\bf 153}$ both have $h^{11}=41$, so a priori could be birational. But no smooth fibre of Arr. 152/198 is birational to a smooth fibre of Arr.153/197. The reason is, that the Picard-Fuchs operator of ${\bf 152}$ has $KCCC$ as singularities, whereas that of ${\bf 153}$ has $ACCC$ singularities, so these operators are not related. We do not know of any other means of distinguishing members of these two families. We see that the Picard-Fuchs operator can be used as a powerful new birational invariant of a variety. We expect this to be applicable in much greater generality.\
Concluding remarks and questions
================================
As for families of $K3$-surfaces or other varieties, there is nothing special about families of Calabi-Yau three-folds that avoid having degenerations with maximal unipotent monodromy points and still have monodromy Zariski dense in $Sp_4({\mathbb{C}})$. One may ask further questions about the possible distribution of types of singularities. For example, the examples with second order Picard-Fuchs operators give variations which have only $K$-points. Do there exist families with only $K$-points and Picard-Fuchs operator of order four? Similarly, do there exists families with only $C$-points? It seems that only the current general lack of examples is the reason for this kind of ignorance.\
As for mirror symmetry, the SYZ-approach via dual torus fibrations (T-duality), [@SYZ], does in no way presupposes the presence of a MUM-point in the moduli space. The point is rather that the absence of a MUM-point does not give a clue where to look for an appropriate torus fibration. However, there is a the well-known idea, going back to [Miles Reid]{} [@Rei], that the different families of Calabi-Yau threefolds may all be connected via geometrical transitions. The most prominent such transition is the [*conifold transition*]{}, where we contract some lines on a Calabi-Yau threefold to form nodes, and smooth these out to produce another Calabi-Yau threefold with different Hodge numbers. As has been suggested by [Morrison]{},[@Mor2], mirror symmetry can, in some sense, be prolongated over such transitions, and in the case of the above mentioned conifold transition, the mirror symmetric process is that of nodal degeneration, followed by a crepant resolution. So if we connect an orphan family to another family that has a MUM-point via a transition, one is tempted to try construct a mirror manifold for orphans using this transition to a family with a MUM-point.\
We have seen in section 1 that the rigid Calabi-Yau from arrangement ${\bf 69}$ appears as a member of the orphan family ${\bf 70}$. But ${\bf 69}$ also is member of the family ${\bf 100}$, which contains even two MUM-points.
{height="5cm"}
[***Family of arrangements 100***]{}
So we have the following picture
{height="3cm"}
In the second part of the paper we will report on the double octics families that contain a MUM-point.\
Appendix A
==========
We will encounter certain modular forms over and over again. For the convenience of the reader we list here the first few Fourier coefficients of these forms.\
[*Weight two modular form:*]{} Associated to the elliptic curve $$y^2=x^3-x$$ of conductor $2^5=32$ there is the unique cusp form, [@HoMi]: $$f_{32} :=q\prod_{n=1}^{\infty} (1-q^{4n})^2(1-q^{8n})^2 \in S_2(\Gamma_0(32))$$
$$\begin{array}{|c||c|c|c|c|c|c|c|c|c|c|}
\hline
Name&2&3&5&7&11&13&17&19&23&29\\
\hline
\hline
f_{32}&0&0&-2&0&0&6&2&0&0&-10\\
\hline
\end{array}$$
[*Weight $3$ modular forms:*]{} The weight three cusp forms for $\Gamma_0(N)$ all will appear with a non-trivial character that will play no role for us. The forms of level $8$ and $16$ are uniquely determined by there level, which we use to name them. $$\begin{array}{|r||r|r|r|r|r|r|r|r|r|r|}
\hline
Name&2&3&5&7&11&13&17&19&23&CM-type\\
\hline
\hline
16&0&0&-6&0&0&10&-30&0&0&{\mathbb{Q}}(\sqrt{-1})\\
\hline
8&-2&-2&0&0&14&0&2&-34&0&{\mathbb{Q}}(\sqrt{-2}) \\
\hline
\end{array}$$ These two forms are $\eta$-products: $$8:=q\prod_{n=1}^{\infty}(1-q^n)^2(1-q^{2n})(1-q^{4n})(1-q^{8n})^2$$ $$16:=q\prod_{n=1}^{\infty}(1-q^{4n})^6$$ The Galois representation associated to the form 8 is the tensor square of the Galois action associated to the form $f_{32}$. For more information on weight $3$ forms we refer to [@Schu1].\
[*Weight $4$ cusp forms*]{}\
For the weight $4$ cusp forms for $\Gamma_0(N)$ we will used the notation used in the book of [Meyer]{} [@Mey].
$$\begin{array}{|r||r|r|r|r|r|r|r|r|r|}
\hline
Name&2&3&5&7&11&13&17&19&23\\
\hline
\hline
6/1 &-2&-3&6&-16&12&38&-126&20&168\\
\hline
8/1 &0&-4&-2&24&-44&22&50&44 &-56\\
\hline
12/1& 0& 3&-18& 8& 36& -10& 18& -100& 72\\
\hline
32/1&0&0& 22& 0& 0&-18&-94&0 &0\\
\hline
32/2&0&8&-10&16&-40&-50&-30&40&48\\
\hline
\end{array}$$
The first two of these forms are also nice $\eta$-products: $$6/1= q\prod_{n=1}^{\infty}(1-q^n)^2(1-q^{2n})^2(1-q^{3n})^2(1-q^{6n})^2$$ $$8/1= q\prod_{n=1}^{\infty}(1-q^{2n})^4(1-q^{4n})^4$$ The Galois representation associated to the form 32/1 is the tensor cube of the Galois action associated to the form $f_{32}$.
There is also one Hilbert modular form $h$ for the field ${\mathbb{Q}}(\sqrt{2})$ that plays a role. It is the unique cuspform of weight $(4,2)$ and level $6\sqrt{2}$; its coefficients for the first inert primes are: $$\begin{array}{|c||c|c|c|c|c|c|c|c|c|}
\hline
Name&2&3&5&7&11&13&17&19&23\\
\hline
h&0&9&10&4\sqrt{2}+16&-726&2938&16\sqrt{2}-62&6650&-8\sqrt{2}+40\\
\hline
\end{array}$$
Appendix B
----------
------------------------------------------------------------------------
\#1\#2[[\#1\#2]{}]{} \#1 \#2 \#3 \#4 \#5 [\#1&\#2&\#3&\#4&\#5]{} \#1 [**\#1.** ]{}
4 $xyzv \left( x+y \right) \left( x+ty+tz-v \right) \left( x+y+tz-v \right) \left( y+z \right) $ ${\Theta}^{2}-t \left( \Theta+\frac12 \right) ^{2}
$
$\left\{
\begin{array}{ccc}
1&0&\infty\\\hline
0&0&1/2\\0&0&1/2
\end{array}\right\}
$
13 $xyzv \left( z+y \right) \left( x-z-v \right) \left( x+y \right) \left( x-z+tv \right)
$ $\Theta^{2}+t(\Theta+\frac12)^{2}$
$\left\{
\begin{array}{ccc}
0&-1&\infty\\\hline
0&0&1/2\\0&0&1/2
\end{array}\right\}
$
34 $xyzv \left( x+y \right) \left( x+z \right) \left( x+y+z+v \right) \left( y-z+tv \right)
$ $\Theta^{2}-t^{2}(\Theta+\frac12)^{2}$
$\left\{
\begin{array}{cccc}
1&0&-1&\infty\\\hline
0&0&0&1/2\\1/2&0&1/2&1/2
\end{array}\right\}
$
72 $xyzv \left( x-y-v \right) \left( x+y+z \right) \left( y+tz+tv
\right) \left( y+z+v \right)
$ ${\Theta}^{2}
+t \left( -3\,{\Theta}^{2}-2\,\Theta-\frac12 \right)
+{t}^{2} \left( \Theta+1 \right) \left( 2\,\Theta+1 \right)
$
$\left\{
\begin{array}{cccc}
1&1/2&0&\infty\\\hline
0&0&0&1/2\\0&1/2&0&1
\end{array}\right\}
$
261 $xyzv \left( x-y-z+v \right) \left( x+y+z+v \right) \left( x-y+tz-tv \right) \left( x+y+tz+tv \right)
$ ${\Theta}^{2}-{t}^{2} \left( \Theta+1 \right) ^{2}$
$\left\{
\begin{array}{cccc}
1&0&-1&\infty\\\hline
0&0&0&1\\0&0&0&1
\end{array}\right\}
$
264 $xyzv \left( x+(2-t)v+2\,y-(2-t)\,z \right)
\left( -x-y+2\,z-(2-t)v \right) \left( x+y+tz \right)
\left( y+2-2\,z \right)$ ${\Theta}^{2}-1/4\,{t}^{2} \left( \Theta+1 \right) ^{2}
$
$\left\{
\begin{array}{cccc}
2&0&-2&\infty\\\hline
0&0&0&1\\0&0&0&1
\end{array}\right\}
$
270 $xyzv \left( x+y+z \right) \left( y+z+v \right) \left(
xt-2\,y+tz+tv \right) \left( -x -2\,y+tz-v \right)
$ ${\Theta}^{2}
+t \left( \frac32\,{\Theta}^{2}+\frac32\,\Theta+\frac12 \right)
+\frac12\,{t}^{2} \left( \Theta+1 \right) ^{2}
$
$\left\{
\begin{array}{cccc}
0&-1&-2&\infty\\\hline
0&0&0&1\\0&0&0&1
\end{array}\right\}
$
Appendix C
----------
------------------------------------------------------------------------
\#1\#2[[\#1\#2]{}]{} \#1 \#2 \#3 \#4 \#5 [\#1&\#2&\#3&\#4&\#5]{} \#1 [**\#1.** ]{}
**33.** $ xyzv \left( x+y \right)\left( y+z \right) \left( x-z+v \right) \left( x-y-z+tv \right)
$\
$\displaystyle {\Theta}^{2} \left( \Theta-1 \right) ^{2}
-\frac1{8}\,t{\Theta}^{2} \left( 20\,{\Theta}^{2}+3 \right)
+\frac1{16}\,{t}^{2} \left( 8\,{\Theta}^{2}+8\,\Theta+3 \right) \left( 2\,\Theta+1 \right) ^{2}
-\frac{1}{32}\,{t}^{3} \left( 2\,\Theta+3 \right) ^{2} \left( 2\,\Theta+1\right) ^{2}
$
$\displaystyle
\left\{ \begin {array}{cccc} 0 & 1& 2& \infty \\\hline
0&0&0&1/2\\
0&1/2&1&1/2\\
1&1/2&1&3/2\\
1&1&2&3/2
\end {array} \right\} $
**35.** $xyz \left( x-v \right)\left( y-v \right) \left( z-v \right) \left( x-y \right)
\left( x+ty+(1-t)z-v \right)
$\
$\Theta\, \left( \Theta-1 \right) \left(\Theta-\frac12 \right) ^{2}
-\frac14\,t{\Theta}^{2} \left( 4\,{\Theta}^{2}+3 \right)
-\frac14\,{t}^{2} \left( {\Theta}^{2}+\Theta+1 \right) \left( 2\,\Theta+1 \right) ^{2}
+\frac14\,{t}^{3} \left( 2\,\Theta+3 \right) \left( 2\,\Theta+1 \right) \left( \Theta+1 \right) ^{2}
$
$\left\{ \begin {array}{cccc}1&0&-1&\infty \\
\hline 0&0&0&1/2\\
0&1/2&1&1\\
1&1/2&1&1\\
1&1&2&3/2
\end {array} \right\} $
**70.** $
yxzv \left( x+ty \right)\left( y-z-v \right) \left( x-y-v \right) \left( x-y+z \right)
$\
$
{\Theta}^{2} \left( \Theta-1 \right) ^{2}
+\frac18\,t{\Theta}^{2} \left( 20\,{\Theta}^{2}+3 \right)
+\frac1{16}\,{t}^{2} \left( 8\,{\Theta}^{2}+8\,\Theta+3 \right) \left( 2\,\Theta+1 \right) ^{2}
+\frac1{32}\,{t}^{3} \left( 2\,\Theta+3 \right) ^{2} \left( 2\,\Theta+1 \right) ^{2}
$
$\left\{ \begin {array}{cccc} 0 & -1& -2& \infty \\
\hline 0&0&0&1/2\\
0&1/2&1&1/2\\
1&1/2&1&3/2\\
1&1&2&3/2
\end {array} \right\} $
**71.** $xyzv \left( x+y \right) \left( x+y+z+v \right) \left( ty-z-v \right) \left( -x+ty-z \right)
$\
$
\Theta\, \left( \Theta-1 \right) \left( \Theta -\frac12 \right) ^{2}
+t{\Theta}^{2} \left( 4\,{\Theta}^{2}+1 \right)
+\frac1{16}\,{t}^{2} \left( 20\,{\Theta}^{2}+20\,\Theta+9 \right) \left( 2\,\Theta+1 \right) ^{2}
+\frac18\,{t}^{3} \left( 2\,\Theta+3 \right) ^{2} \left( 2\,\Theta+1 \right) ^{2}
$
$\left\{
\begin {array}{cccc}
0&-1/2&-1&\infty\\\hline
0&0&0&1/2\\
1/2&1&1/2&1/2\\
1/2&1&1/2&3/2\\
1&2&1&3/2
\end {array} \right\} $
**97.** $xyzv \left( x+y \right) \left( x+y+z+v \right) \left( -x+tz-v \right) \left( y-v-z \right)
$\
$\Theta\, \left( \Theta-1 \right) \left( \Theta -\frac 12\right) ^{2}
+\frac18\,t{\Theta}^{2} \left( 20\,{\Theta}^{2}+7 \right) \
+\frac1{32}\,{t}^{2} \left( 16\,{\Theta}^{2}+16\,\Theta+7 \right) \left( 2\,\Theta+1 \right) ^{2}
+\frac1{32}\,{t}^{3} \left( 2\,\Theta+3 \right) ^{2} \left( 2\,\Theta+1 \right) ^{2}
$
$\left\{
\begin {array}{cccc}0&-1&-2&\infty \\
\hline
0&0&0&1/2\\
1/2&0&1&1/2\\
1/2&1&1&3/2\\
1&1&2&3/2
\end {array} \right\} $
**98.** $xyzv \left( x+z-v \right) \left( x+z+y \right) \left( y+z+v \right) \left( y+tz+tv \right)
$\
$
{\Theta}^{2} \left( \Theta-1 \right) ^{2}
-\frac14\,t{\Theta}^{2} \left( 16\,{\Theta}^{2}+3 \right)
+\frac14\,{t}^{2} \left( 5\,{\Theta}^{2}+5\,\Theta+3 \right) \left( 2\,\Theta+1 \right) ^{2}
-\frac12\,{t}^{3} \left( 2\,\Theta+3 \right) \left( 2\,\Theta+1 \right) \left( \Theta+1 \right) ^{2}
$
$\left\{
\begin {array}{cccc}1&1/2&0&\infty \\
\hline
0&0&0&1/2\\
0&1&0&1\\1&1&1&1\\
1&2&1&3/2
\end {array} \right\} $
152 $
xyzv \left( x+v-y-z \right) \left( x+y+z+v \right) \left(
x-y+tz-tv \right) \left( y+v \right)
$ $
\Theta\, \left( \Theta-1 \right) \left( \Theta -\frac 12 \right) ^{2}
+\frac12\,t\Theta\, \left( 2\,{\Theta}^{3}-8\,{\Theta}^{2}+6\,\Theta-1 \right)
+{t}^{2} \left( -2\,{\Theta}^{4}-4\,{\Theta}^{3}-\frac{11}4\,{\Theta}^{2}-{\frac {17}{4}}\,\Theta-{\frac {11}{16}} \right) \bb
+{t}^{3} \left( -2\,{\Theta}^{4}+\frac14\,{\Theta}^{2}+\frac72\,\Theta+{\frac {9}{8}} \right)
+\frac1{16}\,{t}^{4} \left( 2\,\Theta+1 \right) \left( 8\,{\Theta}^{3}+44\,{\Theta}^{2}+62\,\Theta+25 \right)
\bb+\frac14\,{t}^{5} \left( 2\,\Theta+3 \right) \left( 2\,\Theta+1 \right) \left( \Theta+1 \right) ^{2}
$ $\left\{
\begin {array}{cccc}
1&0&-1&\infty\\\hline
0&0&0&1/2\\
1/2&1/2&0&1\\
1/2&1/2&2&1\\
1&1&2&3/2
\end {array} \right\} $
153 $xyzv \left( x+z+y \right) \left( -x+ty-v \right) \left( -x+ty-z-v \right) \left( y+z+v \right)
$ $\Theta\, \left( \Theta-1 \right) \left(\Theta -\frac12\right) ^{2}
+\frac18\,t\Theta\, \left( 28\,{\Theta}^{3}-16\,{\Theta}^{2}+17\,\Theta-2 \right)
+{t}^{2} \left( {\frac {19}{4}}\,{\Theta}^{4}+\frac72\,{\Theta}^{3}+{\frac {39}{8}}\,{\Theta}^{2}+{\frac {13}{8}}\,\Theta+{\frac {19}{64}} \right) \bb
+{t}^{3} \left( {\frac {25}{8}}\,{\Theta}^{4}+6\,{\Theta}^{3}+{\frac {109}{16}}\,{\Theta}^{2}+\frac72\,\Theta+{\frac {89}{128}} \right)
+{\frac {1}{64}}\,{t}^{4} \left( 2\,\Theta+1 \right) \left( 32\,{\Theta}^{3}+80\,{\Theta}^{2}+82\,\Theta+29 \right)
\bb+\frac1{32}\,{t}^{5} \left( 2\,\Theta+3 \right) \left( 2\,\Theta+1 \right) \left( \Theta+1 \right) ^{2}
$ $\left\{
\begin {array}{cccc}0&-1&-2&\infty \\\hline
0&0&0&1/2\\
1/2&1/2&1/2&1\\
1/2&1/2&3/2&1\\
1&1&2&3/2
\end {array} \right\} $
197 $xyzv \left( x-y-z+v \right) \left( x+tz+v \right) \left( x+ty+tz \right) \left( ty+tz+v \right)
$ ${\Theta}^{2} \left(\Theta - \frac12\right) \left( \Theta+\frac12 \right)
+\frac18\,t \left( 2\,\Theta+1 \right) \left( 32\,{\Theta}^{3}+16\,{\Theta}^{2}+18\,\Theta+5 \right)
+{t}^{2} \left( 25\,{\Theta}^{4}+52\,{\Theta}^{3}+{\frac {121}{2}}\,{\Theta}^{2}+37\,\Theta+{\frac {145}{16}} \right) \bb
+{t}^{3} \left( 38\,{\Theta}^{4}+124\,{\Theta}^{3}+183\,{\Theta}^{2}+133\,\Theta+{\frac {307}{8}} \right)
+{t}^{4} \left( \Theta+1 \right) \left( 28\,{\Theta}^{3}+100\,{\Theta}^{2}+133\,\Theta+63 \right)
\bb+2\,{t}^{5} \left( \Theta+2 \right) \left( \Theta+1 \right) \left( 2\,\Theta+3 \right) ^{2}
$ $\left\{
\begin {array}{cccc}0&-1/2&-1&\infty \\\hline
-1/2&0&0&1\\
0&1/2&1/2&3/2\\
0&3/2&1/2&3/2\\
1/2&2&1&2
\end {array} \right\} $
198 $xyzv \left( x-y-v \right) \left( x+y+z \right) \left( x-y-z+tx \right) \left( y+z+v \right)
$ $\Theta\, \left( \Theta-1 \right) \left( \Theta-\frac12 \right) ^{2}
+\frac18\,t{\Theta}^{2} \left( 24\,{\Theta}^{2}+5 \right)
+{t}^{2} \left( {\frac {13}{4}}\,{\Theta}^{4}+\frac{13}2\,{\Theta}^{3}+{\frac {81}{16}}\,{\Theta}^{2}+{\frac {29}{16}}\,\Theta+{\frac {5}{16}} \right) \bb
+{t}^{3} \left( \frac32\,{\Theta}^{4}+6\,{\Theta}^{3}+8\,{\Theta}^{2}+4\,\Theta+{\frac {25}{32}} \right)
+{\frac {1}{64}}\,{t}^{4} \left( 2\,\Theta+5 \right) ^{2} \left( 2\,\Theta+1 \right) ^{2}
$ $\left\{
\begin {array}{cccc}0&-1&-2&\infty \\\hline
0&0&0&1/2\\
1/2&1/2&1/2&1/2\\
1/2&1/2&1/2&5/2\\
1&1&1&5/2
\end {array} \right\} $
243 $xyzv \left( x+y+v \right) \left( x+y+z \right) \left( x+ty+z+v \right) \left( y+z+v \right)
$ $\Theta\, \left( \Theta-2 \right) \left( \Theta-1 \right) ^{2}
-\frac16\,t\Theta\, \left( \Theta-1 \right) \left( 19\,{\Theta}^{2}-19\,\Theta+9 \right)
+\frac13\,{t}^{2}{\Theta}^{2} \left( 11\,{\Theta}^{2}+4 \right)
-\frac1{24}\,{t}^{3} \left( 11\,{\Theta}^{2}+11\,\Theta+5 \right)
\left( 2\,\Theta+1 \right) ^{2} \bb
+\frac1{48}\,{t}^{4} \left( 2\,\Theta+3 \right) ^{2} \left( 2\,\Theta+1 \right) ^{2}
$ $\left\{
\begin {array}{ccccc}2&3/2&1&0&\infty\\\hline
0&0&0&0&1/2\\
1&1&1/2&1&1/2\\
1&1&1/2&1&3/2\\
2&2&1&2&3/2
\end {array} \right\} $
247 $xyzv \left( x-y-v \right) \left( x+y+z \right) \left( -x+tz-tv \right) \left( y+z+v \right)
$ $
{\Theta}^{2} \left( \Theta-1 \right) ^{2}
+t{\Theta}^{2} \left( 5\,{\Theta}^{2}+1 \right)
+{t}^{2} \left( 2\,{\Theta}^{2}+2\,\Theta+1 \right) \left(
2\,\Theta+1 \right) ^{2}
+{t}^{3} \left( 2\,\Theta+3 \right) \left( 2\,\Theta+1 \right) \left( \Theta+1 \right) ^{2}
$ $\left\{
\begin {array}{cccc}0&-1/2&-1&\infty \\\hline
0&0&0&1/2\\
0&1/2&1&1\\
1&1/2&1&1\\
1&1&2&3/2
\end {array} \right\} $
248 $xyzv \left( x+z+v \right) \left( x+y+z \right) \left( x+(y+1)y-tz+v \right) \left( y-z-v \right)
$ $\Theta\, \left( \Theta-2 \right) \left( \Theta-1 \right) ^{2}
+\frac16\,t\Theta\, \left( \Theta-1 \right) \left( 37\,{\Theta}^{2}-61\,\Theta+36 \right)
+\frac16\,{t}^{2}\Theta\, \left( 91\,{\Theta}^{3}-124\,{\Theta}^{2}+121\,\Theta-36 \right) \bb
+{t}^{3} \left( {\frac {115}{6}}\,{\Theta}^{4}-\frac53\,{\Theta}^{3}+{\frac {107}{6}}\,{\Theta}^{2}+\frac23\,\Theta+\frac12 \right)
+{t}^{4} \left( {\frac {79}{6}}\,{\Theta}^{4}+16\,{\Theta}^{3}+{\frac
{113}{6}}\,{\Theta}^{2}+8\,\Theta+\frac32 \right)\bb
+\frac16\,{t}^{5} \left( 2\,\Theta+1 \right) \left(
14\,{\Theta}^{3}+29\,{\Theta}^{2}+27\,\Theta+9 \right)
+\frac16\,{t}^{6} \left( 2\,\Theta+3 \right) \left( 2\,\Theta+1
\right) \left( \Theta+1 \right) ^{2}
$ $\left\{
\begin {array}{cccccc} 0&-1/2&-1&-3/2&-2&\infty\\\hline
0&0&0&0&0&1/2\\
1&1&0&1&1&1\\
1&1&2&1&1&1\\
2&2&2&2&2&3/2
\end {array} \right\} $
250 $xyzv \left( x+y+z \right) \left( x+ty-z+v \right) \left( x+z+v \right) \left( y+z-v \right)
$ $\Theta\, \left( \Theta-1 \right) \left( \Theta -\frac12\right) ^{2}
+\frac18\,t\Theta\, \left( 44\,{\Theta}^{3}-96\,{\Theta}^{2}+65\,\Theta-12 \right)
+{t}^{2} \left( \frac{19}2\,{\Theta}^{4}-23\,{\Theta}^{3}+{\frac {131}{8}}\,{\Theta}^{2}-{\frac {47}{8}}\,\Theta-\frac14 \right)
\bb+{t}^{3} \left( \frac52\,{\Theta}^{4}-20\,{\Theta}^{3}-{\frac
{23}{4}}\,\Theta-{\frac {17}{32}} \right)
-\frac1{32}\,{t}^{4} \left( 68\,{\Theta}^{2}+100\,\Theta+53 \right) \left( 2\,\Theta+1 \right) ^{2}
\bb-\frac14\,{t}^{5} \left( 8\,{\Theta}^{2}+14\,\Theta+9 \right) \left( 2\,\Theta+1 \right) ^{2}
-\frac18\,{t}^{6} \left( 2\,\Theta+3 \right) ^{2} \left( 2\,\Theta+1 \right) ^{2}
$ $\left\{
\begin {array}{cccccc}1&0&-1/2&-1&-2&\infty \\\hline
0&0&0&0&0&1/2\\
1&1/2&1&1/2&1&1/2\\
1&1/2&3&1/2&1&3/2\\
2&1&4&1&2&3/2
\end {array} \right\} $
252 $xyzv \left( x+y+v \right) \left( x+y+z \right) \left(-x+tz+v \right) \left( -x-2\,y+tz-v \right)
$ ${\Theta}^{2} \left( \Theta-1 \right) ^{2}
+\frac12\,t{\Theta}^{2} \left( 5\,{\Theta}^{2}+1 \right)
+\frac14\,{t}^{2} \left( 2\,{\Theta}^{2}+2\,\Theta+1 \right) \left( 2\,\Theta+1 \right) ^{2}
+\frac18\,{t}^{3} \left( 2\,\Theta+3 \right) \left( 2\,\Theta+1 \right) \left( \Theta+1 \right) ^{2}
$ $\left\{
\begin {array}{cccc} 0&-1&-2&\infty\\\hline
0&0&0&1/2\\
0&1/2&1&1\\
1&1/2&1&1\\
1&1&2&3/2
\end {array} \right\} $
258 $xyzv \left( x-y+2\,z-2v \right) \left( x-y+z-v \right) \left( x+ty+z+tv \right) \left( y-z+2v \right)
$ $\Theta\, \left( \Theta-1 \right) \left(\Theta -\frac12\right) ^{2}
+\frac18\,t\Theta\, \left( 20\,{\Theta}^{3}+48\,{\Theta}^{2}-21\,\Theta+6 \right)
+{t}^{2} \left( -{\Theta}^{4}+19\,{\Theta}^{3}+{\frac {39}{2}}\,{\Theta}^{2}+{\frac {47}{8}}\,\Theta+{\frac {11}{8}} \right) \bb
+{t}^{3} \left( -5\,{\Theta}^{4}-5\,{\Theta}^{3}+{\frac {61}{2}}\,{\Theta}^{2}+{\frac {127}{8}}\,\Theta+{\frac {37}{8}} \right)
+{t}^{4} \left( -{\Theta}^{4}-21\,{\Theta}^{3}-\frac{21}2\,{\Theta}^{2}-{\frac {9}{8}}\,\Theta+{\frac {7}{8}} \right) \bb
+\frac18\,{t}^{5} \left( 2\,\Theta+1 \right) \left( 10\,{\Theta}^{3}-9\,{\Theta}^{2}-27\,\Theta-13 \right)
+\frac14\,{t}^{6} \left( 2\,\Theta+3 \right) \left( 2\,\Theta+1 \right) \left( \Theta+1 \right) ^{2}
$ $\left\{
\begin {array}{cccccc}1&0&-1/2&-1&-2&\infty \\\hline
0&0&0&0&0&1/2\\
1&1/2&1&0&1&1\\
3&1/2&1&1&1&1\\
4&1&2&1&2&3/2
\end {array} \right\} $
266 $xyzv \left( 2x+y+2v \right) \left( x+(t+1)y-z+v \right) \left( x+ty+z \right) \left( y-2\,z+2v \right)
$ $\Theta\, \left( \Theta-1 \right) \left(\Theta -\frac12\right) ^{2}
+\frac14\,t\Theta\, \left( 44\,{\Theta}^{3}-48\,{\Theta}^{2}+37\,\Theta-6 \right)
+{t}^{2} \left( 50\,{\Theta}^{4}-56\,{\Theta}^{3}+40\,{\Theta}^{2}-\frac52\,\Theta+\frac38 \right) \bb
+{t}^{3} \left( 120\,{\Theta}^{4}-288\,{\Theta}^{3}-75\,{\Theta}^{2}-105\,\Theta-21 \right)
+{t}^{4} \left( 112\,{\Theta}^{4}-1008\,{\Theta}^{3}-718\,{\Theta}^{2}-720\,\Theta-{\frac {303}{2}} \right) \bb
+{t}^{5} \left( -224\,{\Theta}^{4}-2464\,{\Theta}^{3}-1924\,{\Theta}^{2}-1628\,\Theta-324 \right)
+{t}^{6} \left( -960\,{\Theta}^{4}-4224\,{\Theta}^{3}-4296\,{\Theta}^{2}-2448\,\Theta-450 \right) \bb
+{t}^{7} \left( -1600\,{\Theta}^{4}-4992\,{\Theta}^{3}-6368\,{\Theta}^{2}-3504\,\Theta-696 \right)
-32\,{t}^{8} \left( 2\,\Theta+1 \right) \left( 22\,{\Theta}^{3}+57\,{\Theta}^{2}+59\,\Theta+21 \right) \bb
-128\,{t}^{9} \left( 2\,\Theta+3 \right) \left( 2\,\Theta+1 \right) \left( \Theta+1 \right) ^{2}
$ $\left\{
\begin {array}{cccccccc}
1/2&0&-1/4&-1/2&-1&(-1+\sqrt{-3})/4&(-1-\sqrt{-3})/4&\infty\\\hline
0&0&0&0&0&0&0&1/2\\
1&1/2&1&1/2&1&1&1&1\\
1&1/2&1&1/2&1&3&3&1\\
2&1&2&1&2&4&4&3/2
\end {array} \right\} $
273 $xyzv \left( x+y+z \right) \left( 2\,x-2\,z-v \right) \left( x+2\,ty-z+tv \right) \left( 2\,y+2\,z+v \right)
$ $\Theta\, \left( \Theta-1 \right) \left(\Theta - \frac12\right) ^{2}
+\frac14\,t\Theta\, \left( 44\,{\Theta}^{3}-48\,{\Theta}^{2}+37\,\Theta-6 \right)
+{t}^{2} \left( 50\,{\Theta}^{4}-56\,{\Theta}^{3}+40\,{\Theta}^{2}-\frac52\,\Theta+\frac38 \right)\bb
+{t}^{3} \left( 120\,{\Theta}^{4}-288\,{\Theta}^{3}-75\,{\Theta}^{2}-105\,\Theta-21 \right)
+{t}^{4} \left( 112\,{\Theta}^{4}-1008\,{\Theta}^{3}-718\,{\Theta}^{2}-720\,\Theta-{\frac {303}{2}} \right)\bb
+{t}^{5} \left( -224\,{\Theta}^{4}-2464\,{\Theta}^{3}-1924\,{\Theta}^{2}-1628\,\Theta-324 \right)
+{t}^{6} \left( -960\,{\Theta}^{4}-4224\,{\Theta}^{3}-4296\,{\Theta}^{2}-2448\,\Theta-450 \right) \bb
+{t}^{7} \left( -1600\,{\Theta}^{4}-4992\,{\Theta}^{3}-6368\,{\Theta}^{2}-3504\,\Theta-696 \right)
-32\,{t}^{8} \left( 2\,\Theta+1 \right) \left( 22\,{\Theta}^{3}+57\,{\Theta}^{2}+59\,\Theta+21 \right) \bb
-128\,{t}^{9} \left( 2\,\Theta+3 \right) \left( 2\,\Theta+1 \right) \left( \Theta+1 \right) ^{2}
$ $\left\{
\begin {array}{cccccccc}
1/2&0&-1/4&-1/2&-1&(-1+\sqrt{-3})/4&(-1-\sqrt{-3})/4&\infty\\\hline
0&0&0&0&0&0&0&1/2\\
1&1/2&1&1/2&1&1&1&1\\
1&1/2&1&1/2&1&3&3&1\\
2&1&2&1&2&4&4&3/2
\end {array} \right\} $
[99]{}
G. Almkvist, C. van Enckevort, D. van Straten, W. Zudilin, [*Tables of Calabi–Yau operators*]{}, [arXiv:math/0507430]{}(2010).
G. Almkvist, D. van Straten, [*Update on Calabi–Yau operators*]{}, in preparation.
G. Almkvist, W. Zudilin, [*Differential equations, mirror maps and zeta values*]{}, in: Mirror symmetry. V, 481515, AMS/IP Stud. Adv. Math., [**38**]{}, Amer. Math. Soc., Providence, RI, (2006).
M. Artin, [*Algebraization of formal muduli: $I$*]{}, 1969 Global Analysis (Papers in Honor of K. Kodaira) pp. 21–71, Univ.Tokyo Press, Tokyo.
V. Batyrev, [*Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties*]{}, Journal of Algebraic Geometry [**3**]{}, (1994), 493 - 535.
V. Batyrev, [*Birational Calabi-Yau n-folds have equal Betti numbers*]{}, in: New trends in algebraic geometry (Warwick, 1996), 1–11, London Math. Soc. Lecture Note Ser., 264, Cambridge Univ. Press, Cambridge, 1999.
P. Candelas, X. de la Ossa, P. Green, L. Parkes, [*An exactly soluble superconformal theory from a mirror pair of Calabi–Yau manifolds*]{}, Phys. Lett. B 258 (1991), no. 1-2, 118 - 126.
Calabi–Yau Database, Version 2.0 [(url: www2.mathematik.uni-mainz.de/CYequations/db/)]{}, Version 3.0 [(url: cydb.mathematik.uni-mainz.de)]{}.
C. Consani, J. Scholten, [*Arithmetic on a quintic threefold*]{}, Internat. J. Math. [**12**]{} (2001), no. 8, 943–972.
S. Cynk, C. Meyer, [*Modularity of some some non-rigid double octic Calabi-Yau threefolds*]{}, Rocky Mountain J. of Math.[**38**]{}(6), (2008), 1937-1958.
S. Cynk, D. van Straten, [*Infinitesimal deformations of double covers of smooth algebraic varieties*]{}, Math. Nachr. 279 (2006), no. 7, 716–726.
S. Cynk, D. van Straten, [*Calabi–Yau conifold expansions*]{}, in: Arithmetic and geometry of K3–surfaces and Calabi–Yau threefolds, 499-515, Fields Inst. Commun., [**67**]{}, Springer, New York, (2013).
S. Cynk, D. van Straten, [*Picard–Fuchs equations for double octics II, (The MUM cases)*]{}, in preparation.
S. Cynk, B. Kocel-Cynk, [*Classification of double octic Calabi-Yau threefolds*]{}, [ArXiv: 1612.04364v1 \[math.AG\]]{}
L. Dieulefait, A. Pacetti, M. Schütt, [*Modularity of the Consani-Scholten quintic.*]{} With an appendix by José Burgos Gil and Pacetti. Doc. Math. 17 (2012), 953–987.
P. Deligne, [*Local behavior of Hodge structures at infinity*]{}, Mirror Symmetry II, Studies in advanced mathematics, vol. 1, AMS/IP, 1997, 683 - 699.
B. Dwork, [*On the Zeta function of a hypersurface III*]{}, Ann. Math.(2) [**83**]{}, 457-519 (1966).
B. Dwork, [*$p$-adic cycles*]{}, Inst. Hautes Etudes Sci. Publ. Math. No. ${\bf 37}$ (1969), 27-115.
A. Garbagnati, B. van Geemen, [*The Picard–Fuchs equation of a family of Calabi–Yau threefolds without maximal unipotent monodromy*]{}, Int. Math. Res. Not. IMRN 2010, no. 16, 3134 - 3143.
B. van Geemen, J. Top, [*An isogeny of K3-surfaces*]{}, Bull. London Math.Soc [**38**]{} (2006) 209-223.
M. Gross, B. Siebert, [*An invitation to toric degenerations*]{}, in: Surveys in differential geometry. Volume XVI. Geometry of special holonomy and related topics, 43 78, Surv. Differ. Geom., 16, Int. Press, Somerville, MA, 2011.
P. Griffiths, [*On the periods of certain rational integrals*]{}, I, II. Ann. of Math. (2) [**90**]{} (1969), 460 - 495; ibid. (2) [**90**]{} (1969) 496 - 541.
M. Gross, D. Huybrechts, D. Joyce [*Calabi–Yau manifolds and related geometries. Lectures at a Summer School in Nordfjordeid, Norway, June 2001*]{}, Springer-Verlag Berlin Heidelberg (2003).
T. Honda, I. Miyawaki, [*Zeta-functions of elliptic curves of $2$-power conductor*]{} J. Math. Soc. Japan Vol.[**26**]{}, No.2, (1974), 362-373.
K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, E. Zaslow, [*Mirror symmetry*]{}, Clay Mathematics Monographs [**1**]{}, AMS, Providence, RI. (2003).
E. Ince, [*Ordinary Differential Equations*]{}, Longmans, Green and Co. Ltd., New York, Toronto, Calcutta, Madras (1927).
K. Kodaira, D. Spencer, [*On deformations of complex analytic structures*]{}, Ann. of Math. [**67**]{} (1958), 328-466.
M. Kuranishi,[*On the locally complete families of complex analytic structures*]{}, Ann. of Math. [**75**]{}, (1962), 536-577.
G. Kurke, G. Pfister, D. Popescu, M. Roczen, T. Most owski,[*Die Approximationseigenschaft lokaler Ringe*]{}, Lecture Notes in Mathematics [**634**]{}, Springer-Verlag, Berlin-New York, (1978).
P. Lairez, [*Computing periods of rational integrals*]{}, Mathematics of Computation, Vol. 85, Number 300 (2016), 1719-1752.
C. Meyer, [*Modular Calabi–Yau threefolds*]{}. Fields Institute Monographs [**22**]{}, American Mathematical Society, Providence, RI, (2005).
D. R. Morrison, [*Geometric aspects of mirror symmetry*]{}, Mathematics unlimited 2001 and beyond, 899 - 918, Springer, Berlin, (2001).
D. R. Morrison, [*Through the looking glass*]{}, in: Mirror symmetry, III (Montreal, PQ, 1995), 263–277, AMS/IP Stud. Adv. Math., 10, Amer. Math. Soc., Providence, RI, 1999.
C. Peters, J. Steenbrink, [*Mixed Hodge Structures*]{}, Ergebnisse der Mathematik und ihre Grenzgebiete, 3. Folge, Springer Verlag, (2008).
M. Reid, [*The moduli space of 3-folds with K=0 may nevertheless be irreducible*]{}. Math. Ann. 278 (1987), 329–334.
J.-C. Rohde, [*Maximal automorphisms of Calabi–Yau manifolds versus maximally unipotent monodromy*]{}, Manuscripta Math. [**131**]{} (2010), no. 3-4, 459 - 474.
J.-C. Rohde, [*Maximal automorphisms of Calabi-Yau manifolds versus maximally unipotent monodromy*]{}, Talk held at the conference [*Number theory and physics at the crossroad*]{} (N. Yui, V. Batyrev, C. Doran, S. Gukov, D. Zagier), Banff (2011).
G. Rond, [*Artin Approximation*]{}, [arXiv]{}:1506.04717v3.
W. Schmid, [*Variations of Hodge Structure: The Singularities of the Period Mapping*]{}, Inventiones Math., no. [**22**]{}, (1973), 211 - 319.
M. Schütt, [*CM newforms with rational coefficients*]{}, Ramanujan J. (2009) [**19**]{}, 187–205.
M. Schütt, [*$K3$ surfaces with Picard rank $20$*]{}, Algebra and Number theory, [**4**]{}(3) (2010), 335-356.
J. Steenbrink, [*Limits of Hodge Structures*]{}, Inv. Math [**31**]{},(1976), 229 - 257.
D. van Straten, [*Conifold Period Expansion*]{}, Report No. 23/2012, Workshop on Singularity Theory and Integrable Systems Oberwolfach (2012), 1392-1394.
A. Strominger, S.-T. Yau, E. Zaslow, [*Mirror Symmetry is T-Duality*]{}, Nucl. Phys. B [**479**]{}, (1996), 243 - 259.
E. Viehweg, [*Quasi-projective moduli for polarized manifolds*]{}. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), [**30**]{}, Springer-Verlag, Berlin, 1995.
E. Vinberg, [*The two most algebraic K3 surfaces*]{}, Math. Ann. 265 (1983), 1-21.
W. Zudilin, private communication, Oberwolfach, April 2012.
[^1]: The first author was partly supported by the Schwerpunkt Polen (Mainz) and NCN grant no. N N201 608040. The second author was partly supported by DFG Sonderforschungsbereich/Transregio 45.
|
---
abstract: |
A wave equation with mass term is studied for all particles and antiparticles of the first generation: electron and its neutrino, positron and antineutrino, quarks $u$ and $d$ with three states of color and antiquarks $\overline{u}$ and $\overline{d}$. This wave equation is form invariant under the $Cl_3^*$ group generalizing the relativistic invariance. It is gauge invariant under the $U(1)\times SU(2) \times SU(3)$ group of the standard model of quantum physics. The wave is a function of space and time with value in the Clifford algebra $Cl_{1,5}$. All features of the standard model, charge conjugation, color, left waves, Lagrangian formalism, are linked to the geometry of this extended space-time.\
\
[**Keywords: invariance group, Dirac equation, electromagnetism, weak interactions, strong interactions, Clifford algebras**]{}
author:
- |
Claude Daviau\
Le Moulin de la Lande\
44522 Pouillé-les-coteaux\
France\
email: claude.daviau@nordnet.fr\
\
Jacques Bertrand\
15 avenue Danielle Casanova\
95210 Saint-Gratien\
France\
email: bertrandjacques-m@orange.fr
title: A wave equation including leptons and quarks for the standard model of quantum physics in Clifford Algebra
---
Introduction {#introduction .unnumbered}
============
We use here all notations of “New insights in the standard model of quantum physics in Clifford Algebra" [@davi:14]. The wave equation for all particles of the first generation is a generalization of the wave equation obtained in 6.7 for the electron and its neutrino. This wave equation has obtained a proper mass term compatible with the gauge invariance in [@dabe:14]. It is a generalization of the homogeneous nonlinear Dirac equation for the electron alone [@davi:93] [@davi:97] [@davi:05] [@davi:11] [@dav2:12] [@davi:12] [@dav:12]. The link with the usual presentation of the standard model is made by the left and right Weyl spinors used for waves of each particle. These right and left waves are parts of the wave with value in $Cl_{1,5}$.
We used previously the same algebra $Cl_{5,1}=Cl_{1,5}$. It is the same algebra, and this explains very well why sub-algebras $Cl_{1,3}$ and $Cl_{3,1}$ have been equally used to describe relativistic physics [@dehe:93] [@hest:86]. But the signature of the scalar product cannot be free, this scalar product being linked to the gravitation in the general relativity. It happens that vectors of $Cl_{1,5}$ are pseudo-vectors of $Cl_{5,1}$ and more generally that $n$-vectors of $Cl_{1,5}$ are $(6-n)$-vectors of $Cl_{5,1}$. The generalization of the wave equation for electron-neutrino is simpler if we use $Cl_{1,5}$. This is the first indication that the signature $+-----$ is the true one. We explain in Appendix A how the reverse in $Cl_{1,5}$ is linked to the reverse in $Cl_{1,3}$, a necessary condition to get the wave equation of all particles of the first generation.
We have noticed, for the electron alone firstly (See [@davi:12] 2.4), next for electron+neutrino [@dabe:14] the double link existing between the wave equation and the Lagrangian density: It is well known that the wave equation may be obtained from the Lagrangian density by the variational calculus. The new link is that the real part of the invariant wave equation is simply $\mathcal{L}=0$. The Lagrangian formalism is then necessary, being a consequence of the wave equation. Next we have extended the double link to electro-weak interactions in the leptonic case (electron + neutrino). Now we are extending the double link to the gauge group of the standard model. The Lagrangian density must then be the real part of the invariant wave equation.
Moreover we generalized the non-linear homogeneous wave equation of the electron, and we got a wave equation with mass term [@dabe:14], form invariant under the $Cl_3^*=GL(2,\mathbb{C})$ group and gauge invariant under the $U(1)\times SU(2)$ gauge group of electro-weak interactions. Our aim is to explain how this may be extended to a wave equation with mass term, both form invariant under $Cl_3^*$ and gauge invariant under the $U(1)\times SU(2)\times SU(3)$ gauge group of the standard model, including both electro-weak and strong interactions.
From the lepton case to the full wave
=====================================
The standard model adds to the leptons (electron $e$ and its neutrino $n$) in the “first generation" two quarks $u$ and $d$ with three states each. Weak interactions acting only on left waves of quarks (and right waves of antiquarks) we read the wave of all fermions of the first generation as follows: $$\begin{aligned}
\Psi &=\begin{pmatrix} \Psi_l & \Psi_r \\ \Psi_g & \Psi_b \end{pmatrix}~;~~\Psi_l=
\begin{pmatrix} \phi_{e} & \phi_{n} \\ \widehat{\phi}_{n} & \widehat{\phi}_{e} \end{pmatrix}
=\begin{pmatrix} \phi_{e} & \phi_{n} \\ \widehat{\phi}_{\overline n}\sigma_1
& \widehat{\phi}_{\overline e}\sigma_1 \end{pmatrix} \\ \Psi_r&=
\begin{pmatrix} \phi_{dr} & \phi_{ur} \\ \widehat{\phi}_{ur} & \widehat{\phi}_{dr} \end{pmatrix}
=\begin{pmatrix} \phi_{dr} & \phi_{ur} \\ \widehat{\phi}_{\overline ur}\sigma_1
& \widehat{\phi}_{\overline dr}\sigma_1 \end{pmatrix};~\Psi_g=
\begin{pmatrix} \phi_{dg} & \phi_{ug} \\ \widehat{\phi}_{ug} & \widehat{\phi}_{dg} \end{pmatrix}
=\begin{pmatrix} \phi_{dg} & \phi_{ug} \\ \widehat{\phi}_{\overline ug}\sigma_1
& \widehat{\phi}_{\overline dg}\sigma_1 \end{pmatrix} \notag\\ \Psi_b&=
\begin{pmatrix} \phi_{db} & \phi_{ub} \\ \widehat{\phi}_{ub} & \widehat{\phi}_{db} \end{pmatrix}
=\begin{pmatrix} \phi_{db} & \phi_{ub} \\ \widehat{\phi}_{\overline ub}\sigma_1
& \widehat{\phi}_{\overline db}\sigma_1 \end{pmatrix}\end{aligned}$$ The electro-weak theory [@wein:67] needs three spinorial waves in the electron-neutrino case: the right $\xi_e$ and the left $\eta_e$ of the electron and the left spinor $\eta_n$ of the electronic neutrino. The form invariance of the Dirac theory imposes to use $\phi_e$ for the electron and $\phi_n$ for its neutrino satisfying $$\begin{aligned}
\phi_e &=\sqrt{2}\begin{pmatrix}
\xi_{1e} & -\eta_{2e}^* \\ \xi_{2e} &\eta_{1e}^*
\end{pmatrix}=\sqrt{2}(\xi_e~~-i\sigma_2 \eta_e^*)~;~~\phi_n=\sqrt{2}\begin{pmatrix}
0 & -\eta_{2n}^* \\0 &\eta_{1n}^* \end{pmatrix}\\
\widehat\phi_e &=\sqrt{2}\begin{pmatrix}
\eta_{1e} & -\xi_{2e}^* \\ \eta_{2e} &\xi_{1e}^*
\end{pmatrix}=\sqrt{2}(\eta_e~~-i\sigma_2\xi_e^*)~;~~\widehat\phi_n=\sqrt{2}\begin{pmatrix}
\eta_{1n} & 0 \\ \eta_{2n}&0 \end{pmatrix}=\sqrt{2}(\eta_n~~0).\end{aligned}$$ Waves $\phi_e$ and $\phi_n$ are functions of space and time with value into the Clifford algebra $Cl_3$ of the physical space. The standard model uses only a left $\eta_n$ wave for the neutrino. We always use the matrix representation (A.1) which allows to see the Clifford algebra $Cl_{1,3}$ as a sub-algebra of $M_4(\mathbb{C})$. Under the dilation $R$ with ratio $r$ induced by any $M$ in $GL(2,\mathbb{C})$ we have (for more details, see [@davi:11]): $$\begin{aligned}
x'&=M x M^\dagger~;~~\det(M)=r e^{i\theta}~;~~x=x^\mu\sigma_\mu~;~~x'={x'}^\mu\sigma_\mu \\
\xi '&= M \xi~;~~\eta '=\widehat{M}\eta~;~~\eta_n '=\widehat{M}\eta_n~;~~\phi_e ' = M\phi_e
~;~~\phi_n '=M \phi_n \\ \Psi_l ' &=\begin{pmatrix}
\phi_e' & \phi_n '\\ \widehat{\phi}_n' &\widehat{\phi}_e'
\end{pmatrix}=\begin{pmatrix}
M & 0 \\0 & \widehat{M} \end{pmatrix}\begin{pmatrix}
\phi_e & \phi_n \\ \widehat{\phi}_n &\widehat{\phi}_e
\end{pmatrix}=N \Psi_l\end{aligned}$$ The form (1.3) of the wave is compatible both with the form invariance of the Dirac theory and with the charge conjugation used in the standard model: the wave $\psi_{\overline{e}}$ of the positron satisfies $$\psi_{\overline{e}}=i\gamma_2\psi^*\Leftrightarrow\widehat{\phi}_{\overline{e}}=\widehat{\phi}_e\sigma_1$$ We can then think the $\Psi_l$ wave as containing the electron wave $\phi_e$, the neutrino wave $\phi_n$ and also the positron wave $\phi_{\overline{e}}$ and the antineutrino wave $\phi_{\overline{n}}$: $$\Psi_l=\begin{pmatrix}
\phi_e & \phi_n \\ \widehat{\phi}_{\overline{n}}\sigma_1 &\widehat{\phi}_{\overline{e}}\sigma_1
\end{pmatrix}~;~~\phi_{\overline{e}} =\sqrt{2}\begin{pmatrix}
\xi_{1\overline{e}} & -\eta_{2\overline{e}}^* \\ \xi_{2\overline{e}} &\eta_{1\overline{e}}^*
\end{pmatrix}~;~~\phi_{\overline{n}}=\sqrt{2}\begin{pmatrix}
\xi_{1\overline{n}} & 0 \\\xi_{2\overline{n}} & 0 \end{pmatrix}$$ And the antineutrino has only a right wave. The multivector $\Psi_l(x)$ is usually an invertible element of the space-time algebra because (See [@davi:14] (6.250)) with: $$\begin{aligned}
a_1&=\det(\phi_e)=\phi_e \overline{\phi}_e=2(\xi_{1e}\eta_{1e}^*+\xi_{2e}\eta_{2e}^*) \\
a_2&=2(\xi_{1\overline{e}}\eta_{1n}^*+\xi_{2\overline{e}}\eta_{2n}^*)=2(\eta_{2e}^*\eta_{1n}^*-
\eta_{1e}^*\eta_{2n}^*)\\
a_3&=2(\xi_{1e}\eta_{1n}^*+\xi_{2e}\eta_{2n}^*)\end{aligned}$$ we got $$\begin{aligned}
\det(\Psi_l)&=a_1 a_1^*+a_2 a_2^*.\end{aligned}$$ Most of the preceding presentation is easily extended to quarks. For each color $c=r,g,b$ the electro-weak theory needs only left waves: $$\Psi_c=\begin{pmatrix}
\phi_{dc}&\phi_{uc}\\ \widehat{\phi}_{uc}& \widehat{\phi}_{dc}
\end{pmatrix}
;~\widehat{\phi}_{dc}=\sqrt{2}\begin{pmatrix}
\eta_{1dc}&0\\ \eta_{2dc}&0
\end{pmatrix}
;~\widehat{\phi}_{uc}=\sqrt{2}\begin{pmatrix}
\eta_{1uc}&0\\ \eta_{2uc}&0
\end{pmatrix}$$ The $\Psi$ wave is now a function of space and time with value into $Cl_{1,5}=Cl_{5,1}$ which is a sub-algebra (on the real field) of $Cl_{5,2}=M_8(\mathbb{C})$: $$\Psi=\begin{pmatrix}
\Psi_l &\Psi_r \\ \Psi_g & \Psi_b
\end{pmatrix};~\widetilde{\Psi}=\begin{pmatrix}
\widetilde\Psi_b & \widetilde\Psi_r \\ \widetilde\Psi_g &\widetilde\Psi_l
\end{pmatrix}$$ The link between the reverse in $Cl_{1,5}$ and the reverse in $Cl_{1,3}$ is not trivial and is detailed in Appendix A. The wave equation for all objects of the first generation reads $$0=(\underline{\mathbf{D}}\Psi)L_{012}+\underline{\mathbf{M}}$$ The mass term reads $$\underline{\mathbf{M}}=\begin{pmatrix}
m_2 \rho_2 \chi_b & m_2 \rho_2 \chi_g \\ m_2 \rho_2\chi_r & m_1 \rho_1 \chi_l
\end{pmatrix}$$ where we use the scalar densities $s_j$ and $\chi$ terms of Appendix B, with $$\rho_1^2=a_1 a_1^*+a_2 a_2^*+a_3 a_3^*~;~~\rho_2^2=\sum_{j=1}^{j=15}s_j s_j^*.$$ The covariant derivative $\underline{\mathbf{D}}$ uses the matrix representation (A.1) and reads $$\begin{aligned}
\underline{\mathbf{D}}&=\underline{\pmb\partial} +\frac{g_1}{2}\underline{\mathbf{B}}\ \underline{P}_0
+\frac{g_2}{2}\underline{\mathbf{W}}^j \underline{P}_j +\frac{g_3}{2} \underline{\mathbf{G}}^k \underline{\mathbf{i}}\Gamma_k \\
\underline{\mathbf{D}}&=\sum_{\mu=0}^{3}L^\mu D_\mu~;~~\underline{\pmb\partial}
=\sum_{\mu=0}^{3}L^\mu \partial_\mu~;~~\underline{\mathbf{B}}
=\sum_{\mu=0}^{3}L^\mu B_\mu\\
\underline{\mathbf{W}}^j&=\sum_{\mu=0}^{3}L^\mu W_\mu^j,~j=1,2,3\\
\underline{\mathbf{G}}^k&=\sum_{\mu=0}^{3}L^\mu G_\mu^k,~k=1,2,\dots, 8.\end{aligned}$$ We use two projectors satisfying $$\underline{P}_\pm(\Psi)=\frac{1}{2}(\Psi\pm \underline{\mathbf{i}}\Psi L_{21})~;~~
\underline{\mathbf{i}}=L_{0123}$$ Three operators act on quarks like on leptons: $$\begin{aligned}
\underline{P}_1(\Psi)&=\underline{P}_+(\Psi)L_{35}\\
\underline{P}_2(\Psi)&=\underline{P}_+(\Psi)L_{5012}\\
\underline{P}_3(\Psi)&=\underline{P}_+(\Psi)(-\underline{\mathbf{i}}).\end{aligned}$$ The fourth operator acts differently on the leptonic and on the quark sector. Using projectors: $$P^+=\frac{1}{2}(I_8+L_{012345})=\begin{pmatrix}
I_4 & 0 \\ 0&0\end{pmatrix};~P^-=\frac{1}{2}(I_8-L_{012345})=\begin{pmatrix}
0 & 0 \\ 0&I_4\end{pmatrix}$$ we can separate the lepton part $\Psi^l$ and the quark part $\Psi^c$ of the wave: $$\Psi^l=P^+\Psi P^+=\begin{pmatrix}\Psi_l & 0 \\ 0&0
\end{pmatrix} ;~\Psi^c=\Psi-\Psi^l=\begin{pmatrix}
0 & \Psi_r \\ \Psi_g & \Psi_b
\end{pmatrix}.$$ and we get (see [@davi:14] (B.4) with $a=b=1$) $$\begin{aligned}
\underline{P}_0(\Psi^l)&=\frac{1}{2}\underline{\mathbf{i}}\Psi^l L_{21}+\frac{3}{2}\Psi^l L_{21} \\
\underline{P}_0(\Psi^c)&=-\frac{1}{3}\Psi^c L_{21}.\end{aligned}$$ This last relation comes from the non-existence of the right part of the $\Psi^c$ waves.
Chromodynamics
==============
We start from generators $\lambda_k$ of the $SU(3)$ gauge group of chromodynamics $$\begin{aligned}
\lambda_1 &=\begin{pmatrix}0&1&0 \\1&0&0 \\0&0&0 \end{pmatrix},~
\lambda_2 =\begin{pmatrix}0&-i&0 \\i&0&0 \\0&0&0 \end{pmatrix},~
\lambda_3 =\begin{pmatrix}1&0&0 \\0&-1&0 \\0&0&0 \end{pmatrix}\notag \\
\lambda_4 &=\begin{pmatrix}0&0&1 \\0&0&0 \\1&0&0 \end{pmatrix},~
\lambda_5 =\begin{pmatrix}0&0&-i \\0&0&0 \\i&0&0 \end{pmatrix},~
\lambda_6 =\begin{pmatrix}0&0&0 \\0&0&1 \\0&1&0 \end{pmatrix}\notag \\
\lambda_7 &=\begin{pmatrix}0&0&0 \\0&0&-i \\0&i&0 \end{pmatrix},~
\lambda_8 =\frac{1}{\sqrt{3}}\begin{pmatrix}1&0&0 \\0&1&0 \\0&0&-2 \end{pmatrix}.\end{aligned}$$ To simplify here notations we use now $l$, $r$, $g$, $b$ instead $\Psi_l$, $\Psi_r$, $\Psi_g$, $\Psi_b$. So we have $\Psi=\begin{pmatrix}l & r \\g & b \end{pmatrix}$. Then (C.1) gives $$\begin{aligned}
\lambda_1 \begin{pmatrix}r\\g\\b \end{pmatrix}&=\begin{pmatrix}g\\r\\0 \end{pmatrix},~
\lambda_2 \begin{pmatrix}r\\g\\b \end{pmatrix}=\begin{pmatrix}-ig\\ir\\0 \end{pmatrix},~
\lambda_3 \begin{pmatrix}r\\g\\b \end{pmatrix}=\begin{pmatrix}r\\-g\\0 \end{pmatrix}\notag\\
\lambda_4 \begin{pmatrix}r\\g\\b \end{pmatrix}&=\begin{pmatrix}b\\0\\r \end{pmatrix},~
\lambda_5 \begin{pmatrix}r\\g\\b \end{pmatrix}=\begin{pmatrix}-ib\\0\\ir \end{pmatrix},~
\lambda_6 \begin{pmatrix}r\\g\\b \end{pmatrix}=\begin{pmatrix}0\\b\\g \end{pmatrix}\\
\lambda_7 \begin{pmatrix}r\\g\\b \end{pmatrix}&=\begin{pmatrix}0\\-ib\\ig \end{pmatrix},~
\lambda_8 \begin{pmatrix}r\\g\\b \end{pmatrix}=\frac{1}{\sqrt{3}}\begin{pmatrix}r\\g\\-2b \end{pmatrix}.\notag\end{aligned}$$ We name $\Gamma_k$ operators corresponding to $\lambda_k$ acting on $\Psi$. We get with projectors $P^+$ and $P^-$ in (1.27): $$\begin{aligned}
\Gamma_1 (\Psi)&=\frac{1}{2}(L_4\Psi L_4+L_{01235}\Psi L_{01235})=
\begin{pmatrix}0 &g\\r &0 \end{pmatrix}\\
\Gamma_2 (\Psi)&=\frac{1}{2}(L_5\Psi L_4 -L_{01234}\Psi L_{01235})=
\begin{pmatrix}0 &-\mathbf{i}g\\\mathbf{i}r &0 \end{pmatrix} \\
\Gamma_3 (\Psi)&=P^+\Psi P^- - P^- \Psi P^+ =
\begin{pmatrix}0 &r\\-g &0 \end{pmatrix}\\
\Gamma_4 (\Psi)&=L_{01253}\Psi P^- = \begin{pmatrix}0 &b\\0 &r \end{pmatrix}~;~
\Gamma_5(\Psi)=L_{01234} \Psi P^-
=\begin{pmatrix}0 &-\mathbf{i}b\\0 &\mathbf{i}r\end{pmatrix}\\
\Gamma_6(\Psi)&= P^- \Psi L_{01253}=\begin{pmatrix}0 &0\\b &g\end{pmatrix}~;~
\Gamma_7(\Psi)=-\underline{\mathbf{i}}P^- \Psi L_4=\begin{pmatrix}0 &0\\-\mathbf{i}b &\mathbf{i}g
\end{pmatrix}\\
\Gamma_8(\Psi)&=\frac{1}{\sqrt{3}}(P^-\Psi L_{012345}+L_{012345}\Psi P^-)
=\frac{1}{\sqrt{3}}\begin{pmatrix}0 &r\\g &-2b\end{pmatrix}.\end{aligned}$$ Everywhere the left up term is $0$, so all $\Gamma_k$ project the wave $\Psi$ on its quark sector.
We can extend the covariant derivative of electro-weak interactions in the electron-neutrino case: $$D\Psi_l= \partial\Psi_l +\frac{g_1}{2} B P_0(\Psi_l)
+\frac{g_2}{2} W^j P_j(\Psi_l)$$ to get the covariant derivative of the standard model $$\underline D (\Psi)= \underline \partial(\Psi) +\frac{g_1}{2}\underline B \ \underline P_0(\Psi)
+\frac{g_2}{2}\underline W^j \underline P_j(\Psi)+\frac{g_3}{2}\underline G^k\underline{\mathbf{i}}
\Gamma_k(\Psi).$$ where $g_3$ is another constant and $\underline G^k$ are eight terms called “gluons”. Since $I_4$ commute with any element of $Cl_{1,3}$ and since $P_j(\mathbf{i} \Psi_{ind})=\mathbf{i} P_j(\Psi_{ind})$ for $j=0,1,2,3$ and $ind=l,r,g,b$ each operator $\underline{\mathbf{i}} \Gamma_k$ commutes with all operators $\underline P_j$.
Now we use 12 real numbers $a^0$, $a^j,~j=1,2,3$, $b^k,~k=1,2,...,8$, we let $$S_0=a^0 \underline P_0;~S_1=\sum_{j=1}^{j=3} a^j \underline P_j;~S_2=\sum_{k=1}^{k=8}b^k
\underline{\mathbf{i}} \Gamma_k;~S=S_0 + S_1 +S_2$$ and we get, using exponentiation $$\begin{aligned}
\exp(S) &=\exp(S_0)\exp(S_1)\exp(S_2)=\exp(S_1)\exp(S_0)\exp(S_2)\notag\\
&=\exp(S_0)\exp(S_2)\exp(S_1)=\dots\end{aligned}$$ in any order. The set of these operators $\exp(S)$ is a $U(1)\times SU(2)\times SU(3)$ Lie group. Only difference with the standard model: the structure of this group is not postulated but calculated. With $$\Psi '=[\exp(S)](\Psi)~;~\underline{\mathbf{D}}=L^\mu \underline D_\mu
~;~\underline{\mathbf{D}}'=L^\mu \underline D_\mu '$$ the gauge transformation reads $$\begin{aligned}
\underline D_\mu ' \Psi '&=[\exp(S)](\underline D_\mu \Psi)\\
B_\mu '&=B_\mu-\frac{2}{g_1}\partial_\mu a^0\\
{W '}_\mu^j \underline P_j&=\Big[\exp(S_1)W_\mu^j \underline P_j-\frac{2}{g_2}\partial_\mu[\exp(S_1)]
\Big] \exp(-S_1)\\
{\underline G '}_\mu^k \underline{\mathbf{i}}\Gamma_k&=\Big[\exp(S_2)\underline{G}_\mu^k
\underline{\mathbf{i}}\Gamma_k-\frac{2}{g_3} \partial_\mu[\exp(S_2)] \Big] \exp(-S_2).\end{aligned}$$ The $SU(3)$ group generated by projectors $\Gamma_k$ acts only on the quark sector of the wave: $$P^+[\exp(b^k\underline{\mathbf{i}} \Gamma_k](\Psi)P^+=
P^+\Psi P^+ = \Psi^l$$ The physical translation is: Leptons do not act by strong interactions. This comes from the structure of the wave itself. It is fully satisfied in experiments. We get then a $U(1)\times SU(2)\times SU(3)$ gauge group for a wave including all fermions of the first generation. This group acts on the lepton sector only by its $U(1)\times SU(2)$ part. Consequently the wave equation is composed of a lepton wave equation and a quark wave equation: $$\begin{aligned}
0&=(\underline{\mathbf{D}}\Psi^l)L_{012}+m_1 \rho_1\begin{pmatrix}
0 & 0 \\ 0 & \chi_l
\end{pmatrix}\\
0&=(\underline{\mathbf{D}}\Psi^c)L_{012}+m_2 \rho_2\chi^c;~\chi^c=\begin{pmatrix}
\chi_b & \chi_g \\ \chi_r & 0
\end{pmatrix}\end{aligned}$$ The wave equation (2.19) is equivalent to the wave equation $$\mathbf{D}\Psi_l\gamma_{012}+m_1\rho_1 \chi_l=0~;~~\gamma_{012}=\gamma_0\gamma_1\gamma_2$$ studied in [@dabe:14] [@dav:14], where $$\begin{aligned}
\chi_l &=\frac{1}{\rho_1^2}\begin{pmatrix}
a_1^*\phi_e+a_2^*\phi_n\sigma_1+a_3^*\phi_n & -a_2^*\phi_{eL}\sigma_1+a_3^*
\phi_{eR} \\ a_2\widehat{\phi}_{eL}\sigma_1+a_3\widehat{\phi}_{eR} &
a_1\widehat{\phi}_e -a_2\widehat{\phi}_n \sigma_1 + a_3\widehat{\phi}_n
\end{pmatrix}\\
\phi_{eR}&=\phi_e\frac{1+\sigma_3}{2}~;~~\phi_{eL}=\phi_e\frac{1-\sigma_3}{2}\end{aligned}$$ This wave equation is equivalent to the invariant equation: $$\widetilde \Psi_l (\mathbf{D}\Psi_l)\gamma_{012}+m\rho_1 \widetilde \Psi_l\chi_l=0~;~~
\widetilde \Psi_l=\begin{pmatrix}
\overline{\phi}_e &\phi_n^\dagger \\ \overline{\phi}_n & \phi_e^\dagger
\end{pmatrix}.$$ This wave equation is form invariant under the Lorentz dilation $R$ induced by any invertible matrix $M$ satisfying (1.5), (1.6), (1.7) [@davi:14]. It is gauge invariant under the $U(1)\times SU(2)$ group [@dabe:14] generated by operators $P_\mu$ which are projections on the lepton sector of the operators defined in (1.23) to (1.29). Therefore we need to study only the quark sector and its wave equation (2.20).
We begin by the double link between wave equation and Lagrangian density that we have remarked firstly in the Dirac equation [@davi:12], next in the lepton case electron+neutrino [@davi:14].
Double link between wave equation and Lagrangian density
========================================================
The existence of a Lagrangian mechanism in optics and mechanics is known since Fermat and Maupertuis. This principle of minimum is everywhere in quantum mechanics from its beginning, it is the main reason of the hypothesis of a wave linked to the move of any material particle made by L. de Broglie [@brog:24]. By the calculus of variations it is always possible to get the wave equation from the Lagrangian density. But another link exists : the Lagrangian density is the real scalar part of the invariant wave equation. This was obtained firstly for the electron alone [@davi:12], next for the pair electron-neutrino [@dabe:14] where the Lagrangian density reads $$\begin{aligned}
\mathcal{L}_l&=\mathcal{L}_0+g_1 \mathcal{L}_1+ g_2 \mathcal{L}_2 +m_1\rho_1\\
\mathcal{L}_0&=\Re[-i(\eta_e^\dagger \sigma^\mu \partial_\mu \eta_e + \xi_e^\dagger
\widehat{\sigma}^\mu \partial_\mu \xi_e +\eta_n^\dagger \sigma^\mu \partial_\mu \eta_n)]\\
\mathcal{L}_1&=B_\mu(\frac{1}{2}\eta_e^\dagger \sigma^\mu \eta_e + \xi_e^\dagger\widehat{\sigma}^\mu \xi_e
+\frac{1}{2}\eta_n^\dagger \sigma^\mu \eta_n )\\
\mathcal{L}_2&=-\Re[(W_\mu^1+iW_\mu^2)\eta_e^\dagger\sigma^\mu \eta_n]+\frac{W_\mu^3}{2}
(\eta_e^\dagger \sigma^\mu \eta_e - \eta_n^\dagger \sigma^\mu \eta_n).\end{aligned}$$ We shall establish the double link now for the wave equation (1.16). It is sufficient to add the property for (2.20). This equation is equivalent to the invariant equation: $$\begin{aligned}
0&=\widetilde{\Psi}^c(\underline{\mathbf{D}}\Psi^c)L_{012}+m_2 \rho_2\widetilde{\Psi}^c
\chi^c\\ \widetilde{\Psi}^c&=\begin{pmatrix}
\widetilde{\Psi}_b & \widetilde{\Psi}_r \\ \widetilde{\Psi}_g &0
\end{pmatrix};~\chi^c= \begin{pmatrix}
\chi_b & \chi_g \\ \chi_r & 0
\end{pmatrix}\end{aligned}$$ We get from the covariant derivative (1.19) with the operators $\underline{P}_j$ in (1.24), (1.25), (1.26) and (1.30) and $\Gamma_k$ in (2.3) to (2.8) and with $\Psi^c$ in (1.28) $$\begin{aligned}
\underline{\mathbf{D}}\Psi^c&=\begin{pmatrix}
A_g & A_b \\0 & A_r \end{pmatrix} \\
A_g &= \pmb\partial \Psi_g-\frac{g_1}{6}\mathbf{B} \Psi_g \gamma_{21}+\frac{g_2}{2}
(\mathbf{W}^1 \Psi_g \gamma_3\mathbf{i}+\mathbf{W}^2 \Psi_g \gamma_3-
\mathbf{W}^3 \Psi_g \mathbf{i})\notag \\
&+\frac{g_3}{2}(\mathbf{G}^1 \mathbf{i}\Psi_r-\mathbf{G}^2 \Psi_r
-\mathbf{G}^3 \mathbf{i}\Psi_g+\mathbf{G}^6 \mathbf{i}\Psi_b+\mathbf{G}^7 \Psi_b
+\frac{1}{\sqrt{3}}\mathbf{G}^8 \mathbf{i}\Psi_g)\end{aligned}$$ $$\begin{aligned}
A_b &= \pmb\partial \Psi_b-\frac{g_1}{6}\mathbf{B} \Psi_b \gamma_{21}+\frac{g_2}{2}
(\mathbf{W}^1 \Psi_b \gamma_3\mathbf{i}+\mathbf{W}^2 \Psi_b \gamma_3-
\mathbf{W}^3 \Psi_b \mathbf{i})\notag \\
&+\frac{g_3}{2}(\mathbf{G}^4 \mathbf{i}\Psi_r-\mathbf{G}^5 \Psi_r
+\mathbf{G}^6 \mathbf{i}\Psi_g-\mathbf{G}^7 \Psi_g
-\frac{2}{\sqrt{3}}\mathbf{G}^8 \mathbf{i}\Psi_b)\\
A_r &= \pmb\partial \Psi_r-\frac{g_1}{6}\mathbf{B} \Psi_r \gamma_{21}+\frac{g_2}{2}
(\mathbf{W}^1 \Psi_r \gamma_3\mathbf{i}+\mathbf{W}^2 \Psi_r \gamma_3-
\mathbf{W}^3 \Psi_r \mathbf{i})\notag \\
&+\frac{g_3}{2}(\mathbf{G}^1 \mathbf{i}\Psi_g+\mathbf{G}^2 \Psi_g
+\mathbf{G}^3 \mathbf{i}\Psi_r+\mathbf{G}^4 \mathbf{i}\Psi_b+\mathbf{G}^5 \Psi_b
+\frac{1}{\sqrt{3}}\mathbf{G}^8 \mathbf{i}\Psi_r)\end{aligned}$$ Next we get $$\begin{aligned}
&\widetilde{\Psi}^c(\underline{\mathbf{D}}\Psi^c)L_{012}+m_2 \rho_2\widetilde{\Psi}^c
\chi^c\\ &=\begin{pmatrix}
\widetilde{\Psi}_b(A_b \gamma_{012}+m_2\rho_2\chi_b)+\widetilde{\Psi}_r(A_r \gamma_{012}+m_2\rho_2\chi_r)
& \widetilde{\Psi}_b(A_g \gamma_{012}+m_2\rho_2\chi_g)\\ \widetilde{\Psi}_g(A_b \gamma_{012}+m_2\rho_2\chi_b)
&\widetilde{\Psi}_g(A_g \gamma_{012}+m_2\rho_2\chi_g)
\end{pmatrix}\notag\end{aligned}$$ The calculation of the Lagrangian density in the general case is similar to the lepton case. We get $$\begin{aligned}
\mathcal{L}&=\mathcal{L}_l+ \mathcal{L}_c \\
\mathcal{L}_c&=\sum_{c=r,g,b}\mathcal{L}_{0c}+g_1\sum_{c=r,g,b}\mathcal{L}_{1c}
+g_2\sum_{c=r,g,b}\mathcal{L}_{2c}+g_3\mathcal{L}_3+m_2\rho_2\end{aligned}$$ The calculation of $\mathcal{L}_{jc},~j=0,1,2$ replaces the pair e-n by the pair dc-uc and suppress the $\xi$ terms, then (3.2) (3.3) (3.4) become $$\begin{aligned}
\mathcal{L}_{0c}&=\Re[-i(\eta_{dc}^\dagger \sigma^\mu \partial_\mu \eta_{dc}
+\eta_{uc}^\dagger \sigma^\mu \partial_\mu \eta_{uc})]\\
\mathcal{L}_{1c}&=-\frac{B_\mu}{6}(\eta_{dc}^\dagger \sigma^\mu \eta_{dc}
+\eta_{uc}^\dagger \sigma^\mu \eta_{uc})\\
\mathcal{L}_{2c}&=-\Re[(W_\mu^1+i W_\mu^2)\eta_{dc}^\dagger\sigma^\mu \eta_{uc}]
+\frac{W_\mu^3}{2}(\eta_{dc}^\dagger\sigma^\mu \eta_{dc}-\eta_{uc}^\dagger\sigma^\mu \eta_{uc})\end{aligned}$$ Since three $SU(2)$ group are included in $SU(3)$ the calculation of $\mathcal{L}_3$ has similarities with the calculation of $\mathcal{L}_2$ and we get $$\begin{aligned}
\mathcal{L}_3=&-\Re[(G_\mu^1+iG_\mu^2)(\eta_{dr}^\dagger\sigma^\mu \eta_{dg}
+\eta_{ur}^\dagger\sigma^\mu \eta_{ug})]\\
&-\Re[(G_\mu^4+iG_\mu^5)(\eta_{dr}^\dagger\sigma^\mu \eta_{db}
+\eta_{ur}^\dagger\sigma^\mu \eta_{ub})]\notag \\
&-\Re[(G_\mu^6+iG_\mu^7)(\eta_{dg}^\dagger\sigma^\mu \eta_{db}
+\eta_{ug}^\dagger\sigma^\mu \eta_{ub})]\notag \\
&+\frac{G_\mu^3}{2}(-\eta_{dr}^\dagger\sigma^\mu \eta_{dr}
-\eta_{ur}^\dagger\sigma^\mu \eta_{ur}+\eta_{dg}^\dagger\sigma^\mu \eta_{dg}
+\eta_{ug}^\dagger\sigma^\mu \eta_{ug})\notag\\
&+\frac{G_\mu^8}{2\sqrt{3}}(-\eta_{dr}^\dagger\sigma^\mu \eta_{dr}
-\eta_{ur}^\dagger\sigma^\mu \eta_{ur}+\eta_{db}^\dagger\sigma^\mu \eta_{db}
+\eta_{ub}^\dagger\sigma^\mu \eta_{ub})\notag\\
&+\frac{G_\mu^8}{2\sqrt{3}}(-\eta_{dg}^\dagger\sigma^\mu \eta_{dg}
-\eta_{ug}^\dagger\sigma^\mu \eta_{ug}+\eta_{db}^\dagger\sigma^\mu \eta_{db}
+\eta_{ub}^\dagger\sigma^\mu \eta_{ub})\notag\end{aligned}$$ This new link between the wave equation and the Lagrangian density is much stronger than the old one, because it comes from a simple separation of the different parts of a multivector in Clifford algebra. The old link, going from the Lagrangian density to the wave equation, supposes a condition of cancellation at infinity which is dubious in the case of a propagating wave. On the physical point of view, there are no difficulties in the case of a stationary wave. Difficulties begin when propagating waves are studied. Our wave equations, since they are compatible with an oriented time and an oriented space, appear as more general, more physical, than Lagrangians. These are only particular consequences of the wave equations.
On the mathematical point of view the old link is always available. It is from the Lagrangian density (3.12) and using Lagrange equations that we have obtained the wave equation (1.16).
Invariances
===========
Form invariance of the wave equation
------------------------------------
Under the Lorentz dilation $R$ induced by an invertible $M$ matrix satisfying $$\begin{aligned}
x'&=M x M^\dagger~;~~\det(M)=r e^{i\theta}~;~~x=x^\mu\sigma_\mu~;~~x'={x'}^\mu\sigma_\mu \\
\eta_{uc} '&=\widehat{M}\eta_{uc}~;~~\eta_{dc} '=\widehat{M}\eta_{dc}~;~~\phi_{dc} ' = M\phi_{dc}
~;~~\phi_{uc} '=M \phi_{uc} \\ \Psi_c ' &=\begin{pmatrix}
\phi_{dc}' & \phi_{uc} '\\ \widehat{\phi}_{uc}' &\widehat{\phi}_{dc}'
\end{pmatrix}=\begin{pmatrix}
M & 0 \\0 & \widehat{M} \end{pmatrix}\begin{pmatrix}
\phi_{dc} & \phi_{uc} \\ \widehat{\phi}_{uc} &\widehat{\phi}_{dc}
\end{pmatrix}=N \Psi_c~;~~c=r,g,b.\end{aligned}$$ We then let $$\underline{N}=\begin{pmatrix}
N & 0 \\ 0 & N \end{pmatrix};~\underline{\pmb{\partial}}=L^\mu \partial_\mu=\begin{pmatrix}
0 & \pmb{\partial} \\ \pmb{\partial} & 0
\end{pmatrix}$$ which implies $${{\Psi}{'}}^c=\underline{N}{\Psi}^c ;~ {{\widetilde{\Psi}}{'}}^c={\widetilde{\Psi}}^c\underline{\widetilde{N}}
;~\underline{\widetilde{N}}=\begin{pmatrix}
\widetilde N & 0 \\ 0 & \widetilde N \end{pmatrix};~\underline{\mathbf{D}}=\underline{\widetilde{N}}\
{\underline{\mathbf{D}}}'\underline{N}.$$ Then we get $$\begin{aligned}
\widetilde{\Psi}^c(\underline{\mathbf{D}}\Psi^c)L_{012}&=\widetilde{\Psi}^c\underline{\widetilde{N}}\
{\underline{\mathbf{D}}}'\underline{N}\Psi^c L_{012}\notag \\ &={\widetilde{\Psi}{'}}^c (
{\underline{\mathbf{D}}}'{\Psi {'}}^c ) L_{012}.\end{aligned}$$ and we shall now study the form invariance of the mass term. All $s_j$ are determinants of a $\phi$ matrix, this implies $$\begin{aligned}
s_j'&=\det(\phi')=\det(M\phi)=\det(M)\det(\phi)=re^{i\theta}s_j\\
{s'}_j^*&=re^{-i\theta}s_j^*;~ \rho_2 '= r \rho_2.\end{aligned}$$ This gives $$\begin{aligned}
{\chi{'}}^c&=\begin{pmatrix}
\chi_b ' & \chi_g ' \\ \chi_r ' & 0
\end{pmatrix}\\
r^2\rho_2^2{\chi{'}}^c&={\rho '}_2^2 {\chi{'}}^c=\begin{pmatrix}
re^{-i\theta}M & 0 \\ 0 & re^{i\theta}\widehat{M}
\end{pmatrix}\rho_2^2\chi^c \\
{\chi{'}}^c &=\begin{pmatrix}
r^{-1}e^{-i\theta}M & 0 \\ 0 & r^{-1}e^{i\theta}\widehat{M}
\end{pmatrix}\chi_c=\widetilde{N}^{-1}\chi^c \\
{\widetilde{\Psi}{'}}^c{\chi{'}}^c&=\widetilde{\Psi}^c\widetilde{N}\widetilde{N}^{-1}\chi^c=\widetilde{\Psi}^c\chi^c\end{aligned}$$ Then the form invariance of the wave equation is equivalent to the condition on the mass term $$\begin{aligned}
m_2'\rho_2'&=m_2\rho_2 \\
m_2'r&=m_2\end{aligned}$$ linked to the existence of the Planck factor [@dav:14].
Gauge invariance of the wave equation
-------------------------------------
Since we have previously proved the gauge invariance of the lepton part of the wave equation, it is reason enough to prove the gauge invariance of the quark part of the wave equation.
### Gauge group generated by $\underline{P}_0$
We have here $$\begin{aligned}
\underline{P}_0(\Psi^c)&=\Psi^c(-\frac{1}{3}L_{21})\\
{\Psi{'}}^c&=[\exp(\theta \underline{P}_0)](\Psi^c)=\Psi^c\exp(-\frac{\theta}{3}L_{21})\\
B_\mu '&=B_\mu-\frac{2}{g_1}B_\mu \end{aligned}$$ To get the gauge invariance of the wave equation we must get $${\chi{'}}^c=\chi^c\exp(-\frac{\theta}{3}L_{21});~\chi_c'=\chi_c\exp(-\frac{\theta}{3}\gamma_{21}),~c=r,g,b.$$ This is satisfied because $$\begin{aligned}
\phi_{dc}'&=\phi_{dc}e^{-i\frac{\theta}{3}\sigma_3};~\phi_{uc}'=\phi_{uc}e^{-i\frac{\theta}{3}\sigma_3}\\
{\eta{'}}_{1dc}^*&=e^{i\frac{\theta}{3}}\eta_{1dc}^*;~{\eta{'}}_{1uc}^*=e^{i\frac{\theta}{3}}\eta_{1uc}^*\notag\\
{\eta{'}}_{2dc}^*&=e^{i\frac{\theta}{3}}\eta_{2dc}^*;~{\eta{'}}_{2uc}^*=e^{i\frac{\theta}{3}}\eta_{2uc}^*\\
s_j'&=e^{2i\frac{\theta}{3}}s_j,~j=1,2,\dots,15.\end{aligned}$$ All up terms in the matrix $\chi_c$ contain $s_j^*\phi_{dc}\sigma_1$ and $s_j^*\phi_{uc}\sigma_1$ terms. We get $$\begin{aligned}
\phi_{dc}'&=\phi_{dc}e^{-i\frac{\theta}{3}\sigma_3}=e^{i\frac{\theta}{3}}\phi_{dc}\\
{s{'}}_j^*\phi_{dc}'\sigma_1 &=e^{-i\frac{\theta}{3}}\phi_{dc}\sigma_1=\phi_{dc}e^{\frac{\theta}{3}\sigma_{12}}
\sigma_1=\phi_{dc}\sigma_1e^{-\frac{\theta}{3}\sigma_{12}}\\
\chi_c'&=\chi_c\exp(-\frac{\theta}{3}\gamma_{21})\\
{\chi{'}}^c&=\chi^c\exp(-\frac{\theta}{3}L_{21}).\end{aligned}$$ And we finally get $$\begin{aligned}
(\underline{\mathbf{D}}'{\Psi{'}}^c)L_{012}+m_2 \rho_2'{\chi{'}}^c
&=[(\underline{\mathbf{D}}\Psi^c)L_{012}+m_2 \rho_2\chi^c]\exp(-\frac{\theta}{3}L_{21})=0\end{aligned}$$ The wave equation with mass term is gauge invariant under the group generated by $\underline{P}_0$.
### Gauge group generated by $\underline{P}_1$
We have here $$\begin{aligned}
\underline{P}_1(\Psi^c)&=\Psi^c L_{35}\\
{\Psi{'}}^c&=[\exp(\theta \underline{P}_1)](\Psi^c)=\Psi^c\exp(\theta L_{35})\\
{W{'}}_\mu^1=W_\mu^1-\frac{2}{g_2}\partial_\mu\theta \end{aligned}$$ We put a more detailed calculation in C.1. We get $$\begin{aligned}
(\underline{\mathbf{D}}'{\Psi{'}}^c)L_{012}+m_2 \rho_2'{\chi{'}}^c
&=(\underline{\mathbf{D}}\Psi^c)\exp(\theta L_{35})L_{012}+m_2 \rho_2'{\chi{'}}^c\notag \\
&=[(\underline{\mathbf{D}}\Psi^c)L_{012}+m_2 \rho_2\chi^c]\exp(\theta L_{35})=0\end{aligned}$$ The wave equation with mass term is then gauge invariant under the group generated by $\underline{P}_1$.
### Gauge group generated by $\underline{P}_2$
We have here $$\begin{aligned}
\underline{P}_2(\Psi^c)&=\Psi^c L_{5012}\\
{\Psi{'}}^c&=[\exp(\theta \underline{P}_2)](\Psi^c)=\Psi^c\exp(\theta L_{5012})\\
{W{'}}_\mu^2=W_\mu^2-\frac{2}{g_2}\partial_\mu\theta \end{aligned}$$ We have put a more detailed calculation in C.2. We get $$\begin{aligned}
(\underline{\mathbf{D}}'{\Psi{'}}^c)L_{012}+m_2 \rho_2'{\chi{'}}^c
&=(\underline{\mathbf{D}}\Psi^c)\exp(\theta L_{5012})L_{012}+m_2 \rho_2'{\chi{'}}^c\notag \\
&=[(\underline{\mathbf{D}}\Psi^c)L_{012}+m_2 \rho_2\chi^c]\exp(-\theta L_{5012})=0\end{aligned}$$ The wave equation with mass term is then gauge invariant under the group generated by $\underline{P}_2$.
### Gauge group generated by $\underline{P}_3$
We have here $$\begin{aligned}
\underline{P}_3(\Psi^c)&=\Psi^c L_{3012}\\
{\Psi{'}}^c&=[\exp(\theta \underline{P}_3)](\Psi^c)=\Psi^c\exp(\theta L_{3012})\\
{W{'}}_\mu^3=W_\mu^3-\frac{2}{g_2}\partial_\mu\theta \end{aligned}$$ We have put a more detailed calculation in C.3. We get $$\begin{aligned}
(\underline{\mathbf{D}}'{\Psi{'}}^c)L_{012}+m_2 \rho_2'{\chi{'}}^c
&=(\underline{\mathbf{D}}\Psi^c)\exp(\theta L_{3012})L_{012}+m_2 \rho_2'{\chi{'}}^c\notag \\
&=[(\underline{\mathbf{D}}\Psi^c)L_{012}+m_2 \rho_2\chi^c]\exp(-\theta L_{3012})=0\end{aligned}$$ The wave equation with mass term is then gauge invariant under the group generated by $\underline{P}_3$.
### Gauge group generated by $\Gamma_1$
We use now the gauge transformation $$\begin{aligned}
\Psi_r'&=C \Psi_r+S\mathbf{i}\Psi_g;~C=\cos(\theta);~S=\sin(\theta)\\
\Psi_g' &=C \Psi_g+S\mathbf{i}\Psi_r\\
\Psi_b'&=\Psi_b\end{aligned}$$ We can then forget here $\Psi_b$. The gauge invariance signifies that the system $$\begin{aligned}
\pmb\partial \Psi_r&=-\frac{g_3}{2}\mathbf{G}^1\mathbf{i}\Psi_g+m_2\rho_2 \chi_r \gamma_{012}\notag\\
\pmb\partial \Psi_g&=-\frac{g_3}{2}\mathbf{G}^1\mathbf{i}\Psi_r+m_2\rho_2 \chi_g \gamma_{012}\end{aligned}$$ must be equivalent to the system $$\begin{aligned}
\pmb\partial \Psi_r'&=-\frac{g_3}{2}{\mathbf{G}{'}}^1\mathbf{i}\Psi_g'+m_2\rho_2' \chi_r' \gamma_{012}\notag\\
\pmb\partial \Psi_g'&=-\frac{g_3}{2}{\mathbf{G}{'}}^1\mathbf{i}\Psi_r'+m_2\rho_2 '\chi_g '\gamma_{012}\end{aligned}$$ Using relations (4.39) and (4.40) the system (4.43) is equivalent to (4.42) if and only if $${\mathbf{G}{'}}^1=\mathbf{G}^1-\frac{2}{g_3}\pmb\partial\theta$$ because we get in C.4 $$\begin{aligned}
\rho '&=\rho \\
\chi_r'&=C\chi_r-S\mathbf{i}\chi_g \\
\chi_g'&=C\chi_g-S\mathbf{i}\chi_r\end{aligned}$$ The change of sign of the phase between (4.39) and (4.46) comes from the anticommutation between $\mathbf{i}$ and $\pmb\partial$.
### Gauge groups generated by $\Gamma_k~,~k>1$
We use with $k=2$ the gauge transformation $$\begin{aligned}
\Psi_r'&=C \Psi_r+S\Psi_g;~C=\cos(\theta);~S=\sin(\theta)\\
\Psi_g' &=C \Psi_g-S\Psi_r\\
\Psi_b'&=\Psi_b\end{aligned}$$ The gauge invariance signifies that the system $$\begin{aligned}
\pmb\partial \Psi_r&=-\frac{g_3}{2}\mathbf{G}^2\Psi_g+m_2\rho_2 \chi_r \gamma_{012}\notag\\
\pmb\partial \Psi_g&=\frac{g_3}{2}\mathbf{G}^2\Psi_r+m_2\rho_2 \chi_g \gamma_{012}\end{aligned}$$ must be equivalent to the system $$\begin{aligned}
\pmb\partial \Psi_r'&=-\frac{g_3}{2}{\mathbf{G}{'}}^2\Psi_g'+m_2\rho_2' \chi_r' \gamma_{012}\notag\\
\pmb\partial \Psi_g'&=\frac{g_3}{2}{\mathbf{G}{'}}^2\Psi_r'+m_2\rho_2 '\chi_g '\gamma_{012}\end{aligned}$$ Using relations (4.48) and (4.49) the system (4.52) is equivalent to (4.51) if and only if $${\mathbf{G}{'}}^2=\mathbf{G}^2-\frac{2}{g_3}\pmb\partial\theta$$ because we get $$\begin{aligned}
\rho '&=\rho \\
\chi_r'&=C\chi_r+S\chi_g \\
\chi_g'&=C\chi_g-S\chi_r.\end{aligned}$$
The case $k=3$ is detailed in C.5 and the case $k=8$ is detailed in C.6. Cases $k=4$ and $k=6$ are similar to $k=1$ and cases $k=5$ and $k=7$ are similar to $k=2$ by permutation of indexes of color.
Concluding remarks
==================
From experimental results obtained in the accelerators physicists have built what is now known as the “standard model". This model is generally thought to be a part of quantum field theory, itself a part of axiomatic quantum mechanics. One of these axioms is that each state describing a physical situation follows a Schrödinger wave equation. Since this wave equation is not relativistic and does not account for the spin 1/2 which is necessary to any fermion, the standard model has evidently not followed the axiom and has used instead a Dirac equation to describe fermions. Our work also starts with the Dirac equation. This wave equation is the linear approximation of our nonlinear homogeneous equation of the electron.
The wave equation presented here is a wave equation for a classical wave, a function of space and time with value into a Clifford algebra. It is not a quantized wave with value into a Hilbertian space of operators. Nevertheless and consequently we get most of the aspects of the standard model, for instance the fact that leptons are insensitive to strong interactions. The standard model is much stronger than generally thought. For instance we firstly did not use the link between the wave of the particle and the wave of the antiparticle, but then we needed a greater Clifford algebra and we could not get the necessary link between reversions[^1] that we use in our wave equation. We also needed the existence of the inverse to build the wave of a system of particles from the waves of its components. And we got two general identities which exist only if all parts of the general wave are left waves, only the electron having also a right wave.
The most important property of the general wave is its form invariance under a group including the covering group of the restricted Lorentz group. Our group does not explain why space and time are oriented, but it respects these orientations. The physical time is then compatible with thermodynamics, and the physical space is compatible with the violation of parity by weak interactions.
The wave accounts for all particles and anti-particles of the first generation. We have also given [@davi:12][@dav:12][@dav:14][@dabe:14] the reason of the existence of three generations, it is simply the dimension of our physical space. Since the $SU(3)$ gauge group of chromodynamics acts independently from the index of generations, the physical quarks may be combinations of quarks of different generations. Quarks composing protons and neutrons are such combinations. Our wave equation allows only two masses at each generation, one for the lepton part of the wave, the other one for the two quarks. The mixing can give a different mass for the two quarks of each generation.
Since the wave equation with mass term is gauge invariant, there is no necessity to use the mechanism of spontaneous symmetry breaking. The scalar boson certainly exists, but it does not explain the masses.
A wave equation is only a beginning. It shall be necessary to study also the boson part of the standard model and the systems of fermions, from this wave equation. A construction of the wave of a system of identical particles is possible and compatible with the Pauli principle [@dav2:12] [@davi:14].
[10]{}
C. Daviau. . PhD thesis, Université de Nantes, 1993.
C. Daviau. Solutions of the [D]{}irac equation and of a nonlinear [D]{}irac equation for the hydrogen atom. , 7((S)):175–194, 1997.
C. Daviau. Interprétation cinématique de l’onde de l’électron. , 30(3-4), 2005.
C. Daviau. . JePublie, Pouillé-les-coteaux, 2011.
C. Daviau. ${C}l_3^*$ invariance of the [D]{}irac equation and of electromagnetism. , 22(3):611–623, 2012.
C. Daviau. . JePublie, Pouillé-les-coteaux, 2012.
C. Daviau. . CISP, Cambridge UK, 2012.
C. Daviau. Gauge group of the standard model in ${C}l_{1,5}$. , http://hal.archives-ouvertes.fr/hal-01055145, 2014.
C. Daviau and J. Bertrand. Relativistic gauge invariant wave equation of the electron-neutrino. , 5:1001–1022, http://dx.doi.org/10.4236/jmp.2014.511102, 2014.
C. Daviau and J. Bertrand. . Je Publie, Pouillé-les-coteaux, 2014 and http://hal.archives-ouvertes.fr/hal-00907848.
Louis de Broglie. Recherches sur la théorie des quantas. , 17(1), 1924.
René Deheuvels. . PUF, Paris, 1993.
D. Hestenes. A unified language for [M]{}athematics and [P]{}hysics and [C]{}lifford [A]{}lgebra and the interpretation of quantum mechanics. In Chisholm and AK Common, editors, [*Clifford Algebras and their applications in Mathematics and Physics*]{}. Reidel, Dordrecht, 1986.
S. Weinberg. A model of leptons. , 19:1264–1266, 1967.
Calculation of the reverse in $Cl_{1,5}$
========================================
Here indexes $\mu, \nu, \rho\dots$ have value $0,1,2,3$ and indexes $a,b,c,d,e$ have value $0,1,2,3,4,5$. We use[^2] the following matrix representation of $Cl_{1,5}$: $$\begin{aligned}
L_\mu&=\begin{pmatrix} 0 & \gamma_\mu \\ \gamma_\mu &0 \end{pmatrix} ;~
L_4=\begin{pmatrix} 0 & -I_4 \\ I_4 & 0 \end{pmatrix} ;~L_5=\begin{pmatrix}
0 &\mathbf{i} \\ \mathbf{i}&0 \end{pmatrix};~\mathbf{i}=\begin{pmatrix}iI_2 &0
\\0 &-iI_2\end{pmatrix}\notag \\
\gamma_0 &=\gamma^0 =\begin{pmatrix} 0 & I_2 \\ I_2 & 0 \end{pmatrix} ;~\gamma_j=
-\gamma^j=\begin{pmatrix} 0 & \sigma_j \\ -\sigma_j & 0 \end{pmatrix}, ~j=1,2,3 \end{aligned}$$ where $\sigma_j$ are Pauli matrices. This gives $$\begin{aligned}
L_{\mu\nu}&=L_\mu L_\nu =\begin{pmatrix}
0 & \gamma_\mu \\ \gamma_\mu &0
\end{pmatrix}\begin{pmatrix}
0 & \gamma_\nu \\ \gamma_\nu &0
\end{pmatrix}=\begin{pmatrix}
\gamma_{\mu\nu}&0 \\0&\gamma_{\mu\nu}
\end{pmatrix} \\ L_{\mu\nu\rho}&=L_{\mu\nu} L_\rho =\begin{pmatrix}
\gamma_{\mu\nu} & 0 \\ 0&\gamma_{\mu\nu}\end{pmatrix}\begin{pmatrix}
0 & \gamma_\rho \\ \gamma_\rho &0
\end{pmatrix}=\begin{pmatrix}
0&\gamma_{\mu\nu\rho} \\ \gamma_{\mu\nu\rho}&0
\end{pmatrix} \\ L_{0123}&=L_{01} L_{23} =\begin{pmatrix}
\gamma_{0123} & 0 \\ 0&\gamma_{0123}\end{pmatrix}=\begin{pmatrix}
\mathbf{i}&0\\0&\mathbf{i}
\end{pmatrix}\end{aligned}$$ We get also $$\begin{aligned}
L_{45}&= L_4 L_5 =\begin{pmatrix}
0 & -I_4 \\I_4 &0 \end{pmatrix}\begin{pmatrix}
0 & \mathbf{i} \\\mathbf{i} &0
\end{pmatrix}=\begin{pmatrix}
-\mathbf{i}&0 \\0&\mathbf{i}
\end{pmatrix}=-L_{54} \\ L_{012345}&=\begin{pmatrix}
\mathbf{i} & 0 \\ 0&\mathbf{i}\end{pmatrix}\begin{pmatrix}
-\mathbf{i}&0 \\0&\mathbf{i}
\end{pmatrix}=\begin{pmatrix}
I_4 & 0 \\ 0&-I_4\end{pmatrix}
\\ L_{01235}&=L_{0123}L_{5} =\begin{pmatrix}
\mathbf{i} & 0 \\ 0&\mathbf{i}\end{pmatrix}\begin{pmatrix}
0&\mathbf{i}\\\mathbf{i}&0
\end{pmatrix}=\begin{pmatrix}
0 & -I_4 \\ -I_4 &0
\end{pmatrix}.\end{aligned}$$ Similarly we get[^3] $$\begin{aligned}
L_{\mu4}&=\begin{pmatrix}
\gamma_\mu & 0 \\0 &-\gamma_\mu \end{pmatrix}
~;~~L_{\mu5}=\begin{pmatrix}
\gamma_\mu\mathbf{i} &0 \\0&\gamma_\mu\mathbf{i}
\end{pmatrix} \\ L_{\mu\nu 4}&=\begin{pmatrix}
0&-\gamma_{\mu\nu} \\ \gamma_{\mu\nu}&0 \end{pmatrix}
~;~~L_{\mu\nu 5}=\begin{pmatrix}
0&\gamma_{\mu\nu}\mathbf{i} \\ \gamma_{\mu\nu}\mathbf{i}&0 \end{pmatrix}
\\ L_{\mu\nu\rho 4}&=\begin{pmatrix}
\gamma_{\mu\nu\rho}&0 \\0&-\gamma_{\mu\nu\rho} \end{pmatrix}
~;~~L_{\mu\nu\rho 5}=\begin{pmatrix}
\gamma_{\mu\nu\rho}\mathbf{i}&0 \\0&\gamma_{\mu\nu\rho}\mathbf{i} \end{pmatrix}
\\ L_{\mu 45}&=\begin{pmatrix}
0&\gamma_{\mu}\mathbf{i} \\-\gamma_{\mu}\mathbf{i}&0 \end{pmatrix}
~;~~L_{\mu\nu 45}=\begin{pmatrix}
-\gamma_{\mu\nu}\mathbf{i}&0 \\0&\gamma_{\mu\nu}\mathbf{i} \end{pmatrix}
\\ L_{\mu\nu\rho 45}&=\begin{pmatrix}
0&\gamma_{\mu\nu\rho}\mathbf{i} \\-\gamma_{\mu\nu\rho}\mathbf{i}&0 \end{pmatrix}
~;~~L_{01234}=\begin{pmatrix}
0&-\mathbf{i} \\\mathbf{i}&0 \end{pmatrix}\end{aligned}$$ Scalar and pseudo-scalar terms read $$\begin{aligned}
\alpha I_8+\omega L_{012345}&=\begin{pmatrix}
(\alpha+\omega)I_4&0 \\ 0&(\alpha-\omega)I_4
\end{pmatrix} \\ \alpha I_8-\omega \Lambda_{012345}&=\begin{pmatrix}
(\alpha-\omega)I_4&0 \\ 0&(\alpha+\omega)I_4
\end{pmatrix}\end{aligned}$$ For the calculation of the 1-vector term $$N^a L_a=N^4 L_4+
N^5 L_5+N^\mu L_\mu$$ we let $$\beta=N^4~;~~ \delta=N^5 ~;~~\mathbf{a}=N^\mu \gamma_\mu.$$ This gives $$N^a L_a=\begin{pmatrix}
0 & -\beta I_4 +\delta \mathbf{i}+\mathbf{a} \\
\beta I_4 +\delta \mathbf{i}+\mathbf{a}&0
\end{pmatrix}.$$ For the calculation of the 2-vector term $$N^{ab} L_{ab}=N^{45} L_{45}+
N^{\mu 4} L_{\mu 4}+N^{\mu 5} L_{\mu 5}+N^{\mu\nu} L_{\mu\nu}$$ we let $$\epsilon=N^{45}~;~~ \mathbf{b}=N^{\mu 4}\gamma_\mu ~;~~\mathbf{c}=N^{\mu 5}\gamma_\mu~;~~
\mathbf{A}=N^{\mu \nu}\gamma_{\mu\nu}$$ This gives $$N^{ab} L_{ab}=\begin{pmatrix}
-\epsilon\mathbf{i}+\mathbf{b}- \mathbf{ic}+\mathbf{A} & 0\\
0 & \epsilon\mathbf{i}-\mathbf{b}- \mathbf{ic}+\mathbf{A}
\end{pmatrix}.$$ For the calculation of the 3-vector term $$N^{abc} L_{abc}=N^{\mu 45} L_{\mu 45}+
N^{\mu\nu 4} L_{\mu\nu 4}+N^{\mu\nu 5} L_{\mu\nu 5}+N^{\mu\nu\rho} L_{\mu\nu\rho}$$ we let $$\mathbf{d}=N^{\mu45}\gamma_\mu~;~~ \mathbf{B}=N^{\mu\nu 4}\gamma_{\mu\nu} ~;~~\mathbf{C}=N^{\mu\nu 5}\gamma_{\mu\nu}~;~~
\mathbf{ie}=N^{\mu \nu\rho}\gamma_{\mu\nu\rho}$$ This gives with (A.3) and (A.9) $$N^{abc}L_{abc}=\begin{pmatrix}
0&\mathbf{di}-\mathbf{B}+ \mathbf{iC}+\mathbf{ie}\\
\mathbf{id}+\mathbf{B}+ \mathbf{iC}+\mathbf{ie}&0
\end{pmatrix}.$$ For the calculation of the 4-vector term $$N^{abcd} L_{abcd}=N^{\mu\nu 45} L_{\mu\nu 45}+
N^{\mu\nu\rho 4} L_{\mu\nu\rho 4}+N^{\mu\nu\rho 5} L_{\mu\nu\rho 5}+N^{0123} L_{0123}$$ we let $$\mathbf{D}=N^{\mu\nu 45}\gamma_{\mu\nu}~;~~ \mathbf{if}=N^{\mu\nu\rho 4}\gamma_{\mu\nu\rho}
~;~~\mathbf{ig}=N^{\mu\nu\rho 5}\gamma_{\mu\nu\rho}~;~~\zeta=N^{0123}$$ This gives with (A.4) and (A.10) $$N^{abcd}L_{abcd}=\begin{pmatrix}
-\mathbf{iD}+\mathbf{if}+ \mathbf{g}+\zeta\mathbf{i}&0\\
0& \mathbf{iD}-\mathbf{if}+ \mathbf{g}+\zeta\mathbf{i}
\end{pmatrix}.$$ For the calculation of the pseudo-vector term $$N^{abcde} L_{abcde}=N^{\mu\nu\rho 45} L_{\mu\nu\rho 45}+
N^{01234} L_{01234}+N^{01235} L_{01235}$$ we let $$\mathbf{ih}=N^{\mu\nu\rho 45}\gamma_{\mu\nu\rho}~;~~ \eta=N^{01234}
~;~~\theta=N^{01235}$$ This gives with (A.7) and (A.12) $$N^{abcde}L_{abcde}=\begin{pmatrix}
0& \mathbf{h}-\eta\mathbf{i}- \theta I_4\\
-\mathbf{h}+\eta\mathbf{i}- \theta I_4
\end{pmatrix}.$$ We then get $$\begin{aligned}
&\Psi= \begin{pmatrix}
\Psi_l & \Psi_r \\ \Psi_g &\Psi_b
\end{pmatrix}\\
&=\begin{pmatrix}
(\alpha+\omega)I_4 +( \mathbf{b}+\mathbf{g})+(\mathbf{A}-\mathbf{iD}) &
-(\beta+\theta)I_4+(\mathbf{a}+\mathbf{h})+(-\mathbf{B}+\mathbf{iC}) \\
+\mathbf{i}(-\mathbf{c}+\mathbf{f}) +(\zeta-\epsilon)\mathbf{i} & +\mathbf{i}(-\mathbf{d}+\mathbf{e})
+(\delta-\eta)\mathbf{i} \\
\\
(\beta-\theta)I_4+(\mathbf{a}-\mathbf{h})+(\mathbf{B}+\mathbf{iC}) &
(\alpha-\omega)I_4 +(-\mathbf{b}+\mathbf{g})+(\mathbf{A}+\mathbf{iD}) \\ +\mathbf{i}(\mathbf{d}+\mathbf{e})
+(\delta+\eta)\mathbf{i} &+\mathbf{i}(-\mathbf{c}-\mathbf{f})
+(\zeta+\epsilon)\mathbf{i}
\end{pmatrix}\notag\end{aligned}$$ This implies $$\begin{aligned}
\Psi_l &= (\alpha+\omega) +(\mathbf{b}+\mathbf{g})+(\mathbf{A}-\mathbf{iD})
+\mathbf{i}(-\mathbf{c}+\mathbf{f}) +(\zeta-\epsilon)\mathbf{i} \\
\Psi_r &=-(\beta+\theta)+(\mathbf{a}+\mathbf{h})+(-\mathbf{B}+\mathbf{iC})
+\mathbf{i}(-\mathbf{d}+\mathbf{e}) +(\delta-\eta)\mathbf{i} \\
\Psi_g &=(\beta-\theta)+(\mathbf{a}-\mathbf{h})+(\mathbf{B}+\mathbf{iC})
+\mathbf{i}(\mathbf{d}+\mathbf{e}) +(\delta+\eta)\mathbf{i} \\
\Psi_b &= (\alpha-\omega) +(-\mathbf{b}+\mathbf{g})+(\mathbf{A}+\mathbf{iD})
+\mathbf{i}(-\mathbf{c}-\mathbf{f}) +(\zeta+\epsilon)\mathbf{i} \end{aligned}$$ In $Cl_{1,3}$ the reverse of $$A=<A>_0+<A>_1+<A>_2+<A>_3+<A>_4$$ is $$\widetilde A=<A>_0+<A>_1-<A>_2-<A>_3+<A>_4$$ we must change the sign of bivectors $\mathbf{A}$, $\mathbf{B}$, $\mathbf{iC}$, $\mathbf{iD}$, and trivectors $\mathbf{ic}$, $\mathbf{id}$, $\mathbf{ie}$, $\mathbf{if}$ and we then get $$\begin{aligned}
\widetilde \Psi_l &= (\alpha+\omega) +(\mathbf{b}+\mathbf{g})+(-\mathbf{A}+\mathbf{iD})
+\mathbf{i}(\mathbf{c}-\mathbf{f}) +(\zeta-\epsilon)\mathbf{i} \\
\widetilde \Psi_r &=-(\beta+\theta)+(\mathbf{a}+\mathbf{h})+(\mathbf{B}-\mathbf{iC})
+\mathbf{i}(\mathbf{d}-\mathbf{e}) +(\delta-\eta)\mathbf{i} \\
\widetilde \Psi_g &=(\beta-\theta)+(\mathbf{a}-\mathbf{h})-(\mathbf{B}+\mathbf{iC})
-\mathbf{i}(\mathbf{d}+\mathbf{e}) +(\delta+\eta)\mathbf{i} \\
\widetilde \Psi_b &= (\alpha-\omega)I_4 +(- \mathbf{b}+\mathbf{g})-(\mathbf{A}+\mathbf{iD})
+\mathbf{i}(\mathbf{c}+\mathbf{f}) +(\zeta+\epsilon)\mathbf{i} \end{aligned}$$ The reverse, in $Cl_{1,5}$ now, of $$A=<A>_0+<A>_1+<A>_2+<A>_3+<A>_4+<A>_5+<A>_6$$ is $$\widetilde A=<A>_0+<A>_1-<A>_2-<A>_3+<A>_4+<A>_5-<A>_6$$ Only terms which change sign, with (A.13), (A.18) and (A.20), are scalars $\epsilon$ and $\omega$, vectors $\mathbf{b}$, $\mathbf{c}$, $\mathbf{d}$, $\mathbf{e}$ and bivectors $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$. These changes of sign are not the same in $Cl_{1,5}$ as in $Cl_{1,3}$. Differences are corrected by the fact that the reversion in $Cl_{1,5}$ also exchanges the place of $\Psi_l$ and $\Psi_b$ terms. We then get from (A.25) $$\begin{aligned}
\widetilde\Psi&=
\begin{pmatrix}
(\alpha-\omega)I_4 +(-\mathbf{b}+\mathbf{g})+(-\mathbf{A}-\mathbf{iD}) &
-(\beta+\theta)I_4+(\mathbf{a}+\mathbf{h})+(\mathbf{B}-\mathbf{iC}) \\
+\mathbf{i}(\mathbf{c}+\mathbf{f}) +(\zeta+\epsilon)\mathbf{i} & +\mathbf{i}(\mathbf{d}-\mathbf{e})
+(\delta-\eta)\mathbf{i} \\
\\
(\beta-\theta)I_4+(\mathbf{a}-\mathbf{h})-(\mathbf{B}+\mathbf{iC}) &
(\alpha+\omega)I_4 +(\mathbf{b}+\mathbf{g})+(-\mathbf{A}+\mathbf{iD}) \\ -\mathbf{i}(\mathbf{d}+\mathbf{e})
+(\delta+\eta)\mathbf{i} &+\mathbf{i}(\mathbf{c}-\mathbf{f})
+(\zeta-\epsilon)\mathbf{i}
\end{pmatrix}\notag\\
&=\begin{pmatrix}
\widetilde\Psi_b & \widetilde\Psi_r \\ \widetilde\Psi_g &\widetilde\Psi_l
\end{pmatrix}.\end{aligned}$$ This link between the reversion in $Cl_{1,3}$ and the reversion in $Cl_{1,5}$ is necessary to get an invariant wave equation. It is not general, for instance the reversion in $Cl_3$ is not linked to the reversion in $Cl_{2,3}$.
Scalar densities and $\chi$ terms
=================================
There are $6\times 5/2=15$ such complex scalar densities: $$\begin{aligned}
s_1&=2(\xi_{1\overline{u}r}\eta_{1ug}^*+\xi_{2\overline{u}r}\eta_{2ug}^*)=
2(\eta_{2ur}^*\eta_{1ug}^*-\eta_{1ur}^*\eta_{2ug}^*)\\
s_2&=2(\xi_{1\overline{u}g}\eta_{1ub}^*+\xi_{2\overline{u}g}\eta_{2ub}^*)=
2(\eta_{2ug}^*\eta_{1ub}^*-\eta_{1ug}^*\eta_{2ub}^*)\\
s_3&=-2(\xi_{1\overline{u}r}\eta_{1ub}^*+\xi_{2\overline{u}r}\eta_{2ub}^*)=
2(\eta_{2ub}^*\eta_{1ur}^*-\eta_{1ub}^*\eta_{2ur}^*)\end{aligned}$$ $$\begin{aligned}
s_4&=2(\xi_{1\overline{d}r}\eta_{1dg}^*+\xi_{2\overline{d}r}\eta_{2dg}^*)=
2(\eta_{2dr}^*\eta_{1dg}^*-\eta_{1dr}^*\eta_{2dg}^*)\\
s_5&=2(\xi_{1\overline{d}g}\eta_{1db}^*+\xi_{2\overline{d}g}\eta_{2db}^*)=
2(\eta_{2dg}^*\eta_{1db}^*-\eta_{1dg}^*\eta_{2db}^*)\\
s_6&=-2(\xi_{1\overline{d}r}\eta_{1db}^*+\xi_{2\overline{d}r}\eta_{2db}^*)=
2(\eta_{2db}^*\eta_{1dr}^*-\eta_{1db}^*\eta_{2dr}^*)\end{aligned}$$ $$\begin{aligned}
s_7&=2(\xi_{1\overline{u}r}\eta_{1dr}^*+\xi_{2\overline{u}r}\eta_{2dr}^*)=
2(\eta_{2ur}^*\eta_{1dr}^*-\eta_{1ur}^*\eta_{2dr}^*)\\
s_8&=2(\xi_{1\overline{u}g}\eta_{1dg}^*+\xi_{2\overline{u}g}\eta_{2dg}^*)=
2(\eta_{2ug}^*\eta_{1dg}^*-\eta_{1ug}^*\eta_{2dg}^*)\\
s_9&=2(\xi_{1\overline{u}b}\eta_{1db}^*+\xi_{2\overline{u}b}\eta_{2db}^*)=
2(\eta_{2ub}^*\eta_{1db}^*-\eta_{1ub}^*\eta_{2db}^*)\end{aligned}$$ $$\begin{aligned}
s_{10}&=2(\xi_{1\overline{u}r}\eta_{1dg}^*+\xi_{2\overline{u}r}\eta_{2dg}^*)=
2(\eta_{2ur}^*\eta_{1dg}^*-\eta_{1ur}^*\eta_{2dg}^*)\\
s_{11}&=2(\xi_{1\overline{u}g}\eta_{1db}^*+\xi_{2\overline{u}g}\eta_{2db}^*)=
2(\eta_{2ug}^*\eta_{1db}^*-\eta_{1ug}^*\eta_{2db}^*)\\
s_{12}&=-2(\xi_{1\overline{d}r}\eta_{1ub}^*+\xi_{2\overline{d}r}\eta_{2ub}^*)=
2(\eta_{2ub}^*\eta_{1dr}^*-\eta_{1ub}^*\eta_{2dr}^*)\end{aligned}$$ $$\begin{aligned}
s_{13}&=2(\xi_{1\overline{u}r}\eta_{1db}^*+\xi_{2\overline{u}r}\eta_{2db}^*)=
2(\eta_{2ur}^*\eta_{1db}^*-\eta_{1ur}^*\eta_{2db}^*)\\
s_{14}&=-2(\xi_{1\overline{d}r}\eta_{1ug}^*+\xi_{2\overline{d}r}\eta_{2ug}^*)=
2(\eta_{2ug}^*\eta_{1dr}^*-\eta_{1ug}^*\eta_{2dr}^*)\\
s_{15}&=-2(\xi_{1\overline{d}g}\eta_{1ub}^*+\xi_{2\overline{d}g}\eta_{2ub}^*)=
2(\eta_{2ub}^*\eta_{1dg}^*-\eta_{1ub}^*\eta_{2dg}^*).\end{aligned}$$ We used in [@dabe:14] $$\chi_l=\frac{1}{\rho_1^2}\begin{pmatrix}
a_1^* \phi_e +a_2^*\phi_n\sigma_1+a_3^* \phi_n & -a_2^*\phi_{eL}\sigma_1+a_3^*\phi_{eR} \\
a_2\widehat \phi_{eL}\sigma_1 + a_3\widehat \phi_{eR} & a_1\widehat \phi_e-a_2\widehat\phi_n
\sigma_1 +a_3\widehat \phi_n
\end{pmatrix}$$ with $\phi_{eR}=\phi_e(1+\sigma_3)/2$ and $\phi_{eL}=\phi_e(1-\sigma_3)/2$, and we need now $$\begin{aligned}
\rho_2^2 \chi_r=
&\begin{pmatrix}
\begin{pmatrix}
s_4^*\phi_{dg}-s_6^*\phi_{db}-s_7^*\phi_{ur}\\-s_{12}^*\phi_{ub}-s_{14}^*\phi_{ug}
\end{pmatrix} \sigma_1 &\begin{pmatrix}s_1^*\phi_{ug}-s_3^*\phi_{ub}+s_7^*\phi_{dr}\\
+s_{10}^*\phi_{dg}+s_{13}^*\phi_{db} \end{pmatrix}
\sigma_1\\ \begin{pmatrix}-s_1\widehat{\phi}_{ug}+s_3\widehat{\phi}_{ub}-s_7\widehat{\phi}_{dr}\\
-s_{10}\widehat{\phi}_{dg}-s_{13}\widehat{\phi}_{db} \end{pmatrix}\sigma_1&
\begin{pmatrix}-s_4\widehat{\phi}_{dg}+s_6\widehat{\phi}_{db}+s_7\widehat{\phi}_{ur} \\
+s_{12}\widehat{\phi}_{ub}+s_{14}\widehat{\phi}_{ug}\end{pmatrix}\sigma_1
\end{pmatrix}\\
\rho_2^2 \chi_g=
&\begin{pmatrix}
\begin{pmatrix}
s_5^*\phi_{db}-s_4^*\phi_{dr}-s_8^*\phi_{ug}\\-s_{10}^*\phi_{ur}-s_{15}^*\phi_{ub}
\end{pmatrix} \sigma_1 &\begin{pmatrix}s_2^*\phi_{ub}-s_1^*\phi_{ur}+s_8^*\phi_{dg}\\
+s_{11}^*\phi_{db}+s_{14}^*\phi_{dr} \end{pmatrix}
\sigma_1\\ \begin{pmatrix}-s_2\widehat{\phi}_{ub}+s_1\widehat{\phi}_{ur}-s_8\widehat{\phi}_{dg}\\
-s_{11}\widehat{\phi}_{db}-s_{14}\widehat{\phi}_{dr} \end{pmatrix}\sigma_1&
\begin{pmatrix}-s_5\widehat{\phi}_{db}+s_4\widehat{\phi}_{dr}+s_8\widehat{\phi}_{ug} \\
+s_{10}\widehat{\phi}_{ur}+s_{15}\widehat{\phi}_{ub}\end{pmatrix}\sigma_1
\end{pmatrix}\\
\rho_2^2 \chi_b=
&\begin{pmatrix}
\begin{pmatrix}
s_6^*\phi_{dr}-s_5^*\phi_{dg}-s_9^*\phi_{ub}\\-s_{11}^*\phi_{ug}-s_{13}^*\phi_{ur}
\end{pmatrix} \sigma_1 &\begin{pmatrix}s_3^*\phi_{ur}-s_2^*\phi_{ug}+s_9^*\phi_{db}\\
+s_{12}^*\phi_{dr}+s_{15}^*\phi_{dg} \end{pmatrix}
\sigma_1\\ \begin{pmatrix}-s_3\widehat{\phi}_{ur}+s_2\widehat{\phi}_{ug}-s_9\widehat{\phi}_{db}\\
-s_{12}\widehat{\phi}_{dr}-s_{15}\widehat{\phi}_{dg} \end{pmatrix}\sigma_1&
\begin{pmatrix}-s_6\widehat{\phi}_{dr}+s_5\widehat{\phi}_{dg}+s_9\widehat{\phi}_{ub} \\
+s_{11}\widehat{\phi}_{ug}+s_{13}\widehat{\phi}_{ur}\end{pmatrix}\sigma_1
\end{pmatrix}\end{aligned}$$
Gauge invariance, details
=========================
Gauge group generated by $\underline{P}_1$
------------------------------------------
Since $\underline{P}_1(\Psi^c)=\Psi^c L_{35}$ we get $$\begin{aligned}
{\Psi{'}}^c&=[\exp(\theta\underline{P}_1)](\Psi^c)=\Psi^c\exp(\theta L_{35}) \\
\Psi_c'&=\Psi_c e^{\theta \gamma_3\mathbf{i}},~c=r,g,b.\end{aligned}$$ We let $$C=\cos(\theta)~;~~S=\sin(\theta)$$ Then (C.2) is equivalent to the system $$\begin{aligned}
\widehat{\phi}_{dc}'&=C\widehat{\phi}_{dc}-iS\widehat{\phi}_{uc}\sigma_3 \\
\widehat{\phi}_{uc}'&=C\widehat{\phi}_{uc}-iS\widehat{\phi}_{dc}\sigma_3 \end{aligned}$$ or to the system $$\begin{aligned}
\eta_{1dc}'&=C\eta_{1dc}-iS\eta_{1uc};~{\eta{'}}_{1dc}^*=C\eta_{1dc}^*+iS\eta_{1uc}^* \\
\eta_{2dc}'&=C\eta_{2dc}-iS\eta_{2uc};~{\eta{'}}_{2dc}^*=C\eta_{2dc}^*+iS\eta_{2uc}^* \\
\eta_{1uc}'&=C\eta_{1uc}-iS\eta_{1dc};~{\eta{'}}_{1uc}^*=C\eta_{1uc}^*+iS\eta_{1dc}^* \\
\eta_{2uc}'&=C\eta_{2uc}-iS\eta_{2dc};~{\eta{'}}_{2uc}^*=C\eta_{2uc}^*+iS\eta_{2dc}^* \end{aligned}$$ We then get $$\begin{aligned}
s_1'&=C^2s_1-S^2s_4+iCS(s_{10}-s_{14})\\
s_4'&=C^2s_4-S^2s_1+iCS(s_{10}-s_{14})\\
s_{10}'&=C^2s_{10}+S^2s_{14}+iCS(s_1+s_4) \\
s_{14}'&=C^2s_{14}+S^2s_{10}-iCS(s_1+s_4).\end{aligned}$$ This implies $$s_{1}'{s{'}}_{1}^* + s_{4}'{s{'}}_{4}^* + s_{10}'{s{'}}_{10}^*+ s_{14}'{s{'}}_{14}^*
=s_1 s_1^* + s_4 s_4^* + s_{10}s_{10}^*+ s_{14}s_{14}^*.$$ Similarly, permuting colors, we get $$\begin{aligned}
s_2'&=C^2s_2-S^2s_5+iCS(s_{11}-s_{15})\\
s_5'&=C^2s_5-S^2s_2+iCS(s_{11}-s_{15})\\
s_{11}'&=C^2s_{11}+S^2s_{15}+iCS(s_2+s_5) \\
s_{15}'&=C^2s_{15}+S^2s_{11}-iCS(s_2+s_5).\end{aligned}$$ This implies $$s_{2}'{s{'}}_{2}^* + s_{5}'{s{'}}_{5}^* + s_{11}'{s{'}}_{11}^*+ s_{15}'{s{'}}_{15}^*
=s_2 s_2^* + s_5 s_5^* + s_{11}s_{11}^*+ s_{15}s_{15}^*.$$ and also $$\begin{aligned}
s_3'&=C^2s_3-S^2s_6+iCS(s_{12}-s_{13})\\
s_6'&=C^2s_6-S^2s_3+iCS(s_{12}-s_{13})\\
s_{12}'&=C^2s_{12}+S^2s_{13}+iCS(s_3+s_6) \\
s_{13}'&=C^2s_{13}+S^2s_{12}-iCS(s_3+s_6).\end{aligned}$$ This implies $$s_{3}'{s{'}}_{3}^* + s_{6}'{s{'}}_{6}^* + s_{12}'{s{'}}_{12}^*+ s_{13}'{s{'}}_{13}^*
=s_3 s_3^* + s_6 s_6^* + s_{12}s_{12}^*+ s_{13}s_{13}^*.$$ Moreover we get $$s_7'=s_7;~s_8'=s_8;~s_9'=s_9.$$ We then get $$\rho'=\rho$$ Next we have $$\begin{aligned}
\chi_r&=\begin{pmatrix}
A & B \\\widehat{B}&\widehat{A}\end{pmatrix};~\chi_r'=\begin{pmatrix}
A' & B' \\\widehat{B}'&\widehat{A}'\end{pmatrix}\\
\widehat{A}&=(-s_4 \widehat{\phi}_{dg}+s_6 \widehat{\phi}_{db}+s_7 \widehat{\phi}_{ur}
+s_{12} \widehat{\phi}_{ub}+s_{14} \widehat{\phi}_{ug})\sigma_1 \\
\widehat{B}&=(-s_1 \widehat{\phi}_{ug}+s_3 \widehat{\phi}_{ub}-s_7 \widehat{\phi}_{dr}
-s_{10} \widehat{\phi}_{dg}-s_{13} \widehat{\phi}_{db})\sigma_1.\end{aligned}$$ and we get $$\begin{aligned}
\widehat{A}'&=C\widehat{A}-iS\widehat{B}\sigma_3 \\
\widehat{B}'&=C\widehat{B}-iS\widehat{A}\sigma_3 \\
\chi_r'&=\chi_r\begin{pmatrix}
C & -iS\sigma_3 \\-iS\sigma_3 & C
\end{pmatrix}=\chi_re^{\theta\gamma_3\mathbf{i}}.\end{aligned}$$ Since we get the same relation for g and b colors we finally get $${\chi{'}}^c=\chi^c \exp(\theta L_{35})$$
Gauge group generated by $\underline{P}_2$
------------------------------------------
Since $\underline{P}_2(\Psi^c)=\Psi^c L_{5012}$ we get $$\begin{aligned}
{\Psi{'}}^c&=[\exp(\theta\underline{P}_2)](\Psi^c)=\Psi^c\exp(\theta L_{5012}) \\
\Psi_c'&=\Psi_c e^{\theta \gamma_3},~c=r,g,b.\end{aligned}$$ We let $$C=\cos(\theta)~;~~S=\sin(\theta)$$ Then (C.35) is equivalent to the system $$\begin{aligned}
\widehat{\phi}_{dc}'&=C\widehat{\phi}_{dc}+S\widehat{\phi}_{uc} \\
\widehat{\phi}_{uc}'&=C\widehat{\phi}_{uc}-S\widehat{\phi}_{dc} \end{aligned}$$ or to the system $$\begin{aligned}
\eta_{1dc}'&=C\eta_{1dc}+S\eta_{1uc};~{\eta{'}}_{1dc}^*=C\eta_{1dc}^*+S\eta_{1uc}^* \\
\eta_{2dc}'&=C\eta_{2dc}+S\eta_{2uc};~{\eta{'}}_{2dc}^*=C\eta_{2dc}^*+S\eta_{2uc}^* \\
\eta_{1uc}'&=C\eta_{1uc}-S\eta_{1dc};~{\eta{'}}_{1uc}^*=C\eta_{1uc}^*-S\eta_{1dc}^* \\
\eta_{2uc}'&=C\eta_{2uc}-S\eta_{2dc};~{\eta{'}}_{2uc}^*=C\eta_{2uc}^*-S\eta_{2dc}^* \end{aligned}$$ We then get $$\begin{aligned}
s_1'&=C^2s_1+S^2s_4-CS s_{10}+CS s_{14}\\
s_4'&=C^2s_4+S^2s_1+CS s_{10}-CS s_{14}\\
s_{10}'&=C^2 s_{10}+S^2 s_{14}+CS s_1-CS s_4 \\
s_{14}'&=C^2 s_{14}+S^2 s_{10}-CS s_1+CS s_4.\end{aligned}$$ This implies $$s_{1}'{s{'}}_{1}^* + s_{4}'{s{'}}_{4}^* + s_{10}'{s{'}}_{10}^*+ s_{14}'{s{'}}_{14}^*
=s_1 s_1^* + s_4 s_4^* + s_{10}s_{10}^*+ s_{14}s_{14}^*.$$ Similarly, permuting colors, we get $$\begin{aligned}
s_2'&=C^2s_2+S^2s_5-CS s_{11}+CS s_{15}\\
s_5'&=C^2s_5+S^2s_2+CS s_{11}-CS s_{15}\\
s_{11}'&=C^2 s_{11}+S^2 s_{15}+CS s_2-CS s_5 \\
s_{15}'&=C^2 s_{15}+S^2 s_{11}-CS s_2+CS s_5.\end{aligned}$$ This implies $$s_{2}'{s{'}}_{2}^* + s_{5}'{s{'}}_{5}^* + s_{11}'{s{'}}_{11}^*+ s_{15}'{s{'}}_{15}^*
=s_2 s_2^* + s_5 s_5^* + s_{11}s_{11}^*+ s_{15}s_{15}^*.$$ and also $$\begin{aligned}
s_3'&=C^2s_3+S^2s_6-CS s_{12}+CS s_{13}\\
s_6'&=C^2s_6+S^2s_3+CS s_{12}-CS s_{13}\\
s_{12}'&=C^2 s_{12}+S^2 s_{13}+CS s_3-CS s_6 \\
s_{13}'&=C^2 s_{13}+S^2 s_{12}-CS s_3+CS s_6.\end{aligned}$$ This implies $$s_{3}'{s{'}}_{3}^* + s_{6}'{s{'}}_{6}^* + s_{12}'{s{'}}_{12}^*+ s_{13}'{s{'}}_{13}^*
=s_3 s_3^* + s_6 s_6^* + s_{12}s_{12}^*+ s_{13}s_{13}^*.$$ Moreover we get $$s_7'=s_7;~s_8'=s_8;~s_9'=s_9.$$ We then get $$\rho'=\rho$$ Next we get with (C.27) $$\begin{aligned}
\widehat{A}'&=C\widehat{A}-S\widehat{B}\sigma_3 \\
\widehat{B}'&=C\widehat{B}+S\widehat{A}\sigma_3 \\
\chi_r'&=\chi_r\begin{pmatrix}
C & -S\sigma_3 \\ S\sigma_3 & C
\end{pmatrix}=\chi_re^{-\theta\gamma_3}.\end{aligned}$$ Since we get the same relation for g and b colors we finally get $${\chi{'}}^c=\chi^c \exp(-\theta L_{5012})$$
Gauge group generated by $\underline{P}_3$
------------------------------------------
Since $\underline{P}_3(\Psi^c)=\Psi^c L_{3012}$ we get $$\begin{aligned}
{\Psi{'}}^c&=[\exp(\theta\underline{P}_3)](\Psi^c)=\Psi^c\exp(\theta L_{3012}) \\
\Psi_c'&=\Psi_c e^{\theta \gamma_{3012}},~c=r,g,b.\end{aligned}$$ Then (C.65) is equivalent to the system $$\begin{aligned}
\widehat{\phi}_{dc}'&=e^{i\theta}\widehat{\phi}_{dc} \\
\widehat{\phi}_{uc}'&=e^{-i\theta}\widehat{\phi}_{uc}\end{aligned}$$ or to the system $$\begin{aligned}
\eta_{1dc}'&=e^{i\theta}\eta_{1dc};~{\eta{'}}_{1dc}^*=e^{-i\theta}\eta_{1dc}^* \\
\eta_{2dc}'&=e^{i\theta}\eta_{2dc};~{\eta{'}}_{2dc}^*=e^{-i\theta}\eta_{2dc}^* \\
\eta_{1uc}'&=e^{-i\theta}\eta_{1uc};~{\eta{'}}_{1uc}^*=e^{i\theta}\eta_{1uc}^* \\
\eta_{2uc}'&=e^{-i\theta}\eta_{2uc};~{\eta{'}}_{2uc}^*=e^{i\theta}\eta_{2uc}^* \end{aligned}$$ We then get $$\begin{aligned}
s_1'&=e^{2i\theta}s_1;~s_2'=e^{2i\theta}s_2;~s_3'=e^{2i\theta}s_3 \\
s_4'&=e^{-2i\theta}s_4;~s_5'=e^{-2i\theta}s_5;~s_6'=e^{-2i\theta}s_6 \\
s_{7}'&=s_{7};~s_{8}'=s_{8};~s_{9}'=s_{9} \\
s_{10}'&=s_{10};~s_{11}'=s_{11};~s_{12}'=s_{12} \\
s_{13}'&=s_{13};~s_{14}'=s_{14};~s_{15}'=s_{15} .\end{aligned}$$ This implies $$\rho'=\rho$$ Next we get with (C.27) $$\begin{aligned}
\widehat{A}'&=e^{-i\theta}\widehat{A}~;~~A'=e^{i\theta}A \\
\widehat{B}'&=e^{i\theta}\widehat{B}~;~~B'=e^{-i\theta}B \\
\chi_r'&=\chi_r\begin{pmatrix}
e^{i\theta} & 0 \\ 0 & e^{-i\theta}
\end{pmatrix}=\chi_re^{\theta\mathbf{i}}.\end{aligned}$$ Since we get the same relation for g and b colors we finally get $${\chi{'}}^c=\chi^c \exp(-\theta L_{3012})$$
Gauge group generated by $\underline{\mathbf{i}}\Gamma_1$
---------------------------------------------------------
We name $f_1$ the gauge transformation $$f_1:\Psi^c \mapsto \underline{\mathbf{i}}\Gamma_1(\Psi^c)=\begin{pmatrix}
0 & \mathbf{i} \Psi_g \\\mathbf{i} \Psi_r & 0
\end{pmatrix}$$ which implies with $C=\cos(\theta)$ and $S=\sin(\theta)$ $$\begin{aligned}
[\exp(\theta f_1)](\Psi^c)&=\begin{pmatrix}
0 & C\Psi_r +S\mathbf{i}\Psi_g \\ C\Psi_g +S\mathbf{i}\Psi_r & \Psi_b
\end{pmatrix}=\begin{pmatrix}
0 & \Psi_r' \\ \Psi_g ' & \Psi_b '
\end{pmatrix}\\
\Psi_r'&=C\Psi_r +S\mathbf{i}\Psi_g \\
\Psi_g '&=C\Psi_g +S\mathbf{i}\Psi_r \\
\Psi_b '&= \Psi_b\end{aligned}$$ The equality (C.84) is equivalent to the system $$\begin{aligned}
{\eta{'}}_{1dr}^*&=C\eta_{1dr}^*+iS\eta_{1dg}^*;~{\eta{'}}_{1ur}^*=C\eta_{1ur}^*+iS\eta_{1ug}^*\\
{\eta{'}}_{2dr}^*&=C\eta_{2dr}^*+iS\eta_{2dg}^*;~{\eta{'}}_{2ur}^*=C\eta_{2ur}^*+iS\eta_{2ug}^*\end{aligned}$$ The equality (C.85) is equivalent to the system $$\begin{aligned}
{\eta{'}}_{1dg}^*&=C\eta_{1dg}^*+iS\eta_{1dr}^*;~{\eta{'}}_{1ug}^*=C\eta_{1ug}^*+iS\eta_{1ur}^*\\
{\eta{'}}_{2dg}^*&=C\eta_{2dg}^*+iS\eta_{2dr}^*;~{\eta{'}}_{2ug}^*=C\eta_{2ug}^*+iS\eta_{2ur}^*\end{aligned}$$ This gives for the invariant scalars $s_j$ $$\begin{aligned}
s_1'&=s_1;~s_4'=s_4;~s_9'=s_9 \\
s_2'&=Cs_2-iSs_3;~s_3'=Cs_3-iSs_2 \\
s_5'&=Cs_5-iSs_6;~s_6'=Cs_6-iSs_5 \\
s_{11}'&=Cs_{11}+iSs_{13};~s_{13}'=Cs_{13}+iSs_{11} \\
s_{12}'&=Cs_{12}+iSs_{15};~s_{15}'=Cs_{15}+iSs_{12}\end{aligned}$$ $$\begin{aligned}
s_7'&=C^2 s_7 - S^2 s_8 + iCS s_{10}+iCS s_{14} \\
s_8'&=C^2 s_8 - S^2 s_7 + iCS s_{14}+iCS s_{10}\\
s_{10}'&=C^2 s_{10}-S^2 s_{14}+iCS s_7 +iCS s_8 \\
s_{14}'&=C^2 s_{14}-S^2 s_{10}+iCS s_8 +iCS s_7 \end{aligned}$$ We then get $$\begin{aligned}
s_2'{s{'}}_2^*+s_3'{s{'}}_3^*&=s_2 s_2^* + s_3 s_3^* \\
s_5'{s{'}}_5^*+s_6'{s{'}}_6^*&=s_5 s_5^* + s_6 s_6^*\\
s_{11}'{s{'}}_{11}^*+s_{13}'{s{'}}_{13}^*&=s_{11} s_{11}^* + s_{13} s_{13}^* \\
s_{12}'{s{'}}_{12}^*+s_{15}'{s{'}}_{15}^*&=s_{12} s_{12}^* + s_{15} s_{15}^* \\
s_7'{s{'}}_7^*+s_8'{s{'}}_8^*+s_{10}'{s{'}}_{10}^*+s_{14}'{s{'}}_{14}^*&=
s_7 s_7^* + s_8 s_8^*+s_{10} s_{10}^* + s_{14} s_{14}^*\\
\rho'&=\rho.\end{aligned}$$ Next we let $$\begin{aligned}
\chi_r&=\begin{pmatrix} A_r & B_r \\ \widehat{B}_r & \widehat{A}_r \end{pmatrix} ;~
\chi_r'=\begin{pmatrix} A_r' & B_r' \\ \widehat{B}_r' & \widehat{A}_r' \end{pmatrix} \\
\chi_g&=\begin{pmatrix} A_g & B_g \\ \widehat{B}_g & \widehat{A}_g \end{pmatrix} ;~
\chi_g'=\begin{pmatrix} A_g' & B_g' \\ \widehat{B}_g' & \widehat{A}_g' \end{pmatrix} \end{aligned}$$ and we get with (B.17) and (B.18) $$\begin{aligned}
A_r'&=C A_r -i S A_g;~B_r'=C B_r -i S B_g \\
A_g'&= CA_g-iS A_r;~B_g'=C B_g -i S B_r.\end{aligned}$$ This gives the awaited result $$\chi_r'=C\chi_r-\mathbf{i} S \chi_g;~\chi_g'=C\chi_g-\mathbf{i}S\chi_r.$$
Gauge group generated by $\underline{\mathbf{i}}\Gamma_3$
---------------------------------------------------------
We name $f_3$ the gauge transformation $$f_3:\Psi^c \mapsto \underline{\mathbf{i}}\Gamma_3(\Psi^c)=\begin{pmatrix}
0 & \mathbf{i} \Psi_r \\-\mathbf{i} \Psi_g & 0
\end{pmatrix}$$ which implies $$\begin{aligned}
[\exp(\theta f_1)](\Psi^c)&=\begin{pmatrix}
0 & e^{\theta\mathbf{i}}\Psi_r \\ e^{-\theta\mathbf{i}}\Psi_g & \Psi_b
\end{pmatrix}=\begin{pmatrix}
0 & \Psi_r' \\ \Psi_g ' & \Psi_b '
\end{pmatrix}\\
\Psi_r'&=e^{\theta\mathbf{i}}\Psi_r \\
\Psi_g '&=e^{-\theta\mathbf{i}}\Psi_g \\
\Psi_b '&= \Psi_b\end{aligned}$$ The equality (C.113) is equivalent to $$\begin{aligned}
\begin{pmatrix}
\phi_{dr}' & \phi_{ur}' \\ \widehat{\phi}_{ur}'& \widehat{\phi}_{dr}'
\end{pmatrix}&=\begin{pmatrix} e^{i\theta}&0 \\0 &e^{-i\theta} \end{pmatrix}
\begin{pmatrix}
\phi_{dr} & \phi_{ur} \\ \widehat{\phi}_{ur}& \widehat{\phi}_{dr}
\end{pmatrix}\end{aligned}$$ The equality (C.114) is equivalent to $$\begin{aligned}
\begin{pmatrix}
\phi_{dg}' & \phi_{ug}' \\ \widehat{\phi}_{ug}'& \widehat{\phi}_{dg}'
\end{pmatrix}&=\begin{pmatrix} e^{-i\theta}&0 \\0 &e^{i\theta} \end{pmatrix}
\begin{pmatrix}
\phi_{dg} & \phi_{ug} \\ \widehat{\phi}_{ug}& \widehat{\phi}_{dg}
\end{pmatrix}\end{aligned}$$ We get $$\begin{aligned}
{\eta{'}}_{1dr}^*&=e^{-i\theta}\eta_{1dr}^*;~{\eta{'}}_{1ur}^*=e^{-i\theta}\eta_{1ur}^*\\
{\eta{'}}_{2dr}^*&=e^{-i\theta}\eta_{2dr}^*;~{\eta{'}}_{2ur}^*=e^{-i\theta}\eta_{2ur}^*\\
{\eta{'}}_{1dg}^*&=e^{i\theta}\eta_{1dg}^*;~{\eta{'}}_{1ug}^*=e^{i\theta}\eta_{1ug}^*\\
{\eta{'}}_{2dg}^*&=e^{i\theta}\eta_{2dg}^*;~{\eta{'}}_{2ug}^*=e^{i\theta}\eta_{2ug}^*\end{aligned}$$ This gives $$\begin{aligned}
s_1'&=s_1~;~~s_2'=e^{-i\theta}s_2~;~~s_3'=e^{i\theta}s_3 \\
s_4'&=s_4~;~~s_5'=e^{-i\theta}s_5~;~~s_6'=e^{i\theta}s_6 \\
s_9'&=s_9~;~~s_8'=e^{-2i\theta}s_8~;~~s_7'=e^{2i\theta}s_7 \\
s_{10}'&=s_{10}~;~~s_{11}'=e^{-i\theta}s_{11}~;~~s_{12}'=e^{i\theta}s_{12} \\
s_{14}'&=s_{14}~;~~s_{15}'=e^{-i\theta}s_{15}~;~~s_{13}'=e^{i\theta}s_{13} \end{aligned}$$ from which we get $$\begin{aligned}
s_j'{s{'}}_j^*&=s_j s_j^*,~j=1,2,\dots,15 \\
\rho'&=\rho \\
\chi_r'&=e^{-\mathbf{i}\theta}\chi_r \\
\chi_g'&=e^{\mathbf{i}\theta}\chi_g\end{aligned}$$ These relations are the awaited ones because $$\begin{aligned}
\pmb\partial \Psi_r'&=\pmb\partial(e^{\mathbf{i}\theta}\Psi_r)\notag \\
&=e^{-\mathbf{i}\theta}(-\mathbf{i}\pmb\partial\theta\Psi_r +\pmb\partial\Psi_r)\\
\pmb\partial \Psi_g'&=\pmb\partial(e^{-\mathbf{i}\theta}\Psi_g)\notag \\
&=e^{\mathbf{i}\theta}( \mathbf{i}\pmb\partial\theta\Psi_g +\pmb\partial\Psi_g)\\
{\mathbf{G}{'}}^3 &=\mathbf{G}^3-\frac{2}{g_3}\pmb\partial\theta.\end{aligned}$$
Gauge group generated by $\underline{\mathbf{i}}\Gamma_8$
---------------------------------------------------------
We name $f_8$ the gauge transformation $$f_8:\Psi^c \mapsto \underline{\mathbf{i}}\Gamma_8(\Psi^c)=\begin{pmatrix}
0 & \frac{\mathbf{i}}{\sqrt{3}} \Psi_r \\\frac{\mathbf{i}}{\sqrt{3}} \Psi_g &
-\frac{2\mathbf{i}}{\sqrt{3}} \Psi_b
\end{pmatrix}$$ which implies $$\begin{aligned}
[\exp(\theta f_1)](\Psi^c)&=\begin{pmatrix}
0 & e^{\frac{\theta\mathbf{i}}{\sqrt{3}}}\Psi_r \\ e^{\frac{\theta\mathbf{i}}{\sqrt{3}}}\Psi_g
& e^{-\frac{2\theta\mathbf{i}}{\sqrt{3}}}\Psi_b
\end{pmatrix}=\begin{pmatrix}
0 & \Psi_r' \\ \Psi_g ' & \Psi_b '
\end{pmatrix}\\
\Psi_r'&=\exp(\frac{\theta\mathbf{i}}{\sqrt{3}})\Psi_r \\
\Psi_g '&=\exp(\frac{\theta\mathbf{i}}{\sqrt{3}})\Psi_g \\
\Psi_b '&= \exp(-\frac{2\theta\mathbf{i}}{\sqrt{3}})\Psi_b\end{aligned}$$ This gives $$\begin{aligned}
\phi_{dr}'&=\exp(\frac{i\theta}{\sqrt{3}})\phi_{dr};~\phi_{ur}'=\exp(\frac{i\theta}{\sqrt{3}})\phi_{ur}\\
\phi_{dg}'&=\exp(\frac{i\theta}{\sqrt{3}})\phi_{dg};~\phi_{ug}'=\exp(\frac{i\theta}{\sqrt{3}})\phi_{ug}\\
\phi_{db}'&=\exp(-\frac{2i\theta}{\sqrt{3}})\phi_{db};~\phi_{ub}'=\exp(-\frac{2i\theta}{\sqrt{3}})\phi_{ub}\end{aligned}$$ We then get $$\begin{aligned}
{\eta{'}}^*_{1dr}&=\exp(\frac{i\theta}{\sqrt{3}})\eta_{1dr}^*;~
{\eta{'}}^*_{1dg}=\exp(\frac{i\theta}{\sqrt{3}})\eta_{1dg}^*;~
{\eta{'}}^*_{1db}=\exp(-\frac{2i\theta}{\sqrt{3}})\eta_{1dg}^* \\
{\eta{'}}^*_{2dr}&=\exp(\frac{i\theta}{\sqrt{3}})\eta_{2dr}^*;~
{\eta{'}}^*_{2dg}=\exp(\frac{i\theta}{\sqrt{3}})\eta_{2dg}^*;~
{\eta{'}}^*_{2db}=\exp(-\frac{2i\theta}{\sqrt{3}})\eta_{2dg}^*\\
{\eta{'}}^*_{1ur}&=\exp(\frac{i\theta}{\sqrt{3}})\eta_{1ur}^*;~
{\eta{'}}^*_{1ug}=\exp(\frac{i\theta}{\sqrt{3}})\eta_{1ug}^*;~
{\eta{'}}^*_{1ub}=\exp(-\frac{2i\theta}{\sqrt{3}})\eta_{1ug}^* \\
{\eta{'}}^*_{2ur}&=\exp(\frac{i\theta}{\sqrt{3}})\eta_{2ur}^*;~
{\eta{'}}^*_{2ug}=\exp(\frac{i\theta}{\sqrt{3}})\eta_{2ug}^*;~
{\eta{'}}^*_{2ub}=\exp(-\frac{2i\theta}{\sqrt{3}})\eta_{2ug}^.\end{aligned}$$ This implies $$\begin{aligned}
s_1'&=\exp(\frac{2i\theta}{\sqrt{3}})s_1;~s_2'=\exp(-\frac{i\theta}{\sqrt{3}})s_2
;~s_3'=\exp(-\frac{i\theta}{\sqrt{3}})s_3 \\
s_4'&=\exp(\frac{2i\theta}{\sqrt{3}})s_4;~s_5'=\exp(-\frac{i\theta}{\sqrt{3}})s_5
;~s_6'=\exp(-\frac{i\theta}{\sqrt{3}})s_6 \\
s_7'&=\exp(\frac{2i\theta}{\sqrt{3}})s_7;~s_8'=\exp( \frac{2i\theta}{\sqrt{3}})s_8
;~s_9'=\exp(-\frac{4i\theta}{\sqrt{3}})s_9 \\
s_{10}'&=\exp(\frac{2i\theta}{\sqrt{3}})s_{10};~s_{11}'=\exp(-\frac{i\theta}{\sqrt{3}})s_{11}
;~s_{12}'=\exp(-\frac{i\theta}{\sqrt{3}})s_{12} \\
s_{13}'&=\exp(-\frac{i\theta}{\sqrt{3}})s_{13};~s_{14}'=\exp(\frac{2i\theta}{\sqrt{3}})s_{14}
;~s_{15}'=\exp(-\frac{i\theta}{\sqrt{3}})s_{15}. \end{aligned}$$ We then get the awaited results $$\begin{aligned}
s_j'{s{'}}_j^*&=s_j s_j^*,~j=1,2,\dots,15 \\
\rho'&=\rho \\
\chi_r'&=\exp(-\frac{\mathbf{i}\theta}{\sqrt{3}})\chi_r \\
\chi_g'&=\exp(-\frac{\mathbf{i}\theta}{\sqrt{3}})\chi_g\\
\chi_b'&=\exp(\frac{2\mathbf{i}\theta}{\sqrt{3}})\chi_b .\end{aligned}$$
[^1]: The reversion is an anti-isomorphism changing the order of any product (see [@davi:14] 1.1). It is specific to each Clifford algebra. The Appendix A explains the link between the reversion in $Cl_{1,3}$ and the reversion in $Cl_{1,5}$
[^2]: $I_2$, $I_4$, $I_8$ are unit matrices. The identification process allowing to include $\mathbb{R}$ in each real Clifford algebra allows to read $a$ instead of $aI_n$ for any complex number $a$.
[^3]: $\mathbf{i}$ anti-commutes with any odd element in space-time algebra and commutes with any even element.
|
---
abstract: 'Recent observations of near supernova show that the acceleration expansion of Universe decreases. This phenomenon is called the transient acceleration. In the second part of work we consider the 3-component Universe composed of a scalar field, interacting with the dark matter on the agegraphic dark energy background. We show that the transient acceleration appears in frame of such a model. The obtained results agree with the latest cosmological observations, namely, the 557 SNIa sample (Union2) was released by the Supernova Cosmology Project (SCP) Collaboration.'
address: 'Akhiezer Institute for Theoretical Physics, National Science Center “Kharkov Institute of Physics and Technology”, Akademicheskaya Str. 1, 61108 Kharkov, Ukraine'
author:
- 'O.A.Lemets'
- 'D.A.Yerokhin'
title: Cosmic acceleration a new review
---
dark energy ,agegraphic ,transient acceleration ,scalar field 98.80.-k ,95.36.+x
Introduction
============
At the begining of 21 century, the standard cosmological model (SCM) has become the dominant model of the universe, replacing the hot model of the universe (Big Bang). SCM is based on two important observational results: accelerated expansion of the universe and the Euclidean spatial geometry. In addition, it is assumed that the early universe is adequately described by the theory of inflation. SCM fixes a number of parameters of the universe and, in particular, its energy structure. According to the SCM the Universe is currently dominated by dark energy (in the form of a cosmological constant $\Lambda$), required to explain the accelerated expansion and dark (non-baryonic) matter (DM). Existence of DM gives a possibility to solve a number of contradictions in the Big Bang model (non-decreasing behavior of rotation curves, the structure of galactic halo, a chronology of the structures formation, etc.).
Attributing the acceleration of the universe expansion exclusively to the negative pressure generated by the cosmological constant drastically reduced the possibilities of the scale factor dynamics and condemned the universe to eternal accelerated expansion.
In fact, the dynamics of the scale factor in SCM is described by Friedmann equations $$\left( {\frac{{\dot a}}
{a}} \right)^2 = H_0^2 \left[ {\Omega _{m0} \left( {\frac{{a_0 }}
{a}} \right)^3 + \Omega _\Lambda } \right]$$ The solution of this equation reads
$$\begin{aligned}
a(t) = a_0 \left( {\frac{{\Omega _{m0} }}
{{\Omega _{\Lambda 0} }}} \right)^{1/3} \left[ {sh\left( {\frac{3}
{2}\sqrt {\Omega _{\Lambda 0} } H_0 t} \right)} \right]^{2/3} , \\
a\left( {t \ll H_0^{ - 1} } \right) \propto t^{2/3} ;\;a\left( {t \gg H_0^{-1}} \right) \propto e^{H_0 t} . \end{aligned}$$
We see that the asymptotic behavior of the solution described the era of matter domination in the early Universe $t\ll H_0^{-1}$ and dark energy domination for the later evolution $t\gg H_0^{-1}$.
We now find the dependence of the deceleration parameter $q\equiv -a\ddot{a}/\dot{a}^2 = -\ddot{a}/\left( aH^2\right)$ on the redshift $z$ for a universe filled with arbitrary components of the state equation $p_i = w_i\rho_i$. In this case, $$q = \frac{3}
{2}\frac{{\sum\limits_i \Omega _i^{(0)} \left( {1 + w_i } \right)(1 + z)^{3\left( {1 + w_i } \right)} }}
{{\sum\limits_i \Omega _i^{(0)} (1 + z)^{3\left( {1 + w_i } \right)} }} - 1$$ For the SCM, this expression takes the form (see Figure 1) $$q = \frac{1}
{2}\frac{{\Omega _M^{(0)} (1 + z)^3 - 2\Omega _\Lambda ^{(0)} }}
{{\Omega _M^{(0)} (1 + z)^3 + \Omega _\Lambda ^{(0)} }}$$ In particular, at the present time
$$q_0 = \frac{{1 - 3\Omega _{\Lambda 0} }}{2}\simeq - 0.6.$$
A characteristic feature of the dependence $q(z)$ - a monotonous tendency to a limiting value of $q(z) = -1$ for $z \to -1$. Physically, this means that once the dark energy became dominant component (at $z \sim 1$), the universe in SCM is doomed to eternal accelerated expansion. Questioned the adequacy of this result is expressed repeatedly [@Andreas], [@Barrow].
Transient acceleration: heuristic arguments
===========================================
Discovery of the transient acceleration is quite logical step to ensure the achievement of observational cosmology. The first step on this thorny path can be regarded as a model of stationary universe. To accomplish the second step, and discover the Hubble expansion of the universe it took almost two thousand years. The next step, though took far less time, still met a lot of objection: even now not every body agree with what was discovered to the accelerated expansion of the universe. We suggest that cosmology now came close to make the next step in this direction.
In this section, we discuss some considerations, both theoretical and observational, indicating that this path.
Theoretical background
----------------------
J. Barrow [@Barrow] was one of the first to rise the question of validity of such scenario. He showed that in many well-motivated scenarios the observed period of vacuum domination is only a transient phenomenon. Soon after acceleration starts, the vacuum energy anti-gravitational properties are reversed, and a matter-dominated decelerating cosmic expansion resumes. Thus, contrary to general expectations, once an acceleration universe does not mean an accelerating forever
To show this, we followed [@Barrow] considering a homogeneous and isotropic Universe with zero spatial curvature that contains two dominant forms of matter: a perfect fluid with pressure $p$ and density $\rho $ linked by an equation of state $p=w\rho $, with $w$ constant, together with a scalar quintessence field $\varphi $ defined by its self-interaction potential $V(\varphi)$.
Many theorists believe that fields with potentials of the form $$V(\varphi) = V_p(\varphi) e^{-\lambda \varphi}.
\label{expVp}$$ are predicted in the low energy limit of $M$-theory , where $V_p(\varphi)$ is a polynomial. Albrecht and Skordis [@Andreas] have proposed a particularly attractive model of quintessence. It is driven by a potential which introduces a small minimum to the exponential potential, which is provided by the simplest polynomial $$V_p(\varphi) = (\varphi -\varphi_0)^{2} + A.
\label{Vp}$$ and the potential takes the form $$V(\varphi )=e^{-\lambda \varphi }\left( A+(\varphi -\varphi_0)^2\right), \label{C}$$ where the constant parameters, $A$ and $\varphi_0$, are of order $1$ in Planck units, so there is also no fine tuning of the potential. In this quintessence models, late-time acceleration is achieved without fine tuning of the initial conditions. Acceleration begins when the field gets trapped in the local minimum of the potential at $\phi =\phi _0+(1\pm \sqrt{1-\lambda ^2A})/\lambda $, which is created by the quadratic factor in eq. (\[C\]) when $1\geq\lambda ^2A$. Once the field gets stuck in the false vacuum its kinetic energy disappears ($\phi \approx $ constant), and the ensuing dominance of $\rho +\rho _\phi $ by an almost constant value of the potential value triggers a period of accelerated expansion that never ends. In the article [@Barrow] it was found that this type of behaviour is by no means generic.
Transient vacuum domination arises in two ways. When $A\lambda ^2<1,$ the $\varphi $ field arrives at the local minimum with enough kinetic energy to roll over the barrier and resume descending the exponential part of the potential where $\varphi \gg\varphi _0$. This kinetic energy is determined by the scaling regime, and so by parameters of the potential and not by initial conditions. Another instance of transient vacuum domination is the whole region $A\lambda ^2>1$. As $A$ increases towards $\lambda ^{-2}$, the potential loses its local minimum, and flattens out into a point of inflexion. This is sufficient to trigger accelerated expansion temporarily, but the field never stops rolling down the potential, and matter-dominated scaling evolution with $a(t)\propto t^{2/3}$ is soon resumed. It is possible for the universe to exit from a period of accelerated expansion and resume decelerated expansion. Moreover, for the well-motivated family of Albrecht-Skordis potentials this is the most likely form of evolution, rather than a state of continuing acceleration.
Observational evidence
----------------------
A. Starobinsky [@Starobinsky] and co-workers investigated the course of cosmic expansion in its recent past using the Constitution SN Ia sample (which includes CfA data at low redshifts), jointly with signatures of baryon acoustic oscillations (BAO) in the galaxy distribution and fluctuations in the cosmic microwave background (CMB). Allowing the equation of state of dark energy (DE) to vary, they find that a coasting model of the universe ($q_0=0$) fits the data about as well as $\Lambda$CDM. This effect, which is most clearly seen using the recently introduced $Om$ diagnostic [@Starobinsky_Om], corresponds to an increase of $Om$ and $q$ at redshifts $z\lesssim 0.3$. In geometrical terms, this suggests that cosmic acceleration may have already peaked and that we are currently witnessing its slowing down.
The case for evolving DE strengthens if a subsample of the Constitution set consisting of SNLS+ESSENCE+CfA SN Ia data is analysed in combination with BAO+CMB using the same statistical methods.
Note also that in article [@Starobinsky] it was shown the impossibility to come to agreement with data obtained in observations SN1a about CMB by using the ansatz CPL. To make this possible, they proposed a new ansatz $$w(z)=- \frac{1+ \tanh\left[(z-z_t)\Delta\right]}{2}.
\label{eq:step}$$ This fit ensures $w = -1$ at early times, and then increases the EOS to a maximum of $w\sim 0$ at low $z$. Figure\[fig\_Srarob\] shows the deceleration parameter $q$ and the $Om$ diagnostic reconstructed using (\[eq:step\]).
In the work [@LiWuYu; @LiWuYu1] for SNIa data,there was used the latest Union2 compilation released by the Supernova Cosmology Project (SCP) Collaboration recently [@Amanullah]. It consists of 557 data points and is the largest published SNIa sample up to now (2010 year).
The authors find that, independent of whether or not the systematic error is considered, there exists a tension between low redshift data (SNIa+BAO) and high redshift ones (CMB), but for the case with the systematic error considered this tension is weaker than without the SNIa. By reconstructing the curves of q(z) and Om(z) from Union2+BAO, we obtain that for both the SNIa with and without the systematic error the cosmic acceleration has already peaked at redshift $z \sim 0.3$ and is decreasing. However, when the CMB data is added in analysis, this result changes dramatically and the observation favors a cosmic expansion with an increasing acceleration, indicating a tension between low redshift data and high redshift.
They find that two different subsamples+BAO+CMB give completely different results on the cosmic expansion history when the systematic error is ignored, with one suggesting a decreasing cosmic acceleration, the other just the opposite, although both of them alone with BAO support that the cosmic acceleration is slowing down.
![At upper panel represent the results reconstructed from Union2+BAO and show the evolutionary behaviors of $q(z)$ at the $68.3\%$ confidence level. The gray regions and the regions between two long dashed lines show the results without and with the systematic errors in the SNIa, respectively. At lower panel represent the $68.3\%$ and $95\%$ confidence level regions for $w_0$ versus $w_1$ in the CPL parametrization, $w=w_0+w_1z/(1+z)$. In the right panel, the system error in the SNIa is considered. The dashed, solid and thick solid lines represent the results obtained from Union2S, Union2S+BAO and Union2S+BAO+CMB, respectively. The point at $w_0=-1$, $w_1=0$ represents the spatially flat $\Lambda$CDM model. []{data-label="fig:TrAcCh"}](picqzu2b.eps "fig:"){width="45.00000%"} ![At upper panel represent the results reconstructed from Union2+BAO and show the evolutionary behaviors of $q(z)$ at the $68.3\%$ confidence level. The gray regions and the regions between two long dashed lines show the results without and with the systematic errors in the SNIa, respectively. At lower panel represent the $68.3\%$ and $95\%$ confidence level regions for $w_0$ versus $w_1$ in the CPL parametrization, $w=w_0+w_1z/(1+z)$. In the right panel, the system error in the SNIa is considered. The dashed, solid and thick solid lines represent the results obtained from Union2S, Union2S+BAO and Union2S+BAO+CMB, respectively. The point at $w_0=-1$, $w_1=0$ represents the spatially flat $\Lambda$CDM model. []{data-label="fig:TrAcCh"}](figu4s.eps "fig:"){width="45.00000%"}
Thus, the evolutional behavior of dark energy reconstructed and the issue of whether the cosmic acceleration is slowing down or even speeding up is highly dependent upon the SNIa data sets, the light curve fitting method of the SNIa, and the parametrization forms of the equation of state. In order to have a definite answer, we must wait for data with more precision and search for the more reliable and efficient methods to analyze these data.
The model of interacting dark energy with a transient acceleration phase {#ISFA_TA}
========================================================================
In this section, we investigate a new quintessence scenario driven by a rolling homogeneous scalar field with exponential potential $V(\varphi)$ interacting with dark matter on the agegraphic background. This scenario predicts transient accelerating phase.
To describe the dynamic properties of such a Universe will adapt the system of equations (\[sys\_xyz\]) for this model. We introduce the modified variables: $$\begin{aligned}
x=\frac{\dot{\varphi}}{\sqrt{6}M_pH},\quad
y=\frac{1}{M_pH}\sqrt{\frac{V(\varphi)}{3}},\quad z=\frac{1}{M_pH}\sqrt{\frac{\rho_{m}}{3}},\quad u=\frac{1}{M_pH}\sqrt{\frac{\rho_{_{q}}}{3}}.\label{var2}\end{aligned}$$
The evolution of the scalar field is described by the Klein–Gordon equation, which in the presence of matter couplings is given by $$\ddot{\varphi}+3 H \dot{\varphi} + \frac{dV}{d\varphi}=- \frac{Q}{\dot{\varphi}}.
\label{eq:kgeqn}$$ In this section, we consider interactions $Q$ that are linear combinations of the scalar field and pressureless matter: $$\label{Q1}
Q= 3H(\alpha\rho_\varphi+\beta \rho_m),$$ where $\alpha$, $\beta$ are constant parameters.
Without specifying the potential of the scalar field $V(\varphi),$ we obtain the system of equations in the form
$$\begin{aligned}
x'&=& \frac{3x}{2}g(x,z,u)-3x + \sqrt{\frac{3}{2}}\lambda y^2-\gamma,\nonumber\hfill\\
\label{sys_xyz}
y'&=& \frac{3y}{2}g(x,z,u)- \sqrt{\frac{3}{2}}\lambda xy, \hfill\\
z'&=& \frac{3z}{2}g(x,z,u)-\frac{3}{2}z + \gamma\frac{x}{z},\nonumber\hfill\\
u'&=& \frac{3u}{2}g(x,z,u)-\frac{u^2}{n},\nonumber\hfill\end{aligned}$$
where $
g(x,z,u) = 2x^2+ z^2+ \frac{2}{3n}u^3
$ and $$\gamma\equiv -\frac{Q}{\sqrt{6}M_pH^2\dot{\varphi}},~\lambda\equiv -\frac{1}{V}\frac{dV}{d\varphi} M_p.$$ In these variables, we obtain
$$\begin{aligned}
\nonumber
Q&=&9H^3M_p^2\left[\alpha(x^2+y^2)+\beta z^3\right],\\
\gamma& = &\frac{\alpha(x^2+y^2)+\beta z^3}{x} \label{Q_gamma_xyz}. \end{aligned}$$
We will consider a scalar field with an exponential potential energy density $$\label{V_exp}
V=V_0\exp\left(\sqrt{\frac{2}{3}}\frac{\mu\varphi}{M_p}\right),$$ where $\mu$ is a constant. In this case we obtain $$\begin{aligned}
x'&=& \frac{3x}{2}\left[g(x,z,u)- \frac{\alpha(x^2+y^2)+\beta z^2}{x^2}\right]-3x - \mu y^2,\nonumber\hfill\\
\label{sys_xyz_V}
y'&=& \frac{3y}{2}g(x,z,u)+\mu xy, \hfill\\
z'&=& \frac{3z}{2}\left[g(x,z,u)+\frac{\alpha(x^2+y^2)+\beta z^2}{z^2}\right]-\frac{3}{2}z ,\nonumber\hfill\\
u'&=& \frac{3u}{2}g(x,z,u)-\frac{u^2}{n}.\nonumber\hfill\end{aligned}$$ In this model the deceleration parameter has the form $$q=-1+\frac 32\left[2x^2+z^2+\frac{2}{3n}u^3\right].$$ Note that in this model, the cosmological parameters are not explicitly depend of the parameters of interaction, but only determine by the behaviors of dynamical variables. This fact complicates the analysis of our model.
The fact that $q$ is independent of the interaction term implies that the region of phase space is the same for all of the models considered. It is possible to make some qualitative comments about the system (\[sys\_xyz\_V\]) for some special cases.
Initially, considering the case, $y = 0,$ we obtain the constraining relation, which should be imposed on the parameters of interaction for the emergence of critical points. It is easy to show that in this case, the condition of real energy density requires $$\label{cond}
2\sqrt{\frac{\beta}{\alpha}}>1+\alpha+\beta,$$ which necessarily requires $0>\beta>\alpha,|\alpha|+|\beta|<1.$ This critical point corresponds to the matter-dominated Universe and is unstable. The cases of several such points are possible, but they are of no interest. Finally, it can be shown that any of this equilibrium point within (but not on) the boundary will exist for $x_0<0.$ For $z\neq 0$ and constrain satisfied (\[cond\]), it $$x_c=\left[\left(a+\sqrt{{\beta}/{\alpha}}\right)^{1/2}+a\right]z_c,$$ where $a=\left(2\sqrt{{\beta}/{\alpha}}-(1+\alpha+\beta)\right)^{1/4}.$
Review of the case $Q=0$
------------------------
In this subsection we consider in more detail the case of absence of interaction between the scalar field and dark matter. Critical points of the system (\[sys\_xyz\_V\]) in this $(\alpha=\beta=0)$ case are given in the Table \[tab:2\]. As can be seen from the table, the system (\[sys\_xyz\_V\]), there are six physically admissible critical points, the latter of which is the attractor. The first critical point, $(1,0,0,0),$ is unstable and corresponds to the scalar field dominated era with extremely rigid equation of state, the second critical point corresponds to the period of evolution when the scalar field behaves as a cosmological constant. The other point, $(0,0,1,0),$ is physically unrealistic. It corresponds to a Universe filled with dark matter (it contains neither scalar field, nor agegraphic dark energy) and it is also unstable. Fourth critical point $(0,0,0,1)$ correspond to the Universe consisting only of the agegraphic dark energy and has already been discussed in detail.
The physical interest present the final sixth critical point, which is the an attractor. It corresponds to the Universe consisting of a scalar field and agegraphic dark energy. The location of this critical point is completely determined by the parameter of the potential $\mu$ and the value of $n:$ $$\begin{array}{cccccl}
x_{*}&=&\frac{2}{3n\mu}u_{*}, \quad& y_{*}&=&\sqrt{1-\left(1+\frac{4}{9n^2\mu^2}\right)u_{*}^2},\\
z_{*}&=&0, \quad & u_{*}&=&\frac{3}{2n\mu^2}\left(-1+\sqrt{1+\frac{4n^2\mu^4}{9}}\right).
\end{array}$$
The fact that $x_*\propto u_*$ is the characteristic feature of the tracking solutions. Note also that between the scalar field and agegraphic dark energy, evidence of background interactions. There is because the dynamics of the scalar field affects agegraphic dark energy. This effect is that agegraphic dark energy, having a negative pressure, affect on the rate of expansion of the Universe, which affects the Hubble parameter, which is included in the Klein-Gordon equation for a scalar field.
[ccccc]{} $(x_c,y_c,z_c,u_c)$ & Stability & $q$ & $w_{\varphi}$& $w_{tot}$\
coordinates & character & & &\
$\;(1,0,0,0)$ & unstable & $2$ &$ 1$& $1$\
$\;(0,1,0,0)$ & unstable & $-1$ &$-1$& $-1$\
$\;(0,0,1,0)$ &unstable &$\frac{1}{2}$&$\nexists$&0\
$\;(0,0,0,1)$ & stable & $-1+\frac{1}{n}$ &$ \nexists$& $-1+\frac{2}{3n}$\
$(-\frac{3}{2\mu},\frac{3}{2\mu},\sqrt{1-\frac{3}{2\mu^2}},0)$ & unstable &$\frac{1}{2}$ &$\nexists$&0\
$(x_*,y_*,0,u_*)$ & attractor &$q_*<0$&$w_{\varphi*}$&$w_{tot*}$\
The attractor, once reached, brings to zero the matter density. To allow for the observed matter content of the Universe, we have to select the initial conditions, if they exist, in such a way that the attractor is not yet reached at the present time, but the expansion is already accelerated.
For example, consider the case when $\mu = -3, n=3.$ The dynamics of such a Universe is consistent with the observed data, which will be shown in the following section. We note only that in this model the values of deceleration parameter and the scalar field state parameter in the attractor are equal respectively $q_*\approx -0.68,~w_{\varphi*}\approx -0.78.$
Review of the case $Q=3H\alpha\rho_\varphi$
-------------------------------------------
In the above case, the phenomenon of transient acceleration that occurs in such a Universe does not match the observations. However, it is possible to fit the observational data with this model in the case when dark matter and scalar field interact. In this subsection we consider the special case of interaction (\[Q1\]) when $ \beta = 0.$
In figure \[fig:4\], we show the evolution of $\Omega_q, \Omega_{m}$ and $\Omega_{\varphi}$ for cosmological model in the case when $\alpha=0.005$, $\mu=-5$ and $n = 3.$
![Behavior of $\Omega_{\varphi}$ (dot line), $\Omega_{q}$ (dash line) and $\Omega_{m}$ (solid line) as a function of $N=\ln a$ for $n=3,\;\alpha = 0.005$ and $\mu = -5$ (upper side). Evolution of deceleration parameter for this model (lower side).[]{data-label="fig:4"}](Omega_lna.eps "fig:"){width="45.00000%"} ![Behavior of $\Omega_{\varphi}$ (dot line), $\Omega_{q}$ (dash line) and $\Omega_{m}$ (solid line) as a function of $N=\ln a$ for $n=3,\;\alpha = 0.005$ and $\mu = -5$ (upper side). Evolution of deceleration parameter for this model (lower side).[]{data-label="fig:4"}](q_z.eps "fig:"){width="45.00000%"}
From the form of equations and the character of the interaction it can be easily understood that neither the nature nor the location of the attractor, that have been found in the previous subsection dos not change with the inclusion of the interaction. Interaction only affects the behavior of dynamical variables that are the correspond to trajectory in phase space between critical points. This is a consequence of the above degeneracy from the parameters of interaction.
With these values of the parameters of interaction, transient acceleration begins almost in the present era. In our model as in most cosmological models, where the scalar field plays the role of dark energy it begins to dominate causing a period of accelerated expansion of the Universe. Its consequence of the accelerated expansion of the Universe contribution of ADE increases, resulting that the background (space) is changing faster than the field and it becomes asymptotically free. This field has extremely rigid equation of state that leads to the fact that the accelerated expansion of the Universe is slowing down. Soon, however, when the contribution of ADE has grows enough so that the scalar field cannot longer impede the expansion of the Universe, it begins to accelerate again.
Observational data {#OBS}
==================
In the present section, we will consider the latest cosmological observations, namely, the 557 SNIa sample (Union2) was released by the Supernova Cosmology Project (SCP) Collaboration [@Amanullah].
The data points of the 557 Constitution SNIa compiled in [@Amanullah] are given in terms of the distance modulus $\mu_{obs}(z_i)$. On the other hand, the theoretical distance modulus is defined as $$\mu_{th}(z_i)\equiv 5\log_{10}D_L(z_i)+\mu_0\,,$$ where $\mu_0\equiv 42.38-5\log_{10}h$ and $h$ is the Hubble constant $H_0$ in units of $100~{\rm km\,s^{-1}\,Mpc^{-1}}$, whereas $$D_L(z)=(1+z)\int_0^z \frac{d\tilde{z}}{E(\tilde{z};{\bf p})}\,,$$ in which $E\equiv H/H_0$, and ${\bf p}$ denotes the model parameters.
Theoretical distance modulus will be different for the different models and comparing $\mu_{th}(z_i)$ with $\mu_{obs} (z_i) $, one can judge the plausibility of an cosmological model. So to understand whether the theoretical model corresponds to the observational data is enough to know the value of $E\equiv H/H_0$, which is easy to find through a system of equations (\[sys\_xyz\_V\]). As seen from figure \[fig:mu\_z\_1\] our models are in accordance with the observational data.
![The dependence of the modulus distance from the redshift, theoretically calculated (solid red line) by the model with $Q=3H\alpha \rho_\varphi$, $\alpha=0.005$, $\mu=-5$ and $n=3$. $ h = 0.70 $ $\rm \, km \,c^{-1}\,Mpc^{-1} $ points found in the observations of supernovae type 1a[@Amanullah]](mu.eps){width="45.00000%"}
. \[fig:mu\_z\_1\]
Conclusion
==========
The original agegraphic dark energy model was proposed in [@0707.4049] based on the Károlyházy uncertainty relation, which arises from quantum mechanics together with general relativity. The interacting agegraphic dark energy model has certain advantages compared to the original agegraphic or holographic dark energy model. Many studies show that this model gives an opportunity to explain the accelerated expansion of the Universe without a cosmological constant or some form of the scalar field. All the three models give dynamics of the Universe which are virtually indistinguishable from SCM, but without most of its problems, such as the cosmological constant, fine tuning and coincidence problems.
Some authors have recently suggested that the cosmic acceleration have already peaked and that we are currently observing its slowing down [@Barrow; @Starobinsky; @Lima]. Under a kinematic analysis of the most recent SNe Ia compilations, the paper [@Lima; @LiWuYu; @LiWuYu1] shows the existence of a considerable probability in the relevant parameter space that the Universe is already in a decelerating expansion regime.
One of the deficiencies of original ADE model is the inability to explain the phenomenon of transient acceleration.
Density of holographic dark energy is determined by the surface terms in action, while volume terms are usually ignored. We take into account both surface and volume terms, where the latter correspond to (described by) homogeneous scalar field with exponential potential $V(\varphi)$.
We consider a model of Universe consisting of dark matter interacting with a scalar field on the agegraphic background. It is shown that this model can explain the transient acceleration. This model also is in accordance with the observational data.
Acknowledgements {#acknowledgements .unnumbered}
================
We are grateful to Prof. Yu.L. Bolotin for kind help and discussions. We also thank V.A. Cherkaskiy for careful reading and editing of this article.
[999]{} A. Albrecht and C. Skordis, Phenomenology of a realistic accelerating universe using only Planck-scale physics, Phys. Rev. Lett. [**84**]{}, 2076 (2000) \[astro-ph/9908085\].
J. Barrow, R. Bean and J. Magueijo,Can the Universe escape eternal acceleration?, Mon. Not. Roy. Astron. Soc. [**316**]{}, 41 (2000) \[astro-ph/0004321v1\].
Arman Shafieloo, Varun Sahni, Alexei A. Starobinsky,Is cosmic acceleration slowing down?, Phys. Rev. D [**80**]{}, 101301 (2009), \[arXiv:0903.5141v4\]
Varun Sahni, Arman Shafieloo, Alexei A. Starobinsky, Two new diagnostics of dark energy, Phys. Rev. [**D 78**]{}, 103502 (2008) \[arXiv:0807.3548v3\].
Antonio C. C. Guimara es, Jose’ Ademir S. Lima, Could the cosmic acceleration be transient? A cosmographic evaluation, \[arXiv:1005.2986\]. Zhengxiang Li, Puxun Wu and Hongwei Yu, Examining the cosmic acceleration with the latest Union2 supernova data, \[arXiv:1011.1982v1\]. Zhengxiang Li, Puxun Wu and Hongwei Yu, Probing the course of cosmic expansion with a combination of observational data, \[arXiv:1011.2036v1\].
R. Amanullah *et al.*, Spectra and Hubble Space Telescope light curves of six type 1a supernovae at $0.511 < z < 1.12$ and the Union2 compilation, The Astrophysical Journal [**716**]{} 712 (2010). L. Amendola, Coupled quintessence, Phys. Rev. [**D 62**]{}, 043511 (2000) \[astro-ph/9908023\]. M. Hicken [*et al.*]{}, Improved dark energy constraints from 100 new CfA supernova type Ia light curves, Astrophys. J. [**700**]{}, 1097 (2009) \[arXiv:0901.4804\]. R. G. Cai, A dark energy model characterized by the age of the universe, Phys. Lett. [**B 657**]{}, 228 (2007) \[arXiv:0707.4049\].
|
---
abstract: 'Relativistic heavy-ion collisions provide an ideal environment to study the emergent phenomena in quantum chromodynamics (QCD). The chiral magnetic effect (CME) is one of the most interesting, arising from the topological charge fluctuations of QCD vacua, immersed in a strong magnetic field. Since the first measurement nearly a decade ago of the possibly CME-induced charge correlation, extensive studies have been devoted to background contributions to those measurements. Many new ideas and techniques have been developed to reduce or eliminate the backgrounds. This article reviews these developments and the overall progress in the search for the CME.'
address: |
Department of Physics and Astronomy, Purdue University,\
West Lafayette, IN 47907, US\
zhao656@purdue.edu
author:
- Jie Zhao
bibliography:
- 'ref.bib'
title: 'Search for the Chiral Magnetic Effect in Relativistic Heavy-Ion Collisions'
---
Introduction
============
Quark interactions with topological gluon configurations can induce chirality imbalance and local parity violation in quantum chromodynamics (QCD)[@Lee:1973iz; @Lee:1974ma; @Morley:1983wr; @Kharzeev:1998kz; @Kharzeev:2004ey; @Kharzeev:2007jp; @Fukushima:2008xe; @Kharzeev:2015znc]. In relativistic heavy-ion collisions, this can lead to observable electric charge separation along the direction of the strong magnetic field produced by spectator protons[@Kharzeev:2004ey; @Kharzeev:2007jp; @Fukushima:2008xe; @Muller:2010jd; @Kharzeev:2015znc]. This phenomenon is called the chiral magnetic effect (CME). An observation of the CME-induced charge separation would confirm several fundamental properties of QCD, namely, approximate chiral symmetry restoration, topological charge fluctuations, and local parity violation. Extensive theoretical efforts have been devoted to characterize the CME, and intensive experimental efforts have been invested to search for the CME in heavy-ion collisions at BNL’s Relativistic Heavy Ion Collider (RHIC) and CERN’s Large Hadron Collider (LHC)[@Kharzeev:2015znc].
Transitions between gluonic configurations of QCD vacua can be described by instantons/sphelarons and characterized by the Chern-Simons topological charge number. Quark interactions with such topological gluonic configurations can change their chirality, leading to an imbalance in left- and right-handed quarks (nonzero axial chemical potential $\mu_{5}$); $N_{L} - N_{R} = 2n_{f}Q_{w} \propto \mu_{5}$, $n_{f}$ is the number of light quark flavors and $Q_{w}$ is the topological charge of the gluonic configuration. Thus, gluonic field configurations with nonzero topological charges induce local parity violation[@Kharzeev:1998kz; @Kharzeev:2004ey; @Kharzeev:2007jp; @Fukushima:2008xe; @Kharzeev:2015znc]. It was suggested that in relativistic heavy-ion collisions, where the deconfinement phase transition and an extremely strong magnetic field are present, The chirality imbalance of quarks in the local metastable domains will generate an electromagnetic current, $\vec{J} \propto \mu_{5}\vec{B}$, along the direction of the magnetic field. Quarks hadronize into charged hadrons, leading to an experimentally observable charge separation. The measurements of this charge separation provide a means to study the non-trivial QCD topological structures in relativistic heavy-ion collisions[@Lee:1973iz; @Lee:1974ma; @Morley:1983wr; @Kharzeev:1998kz; @Kharzeev:1999cz].
In heavy-ion collisions, particle azimuthal angle distribution in momentum space is often described with a Fourier decomposition: $$\begin{split}
\frac{\rm{d}N}{\rm{d}\phi} \propto 1& + 2v_{1}\cos(\Delta\phi) + 2v_{2}\cos(2\Delta\phi) + ... \\
& + 2a_{1}\sin(\Delta\phi) + 2a_{2}\sin(2\Delta\phi) + ...
\end{split}
\label{eqThreeCtor3}$$ where $\Delta\phi=\phi - {\psi_{\rm RP}}$, and ${\psi_{\rm RP}}$ is the reaction-plane direction, defined to be the direction of the impact parameter vector and expected to be perpendicular to the magnetic field direction on average. The parameters $v_{1}$ and $v_{2}$ account for the directed flow and elliptic flow. The parameters $a_{1,2}$ can be used to describe the charge separation effects. Usually only the first harmonic coefficient $a_{1}$ is considered. Positively and negatively charged particles have opposite $a_{1}$ values, $a_{1}^{+}=-a_{1}^{-}$, and are proportional to $Q_{w}B$. However, they average to zero because of the random topological charge fluctuations from event to event, making a direct observation of this parity violation effect impossible. Indeed, the measured ${\langle a_{1}\rangle}$ of both positive and negative charges are less than $5\times10^{-4}$ at the 95% confidence level in Au+Au collisions at [$\sqrt{s_{\rm NN}}$ ]{}= 200 GeV[@Adamczyk:2013hsi]. The observation of this parity violation effect is possible only via correlations, e.g. measuring ${\langle a_{\alpha}a_{\beta}\rangle}$ with the average taken over all events in a given event sample. The $\gamma$ correlator is designed for this propose: $$\begin{split}
\gamma &= {\langle \cos(\phi_{\alpha}+\phi_{\beta}-2\psi_{RP})\rangle} \\
&= {\langle \cos\Delta\phi_{\alpha}\cos\Delta\phi_{\beta}\rangle} - {\langle \sin\Delta\phi_{\alpha}\sin\Delta\phi_{\beta}\rangle} \\
&= [{\langle v_{1,\alpha}v_{1,\beta} + B_{in}\rangle}] - [{\langle a_{\alpha}a_{\beta} + B_{out}\rangle}].
\end{split}
\label{eqThreeCtor4}$$ $B_{in}$ and $B_{out}$ are the reaction plane dependent backgrounds in in-plane and out-plane directions, which are assumed to largely cancel out in their difference, while there are still residual background contributions (e.g. momentum conservation effect [@Pratt:2010zn; @Bzdak:2010fd]). At mid-rapidity, the $v_{1}$ is averaged to zero, and the $v_{1}$ contribution (${\langle v_{1,\alpha}v_{1,\beta}\rangle}$) is expected to be small. Moreover, the $v_{1}$ background is expected to be charge independent. By taking the opposite-sign (OS) and same-sign (SS) $\gamma$ difference those charge independent backgrounds can be further cancelled out. Thus, usually the $\Delta\gamma$ correlator is used: $$\begin{split}
\Delta\gamma = \gamma_{OS} -\gamma_{SS}
\end{split}
\label{eqThreeCtor5}$$ where OS and SS describe the charge sign combinations between the $\alpha$ and $\beta$ particle.
The $\gamma$ correlator can be calculated by the three-particle correlation method without an explicit determination of the reaction plane; instead, the role of the reaction plane is played by the third particle, $c$. Under the assumption that particle $c$ is correlated with particles $\alpha$ and $\beta$ only via common correlation to the reaction plane, we have:
$$\begin{split}
{\langle \cos(\phi_{\alpha}+\phi_{\beta}-2\psi_{RP})\rangle} = {\langle \cos(\phi_{\alpha}+\phi_{\beta}-2\phi_c)\rangle}/v_{2,c}
\end{split}
\label{eqThreeCtor0}$$
where $v_{2,c}$ is the elliptic flow parameter of the particle $c$, and $\phi_{\alpha}$, $\phi_{\beta}$ and $\phi_{c}$ are the azimuthal angles of particle $\alpha$, $\beta$ and $c$, respectively.
Challenges and Strategies
=========================
A significant ${\Delta\gamma}$ has indeed been observed in heavy-ion collisions at RHIC and LHC[@Abelev:2009ad; @Abelev:2009ac; @Adamczyk:2013hsi; @Adamczyk:2013kcb; @Adamczyk:2014mzf; @Abelev:2012pa; @Ajitanand:2010]. The first $\gamma$ measurement was made by the STAR collaboration at RHIC in 2009 [@Abelev:2009ad]. Fig. \[FG\_FirstCME\] shows their $\gamma$ correlator as a function of the collision centrality in Au+Au and Cu+Cu collisions at [$\sqrt{s_{\rm NN}}$ ]{}= 200 GeV. Charge dependent signal of the same-sign and opposite-sign charge $\gamma$ correlators have been observed. Similarly, Fig. \[FG\_BESCME\] shows the ${\gamma_{\rm OS}}$ and ${\gamma_{\rm SS}}$ correlator as a function of the collision centrality in Au+Au collisions at [$\sqrt{s_{\rm NN}}$ ]{}= 7.7-200 GeV from STAR[@Adamczyk:2014mzf] and in Pb+Pb collisions at 2.76 TeV from ALICE[@Abelev:2012pa]. At high collision energies, charge dependent signals are observed, and ${\gamma_{\rm OS}}$ is larger than ${\gamma_{\rm SS}}$. The difference between ${\gamma_{\rm OS}}$ and ${\gamma_{\rm SS}}$, ${\Delta\gamma}$, decreases with increasing centrality, which would be consistent with expectation of the magnetic field strength to decrease with increasing centrality. At the low collision energy of [$\sqrt{s_{\rm NN}}$ ]{}=7.7 GeV, the difference between the ${\gamma_{\rm OS}}$ and ${\gamma_{\rm SS}}$ disappears, which could be consistent with the disappearance of the CME in the hadronic dominant stage at this energy. Thus, these results are qualitatively consistent with the CME expectation.
![(Color online) $\gamma$ correlator in Au+Au and Cu+Cu collisions at [$\sqrt{s_{\rm NN}}$ ]{}= 200 GeV. Shaded bands represent uncertainty from the measurement of $v_{2}$. The thick solid (Au+Au) and dashed (Cu+Cu) lines represent HIJING calculations of the contributions from three-particle correlations. Collision centrality increases from left to right[@Abelev:2009ad]. []{data-label="FG_FirstCME"}](./STAR_FirstCME.pdf){width="7.0cm"}
![(Color online) $\gamma$ correlator as a function of centrality for Au+Au collisions at 7.7-200 GeV[@Abelev:2009ac; @Abelev:2009ad; @Adamczyk:2014mzf], and for Pb+Pb collisions at 2.76 TeV[@Abelev:2012pa]. Grey curves are the charge independent results from MEVSIM calculations[@Adamczyk:2014mzf]. []{data-label="FG_BESCME"}](./STAR_BES.pdf){width="13.0cm"}
There are, however, mundane physics that could generate the same effect as the CME in the ${\Delta\gamma}$ variable, which contribute to the background in the ${\Delta\gamma}$ measurements. An example is the resonance or cluster decay (coupled with $v_{2}$) background[@Wang:2009kd; @Wang:2016iov; @Voloshin:2004vk]; the ${\Delta\gamma}$ variable is ambiguous between a back-to-back OS pair from the CME perpendicular to $\psi_2$ and an OS pair from a resonance decay along $\psi_2$. Calculations with local charge conservation and momentum conservation effects can almost fully account for the measured ${\Delta\gamma}$ signal at RHIC[@Pratt:2010zn; @Schlichting:2010qia; @Bzdak:2010fd]. A Multi-Phase Transport (AMPT)[@Zhang:1999bd; @Lin:2001zk; @Lin:2004en] model simulations can also largely account for the measured ${\Delta\gamma}$ signal[@Ma:2011uma; @Shou:2014zsa]. In general, these backgrounds are generated by two particle correlations coupled with elliptic flow ($v_{2}$): $$\begin{split}
{\langle \cos(\phi_{\alpha}+\phi_{\beta}-2\psi_{RP})\rangle} &= {\langle \cos(\phi_{\alpha}+\phi_{\beta}-2\phi_{reso.} + 2\phi_{reso.} -2\psi_{RP})\rangle}, \\
&\approx {\langle \cos(\phi_{\alpha}+\phi_{\beta}-2\phi_{reso.}\rangle}\times v_{2,reso.}.
\end{split}
\label{EQ_2}$$ Thus, a two particle correlation of ${\langle \cos(\phi_{\alpha}+\phi_{\beta}-2\phi_{reso.}\rangle}$ from resonance (cluster) decays, coupled with the $v_{2}$ of the resonance (cluster), will lead to a ${\Delta\gamma}$ signal.
Experimentally, various proposals and attempts have been put forward to reduce or eliminate backgrounds, exploiting their dependences on $v_{2}$ and two particle correlations. (1) Using the event shape selection, by varying the event-by-event $v_{2}$ exploiting statistical (event-by-event $v_{2}, q_{2}$ methods)[@Adamczyk:2013kcb; @Wen:2016zic] and dynamical fluctuations (event shape engineering method)[@Acharya:2017fau; @Sirunyan:2017quh], it is expected that the $v_{2}$ independent contribution to the ${\Delta\gamma}$ can be extracted. (2) Isobaric collisions and Uranium+Uranium collisions have been proposed[@Voloshin:2010ut] to take advantage of the different nuclear properties (such as proton number, shape). (3) Control experiments of small system p+A or d+A collisions are used to study the background behavior[@Khachatryan:2016got; @Zhao:2017wck], where backgrounds and possible CME signals are expected to be uncorrelated because the participant plane[@Alver:2008zza] and the magnetic field direction are uncorrelated due to geometry fluctuations in these small system collisions. (4) A new idea of differential measurements with respect to reaction plane and participant plane are proposed[@Xu:2017qfs; @Xu:2017zcn], which takes advantage of the geometry fluctuation effects on the participant plane and the magnetic field direction in A+A collisions. (5) A new method exploiting the invariant mass dependence of the ${\Delta\gamma}$ measurements is devised, which identifies and removes the resonance decay backgrounds, to enhance the sensitivity of CME measurement[@Zhao:2017nfq; @Zhao:2017wck]. (6) New $R(\Delta S)$ correlator is designed to detect the CME-driven charge separation[@Magdy:2017yje; @Ajitanand:2010rc]. In the following sections we will review these proposals and attempts in more detail.
Event-by-event selection methods
================================
The main background sources of the ${\Delta\gamma}$ measurements are from the elliptic flow ($v_{2}$) induced effects. These backgrounds are expected to be proportional to the $v_{2}$. One possible way to eliminate or suppress these $v_{2}$ induced backgrounds is to select “spherical” events with $v_{2}=0$ exploiting the statistical and dynamical fluctuations of the event-by-event $v_{2}$. Due to finite multiplicity fluctuations, one can easily vary the shape of the final particle momentum space, which is directly related to the $v_{2}$ backgrounds[@Adamczyk:2013kcb].
By using the event-by-event $v_{2}$, STAR has carried out the first attempt to remove the backgrounds[@Adamczyk:2013kcb]. Fig. \[FG\_STARese\] shows the charge multiplicity asymmetry correlator ($\Delta$) as a function of the event-by-event $v_{2}$. The event-by-event $v_{2}$ (${{v_{2,{\rm ebye}}}^{\rm obs}}$) can be measured by the $Q$ vector method: $$\begin{split}
&{{v_{2,{\rm ebye}}}^{\rm obs}}= Q^{*}\it{q}_{EP}, \\
&Q=\frac{1}{N}\sum_{j=1}^{N} e^{2i\phi_j}, \ \it{q}_{EP}=e^{2i{\psi_{\rm EP}}}, \\
\end{split}
\label{EQ_ESE1}$$ where $Q$ sums over particles (used for the $\Delta$ correlator) in each event; ${\psi_{\rm EP}}$ is the event plane (EP) azimuthal angle, reconstructed from final-state particles, as a proxy for participant plane (${\psi_{\rm PP}}$) that is not experimentally accessible. To avoid self-correlation, particles used for the EP calculations are exclusive to the particles used for $Q$ and $\Delta$ correlator. The results show strong correlation between the $\Delta$ correlator and the ${{v_{2,{\rm ebye}}}^{\rm obs}}$. By selecting the events with ${{v_{2,{\rm ebye}}}^{\rm obs}}=0$, the $\Delta$ correlator is largely reduced[@Adamczyk:2013kcb; @Tu:2015qm]. The ${\Delta\gamma}$ correlator shows similar correlation with ${{v_{2,{\rm ebye}}}^{\rm obs}}$ from the preliminary STAR data[@Zhao:2017ckp].
![(Color online) charge multiplicity asymmetry correlations ($\Delta$) as a function of ${{v_{2,{\rm ebye}}}^{\rm obs}}$ from Au+Au collisions at [$\sqrt{s_{\rm NN}}$ ]{}= 200 GeV[@Adamczyk:2013kcb]. []{data-label="FG_STARese"}](./STAR_ESE.pdf){width="6.2cm"}
A similar method selecting events with the $q_{2}$ (see Eq. \[EQ\_ESE2\]) variable has been proposed recently[@Wen:2016zic]. To suppress the $v_{2}$ related background, a tighter cut, $q_{2} = 0$, is proposed to extract signal. The cut is tighter because $q_{2} = 0$ corresponds to a zero $2^{nd}$ harmonic to any plane, while ${{v_{2,{\rm ebye}}}^{\rm obs}}= 0$ corresponds to zero $2^{nd}$ harmonic with respect to the reconstructed EP in the event.
These methods assume the background to be linear in $v_{2}$ of the final-state particles. However, the background arises from the correlated pairs (resonance/cluster decay) coupled with the $v_{2}$ of the parent sources, not the final-state particles. In case of resonance decays: ${\Delta\gamma}= {\langle \cos(\phi_{\alpha}+\phi_{\beta}-2\psi_{RP})\rangle} = {\langle \cos(\phi_{\alpha}+\phi_{\beta}-2\phi_{\rm reso.}\rangle}v_{2,\rm reso.}$, where ${\langle \cos(\phi_{\alpha}+\phi_{\beta}-2\phi_{\rm reso.}\rangle}$ depends on the resonance decay kinematics, and $v_{2,\rm{reso.}}$ is the $v_{2}$ of the resonances, not the decay particles’. It is difficult, if not at all impossible, to ensure the $v_{2}$ of all the background sources to be zero. Thus, it is challenging to completely remove flow background by using the event-by-event $v_{2}$ or $q_{2}$ methods[@Wang:2016iov].
Event shape engineering
=======================
Because of dynamical fluctuations of the event-by-event $v_{2}$, one could possibly select events with different initial participant geometries (participant eccentricities) even with the same impact parameter[@Voloshin:2010ut; @Bzdak:2011np; @Schukraft:2012ah]. By restricting to a narrow centrality, while varying event-by-event $v_{2}$, one is presumably still fixing the magnetic filed (mainly determined by the initial distribution of the spectator protons)[@Bzdak:2011np]. This provides a way to decouple the magnetic field and the $v_{2}$, and thus a possible way to disentangle background contributions from potential CME signals. This is usually called the event shape engineering (ESE) method[@Schukraft:2012ah].
In ESE, instead of selecting on ${{v_{2,{\rm ebye}}}^{\rm obs}}$, one use the flow vector to possibly access the initial participant geometry, selecting different event shapes by making use of the dynamical fluctuations of $v_{2}$[@Schukraft:2012ah; @Voloshin:2010ut; @Bzdak:2011np]. The ESE method is performed based on the magnitude of the second-order reduced flow vector, $q_{2}$[@Adler:2002pu], defined as: $$\begin{split}
&q_{2} = \frac{|Q_{2}|}{\sqrt{M}}, \\
&|Q_{2}| = \sqrt{Q_{2,x}^{2}+Q_{2,y}^{2}}, \\
&Q_{2,x} = \sum_{i}w_{i}\cos(2\phi_{i}), \ Q_{2,y} = \sum_{i}w_{i}\sin(2\phi_{i}), \\
\end{split}
\label{EQ_ESE2}$$ where $|Q_{2}|$ is the magnitude of the second order harmonic flow vector and M is the multiplicity. The sum runs over all particles/hits, $\phi_{i}$ is the azimuthal angle of the $i$-th particle/hit, and $w_{i}$ is the weight (usually taken to be the ${p_{T}}$ of the particle or energy deposition of the hit)[@Acharya:2017fau; @Sirunyan:2017quh].
![(Color online) Schematic comparison of the event-by-event selection (A) and the ESE (B) methods. Different boxes represent different phase spaces, usually displaced in $\eta$. The green color for ${{v_{2,{\rm ebye}}}^{\rm obs}}$ in (A) reflects that the ${{v_{2,{\rm ebye}}}^{\rm obs}}$ is calculated with respect to the event-plane. The ${\Delta\gamma}$ correlator is usually calculated from the correlation between the particle of interest (POI, here the POI refers to the $\alpha$ and $\beta$ particles in Eq. \[eqThreeCtor0\]) and the event-plane. []{data-label="FG_ESEcom"}](./ESEcomparison.pdf){width="10cm"}
Figure \[FG\_ESEcom\] is a schematic comparison of the event-by-event selection and the ESE methods. Basically, the most important difference between these two groups of methods lies in which phase space to calculate the ${{v_{2,{\rm ebye}}}^{\rm obs}}$ or $q_{2}$ variables for event selection. In the event-by-event selection methods, the same phase space of the particle of interest (POI) is used for event selections, thus these methods take advantage of statistical as well as dynamical fluctuations of the POI. In the ESE method, a different phase space is used (often displaced in $\eta$), so that the event selection is dominated by the dynamical fluctuations, because statistical fluctuations of POI and event selection are independent. The dynamical fluctuations stem out of the common origin of the initial participant geometries. Thus a zero $q_{2}$ should correspond to an average zero $v_{2}$ of the background sources of the POI. However, a zero $q_{2}$ is unlikely accessible directly from data, so extrapolation is often involved.
![(Color online) (Left) The $q_{2}$ distributions in multiplicity range $185 \leq N^{\rm offline}_{\rm trk} < 250$ in Pb+Pb collisions. Red dash line represents the selection used to divide the events into multiple $q_{2}$ classes. (Right) The correlation between $v_{2}$ and $q_{2}$ in p+Pb and Pb+Pb collisions based on the $q_{2}$ selections of the events[@Sirunyan:2017quh]. []{data-label="FG_CMSeseA"}](./CMS_ESE0.pdf "fig:"){width="6.2cm"} ![(Color online) (Left) The $q_{2}$ distributions in multiplicity range $185 \leq N^{\rm offline}_{\rm trk} < 250$ in Pb+Pb collisions. Red dash line represents the selection used to divide the events into multiple $q_{2}$ classes. (Right) The correlation between $v_{2}$ and $q_{2}$ in p+Pb and Pb+Pb collisions based on the $q_{2}$ selections of the events[@Sirunyan:2017quh]. []{data-label="FG_CMSeseA"}](./CMS_ESE1.pdf "fig:"){width="6.2cm"}
Figure \[FG\_CMSeseA\](left) shows the $q_{2}$ distribution in Pb+Pb collisions from CMS[@Sirunyan:2017quh]. The events of a narrow multiplicity bin are divided into several classes with each corresponding to a fraction of the full distribution, where 0-1% represents the events with the largest $q_{2}$ value, and 95-100% corresponds to the events with smallest $q_{2}$ value, and so on. Fig. \[FG\_CMSeseA\](right) shows that the $v_{2}$ is closely proportional to $q_{2}$, suggesting those two quantities are strongly correlated because of the common initial-state geometry[@Sirunyan:2017quh]. One could thus use the $q_{2}$ to select events with different $v_{2}$, and study the $v_{2}$ dependence of the ${\Delta\gamma}$ correlator. In a similar way, the ${\Delta\gamma}$ correlator is also calculated in each $q_{2}$ class.
Fig. \[FG\_ALICEeseB\](upper left) shows ${\Delta\gamma}$ correlator as a function of $v_{2}$ in different centralities in Pb+Pb collisions from ALICE[@Acharya:2017fau]. To compensate for the dilution effect, ${\Delta\gamma}$ correlator was multiplied by the charged-particle density in a given centrality bin ($dN_{ch}/d\eta$) in the lower left panel. The results show strong dependence on $v_{2}$, and the $dN_{ch}/d\eta$ scaled ${\Delta\gamma}$ correlator falls approximately onto the same linear trend for different centralities. This is qualitatively consistent with the expectation from background effects, such as resonance decay coupled with $v_{2}$[@Hori:2012kp; @Wang:2009kd; @Wang:2016iov]. Therefore, the observed dependence on $v_{2}$ indicates a large background contribution to ${\Delta\gamma}$ correlator[@Acharya:2017fau].
By restricting to a given narrow centrality, the event shape selection is expected to be less affected by the magnetic field[@Bzdak:2011np]. The different dependences of the CME signal and background on $v_{2}$ ($q_{2}$) could possibly be used to disentangle the CME signal from background. Fig. \[FG\_ALICEeseB\](right) shows the $v_{2}$ dependence of the ${\langle |\textbf B|^{2} \cos2(\psi_{B} - \psi_{2})\rangle}$ from Monte Carlo Glauber calculation[@Acharya:2017fau]. The CME signal is assumed to be proportional to ${\langle |\textbf B|^{2} \cos2(\psi_{B} - \psi_{2})\rangle}$, where $|\textbf B|$ and $\psi_{B}$ are the magnitude and azimuthal direction of the magnetic field. The calculation shows that the CME signal weakly depends on $v_{2}$ within each given centrality (Fig. \[FG\_ALICEeseB\] right panel) and approximately linear. To extract the contribution of the possible CME signal to the current ${\Delta\gamma}$ measurements, a linear function is fit to the data: $$\begin{split}
F_{1}(v_{2}) = p_{0}(1+p_{1}(v_{2}-{\langle v_{2}\rangle})/{\langle v_{2}\rangle}).
\end{split}
\label{EQ_ESE3}$$ Here $p_{0}$ accounts for a overall scale. $p_{1}$ is the normalised slope, reflecting the dependence on $v_{2}$. In a pure background scenario, the ${\Delta\gamma}$ correlator is linearly proportional to $v_{2}$ and the $p_{1}$ parameter is equal to unity, Eq. \[EQ\_ESE3\] is reduced to $F_{1}(v_{2}) = p_{0}v_{2}/{\langle v_{2}\rangle} \propto v_{2} $. On the other hand, a significant CME contribution would result in non-zero intercepts at $v_{2}$ = 0 of the linear functional fits shown in Fig. \[FG\_ALICEeseB\](top left).
![(Color online) (Left top) The ${\Delta\gamma}$ correlator and (Left bottom) the charged-particle density scaled correlator ${\Delta\gamma}\cdot dN_{ch}/d\eta$ as a function of $v_{2}$ for shape selected events with $q_{2}$ for various centrality classes. Error bars (shaded boxes) represent the statistical (systematic) uncertainties. (Right) The expected dependence of the CME signal on $v_{2}$ for various centrality classes from a MC-Glauber simulation[@Miller:2007ri]. The solid lines depict linear fits based on the $v_{2}$ variation observed within each centrality interval[@Acharya:2017fau]. []{data-label="FG_ALICEeseB"}](./ALICE_ESE1.pdf "fig:"){width="6.2cm"} ![(Color online) (Left top) The ${\Delta\gamma}$ correlator and (Left bottom) the charged-particle density scaled correlator ${\Delta\gamma}\cdot dN_{ch}/d\eta$ as a function of $v_{2}$ for shape selected events with $q_{2}$ for various centrality classes. Error bars (shaded boxes) represent the statistical (systematic) uncertainties. (Right) The expected dependence of the CME signal on $v_{2}$ for various centrality classes from a MC-Glauber simulation[@Miller:2007ri]. The solid lines depict linear fits based on the $v_{2}$ variation observed within each centrality interval[@Acharya:2017fau]. []{data-label="FG_ALICEeseB"}](./ALICE_ESE2.pdf "fig:"){width="6.2cm"}
In a naive two components model with signal and background, a measured observable ($O_{m}$) can be expressed as: $$\begin{split}
\frac{S}{S+B}\times O_{S} + \frac{B}{S+B}\times O_{B} = O_{m}, \\
\end{split}
\label{EQ_ESE4}$$ $O_{S}$ and $O_{B}$ are the values of the observable $O$ from signal and background, $\frac{S}{S+B}$ represents the fraction of signal contribution in the measurement. The $p_{1}$ from the fit to the measured data is thus a combination of CME signal slope ($p_{1,Sig} = p_{1,MC}$) and the background slope ($ p_{1,Bkg} \equiv 1$): $$\begin{split}
&f_{CME}\times p_{1,Sig} + (1-f_{CME})\times p_{1,Bkg} = p_{1,data},
\end{split}
\label{EQ_ESE5}$$ where $f_{CME} = \frac{{\Delta\gamma}_{CME}}{{\Delta\gamma}_{CME}+{\Delta\gamma}_{Bkg}}$ represents the CME fraction to the ${\Delta\gamma}$ correlator from the measurements, and $p_{1,MC}$ is the slope parameter from the MC calculations in Fig. \[FG\_ALICEeseB\] right panel. Figure \[FG\_ALICEeseC\](left) shows the centrality dependence of $p_{1,data}$ from fits to data and $p_{1,MC}$ to the signal expectations based on MC-Glauber, MC-KLN CGC and EKRT models[@Acharya:2017fau]. Fig. \[FG\_ALICEeseC\](right) presents the estimated $f_{CME}$ from the three models. The $f_{CME}$ extracted for central (0-10%) and peripheral (50-60%) collisions have currently large uncertainties. Combining the points from 10-50% neglecting a possible centrality dependence gives $f_{CME} = 0.10 \pm 0.13$, $f_{CME} = 0.08 \pm 0.10$ and $f_{CME} = 0.08 \pm 0.11$ for the MC-Glauber, MC-KLN CGC and EKRT models inputs of $p_{1,MC}$, respectively. These results are consistent with zero CME fraction and correspond to upper limits on $f_{CME}$ of 33%, 26% and 29%, respectively, at 95% confidence level for the 10-50% centrality interval[@Acharya:2017fau].
![(Color online) (Left) Centrality dependence of the $p_{1}$ parameter from a linear fit to the ${\Delta\gamma}$ correlator in Pb+Pb collisions from ALICE and from linear fits to the CME signal expectations from MC-Glauber[@Miller:2007ri], MC-KLN CGC[@Drescher:2007ax; @ALbacete:2010ad] and EKRT[@Niemi:2015qia] models. (Right) Centrality dependence of the CME fraction extracted from the slope parameter of fits to data and different models. Points from MC simulations are slightly shifted along the horizontal axis for better visibility. Only statistical uncertainties are shown[@Acharya:2017fau]. []{data-label="FG_ALICEeseC"}](./ALICE_ESE3.pdf "fig:"){width="6.2cm"} ![(Color online) (Left) Centrality dependence of the $p_{1}$ parameter from a linear fit to the ${\Delta\gamma}$ correlator in Pb+Pb collisions from ALICE and from linear fits to the CME signal expectations from MC-Glauber[@Miller:2007ri], MC-KLN CGC[@Drescher:2007ax; @ALbacete:2010ad] and EKRT[@Niemi:2015qia] models. (Right) Centrality dependence of the CME fraction extracted from the slope parameter of fits to data and different models. Points from MC simulations are slightly shifted along the horizontal axis for better visibility. Only statistical uncertainties are shown[@Acharya:2017fau]. []{data-label="FG_ALICEeseC"}](./ALICE_ESE4.pdf "fig:"){width="6.2cm"}
The above analysis is model-dependent, relying on precise modeling of the magnetic field with a given centrality. The CMS collaboration took a different approach, cutting on very narrow centrality bins and assuming the magnetic field to be constant within each centrality bin[@Acharya:2017fau]. The background contribution to the $\gamma$ correlator is approximated to be[@Bzdak:2012ia]: $$\begin{split}
\gamma & \equiv {\langle \cos(\phi_{\alpha}-\phi_{\beta}-2\psi_{RP})\rangle}, \\
\gamma^{bkg} & =\kappa_{2}{\langle \cos(\phi_{\alpha}-\phi_{\beta})\rangle}{\langle \cos2(\phi_{\beta}-\psi_{RP})\rangle} \\
& =\kappa_{2} \delta v_{2}, \\
\delta &\equiv {\langle \cos(\phi_{\alpha}-\phi_{\beta})\rangle}. \\
\end{split}
\label{EQ_ESE6}$$ Here, $\delta$ represents the charge-dependent two-particle azimuthal correlator and $\kappa_{2}$ is a constant parameter, independent of $v_{2}$, but mainly determined by the kinematics and acceptance of particle detection[@Bzdak:2012ia]. The $\delta$, $\gamma$ and $v_{2}$ are experimental measured observables. With event shape engineering to select event with different $v_{2}$, the above Eq.\[EQ\_ESE6\] can be tested. The charge-independent background sources are eliminated by taking the difference of the correlators ($\gamma, \delta$) between same- and opposite-sign pairs. In the background scenario, the ${\Delta\gamma}$ is expected to be: $$\begin{split}
{\Delta\gamma}=\kappa_{2} \Delta\delta v_{2}. \\
\end{split}
\label{EQ_ESE7}$$ A linear function was used to extract the $v_{2}$-independent fraction of the ${\Delta\gamma}$ correlator: $$\begin{split}
{\Delta\gamma}/\Delta\delta = a_{norm} v_{2} + b_{norm}, \\
\end{split}
\label{EQ_ESE8}$$ where $b_{norm}$ could be possibly the contribution from CME signal.
![(Color online) The ratio between ${\Delta\gamma}$ (${\Delta\gamma}_{112}$) and $\Delta\delta$ correlators, ${\Delta\gamma}/\Delta\delta$, averaged over $|\Delta\eta|< 1.6$ as a function of $v_{2}$ evaluated in each $q_{2}$ class, for different multiplicity ranges in p+Pb (left) and Pb+Pb (right) collisions[@Sirunyan:2017quh]. []{data-label="FG_CMSESEA"}](./CMS_ESE2a.pdf "fig:"){width="6.2cm"} ![(Color online) The ratio between ${\Delta\gamma}$ (${\Delta\gamma}_{112}$) and $\Delta\delta$ correlators, ${\Delta\gamma}/\Delta\delta$, averaged over $|\Delta\eta|< 1.6$ as a function of $v_{2}$ evaluated in each $q_{2}$ class, for different multiplicity ranges in p+Pb (left) and Pb+Pb (right) collisions[@Sirunyan:2017quh]. []{data-label="FG_CMSESEA"}](./CMS_ESE2b.pdf "fig:"){width="6.2cm"}
![(Color online) (Left) Extracted intercept parameter $b_{norm}$ and (Right) corresponding upper limit of the fraction of $v_{2}$-independent ${\Delta\gamma}$ correlator component, averaged over $|\Delta\eta|< 1.6$, as a function of $N^{\rm offline}_{\rm trk}$ in p+Pb and Pb+Pb collisions from CMS[@Sirunyan:2017quh]. []{data-label="FG_CMSESEB"}](./CMS_ESE4.pdf "fig:"){width="6.2cm"} ![(Color online) (Left) Extracted intercept parameter $b_{norm}$ and (Right) corresponding upper limit of the fraction of $v_{2}$-independent ${\Delta\gamma}$ correlator component, averaged over $|\Delta\eta|< 1.6$, as a function of $N^{\rm offline}_{\rm trk}$ in p+Pb and Pb+Pb collisions from CMS[@Sirunyan:2017quh]. []{data-label="FG_CMSESEB"}](./CMS_ESE3.pdf "fig:"){width="6.2cm"}
Figure \[FG\_CMSESEA\] shows the ratio of $\Delta\gamma/\Delta\delta$ as function of $v_{2}$ for different multiplicity ranges in p+Pb (left) and Pb+Pb (right) collisions[@Sirunyan:2017quh]. The values of the intercept parameter $b_{norm}$ are shown as a function of event multiplicity in Fig. \[FG\_CMSESEB\](left). Within statistical and systematic uncertainties, no significant positive value for $b_{norm}$ is observed. Result suggests that the $v_{2}$-independent contribution to the ${\Delta\gamma}$ correlator is consistent with zero, and the ${\Delta\gamma}$ results are consistent with the background-only scenario of charge-dependent two-particle correlations[@Sirunyan:2017quh]. Based on the assumption of a nonnegative CME signal, the upper limit of the $v_{2}$-independent fraction in the ${\Delta\gamma}$ correlator is obtained from the Feldman-Cousins approach[@Feldman:1997qc] with the measured statistical and systematic uncertainties. Fig. \[FG\_CMSESEB\](right) shows the upper limit of the fraction $f_{norm}$, the ratio of the $b_{norm}$ value to the value of ${\langle {\Delta\gamma}\rangle}/{\langle \Delta\delta\rangle}$, at 95% CL as a function of event multiplicity. The $v_{2}$-independent component of the ${\Delta\gamma}$ correlator is less than 8-15% for most of the multiplicity or centrality range. The combined limits from all presented multiplicities and centralities are also shown in p+Pb and Pb+Pb collisions. An upper limit on the $v_{2}$-independent fraction of the three-particle correlator, or possibly the CME signal contribution, is estimated to be 13% in p+Pb and 7% in Pb+Pb collisions, at 95% CL. The results are consistent with a $v_{2}$-dependent background-only scenario, posing a significant challenge to the search for the CME in heavy ion collisions using three-particle azimuthal correlations[@Sirunyan:2017quh].
![(Color online) The ${\Delta\gamma}_{112}$ (left top), ${\Delta\gamma}_{123}$ (left middle), $\Delta\delta$ (left bottom), averaged over $|\Delta\eta|< 1.6$ as a function of $N^{\rm offline}_{\rm trk}$ in p+Pb and Pb+Pb collisions. The p+Pb results are obtained with particle $c$ from Pb- and p-going sides separately. The ratio of ${\Delta\gamma}_{112}$ and ${\Delta\gamma}_{123}$ to the product of $v_{n}$ and $\Delta\delta$ in p+Pb collisions for the Pb-going direction (right top) and Pb+Pb collisions (right bottom). Statistical and systematic uncertainties are indicated by the error bars and shaded regions, respectively[@Sirunyan:2017quh]. []{data-label="FG_CMSgm123"}](./CMS_gm123A.pdf "fig:"){width="6.2cm"} ![(Color online) The ${\Delta\gamma}_{112}$ (left top), ${\Delta\gamma}_{123}$ (left middle), $\Delta\delta$ (left bottom), averaged over $|\Delta\eta|< 1.6$ as a function of $N^{\rm offline}_{\rm trk}$ in p+Pb and Pb+Pb collisions. The p+Pb results are obtained with particle $c$ from Pb- and p-going sides separately. The ratio of ${\Delta\gamma}_{112}$ and ${\Delta\gamma}_{123}$ to the product of $v_{n}$ and $\Delta\delta$ in p+Pb collisions for the Pb-going direction (right top) and Pb+Pb collisions (right bottom). Statistical and systematic uncertainties are indicated by the error bars and shaded regions, respectively[@Sirunyan:2017quh]. []{data-label="FG_CMSgm123"}](./CMS_gm123B.pdf "fig:"){width="6.2cm"}
The CME-driven charge separation are expected along the magnetic field direction normal to the reaction plane, estimated by the second-order event plane ($\psi_{2}$). The third-order event plane ($\psi_{3}$) are expected to be weakly correlated with $\psi_{2}$[@Aad:2014fla], thus the CME-driven charge separation effect with respect to $\psi_{3}$ is expected to be negligible. In light of $v_{2}$-dependent background-only scenario, where background can be expressed as Eq \[EQ\_ESE6\]. A similar correlator ($\gamma_{123}$) with respect to third-order event plane ($\psi_{3}$) are constructed to study the background effects[@Sirunyan:2017quh]: $$\begin{split}
\gamma_{123} & \equiv {\langle \cos(\phi_{\alpha}-2\phi_{\beta}-3\psi_{3})\rangle} \\
\gamma_{123}^{bkg} & =\kappa_{3}{\langle \cos(\phi_{\alpha}-\phi_{\beta})\rangle}{\langle \cos3(\phi_{\beta}-\psi_{3})\rangle} \\
& =\kappa_{3} \delta v_{3}. \\
\end{split}
\label{EQ_ESE9}$$ In the flow-dependent background-only scenario, the $\kappa_{2}$ and $\kappa_{3}$ mainly depend on particle kinematics and detector acceptance effects, and are expected to be similar, largely independent of harmonic event plane order. Fig. \[FG\_CMSgm123\](left) shows the ${\Delta\gamma}$ ($\Delta\gamma_{112}$), $\Delta\gamma_{123}$, $\Delta\delta$ correlator as a function of multiplicity in p+Pb and Pb+Pb collisions. Fig. \[FG\_CMSgm123\](right) shows the ratio of the ${\Delta\gamma}$ ($\Delta\gamma_{112}$) and $\Delta\gamma_{112}$ to the product of $v_{n}$ and $\Delta\delta$. The results show that the ratio is similar for $n$=2 and 3, and also similar between p+Pb and Pb+Pb collisions, indicating that the $\kappa_{3}$ is similar to $\kappa_{2}$. These results are consistent with the flow-dependent background-only scenario.
The event shape selection provides a very useful tool to study the background behavior of the ${\Delta\gamma}$. All current experimental results at LHC suggest that the ${\Delta\gamma}$ are strongly dependent on the $v_{2}$ and consistent with the flow-background only scenario. In summary, the $v_{2}$ independent contribution are estimated by different methods from STAR, ALICE and CMS, and current results indicate that a large contribution of the ${\Delta\gamma}$ correlator is from the $v_{2}$ related background.
Isobaric collisions
===================
The CME is related to the magnetic field while the background is produced by $v_{2}$-induced correlations. In order to gauge differently the magnetic field relative to the $v_{2}$, isobaric collisions and Uranium+Uranium collisions have been proposed[@Voloshin:2010ut]. The isobaric collisions are proposed to study the two systems with similar $v_{2}$ but different magnetic field strength[@Voloshin:2010ut], such as ${^{96}_{44}\rm{Ru}}$ and ${^{96}_{40}\rm{Zr}}$, which have the same mass number, but differ by charge (proton) number. Thus one would expected very similar $v_{2}$ at mid-rapidity in ${^{96}_{44}\rm{Ru} + ^{96}_{44}\rm{Ru}}$ and ${^{96}_{40}\rm{Zr} + ^{96}_{40}\rm{Zr}}$ collisions, but the magnetic field, proportional to the nuclei electric charge, could vary by 10%. If the measured ${\Delta\gamma}$ is dominated by the CME-driven charge separation, then the variation of the magnetic field strength between ${^{96}_{44}\rm{Ru} + ^{96}_{44}\rm{Ru}}$ and ${^{96}_{40}\rm{Zr} + ^{96}_{40}\rm{Zr}}$ collision provides an ideal way to disentangle the signal of the chiral magnetic effect from $v_{2}$ related background, as the $v_{2}$ related backgrounds are expected to be very similar between these two systems.
To test the idea of the isobaric collisions, Monte Carlo Glauber calculations[@Deng:2016knn; @Huang:2017azw; @Xu:2017zcn] of the spatial eccentricity ($\epsilon_{2}$) and the magnetic field strength[@Deng:2016knn; @Xu:2017zcn] form ${^{96}_{44}\rm{Ru} + ^{96}_{44}\rm{Ru}}$ and ${^{96}_{40}\rm{Zr} + ^{96}_{40}\rm{Zr}}$ collisions have been carried out. The Woods-Saxon spatial distribution is used[@Deng:2016knn]: $$\begin{split}
\rho(r,\theta) = \frac{1}{1+\rm{exp}\{[r-R_{0}-\beta_{2}R_{0}Y_{2}^{0}(\theta)]/ \it{a} \}},
\end{split}
\label{EQ_ISO1}$$ where $\rho^{0} = 0.16 fm^{-3}$ is the normal nuclear density, $R^{0}$ is the charge radius of the nucleus, $a$ represent the surface diffuseness parameter. $Y_{2}^{0}$ is the spherical harmonic. The parameter $a$ is almost identical for ${^{96}_{44}\rm{Ru}}$ and ${^{96}_{40}\rm{Zr}}$: $a\approx0.46$ fm. $R_{0}=5.085$ fm and 5.020 fm are used for ${^{96}_{44}\rm{Ru}}$ and ${^{96}_{40}\rm{Zr}}$, and are used for both the proton and nucleon densities. The deformity quadrupole parameter $\beta_{2}$ has large uncertainties; there are two contradicting sets of values from current knowledge[@Deng:2016knn], $\beta_{2}({^{96}_{44}\rm{Ru}})=0.158$ and $\beta_{2}({^{96}_{40}\rm{Zr}})=0.080$ (referred to as case 1) vis a vis $\beta_{2}({^{96}_{44}\rm{Ru}})=0.053$ and $\beta_{2}({^{96}_{40}\rm{Zr}})=0.217$ (referred to as case 2). These would yield less than 2% difference in $\epsilon_{2}$, hence a less than 2% residual $v_{2}$ background, between ${^{96}_{44}\rm{Ru} + ^{96}_{44}\rm{Ru}}$ and ${^{96}_{40}\rm{Zr} + ^{96}_{40}\rm{Zr}}$ collisions in the 20-60% centrality range[@Deng:2016knn]. In that centrality range, the mid-rapidity particle multiplicities are almost identical for ${^{96}_{44}\rm{Ru} + ^{96}_{44}\rm{Ru}}$ and ${^{96}_{40}\rm{Zr} + ^{96}_{40}\rm{Zr}}$ collisions at the same energy[@Deng:2016knn; @Xu:2017zcn].
The magnetic field strengths in ${^{96}_{44}\rm{Ru} + ^{96}_{44}\rm{Ru}}$ and ${^{96}_{40}\rm{Zr} + ^{96}_{40}\rm{Zr}}$ collisions are calculated by using Lienard-Wiechert potentials alone with HIJING model taking into account the event-by-event azimuthal fluctuation of the magnetic field orientation[@Deng:2016knn; @Deng:2012pc]. HIJING model with the above two sets (case 1 and 2) of Woods-Saxon densities are simulated. Fig. \[FG\_ISO1\](a) shows the calculation of the event-averaged initial magnetic field squared with correction from the event-by-event azimuthal fluctuation of the magnetic field orientation, $$B_{sq} \equiv {\langle (eB/m_{\pi}^{2})^{2} \cos[2(\psi_{B} - \psi_{\rm{RP}})]\rangle},$$ for the two collision systems at 200 GeV. Fig. \[FG\_ISO1\](b) shows that the relative difference in $B_{sq}$, defined as $R_{B_{sq}}=2(B_{sq}^{Ru+Ru}-B_{sq}^{Zr+Zr})/(B_{sq}^{Ru+Ru}+B_{sq}^{Zr+Zr})$, between ${^{96}_{44}\rm{Ru} + ^{96}_{44}\rm{Ru}}$ and ${^{96}_{40}\rm{Zr} + ^{96}_{40}\rm{Zr}}$ collisions is approaching 15% (case 1) or 18% (case 2) for peripheral events, and reduces to about 13% (cases 1 and 2) for central events. Fig. \[FG\_ISO1\](b) also shows the relative difference in the initial eccentricity ($R_{\epsilon_{2}}=2(\epsilon_{2}^{Ru+Ru}-\epsilon_{2}^{Zr+Zr})/(\epsilon_{2}^{Ru+Ru}+\epsilon_{2}^{Zr+Zr})$), obtained from the Monte-Carlo Glauber calculation. The relative difference in $\epsilon_{2}$ is practically zero for peripheral events, and goes above (below) 0 for the parameter set of case 1 (case 2) in central collisions. The relative difference in $v_{2}$ from ${^{96}_{44}\rm{Ru} + ^{96}_{44}\rm{Ru}}$ and ${^{96}_{40}\rm{Zr} + ^{96}_{40}\rm{Zr}}$ collisions is expected to closely follow that in eccentricity, indicating the $v_{2}$-related backgrounds are almost the same (different within 2%) for ${^{96}_{44}\rm{Ru} + ^{96}_{44}\rm{Ru}}$ and ${^{96}_{40}\rm{Zr} + ^{96}_{40}\rm{Zr}}$ collisions in 20-60% centrality range.
![(Color online) (a) Event-averaged initial magnetic field squared at the center of mass of the overlapping region, with correction from event-by-event fluctuation of magnetic field azimuthal orientation, for ${^{96}_{44}\rm{Ru} + ^{96}_{44}\rm{Ru}}$ and ${^{96}_{40}\rm{Zr} + ^{96}_{40}\rm{Zr}}$ collisions at 200 GeV, and (b) their relative difference versus centrality. Also shown in (b) is the relative difference in the initial eccentricity. The solid (dashed) curves correspond to the parameter set of case 1 (case 2)[@Deng:2016knn]. []{data-label="FG_ISO1"}](./Isobar_B.pdf "fig:"){width="6.2cm"} ![(Color online) (a) Event-averaged initial magnetic field squared at the center of mass of the overlapping region, with correction from event-by-event fluctuation of magnetic field azimuthal orientation, for ${^{96}_{44}\rm{Ru} + ^{96}_{44}\rm{Ru}}$ and ${^{96}_{40}\rm{Zr} + ^{96}_{40}\rm{Zr}}$ collisions at 200 GeV, and (b) their relative difference versus centrality. Also shown in (b) is the relative difference in the initial eccentricity. The solid (dashed) curves correspond to the parameter set of case 1 (case 2)[@Deng:2016knn]. []{data-label="FG_ISO1"}](./Isobar_R.pdf "fig:"){width="6.2cm"}
Based on the available experimental ${\Delta\gamma}$ measurements in Au+Au collisions at 200 GeV and the calculated magnetic field strength and eccentricity difference between ${^{96}_{44}\rm{Ru} + ^{96}_{44}\rm{Ru}}$ and ${^{96}_{40}\rm{Zr} + ^{96}_{40}\rm{Zr}}$ collisions, expected signals from the isobar collisions are estimated[@Deng:2016knn]: $$\begin{split}
&S = {\Delta\gamma}\times N_{part}, \\
&S^{Ru+Ru} = \bar{S} \big[ (1-bg)\big(1+\frac{R_{B_{sq}}}{2}\big) +bg\big(1+\frac{R_{\epsilon_{2}}}{2} \big) \big], \\
&S^{Zr+Zr} = \bar{S} \big[ (1-bg)\big(1-\frac{R_{B_{sq}}}{2}\big) +bg\big(1-\frac{R_{\epsilon_{2}}}{2} \big) \big], \\
&R_{S} = (1-bg)R_{B_{sq}} + bgR_{\epsilon_{2}}, \ bg \in [0,1], \\
\end{split}
\label{EQ_ISO2}$$ where $S$ represents the $N_{part}$ scaled ${\Delta\gamma}$ correlator ($N_{part}$ account for the dilution effect[@Abelev:2009ad; @Ma:2011uma]). The $bg$ is the $v_{2}$ related background fraction of the ${\Delta\gamma}$ correlator. Fig. \[FG\_ISO2\](left) shows the $R_{S}$ with $400\times10^{6}$ events for each of the two collisions types, assuming $\frac{2}{3}$ of the ${\Delta\gamma}$ comes from the $v_{2}$ related background, and compared with $R_{\epsilon_{2}}$. Fig. \[FG\_ISO2\](right) shows the magnitude and significance of the projected relative difference between ${^{96}_{44}\rm{Ru} + ^{96}_{44}\rm{Ru}}$ and ${^{96}_{40}\rm{Zr} + ^{96}_{40}\rm{Zr}}$ collisions as a function of the background level. With the given event statistics and assumed background level, the isobar collisions will give 5$\sigma$ significance.
![(Color online) (Left) Estimated relative difference in $S = {\Delta\gamma}\times N_{part}$ and in the initial eccentricity for ${^{96}_{44}\rm{Ru} + ^{96}_{44}\rm{Ru}}$ and ${^{96}_{40}\rm{Zr} + ^{96}_{40}\rm{Zr}}$ collisions at 200 GeV. (Right) Magnitude and significance of the relative difference in the CME signal between ${^{96}_{44}\rm{Ru} + ^{96}_{44}\rm{Ru}}$ and ${^{96}_{40}\rm{Zr} + ^{96}_{40}\rm{Zr}}$, $R_{S} - R_{\epsilon_{2}}$ as a function of the background level[@Deng:2016knn]. []{data-label="FG_ISO2"}](./Isobar_df.pdf "fig:"){width="5.8cm"} ![(Color online) (Left) Estimated relative difference in $S = {\Delta\gamma}\times N_{part}$ and in the initial eccentricity for ${^{96}_{44}\rm{Ru} + ^{96}_{44}\rm{Ru}}$ and ${^{96}_{40}\rm{Zr} + ^{96}_{40}\rm{Zr}}$ collisions at 200 GeV. (Right) Magnitude and significance of the relative difference in the CME signal between ${^{96}_{44}\rm{Ru} + ^{96}_{44}\rm{Ru}}$ and ${^{96}_{40}\rm{Zr} + ^{96}_{40}\rm{Zr}}$, $R_{S} - R_{\epsilon_{2}}$ as a function of the background level[@Deng:2016knn]. []{data-label="FG_ISO2"}](./Isobar_sg.pdf "fig:"){width="6.6cm"}
The above estimates assume Woods-Saxon densities, identical for proton and neutron distributions. Using the energy density functional (EDF) method with the well-known SLy4 mean field [@Chabanat:1997un] including pairing correlations (Hartree-Fock-Bogoliubov, HFB approach)[@Wang:2016rqh; @Bender:2003jk; @ring1980nuclear], assumed spherical, the ground-state density distributions for ${^{96}_{44}\rm{Ru}}$ and ${^{96}_{40}\rm{Zr}}$ are calculated. The results are shown in Fig. \[FG\_ISO3\](left)[@Xu:2017zcn]. They show that protons in Zr are more concentrated in the core, while protons in Ru, 10% more than in Zr, are pushed more toward outer regions. The neutrons in Zr, four more than in Ru, are more concentrated in the core but also more populated on the nuclear skin. Fig. \[FG\_ISO3\](right) shows the relative differences between ${^{96}_{44}\rm{Ru} + ^{96}_{44}\rm{Ru}}$ and ${^{96}_{40}\rm{Zr} + ^{96}_{40}\rm{Zr}}$ collisions as functions of centrality in $v_{2}\{\psi\}$ and $B_{sq}\{\psi\}$ with respect to ${\psi_{\rm RP}}$ and ${\psi_{\rm EP}}$ from AMPT simulation with the densities calculated by the EDF method. Results suggest that with respect to ${\psi_{\rm EP}}$, the relative difference in $\epsilon_{2}$ and $v_{2}$ are as large as $\sim$3%. With respect to ${\psi_{\rm RP}}$, the difference in $\epsilon_{2}$ and $v_{2}$ becomes even larger ($\sim$10%), and the difference in $B_{sq}$ is only 0-15%[@Xu:2017zcn]. These studies suggest that the premise of isobaric sollisions for the CME search may not be as good as originally anticipated, and could provide additional important guidance to the experimental isobaric collision program.
![(Color online) (Left) Proton and neutron density distributions of the ${^{96}_{44}\rm{Ru}}$ and ${^{96}_{40}\rm{Zr}}$ nuclei, assumed spherical, calculated by the EDF method[@Xu:2017zcn]. (Right) Relative differences between ${^{96}_{44}\rm{Ru} + ^{96}_{44}\rm{Ru}}$ and ${^{96}_{40}\rm{Zr} + ^{96}_{40}\rm{Zr}}$ collisions as functions of centrality in $v_{2}\{\psi\}$ and $Bsq\{\psi\}$ with respect to ${\psi_{\rm RP}}$ and ${\psi_{\rm EP}}$ from AMPT simulation with the densities from the left plot[@Xu:2017zcn]. []{data-label="FG_ISO3"}](./Isobar_density1.pdf "fig:"){width="5.8cm"} ![(Color online) (Left) Proton and neutron density distributions of the ${^{96}_{44}\rm{Ru}}$ and ${^{96}_{40}\rm{Zr}}$ nuclei, assumed spherical, calculated by the EDF method[@Xu:2017zcn]. (Right) Relative differences between ${^{96}_{44}\rm{Ru} + ^{96}_{44}\rm{Ru}}$ and ${^{96}_{40}\rm{Zr} + ^{96}_{40}\rm{Zr}}$ collisions as functions of centrality in $v_{2}\{\psi\}$ and $Bsq\{\psi\}$ with respect to ${\psi_{\rm RP}}$ and ${\psi_{\rm EP}}$ from AMPT simulation with the densities from the left plot[@Xu:2017zcn]. []{data-label="FG_ISO3"}](./Isobar_density2.pdf "fig:"){width="6.6cm"}
Uranium+Uranium collisions
==========================
Isobaric collisions produce different magnetic field but similar $v_{2}$. One may produce on average different $v_{2}$ but same magnetic fields, this may be achieved by uranium+uranium collisions[@Voloshin:2010ut]. Unlike the nearly spherical nuclei of gold (Au), uranium (U) nuclei have a highly ellipsoidal shape. By colliding two uranium nuclei, there would be various collision geometries, such as the tip-tip or body-body collisions. In very central collisions, due to the particular ellipsoidal shape of the uranium nuclei, the overlap region would still be ellipsoidal in the body-body U+U collisions. This ellipsoidal shape of the overlap region would generate a finite elliptic flow, giving rise to the background in the ${\Delta\gamma}$ measurements. On the other hand, the magnetic field are expected to vanish in the overlap region in those central body-body collisions. Thus in general the magnetic field driven CME signal will vanish in these very central collisions. By comparing central Au+Au collisions of different configurations, it may be possible to disentangle CME and background correlations contributing to the experimental measured ${\Delta\gamma}$ signal[@Voloshin:2010ut]. In 2012 RHIC ran U+U collisions. Preliminary experimental results in central U+U have been compared with the results from central Au+Au[@Wang:2012qs; @Tribedy:2017hwn]. However, the geometry of the overlap region is much more complicated than initially anticipated, and the experimental systemic uncertainties are under further detailed investigation. So far there is no clear conclusion in term of the disentangle of the CME and $v_{2}$ related background from the preliminary experimental data yet.
Small system p+A or d+A collisions
==================================
The small system p+A or d+A collisions provides a control experiment, where the CME signal can be “turned off”, but the $v_{2}$ related backgrounds still persist. In non-central heavy-ion collisions, the ${\psi_{\rm PP}}$, although fluctuating[@Alver:2006wh], is generally aligned with the reaction plane, thus generally perpendicular to ${\vec{B}}$. The ${\Delta\gamma}$ measurement is thus $\emph{entangled}$ by the two contributions: the possible CME and the $v_2$-induced background. In small-system p+A or d+A collisions, however, the ${\psi_{\rm PP}}$ is determined purely by geometry fluctuations, uncorrelated to the impact parameter or the ${\vec{B}}$ direction[@Khachatryan:2016got; @Belmont:2016oqp; @Tu:2017kfa]. As a result, any CME signal would average to zero in the ${\Delta\gamma}$ measurements with respect to the ${\psi_{\rm PP}}$. Background sources, on the other hand, contribute to small-system p+A or d+A collisions similarly as to heavy-ion collisions. Comparing the small system p+A or d+A collisions to A + A collisions could thus further our understanding of the background issue in the ${\Delta\gamma}$ measurements.
![(Color online) Single-event display from a Monte-Carlo Glauber event of a peripheral Pb+Pb (a) and a central p+Pb (b) collision at 5.02 TeV. The open gray \[solid green (light gray)\] circles indicate spectator nucleons (participating protons) traveling in the positive z direction, and the open gray \[solid red (dark gray)\] circles with crosses indicate spectator nucleons (participating protons) traveling in the negative z direction. In each panel, the calculated magnetic field vector is shown as a solid magenta line and the long axis of the participant eccentricity is shown as a solid black line[@Belmont:2016oqp]. []{data-label="FG_SM1"}](./EventPlane_pAAA.pdf){width="12.0cm"}
Figure \[FG\_SM1\] shows a single-event display from a Monte Carlo-Glauber event of a peripheral Pb+Pb (a) and a central p+Pb (b) collision at 5.02 TeV[@Belmont:2016oqp]. In A+A collisions, due to the geometry of the overlap region, the eccentricity long axis are highly correlated with the impact parameter direction. Meanwhile the magnetic field direction is mainly determined by the positions of the protons in the two colliding nucleus, which is also generally perpendicular to the impact parameter direction. Thus in A+A collisions, these two direction are highly correlated with each other. Consequently, the ${\Delta\gamma}$ measurements are entangled with the $v_{2}$ background and possible CME signal. While in p+A (Fig. \[FG\_SM1\], b) due to fluctuations in the positions of the nucleons, the eccentricity long axis and magnetic field direction are no longer correlated with each other. So the ${\Delta\gamma}$ measurements in p+A collisions with respected to the eccentricity long axis (estimated by ${\psi_{\rm PP}}$) will lead to zero CME signal on average, and similarly for d+A collisions.
The recent ${\Delta\gamma}$ measurements in small system p+Pb collisions from CMS have triggered a wave of discussions about the interpretation of the CME in heavy-ion collisions[@Khachatryan:2016got]. The ${\Delta\gamma}$ correlator signal from p+Pb is comparable to the signal from Pb+Pb collisions at similar multiplicities, which indicates significant background contributions in Pb+Pb collisions at LHC energy.
![(Color online) The opposite-sign and same-sign three-particle correlator averaged over $|\eta_{\alpha}-\eta_{\beta}| < 1.6$ as a function of $N^{\rm offline}_{\rm trk}$ in p+Pb and Pb+Pb collisions at [$\sqrt{s_{\rm NN}}$ ]{}= 5.02 TeV from CMS collaboration. Statistical and systematic uncertainties are indicated by the error bars and shaded regions, respectively[@Khachatryan:2016got]. []{data-label="FG_SM2"}](./CMS_PRL1.pdf){width="7.0cm"}
![(Color online) The difference of the opposite-sign and same-sign three-particle correlators (a) as a function of $|\eta_{\alpha}-\eta_{\beta}|$ for $185 \leq N^{\rm offline}_{\rm trk} < 220$ and (b) as a function of $N^{\rm offline}_{\rm trk}$, averaged over $|\eta_{\alpha}-\eta_{\beta}| < 1.6$, in p+Pb and Pb+Pb collisions at [$\sqrt{s_{\rm NN}}$ ]{}= 5.02 TeV from CMS collaboration. The p-Pb results are obtained with particle $c$ from Pb- and p-going sides separately. Statistical and systematic uncertainties are indicated by the error bars and shaded regions, respectively[@Khachatryan:2016got]. []{data-label="FG_SM3"}](./CMS_PRL2.pdf "fig:"){width="6.2cm"} ![(Color online) The difference of the opposite-sign and same-sign three-particle correlators (a) as a function of $|\eta_{\alpha}-\eta_{\beta}|$ for $185 \leq N^{\rm offline}_{\rm trk} < 220$ and (b) as a function of $N^{\rm offline}_{\rm trk}$, averaged over $|\eta_{\alpha}-\eta_{\beta}| < 1.6$, in p+Pb and Pb+Pb collisions at [$\sqrt{s_{\rm NN}}$ ]{}= 5.02 TeV from CMS collaboration. The p-Pb results are obtained with particle $c$ from Pb- and p-going sides separately. Statistical and systematic uncertainties are indicated by the error bars and shaded regions, respectively[@Khachatryan:2016got]. []{data-label="FG_SM3"}](./CMS_PRL3.pdf "fig:"){width="6.2cm"}
Figure \[FG\_SM2\] shows the first ${\Delta\gamma}$ measurements in small system p+A collisions from CMS, by using p+Pb collisions at 5.02 TeV compared with Pb+Pb at same energy. The results are plotted as a function of event charged-particle multiplicity ($N^{\rm offline}_{\rm trk}$). The p+Pb and Pb+Pb results are measured in the same $N^{\rm offline}_{\rm trk}$ ranges up to 300. The p+Pb results obtained with particle c in Pb-going forward direction. Within uncertainties, the SS and OS correlators in p+Pb and Pb+Pb collisions exhibit the same magnitude and trend as a function of event multiplicity. By taking the difference between SS and OS correlators, Fig \[FG\_SM3\] shows the $|\Delta\eta|=|\eta_{\alpha}-\eta_{\beta}|$ and multiplicity dependence of ${\Delta\gamma}$ correlator. The p+Pb and Pb+Pb data show similar $|\Delta\eta|$ dependence, decreasing with increasing $|\Delta\eta|$. The distributions show a traditional short range correlation structure, indicating the correlations may come from the hadonic stage of the collisions, while the CME is expected to be a long range correlation arising from the early stage. The multiplicity dependence of ${\Delta\gamma}$ correlator are also similar between p+Pb and Pb+Pb, decreasing as a function of $N^{\rm offline}_{\rm trk}$, which could be understood as a dilution effect that falls with the inverse of event multiplicity[@Abelev:2009ad]. There is a hint that slopes of the $N^{\rm offline}_{\rm trk}$ dependence in p+Pb and Pb+Pb are slightly different in Fig. \[FG\_SM3\](b), which might be worth further investigation. The similarity seen between high-multiplicity p+Pb and peripheral Pb+Pb collisions strongly suggests a common physical origin, challenges the attribution of the observed charge-dependent correlations to the CME[@Khachatryan:2016got].
It is predicted that the CME would decrease with the collision energy due to the more rapidly decaying $B$ at higher energies[@Kharzeev:2015znc; @Deng:2012pc]. Hence, the similarity between small-system and heavy-ion collisions at the LHC may be expected, and the situation at RHIC could be different[@Kharzeev:2015znc]. Similar control experiments using p+Au and d+Au collisions are also performed at RHIC[@Zhao:2017wck; @Zhao:2018pnk].
![(Color online) The preliminary ${\gamma_{\rm SS}}$, ${\gamma_{\rm OS}}$ (Left panel) and ${\Delta\gamma}$ (Right panel) correlators in p+Au and d+Au collisions as a function of multiplicity, compared to those in Au+Au collisions at [$\sqrt{s_{\rm NN}}$ ]{}= 200 GeV from STAR collaboration. Particles $\alpha$, $\beta$ and $c$ are from the TPC pseudorapidity coverage of $|\eta|<1$ with no $\eta$ gap applied. The $v_{2,c}\{2\}$ is obtained by two-particle cumulant with $\eta$ gap of $\Delta\eta > 1.0$. Statistical uncertainties are shown by the vertical bars and systematic uncertainties are shown by the caps[@Zhao:2017wck; @Zhao:2018pnk]. []{data-label="FG_SM4"}](./STAR_small_gamma.pdf "fig:"){width="6.2cm"} ![(Color online) The preliminary ${\gamma_{\rm SS}}$, ${\gamma_{\rm OS}}$ (Left panel) and ${\Delta\gamma}$ (Right panel) correlators in p+Au and d+Au collisions as a function of multiplicity, compared to those in Au+Au collisions at [$\sqrt{s_{\rm NN}}$ ]{}= 200 GeV from STAR collaboration. Particles $\alpha$, $\beta$ and $c$ are from the TPC pseudorapidity coverage of $|\eta|<1$ with no $\eta$ gap applied. The $v_{2,c}\{2\}$ is obtained by two-particle cumulant with $\eta$ gap of $\Delta\eta > 1.0$. Statistical uncertainties are shown by the vertical bars and systematic uncertainties are shown by the caps[@Zhao:2017wck; @Zhao:2018pnk]. []{data-label="FG_SM4"}](./STAR_small_dg.pdf "fig:"){width="6.2cm"}
Fig. \[FG\_SM4\](left) shows the ${\gamma_{\rm SS}}$ and ${\gamma_{\rm OS}}$ results as functions of particle multiplicity (${N}$) in [p+A]{} and [d+A]{} collisions at ${\sqrt{s_{_{\rm NN}}}}=200$ GeV. Here ${N}$ is taken as the geometric mean of the multiplicities of particle $\alpha$ and $\beta$. The corresponding [Au+Au]{} results are also shown for comparison. The trends of the correlator magnitudes are similar, decreasing with increasing ${N}$. The ${\gamma_{\rm SS}}$ results seem to follow a smooth trend in ${N}$ over all systems. The ${\gamma_{\rm OS}}$ results are less so; the small system data appear to differ somewhat from the heavy-ion data over the range in which they overlap in ${N}$. Similar to LHC, the small system ${\Delta\gamma}$ results at RHIC are found to be comparable to Au+Au results at similar multiplicities (Fig. \[FG\_SM4\], right). While in the overlapping ${N}$ range between [p(d)+Au]{} and [Au+Au]{} collisions, the ${\Delta\gamma}$ data differ by $\sim$20-50%. This seems different from the LHC results where the [p+Pb]{} and [Pb+Pb]{} data are found to be highly consistent with each other in the overlapping ${N}$ range[@Khachatryan:2016got]. However, the CMS [p+Pb]{} data are from high multiplicity collisions, overlapping with [Pb+Pb]{} data in the 30-50% centrality range, whereas the RHIC [p(d)+Au]{} data are from minimum bias collisions, overlapping with [Au+Au]{} data only in peripheral centrality bins. Since the decreasing rate of ${\Delta\gamma}$ with ${N}$ is larger in [p(d)+Au]{} than in [Au+Au]{} collisions, the [p(d)+Au]{} data could be quantitatively consistent with the [Au+Au]{} data at large ${N}$ in the range of the 30-50% centrality. It is interesting to note that this is similar to the observed difference in the slope of the $N^{\rm offline}_{\rm trk}$ dependence in p+Pb and Pb+Pb by CMS [@Khachatryan:2016got] as mentioned previously. Considering these observations, the similarities in the RHIC and LHC data regarding the comparisons between small-system and heavy-ion collisions are astonishing.
Since the [p+A]{} and [d+A]{} data are all backgrounds, the ${\Delta\gamma}$ should be approximately proportional to the averaged $v_2$ of the background sources, and in turn, the $v_2$ of final-state particles. It should also be proportional to the number of background sources, and, because ${\Delta\gamma}$ is a pair-wise average, inversely proportional to the total number of pairs as the dilution effect. The number of background sources likely scales with multiplicity, so the ${\Delta\gamma}\propto v_2/{N}$. Therefore, to gain more insight, the ${\Delta\gamma}$ was scaled by ${N}/v_2$: $${{\Delta\gamma}_{\rm scaled}}={\Delta\gamma}\times{N}/v_2\,.
\label{eq:scale}$$ Fig. \[FG\_SM5\] shows the scaled ${{\Delta\gamma}_{\rm scaled}}$ as a function of ${N}$ in [p+A]{} and [d+A]{} collisions, and compares that to in [Au+Au]{} collisions. AMPT simulation results for d+Au and Au+Au are also plotted for comparison. The AMPT simulations can account for about $2/3$ of the STAR data, and are approximately constant over ${N}$. The ${{\Delta\gamma}_{\rm scaled}}$ in [p+A]{} and [d+A]{} collisions are compatible or even larger than that in [Au+Au]{} collisions. Since in [p+A]{} and [d+A]{} collisions only the background is present, the data suggest that the peripheral [Au+Au]{} measurement may be largely, if not entirely, background. For both small-system and heavy-ion collisions, the ${{\Delta\gamma}_{\rm scaled}}$ is approximately constant over ${N}$. It may not be strictly constant because the correlations caused by decays (${\Delta\gamma}_{\rm bkgd}\propto{\langle \cos(\alpha+\beta-2{\phi_{\rm res}})\rangle}\times{v_{2,{\rm res}}}$), depends on the ${\langle \cos(\alpha+\beta-2{\phi_{\rm res}})\rangle}$ which is determined by the parent kinematics and can be somewhat ${N}$-dependent. Given that the background is large, suggested by the [p+A]{} and [d+A]{} data, the approximate ${N}$-independent ${{\Delta\gamma}_{\rm scaled}}$ in [Au+Au]{} collisions is consistent with the background scenario.
![(Color online) The scaled three-particle correlator difference in p+Au and d+Au collisions as a function of ${N}$, compared to those in Au+Au collisions at [$\sqrt{s_{\rm NN}}$ ]{}= 200 GeV from the preliminary STAR data. AMPT simulation results for d+Au and Au+Au are also plotted for comparison[@Zhao:2017wck; @Zhao:2018pnk]. []{data-label="FG_SM5"}](./STAR_small_scaled.pdf){width="7.0cm"}
Due to the decorrelation of the ${\psi_{\rm PP}}$ and the magnetic field direction in small system [p(d)+A]{} collisions, the comparable ${\Delta\gamma}$ measurements (with respect to the ${\psi_{\rm PP}}$) in small system [p(d)+A]{} collisions and in A+A collisions at the same energy from LHC/RHIC suggests that there is significant background contribution in the ${\Delta\gamma}$ measurements in A+A collisions, where the ${\Delta\gamma}$ measurements (with respect to the ${\psi_{\rm PP}}$) in small system [p(d)+A]{} collisions are all backgrounds. While, by considering the fluctuating proton size, Monte Carlo Glauber model calculation shows that there could be significant correlation between the magnetic field direction and ${\psi_{\rm PP}}$ direction in high multiplicity p+A collisions, even though the magnitude of the correlation is still much smaller than in A+A collisions. Those calculations may indicate possibilities of studying the chiral magnetic effect in small systems[@Kharzeev:2017uym; @Zhao:2017rpf].
The decorrelation of the ${\psi_{\rm PP}}$ and the magnetic field direction in small system [p(d)+A]{} collisions provides not only a way to “turn off” the CME signal, but also a way to “turn off” the $v_{2}$-related background. The background contribution to the ${\Delta\gamma}$ measurement with respect to the magnetic field direction would average to zero due to this decorrelation effect in system [p(d)+A]{} collisions. So the key question is weather we could measure a direction that possibly accesses the magnetic field direction. The magnetic field is mainly generated by spectator protons and therefore experimentally best measured by the 1st-order harmonic plane ($\psi_{1}$) using the spectator neutrons.
![(Color online) The preliminary ${\Delta\gamma}$ correlator in p+Au collisions with respect to $\psi_{1}$ of spectator neutrons measured by the ZDC-SMD, compared to the ${\Delta\gamma}$ measured with respected to $\psi_{2}$ in [p(d)+Au]{} and Au+Au collisions at [$\sqrt{s_{\rm NN}}$ ]{}= 200 GeV from STAR[@Zhao:2017p1]. []{data-label="FG_SM6"}](./STAR_ZDCpA.pdf){width="7.0cm"}
Fig. \[FG\_SM6\] shows the preliminary ${\Delta\gamma}$ measurement in p+Au collisions with respect to $\psi_{1}$ of spectator neutrons measured by the shower maximum detectors of zero-degree calorimeters (ZDC-SMD) from STAR. The measurement is currently consistent with zero with large uncertainty[@Zhao:2017p1]. In the future with improved experimental precision, this could possibly provide an excellent way to search for CME in small systems.
Measurement with respect to reaction plane
==========================================
Again, one important point is that the CME-driven charge separation is along the magnetic field direction ($\psi_{B}$), different from the participant plane (${\psi_{\rm PP}}$). The major background to the CME is related to the elliptic flow anisotropy ($v_{2}$), determined by the participant geometry, therefore the largest with respect to the ${\psi_{\rm PP}}$. The $\psi_{B}$ and ${\psi_{\rm PP}}$ in general correlate with the ${\psi_{\rm RP}}$, the impact parameter direction, therefore correlate to each other. While the magnetic field is mainly produced by spectator protons, their positions fluctuate, thus $\psi_{B}$ is not always perpendicular to the ${\psi_{\rm RP}}$. The position fluctuations of participant nucleons and spectator protons are independent, thus ${\psi_{\rm PP}}$ and $\psi_{B}$ fluctuate independently about ${\psi_{\rm RP}}$. Fig. \[FG\_RP1\] depicts the display from a single Monte-Carlo Glauber event in mid-central Au+Au collision at 200 GeV.
![(Color online) Single-event display from a Monte-Carlo Glauber event of a mid-central Au+Au collision at 200 GeV. The gray markers indicate participating nucleons, and the red (green) markers indicate the spectator nucleons traveling in positive (negative) z direction The blue arrow indicates the magnetic field direction. The long axis of the participant eccentricity is shown as the black arrow. The magenta arrow shows the direction determined by spectator nucleons. []{data-label="FG_RP1"}](./AuAu_psiSP.pdf){width="7.0cm"}
The eccentricity of the transverse overlap geometry is by definition $\epsilon_{2}\{{\psi_{\rm PP}}\} \equiv {\langle \epsilon_{2}\{{\psi_{\rm PP}}\}_{evt}\rangle}$. The overlap geometry averaged over many events is an ellipse with its short axis being along the ${\psi_{\rm RP}}$; its eccentricity is ${\langle \epsilon_{2}\{{\psi_{\rm PP}}\}_{evt} \cos2({\psi_{\rm PP}}- {\psi_{\rm RP}})\rangle}$ and $$\begin{split}
& a^{PP}(\epsilon_{2}) \equiv \epsilon_{2}\{{\psi_{\rm RP}}\}/\epsilon_{2}\{{\psi_{\rm PP}}\} \approx a^{PP}, \\
& a^{PP} \equiv {\langle \cos2({\psi_{\rm PP}}- {\psi_{\rm RP}})\rangle}.
\end{split}
\label{EQ_RP1}$$ The magnetic field strength with respect to a direction $\psi$ is: $B_{sq}\{\psi\} \equiv {\langle (eB/m_{\pi}^{2})^{2}\cos2(\psi_{B}-\psi)\rangle}$. And $$\begin{split}
a^{PP}_{B_{sq}} \equiv B_{sq}\{{\psi_{\rm PP}}\}/B_{sq}\{{\psi_{\rm RP}}\} \approx a^{PP}.
\end{split}
\label{EQ_RP2}$$
The relative difference of the eccentricity ($\epsilon_{2}$) or magnetic field strength ($B_{sq}$) with respect to ${\psi_{\rm PP}}$ and ${\psi_{\rm RP}}$ are defined below:
$$\begin{split}
R^{PP}(X) & \equiv 2\cdot \frac{X\{{\psi_{\rm RP}}\}-X\{{\psi_{\rm PP}}\}}{X\{{\psi_{\rm RP}}\}-X\{{\psi_{\rm PP}}\}}, \ X = \epsilon_{2}, or B_{sq}, \\
R^{PP}(\epsilon_{2}) & \equiv -2(1-a^{PP}_{\epsilon_{2}})/(1+a^{PP}_{\epsilon_{2}}) \approx -R_{PP}, \\
R^{PP}(B_{sq}) & \equiv 2(1-a^{PP}_{B_{sq}})/(1+a^{PP}_{B_{sq}}) \approx R_{PP},
\end{split}
\label{EQ_RP3}$$
where $$\begin{split}
R^{PP} \equiv 2(1-a^{PP})/(1+a^{PP}).
\end{split}
\label{EQ_RP4}$$
The ${\psi_{\rm PP}}$ and $\epsilon_{2}$ are not experimentally measured. Usually the event plane (${\psi_{\rm EP}}$) reconstructed from final-state particles is used as a proxy for ${\psi_{\rm PP}}$. $v_{2}$ can be used as a proxy for $\epsilon_{2}$: $$\begin{split}
a^{EP}_{v_{2}} \equiv v_{2}\{{\psi_{\rm RP}}\}/v_{2}\{{\psi_{\rm EP}}\} \approx a^{EP}.
\end{split}
\label{EQ_RP5}$$ Although a theoretical concept, the ${\psi_{\rm RP}}$ may be assessed by Zero-Degree Calorimeters (ZDC) measuring spectator neutrons[@Reisdorf:1997flow; @Abelev:2013cva; @Adamczyk:2016eux]. Similar to Eq.\[EQ\_RP1\],\[EQ\_RP2\],\[EQ\_RP3\],\[EQ\_RP4\], these relations hold by replacing the ${\psi_{\rm PP}}$ with ${\psi_{\rm EP}}$. For example, $$\begin{split}
R^{EP} \equiv 2(1-a^{EP})/(1+a^{EP}).
\end{split}
\label{EQ_RP6}$$
![(Color online) Relative differences $R^{PP}(\epsilon_{2})$, $R^{PP}(B_{sq})$ from Monte Carlo Glauber model (upper panel) and $R^{EP}(v_{2})$, $R^{EP}(B_{sq})$ from AMPT (lower panel) for (a,f) Au+Au, (b,g) Cu+Cu, (c,h) Ru+Ru, and (d,i) Zr+Zr at RHIC, and (e,j) Pb+Pb at the LHC. Both the Woods-Saxon and EDF-calculated[@Xu:2017zcn] densities are shown for the Monte Carlo Glauber calculations, while the used density profiles are noted for the AMPT results[@Xu:2017qfs]. []{data-label="FG_RP2"}](./RRPvsPP.pdf){width="12.5cm"}
Figure \[FG\_RP2\](upper panel) shows $R^{PP}(\epsilon_{2})$ and $R^{PP}(B_{sq})$ calculated by a Monte Carlo Glauber model[@Xu:2014ada; @Zhu:2016puf] for Au+Au, Cu+Cu, Ru+Ru, Zr+Zr collisions at RHIC and Pb+Pb collisions at the LHC. The results are compared to the corresponding $\pm R^{PP}$. These numbers agree with each other, indicating good approximations used in Eq. \[EQ\_RP1\],\[EQ\_RP2\]. Fig. \[FG\_RP2\](lower panel) shows $R^{EP}(v_{2})$ and $R^{EP}(B_{sq})$ calculated from AMPT simulation[@Lin:2004en; @Lin:2001zk]. Again, good agreements are found between $R^{EP}(v_{2})$, $R^{EP}(B_{sq})$ and $\pm R^{EP}$. Both show the opposite behavior of $R^{PP(EP)}(\epsilon_{2}(v_{2}))$ and $R^{PP(EP)}(B_{sq})$, which approximately equal to $\pm R_{PP(EP)}$.
![(Color online) (Left) $\gamma$ measured with $\psi_{1}$ and $\psi_{2}$ vs. centrality in Au+Au collisions at [$\sqrt{s_{\rm NN}}$ ]{}= 200 GeV from STAR[@Adamczyk:2013hsi; @Abelev:2009ac; @Abelev:2009ad]. The Y4 and Y7 represent the results from the 2004 and 2007 RHIC run. Shaded areas for the results measured with $\psi_{2}$ represent the systematic uncertainty of the event plane determination. Systematic uncertainties for the results with respect to $\psi_{1}$ are negligible compared to the statistical ones shown. (Right) The extracted fraction of CME contribution[@Xu:2017qfs] in the ${\Delta\gamma}\{\psi_{2}\}$[@Abelev:2009ac; @Abelev:2009ad; @Adamczyk:2013hsi] measurement in the 20-60% centrality Au+Au collisions vs. “true” $v_{2}$; the gray area indicates the $\pm1\sigma$ statistical uncertainty, dominated by that in ${\Delta\gamma}\{\psi_{1}\}$[@Adamczyk:2013hsi]. The dashed curves would be the new $\pm1\sigma$ uncertainty with ten-fold increase in statistics. []{data-label="FG_RP3"}](./STAR_ZDC.pdf "fig:"){width="6.7cm"} ![(Color online) (Left) $\gamma$ measured with $\psi_{1}$ and $\psi_{2}$ vs. centrality in Au+Au collisions at [$\sqrt{s_{\rm NN}}$ ]{}= 200 GeV from STAR[@Adamczyk:2013hsi; @Abelev:2009ac; @Abelev:2009ad]. The Y4 and Y7 represent the results from the 2004 and 2007 RHIC run. Shaded areas for the results measured with $\psi_{2}$ represent the systematic uncertainty of the event plane determination. Systematic uncertainties for the results with respect to $\psi_{1}$ are negligible compared to the statistical ones shown. (Right) The extracted fraction of CME contribution[@Xu:2017qfs] in the ${\Delta\gamma}\{\psi_{2}\}$[@Abelev:2009ac; @Abelev:2009ad; @Adamczyk:2013hsi] measurement in the 20-60% centrality Au+Au collisions vs. “true” $v_{2}$; the gray area indicates the $\pm1\sigma$ statistical uncertainty, dominated by that in ${\Delta\gamma}\{\psi_{1}\}$[@Adamczyk:2013hsi]. The dashed curves would be the new $\pm1\sigma$ uncertainty with ten-fold increase in statistics. []{data-label="FG_RP3"}](./RPvsPP.pdf "fig:"){width="5.7cm"}
The ${\Delta\gamma}$ variable contains CME signal and the $v_{2}$-induced background: $$\begin{split}
{\Delta\gamma}\{\psi\} = \rm{CME}(B_{sq}\{\psi\}) + \rm{BKG}(v_{2}\{\psi\}).
\end{split}
\label{EQ_RP7}$$ By using the ZDC measured 1st order event plane $\psi_{1}$ as a estimation of the ${\psi_{\rm RP}}$, and 2nd order event plane $\psi_{2}$ reconstructed from final-state particles as a proxy of the ${\psi_{\rm EP}}$, we can measure the ${\Delta\gamma}\{{\psi_{\rm RP}}\}$ and ${\Delta\gamma}\{{\psi_{\rm PP}}\}$. Assuming the $\rm{CME}(B_{sq}\{\psi\})$ are expected to be proportional to $B_{sq}$ and $\rm{BKG}(v_{2}\{\psi\})$ proportional to $v_{2}$, we have:
$$\begin{split}
R_{EP}({\Delta\gamma}) = 2\frac{r(1-a^{EP}_{B_{sq}}) -(1-a^{EP}_{v_{2}})}{r(1+a^{EP}_{B_{sq}}) +(1+a^{EP}_{v_{2}})} \approx \frac{1-r}{1+r}R^{EP}(v_{2}).
\end{split}
\label{EQ_RP8}$$
Here $r \equiv \rm{CME}(B_{sq}\{{\psi_{\rm RP}}\})/\rm{BKG}(v_{2}\{{\psi_{\rm EP}}\})$ can be considered as the relative CME signal to background contribution, $$\begin{split}
r = \frac{1+a^{EP}_{v_{2}}}{1+a^{EP}_{B_{sq}}} \frac{R^{EP}({\Delta\gamma})-R^{EP}(v_{2})}{R^{EP}(B_{sq})-R^{EP}({\Delta\gamma})} \approx \frac{R^{EP}(v_{2})-R^{EP}({\Delta\gamma})}{R^{EP}(v_{2})+R^{EP}({\Delta\gamma})}.
\end{split}
\label{EQ_RP9}$$ With respect to ${\psi_{\rm RP}}$ and ${\psi_{\rm EP}}$, the CME signal fractions are, respectively, $$\begin{split}
&f_{RP}(\rm{CME}) = \rm{CME}(B_{sq}\{{\psi_{\rm RP}}\})/{\Delta\gamma}\{{\psi_{\rm RP}}\}=r/(r+\it{a}^{EP}_{v_{2}}), \\
&f_{EP}(\rm{CME}) = \rm{CME}(B_{sq}\{{\psi_{\rm EP}}\})/{\Delta\gamma}\{{\psi_{\rm EP}}\}=r/(r+1/\it{a}^{EP}_{B_{sq}}).
\end{split}
\label{EQ_RP10}$$
Experimentally, $R^{EP}(v_{2})$ can be estimated by $v_{2}$ measurements with respect to ZDC $\psi_{1}$ and second order event plane $\psi_{2}$ (such as the forward time projection chamber, FTPC). $R^{EP}(B_{sq})$ cannot but may be approximated by $−R^{EP}(v_{2})$, as demonstrated by the Monte Carlo Glauber calculations and AMPT (Fig. \[FG\_RP2\]). Fig. \[FG\_RP3\](left) shows the STAR measured ${\Delta\gamma}$ with respect to $\psi_{1}$ by ZDC and $\psi_{2}$ TPC[@Adamczyk:2013hsi; @Abelev:2009ac; @Abelev:2009ad]. Their relative difference can be used as a experimental estimation of the $R^{EP}({\Delta\gamma}) = 2({\Delta\gamma}\{\psi_{1}\} - {\Delta\gamma}\{\psi_{2}\})/({\Delta\gamma}\{\psi_{1}\} - {\Delta\gamma}\{\psi_{2}\} )$. By assuming $a^{EP}_{v_{2}} = a^{EP}_{B_{sq}}$ and $R^{EP}(B_{sq}) = −R^{EP}(v_{2})$, the extracted fraction $f^{EP}_{CME}$ by Eq. \[EQ\_RP9\],\[EQ\_RP10\] as a function of “ture” $v_{2}$ is shown in Fig. \[FG\_RP3\](right) by the thick curve as a function of the “true” $v_{2}$. The gray area is the uncertainty. The vertical lines indicate the various measured $v_{2}$ values. At present the data precision does not allow a meaningful constraint on $f_{CME}$; the limitation comes from the ${\Delta\gamma}\{\psi_{1}\}$ measurement which has an order of magnitude larger statistical error than that of ${\Delta\gamma}\{\psi_{2}\}$. With tenfold increase in statistics, the constraint would be the dashed curves. This is clearly where the future experimental emphasis should be placed: larger Au+Au data samples are being analyzed and more Au+Au statistics are to be accumulated; ZDC upgrade is ongoing in the CMS experiment at the LHC; fixed target experiments at the SPS may be another viable venue where all spectator nucleons are measured in the ZDC allowing possibly a better determination of $\psi_{1}$[@Xu:2017qfs].
Invariant mass method
=====================
It has been known since the very beginning that the ${\Delta\gamma}$ could be contaminated by background from resonance decays coupled with the elliptic flow ($v_{2}$)[@Voloshin:2004vk]. Only recently, a toy-model simulation estimate was carried out which indicates that the resonance decay background can indeed largely account for the experimental measured ${\Delta\gamma}$[@Wang:2016iov], contradictory to early claims [@Voloshin:2004vk]. The pair invariant mass would be the first thing to examine in terms of resonance background, however, the invariant mass ($m_{inv}$) dependence of the ${\Delta\gamma}$ has not been studied until recently[@Zhao:2017nfq]. The invariant mass method of the ${\Delta\gamma}$ measurements provides the ability to identify and remove resonance decay background, enhancing the sensitivity of the measured CME signal.
CME-driven charge separation refers to the opposite-sign charge moving in opposite directions along the magnetic field (${\vec{B}}$). Because of resonance elliptic anisotropy (${v_{2,{\rm res}}}$), more OS pairs align in the ${\psi_{\rm RP}}$ than ${\vec{B}}$ direction, and it is an anti-charge separation along ${\psi_{\rm RP}}$. This would mimic the same effect as the CME on the ${\Delta\gamma}$ variable[@Voloshin:2004vk; @Abelev:2009ac; @Abelev:2009ad]. In term of the ${\Delta\gamma}$ variable, these backgrounds can be expressed by:
$$\begin{aligned}
{\Delta\gamma}&\propto&{\langle \cos(\alpha+\beta-2{\phi_{\rm res}})\cos2({\phi_{\rm res}}-{\psi_{\rm RP}})\rangle}\nonumber\\
&\approx&{\langle \cos(\alpha+\beta-2{\phi_{\rm res}})\rangle}{v_{2,{\rm res}}}\,.
\label{EQ_IM1}\end{aligned}$$
where ${\langle \cos(\alpha+\beta-2{\phi_{\rm res}})\rangle}$ is the angular correlation from the resonance decay, ${v_{2,{\rm res}}}$ is the $v_{2}$ of the resonance. The factorization of ${\langle \cos(\alpha+\beta-2{\phi_{\rm res}})\rangle}$ with ${v_{2,{\rm res}}}$ is only approximate, because both ${\langle \cos(\alpha+\beta-2{\phi_{\rm res}})\rangle}$ and ${v_{2,{\rm res}}}$ depend on ${p_{T}}$ of the resonance.
Many resonances have broad mass distributions[@Agashe:2014kda]. Experimentally, they are hard to identify individually in relativistic heavy-ion collisions. Statistical identification of resonances does not help eliminate their contribution to the ${\Delta\gamma}$ variable. However, most of the $\pi$-$\pi$ resonances contributions are dominated at low invariant mass region (Fig. \[FG\_IM1\], left)[@Adams:2003cc], It is possible to exclude them entirely by applying a lower cut on the invariant mass, for example $m_{inv}>2.0$ [GeV/$c^2$ ]{}. Results from AMPT model show that with such a $m_{inv}$ cut, although significantly reducing the statistics, can eliminate essentially all resonance decay backgrounds[@Zhao:2017nfq]. The preliminary experimental data from STAR show similar results as AMPT. Fig. \[FG\_IM1\](right) shows the results with and without such an invariant mass cut. By applying the mass cut, the ${\Delta\gamma}$ is consistent with zero with current uncertainty in Au+Au collisions at 200 GeV[@Zhao:2017wck]. The results are summarized in Table \[TB\_IM1\].
![ (Left panel) The raw $\pi$-$\pi$ invariant mass ($m_{inv}$) distributions after subtraction of the like-sign reference distribution for minimum bias p+p (top) and peripheral Au+Au (bottom) collisions. The insert plot corresponds to the raw $\pi^{+}$-$\pi^{-}$ $m_{inv}$ (solid line) and the like-sign reference distributions (open circles) for peripheral Au+Au collisions[@Adams:2003cc]. (Right panel) The inclusive ${\Delta\gamma}$ over all mass (black) and at $m_{inv} > 1.5$ [GeV/$c^2$ ]{}(green) as a function of centrality in Au+Au collisions at 200 GeV[@Zhao:2017wck]. []{data-label="FG_IM1"}](./STAR_mass0.pdf){width="13cm"}
While CME is generally expected to be a low $\pT$ phenomenon[@Kharzeev:2007jp; @Abelev:2009ad]; its contribution to high mass may be small. In order to extract CME at low mass, resonance contributions need to be subtracted. The invariant mass ${\Delta\gamma}$ measurement provides such a tool that could possibly isolate the CME from the resonance background, by taking advantage of their different dependences on $m_{inv}$.
For example, the $\rho$ decay background contribution to the ${\Delta\gamma}$ is: $$\begin{aligned}
({\Delta\gamma})_{\rho}& = \frac{N_{\rho}}{(N_{\pi^{+}+\pi^{-}})} \gamma_{\rho} = r_{\rho} \gamma_{\rho}
\label{EQ_IM2}\end{aligned}$$ where $r_{\rho}$ is the relative abundance of $\rho$ decay pairs over all OS pairs, and $\gamma_{\rho} \equiv {\langle \cos(\alpha+\beta-2{\phi_{\rm res}}){v_{2,{\rm res}}}\rangle}$ quantifies the $\rho$ decay angular correlations coupled with its $v_{2}$. Consider the event to be composed of primordial pions containing CME signals (CME) and common (charge-independent) background, such as momentum conservation ($\gamma_{m.c.}$)[@Bzdak:2010fd; @Pratt:2010zn], and the resonance ($\rho$ for instance) decay pions containing correlations from the decay[@Voloshin:2004vk; @Schlichting:2010qia; @Wang:2016iov]. The ${m_{inv}}$ dependency of the ${\Delta\gamma}$ can be expressed as: $$\begin{split}
{\Delta\gamma}({m_{inv}}) & = \frac{N_{SS}(\gamma_{CME}+\gamma_{m.c.}+N_{\rho}\gamma_{\rho})}{N_{SS}+N_{\rho}} - (-\gamma_{CME} + \gamma_{m.c.}) \\
& = r({m_{inv}})(\gamma_{\rho} - \gamma_{m.c.}) + (1-r({m_{inv}})/2){\Delta\gamma}_{CME} \\
& \approx r({m_{inv}})R({m_{inv}}) + {\Delta\gamma}_{CME}({m_{inv}}).
\end{split}
\label{EQ_IM2}$$ The first term is resonance contributions, where the response function $R({m_{inv}}) = \gamma_{\rho} - \gamma_{m.c.} = {\langle f({m_{inv}})v_{2}({m_{inv}})\rangle} - \gamma_{m.c.}$ is likely a smooth function of ${m_{inv}}$, while $r({m_{inv}})$ contains resonance spectral profile. Consequently, the first term is not smooth but a peaked function of ${m_{inv}}$. The second term in Eq. \[EQ\_IM2\] is the CME signal which should be a smooth function of ${m_{inv}}$ (here the negligible $r/2$ term was dropped). However, the exact functional form of CME(${m_{inv}}$) is presently unknown and needs theoretical input. The different dependences of the two terms can be exploited to identify CME signals at low ${m_{inv}}$. The possibility of the this method was studied by a toy-MC simulation along with the AMPT models[@Zhao:2017nfq].
Figure \[FG\_IM2\] shows the preliminary results in mid-central Au+Au collisions from STAR experiments[@Zhao:2017wck]. Fig. \[FG\_IM2\](top) shows the relative OS and SS pair difference ($r=(N_{OS.}-N_{SS.})/N_{OS.}$) as a function of invariant mass. Fig. \[FG\_IM2\](middle) shows the ${\Delta\gamma}$ correlator as function of $\pi$-$\pi$ invariant mass. The data shows resonance structure in ${\Delta\gamma}$ as function of mass; a clear resonance peak from $K_{s}^{0}$ decay are observed, and possible $\rho$ and $f^{0}$ peaks are also visible. The ${\Delta\gamma}$ correlator traces the distribution of those resonances. ${\Delta\gamma}$ decreases as $r$ decreases with increasing mass, In a two components model of resonances background plus CME signal. The ${\Delta\gamma}(m)=r\times(a+b\times m)$+f(CME), where f(CME) represents the CME contribution. The background contribution will follow the distribution of $r$, while the f(CME) is most likely a smooth distribution in $m_{inv}$. Fig. \[FG\_IM2\](bottom) shows the ratio of the ${\Delta\gamma}$/$r$ as function of mass. No evidence of inverse shape of the resonance mass distribution is in the ratio of ${\Delta\gamma}$/$r$, suggesting insignificant CME signal contributions.
![(Color online) Pair invariant mass ($m_{inv}$) dependence of the relative excess of OS over SS charged $\pi$ pair multiplicity, $r=(N_{OS}-N_{SS})/N_{OS}$ (top panel), event plane dependent azimuthal correlator difference, ${\Delta\gamma}={\gamma_{\rm OS}}-{\gamma_{\rm SS}}$ (middle panel), and the ratio of ${\Delta\gamma}/r$ (bottom panel) in 20-50% Au+Au collisions at 200 GeV. Errors shown are statistical. The red curve in the middle panel shows the two-component model fit assuming a constant CME contribution independent of $m_{inv}$; The blue curve in the bottom panel shows the corresponding resonance response function[@Zhao:2017wck]. []{data-label="FG_IM2"}](./STAR_mass.pdf){width="7.0cm"}
In order to isolate the possible CME from the resonances contributions, the two components model is used to fit the ${\Delta\gamma}$ as function of invariant mass (Fig. \[FG\_IM2\] (middle)). Currently, there is no available theoretical calculation on the mass dependence of the CME contribution, therefore two functional forms are considered: ([i]{}) a constant CME distribution independent of mass, and ([ii]{}) a exponential CME distribution as function of mass. The extracted ${\Delta\gamma}$ from CME contribution is $(5.9\pm9.0)\times10^{-6}$ from the constant CME fit, and $(3.0\pm2.0)\times10^{-5}$ from the exponential CME fit, which correspond to $(3.2\pm4.9)$% (constant CME) and $(16\pm11)$% (exponential CME) of the inclusive ${\Delta\gamma}$ ($(1.82\pm0.03)\times10^{-4}$) measurement. The results are also summarized in Table \[TB\_IM2\]. Future theoretical calculations of the CME mass dependence would help to understand the results more precisely.
Invariant mass method provides for the first time a useful tool to identify the background sources for the CME ${\Delta\gamma}$ measurements, and provides a possible way to isolate the CME signal from the backgrounds. There are still debates weather the CME should be a low $\pT$/${m_{inv}}$ phenomenon, and their ${m_{inv}}$ dependence is also not clear currently. Recent study[@Shi:2017cpu] indicates that the CME signal is rather independent of $\pT$ at $\pT>0.2$ [GeV/$c$ ]{}, suggesting that the signal may persist to high ${m_{inv}}$. Nevertheless, a lower ${m_{inv}}$ cut will eliminate resonance contributions to ${\Delta\gamma}$, and a measured positive ${\Delta\gamma}$(${m_{inv}}$) signal would point to the possible existence of the CME at high ${m_{inv}}$. A null measurement at high ${m_{inv}}$, however, does not necessarily mean no CME also at low ${m_{inv}}$. Further theoretical calculation on the CME ${m_{inv}}$ dependence could help to extract the CME signal more precisely. On the other side, using ESE method to select events with different $v_{2}$ might be able to help to extract the background ${m_{inv}}$ distributions by comparing their ${m_{inv}}$ dependences of the ${\Delta\gamma}$ distributions. In the upcoming isobar run at RHIC, it is also worthwhile to compare the ${\Delta\gamma}$(${m_{inv}}$) dependences between the two systems, which could help to understand where the possible CME ${\Delta\gamma}$ signal comes from, for example the resonance abundance difference due to isospin difference between Zr and Ru or other effects[@Wang:2016iov]. Further more it could also help to locate ${m_{inv}}$ position of the possible CME ${\Delta\gamma}$ signal and possibly provide the only way to study the ${m_{inv}}$ property of the sphaleron or instanton mechanism for transitions between QCD vacuum states.
$R(\Delta S)$ correlator
========================
Recently a new observable, ${R_{\psi_{m}}(\Delta{S})}$ (m=2, 3, refer to $\psi_{2}, \psi_{3}$), has been proposed to measure the CME-driven charge separation in heavy-ion collisions[@Magdy:2017yje; @Ajitanand:2010rc]. $$\begin{split}
&\Delta S = \langle S_{p}^{h+} \rangle - \langle S_{n}^{h-} \rangle, \\
&\langle S_{p}^{h+} \rangle = \frac{\displaystyle\sum_{1}^{N_{p}}\sin( \frac{m}{2} \Delta\phi_{m})}{N_{p}}, \langle S_{n}^{h-} \rangle = \frac{\displaystyle\sum_{1}^{N_{n}}\rm{sin}( \frac{m}{2} \Delta\phi_{m})}{N_{n}}, \\
&\Delta\phi_{m} = \phi - \psi_{m}, m=2,3,
\end{split}$$ where $\phi$ is the azimuthal angle of the positively (p) or negatively (n) charged hadrons. $\Delta S$ quantifies the charge separation along a certain direction. The correlation functions ${C_{\psi_{m}}(\Delta{S})}$ were constructed from the ratio of the $N_{real}(\Delta S)$ distribution to the charge-shuffled $N_{shuffled}(\Delta S)$ distribution. $$\begin{split}
{C_{\psi_{m}}(\Delta{S})} = \frac{N_{real}(\Delta S)}{N_{\rm{shuffled}}(\Delta S)}.
\end{split}$$ The $N_{\rm{shuffled}}(\Delta S)$ distribution was obtained by randomly shuffling the charges of the positively and negatively charged particles in each event. By replacing the $\psi_{m}$ with $\psi_{m} + \pi/m$, the same procedures were carried out to obtain the ${C_{\psi_{m}}^{\perp}(\Delta{S})}$. The $\pi/m$ rotation of the event planes, guarantees that a possible CME-driven charge separation does not contribute to ${C_{\psi_{m}}^{\perp}(\Delta{S})}$. In the end, the ${R_{\psi_{m}}(\Delta{S})}$ correlator was obtained by taken the ratio between ${C_{\psi_{m}}(\Delta{S})}$ and ${C_{\psi_{m}}^{\perp}(\Delta{S})}$: $$\begin{split}
{R_{\psi_{m}}(\Delta{S})} = {C_{\psi_{m}}(\Delta{S})} / {C_{\psi_{m}}^{\perp}(\Delta{S})}.
\end{split}$$ The ${C_{\psi_{m}}(\Delta{S})}$ measures the combined effects of CME-driven charge separation and the background, and the ${C_{\psi_{m}}^{\perp}(\Delta{S})}$ provides the reference for the background. The ratio between the ${C_{\psi_{m}}(\Delta{S})}$ and ${C_{\psi_{m}}^{\perp}(\Delta{S})}$ are designed to detect the CME-driven charged separation.
The CME-driven charge separation is along the magnetic field direction, which is perpendicular to the ${\psi_{\rm RP}}$. By using the $\psi_{2}$ as a proxy of the ${\psi_{\rm RP}}$, the ${R_{\psi_{2}}(\Delta{S})}$ are designed to provide the sensitivity to detect the CME-driven charged separation. Since there is little, if any, correlation between ${\psi_{\rm RP}}$ and $\psi_{3}$, the ${R_{\psi_{3}}(\Delta{S})}$ measurements are insensitive to CME-driven charge separation, but still sensitive to background[@Magdy:2017yje].
Figure \[FG\_RS1\] shows the initial studies with A Multi-Phase Transport (AMPT) and Anomalous Viscous Fluid Dynamics (AVFD) models[@Magdy:2017yje]. The AMPT[@Zhang:1999bd; @Lin:2004en] has been quite successful in describing the experimentally measured data (particle yields, flow) in heavy ion collisions. Therefore it provides a good reference for the background response of the ${R_{\psi_{m}}(\Delta{S})}$ correlator, especially the resonance decay and the flow related background. In additional to the background, the AVFD model[@Shi:2017cpu] could include the evolution of chiral fermion currents in the hot dense medium during the bulk hydrodynamic evolution. which can be used to study the ${R_{\psi_{m}}(\Delta{S})}$ response to the CME-driven charge separation. Both the AMPT and AVFD shows the convex shapes of ${R_{\psi_{2}}(\Delta{S})}, {R_{\psi_{3}}(\Delta{S})}$ for typical resonance backgrounds (Fig. \[FG\_RS1\] panel (a,c)). With implementing anomalous transport from the CME, the AVFD model simulation shows a concave ${R_{\psi_{2}}(\Delta{S})}$ distribution (Fig. \[FG\_RS1\] panel (b)), which is consistent with the expectation of the ${R_{\psi_{2}}(\Delta{S})}$ correlator response to the CME-driven charge separation. Preliminary experimental data from STAR, reveal concave ${R_{\psi_{2}}(\Delta{S})}$ distributions in 200 GeV Au+Au collisions[@Roy:2017rs].
![(Color online) Comparison of the ${R_{\psi_{2}}(\Delta{S})}$ correlators for (a) background-driven charge separation ($a_{1}$ = 0) in 30-40% Au+Au collisions ([$\sqrt{s_{\rm NN}}$ ]{}= 200 GeV) obtained with the AMPT and AVFD models, and (b) the combined effects of background- and CME-driven ($a_{1} = 1.0\%)$ charge separation in Au+Au collisions obtained with the AVFD model at the same centrality and beam energy. (c) Comparison of the ${R_{\psi_{3}}(\Delta{S})}$ correlators for background-driven charge separation ($a_{1}$ = 0) in 30-50% Au+Au collisions ([$\sqrt{s_{\rm NN}}$ ]{}= 200 GeV) obtained with the AMPT model[@Magdy:2017yje]. []{data-label="FG_RS1"}](./newRs.pdf){width="12.5cm"}
A 3+1-dimensional hydrodynamic study[@Bozek:2017plp], however, indicates concave ${R_{\psi_{2}}(\Delta{S})}$ shapes for backgrounds as well, it also shows a concave shapes of ${R_{\psi_{3}}(\Delta{S})}$ distribution for the background, which is different from the expectation of convex shape of background.
To better understand those results from different models, hence gain more information from the experimental data, a more detailed and systematic study of ${R_{\psi_{m}}(\Delta{S})}$ correlator responses to the background seems important. For example, the ${R_{\psi_{3}}(\Delta{S})}$ response in AVFD model with and without CME-driven charge separation. And the resonance $v_{n}$, and $p_{T}$ dependences of the ${R_{\psi_{m}}(\Delta{S})}$ behavior[@Feng:2018chm]. The resonance $v_{2}$ introduces different numbers of decay $\pi^+\pi^-$ pairs in the in-plane and out-of-plane directions. The resonance $p_{T}$ affects the opening angle of the decay $\pi^+\pi^-$ pair. Low $p_{T}$ resonances decay into large opening-angle pairs, and result in more “back-to-back” pairs out-of-plane because of the more in-plane resonances, mimicking a CME charge separation signal perpendicular to the reaction plane, or a concave ${R_{\psi_{2}}(\Delta{S})}$. High $p_{T}$ resonances, on the other hand, decay into small opening-angle pairs, and result in a background behavior of convex ${R_{\psi_{2}}(\Delta{S})}$.
Other than the ${\Delta\gamma}$ correlator, it is worth developing new methods and/or observables to search for the CME, such as the ${R_{\psi_{m}}(\Delta{S})}$ correlator. Currently more detailed investigations are needed to understand how the ${R_{\psi_{m}}(\Delta{S})}$ correlator is compared with other correlators, what the advantage and disadvantage of the ${R_{\psi_{m}}(\Delta{S})}$ correlator is, and possibly what the connections are between these correlators. More detailed studies could help gain a better understanding of the experimental results and more clear interpretation in term of CME, and future study of the RHIC isobaric data [@Magdy:2018lwk; @Sun:2018idn].
summary
=======
The non-trivial topological structures of the QCD have wide ranging implications. Relativistic heavy-ion collisions provide an ideal environment to study the novel phenomena induced by those topological structures, such as the chiral magnetic effect (CME). Since the first $\gamma$ measurements in 2009, experimental results have been abundant in relativistic heavy-ion as well as small system collisions. In this review, several selected recent progresses on the experimental search for the CME in relativistic heavy-ion collisions are summarized. Major conclusions are as follows:
**Event shape selection**: Using the event shape selection, by varying the event-by-event $v_{2}$, exploiting statistical (event-by-event $v_{2}, q_{2}$ methods) and dynamical fluctuations (ESE method), experimental results suggest that the ${\Delta\gamma}$ correlator is strongly dependent on the $v_{2}$. The $v_{2}$ independent contribution are estimated by different methods from STAR, ALICE and CMS collaboration; results indicate that a large contribution of the ${\Delta\gamma}$ correlator is from the $v_{2}$ related background.
**Isobaric collisions and Uranium+Uranium collisions**: By taking advantage of the nuclear property (such as proton number, shape), isobaric collisions of ${^{96}_{40}\rm{Zr} + ^{96}_{40}\rm{Zr}}$, ${^{96}_{44}\rm{Ru} + ^{96}_{44}\rm{Ru}}$ collisions and Uranium+Uranium collisions have been proposed. So far there is no clear conclusion in term of the disentangle of the CME and $v_{2}$ related background from the preliminary experimental Uranium+Uranium results yet. Theoretical calculations suggest that the upcoming isobaric collisions at RHIC in 2018 will provide a powerful tool to disentangle the CME signal from the $v_{2}$ related backgrounds. While there could be non-negligible deviations of the ${^{96}_{44}\rm{Ru}}$ and ${^{96}_{40}\rm{Zr}}$ nuclear densities from Woods-Saxon which could introduce extra uncertainty.
**Small system collisions**: The recent ${\Delta\gamma}$ measurements in small system p+Pb collisions from CMS have triggered a wave of discussions about the interpretation of the CME in heavy-ion collisions. Preliminary results from STAR also show comparable ${\Delta\gamma}$ in small system [p(d)+Au]{} collisions with that in [Au+Au]{} collisions. These results indicate significant background contributions in the ${\Delta\gamma}$ measurements in heavy-ion collisions. On other hand, theoretical calculation shows a possibility that CME may contribute to the ${\Delta\gamma}$ in p+Pb collisions with respect to $\psi_{2}$. The ${\Delta\gamma}$ measurements in small system [p(d)+A]{} collisions with respect to $\psi_{1}$ using the spectator neutrons are worth to follow in the future.
**Measurement with respect to the reaction plane**: New idea of differential measurements with respect to the reaction plane (${\psi_{\rm RP}}$) and participant plane (${\psi_{\rm PP}}$) are proposed, where the ${\psi_{\rm RP}}$ could possibly be assessed by spectator neutrons measured by the zero-degree calorimeters (ZDC). The $v_{2}$ is stronger along ${\psi_{\rm PP}}$ and weaker along ${\psi_{\rm RP}}$; in contrast, the magnetic field, being from spectator protons, is weaker along ${\psi_{\rm PP}}$ and stronger along ${\psi_{\rm RP}}$. The ${\Delta\gamma}$ measured with respect to ${\psi_{\rm RP}}$ and ${\psi_{\rm PP}}$ contain different amounts of CME and background, and can thus determine these two contributions.
**Invariant mass method**: New method exploiting the invariant mass dependence of the ${\Delta\gamma}$ measurements provides a useful tool to identify the background sources, and provides a possible way to isolate the CME signal from the backgrounds. Preliminary results from STAR show that by applying a mass cut to remove the resonance background, the ${\Delta\gamma}$ is consistent with zero with current uncertainty in Au+Au collisions. In the low mass region, resonance peaks are observed in ${\Delta\gamma}$ as a function of ${m_{inv}}$. By assuming smooth CME ${m_{inv}}$ distribution, it’s possible to extract the CME signal. While there are debates wheather the CME should be a low $\pT$/${m_{inv}}$ phenomenon, their ${m_{inv}}$ dependence is also not clear currently. In the upcoming isobar run at RHIC, the comparison of the ${\Delta\gamma}$(${m_{inv}}$) dependences between the two systems would help to further our understanding. and will provide a possible way to study the ${m_{inv}}$ property of the sphaleron or instanton mechanism for transitions between QCD vacuum states.
**$R(\Delta S)$ correlator**: New $R(\Delta S)$ correlator has been proposed to measure the CME-driven charge separation. Preliminary experimental results indicate a CME dominated scenario. To gain better understanding of the experimental results and more clear implications in term of its CME interpretation, more detailed investigations are needed, such as, the resonance $v_{n}$, and $p_{T}$ dependences of the $R(\Delta S)$ behavior.
While the physics behind CME is of paramount importance, the present experimental evidences for the existence of the CME are rather ambiguous. Most of the results indicate that there are significant background contributions in the ${\Delta\gamma}$ measurements, the CME signal might be small fraction, while there is no doubt that the unremitting pursuit is encouraging and will be rewarded. Toward the discovery of the CME, new ideas, new methods, new technologies are called for. The author is hopeful that this day will come soon.
Acknowledgments {#acknowledgments .unnumbered}
===============
I greatly thank Prof. Fuqiang Wang, Prof. Wei Xie and other members of the Purdue High Energy Nuclear Physics Group for discussions and comments. This work was supported by the U.S. Department of Energy (Grant No. de-sc0012910).
|
1 truecm ****
Path-integral over non-linearly realized groups
and Hierarchy solutions
1.3 truecm
D. Bettinelli[^1], R. Ferrari[^2], A. Quadri[^3]
Dip. di Fisica, Università degli Studi di Milano\
and INFN, Sez. di Milano\
via Celoria 16, I-20133 Milano, Italy
0.8 truecm ****
Abstract
0.5 truecm
> The technical problem of deriving the full Green functions of the elementary pion fields of the nonlinear sigma model in terms of ancestor amplitudes involving only the flat connection and the nonlinear sigma model constraint is a very complex task. In this paper we solve this problem by integrating, order by order in the perturbative loop expansion, the local functional equation derived from the invariance of the SU(2) Haar measure under local left multiplication. This yields the perturbative definition of the path-integral over the non-linearly realized SU(2) group.
Introduction
============
The perturbative quantization of the nonlinear sigma model in $D=4$ requires a strategy for the definition of the path-integral over the Haar measure of non-linearly realized groups.
It has been recently pointed out [@Ferrari:2005ii]- [@Ferrari:2005fc] that such a definition can be implemented through the local functional equation which expresses the invariance of the Haar measure under local left group multiplication. The subtraction procedure is required to be symmetric, thus preserving the validity of the local functional equation to all orders in the loop expansion [@Ferrari:2005fc]. The local functional equation fixes the Green functions of the quantized pion fields parameterizing the SU(2) group element (over which the path-integral is performed) in terms of those of the SU(2) flat connection and the order parameter (ancestor composite operators). This goes under the name of hierarchy principle [@Ferrari:2005ii]. Moreover there is only a finite number of divergent ancestor amplitudes at every loop order (weak power-counting theorem [@Ferrari:2005va],[@Ferrari:2005fc]).
The local solutions of the linearized functional equation (relevant for the classification of the allowed finite renormalizations order by order in the loop expansion) were obtained in [@Ferrari:2005va]. In the one-loop approximation these results have been shown [@Ferrari:2005va] to reproduce those of Ref. [@Gasser:1983yg].
In this paper we show how to explicitly solve the local functional equation by reconstructing the full Green functions of the quantized fields once the relevant ancestor amplitudes are known, to every order in the loop expansion.
In the one-loop approximation (linearized functional equation) this is achieved by group-theoretical methods allowing to introduce a suitable set of invariant variables in one-to-one correspondence with the external sources $J_{a\mu}$ (coupled in the classical action to the flat connection) and $K_0$ (coupled to the order parameter). These invariant variables give rise to the whole dependence of the one-loop vertex functional on the quantized fields.
As a special case one can apply this algorithm to the space of local functionals. We then show that the results of Ref. [@Ferrari:2005va] are recovered.
At higher orders one has to solve an inhomogeneous equation. For that purpose we make use of algebraic BRST techniques originally developed in the context of gauge theories [@Ferrari:1998jy]-[@Quadri:2005pv] in order to invert the linearized operator in the relevant sector at ghost number one.
The main result is that starting from two loops on the dependence of the vertex functional on the quantized fields $\phi_a$ is two-fold: the $n$-th loop ancestor amplitudes induce the dependence on the $\phi$’s through the invariant variables solution of the linearized functional equation (implicit dependence). The lower-order contributions (giving rise to the inhomogeneous term as a consequence of the bilinearity of the functional equation) account for the explicit dependence of the $n$-th order vertex functional on the quantized fields.
We stress that in this approach the functional equation is recursively solved order by order in the loop expansion. This allows to obtain the full dependence of the vertex functional on the quantized fields (which is uniquely determined once the ancestor amplitudes are known) to all loop orders.
This algorithm can be applied to many problems arising in the quantization of nonrenormalizable theories based on the hierarchy principle. We just mention two of them here. The technique discussed in this paper can be applied to higher loops Chiral Perturbation theory [@Bijnens:1999hw] in order to determine the full dependence of the vertex functional on the pion fields (including those terms which are on-shell vanishing).
Moreover this method is expected to provide a very useful tool in the program of the consistent quantization of the Stueckelberg model [@Stueckelberg]-[@Ferrari:2004pd] for massive non-abelian gauge bosons.
The paper is organized as follows. In Sect. \[sec.1\] we briefly review the subtraction procedure based on the hierarchy principle in the flat connection formalism. In Sect. \[sec.2\] we solve the local functional equation in the one-loop approximation in full generality. We do not impose any locality restrictions on the space of the solutions. In Sect. \[sec:oneloop\] we discuss some one-loop examples. We show that by applying the algorithm of Sect. \[sec.2\] to the space of local functionals the results of Ref. [@Ferrari:2005va] are recovered. We also solve explicitly the hierarchy for the four-point pion amplitudes (one loop). In Sect. \[sec.3\] the technique for the determination of the higher order solution is developed. In Sect. \[sec:2loop\] we apply this technique on some examples at the two loop level. In particular we obtain the solution of the hierarchy for the four point pion functions at two loops. In Sect. \[sec:fr\] we comment on the possible finite renormalizations which are allowed from a mathematical point of view by the weak power-counting, order by order in the loop expansion, and we show that they can be interpreted as a redefinition of the external sources $J_{a\mu}$ and $K_0$ by finite quantum corrections. Conclusions are finally given in Sect. \[sec.5\].
The flat connection formalism {#sec.1}
=============================
In the flat connection formalism [@Ferrari:2005ii] the pion fields are embedded into the SU(2) flat connection $$\begin{aligned}
&& F_\mu = i \Omega \partial_\mu \Omega^\dagger = \frac{1}{2} F_{a\mu} \tau_a \, .
\label{eq.flat}\end{aligned}$$ In the above equation $\tau_a$ are the Pauli matrices and $\Omega$ denotes the SU(2) group element. $\Omega$ is parameterized in terms of the pion fields $\phi_a$ as follows: $$\begin{aligned}
&& \Omega = \frac{1}{v_D} (\phi_0 + i \tau_a \phi_a) \, , ~~~
\Omega^\dagger \Omega =1 \, , ~~~ {\rm det} ~ \Omega = 1\, ,
\nonumber \\
&&
\phi_0^2 + \phi_a^2 = v_D^2 \, .
\label{sec.1:1}\end{aligned}$$ $v_D$ is the $D$-dimensional mass scale $$v_D = v^{D/2-1}$$ and $v$ has mass dimension one.
The $D$-dimensional action of the nonlinear sigma model is written in the presence of an external vector source $J_{a\mu}$ [^4] and of a scalar source $K_0$ coupled to the solution of the nonlinear sigma model constraint $\phi_0$: $$\begin{aligned}
\G^{(0)} = \int d^Dx \,
\Big ( \frac{v_D^2}{8} (F_{a\mu} - J_{a\mu})^2
+ K_0 \phi_0 \Big ) \, .
\label{sec.1:2}\end{aligned}$$ The invariance of the Haar measure in the path-integral under the local gauge transformations $$\begin{aligned}
&& \Omega ' = U \Omega \, , \nonumber \\
&& F'_\mu = U F_\mu U^\dagger + i U \partial_\mu U^\dagger \, ,
\label{sec.1:3}\end{aligned}$$ where $U$ is an element of SU(2) allows to derive the following local functional equation for the 1-PI vertex functional $\G$ [@Ferrari:2005ii] $$\begin{aligned}
\Big ( - \partial_\mu \frac{\delta \G}{\delta J_{a \mu}}
+ \epsilon_{abc} J_{c\mu} \frac{\delta \G}{\delta J_{b \mu}}
+ \frac{1}{2} K_0 \phi_a +
\frac{1}{2} \frac{\delta \G}{\delta K_0} \frac{\delta \G}{\delta \phi_a}
+ \frac{1}{2} \epsilon_{abc} \phi_c \frac{\delta \G}{\delta \phi_b}
\Big ) (x) = 0 \, .
\label{sec.1:4}\end{aligned}$$ Moreover one requires that the vacuum expectation value of the order parameter is fixed by the condition $$\begin{aligned}
\left . \frac{\delta \G}{\delta K_0(x)} \right |_{\vec{\phi} = K_0 = J_{a\mu} = 0} = v_D \, .
\label{sec.norm.cond}\end{aligned}$$
A weak power-counting theorem [@Ferrari:2005va] exists for the loop-wise perturbative expansion. Accordingly at any given loop order the number of divergent ancestor amplitudes (i.e. those only involving the insertion of the ancestor composite operators) is finite. On the contrary, already at one loop level there is an infinite number of divergent 1-PI amplitudes involving the $\phi_a$ fields (descendant amplitudes). The latter can be fixed in terms of the ancestor ones by recursively differentiating the local functional equation (\[sec.1:4\]).
One-loop solution {#sec.2}
=================
In the one-loop approximation eq.(\[sec.1:4\]) becomes $$\begin{aligned}
\!\!\!\!\!\!
{\cal S}_a (\G^{(1)}) & = &
\Big ( - \partial_\mu \frac{\delta \G^{(1)}}{\delta J_{a \mu}}
+ \epsilon_{abc} J_{c\mu} \frac{\delta \G^{(1)}}{\delta J_{b \mu}}
+ \frac{1}{2} \frac{\delta \G^{(0)}}{\delta K_0} \frac{\delta \G^{(1)}}{\delta \phi_a}
+ \frac{1}{2} \frac{\delta \G^{(1)}}{\delta K_0} \frac{\delta \G^{(0)}}{\delta \phi_a}
\nonumber \\
& &
+ \frac{1}{2} \epsilon_{abc} \phi_c \frac{\delta \G^{(1)}}{\delta \phi_b}
\Big ) (x) = 0 \, .
\label{sec.1:6}\end{aligned}$$ In order to solve the above equation we construct invariant variables in one-to-one correspondence with the external sources. For that purpose we remark that the combination $$\begin{aligned}
\overline{K}_0 =
\frac{v_D^2 K_0}{\phi_0} - \phi_a \frac{\delta S_0}{\delta \phi_a}
\label{sec.1:9}\end{aligned}$$ with $$\begin{aligned}
S_0 = \frac{v_D^2}{8} \int d^Dx \, \Big ( F_{a\mu} - J_{a\mu} \Big )^2
\label{sec.1:10}\end{aligned}$$ is an invariant [@Ferrari:2005va]. Moreover the transformation $K_0 \rightarrow \overline{K}_0$ is invertible.
On the other hand eq.(\[sec.1:6\]) implies that $J_{a\mu}$ transforms as a background connection.
The transformation properties of $\phi_a$ implement the non-linearly realized SU(2) local transformation in eq.(\[sec.1:3\]). Hence $F_{a\mu}$ transforms as a gauge connection and therefore the combination $$\begin{aligned}
I_\mu = I_{a\mu} \frac{\tau_a}{2} = F_{\mu} - J_{\mu}
\label{sec.1:8}\end{aligned}$$ transforms in the adjoint representation (being the difference of two connections): $$\begin{aligned}
I'_\mu = U I_\mu U^\dagger \, .
\label{sec.1:8t}\end{aligned}$$ As a consequence the conjugate of $I_\mu$ w.r.t. $\Omega$ $$\begin{aligned}
j_\mu = j_{a \mu} \frac{\tau_a}{2} = \Omega^\dagger I_\mu \Omega
\label{sec.1:n1}\end{aligned}$$ is an invariant under the transformations in eqs.(\[sec.1:3\]) and (\[sec.1:8t\]).
By direct computation one finds that $j_{a \mu}$ in eq.(\[sec.1:n1\]) is given by $$\begin{aligned}
v_D^2 ~ j_{a\mu} & = & v_D^2 I_{a\mu} - 2 \phi_b^2 I_{a\mu} +
2 \phi_b I_{b\mu} \phi_a
+ 2 \phi_0 \epsilon_{abc} \phi_b I_{c\mu} \nonumber \\
& \equiv & v_D^2 ~ R_{ba} I_{b\mu}
\, .
\label{sec.1:n2}\end{aligned}$$ The matrix $R_{ba}$ in the above equation is an element of the adjoint representation of the SU(2) group. Hence the transformation $J_{a\mu} \rightarrow j_{a\mu}$ is invertible.
The linearized functional equation (\[sec.1:6\]) has a very simple form in the variables $\{ \phi_a, \overline{K}_0, j_{a\mu} \}$. In fact, by taking into account the invariance of $\overline{K}_0$ and $j_{a\mu}$ under ${\cal S}_a$, eq.(\[sec.1:6\]) reduces to $$\begin{aligned}
\Theta_{ab}\frac{\delta\G^{(1)}[\phi_a, \overline K_0, j_{a\mu}]}{\delta \phi_b} = 0 \, ,
\label{sec.1:n3}\end{aligned}$$ where the matrix $\Theta_{ab}$ gives the variation of $\phi_b$: $$\begin{aligned}
\Theta_{ab} = \frac{1}{2} \phi_0 \delta_{ab} + \frac{1}{2} \epsilon_{abc} \phi_c \, .
\label{sec.1:n4}\end{aligned}$$ $\Theta_{ab}$ is invertible as a consequence of the nonlinear constraint in the second line of eq.(\[sec.1:1\]) and thus eq.(\[sec.1:n3\]) is equivalent to $$\begin{aligned}
\frac{\delta\G^{(1)}[\phi_a, \overline K_0, j_{a\mu}]}{\delta \phi_b} = 0 \, .
\label{sec.1:n5}\end{aligned}$$ That means that the only dependence of the symmetric vertex functional $\G^{(1)}$ on the pion fields is through the variables $\overline{K}_0$ and $j_{a\mu}$.
This in turn allows to integrate the linearized functional equation (\[sec.1:6\]). For that purpose one has to replace in the ancestor amplitudes 1-PI functional the source $K_0$ with $\frac{1}{v_D} \overline{K}_0$ and $J_{a\mu}$ with $-j_{a\mu}$. The normalization of $\overline{K}_0$ and $j_{a\mu}$ is fixed by the boundary conditions $$\begin{aligned}
&&
\overline K_0 |_{\vec\phi=0} = v_D K_0
\nonumber\\&&
- j_{a\mu}|_{\vec\phi=0} = J_{a\mu} \, .
\label{sec.1:n5.1}\end{aligned}$$
By eq.(\[sec.1:n5\]) this algorithm gives rise to the full dependence on the pion fields at the one loop level. Thus we can state the following Proposition:
[**Proposition 1**]{}. Given the ancestor amplitudes 1-PI functional ${\cal A}^{(1)}[K_0,J_{a\mu}]$ the solution of the linearized local functional equation (\[sec.1:6\]) is obtained through the replacement rule $$\begin{aligned}
\G^{(1)}[\phi_a,K_0,J_{a\mu}] =
\left .
{\cal A}^{(1)}[K_0,
J_{a\mu}] \right |_{K_0 \rightarrow
\frac{1}{v_D} \overline{K}_0, J_{a\mu} \rightarrow -j_{a \mu}}
\,
\label{sec4.sol.NL}\end{aligned}$$ where in the R.H.S. of the above equation $\overline{K}_0$ is given by eq.(\[sec.1:9\]) and $j_{a \mu}$ by eq.(\[sec.1:n2\]).
In view of this result we say that $\G^{(1)}$ depends on the $\phi$’s only implicitly (i.e. through $\overline{K}_0$ and $j_{a\mu}$). This terminology will prove convenient when studying the dependence of the vertex functional on the $\phi$’s at higher orders.
We stress that no restriction to the space of local functionals is used in the above derivation. Eq.(\[sec4.sol.NL\]) thus provides the full set of Green functions involving at least one pion in terms of the ancestor amplitudes. This solves the hierarchy at the one loop level.
One-loop examples {#sec:oneloop}
=================
When restricted to the local (in the sense of formal power series) functionals, the prescription in eq.(\[sec4.sol.NL\]) gives back the results of [@Ferrari:2005va]. This follows from the uniqueness of the hierarchy solution once the ancestor amplitudes are fixed.
As an example we derive the local invariants ${\cal I}_1, \dots,{\cal I}_7$ parameterizing the one-loop divergences of the nonlinear sigma model in $D=4$ (see Appendix \[app:B\]) by performing the substitution $K_0 \rightarrow \frac{1}{v_D} \overline{K}_0$ , $J_{a\mu} \rightarrow -j_{a\mu}$ in the relevant ancestor monomials $$\begin{aligned}
&
\int \, \partial_\mu J_{a\nu} \partial^\mu J^\nu_a\, , ~~~
\int \, \partial J_a \partial J_a \, , ~~~
\int \, \epsilon_{abc} \partial_\mu J_{a \nu} J^\mu_b J^\nu_c \, , &
\nonumber \\
& \int \, K_0^2 \, , ~~~
\int \, K_0 J^2 \, , ~~~
\int \, (J^2)^2 \, , ~~~
\int \, J_{a\mu} J^\mu_b J_{a \nu} J^\nu_b \, .
&
\label{ex.10}\end{aligned}$$ The monomials in the second line of the above equation do not contain derivatives. By using the SU(2) constraint $$\begin{aligned}
R_{ba} R_{ca} = \delta_{bc}
\label{ex.12}\end{aligned}$$ we get $$\begin{aligned}
&& j_{a\mu}^2 = I_{a\mu}^2 \, , ~~~~ j_{a\mu} j^\mu_b j_{a \nu} j^\nu_b = I_{a\mu} I_{b}^\mu I_{a\nu} I^\nu_b \, .
\label{ex.12.1}\end{aligned}$$ Therefore $$\begin{aligned}
&& \int d^Dx \, K_0^2 \rightarrow \frac{1}{v_D^2} \int d^D x \, \overline{K}_0^2 = \frac{1}{v_D^2} {\cal I}_4 \, ,
\nonumber \\
&& \int d^Dx \, K_0 J^2 \rightarrow
\frac{1}{v_D} \int d^Dx \, \overline{K}_0 j^2 = \frac{1}{v_D} {\cal I}_5 \, ,
\nonumber \\
&& \int d^Dx
\, (J^2)^2 \, \rightarrow \int d^Dx \, (j^2)^2 = {\cal I}_6 \, ,
\nonumber \\
&& \int d^Dx
\, J_{a\mu} J^\mu_b J_{b \nu} J^\nu_b
\rightarrow \int d^Dx \, j_{a\mu} j^\mu_b j_{b \nu} j^\nu_b = {\cal I}_7 \, .
\label{ex.13}\end{aligned}$$
In order to establish the matching for the ancestor monomials involving derivatives in the first line of eq.(\[ex.10\]), we notice that the flat connection $F_{a\mu}$ can be computed in terms of $R_{ba}$ as well (since $R_{ba}$ belongs to the adjoint representation of the SU(2) group). In fact one finds $$\begin{aligned}
i R_{bc} \partial_\mu R^\dagger_{ca} = i R_{bc} \partial_\mu R_{ac} =
(T_c)_{ba} F_{c\mu}
\label{ex.1}\end{aligned}$$ where $(T_c)_{ba} = i \epsilon_{cab}$ are the generators of the adjoint representation satisfying the commutation relation $$\begin{aligned}
[T_a, T_b] = i \epsilon_{abc} T_c \, .
\label{ex.2}\end{aligned}$$ Eq.(\[ex.1\]) can be checked as follows. We set $$\begin{aligned}
&&
R_a \equiv \Omega^\dagger \tau_a\Omega = \tau_bR_{ab}
\nonumber \\&&
R_{ab}= \frac{1}{2} T_r\left( \tau_b \Omega^\dagger \tau_a\Omega
\right) \, .
\label{or.1}\end{aligned}$$ By using the following identities $$\begin{aligned}
&&
T_r\left( \tau_a F_\mu \tau_b
\right)
= T_r\left( \Omega^\dagger\tau_a F_\mu \tau_b \Omega
\right)
\nonumber \\&&
=i T_r\left( \Omega^\dagger\tau_a\Omega \partial_\mu \Omega^\dagger\tau_b \Omega
\right)
\nonumber \\&&
=i T_r\left( \Omega^\dagger\tau_a\Omega \partial_\mu \left[\Omega^\dagger\tau_b
\Omega\right]\right)
-i T_r\left( \Omega^\dagger\tau_a\Omega \Omega^\dagger\tau_b \partial_\mu\Omega
\right)
\nonumber \\&&
=i T_r\left( \Omega^\dagger\tau_a\Omega \partial_\mu \left[\Omega^\dagger\tau_b
\Omega\right]\right)
+i T_r\left( \tau_a\Omega \Omega^\dagger\tau_b \Omega\partial_\mu\Omega^\dagger
\right)
\label{or.2}\end{aligned}$$ we find $$\begin{aligned}
T_r\left( \tau_a F_\mu \tau_b
\right)-T_r\left( \tau_b F_\mu \tau_a
\right)
=i T_r\left( \Omega^\dagger\tau_a\Omega \partial_\mu \left[\Omega^\dagger\tau_b
\Omega\right]\right)
\label{or.3}\end{aligned}$$ which gives directly eq.(\[ex.1\]): $$\begin{aligned}
-i\epsilon_{abc} F_{c\mu}
=i R_{ac}\partial_\mu R_{bc}
= - i R_{bc} \partial_\mu R_{ac} \, .
\label{or.4}\end{aligned}$$
By repeated application of eq.(\[ex.12\]) and eq.(\[ex.1\]) we then get $$\begin{aligned}
&& \int d^Dx \, \partial_\mu J_{a\nu} \partial^\mu J^\nu_a \rightarrow
\int d^Dx \, \partial_\mu j_{a\nu} \partial^\mu j^\nu_a = \int d^Dx \, \partial_\mu \Big ( R_{ba} I_{\nu b} \Big ) \partial^\mu \Big ( R_{ca} I^\nu_c \Big ) \nonumber \\
&& ~~~~~~~~~~~~ = \int d^Dx \, (D_\mu[F] I_\nu)_a (D^\mu[F] I^\nu)_a = {\cal I}_1 \, ,
\label{ex.15}\end{aligned}$$ where $D_\mu[F]$ is the covariant derivative w.r.t. $F_{a\mu}$: $$\begin{aligned}
(D_\mu[F] I_{\nu})_a = \partial_\mu I_{a\nu} + \epsilon_{abc} F_{b \mu} I_{c\nu} \, .
\label{ex.16}\end{aligned}$$ In a similar way we get $$\begin{aligned}
&& \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\int d^Dx \, \partial J_a \partial J_a \rightarrow \int d^Dx \, \partial j_a \partial j_a = \int d^Dx \, (D_\mu[F] I^\mu)_a (D_\nu[F] I^\nu)_a = {\cal I}_2 \, . \nonumber \\
\label{ex.17}\end{aligned}$$ Moreover $$\begin{aligned}
&&
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\int d^Dx \, \epsilon_{abc} \partial_\mu J_{a \nu} J^\mu_b J^\nu_c \rightarrow
- \int d^Dx \, \epsilon_{abc} \partial_\mu j_{a \nu} j^\mu_b j^\nu_c \nonumber \\
&& \!\!\!\!\!\!\!\!\!\! = - \int d^Dx \, \epsilon_{abc} \Big ( \partial_\mu R_{qa} I_{q \nu}
R_{pb} I^\mu_p R_{rc} I^\nu_r + R_{qa} \partial_\mu I_{q\nu} R_{pb} I^\mu_p R_{rc} I^\nu_r \Big ) \, .
\label{ex.18}\end{aligned}$$ By noticing that $$\begin{aligned}
\epsilon_{abc} R_{qa} R_{pb} R_{rc} = \epsilon_{qpr}
\label{ex.19}\end{aligned}$$ and by using eqs.(\[ex.12\]) and (\[ex.1\]) into eq.(\[ex.18\]) we finally get $$\begin{aligned}
- \int d^Dx \, \epsilon_{abc} \partial_\mu j_{a \nu} j^\mu_b j^\nu_c =
- \int d^Dx \, \epsilon_{abc} (D_\mu[F] I_\nu)_a I^\mu_b I^\nu_c = - {\cal I}_3 \, .
\label{ex.20}\end{aligned}$$
As we have mentioned several times, the algorithm for solving the hierarchy based on Proposition 1 can be applied in order to derive the full Green functions involving at least one pion field in terms of the ancestor amplitudes. As an example, we obtain here the full one-loop four point pion amplitude in terms of the relevant ancestor amplitudes $\G^{(1)}_{JJ}, \G^{(1)}_{JJJ}, \G^{(1)}_{JJJJ}, \G^{(1)}_{K_0 K_0}$ and $\G^{(1)}_{K_0 JJ}$. For that purpose one has to perform the substitution $J_{a\mu} \rightarrow -j_{a\mu}$ and $K_0 \rightarrow \frac{1}{v_D} \overline{K}_0$ in the relevant part of the ancestor functional $$\begin{aligned}
&&
\!\!\!\!\!\!\!\!\!
{\cal A}^{(1)}[K_0, J_{a\mu} ] = \frac{1}{2} \int \, \G^{(1)}_{J_{a\mu}(x) J_{b\nu}(y)} J_{a\mu}(x) J_{b\nu}(y)
+ \nonumber \\
&& + \frac{1}{3!} \int \, \G^{(1)}_{J_{a\mu}(x) J_{b\nu}(y) J_{c\rho}(z)} J_{a\mu}(x) J_{b\nu}(y) J_{c\rho}(z)
\nonumber \\
&& + \frac{1}{4!} \int \, \G^{(1)}_{J_{a\mu}(x) J_{b\nu}(y) J_{c\rho}(z) J_{d\sigma}(w)} J_{a\mu}(x) J_{b\nu}(y) J_{c\rho}(z) J_{c\sigma}(w) \nonumber \\
&& + \frac{1}{2} \int \, \G^{(1)}_{J_{a\mu}(x) J_{b\nu}(y) K_0(z)} J_{a\mu}(x) J_{b\nu}(y) K_0(z) \nonumber \\
&& + \frac{1}{2} \int \, \G^{(1)}_{K_0(x) K_0(y)} K_0(x) K_0(y) + \dots
\label{ex.20.1}\end{aligned}$$ by keeping all terms contributing up to four pion fields. This amounts to truncate the expansion of $\overline{K}_0$ up to two $\phi$’s and the expansion of $j_{a\mu}$ up to three $\phi$’s: $$\begin{aligned}
&&
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\overline{K}_0 = \frac{1}{v_D} \phi_a \square \phi_a + \dots \, , \nonumber \\
&&
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
j_{a\mu} = \frac{2}{v_D} \partial_\mu \phi_a - \frac{2}{v_D^2} \epsilon_{abc} \partial_\mu \phi_b \phi_c
+ \frac{1}{v_D^3} \Big ( - \phi_b^2 \partial_\mu \phi_a + 2 \phi_b \partial_\mu \phi_b \phi_a \Big ) + \dots
\label{ex.21}\end{aligned}$$ Then one gets $$\begin{aligned}
&&
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\G^{(1)}[\phi\phi\phi\phi] = \frac{2}{v_D^4} \int \, \G^{(1)}_{J_{a\mu}(x) J_{b\nu}(y)}
\Big ( \partial_\mu \phi_a(x) ( - \phi_c^2(y) \partial_\nu \phi_b(y) + 2 \phi_c(y) \partial_\nu \phi_c(y) \phi_b(y) )
\nonumber \\
&& ~~~~~~~~~~~~~~~~~~~~~~~ + \epsilon_{apq} \epsilon_{brs} \partial_\mu \phi_p(x) \phi_q(x) \partial_\nu \phi_r(y) \phi_s(y) \Big ) \nonumber \\
&& \!\!\!\!\!\!
+ \frac{4}{v_D^4} \int \, \G^{(1)}_{J_{a\mu}(x) J_{b\nu}(y) J_{c\rho}(z)}
\epsilon_{apq} \partial_\mu \phi_p(x) \phi_q(x) \partial_\nu \phi_b(y) \partial_\rho \phi_c(z) \nonumber \\
&& \!\!\!\!\!\!
+ \frac{2}{3 v_D^4} \int \, \G^{(1)}_{J_{a\mu}(x) J_{b\nu}(y) J_{c\rho}(z) J_{d\sigma}(w)}
\partial_\mu \phi_a(x) \partial_\nu \phi_b(y) \partial_\rho \phi_c(z) \partial_\sigma \phi_d(w) \nonumber \\
&& \!\!\!\!\!\!
+ \frac{2}{v_D^3} \int \, \G^{(1)}_{J_{a\mu}(x) J_{b\nu}(y) K_0(z)}\partial_\mu \phi_a(x) \partial_\nu \phi_b(y)
(\phi_c \square \phi_c)(z) \nonumber \\
&& \!\!\!\!\!\!
+ \frac{1}{2v_D^2} \int \, \G^{(1)}_{K_0(x) K_0(y)} (\phi_a \square \phi_a)(x) (\phi_b \square \phi_b)(y)
\, .
\label{ex.4pphi}\end{aligned}$$
Higher orders {#sec.3}
=============
At higher orders the functional equation (\[sec.1:4\]) yields an inhomogeneous equation for $\G^{(n)}$, $n>1$: $$\begin{aligned}
{\cal S}_a(\G^{(n)}) = - \frac{1}{2} \sum_{j=1}^{n-1}
\frac{\delta \G^{(j)}}{\delta K_0} \frac{\delta \G^{(n-j)}}{\delta \phi_a} \, .
\label{ho.0}\end{aligned}$$ In order to recursively integrate eq.(\[ho.0\]) order by order in the loop expansion it is convenient to introduce a BRST formulation for the linearized functional operator ${\cal S}_a$. For that purpose we define the BRST differential [@Ferrari:2005va] $$s = \int d^Dx \, \omega_a {\cal S}_a$$ where $\omega_a$ are classical anticommuting local parameters. The variables $\overline{K}_0$ and $j_{a\mu}$ are $s$-invariant while $$\begin{aligned}
s \phi_a = \frac{1}{2} \phi_0 \omega_a + \frac{1}{2} \epsilon_{abc} \phi_b \omega_c \equiv \Theta_{ab} \omega_b \, , ~~~~ s\omega_a = -\frac{1}{2} \epsilon_{abc} \omega_b \omega_c \, .
\label{ho.1}\end{aligned}$$ The BRST transformation of $\omega_a$ is dictated by nilpotency. $\omega_a$ have ghost number one, while all the remaining variables have ghost number zero. In view of the fact that there are no variables with negative ghost number and that the vertex functional $\G$ has ghost number zero, $\G$ cannot depend on $\omega_a$.
The introduction of a BRST differential allows to make use of the technique of the Slavnov-Taylor (ST) parameterization of the effective action [@Quadri:2003ui]-[@Quadri:2005pv] (originally developed in order to restore the ST identities for power-counting renormalizable gauge theories in the absence of a symmetric regularization) in order to solve the local functional equation at orders $\geq 1$.
For that purpose we remark that, since the matrix $\Theta_{ab}$ in eq.(\[ho.1\]) is invertible, we can perform a further change of variables by setting $$\begin{aligned}
\overline{\omega}_a = \Theta_{ab} \omega_b \, .
\label{ho.2}\end{aligned}$$ The inverse matrix $\Theta^{-1}_{ca}$ is given by $$\begin{aligned}
&& \Theta^{-1}_{ca} = \frac{2\phi_0}{v_D^2} \delta_{ca}
+ \frac{2}{v_D^2 \phi_0} \phi_c \phi_a
- \frac{2}{v_D^2} \epsilon_{cpa} \phi_p \, .
\label{ho.3.1}\end{aligned}$$ The action of $s$ on the variables $\{ \overline{K}_0, j_{a\mu}, \phi_a, \overline{\omega}_a \}$ is finally given by $$\begin{aligned}
&&
s \overline{K}_0 = s j_{a \mu} = 0 \, ,
\nonumber \\
&&
s \phi_a = \overline{\omega}_a \, , ~~~~ s \overline{\omega}_a = 0 \, ,
\label{ho.3}\end{aligned}$$ i.e. $s$ has been cohomologically trivialized: $(\phi_a, \overline{\omega}_a)$ form a BRST doublet [@Piguet:1995er]-[@Quadri:2002nh], while $\overline{K}_0$ and $j_{a\mu}$ are invariant.
We are now in a position to recursively solve the local functional equation at higher orders in perturbation theory. By using the BRST differential $s$ eq.(\[ho.0\]) reads $$\begin{aligned}
s \G^{(n)} = \Delta^{(n)} \equiv
- \frac{1}{2} \sum_{j=1}^{n-1} \int d^Dx \, \omega_a
\frac{\delta \G^{(j)}}{\delta K_0(x)} \frac{\delta \G^{(n-j)}}{\delta \phi_a(x)}
\, ,
\label{sho.1}\end{aligned}$$ where $\Delta^{(n)}$ depends only on known lower order terms. Nilpotency of $s$ implies that $\Delta^{(n)}$ is invariant: $$\begin{aligned}
s \Delta^{(n)} = 0 \, .
\label{ho.cc}\end{aligned}$$ This consistency condition can be checked to hold as a consequence of the fulfillment of the functional equation up to order $n-1$, as shown in Appendix \[appA\].
By using eq.(\[ho.3\]) into eq.(\[sho.1\]) we find $$\begin{aligned}
\int d^Dx \, \overline{\omega}_a \frac{\delta \G^{(n)}}{\delta \phi_a}
= \Delta^{(n)}[\overline{\omega}_a, \phi_a, \overline{K}_0, j_{a\mu}] \, .
\label{new.ho.2} \end{aligned}$$ We remark that $\Delta^{(n)}$ is linear in $\overline{\omega}_a$. By differentiating eq.(\[new.ho.2\]) and by setting $\overline{\omega}_a=0$ we get $$\begin{aligned}
\frac{\delta \G^{(n)}}{\delta \phi_a(x)} =
\frac{\delta \Delta^{(n)}}{\delta \overline{\omega}_a(x)}
\label{new.ho.3}\end{aligned}$$ which fixes the explicit dependence of the symmetric vertex functional $\G^{(n)}$ on $\phi_a(x)$ ($\G^{(n)}$ depends on $\phi$ also implicitly through the invariant variables $j_{a\mu}$ and $\overline{K}_0$). By successive differentiation of eq.(\[new.ho.3\]) we obtain $$\begin{aligned}
\!\!\!\!\!\!\!\!\!\!\!\!\!
\G^{(n)}_{\phi_{a_1} \dots \phi_{a_m} \zeta_{b_1} \dots \zeta_{b_n}}
& = &
\Delta^{(n)}_{\overline{\omega}_{a_1} \phi_{a_2} \dots \phi_{a_m}
\zeta_{b_1} \dots \zeta_{b_n}}
\nonumber \\
& = & \frac{1}{m!} \sum_{\sigma \in S_m}
\Delta^{(n)}_{\overline{\omega}_{a_{\sigma(1)}}
\phi_{a_{\sigma(2)}} \dots
\phi_{a_{\sigma(m)}}
\zeta_{b_1} \dots \zeta_{b_n} }
\label{new.ho.3.1}\end{aligned}$$ where $\zeta$ is a collective notation standing for $j_{a\mu}$ and $\overline{K}_0$.
The equality in the second line of the above equation is a consequence of the Bose statistics of the $\phi$’s. We point out that eq.(\[new.ho.3.1\]) imposes a consistency condition on $\Delta^{(n)}$, i.e. $$\begin{aligned}
\!\!
\Delta^{(n)}_{\overline{\omega}_{a_1} \phi_{a_2} \dots \phi_{a_m}
\zeta_{b_1} \dots \zeta_{b_n} }
= \frac{1}{m!} \sum_{\sigma \in S_m}
\Delta^{(n)}_{\overline{\omega}_{a_{\sigma(1)}}
\phi_{a_{\sigma(2)}} \dots
\phi_{a_{\sigma(m)}}
\zeta_{b_1} \dots \zeta_{b_n}} \, .
\label{new.ho.3.2}\end{aligned}$$ This condition holds as a consequence of eq.(\[ho.cc\]), as is proven in Appendix \[app:int\].
Eq.(\[new.ho.3\]) shows that at order $n\geq 2$ the vertex functional exhibits a further dependence on the $\phi$’s (in addition to the implicit one through the variables $\overline{K}_0$ and $j_{a\mu}$). We refer to it as the explicit dependence of $\G^{(n)}$ on $\phi_a$. It is a remarkable fact that this latter dependence on the pion fields comes from amplitudes involving the pion field of lower order strictly. In particular, they do not affect the $n$-th loop ancestor amplitudes.
In order to recover the full $n$-th loop vertex functional one also needs to take into account the implicit dependence on the pion fields through $\overline{K}_0$ and $j_{a\mu}$. In fact we can state the following 0.5 truecm [**Proposition 2.**]{} Given the functional ${\cal A}^{(n)}[K_0,J_{a\mu}]$ collecting the $n$-th order ancestor amplitudes, $n\geq2$, the full $n$-th loop vertex functional is given by $$\begin{aligned}
&&
\G^{(n)}[\phi_a, K_0, J_{a\mu} ] =
\left .
{\cal A}^{(n)}[K_0,
J_{a\mu}] \right |_{K_0 \rightarrow
\frac{1}{v_D} \overline{K}_0, J_{a\mu} \rightarrow -j_{a \mu}}
\nonumber \\
&& ~~~~~~~~~~~~~~~~~~~~~~~~
+ \int d^Dx \, \int_0^1 dt \, \phi_a(x) \lambda_t \frac{\delta \Delta^{(n)}}{\delta \overline{\omega}_a(x)}
\label{new.ho.4}\end{aligned}$$ where $\lambda_t$ acts as follows on a functional $X[\phi_a, \overline{K}_0,
j_{a\mu}]$: $$\begin{aligned}
\lambda_t X[\phi_a, \overline{K}_0, j_{a\mu}] =
X[t \phi_a, \overline{K}_0, j_{a\mu}] \, .
\label{new.ho.5}\end{aligned}$$ The first term in the R.H.S. of eq.(\[new.ho.4\]) accounts for the implicit dependence on $\phi_a$ through $\overline{K}_0$ and $j_{a\mu}$. It is of the same form as in the one loop approximation eq.(\[sec4.sol.NL\]).
The second term in the R.H.S. of eq.(\[new.ho.4\]) is present only from two loops on. It arises as a consequence of the bilinearity of the local functional equation (\[sec.1:4\]). It gives rise to the explicit dependence of $\G^{(n)}$ on $\phi_a$ dictated by eq.(\[new.ho.3\]). This can be checked by taking derivatives w.r.t. $\phi_{a_1}, \dots, \phi_{a_m}$ of eq.(\[new.ho.4\]) and then setting $\phi=0$ (derivatives w.r.t. $\zeta_{b_1}, \dots, \zeta_{b_n}$ do not play any role in the following argument). The only contribution comes from the second term and yields $$\begin{aligned}
&&
\!\!\!\!\!\!\!
\frac{\delta^m}{\delta \phi_{a_1} \dots \delta \phi_{a_m}}
\int d^Dx \int_0^1 dt \, \phi_a(x) \lambda_t \frac{\delta \Delta^{(n)}}{\delta \overline \omega_a(x)} \nonumber \\
&& =
\frac{\delta^m}{\delta \phi_{a_1} \dots \delta \phi_{a_m}} \int d^Dx d^Dy_1 \dots d^Dy_{m-1}\nonumber \\
&& ~~~~~~
\frac{1}{(m-1)!} \int_0^1 dt~ t^{m-1} \Delta^{(n)}_{\overline \omega_a(x) \phi_{b_1}(y_1) \dots \phi_{b_{m-1}}(y_{m-1})}
\phi_a(x) \phi_{b_1}(y_1) \dots \phi_{b_{m-1}}(y_{m-1}) \nonumber \\
&& = \frac{1}{m!} \sum_{\sigma \in S_m}
\Delta^{(n)}_{\overline{\omega}_{a_{\sigma(1)}}
\phi_{a_{\sigma(2)}} \dots
\phi_{a_{\sigma(m)}}}
= \G^{(n)}_{\phi_{a_1} \dots \phi_{a_m}}
\label{new.ho.bis}\end{aligned}$$ where in the last line we have used eq.(\[new.ho.3.1\]).
Eq.(\[new.ho.4\]) provides the full set of $n$-th order Green functions in terms of $n$-th order ancestor amplitudes and known lower order terms, thus solving the hierarchy.
Two-loop examples {#sec:2loop}
=================
In this Section we apply the method developed in Sect. \[sec.3\] at two loop order. The two-loop inhomogeneous term is $$\begin{aligned}
\Delta^{(2)} = - \int d^Dx \, \frac{1}{2} \omega_a(x) \frac{\delta \G^{(1)}}{\delta K_0(x)} \frac{\delta \G^{(1)}}{\delta \phi_a(x)} \, .
\label{EX.2l.1}\end{aligned}$$ In order to apply eq.(\[new.ho.4\]) we need to express the R.H.S. in terms of the variables $\{ \overline{K}_0,
j_{a\mu}, \phi_a \}$. For that purpose we write $$\begin{aligned}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\Delta^{(2)} & = & - \int d^Dx \,
\frac{1}{2} \Theta^{-1}_{ab} \overline{\omega}_b \int d^Dy \,
\frac{\delta \overline{K}_0(y)}{\delta K_0(x)}
\frac{\delta \G^{(1)}}{\delta \overline{K}_0(y)} \nonumber \\
& &
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\int d^Dz \, \Big (
\frac{\delta \overline{K}_0(z)}{\delta \phi_a(x)}
\frac{\delta}{\delta \overline{K}_0(z)}
+
\frac{\delta j_{c\mu}(z)}{\delta \phi_a(x)}
\frac{\delta}{\delta j_{c\mu}(z)}
+
\delta^D(x-z) \frac{\delta}{\delta \phi_a(z)} \Big )
\G^{(1)} \, .
\label{EX.2l.2}\end{aligned}$$ where the matrix $ \Theta^{-1}_{ab}$ is given in eq.(\[ho.3.1\]).
Moreover $$\begin{aligned}
\frac{\delta \overline{K}_0(y)}{\delta K_0(x)} =
\frac{v_D^2}{\phi_0} \delta^D(y-x)
\label{EX.2l.3}\end{aligned}$$ while by eq.(\[sec.1:n5\]) one has in the variables $\{ \overline{K}_0, j_{a\mu}, \phi_a \}$ $$\begin{aligned}
\frac{\delta \G^{(1)}}{\delta \phi_a(x)} = 0 \, .
\label{EX.2l.3.1}\end{aligned}$$ Therefore $$\begin{aligned}
\Delta^{(2)} & = & - \int d^Dx \,
\frac{1}{2} \frac{v_D^2}{\phi_0}
\Theta^{-1}_{ab} \overline{\omega}_b
\frac{\delta \G^{(1)}}{\delta \overline{K}_0(x)} \nonumber \\
& &
\int d^Dz \, \Big (
\frac{\delta \overline{K}_0(z)}{\delta \phi_a(x)}
\frac{\delta}{\delta \overline{K}_0(z)}
+
\frac{\delta j_{c\mu}(z)}{\delta \phi_a(x)}
\frac{\delta}{\delta j_{c\mu}(z)}
\Big )
\G^{(1)}
\label{EX.2l.3.2}\end{aligned}$$ It is useful to introduce two transition functions (encoding the effect of the change of variables from $\{ K_0, J_{a\mu}, \phi_a \}$ to $\{ \overline{K}_0, j_{a\mu}, \phi_a \}$): $$\begin{aligned}
&& G_b(x,z) = \frac{1}{2} \frac{v_D^2}{\phi_0(x)}
\Theta^{-1}_{ab}(x)
\frac{\delta \overline{K}_0(z)}{\delta \phi_a(x)} \, ,
\nonumber \\
&& H_{bc,\mu}(x,z) = \frac{1}{2} \frac{v_D^2}{\phi_0(x)}
\Theta^{-1}_{ab}(x)
\frac{\delta j_{c\mu}(z)}{\delta \phi_a(x)}
\label{EX.2l.4}\end{aligned}$$ so that eq.(\[EX.2l.3\]) reads $$\begin{aligned}
\Delta^{(2)} & = & - \int d^Dx \int d^Dz \,
\overline{\omega}_b(x) \frac{\delta \G^{(1)}}{\delta \overline{K}_0(x)} \nonumber \\
&& ~~~~ \Big ( G_b(x,z) \frac{\delta}{\delta \overline{K}_0(z)}
+ H_{bc,\mu}(x,z) \frac{\delta}{\delta j_{c\mu}(z)} \Big )
\G^{(1)} \, .
\label{EX.2l.5}\end{aligned}$$ In the two-loop approximation eq.(\[new.ho.3\]) is finally $$\begin{aligned}
&& \!\!\!\!\!\!\!\!\!\!\!\!\!\!
\frac{\delta \G^{(2)}}{\delta \phi_b(x)} =
\frac{\delta \Delta^{(2)}}{\delta \overline{\omega}_b(x)}
\nonumber \\
&& \!\!\!\!\!\!\!\!
= - \int d^Dz \,
\frac{\delta \G^{(1)}}{\delta \overline{K}_0(x)} \Big ( G_b(x,z) \frac{\delta}{\delta \overline{K}_0(z)}
+ H_{bc,\mu}(x,z) \frac{\delta}{\delta j_{c\mu}(z)} \Big )
\G^{(1)}
\label{EX.2l.6}\end{aligned}$$ while eq.(\[new.ho.4\]) consequently reads $$\begin{aligned}
&& \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\G^{(2)}[\phi_a, K_0, J_{a\mu} ] =
\left .
{\cal A}^{(2)}[K_0,
J_{a\mu}] \right |_{K_0 \rightarrow
\frac{1}{v_D} \overline{K}_0, J_{a\mu} \rightarrow -j_{a \mu}}
\nonumber \\
&& \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
- \int d^Dx \, \int_0^1 dt \, \phi_b(x) \lambda_t
\int d^Dz \,
\frac{\delta \G^{(1)}}{\delta \overline{K}_0(x)} \Big ( G_b(x,z) \frac{\delta}{\delta \overline{K}_0(z)}
+ H_{bc,\mu}(x,z) \frac{\delta}{\delta j_{c\mu}(z)} \Big )
\G^{(1)} \, .
\nonumber \\
\label{EX.new.ho.4}\end{aligned}$$ The second line encodes the effects of the nonlinearity of the local functional equation at two loop order.
It should be noticed that, due to the peculiar structure of the dependence of the one-loop vertex functional on the pions given by eq.(\[sec.1:n5\]), one finds some special simplifications at two loop level. In particular the second line of eq.(\[EX.new.ho.4\]) does not contribute to the four point pion Green function in the two loop approximation.
In order to show this property we remark that the expansion of $\overline{K}_0$ starts with two $\phi$’s while $j_{a\mu}$ starts with one $\phi$. Hence the term with two derivatives w.r.t. $\overline{K}_0$ in the second line of eq.(\[EX.new.ho.4\]) gives contributions of order $O(\phi^5)$.
In order to obtain the contribution to the four point pion function of the term involving one derivative w.r.t. $j_{c\mu}$ in the second line it is sufficient to keep $H_{bc ,\mu}$ at order zero: $$\begin{aligned}
H_{bc;\mu}(x,z) =
\frac{2}{v_D} \delta_{bc} \partial_{z_\mu} \delta^D(x-z)
+ O(\phi) \, .
\label{EX.new.1.1}\end{aligned}$$ This yields $$\begin{aligned}
- \frac{2}{v_D} \int d^Dx \, \phi_b(x)
\Big [ \frac{\delta \G^{(1)}}{\delta \overline{K}_0(x)} \Big ]_{\phi\phi} \partial^\mu \Big [ \frac{\delta \G^{(1)}}{\delta j_{b\mu}(x)} \Big]_{\phi} \,
\label{EX.new.2}\end{aligned}$$ where the subscript denotes the order of the projection for the $\phi$’s. Moreover the derivative $$\begin{aligned}
\Big [ \frac{\delta \G^{(1)} }{\delta j_{b\mu}(x)} \Big ]_{\phi}
\label{EX.new.4}\end{aligned}$$ receives contributions only from the amplitude $\G^{(1)}_{J_{a\mu} J_{b\nu}}$ [@Ferrari:2005ii] through $$\begin{aligned}
&&
\!\!\!\!\!\!\!\!\!\!\!
\frac{1}{2}\int d^Dx d^Dy \, \G^{(1)}_{J_{a\mu}(x) J_{b\nu}(y)}
j_{a\mu}(x) j_{b\nu}(y) \nonumber \\
&& \!\!\!\!\!\!\! =
- \frac{1}{2} \int d^Dx d^Dy \,
(\square g^{\mu \nu} - \partial^\mu \partial^\nu) j_{a\mu}(x)
j_{b\nu}(y) \nonumber \\
&& ~~~~~~
\int d^D p \, \frac{4i}{m_D^4} \frac{1}{D-1}
e^{i p (x-y)} I_2(p)
\label{EX.new.5}\end{aligned}$$ where $$\begin{aligned}
I_2(p) = \int \frac{d^Dp}{(2\pi)^D} \,
\frac{1}{k^2 (k+p)^2} \, .
\label{EX.new.6}\end{aligned}$$ By taking the gradient according to eq.(\[EX.new.2\]) one finds zero as a consequence of the transversality of $\G^{(1)}_{J_{a\mu}(x) J_{b\nu}(y)}$. Therefore the second line of eq.(\[EX.new.ho.4\]) does not give any contribution to the four point pion function at two loop level. The contribution from the first line can be derived according to the methods discussed in Sect. \[sec:oneloop\]. So we get finally $$\begin{aligned}
&&
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\G^{(2)}[\phi\phi\phi\phi] = \frac{2}{v_D^4} \int \, \G^{(2)}_{J_{a\mu}(x) J_{b\nu}(y)}
\Big ( \partial_\mu \phi_a(x) ( - \phi_c^2(y) \partial_\nu \phi_b(y) + 2 \phi_c(y) \partial_\nu \phi_c(y) \phi_b(y) )
\nonumber \\
&& ~~~~~~~~~~~~~~~~~~~~~~~ + \epsilon_{apq} \epsilon_{brs} \partial_\mu \phi_p(x) \phi_q(x) \partial_\nu \phi_r(y) \phi_s(y) \Big ) \nonumber \\
&& \!\!\!\!\!\!
+ \frac{4}{v_D^4} \int \, \G^{(2)}_{J_{a\mu}(x) J_{b\nu}(y) J_{c\rho}(z)}
\epsilon_{apq} \partial_\mu \phi_p(x) \phi_q(x) \partial_\nu \phi_b(y) \partial_\rho \phi_c(z) \nonumber \\
&& \!\!\!\!\!\!
+ \frac{2}{3 v_D^4} \int \, \G^{(2)}_{J_{a\mu}(x) J_{b\nu}(y) J_{c\rho}(z) J_{d\sigma}(w)}
\partial_\mu \phi_a(x) \partial_\nu \phi_b(y) \partial_\rho \phi_c(z) \partial_\sigma \phi_d(w) \nonumber \\
&& \!\!\!\!\!\!
+ \frac{2}{v_D^3} \int \, \G^{(2)}_{J_{a\mu}(x) J_{b\nu}(y) K_0(z)}\partial_\mu \phi_a(x) \partial_\nu \phi_b(y)
(\phi_c \square \phi_c)(z) \nonumber \\
&& \!\!\!\!\!\!
+ \frac{1}{2v_D^2} \int \, \G^{(2)}_{K_0(x) K_0(y)} (\phi_a \square \phi_a)(x) (\phi_b \square \phi_b)(y)
\, .
\label{ex.4pphi.2L}\end{aligned}$$ This formula exhibits a functional dependence of $\G^{(2)}_{\phi_{a_1} \phi_{a_2} \phi_{a_3} \phi_{a_4}}$ on the ancestor amplitudes as in the one loop approximation (see eq.(\[ex.4pphi\])). This is a rather surprising result which holds as a consequence of the transversality of the one-loop $JJ$ ancestor amplitude.
Hierarchy and Finite Renormalizations {#sec:fr}
=====================================
From the results of Sects. \[sec.2\] and \[sec.3\] it is clear that for any solution of the local functional equation (\[sec.1:4\]) the knowledge of the ancestor amplitudes order by order in the loop expansion completely determines the dependence on the pion fields. One important consequence of this result is that it has been obtained without relying on the specific subtraction procedure. In particular if we want to perform any subtraction in order to define the theory in $D=4$, it is sufficient to operate on the ancestor amplitudes. The subtractions on the amplitudes involving any number of pions are induced by the integration of the functional equation which has been developed in the previous Sections.
In this Section we exploit this property in order to shed light on the finite renormalizations allowed from a mathematical point of view by the local symmetry and the weak power-counting theorem.
For that purpose we remark that a sufficient condition for the fulfillment of the local functional equation (\[sec.1:4\]) is conjectured to be (in the presence of a symmetric regularization like Dimensional Regularization [@Ferrari:2005fc]) the validity of the same functional equation (\[sec.1:4\]) for the functional $$\begin{aligned}
\widehat \G = \G^{(0)} + \sum_{k=1}^\infty \widehat \G^{(k)}
\label{fr.1}\end{aligned}$$ where $\G^{(0)}$ is the classical action in eq.(\[sec.1:2\]) (giving rise to the tree-level Feynman rules) while $\widehat \G^{(k)}$ collects the $k$-th order counterterms. From the mathematical point of view the latter may contain $k$-th order finite renormalizations compatible with the symmetry properties and the weak power-counting bounds [@Ferrari:2005va]. This conjecture is supported by formal arguments [@paper_prep] and by some explicit two-loop examples [@Ferrari:2005fc].
We will now prove that the ancestor amplitudes of $\widehat \G$ can be obtained from the tree-level ancestor amplitudes through a suitable redefinition of the classical sources $J_{a\mu}$ and $K_0$: $$\begin{aligned}
&& J_{a\mu} \rightarrow J_{a\mu} + A_{1,a\mu}(J) + A_{2,a\mu}(J) + \dots \, , \nonumber \\
&& K_0 \rightarrow K_0(1 + B_1(K_0,J) + B_2(K_0,J) + \dots)
\label{fr.2}\end{aligned}$$ where $A_{j,a\mu}, B_j$ are of order $\hbar^j$. $A_{j,a\mu}$ does not depend on $K_0$. We also set $A_{0,a\mu} = J_{a\mu}, B_0=1$.
First we notice that by using integration by parts it is always possible to decompose in a unique way an integrated local functional $\int d^Dx \, X(J,K_0)$ according to $$\begin{aligned}
\int d^Dx \, X(J,K_0) = \int d^Dx \, \Big ( J_{a\mu} {\cal P}_a^\mu[X] + K_0 {\cal Q}[X] \Big )
\label{fr.3}\end{aligned}$$ where ${\cal P}_a^\mu[X]$ is the result of the projection of $X$ into a local function of $J$ and its derivatives while ${\cal Q}[X]$ includes also local dependence on $K_0$ and its derivatives. In order to determine the unknown functions $A_{j,a\mu}$ and $B_j$ in eq.(\[fr.2\]) we perform the substitution (\[fr.2\]) into $$\begin{aligned}
&& \G^{(0)}[0,J_{a\mu},K_0] = \int d^Dx \, \Big ( \frac{v_D^2}{8} J^2 + v_D K_0 \Big ) \nonumber \\
&& ~~~~~~~~~ \rightarrow \sum_{l=0}^\infty
\int d^Dx \, \Big ( \frac{v_D^2}{8} \sum_{j=0}^l A_{j,a\mu} A_{l-j,a}^\mu + v_D K_0 B_l \Big )
\label{fr.5}\end{aligned}$$ and then compare the second line of the above equation with the ancestor counterterms $$\widehat \G^{(l)}[0,K_0,J_{a\mu}] \equiv \int d^Dx \ \widehat{\cal L}_l(J,K_0) \, .$$ This gives $$\begin{aligned}
&&
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\int d^Dx \, \widehat {\cal L}_l = \frac{v_D^2}{8} \int d^Dx \sum_{j=0}^l A_{j,a\mu} A_{l-j,a}^\mu
+ \int d^Dx \, v_D K_0 B_l
\nonumber \\
&&
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
= \int d^Dx \, \Big ( \frac{v_D^2}{4} J_{a\mu} A_{l,a}^\mu + \frac{v_D^2}{8} \sum_{j=1}^{l-1} A_{j,a\mu} A_{l-j,a}^\mu + v_D K_0 B_l \Big ) \, ,
~~ l=1,2,3,\dots
\label{fr.6}\end{aligned}$$ and hence we derive the recursive solution $$\begin{aligned}
&&
\!\!\!\!\!\!\!\!\!\!\!\!
B_0 = 1 \, , ~~~~ B_l = \frac{1}{v_D} {\cal Q} [\widehat {\cal L}_l] \, , \nonumber \\
&&
\!\!\!\!\!\!\!\!\!\!\!\!
A_{0,a\mu} = J_{a\mu} \, , \nonumber \\
&&
\!\!\!\!\!\!\!\!\!\!\!\!
A_{l,a\mu} = \frac{4}{v_D^2} {\cal P}_{a\mu} [\widehat {\cal L}_l] - \frac{1}{2} {\cal P}_{a\mu}
\Big [ \sum_{j=1}^{l-1} A_{j,b\nu} A_{l-j,b}^\nu \Big ] \, , ~~~~ l=1,2,3,\dots
\label{fr.7}\end{aligned}$$ This result states that all possible finite renormalizations in $\widehat \G^{(k)}$, $k>1$, compatible with the local symmetry and the weak power-counting, can in fact be interpreted as a redefinition of the sources $J_{a\mu}$ and $K_0$ by finite quantum corrections. The latter correspond to the ambiguities allowed in the effective field theory approach discussed in [@paper_prep].
Conclusions {#sec.5}
===========
The requirement of the invariance of the group Haar measure under local left multiplication can be implemented by a local functional equation for the 1-PI vertex functional of the nonlinear sigma model. This equation can be preserved by the subtraction procedure and completely fixes the dependence of the vertex functional on the pion fields in terms of the ancestor amplitudes (i.e. amplitudes only involving the flat connection and the nonlinear sigma model constraint).
Very remarkably the recursive solution can be written in a very compact form in terms of invariant variables (inducing an implicit dependence of the vertex functional on the quantized field) plus (at order $n \geq 2$) a contribution yielding an explicit dependence on $\phi_a$. The latter is fixed by lower order terms (see eq.(\[new.ho.4\])) and does not affect the $n$-th loop ancestor amplitudes. This solution provides the full dependence of the 1-PI symmetric amplitudes on the pion fields.
From a technical point of view the method which has been developed in order to integrate the local functional equation extends the cohomological techniques originally developed in the context of gauge theories. In particular it deals with the full Green functions of the theory (no locality restrictions) and it solves explicitly the inhomogeneous equation (arising from the loop expansion of the bilinear local functional equation) in the absence of multiplicative renormalization (as it happens for the subtraction procedure of the nonlinear sigma model).
The integration of the local functional equation at higher orders in the loop expansion allows to treat a new class of problems which could not be addressed by the knowledge of the solutions of the linearized functional equation only.
Among them we think that two issues are worthwhile to be pointed out. The first one is that our method allows the determination of all pion amplitudes at higher orders in Chiral Perturbation Theory.
The second one is the possibility to investigate the use of the techniques discussed in this paper in order to set up a consistent framework for the study of the structure of the higher order divergences within the program of the quantization of the Stückelberg model for non-abelian massive gauge bosons.
Acknowledgments {#acknowledgments .unnumbered}
===============
One of us (A.Q.) would like to thank G. Barnich and G. Colangelo for useful discussions. He also acknowledges the warm hospitality at the Institut für Theoretische Physik in Bern, Switzerland.
Consistency condition {#appA}
=====================
In this appendix we verify eq.(\[ho.cc\]) as a consequence of the recursive validity of the functional equation at lower orders. The technique is a variant of the general proof of the consistency condition in the Batalin-Vilkovisky (BV) formalism [@Gomis:1994he]. One should notice that in the present case the introduction of the antifield $J_{a\mu}^*$ for the background source $J_{a\mu}$ is forbidden (since this would lead to an empty cohomology [@Henneaux:1998hq]). Therefore one cannot use the standard BV bracket.
The local functional equation at order $n$ in the loop expansion reads $$\begin{aligned}
s \G^{(n)} = - \frac{1}{2} \sum_{j=1}^{n-1} \int d^Dx \, \omega_a \frac{\delta \G^{(j)}}{\delta K_0}
\frac{\delta \G^{(n-j)}}{\delta \phi_a} \, ,
\label{wz.1}\end{aligned}$$ which is useful to rewrite in the more symmetric form $$\begin{aligned}
s \G^{(n)} & = &
- \frac{1}{2} \sum_{j=1}^{n-1} \langle \G^{(j)}, \G^{(n-j)} \rangle
\, .
\label{wz.2}\end{aligned}$$ In the above equation we have adopted the notation $$\begin{aligned}
\langle X, Y \rangle & = & \int d^Dx \, \frac{1}{2} \omega_a \Big (
\frac{\delta X}{\delta K_0}
\frac{\delta Y}{\delta \phi_a}
+ \frac{\delta Y}{\delta K_0}
\frac{\delta X}{\delta \phi_a}
\Big ) \, .
\label{wz.3}\end{aligned}$$ The following properties hold for $\langle X, Y \rangle$: $$\begin{aligned}
&& \langle X, Y \rangle = \langle Y, X \rangle \, , \nonumber \\
&& s \langle X, Y \rangle = - \langle s X, Y \rangle
- \langle X, s Y \rangle ~~~~~ X,Y ~~~ \mbox{bosonic} \, .
\label{wz.4}\end{aligned}$$
We denote by $\Delta^{(n)}$ the R.H.S. of eq.(\[wz.2\]), i.e. we set $$\begin{aligned}
\Delta^{(n)} = - \frac{1}{2} \sum_{j=1}^{n-1} \langle \G^{(j)}, \G^{(n-j)} \rangle \, .
\label{wz.5}\end{aligned}$$ If a solution to eq.(\[wz.2\]) exists, by the nilpotency of $s$ the following consistency condition has to be verified: $$\begin{aligned}
s \Delta^{(n)} = 0 \, .
\label{wz.n.1}\end{aligned}$$ Let us verify that this is indeed the case under the recursive assumption that the master equation has been fulfilled up to order $n-1$. By using eq.(\[wz.4\]) we get $$\begin{aligned}
s \Delta^{(n)} & = & s \Big ( -\frac{1}{2} \sum_{j=1}^{n-1}
\langle \G^{(j)}, \G^{(n-j)} \rangle \Big ) \nonumber \\
& = & + \frac{1}{2} \sum_{j=1}^{n-1}
\Big ( \langle s \G^{(j)}, \G^{(n-j)} \rangle
+ \langle \G^{(j)}, s \G^{(n-j)} \rangle \Big )
\nonumber \\
& = & + \frac{1}{2} \sum_{j=1}^{n-1}
\Big ( \langle s \G^{(j)}, \G^{(n-j)} \rangle
+ \langle s \G^{(n-j)}, \G^{(j)} \rangle \Big )
\nonumber \\
& = & \sum_{j=1}^{n-1} \langle s \G^{(j)}, \G^{(n-j)} \rangle
\label{wz.6}\end{aligned}$$ Now we use the recursive assumption that $$\begin{aligned}
s \G^{(j)} & = & - \frac{1}{2} \sum_{k=1}^{j-1} \langle \G^{(k)}, \G^{(j-k)} \rangle
\label{wz.7}\end{aligned}$$ so that $$\begin{aligned}
s \Delta^{(n)} & = & -\frac{1}{2} \sum_{j=1}^{n-1} \sum_{k=1}^{j-1}
\langle \langle \G^{(k)}, \G^{(j-k)} \rangle , \G^{(n-j)} \rangle \nonumber \\
& = & -\frac{1}{2} \cdot \frac{1}{3} \sum_{j=1}^{n-1} \sum_{k=1}^{j-1} \Big ( \langle \langle \G^{(k)}, \G^{(j-k)} \rangle , \G^{(n-j)} \rangle
+\langle \langle \G^{(j-k)}, \G^{(n-j)} \rangle , \G^{(k)} \rangle
\nonumber \\
&& ~~~~~~~~~~~~~~~~~~~~~~
+\langle \langle \G^{(n-j)}, \G^{(k)} \rangle , \G^{(j-k)} \rangle
\Big ) \, .
\label{wz.8}\end{aligned}$$ It turns out that the symmetrized bracket enjoys the following Jacobi identity ($X,Y,Z$ are assumed to be bosonic): $$\begin{aligned}
\langle \langle X,Y \rangle, Z \rangle +
\langle \langle Z,X \rangle, Y \rangle +
\langle \langle Y,Z \rangle, X \rangle = 0 \, .
\label{wz.9}\end{aligned}$$ The proof of the above equation is provided in the next subsection. By using eq.(\[wz.9\]) into eq.(\[wz.8\]) we finally get $$\begin{aligned}
s \Delta^{(n)} = 0 \, .
\label{wz.10}\end{aligned}$$
Proof of the Jacobi identity for the symmetrized bracket
--------------------------------------------------------
We assume $X,Y,Z$ to be bosonic. We write explicitly $\langle \langle X,Y \rangle , Z \rangle$: $$\begin{aligned}
\langle \langle X,Y \rangle , Z \rangle & = &
\int d^Dx \, \frac{1}{2} \omega_a(x) \frac{\delta}{\delta K_0(x)}
( \langle X,Y \rangle ) \frac{\delta Z}{\delta \phi_a(x)}
\nonumber \\
& &
+ \int d^Dx \, \frac{1}{2} \omega_a(x) \frac{\delta Z}{\delta K_0(x)}
\frac{\delta}{\delta \phi_a(x)} (\langle X,Y \rangle)
\nonumber \\
& = & \int d^Dx \frac{1}{2} \omega_a(x) \frac{\delta}{\delta K_0(x)}
\Big [ \int d^Dy \, \frac{1}{2} \omega_b(y) \frac{\delta X}{\delta K_0(y)}
\frac{\delta Y}{\delta \phi_b(y)} \nonumber \\
& & ~~~~~ +\int d^Dy \, \frac{1}{2} \omega_b(y) \frac{\delta Y}{\delta K_0(y)}
\frac{\delta X}{\delta \phi_b(y)} \Big ] \frac{\delta Z}{\delta \phi_a(x)} \nonumber \\
& & + \int d^Dx \, \frac{1}{2} \omega_a(x) \frac{\delta Z}{\delta K_0(x)}
\frac{\delta}{\delta \phi_a(x)}
\Big [ \int d^Dy \, \frac{1}{2} \omega_b(y)
\frac{\delta X}{\delta K_0(y)}\frac{\delta Y}{\delta \phi_b(y)} \nonumber \\
& & ~~~~~ + \int d^Dy \, \frac{1}{2} \omega_b(y)
\frac{\delta Y}{\delta K_0(y)}\frac{\delta X}{\delta \phi_b(y)} \Big ]
\label{wz.11}\end{aligned}$$ We notice that the following terms in the R.H.S. of eq.(\[wz.11\]) $$\begin{aligned}
&& \int d^Dx d^Dy \, \frac{1}{2} \omega_a(x) \frac{1}{2} \omega_b(y)
\frac{\delta Z}{\delta K_0(x)} \frac{\delta X}{\delta K_0(y)} \frac{\delta^2 Y}{\delta \phi_a(x) \delta \phi_b(y)} \, , \nonumber \\
&& \int d^Dx d^Dy \, \frac{1}{2} \omega_a(x) \frac{1}{2} \omega_b(y)
\frac{\delta Z}{\delta K_0(x)} \frac{\delta Y}{\delta K_0(y)} \frac{\delta^2 X}{\delta \phi_a(x) \delta \phi_b(y)}
\label{wz.12}\end{aligned}$$ are zero since $\omega_a(x)$ and $\omega_b(y)$ are anticommuting.
We make use of eq.(\[wz.11\]) in order to write the sum $ \langle \langle X, Y \rangle, Z \rangle + \mbox{cyclic}$. We organize the terms according to the number of derivatives w.r.t $K_0$ acting on a single functional. We obtain $$\begin{aligned}
\langle \langle X, Y \rangle, Z \rangle + \mbox{cyclic} & = & \int d^Dx \int d^Dy ~
\frac{1}{2} \omega_a(x) \frac{1}{2} \omega_b(y) \nonumber \\
&& ~~~~~ \times \Big [
\frac{\delta^2 X}{\delta K_0(x) \delta K_0(y)}
\Big ( \frac{\delta Y}{\delta \phi_b(y)} \frac{\delta Z}{\delta \phi_a(x)}
+\frac{\delta Z}{\delta \phi_b(y)} \frac{\delta Y}{\delta \phi_a(x)} \Big ) \nonumber \\
&& ~~~~~~~ +
\frac{\delta^2 Y}{\delta K_0(x) \delta K_0(y)}
\Big ( \frac{\delta Z}{\delta \phi_b(y)} \frac{\delta X}{\delta \phi_a(x)}
+\frac{\delta X}{\delta \phi_b(y)} \frac{\delta Z}{\delta \phi_a(x)} \Big ) \nonumber \\
&& ~~~~~~~ +
\frac{\delta^2 Z}{\delta K_0(x) \delta K_0(y)}
\Big ( \frac{\delta X}{\delta \phi_b(y)} \frac{\delta Y}{\delta \phi_a(x)}
+\frac{\delta Y}{\delta \phi_b(y)} \frac{\delta X}{\delta \phi_a(x)} \Big ) \Big ] \nonumber \\
&& + \int d^Dx \int d^Dy ~
\frac{1}{2} \omega_a(x) \frac{1}{2} \omega_b(y) \nonumber \\
&& ~~~~~ \times \Big [ \frac{\delta X}{\delta K_0(y)} \Big ( \frac{\delta^2 Y}{\delta K_0(x) \delta \phi_b(y)}
\frac{\delta Z}{\delta \phi_a(x)} + \frac{\delta^2 Z}{\delta K_0(x) \delta \phi_b(y)} \frac{\delta Y}{\delta \phi_a(x)} \Big ) \nonumber \\
&& ~~~~~~~~~ + \frac{\delta X}{\delta K_0(x)} \Big ( \frac{\delta^2 Y}{\delta \phi_a(x) \delta K_0(y)}
\frac{\delta Z}{\delta \phi_b(y)} + \frac{\delta^2 Z}{\delta \phi_a(x) \delta K_0(y)}
\frac{\delta Y}{\delta \phi_b(y)} \Big ) \nonumber \\
&& ~~~~~~~~~ + \mbox{cyclic} \Big ]
\label{wz.13}\end{aligned}$$ The terms in the first block between square brackets in the above equation vanish by symmetry once the anticommutativity of $\omega_a(x), \omega_b(y)$ is taken into account.
The second block requires some manipulations. If one exchanges $y \leftrightarrow x$ and $a \leftrightarrow b$ in the second line of the second block, the latter becomes $$\begin{aligned}
&& + \int d^Dx \int d^Dy ~
\frac{1}{2} \omega_a(x) \frac{1}{2} \omega_b(y) \nonumber \\
&& ~~~~~ \times \frac{\delta X}{\delta K_0(y)} \Big ( \frac{\delta^2 Y}{\delta K_0(x) \delta \phi_b(y)}
\frac{\delta Z}{\delta \phi_a(x)} + \frac{\delta^2 Z}{\delta K_0(x) \delta \phi_b(y)} \frac{\delta Y}{\delta \phi_a(x)} \Big ) \nonumber \\
&& + \int d^Dx \int d^Dy ~
\frac{1}{2} \omega_b(y) \frac{1}{2} \omega_a(x) \nonumber \\
&& ~~~~~ \times \frac{\delta X}{\delta K_0(y)} \Big ( \frac{\delta^2 Y}{\delta K_0(x) \delta \phi_b(y)}
\frac{\delta Z}{\delta \phi_a(x)} + \frac{\delta^2 Z}{\delta K_0(x) \delta \phi_b(y)} \frac{\delta Y}{\delta \phi_a(x)} \Big ) \nonumber \\
&& ~~~~~~~~~ + \mbox{cyclic}
\label{wz.14}\end{aligned}$$ The above expression is zero since $\omega_a(x), \omega_b(y)$ anticommute.
Therefore we establish the Jacobi identity for the symmetrized bracket in the form $$\begin{aligned}
\langle \langle X,Y \rangle, Z \rangle +
\langle \langle Z,X \rangle, Y \rangle +
\langle \langle Y,Z \rangle, X \rangle = 0
\label{wz.15}\end{aligned}$$ with $X,Y,Z$ bosonic.
Integrability condition {#app:int}
=======================
In this Appendix we check that eq.(\[new.ho.3.2\]) holds as a consequence of eq.(\[ho.cc\]). Eq.(\[ho.cc\]) reads in the variables $\{ \overline{K}_0, j_{a\mu}, \phi_a, \overline{\omega}_a\}$ $$\begin{aligned}
\int d^Dx \, \overline{\omega}_a(x) \frac{\delta \Delta^{(n)}}{\delta \phi_a(x)} = 0 \, .
\label{app:int.1}\end{aligned}$$ By differentiating the above equation w.r.t. $\overline{\omega}_a(x), \overline{\omega}_b(y)$ and by setting $\overline{\omega}=0$ we get $$\begin{aligned}
\frac{\delta^2 \Delta^{(n)}}{\delta \overline \omega_b(y) \delta \phi_a(x)}
= \frac{\delta^2 \Delta^{(n)}}{\delta \overline \omega_a(x) \delta \phi_b(y)} \, .
\label{app:int.2}\end{aligned}$$ Let us now consider the R.H.S. of eq.(\[new.ho.3.2\]). For each permutation $\sigma \in S_m$ there exists a unique integer $1 \leq K \leq m$ such that $\sigma(K)=1$. Therefore (we drop here the dependence on $\zeta_1, \dots, \zeta_n$ since the latter does not play any role in the following argument) $$\begin{aligned}
&&
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\frac{1}{m!} \sum_{\sigma \in S_m} \Delta^{(n)}_{\overline{\omega}_{a_{\sigma(1)}} \phi_{a_{\sigma(2)}} \dots \phi_{a_{\sigma(m)}}}
= \frac{1}{m!} \sum_{\sigma \in S_{m-1}[{2,\dots, m}]} \Delta^{(n)}_{\overline{\omega}_{a_1} \phi_{a_{\sigma(2)}} \dots \phi_{a_{\sigma(m)}}}
\nonumber \\
&&
+ \frac{1}{m!} \sum_{K=2}^m \sum_{\sigma \in S_{m-1}[{1,2,\dots,\widehat{K}, \dots, m}]} \Delta^{(n)}_{\overline{\omega}_{a_{\sigma(1)}}
\phi_{a_{\sigma(K)}} \phi_{a_{\sigma(2)}} \dots \widehat{\phi}_{a_{\sigma(K)}} \dots \phi_{a_{\sigma(m)}}} \, .
\label{app:int.3}\end{aligned}$$ In the above equation a hat over a variable denotes omission of the latter from the relevant list and $S_{m-1}[a,b,\dots,c]$ denotes the group of permutations over the $m-1$ elements $\{ a,b,\dots,c \}$.
We now use eq.(\[app:int.2\]) in the second line of eq.(\[app:int.3\]) as well as the fact that $\sigma(K)=1$ and we get $$\begin{aligned}
&&
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\frac{1}{m!} \sum_{\sigma \in S_m} \Delta^{(n)}_{\overline{\omega}_{a_{\sigma(1)}} \phi_{a_{\sigma(2)}} \dots \phi_{a_{\sigma(m)}}}
= \frac{1}{m!} \sum_{\sigma \in S_{m-1}[{2,\dots, m}]} \Delta^{(n)}_{\overline{\omega}_{a_1} \phi_{a_{\sigma(2)}} \dots \phi_{a_{\sigma(m)}}}
\nonumber \\
&&
+ \frac{1}{m!} \sum_{K=2}^m \sum_{\sigma \in S_{m-1}[{1,2,\dots,\widehat{K}, \dots, m}]} \Delta^{(n)}_{\overline{\omega}_{a_1}
\phi_{a_{\sigma(1)}} \phi_{a_{\sigma(2)}} \dots \widehat{\phi}_{a_{\sigma(K)}} \dots \phi_{a_{\sigma(m)}}} \, .
\label{app:int.4}\end{aligned}$$ By the Bose statistics of the $\phi$’s we also get $$\begin{aligned}
&&
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\frac{1}{m!} \sum_{\sigma \in S_m} \Delta^{(n)}_{\overline{\omega}_{a_{\sigma(1)}} \phi_{a_{\sigma(2)}} \dots \phi_{a_{\sigma(m)}}}
= \frac{1}{m} \Delta^{(n)}_{\overline{\omega}_{a_1} \phi_{a_2} \dots \phi_{a_m}}
+ \frac{m-1}{m} \Delta^{(n)}_{\overline{\omega}_{a_1}
\phi_{a_2} \phi_{a_3} \dots \dots \phi_{a_m}} \nonumber \\
&& = \Delta^{(n)}_{\overline{\omega}_{a_1} \phi_{a_2} \dots \phi_{a_m}} \, ,
\label{app:int.5}\end{aligned}$$ which proves eq.(\[new.ho.3.2\]).
A comment is in order here. It is a well-known fact in cohomological algebra [@Piguet:1995er; @Barnich:2000zw; @Quadri:2002nh] that if a local functional with ghost number one satisfies the consistency condition in eq.(\[ho.cc\]) (i.e. it is BRST closed) and the BRST differential $s$ has been trivialized by reduction to a doublet pair $$s\phi_a = \overline{\omega}_a, ~~~~ s\overline{\omega}_a=0$$ then that functional is also BRST-exact.
The present analysis generalizes this result to the case of arbitrary functionals, the locality property being nowhere used in the above construction.
One-loop invariants {#app:B}
===================
We report here the invariants parameterizing the one-loop divergences of the nonlinear sigma model in $D=4$ [@Ferrari:2005va]. The background connection is denoted by $J_{a\mu}$. $$\begin{aligned}
&& {\cal I}_1 = \int d^Dx \, \Big [ D_\mu ( F -J )_\nu \Big ]_a \Big [ D^\mu ( F -J )^\nu \Big ]_a \, ,
\nonumber \\
&& {\cal I}_2 = \int d^Dx \, \Big [ D_\mu ( F -J )^\mu \Big ]_a \Big [ D_\nu ( F -J )^\nu \Big ]_a \, ,
\nonumber \\
&& {\cal I}_3 = \int d^Dx \, \epsilon_{abc} \Big [ D_\mu ( F -J )_\nu \Big ]_a \Big ( F^\mu_b -J^\mu_b \Big ) \Big ( F^\nu_c -J^\nu_c \Big ) \, , \nonumber \\
&& {\cal I}_4 = \int d^Dx \, \Big ( \frac{m_D^2 K_0}{\phi_0} - \phi_a \frac{\delta S_0}{\delta \phi_a} \Big )^2 \, , \nonumber \\
&& {\cal I}_5 = \int d^Dx \, \Big ( \frac{m_D^2 K_0}{\phi_0} - \phi_a \frac{\delta S_0}{\delta \phi_a} \Big ) \Big ( F^\mu_b -J^\mu_b \Big )^2 \, ,
\nonumber \\
&& {\cal I}_6 = \int d^Dx \, \Big ( F^\mu_a -J^\mu_a\Big )^2
\Big ( F^\nu_b -J^\nu_b \Big )^2 \, , \nonumber \\
&& {\cal I}_7 = \int d^Dx \, \Big ( F^\mu_a -J^\mu_a\Big )
\Big ( F^\nu_a -J^\nu_a\Big )
\Big ( F_{b\mu} -J_{b\mu} \Big )
\Big ( F_{b\nu} -J_{b\nu} \Big ) \, .
\label{appE:4}\end{aligned}$$ In the above equation $D_\mu[F]$ stands for the covariant derivative w.r.t. $F_{a\mu}$ $$\begin{aligned}
D_\mu[F]_{ab} = \delta_{ab} \partial_\mu + \epsilon_{acb} F_{c \mu} \, .
\label{appE.16}\end{aligned}$$
[99]{}
R. Ferrari, JHEP [**0508**]{} (2005) 048 \[arXiv:hep-th/0504023\].
R. Ferrari and A. Quadri, Int. J. Theor. Phys. [**45**]{} (2006) 2497 \[arXiv:hep-th/0506220\].
R. Ferrari and A. Quadri, JHEP [**0601**]{} (2006) 003 \[arXiv:hep-th/0511032\].
J. Gasser and H. Leutwyler, Annals Phys. [**158**]{} (1984) 142.
R. Ferrari and P. A. Grassi, Phys. Rev. D [**60**]{} (1999) 065010 \[arXiv:hep-th/9807191\]. R. Ferrari, P. A. Grassi and A. Quadri, Phys. Lett. B [**472**]{} (2000) 346 \[arXiv:hep-th/9905192\].
A. Quadri, JHEP [**0304**]{} (2003) 017 \[arXiv:hep-th/0301211\]. A. Quadri, J. Phys. G [**30**]{} (2004) 677 \[arXiv:hep-th/0309133\]. A. Quadri, JHEP [**0506**]{} (2005) 068 \[arXiv:hep-th/0504076\].
J. Bijnens, G. Colangelo and G. Ecker, Annals Phys. [**280**]{} (2000) 100 \[arXiv:hep-ph/9907333\]. E. C. G. Stückelberg, Helv. Phys. Helv. Acta [**11**]{} (1938) 299.
H. Ruegg and M. Ruiz-Altaba, Int. J. Mod. Phys. A [**19**]{} (2004) 3265 \[arXiv:hep-th/0304245\].
R. Ferrari and A. Quadri, JHEP [**0411**]{} (2004) 019 \[arXiv:hep-th/0408168\]. O. Piguet and S. P. Sorella, “Algebraic renormalization: Perturbative renormalization, symmetries and anomalies,” Lect. Notes Phys. [**M28**]{}, 1 (1995). G. Barnich, F. Brandt and M. Henneaux, Phys. Rept. [**338**]{}, 439 (2000) \[arXiv:hep-th/0002245\]. A. Quadri, JHEP [**0205**]{} (2002) 051 \[arXiv:hep-th/0201122\]. J. Gomis, J. Paris and S. Samuel, Phys. Rept. [**259**]{} (1995) 1 \[arXiv:hep-th/9412228\].
D. Bettinelli, R. Ferrari, A. Quadri, “A Comment on the Renormalization of the Nonlinear Sigma Model”, arXiv:hep-th/0701197.
M. Henneaux and A. Wilch, Phys. Rev. D [**58**]{} (1998) 025017 \[arXiv:hep-th/9802118\].
[^1]: e-mail: [daniele.bettinelli@mi.infn.it]{}
[^2]: e-mail: [ruggero.ferrari@mi.infn.it]{}
[^3]: e-mail: [andrea.quadri@mi.infn.it]{}
[^4]: In this paper we denote by $J_{a\mu}$ the background connection. The classical action of Ref. [@Ferrari:2005ii] differs by a term $\frac{v_D^2}{8} J_{a\mu}^2$ w.r.t. the action in eq.(\[sec.1:2\]). The source coupled to the flat connection is given by $- \frac{v_D^2}{4} J_{a\mu}$. Moreover we set the gauge coupling constant $g$ to $1$.
|
**Relativistic Warning to Space Missions Aimed to Reach Phobos** [^1]\
**[Alexander P. Yefremov]{}\
*Institute of Gravitation and Cosmology of Peoples’ Friendship University of Russia\
E-mail: a.yefremov@rudn.ru***
**Abstract**\
[Disagreement in estimations of the observed acceleration of Phobos yields several theories em-pirically modifying classical description of motion of the satellite, but its orbital positions de-tected by Mars-aimed spacecraft differ from predictions. It is shown that the satellite’s orbital perturbations can be explained as manifestations of the relativistic time-delay effect ignored in classical models. So computed limits of Phobos’ acceleration essentially exceed the experimental values. The satellite’s expected orbital shift is calculated for the moment of contact with a land-ing module of the Phobos-Grunt project; the shift assessed in kilometers may prevent the mission success. Limits of the apparent relativistic accelerations are predicted for fast satellites of Jupiter.]{}\
**Keywords**: planet, satellite, Earth, Mars, Phobos, acceleration, shift, quaternion, relativity.
1.5em
1. Introduction: Phobos’ irregular motion, theory and practice {#introduction-phobos-irregular-motion-theory-and-practice .unnumbered}
==============================================================
A satellite of the planet Mars, Phobos, discovered by A.Hall in 1877 still attracts great attention. In 1911 after years of observations H.Struve offered a classical theory of Martian satellites’ motion taking into account the planet’s oblate shape and solar gravity. In 1945 B.Sharpless discovered a secular increase of Phobos velocity${}_{ }$\[1\] and surmised that the moon was spiraling in toward Mars. I.Shklovski ascribing the orbit’s decay to atmospheric friction concluded that the moon could be hollow \[2\], maybe artificial, but the orbit’s evolution was also referred to influence of tidal forces \[3\]. Later mathematical models \[4-6\] were developed in attempts to better explain observational results, some of conclusions though uncertain about the acceleration value and even sign.
Cosmic era made Phobos a desired but hardly accessible goal. In 1988 Russian Phobos-1 (said badly operated) passed by the target while Phobos-2 disappeared at 50 *m* from the moon’s surface. In 1999 Mars Climate Orbiter (NASA, also said badly operated) was lost near Phobos’ orbit, and Mars Polar Lander vanished hardly touching the Martian atmosphere. In 2003 Beagle-2 (UK) shared the destiny without any firm conclusion of the loss. Survivors showed deficiency of existing theories: Mariner 9 (NASA, 1971) \[7, 8\] and Mars Express (ESA, 2004) \[9\] found Phobos in kilometers ahead of its expected position. A new space mission Phobos-Grunt (Russia) is planned soon \[10\]; if its computer program determines the target position using old models, the project may have problems.
Another reason of the Phobos’ motion irregularity is considered here on the base of relativity theory. Section 2 comprises deduction of formulae for apparent acceleration and for relativistic shift of a solar system planet’s satellite observed from the Earth, using methods of quaternion model of relativity. In Section 3 calculated and experimental values of the Phobos’ acceleration are compared, and the shift value is assessed for the Phobos-Grunt space mission. A compact discussion is found in Section 4 with prediction of relativistic shift of fast satellites of Jupiter potentially observed from the Earth.
2. Relativistic explanation {#relativistic-explanation .unnumbered}
===========================
Let the Earth (frame of reference $\Sigma $) and a planet of the solar system ($\Sigma '$, e.g. Mars) have circular trajectories (for simplicity) in ecliptic plane and revolve about the Sun with velocities of constant values $V_{E} ,\, \, V_{P} $. A planet’s satellite with orbital period $T'={\rm const}$ (from viewpoint of $\Sigma '$) can be regarded as a clock. The value of $\Sigma-\Sigma '$ relative velocity is found as $$\label{GrindEQ__1_}
V=\sqrt{V_{P}^{2} +V_{E}^{2} -2\, V_{P} V_{E} \cos \Omega t} ,$$ where $\Omega $ is the difference of orbital angular velocities of the Earth and the planet, $\Omega t$ is a $({\bf V}_{E} ,{\bf V}_{P} )$-angle, its zero initial value is chosen at the planets’ opposition point (where the Martian satellites are usually observed optically \[6\]). The relative velocity value is always different from zero, $V\ne 0$, hence a relativistic time-delay effect exists. A clock belonging to $\Sigma '$ should be slow in $\Sigma $, i.e. the satellite, as a point of $\Sigma '$-clock’s arrow, should be seen in $\Sigma $ at earlier position on its orbit than it is in $\Sigma '$. Emphasize two features of the effect. First, it is accumulated with time since the satellite’s apparent shift increases, so the effect is potentially detected. Second, the $\Sigma $-observer will find the satellite’s motion non-uniform, since the relative velocity is variable, $V=V(t)$, the frames $\Sigma $, $\Sigma '$ being non-inertial. This hampers computation of the shift-effect by means of Special Relativity (SR) valid for inertial frames of reference, though SR can be applied locally as it is done in \[11\] in the deduction of formula for the Thomas precession. But here assessment of the relativistic shift is done with the help of a more “technological” approach based on quaternion square root from SR space-time interval, the method admitting computation of relativistic effects for arbitrary frames without addressing tensor calculus of general relativity. The quaternion model of relativity theory is described in detail in Ref. \[12\], below its very short description is given.
It is straightforwardly verified that multiplication of quaternions, the hypercomplex numbers built on one scalar (ordinary) unit and three non-commutative vector units ${\bf q}_{k} $, is invariant under rotations of the vector units by matrices belonging to special orthogonal group with complex parameters $$\label{GrindEQ__2_}
\, {\bf q}_{n'} =O_{n'k} {\bf q}_{k}$$ (summation in the repeating indices is assumed here and further on), $\, O_{n'k} \in SO(3,\mathbb{C})$, the group being 1:1 isomorphic to the Lorentz group. It is also proved that similar rotations keep form of the vector-quaternion $$\label{GrindEQ__3_}
\, d{\bf z}=(dx_{k} +idt\, e_{k} )\, {\bf q}_{k} ,$$ under the space-time orthogonality condition $\, dx_{k} e_{k} =0$, where$\, \{ dx_{k} ,\, dt\} $ are differentials of a particle’s space-time coordinates in a frame $\, \Sigma \equiv \{ {\bf q}_{k} \} $,$\, e_{k} $ is any unit vector. The square of the vector-interval yields $\, (d{\bf z})^{2} =dt^{2} -dr^{2} $, the Minkowski-type space-time interval of SR, so instead of invariance of this interval one can analyze form-invariance of $\, d\textbf{z}$ thus obtaining all cinematic effects of SR with an additional advantage to consider non-inertial frames of reference \[12, 13\]. Apply the method for computation of characteristics of the satellite’s motion estimated by an Earth’s observer.
The form-invariant vector-interval describing the relativistic system “Earth-planet (satellite)” is chosen in the form automatically satisfying the space-time orthogonality condition $$\label{GrindEQ__4_}
\, d\textbf{z}=i\,cdt'{\bf q}_{1'} =i\, cdt({\bf q}_{1} +\frac{V}{c} \, {\bf q}_{2} )$$ the fundamental velocity $c$ is constant, $dt'$ is a proper time interval in $\, \Sigma '$, $\, dt$ is respective time interval of the observer. The cinematic situation described by is equivalent to the $\, \Sigma -\Sigma '$ transformation of the type with the matrix $$\label{GrindEQ__5_}
O_{k'n} =\left(\begin{array}{ccc} {\cosh \psi } & {-i\sinh \psi } & {0} \\ {i\sinh \psi } & {\cosh \psi } & {0} \\ {0} & {0} & {1} \end{array}\right)$$ what leads to standard expression for relative velocity as a function of hyperbolic parameter $V/c=\tanh \psi $, and to the time-delay relation $$\label{GrindEQ__6_}
dt'=\, dt/\cosh \psi$$ apparently the same as in SR but valid for the non-inertial case. Now let the time-interval $dt'\to \, T'$ be period of the satellite’s revolution measured in$\, \Sigma '$ (in fact, a physically real period, in this case is small compared to $t$, time of observation), and $dt\to \, T$ be the similar “period” (here a variable magnitude) observed from$\, \Sigma $. Then acquire the form $$\label{GrindEQ__7_}
T'=T/\cosh \psi =T\, \sqrt{1-V^{2} (t)/c^{2} }$$ so the period observed from the Earth is always greater that the real one $T>T'$. But search for the period’s difference, whatever desirable it could be, is of no use since relativistic corrections, important as will be shown below, would be slurred over by uncertainty of our knowledge of the involved magnitudes: gravitational constant \[14\], the planet’s and the satellite’s physical parameters \[15\]. So only the limits of the satellite’s acceleration value and an integral apparent shift will be assessed with possible accuracy.
Making future expressions compact, denote $$A\equiv (V_{P}^{2} +V_{E}^{2} )/2c^{2} <<1, B\equiv \, V_{P} V_{E} /c^{2} <<1$$ and using (1, 7) find (up to the small *A*, *B*) the difference between the values of satellite’s real orbital velocity $V'_{S} $ estimated in $\Sigma '$, and observed velocity $V_{S} (t)$ estimated in $\Sigma $ $$\label{GrindEQ__8_}
V'_{S} -V_{S} (t)=2\pi \, r\left(\frac{1}{T'} -\frac{1}{T} \right)=V'_{S} (A-B\cos \Omega t)$$ *r* being the radius of the satellite’s orbit. Differentiation of with respect to time of observation leads to formula for apparent satellite’s acceleration $$\label{GrindEQ__9_}
a=\frac{dV_{S} (t)}{dt} =-V'_{S} \Omega B\sin \Omega t$$ Now using find the satellite’s orbital shift, the difference between its real position and that observed from the Earth $$\label{GrindEQ__10_}
\Delta l\equiv \int (V'_{S} -V_{S} )\, dt =V'_{S} \, (\, At-B\, \sin \Omega t\, /\Omega )$$ the integration constant is chosen zero assuming no shift at the beginning of the observation. As expected the shift’s value monotonously grows linearly in time with imposed cyclic displacements having period of the oppositions.
3. Computations of the effects for the relativistic Earth-Mars (Phobos) system {#computations-of-the-effects-for-the-relativistic-earth-mars-phobos-system .unnumbered}
==============================================================================
Let the planet and satellite be Mars and Phobos. Observed secular acceleration of Phobos cited in literature varies from \[1\] $a_{\exp } =+1.88\times 10^{-3} \deg yr^{-2} $ to zero \[3\] and to the negative value \[4, 6\] $a_{\exp } =-0.83\times 10^{-3} \deg yr^{-2} $, in degrees of the satellite’s longitude in a year, one degree of Phobos’ orbit (1/180 of the orbit length) equal to $327\, km$. Calculation of respective relativistic values requires the following (conventional) data available in many sources, e.g. in \[15\] $$\begin{array}{ll}
\mbox{fundamental velocity} & c=2.997\times 10^{10} \, cm\, \, s^{-1} \\
\mbox{Earth's orbital velocity} & V_{E} =2.978\times 10^{6} \, cm\, \, s^{-1}\\
\mbox{Mars' mean orbital velocity} & V_{P} =2.413\times 10^{6} \, cm\, \, s^{-1}\\
\mbox{Earth-Mars angular velocity difference} & \Omega =0.932\times 10^{-7} \, s^{-1}\\
\mbox{Phobos' orbital velocity} & V'_{S} =2.14\times 10^{5} cm\, \, s^{-1}
\end{array}$$ then the small unit-free coefficients are found as $A=8.174\times 10^{-9} $, $B=7.997\times 10^{-9} $.
Using formula find upper limit (amplitude) of the apparent acceleration caused by relativistic reasons $$\label{GrindEQ__11_}
a_{\max } =V'_{S} \Omega \, B=1.59\times 10^{-10} cm\, \, s^{-2} =4.84\times 10^{-3} \deg yr^{-2}$$ Thus the experimental acceleration values are found well inside the limits of the apparent acceleration $-a_{\max } <a_{\exp } <a_{\max } $, its sign depending on the observational data obtained before or after the opposition point. In particular, if the data fixed in a month before the Mars-Earth opposition is compared with that obtained at the opposition point, then the conclusion should be made that the satellite moves with an acceleration having the value cited in Ref. \[1\]. Vice versa, the satellite’s deceleration \[4, 6\] should be detected if the observation is done a couple of weeks after the opposition peak.
Now turn to Eq. to assess the Phobos’ apparent orbital shift for parameters of the mission Phobos-Grunt. The mission is planned to start at the end of 2011, and it is expected to reach Phobos within $1.25\, yr$ of flight[^2]. The satellite’s orbital parameters for the spacecraft’s computer program could be obtained at the last opposition in Jan. 2010 (i.e. plus $1.75\, yr$ to the flight time) or, for higher accuracy, in the last perihelion opposition in Aug. 2003 (plus $8.25\, yr$), thus the time intervals between the observations and the spacecraft-moon contact are $t_{1} =3yr$ or $t_{2} =9.5yr$. So if the relativistic effect is ignored, the mission can find Phobos in $\Delta l_{1} =1.55\, km$ or $\Delta l_{2} =5.18\, km$ ahead of its expected position, as earlier missions did. These shifts appear to be not too great compared to the moon’s size ($20\, km$), moreover, corrections of spacecraft’s Martian orbits are foreseen. But the shift-effect seems worth to be taken into account in advance since a light signal correcting the spacecraft position will have to cover twice the Earth-Mars distance of $2.15\times 10^{8} \, km$ (at the planned contact moment), and it will do it within 24 minutes, a time sufficient for $3000\, km$ displacement of the spacecraft on its Martian orbit (recall for comparison the last $50\, m$ of Phobos-2).
An independent effect of apparent replacement of Phobos arises when distance between the satellite and an observer changes, light velocity being finite. Elliptic shape of the Mars’ orbit makes this phenomenon essential for the Earth’s observer; as well a Phobos’ virtual acceleration must be detected by a spacecraft as it approaches the moon. But this effect is obvious and hopefully is taken into account in any space mission.
4. Discussion and a prediction {#discussion-and-a-prediction .unnumbered}
==============================
The above given formulas and numbers should of course be regarded only as a zero-iteration to a mathematical job good enough for engineering purposes. Strict computational technology must take into account a series of essential details, among them eccentricity of the planets’ orbits, dependence on time of the velocities, and certainly reliable values of the satellite’s dynamic parameters refined from synthesis of observations and theoretic considerations, e.g. solution of the moon’s equation of motion in Schwarzschild (or even Kerr) gravity as well as gravitational influence of other moons. But realization of these improvements is technologically clear, and if necessary it can be successfully performed. Nonetheless the shift-effect is noted. In reality its existence will hardly cause troubles for spacecraft aimed to explore a planet due to tiny probability to meet a small moon. But if the goal is the moon itself the effect may become important. Hence to a certain extent it must be taken into consideration, and in particular in the planned Phobos-Grunt project; otherwise the mission will be under a noticeable “relativistic danger”.
Note in conclusion that the relativistic shift-effect potentially can be detected in the motion of other satellites of the solar system planets; e.g. the Earth-Jupiter relative motion should cause apparent acceleration of fast satellites of Jupiter. Assess the range of the acceleration values for the fastest Jovian moons Metis and Adrastea, necessary data given below $$\begin{array}{ll}
\mbox{mean orbital velocity of Jupiter}& V_{P} =1.307\times 10^{6} \, cm\, \, s^{-1} \\
\mbox{Earth-Jupiter angular velocity difference}& \Omega =1.823\times 10^{-7} \, s^{-1}\\
\mbox{mean orbital velocity of Metis} &V'_{M} =3.150\times 10^{6} cm\, \, s^{-1} \\
\mbox{mean orbital velocity of Adrastea}& V'_{A} =3.138\times 10^{6} cm\, \, s^{-1}
\end{array}$$ the unit-free coefficient being $B=4.331\times 10^{-9} $. As is done above for Phobos, the range of the acceleration values is calculated with the help of formula . So it is predicted here that precise experimental measurement of parameters of Jovian satellites’ motion may lead to descovery of the moons acceleration inside the limits for Metis $a_{M} \le 2.49\times 10^{-9} cm\, \, s^{-2} =5.54\times 10^{-3} \deg yr^{-2} $, one degree of Metis’ orbit equal to $4,468\, km$, and for Adrastea
$a_{A} \le 2.48\times 10^{-9} cm\, \, s^{-2} =5.47\times 10^{-3} \deg yr^{-2} $, one degree of Adrastea’s orbit equal to $4,503\, km$.
[99]{} B.P.Sharpless, “Secular acceleration in the longitudes of the satellites of Mars,” *Astron. J*. Vol. 51, 1945, pp.185-186
**DOI:** 10.1086/105871 I.S.Shklovsky, “The Universe, Life, and Mind”, Academy of Sciences USSR, Moscow, 1962, pp.156-165
G.A.Wilkins, “Meeting of Royal Astronomical Society,” *The Observatory* Vol. 90, No.976, 1970, pp.37-38
A.T.Sinclair, “The motions of the satellites of Mars,” *Monthly Notices Roy.Astron. Soc*., Vol. 155, 1972, pp.249-274
G.A.Wilkins, A.T.Sinclair, “The dynamic of the planets and their satellites”, *Proc. R. Soc. Lond*. Vol. A 386, 1974, pp.85-104
**DOI:** 10.1098/rspa.1974.0008 V.A.Shor, “The motion of the Martian satellites,” *Celestial Mechanics* Vol. 12, 1975, pp.61-75;
**DOI:** 10.1007/BF01228625 G.H.Born, T.C.Duxbury, “The methods of Phobos and Deimos from Mariner 9 TV data,” *Celestial Mechanics* Vol. 12, 1975, pp.77-88
**DOI:** 10.1007/BF01228626 A.T.Sinclair, “The orbits of the satellites of Mars determined from Earth based and spacecraft observations,” *Astron. Astrophys*. Vol. 220, 1989, pp.321-328
Web site: European Space Agency, Mars Express,
, 2009 Web site: RussianSpaceWeb, , 2011 E.F.Taylor, J.A.Wheeler, “Space-Time Physics,” W.H.Freeman & C, San Francisco, London, 1966 A.P.Yefremov, “Quaternion model of relativity: solutions for non-inertial motions and new effects,” *Adv. Sci. Lett*., Vol. 1, No. 2, 2008, pp.79–86
A.P.Yefremov, “Algebra, Geometry and Physical Theories,” *Hypercomplex Numbers in Geometry and Phys*. Vol. 1, No. 1, 2004, pp.105-120
. G.T.Gilles, “The Newtonian gravitational constant, resent measurements and related studies,” *Rep. Prog. Phys*. Vol. 60, 1997, pp.151-225
**DOI:** 10.1088/0034-4885/60/2/001 H.H.Kieffer, B.M.Jakosky, C.W.Snyder, and M.S.Matthews, eds, “Mars,” Tucson: Univ. of Arizona Press, 1993
[^1]: Video report of this topic on the scientific seminar in PFUR:\
`http://www.youtube.com/watch?v=VG3Z_j97Y0Y` (in English)\
`http://www.youtube.com/watch?v=fNqvcsYkix0` (in Russian)
[^2]: Unfortunately the mission failed at the launching stage (9 Nov. 2011), as reported due to technical reasons.
|
---
abstract: 'In dose-finding trials, due to staggered enrollment, it might be desirable to make dose assignment decisions in real-time in the presence of pending toxicity outcomes, for example, when patient accrual is fast or the dose-limiting toxicity is late-onset. Patients’ time-to-event information may be utilized to facilitate such decisions. We propose a unified statistical framework for time-to-event modeling in dose-finding trials, which leads to two classes of time-to-event designs: TITE deigns and POD designs. TITE designs are based on inference on toxicity probabilities, while POD designs are based on inference on dose-finding decisions. These two classes of designs contain existing designs as special cases and also give rise to new designs. We discuss and study theoretical properties of these designs, including large-sample convergence properties, coherence principles, and the underlying decision rules. To facilitate the use of time-to-event designs in practice, we introduce efficient computational algorithms and review common practical considerations, such as safety rules and suspension rules. Finally, the operating characteristics of several designs are evaluated and compared through computer simulations.'
address:
- 'Department of Public Health Sciences, University of Chicago'
- 'Department of Public Health Sciences, University of Chicago'
author:
- Tianjian Zhou
- |
Yuan Ji\
Department of Public Health Sciences, University of Chicago
bibliography:
- 'ref-TITE-TPI.bib'
title: 'A Unified Framework for Time-to-Event Dose-Finding Designs'
---
\#1
Introduction {#sec:intro}
============
The goal of dose-finding trials is to find the maximum tolerated dose (MTD), the highest dose with toxicity probability close to or lower than a pre-specified target rate $p_{\text{T}}$. The type of toxicity is usually severe, like organ failure, and is called dose-limiting toxicity (DLT). The premise behind dose-finding trials is that both the toxicity and efficacy of a treatment monotonically increase with the dose level. A dose level that is too low can not provide needed efficacy, e.g. anti-tumor activity, while a dose level that is too high might have severe toxicity. Therefore, it is crucial to find an appropriate dose that has the highest possible efficacy while maintains tolerable toxicity. Usually, a grid of discrete dose levels are investigated, and cohorts of patients are sequentially enrolled and adaptively treated at dose levels based on the previously observed data. The trial objectives include the identification of the MTD and the estimation of the dose-toxicity curve, as well as maximizing the chance of treating patients at safe and efficacious doses.
The evaluation of DLT is conducted by following patients post-treatment within a time window. During the time window, DLT events are recorded, if any. If a patient does not experience any DLT during the follow-up window, the patient is declared having no DLT. Most existing designs require the DLT evaluation of all the previously enrolled cohorts to be completed before they can make a treatment assignment for the next cohort. Consequently, we refer to this type of designs as *complete-data designs*. Examples of complete-data designs include the 3+3 design [@storer1989design], continual reassessment method (CRM, [@o1990continual; @goodman1995some; @shen1996consistency; @o1996continual]), escalation with overdose control (EWOC, [@babb1998cancer]), cumulative cohort design (CCD, [@ivanova2007cumulative]), Bayesian logistic regression model (BLRM, [@neuenschwander2008critical]), modified toxicity probability interval design (mTPI, [@ji2010modified]; [@ji2013modified]), Bayesian optimal interval design (BOIN, [@liu2015bayesian]; [@yuan2016bayesian]), mTPI-2 design ([@guo2017bayesian]), keyboard design ([@yan2017keyboard]), semiparametric dose finding method (SPM, [@clertant2017semiparametric]; [@clertant2018semiparametric]) and i3+3 design ([@liu2019design]), among many others. For therapies of which the toxicity is acute and can be ascertained in early cycles, such as cytotoxic therapies, waiting for the DLT evaluation of previous patients may not be a concern, as the DLT assessment window can be short. However, for therapies that usually have late-onset toxicity, such as immunotherapies [@weber2015toxicities; @kanjanapan2019delayed], it is more sensible to use a relatively long assessment window. This may cause difficulty for these designs to operate, since patient enrollment needs to be frequently suspended until the previous patients have finished their assessment. The same difficulty arises when patient accrual is fast compared to the length of the assessment window. For example, in Figure \[fig:dose\_finding\](a), while waiting for the DLT outcomes of the first 3 patients, the trial needs to be suspended, and 3 eligible patients have to be turned away. Trial suspension is undesirable in practice for two reasons. First, trial duration is prolonged, which delays scientific research and drug development. Second, subsequent patients that are available for enrollment need to be turned away, which results in a delay in their cancer care. Many patients participating in the trial do not have alternative choices for treatment, and the trial may be their last treatment option. Their diseases may also be in rapid deterioration, thus they are in need of immediate treatment.
---------------------------- -- ----------------------------
\(a) Complete-data designs \(b) Time-to-event designs
---------------------------- -- ----------------------------
To address these practical concerns, several designs have been proposed to allow for consecutive patient accrual even if some enrolled patients are still pending for DLT assessment. These include the time-to-event CRM (TITE-CRM, [@cheung2000sequential; @normolle2006designing]), rolling six design (R6, [@skolnik2008shortening]), expectation-maximization CRM (EM-CRM, [@yuan2011robust]), data augmentation CRM (DA-CRM, [@liu2013bayesian]), time-to-event BOIN design (TITE-BOIN, [@yuan2018time]), time-to-event keyboard design (TITE-keyboard, [@lin2018time]), rolling TPI design (R-TPI, [@guo2019rtpi]) and probability-of-decision TPI design (POD-TPI, [@zhou2019pod]). Except for R6 and R-TPI, these designs utilize time-to-event information to make treatment assignments thus are referred to as *time-to-event designs*. As an example, in Figure \[fig:dose\_finding\](b), when the 4th patient is available for enrollment, patients 2 and 3 are still being followed without definitive outcomes. Based on the DLT outcome of patient 1, time-to-DLT information of patient 1 and follow-up time information of patients 2 and 3, a time-to-event design may enroll the patient and de-escalate the dose level, which avoids the trial suspension.
In this article, we aim to propose a unified statistical framework for time-to-event modeling in dose-finding trials. See Figure \[fig:illustration\] for a summary. The key component is the construction of the likelihood function with time-to-event data, and the primary interest is inference on toxicity probabilities. Specifically, two equivalent modeling approaches can be taken for the likelihood construction. The statistical framework gives rise to two classes of time-to-event designs, which contain the existing time-to-event designs as special cases and also lead to new time-to-event designs. The first class of time-to-event designs, called TITE designs, make dose-finding decisions based on inference on toxicity probabilities. The second class of time-to-event designs, called POD (probability of decision) designs, is a new type of designs that directly make inference on dose-finding decisions when DLT outcomes may be pending. The POD designs directly reflect the confidence of possible decisions and offer the investigators and regulators a way to properly assess and control the chance of making incompatible decisions when not all patients have been completely followed. As a result, we argue that the POD designs might be more suitable for practical applications.
Along with the statistical framework, we introduce several computational algorithms to facilitate the use of time-to-event designs in practice. We discuss and study theoretical properties of time-to-event designs, such as large-sample convergence properties, coherence principles and the underlying decision rules, with a focus on interval-based nonparametric designs. We also review practical considerations of time-to-event designs, which are important in the execution of clinical trials. Usually, ad-hoc rules need to be imposed to ensure that the designs satisfy safety concerns and ethical constraints. Lastly, we examine finite-sample operating characteristics of some designs through computer simulations.
The remainder of the paper is structured as follows. In Section \[sec:complete\_data\_design\], we give a brief review of complete-data designs. In Section \[sec:framework\], we describe a framework for time-to-event modeling in dose-finding trials. In Section \[sec:two\_class\_designs\], we introduce the two classes of time-to-event designs, including the TITE and POD designs. In Section \[sec:property\], we study theoretical properties of time-to-event designs. In Section \[sec:practical\], we discuss practical considerations. We assess the operating characteristics of several existing and newly proposed dose-finding designs via simulation studies in Section \[sec:simulation\]. Finally, we conclude with a discussion in Section \[sec:discussion\]. Technical details, including the computational algorithms and proof of the theoretical results, are provided in the Appendix.
Review of Complete-Data Designs {#sec:complete_data_design}
===============================
We start with a brief review of complete-data designs. At a certain moment in a dose-finding trial, suppose $N$ patients have been treated, and the $(N+1)$th patient is eligible for enrollment. Let $Z_i \in \{1, \ldots, D \}$ denote the dose assigned to patient $i$, where $D$ is the number of available doses in the trial. Each patient is supposed to be followed for a fixed period of time $W$, and we use $Y_i = 1$ or $0$ to represent whether or not patient $i$ experiences DLT within the time window, respectively. For example, in many oncology trials, $W = 28$ days. The conditional distribution of $Y_i$ given $Z_i$ is commonly modeled with a Bernoulli distribution, $$\begin{aligned}
\Pr(Y_i = y \mid Z_i = z, p_z) = p_z^y (1 - p_z)^{1-y}, \quad
y \in \{ 0, 1 \}.
\label{eq:bernoulli_dist}\end{aligned}$$ Here, $p_z$ is the probability of DLT at dose $z$ within the assessment window. A widely recognized assumption is that the DLT probability is monotone with the dose level, i.e. $p_1 \leq p_2 \leq \cdots \leq p_D$.
Suppose $Y_i$’s are fully observed for the first $N$ patients, and denote by ${\mathcal{H}}_{N}^* = \{ (Y_i, Z_i) : i \leq N \}$ the previous history of observations. A complete-data design ${\mathcal{A}}^*$ can be viewed as a function of ${\mathcal{H}}_{N}^*$, which prescribes a dose ${\mathcal{A}}^*({\mathcal{H}}_{N}^*)$ for the new patient through two steps: (1) making inference about $p_z$’s, and (2) translating such inference to a dose-finding decision. Inference on $p_z$’s can be based on the likelihood, $$\begin{aligned}
L( {\bm{p}}\mid {\mathcal{H}}_N^* ) = \prod_{i = 1}^N p_{z_i}^{y_i} (1 - p_{z_i})^{1 - y_i},
\label{eq:likelihood_complete}\end{aligned}$$ where ${\bm{y}}= (y_1, \ldots, y_N)$ and ${\bm{z}}= (z_1, \ldots, z_N)$ are the observed outcomes and dose assignments for the $N$ patients, respectively, and ${\bm{p}}= (p_1, \ldots, p_D)$ is the vector of toxicity probabilities. As mentioned in Section \[sec:intro\], there is a rich literature on complete-data dose-finding designs. We provide a brief review of several designs in Appendix \[supp:sec:review\].
For the following discussion, it is helpful to categorize the existing complete-data designs into several classes. First, we can categorize the designs according to how they make inference about ${\bm{p}}$. A *parametric* design (e.g., CRM and BLRM) models the toxicity probabilities with a parametric curve $p_z = \phi(z, \bm \alpha)$, which is monotonically increasing in $z$. A *nonparametric* design (e.g., BOIN and mTPI-2) does not make parametric assumptions for estimating ${\bm{p}}$. A *semiparametric* design (e.g., SPM) does not make parametric assumptions about ${\bm{p}}$ but imposes some constraint on ${\bm{p}}$ to ensure its (partial) ordering. Second, we can also categorize the designs based on how they translate inference on ${\bm{p}}$ to a dose assignment decision. Generally, a design starts at a low dose. At each subsequent step, a *point-based* design (e.g., CRM) allocates the next cohort to $d^* = \operatorname*{arg\,min}_{z} | \hat{p}_{z} - p_{{\text{T}}} |$, where $\hat{p}_{z}$ is a point estimate of $p_z$ (e.g., MLE or posterior mean) based on . On the other hand, suppose the currently-administered dose is $d$, and $\epsilon_1 > 0$ and $\epsilon_2 > 0$ are pre-determined constants. *Interval-based* designs make dose-finding decisions based on the interval $I_{{\text{E}}} = [p_{{\text{T}}} - \epsilon_1, p_{{\text{T}}} + \epsilon_2]$. One class of interval-based designs (e.g., CCD and BOIN) make stay (at $d$), escalation (to $d+1$) or de-escalation (to $d-1$) decisions based on whether $\hat{p}_{d}$ is within, below or above $I_{{\text{E}}}$, respectively. Another class of interval-based designs (e.g., mTPI-2 and keyboard) consider a partition of the $[0, 1]$ interval into sub-intervals (from left to right) $\{ I_{{\text{U}}_0}, \ldots, I_{{\text{U}}_{K_1}} \}$, $I_{{\text{E}}}$ and $\{ I_{{\text{O}}_0}, \ldots, I_{{\text{O}}_{K_2}} \}$, where $I_{{\text{E}}} = [p_{{\text{T}}} - \epsilon_1, p_{{\text{T}}} + \epsilon_2]$ is the only sub-interval that contains $p_{{\text{T}}}$. The dose-finding decision is stay (at $d$), escalation (to $d+1$) or de-escalation (to $d-1$) if $ \arg \max_{j}$ $\Pr(p_d \in I_j \mid {\mathcal{H}}_N^*)$ equals ${\text{E}}$, belongs to $\{ {\text{U}}_0, \ldots, {\text{U}}_{K_1} \}$ or belongs to $\{ {\text{O}}_0, \ldots, {\text{O}}_{K_2} \}$, respectively. Here, $\Pr(p_d \in I_j \mid {\mathcal{H}}_N^*)$ is the posterior probability of $p_d$ falling within the interval $I_j$. Although the existing complete-data designs differ in many aspects, they can be extended to time-to-event designs based on the same strategy, which will be elaborated in the next sections.
Framework for Time-to-Event Modeling in Dose-Finding Trials {#sec:framework}
===========================================================
Setup {#sec:notation}
-----
Since patients enter clinical trials sequentially at random time, it is often the case that when a new patient is eligible for enrollment, some previously enrolled patients are still being followed without DLT, thus their DLT outcomes $Y_i$’s remain unknown. As discussed in Section \[sec:intro\], even when some outcomes are pending, it is still desirable to enroll the patient and assign an appropriate dose. Complete-data designs do not allow this, and time-to-event designs attempt to address this problem. The key is to develop inference on ${\bm{p}}$ and a decision rule. With pending outcomes, inference on ${\bm{p}}$ becomes less straightforward and ideally requires modeling time-to-event data, because these data provide information regarding the likelihood of the pending patients experiencing DLT in the future [@cheung2000sequential; @yuan2018time]. For example, a patient followed for 21 days without DLT provides different information from another followed for 2 days without DLT. Such difference can be exploited for better inference and decision making.
Define *trail time* as the number of days since the enrollment of the first patient. Let $\tau_i^*$ denote the trial time when patient $i$ is enrolled. By definition, $\tau_1^* = 0$. A patient will be followed for a duration of $W$ days. Call $W$ the follow-up window. We denote by $(\tau_i^* + T_i)$ the trial time when patient $i$ experiences DLT. Note that $T_i$ can be greater than $W$ in reality. At any trial time $\tau$, patient $i$ may or may not have experienced DLT. If s/he has experienced DLT, then $(\tau_i^* + T_i) \le \tau$. If s/he has not experienced DLT, s/he either is still being followed or has completed $W$ days of follow-up, but without experiencing DLT in either case, and we call the patient is “censored”. At trial time $\tau$, a patient $i$ who is censored has a censoring time $U_i (\tau) = \min \{ \max(\tau - \tau^*_i, 0),
W \}$. Let $Y_i$ be the indicator of whether patient $i$ experiences DLT within the follow-up window $W$, i.e., $Y_i
= {\mathbbm{1}}(T_i \leq W)$. We do not observe $T_i$ at trial time $\tau$ if $T_i > U_i(\tau)$, but we always observe the follow-up time of the patient, given by $V_i = T_i \wedge U_i(\tau)$. Similarly, we do not observe indicator $Y_i$ at trial time $\tau$ if $T_i > U_i(\tau)$ and $U_i(\tau) < W$, but we always know the current DLT status of the patient, given by $\tilde{Y}_i = {\mathbbm{1}}[T_i \leq U_i(\tau)]$. For example, in Figure \[fig:dose\_finding\](b), we have $\tau_1^* = 0$ and $\tau_2^* = 7$ for patients 1 and 2, respectively. On day $\tau = 22$ since trial start, for patient 1, we have $T_1 = 21$, $U_1 = 22$, $V_1 = 21$, and $Y_1 = {\tilde{Y}}_1 = 1$; for patient 2, we have $U_2 = 15$, $V_2 = 15$, ${\tilde{Y}}_2 = 0$, and $T_2$ and $Y_2$ are unknown. The available information at study time $\tau$ can be summarized by ${\mathcal{H}}(\tau) = \{ (\tilde{Y}_i(\tau), V_i(\tau), Z_i) : i \leq N(\tau) \}$, where $N(\tau)$ is the total number of treated patients just prior to $\tau$. A time-to-event design ${\mathcal{A}}$ can be viewed as a function of ${\mathcal{H}}(\tau)$. That is, if a new patient is enrolled at time $\tau$, the design would assign a dose ${\mathcal{A}}[{\mathcal{H}}(\tau)]$ for the patient.
We introduce some more notation to facilitate the upcoming discussion. Denote by $$\begin{aligned}
B_i(\tau) =
\begin{cases}
0, & \text{if } {\tilde{Y}}_i(\tau) = 0 \text{ and } V_i(\tau) < W; \\
1, & \text{if } {\tilde{Y}}_i(\tau) = 1 \text{, or } {\tilde{Y}}_i(\tau) = 0 \text{ and } V_i(\tau) = W.
\end{cases}\end{aligned}$$ In other words, $B_i(\tau) = 0$ or 1 represents patient $i$’s DLT outcome $Y_i$ has not or has been fully assessed, respectively. Therefore, $Y_i = {\tilde{Y}}_i(\tau)$ if $B_i(\tau) = 1$. Following the convention in the missing data literature, we use ${\bm{Y}}_{{\text{obs}}} (\tau) = \{ Y_i : B_i(\tau) = 1, i \leq N(\tau)\}$ or ${\bm{Y}}_{{\text{mis}}} (\tau) = \{ Y_i : B_i(\tau) = 0, i \leq N(\tau) \}$ to represent the sets of DLT outcomes that have been observed or are pending at time $\tau$, respectively. Lastly, let $N_z(\tau) = \sum_{i = 1}^{N(\tau)} {\mathbbm{1}}(Z_i = z)$ denote the number of patients that have been treated at dose $z$ just prior to $\tau$. Among the $N_z(\tau)$ patients, let $n_z(\tau)$, $m_z(\tau)$ and $r_z(\tau)$ denote the number of patients having DLT, non-DLT and pending outcomes, respectively. Mathematically, these can be written as $n_z(\tau) = \sum_{i = 1}^{N(\tau)} {\mathbbm{1}}[Z_i = z, Y_i = 1, B_i (\tau) = 1]$, $m_z(\tau) = \sum_{i = 1}^{N(\tau)} {\mathbbm{1}}[Z_i = z, Y_i = 0, B_i (\tau) = 1]$, and $r_z(\tau) = \sum_{i = 1}^{N(\tau)} {\mathbbm{1}}[Z_i = z, B_i (\tau) = 0]$.
In the next sections, we will describe how to use the observed data ${\mathcal{H}}(\tau)$ to make inference on ${\bm{p}}$.
Modeling Time-to-Toxicity Data {#sec:model_tite}
------------------------------
In the first step, we specify a model for the time-to-toxicity data. For the following discussion, patient index $i$ is suppressed from the subscript to simplify notation when no confusion is likely. Recall that $\{ Y= 1 \}$ is equivalent to $\{ T \leq W \}$. We still model $Y$ with a Bernoulli distribution as in Equation , which is equivalent to assuming $\Pr(T \leq W \mid Z = z, p_z) = p_z$. Next, we generally write $f_{T \mid Z}(t \mid z, {\bm{p}}, {\bm{\xi}})$ the probability density function (pdf) of $T$ at dose level $Z = z$, where ${\bm{\xi}}$ denotes additional and nuisance parameters that characterize the distribution of $T$. For $t \leq W$, $$\begin{aligned}
f_{T \mid Z}(t \mid z, {\bm{p}}, {\bm{\xi}})
&= \Pr(Y = 1 \mid Z = z, {\bm{p}}, {\bm{\xi}}) \cdot f_{T \mid Z, Y}(t \mid z, Y = 1, {\bm{p}}, {\bm{\xi}}) \\
&= p_z \cdot f_{T \mid Z, Y}(t \mid z, Y = 1, {\bm{\xi}}).\end{aligned}$$ The first equation is true since $f_{T \mid Z, Y}(t \mid z, Y = 0, {\bm{p}}, {\bm{\xi}}) = 0$ for $t \leq W$. The second equation assumes that the conditional distribution $f_{T \mid Z, Y}(t \mid z, Y = 1, {\bm{p}}, {\bm{\xi}})$ does not depend on ${\bm{p}}$. That is, given that a patient experiences DLT within the assessment window at a dose $z$, when the patient experiences the DLT does not depend on the toxicity probability $p_z$ of the dose. This assumption is implicitly made by all existing methods (both complete-data designs and time-to-event designs). Note that if $f_{T \mid Z, Y}(t \mid z, Y = 1, {\bm{p}}, {\bm{\xi}})$ does depend on ${\bm{p}}$, it should be included in the complete data likelihood according to the likelihood principle.
To specify $f_{T \mid Z}(t \mid z, {\bm{p}}, {\bm{\xi}})$, it suffices to specify $f_{T \mid Z, Y}(t \mid z, Y = 1, {\bm{\xi}})$, which can be any probability density with support in $(0, W]$. Examples of possible specifications of $f_{T \mid Z, Y}(t \mid z, Y = 1, {\bm{\xi}})$ include a uniform distribution, a piecewise uniform distribution, a discrete hazard model, a piecewise constant hazard model, and a rescaled beta distribution. See Appendix \[supp:sec:model\_tite\] for details.
The survival function of $T$ is given by $$\begin{aligned}
S_{T \mid Z} (t \mid z, {\bm{p}}, {\bm{\xi}}) = \Pr(T > t \mid Z = z, {\bm{p}}, {\bm{\xi}}) = \int_{t}^{\infty} f_{T \mid Z}(v \mid z, {\bm{p}}, {\bm{\xi}}) {\text{d}}v.\end{aligned}$$ The survival function must satisfy $S_{T \mid Z} (W \mid z, {\bm{p}}, {\bm{\xi}}) = 1 - p_z$. This is important, since $p_z$ only represents the probability of DLT within time window $(0, W]$. For $t < W$, $$\begin{aligned}
S_{T \mid Z} (t \mid z, {\bm{p}}, {\bm{\xi}}) = 1 - p_z \int_{0}^t f_{T \mid Z, Y}(v \mid z, Y = 1, {\bm{\xi}}) {\text{d}}v \triangleq 1 - p_z \rho(t \mid z, {\bm{\xi}}),\end{aligned}$$ where we denote by $$\begin{aligned}
\rho(t \mid z, {\bm{\xi}}) = \int_{0}^t f_{T \mid Z, Y}(v \mid z, Y = 1, {\bm{\xi}}) {\text{d}}v.
\label{eq:rho}\end{aligned}$$
\[rmk:rho\] The function $\rho(t \mid z, {\bm{\xi}})$ must satisfy: (1) $\rho(t \mid z, {\bm{\xi}}) \in [0, 1]$ for $t \in (0, W]$, and (2) $\rho(W \mid z, {\bm{\xi}}) = 1$. Also, by definition, $\rho(t \mid z, {\bm{\xi}})$ is non-decreasing in $t$.
Survival Likelihood {#sec:likelihood_tite}
-------------------
The likelihood of ${\bm{p}}$ and ${\bm{\xi}}$ at time $\tau$ can be constructed based on survival modeling by treating the unknown times-to-toxicities (i.e., $\{ T_i: {\tilde{Y}}_i(\tau) = 0, i \leq N(\tau) \}$) as censored observations. See, for example, Section 3.5 in [@klein2006survival]. To simplify notation, we omit the time index $\tau$ in the following discussion. For the patients with observed toxicities ($\tilde{Y}_i = 1$), their contributions to the likelihood are the pdfs of the times-to-toxicities. For the patients without observed toxicities ($\tilde{Y}_i = 0$), their times-to-toxicities are right censored, and their contributions to the likelihood are the survival functions at the censoring times. In particular, $$\begin{aligned}
L( {\bm{p}}, {\bm{\xi}}\mid {\mathcal{H}})
=
\prod_{i = 1}^N \Big[ f_{T \mid Z}(v_i \mid z_i, {\bm{p}}, {\bm{\xi}})^{{\mathbbm{1}}({\tilde{y}}_i = 1)}
S_{T \mid Z}(v_i \mid z_i, {\bm{p}}, {\bm{\xi}})^{{\mathbbm{1}}({\tilde{y}}_i = 0)} \Big].\end{aligned}$$ The likelihood can be further written as $$\begin{gathered}
L ( {\bm{p}}, {\bm{\xi}}\mid {\mathcal{H}})
= \prod_{i = 1}^N \Big\{ p_{z_i}^{{\mathbbm{1}}({\tilde{y}}_i = 1)} f_{T \mid Z, Y}(v_i \mid z_i, Y = 1, {\bm{\xi}})^{{\mathbbm{1}}({\tilde{y}}_i = 1)} \times \\
\left[ 1 - \rho(v_i \mid z_i, {\bm{\xi}}) p_{z_i} \right]^{{\mathbbm{1}}({\tilde{y}}_i = 0)}
\Big\}.
\label{eq:likelihood_tite}\end{gathered}$$
Here, due to Remark \[rmk:rho\], $\rho_i \triangleq \rho(v_i \mid z_i, {\bm{\xi}})$ can be interpreted as the weight of a patient who is still being followed within the assessment window. The likelihood can be considered as a weighted likelihood as in [@cheung2000sequential], where a patient with a complete outcome (${\tilde{y}}_i = 1$, or ${\tilde{y}}_i = 0$ and $v_i = W$) receives full weight, and a patient with a pending outcome (${\tilde{y}}_i = 0$ and $v_i < W$) receives a weight of $\rho_i$. The longer the follow-up time, the larger the weight. The last term $f_{T \mid Z, Y}(v_i \mid z_i, Y = 1, {\bm{\xi}})$ is not related to ${\bm{p}}$ but provides information about the time-to-toxicity.
From a Bayesian perspective, with the likelihood and prior distributions $\pi_0({\bm{p}})$ and $\pi_0({\bm{\xi}})$, inference on ${\bm{p}}$ and ${\bm{\xi}}$ is realized using the posterior distribution, $$\begin{aligned}
\pi({\bm{p}}, {\bm{\xi}}\mid {\mathcal{H}}) \propto \pi_0({\bm{p}}) \pi_0({\bm{\xi}}) L ( {\bm{p}}, {\bm{\xi}}\mid {\mathcal{H}}).\end{aligned}$$ In general, the posterior is not available in closed form, and Monte Carlo simulation is applied to approximate the posterior. We provide a simple computational algorithm in Appendix \[supp:sec:inference\_sur\]. From a frequentist perspective, the maximum likelihood estimate (MLE) $\hat{{\bm{p}}}$ can be used as an estimate for ${\bm{p}}$. One can calculate $(\hat{{\bm{p}}}, \hat{{\bm{\xi}}}) = \operatorname*{arg\,max}_{{\bm{p}}, {\bm{\xi}}} L( {\bm{p}}, {\bm{\xi}}\mid {\mathcal{H}})$ by taking partial derivatives of the log-likelihood with respect to all the parameters or using other optimization techniques. Again, see more details in Appendix \[supp:sec:inference\_sur\]. In the CRM and BLRM designs, ${\bm{p}}$ is modeled by a parametric curve, $p_z = \phi(z, \bm \alpha)$, where $\bm \alpha$ denotes unknown parameters. In such cases, the likelihood is re-parameterized with respect to $\bm \alpha$, and a prior distribution $\pi_0(\bm \alpha)$ would be specified for $\bm \alpha$ (instead of ${\bm{p}}$).
Augmented Likelihood with Missing Data {#sec:likelihood_missing}
--------------------------------------
The likelihood of ${\bm{p}}$ and ${\bm{\xi}}$ can be alternatively constructed based on modeling of missing data by treating the pending DLT outcomes (i.e., ${\bm{Y}}_{{\text{mis}}}$) as missing and augmenting the likelihood function that incorporates the unknown ${\bm{Y}}_{{\text{mis}}}$ as a vector of latent variables. Specifically, a patient having an observed toxic outcome ($Y_i = 1$ and $B_i = 1$) and a known DLT time $v_i$ contributes $p_{z_i} f_{T \mid Z, Y}(v_i \mid z_i, Y = 1, {\bm{\xi}})$ to the likelihood. A patient having a latent toxic outcome ($Y_i = 1$ and $B_i = 0$) and a follow-up time $v_i$ contributes $p_{z_i} \int_{v_i}^W f_{T \mid Z, Y}(t \mid z_i, Y = 1, {\bm{\xi}}) {\text{d}}t$ to the likelihood, because the DLT will occur in the interval $(v_i, W]$. Finally, a patient with an observed or latent non-DLT outcome ($Y_i = 0$) contributes $(1 - p_{z_i})$ to the likelihood. Therefore, using , the augmented likelihood is given by $$\begin{gathered}
L( {\bm{p}}, {\bm{\xi}}, {\bm{y}}_{{\text{mis}}} \mid {\mathcal{H}})
= \prod_{i = 1}^N \Big \{ p_{z_i}^{{\mathbbm{1}}(y_i = 1)} (1 - p_{z_i})^{{\mathbbm{1}}(y_i = 0)} \times \\
f_{T \mid Z, Y}(v_i \mid z_i, Y = 1, {\bm{\xi}})^{{\mathbbm{1}}(y_i = 1, b_i = 1)} \left[1 - \rho(v_i \mid z_i, {\bm{\xi}}) \right]^{{\mathbbm{1}}(y_i = 1, b_i = 0)} \Big \}.
\label{eq:likelihood_missing}\end{gathered}$$
Although the augmented likelihood involves additional parameters compared to the survival likelihood, the following proposition shows inference under both approaches is the same.
\[prop:eq\_mis\_sur\] The derived likelihood by marginalizing over ${\bm{y}}_{{\text{mis}}}$ is the same as the survival likelihood for ${\bm{p}}$ and ${\bm{\xi}}$.
The proof is given in Appendix \[supp:sec:eq\_sur\_mis\]. The augmented likelihood opens the door to a set of flexible computational algorithms for making inference on ${\bm{p}}$. For example, the posterior distribution $\pi({\bm{p}}\mid {\mathcal{H}})$ can be simulated using the data augmentation method [@tanner1987calculation]. The MLE of ${\bm{p}}$ can be calculated through the expectation-maximization algorithm [@dempster1977maximum]. We elaborate these methods in Appendix \[supp:sec:inference\_mis\]. The inference involves the conditional probability, $$\begin{aligned}
&{} \Pr(Y_{{\text{mis}}, i} = 1 \mid Z_i = z_i, T_i > v_i, {\bm{p}}, {\bm{\xi}}) \label{eq:dist_missing} \\
= &{} \frac{\Pr(T_i > v_i \mid Z_i = z_i, Y_{{\text{mis}}, i} = 1, {\bm{\xi}}) \cdot \Pr(Y_{{\text{mis}}, i} = 1 \mid Z_i = z_i, {\bm{p}})}{\sum_{y \in \{0, 1\}} \Pr(T_i > v_i \mid Z_i = z_i, Y_{{\text{mis}}, i} = y, {\bm{\xi}}) \cdot \Pr(Y_{{\text{mis}}, i} = y \mid Z_i = z_i, {\bm{p}})} \nonumber \\
= &{} \frac{ [1 - \rho(v_i \mid z_i, {\bm{\xi}})] \cdot p_{z_i} }{ [1 - \rho(v_i \mid z_i, {\bm{\xi}})] \cdot p_{z_i} + (1 - p_{z_i})}. \nonumber\end{aligned}$$ That is, the probability of a pending patient experiencing DLT within the assessment window given the patient is treated at dose $z_i$ and has been followed for $v_i$ units of time.
Two Classes of Time-to-Event Designs {#sec:two_class_designs}
====================================
TITE Designs {#sec:tite_design}
------------
In Section \[sec:framework\], we have proposed a statistical framework for time-to-event modeling in dose-finding trials. Based on this framework and using each of the complete-data designs (Section \[sec:complete\_data\_design\]), one can easily generate a corresponding time-to-event design. Following the literature, we call this class of designs TITE designs. Below, we illustrate this idea through reviewing five existing TITE designs: TITE-CRM, EM-CRM, DA-CRM, TITE-BOIN and TITE-keyboard, and proposing three new TITE designs: TITE-TPI, TITE-SPM and TITE-i3.
### Existing TITE Designs
TITE-CRM & EM-CRM & DA-CRM
: The TITE-CRM design [@cheung2000sequential] is a TITE extension of the CRM design. It assumes a dose-toxicity curve $p_z = \phi(z, \alpha)$, such that $\phi$ monotonically increases with $z$, and $\alpha$ is an unknown parameter. For example, a commonly used dose-toxicity curve is $\phi(z, \alpha) = p_{0z}^{\exp(\alpha)}$, where $p_{0z}$’s are pre-specified constants (skeletons) satisfying $p_{01} < \cdots < p_{0D}$. The likelihood is re-parameterized with respect to $\alpha$ and becomes $$\begin{gathered}
L ( \alpha, {\bm{\xi}}\mid {\mathcal{H}})
= \prod_{i = 1}^N \Big\{ \phi(z_i, \alpha)^{{\mathbbm{1}}({\tilde{y}}_i = 1)} \left[ 1 - \rho(v_i \mid z_i, {\bm{\xi}}) \phi(z_i, \alpha) \right]^{{\mathbbm{1}}({\tilde{y}}_i = 0)} \times \\
f_{T \mid Z, Y}(v_i \mid z_i, Y = 1, {\bm{\xi}})^{{\mathbbm{1}}({\tilde{y}}_i = 1)} \Big\}.\end{gathered}$$ By default, the conditional distribution of $[T \mid Z, Y = 1]$ is modeled by a uniform distribution, thus $\rho(v_i \mid z_i, {\bm{\xi}}) \equiv v_i / W$ and $f_{T \mid Z, Y}(v_i \mid z_i, Y = 1, {\bm{\xi}}) \equiv 1 / W$. Alternatively, one can model $[T \mid Z, Y = 1]$ with a piecewise-uniform distribution, which can be calibrated to match the adaptive weighting scheme described in [@cheung2000sequential]. Inference on ${\bm{p}}$ can be Bayesian or based on MLE. The dose $d^* = \operatorname*{arg\,min}_{z} | \hat{p}_{z} - p_{{\text{T}}} |$ is recommended for the next patient, subject to some practical safety restrictions [@goodman1995some; @cheung2005coherence]. Here $\hat{p}_{z}$ is an appropriate point estimate of $p_z$.
The EM-CRM [@yuan2011robust] and DA-CRM [@liu2013bayesian] are two alternative TITE extensions of the CRM design. They take the missing data modeling approach and consider the augmented likelihood . EM-CRM models $[T \mid Z, Y = 1]$ with a discrete hazard model and estimates ${\bm{p}}$ using the expectation-maximization algorithm. It also has an additional layer of model averaging over different choices of skeletons to improve its robustness. DA-CRM models $[T \mid Z, Y = 1]$ with a piecewise constant hazard model and estimates ${\bm{p}}$ using the data augmentation method. According to Proposition \[prop:eq\_mis\_sur\], TITE-CRM, EM-CRM and DA-CRM would yield identical inference if the same specification of $f_{T \mid Z, Y}(t \mid z, Y = 1, {\bm{\xi}})$ was used.
TITE BOIN
: The TITE-BOIN design [@yuan2018time] is a TITE extension of the BOIN design. See Appendix \[supp:sec:review\_boin\] for more details about BOIN. To maintain the transparent and simple decision rules in BOIN, TITE-BOIN uses single imputation, substituting ${\bm{Y}}_{{\text{mis}}}$ with their expected values $\hat{{\bm{y}}}_{{\text{mis}}}$. Specifically, $\hat{y}_{{\text{mis}}, i} = \text{E}(Y_{{\text{mis}}, i} = 1 \mid Z_i = z_i, T_i > v_i, {\bm{p}}, {\bm{\xi}}) = \Pr(Y_{{\text{mis}}, i} = 1 \mid Z_i = z_i, T_i > v_i, {\bm{p}}, {\bm{\xi}})$, given in . A uniform distribution is assumed for $[T \mid Z, Y = 1]$, thus $\rho(v_i \mid z_i, {\bm{\xi}}) \equiv v_i / W$. The imputation involves the unknown parameter ${\bm{p}}$, for which an estimate based on an approximation procedure is plugged in. Finally, the decision rule of BOIN is applied to the imputed dataset $({\bm{y}}_{{\text{obs}}}, \hat{{\bm{y}}}_{{\text{mis}}})$. We note that another way of extending the BOIN design is to consider the BOIN hypothesis test (Appendix Equation \[eq:boin\_hypothesis\]) directly under the likelihood , although this would be more complicated than the single imputation approach.
TITE-keyboard
: The TITE-keyboard design [@lin2018time] is a TITE extension of the keyboard design. It considers a partition of the $[0, 1]$ interval into sub-intervals $\{ I_{{\text{U}}_0}, I_{{\text{U}}_1}, \ldots, I_{{\text{U}}_{K_1}} \}$, $I_{{\text{E}}}$ and $\{ I_{{\text{O}}_0}, I_{{\text{O}}_1}, \ldots, I_{{\text{O}}_{K_2}} \}$, such that all the sub-intervals have the same length except for the intervals reaching the boundary of $[0, 1]$. Here, $I_{{\text{E}}} = [p_{{\text{T}}} - \epsilon_1, p_{{\text{T}}} + \epsilon_2]$ is the only sub-interval that contains $p_{{\text{T}}}$, $I_{{\text{U}}_k}$’s are on the left of $I_{{\text{E}}}$, and $I_{{\text{O}}_k}$’s are on the right of $I_{{\text{E}}}$. Except for the two boundary sub-intervals (denoted by $I_{{\text{U}}_0}$ and $I_{{\text{O}}_0}$), the equal-length sub-intervals are referred to as keys, and $I_{{\text{E}}}$ is referred to as the target key. By default, TITE-keyboard assumes independent ${\text{Beta}}(1, 1)$ priors on $p_z$’s, $\pi_0(p_z) = {\mathbbm{1}}_{[0, 1]}$. Suppose the current dose is $d$. With a model for $[T \mid Z, Y = 1]$ and an additional prior $\pi_0({\bm{\xi}})$, the posterior $\pi (p_d, {\bm{\xi}}\mid {\mathcal{H}}) \propto \pi_0 (p_d) \pi_0({\bm{\xi}}) L( {\bm{p}}, {\bm{\xi}}\mid {\mathcal{H}})$. Let $j^* = \operatorname*{arg\,max}_j \Pr(p_d \in I_j \mid {\mathcal{H}})$. The dose assignment decision for the next patient follows the keyboard design. That is, to escalate, stay or de-escalate, if $j^*$ belongs to $\{ {\text{U}}_1, \ldots, {\text{U}}_{K_1} \}$, equals E or belongs to $\{ {\text{O}}_1, \ldots, {\text{O}}_{K_2} \}$, respectively.
### New TITE Designs {#sec:tite_new}
TITE-TPI
: We propose the TITE-TPI design as a TITE extension of the mTPI-2 design. Similar to the keyboard design, mTPI-2 considers a partition of the $[0, 1]$ interval into equal-length sub-intervals. The sub-interval $I_{{\text{E}}}$ is referred to as the equivalence interval, $I_{{\text{U}}_k}$’s are referred to as underdosing intervals, and $I_{{\text{O}}_k}$’s are referred to as overdosing intervals. Let $d$ denote the current dose level, and let model $\mathcal{M}_d = j$ represent $\{ p_d \in I_j \}$, $j = {\text{U}}_0, \ldots, {\text{U}}_{K_1}, {\text{E}}, {\text{O}}_0, \ldots, {\text{O}}_{K_2}$. We consider the following hierarchical prior model for $p_d$, $$\begin{aligned}
\begin{split}
&\Pr(\mathcal{M}_d = j) = 1/(K_1 + K_2 + 3), \; \text{for $j = {\text{U}}_0, \ldots, {\text{U}}_{K_1}, {\text{E}}, {\text{O}}_0, \ldots, {\text{O}}_{K_2}$}; \\
&p_d \mid \mathcal{M}_d \sim {\text{TBeta}}(1, 1; I_{\mathcal{M}_d}),
\end{split}\end{aligned}$$ where ${\text{TBeta}}(\cdot, \cdot; I)$ represents a truncated beta distribution restricted to interval $I$. For TITE-TPI, with a model for $[T \mid Z, Y = 1]$ and an additional prior $\pi_0({\bm{\xi}})$, the posterior $\pi (\mathcal{M}_d, p_d, {\bm{\xi}}\mid {\mathcal{H}}) \propto \pi_0 (\mathcal{M}_d) \pi_0 (p_d \mid \mathcal{M}_d) \pi_0({\bm{\xi}}) L( {\bm{p}}, {\bm{\xi}}\mid {\mathcal{H}})$. The dose assignment decision for TITE-TPI follows the mTPI-2 design. That is, to escalate, stay or de-escalate, if $\arg \max_{j} \Pr(\mathcal{M}_d = j \mid {\mathcal{H}})$ belongs to $\{ {\text{U}}_0, \ldots, {\text{U}}_{K_1} \}$, equals E or belongs to $\{ {\text{O}}_0, \ldots, {\text{O}}_{K_2} \}$, respectively.
TITE-SPM
: We propose the TITE-SPM design as a TITE extension of the SPM design. The SPM directly models the the location of the MTD $\gamma$, $\gamma \in \{1, \ldots, D\}$. That is, $\{ \gamma = d \}$ means dose level $d$ is the MTD. Conditional on $\gamma$, the support of $p_z$ is restricted to $$\begin{aligned}
\text{supp}(p_z) =
\begin{cases}
I_{{\text{U}}} = [0, p_{{\text{T}}} - \epsilon_1), & \text{if } z < \gamma; \\
I_{{\text{E}}} = [p_{{\text{T}}} - \epsilon_1, p_{{\text{T}}} + \epsilon_2], & \text{if } z = \gamma; \\
I_{{\text{O}}} = (p_{{\text{T}}} + \epsilon_2, 1], & \text{if } z > \gamma.
\end{cases}\end{aligned}$$ This restriction guarantees the partial ordering of the $p_z$’s. The priors on $\gamma$ and $p_z$’s can be specified as follows, $$\begin{aligned}
\begin{split}
&\Pr(\gamma = z^*) = \kappa_{z^*}, \\
&p_z \mid \gamma \sim {\text{TBeta}}(c \theta_z^{\gamma} + 1, c(1 - \theta_z^{\gamma}) + 1; I_{z}^{\gamma}).
\end{split}\end{aligned}$$ Here $\kappa_{z^*}$, $c$ and $\theta_{z}^{\gamma}$ are hyperparameters, and $I_z^{\gamma} = I_{{\text{U}}}, I_{{\text{E}}}$ or $I_{{\text{O}}}$ for $z < \gamma$, $z = \gamma$ or $z > \gamma$, respectively. The hyperparameter $\theta_{z}^{\gamma}$ is the prior mode of $p_z$ if $\gamma$ is the assumed MTD, and is specified in a similar fashion as CRM. For TITE-SPM, with a model for $[T \mid Z, Y = 1]$ and an additional prior $\pi_0({\bm{\xi}})$, the posterior $\pi(\gamma, {\bm{p}}, {\bm{\xi}}\mid {\mathcal{H}}) \propto \pi_0(\gamma) \pi_0({\bm{p}}\mid \gamma) \pi_0({\bm{\xi}}) L( {\bm{p}}, {\bm{\xi}}\mid {\mathcal{H}})$. The dose assignment decision for TITE-SPM follows the SPM design. That is, to assign the dose $\hat{\gamma} = \operatorname*{arg\,max}_{\gamma} \pi(\gamma \mid {\mathcal{H}})$ to the next patient, subject to some safety restrictions.
TITE-i3
: We propose the TITE-i3 design as a TITE extension of the i3+3 design. The i3+3 design consists of a set of algorithmic decision rules, that is, model free. Again, it considers a partition of the $[0, 1]$ interval into $I_{{\text{E}}}$, $I_{{\text{U}}}$ and $I_{{\text{O}}}$. Suppose the current dose is $d$. If $n_d / N_d \in I_{{\text{U}}}$, the decision is to escalate. If $n_d / N_d \in I_{{\text{E}}}$, the decision is to stay. If $n_d / N_d \in I_{{\text{O}}}$, the decision is to stay when $(n_d - 1) / N_d \in I_{{\text{U}}}$ and is to de-escalate otherwise. For TITE-i3, we replace $n_d$ in the i3+3 design with $N_d \hat{p}_d$, where $\hat{p}_d$ is the MLE under the likelihood . To maintain the simple algorithmic rules in i3+3, we model $[T \mid Z, Y = 1]$ with a uniform distribution, and the MLE is easy to solve.
According to how a design makes inference about ${\bm{p}}$ and translates such inference to a dose-finding decision, we can categorize the TITE designs in the same way as the complete-data designs (Section \[sec:complete\_data\_design\]). For example, TITE-CRM is a point-based parametric TITE design, and TITE-BOIN, TITE-keyboard, TITE-TPI and TITE-i3 are interval-based nonparametric TITE designs.
POD Designs {#sec:pod_design}
-----------
Taking one step further of the TITE designs, one can directly make inference on possible dose-finding decisions when some DLT outcomes are pending. This leads to a new class of POD (probability-of-decision) designs. We discuss the details next.
As mentioned in Section \[sec:complete\_data\_design\], the dose assignment decision for any complete-data design can be written as a deterministic function of the previous (complete) DLT outcomes ${\bm{y}}$ and dose assignments ${\bm{z}}$, denoted by ${\mathcal{A}}^*({\bm{y}}, {\bm{z}})$. In other words, the dose level ${\mathcal{A}}^*({\bm{y}}, {\bm{z}}) \in \{ 1, \ldots, D \}$ will be used to treat the next patient. For example, when $p_{{\text{T}}} = 0.2$, for CRM with default prior hyperparameters, ${\mathcal{A}}_{\text{CRM}}^*[(0, 0, 0, 0, 0, 1),$ $(1, 1, 1, 2, 2,$ $2)] = 2$; for mTPI-2 with $\epsilon_1 = \epsilon_2 = 0.05$, ${\mathcal{A}}_{\text{mTPI-2}}^*[(0, 0,$ $0, 0, 0, 1), (1, 1, 1 , 2, 2, 2)] = 1$.
In the presence of pending outcomes, let $A = {\mathcal{A}}^*[({\bm{y}}_{{\text{obs}}}, {\bm{Y}}_{{\text{mis}}}), {\bm{z}}]$ denote the dose assignment decision. Since ${\bm{Y}}_{{\text{mis}}}$ is a vector of latent variables and $A$ is a function of ${\bm{Y}}_{{\text{mis}}}$, $A$ is essentially a random variable. Under the Bayesian paradigm, the posterior distribution of $A$ is given by $$\begin{aligned}
\Pr(A = a \mid {\mathcal{H}}) = \sum_{{\bm{y}}_{{\text{mis}}}: {\mathcal{A}}^*[({\bm{y}}_{{\text{obs}}}, {\bm{y}}_{{\text{mis}}}), {\bm{z}}] = a} \Pr({\bm{Y}}_{{\text{mis}}} = {\bm{y}}_{{\text{mis}}} \mid {\mathcal{H}}).
\label{eq:post_decision}\end{aligned}$$ Here, $$\begin{aligned}
\Pr({\bm{Y}}_{{\text{mis}}} = {\bm{y}}_{{\text{mis}}} \mid {\mathcal{H}}) = \int_{{\bm{\xi}}} \int_{{\bm{p}}} \Pr({\bm{Y}}_{{\text{mis}}} = {\bm{y}}_{{\text{mis}}} \mid {\mathcal{H}}, {\bm{p}}, {\bm{\xi}}) \pi({\bm{p}}, {\bm{\xi}}\mid {\mathcal{H}}) {\text{d}}{\bm{p}}{\text{d}}{\bm{\xi}}\end{aligned}$$ is the posterior predictive distribution of ${\bm{Y}}_{{\text{mis}}}$, and $\Pr({\bm{Y}}_{{\text{mis}}} = {\bm{y}}_{{\text{mis}}} \mid {\mathcal{H}}, {\bm{p}}, {\bm{\xi}})$ is given in . From a frequentist perspective, instead of marginalizing over the posterior distribution of ${\bm{p}}$ and ${\bm{\xi}}$, one could plug in the MLE of ${\bm{p}}$ and ${\bm{\xi}}$. The probability is referred to as the POD, which accounts for the variability in the missing data and directly reflects the confidence of every possible decision. The dose assignment for the next patient can be guided by the POD. For example, one may make the decision with the highest POD, $a^* = \operatorname*{arg\,max}_a \Pr(A = a \mid {\mathcal{H}})$.
### Existing and New POD designs {#sec:pod_example}
POD-TPI
: We illustrate the POD designs through reviewing the POD-TPI design [@zhou2019pod]. POD-TPI assumes independent ${\text{Beta}}(1, 1)$ priors on $p_z$’s. With a model for $[T \mid Z, Y = 1]$ and an additional prior $\pi_0({\bm{\xi}})$, the posterior $\pi ({\bm{p}}, {\bm{\xi}}\mid {\mathcal{H}}) \propto \pi_0 ({\bm{p}}) \pi_0({\bm{\xi}}) L( {\bm{p}}, {\bm{\xi}}\mid {\mathcal{H}})$. By default, POD-TPI models $[T \mid Z, Y = 1]$ with a piecewise uniform distribution. The posterior predictive distribution of ${\bm{Y}}_{{\text{mis}}}$ is then computed. Finally, suppose the current dose is $d$, and let $A = {\mathcal{A}}^*[({\bm{y}}_{{\text{obs}}}, {\bm{Y}}_{{\text{mis}}}), {\bm{z}}] \in \{ d-1, d, d+1 \}$, where ${\mathcal{A}}^*$ is the decision function of mTPI-2. The PODs of possible decisions are calculated, and the decision with the highest POD is executed, subject to additional safety restrictions.
Apparently, the decision function ${\mathcal{A}}^*$ can be based on any complete-data design. The model for $[T \mid Z, Y = 1]$ and priors on ${\bm{p}}$ and ${\bm{\xi}}$ can also be adjusted if desired. In this way, we obtain a new class of POD designs, such as POD-CRM, POD-BOIN, POD-keyboard, POD-SPM and POD-i3. We can categorize the POD designs according to the corresponding complete-data designs ${\mathcal{A}}^*$. For example, POD-TPI is an interval-based nonparametric POD design.
Design Properties {#sec:property}
=================
In this section, we study large- and finite-sample properties of the aforementioned time-to-event designs, with an emphasis on interval-based nonparametric designs.
Large-Sample Convergence Properties {#sec:convergence}
-----------------------------------
Dose-finding studies are usually carried out with relatively small sample sizes (10 to 50 subjects). Still, as noted in [@oron2011dose], large-sample convergence properties should be viewed as a necessary quality criterion for dose-finding designs. In general, the large-sample properties for a particular complete-data design should also hold for its time-to-event version, as long as the DLT assessment window $W$ and the patient accrual rate are both finite. Intuitively, at time $\tau$, all patients enrolled before $(\tau - W)$ have finished their DLT assessments, and only the patients enrolled within $(\tau - W, \tau]$ can have pending outcomes. As $\tau \rightarrow \infty$, the number of complete outcomes goes to infinity too, and the number of pending outcomes is finite with probability one, making the contribution of the pending outcomes negligible in the likelihood .
In what follows, we present some general large-sample results for interval-based nonparametric time-to-event designs. First, the following lemma establishes the consistency of the posterior distribution and MLE of $p_z$ in a time-to-event setting when (1) the number of patients treated by dose $z$, $N_z$, goes to infinity, and (2) the number of pending outcomes at dose $z$, $r_z$, is small compared to $N_z$.
\[lem:consistency\] Suppose the true distribution of the DLT outcome is $\Pr(Y_i = 1 \mid Z_i = z) = p_{0z}$, and $r_z = o(N_z)$. (1) Let ${\mathcal{C}}_{\varepsilon} = \{ p_z : | p_z - p_{0z} | < \varepsilon \}$. Let $\pi_0(p_z)$ be a prior distribution for $p_z$ such that $\pi_0(p_z \in {\mathcal{C}}_{\varepsilon}) > 0$ for every $\varepsilon > 0$, and the likelihood of $p_z$ is as in . Then, for every $\varepsilon > 0$, the posterior distribution $\pi(p_z \in {\mathcal{C}}_{\varepsilon} \mid {\mathcal{H}}) \rightarrow 1$ almost surely as $N_z \rightarrow \infty$. (2) The maximum likelihood estimator $\hat{p}_z \rightarrow p_{0z}$ almost surely as $N_z \rightarrow \infty$.
The proof is given in Appendix \[supp:sec:convergence\]. As a consequence of Lemma \[lem:consistency\], we have the following convergence theorem for interval-based nonparametric TITE designs.
\[thm:convergence\] Suppose the conditions in Lemma \[lem:consistency\] are met. If there is a dose $d^*$ satisfying $p_{0d^*} \in (p_{{\text{T}}} - \epsilon_1, p_{{\text{T}}} + \epsilon_2)$, and $d^*$ is also the only dose such that $p_{0d^*} \in [p_{{\text{T}}} - \epsilon_1, p_{{\text{T}}} + \epsilon_2]$, then dose allocations in interval-based nonparametric TITE designs converge almost surely to $d^*$.
Again, the proof is given in Appendix \[supp:sec:convergence\]. When the condition about $d^*$ is violated, other convergence or oscillation results can be obtained as in [@oron2011dose].
Next, the following lemma establishes the consistency of the dose-finding decisions in interval-based nonparametric POD designs.
\[lem:consistency\_POD\] Suppose $d$ is the current dose, which is neither the lowest dose nor the highest dose. Suppose ${\mathcal{A}}^*$ is the dose decision function of an interval-based nonparametric complete-data design, and the conditions in Lemma \[lem:consistency\] are met. (1) If $p_{0d} \in (p_{{\text{T}}} - \epsilon_1, p_{{\text{T}}} + \epsilon_2)$, then $\exists N_{0d} > 0$, when $N_d > N_{0d}$, $\Pr(A = d \mid {\mathcal{H}}) = 1$ almost surely. (2) If $p_{0d} < p_{{\text{T}}} - \epsilon_1$, then $\exists N_{0d} > 0$, when $N_d > N_{0d}$, $\Pr(A = d+1 \mid {\mathcal{H}}) = 1$ almost surely. (3) If $p_{0d} > p_{{\text{T}}} + \epsilon_2$, then $\exists N_{0d} > 0$, when $N_d > N_{0d}$, $\Pr(A = d-1 \mid {\mathcal{H}}) = 1$ almost surely.
See Appendix \[supp:sec:convergence\] for the proof. As a result, the convergence of dose allocations (Theorem \[thm:convergence\]) also holds for interval-based nonparametric POD designs.
For point-based designs or parametric designs, the consistency and convergence results require additional assumptions. We direct the readers to [@cheung1999sequential] for an example under the TITE-CRM setting.
Coherence Principles {#sec:coherence}
--------------------
The coherence principles are another quality criterion for dose-finding designs motivated by ethical concerns in trial conduct. [@cheung2005coherence] introduced a coherence condition for time-to-event designs, which states that a time-to-event design should not de-escalate from time $\tau$ to $\tau+ \tau'$ if no toxicity occurs during $[\tau, \tau+ \tau')$, and it should not escalate from time $\tau$ to $\tau+\tau'$ if a toxicity occurs within $[\tau, \tau+ \tau')$ (for $\tau' \rightarrow 0^+$). The formal definition is given below.
\[def:coherence\] A time-to-event design ${\mathcal{A}}$ is *coherent* if (1) for any $\tau, \tau' > 0$, $$\begin{aligned}
\text{Pr}_{{\mathcal{A}}} \big\{ {\mathcal{A}}[{\mathcal{H}}(\tau + \tau')] < {\mathcal{A}}[{\mathcal{H}}(\tau)] \mid {\tilde{Y}}_i(\tau + \tau') - {\tilde{Y}}_i(\tau) = 0 \text{ for all $i$} \big\} = 0;\end{aligned}$$ and (2) for any $\tau > 0$, $$\begin{aligned}
\lim_{\tau' \rightarrow 0^+} \text{Pr}_{{\mathcal{A}}} \big\{ {\mathcal{A}}[{\mathcal{H}}(\tau + \tau')] > {\mathcal{A}}[{\mathcal{H}}(\tau)] \mid {\tilde{Y}}_i(\tau + \tau') - {\tilde{Y}}_i(\tau) = 1 \text{ for some $i$} \big\} = 0.\end{aligned}$$
[@cheung2005coherence] showed that the TITE-CRM design is coherent if the weight $\rho(v_i \mid z_i, {\bm{\xi}})$ is continuous and nondecreasing in $v_i$, which is automatically satisfied under the proposed framework (see Equation \[eq:rho\]). In contrast, interval-based nonparametric designs only use observations at the current dose to make dose-finding decisions thus may be incoherent in the sense of Definition \[def:coherence\]. For example, consider target DLT rate $p_{{\text{T}}} = 0.2$. Assume for two adjacent patients, the sequences of dose assignments ${\bm{z}}= (2, 1)$ and DLT outcomes ${\bm{y}}= (1, 0)$. Using the BOIN or TITE-BOIN design with default hyperparameters, the 3rd patient is assigned to dose level 2. Suppose by the time the 4th patient is enrolled, the 3rd patient has finished DLT assessment with no event. However, since the empirical DLT rate at dose 2 is 0.5, the 4th patient would be assigned to dose 1. In other words, no toxicity occurs between the enrollment of patients 3 and 4, but the dose level de-escalates from 2 to 1, which is incoherent. This is because information at different dose levels is used to make the dose assignments for patients 3 and 4. To avoid incoherent dose-finding decisions for interval-based nonparametric designs, one may impose ad-hoc rules (such as those in Section \[sec:practical\]). On the other hand, one may still think such decisions are reasonable and consider alternative coherence conditions for interval-based nonparametric designs, such as the condition given below.
\[supp:def:coherence\_interval\] An interval-based nonparametric time-to-event design is interval coherent if (1) for any $\tau, \tau' > 0$, if the currently-administrated doses just prior to $\tau$ and $\tau + \tau'$ are the same (denoted by $d$), then $$\begin{aligned}
\text{Pr}_{{\mathcal{A}}} \big\{ {\mathcal{A}}[{\mathcal{H}}(\tau + \tau')] < {\mathcal{A}}[{\mathcal{H}}(\tau)] \mid
\tilde{Y}_i(\tau + \tau') - \tilde{Y}_i(\tau) = 0 \text{ for all $i$ s.t. $Z_i = d$} \big\} = 0;\end{aligned}$$ and (2) for any $\tau > 0$, suppose the currently-administrated doses just prior to $\tau$ is $d$, then $$\begin{gathered}
\lim_{\tau' \rightarrow 0^+} \text{Pr}_{{\mathcal{A}}} \big\{ {\mathcal{A}}[{\mathcal{H}}(\tau + \tau')] > {\mathcal{A}}[{\mathcal{H}}(\tau)] \mid \\
{\tilde{Y}}_i(\tau + \tau') - {\tilde{Y}}_i(\tau) = 1 \text{ for some $i$ s.t. $Z_i = d$} \big\} = 0,\end{gathered}$$ if the currently-administrated doses just prior to $\tau + \tau'$ is also $d$.
In Appendix \[supp:sec:coherence\], we show for a simple case that an interval-based nonparametric TITE design is interval coherent in the sense of Definition \[supp:def:coherence\_interval\].
[@liu2015bayesian] defined another coherence condition for dose-finding designs, which states that a dose-finding design is *long-term memory coherent* if it does not de-escalate (or escalate) when the observed toxicity rate in the accumulative cohorts at the current dose is lower (or higher) than the target toxicity rate. In other words, suppose the current dose is $d$, then a design is long-term memory coherent if it does not de-escalate (or escalate) when $n_{d} / (n_d + m_d) < p_{{\text{T}}}$ (or $> p_{{\text{T}}}$). Under this definition, time-to-event designs may be incoherent in escalation because the pending outcomes may contribute additional evidence to counteract the toxic outcomes. If we think such an escalation is reasonable, we may define that an escalation is incoherent only if $n_{d} / (n_d + m_d + r_d) > p_{{\text{T}}}$. Alternatively, as in [@lin2018time], we may also assign each pending outcome a weight $\rho$ and calculate an adjusted toxicity rate $\tilde{p}_d = n_{d} / [n_d + m_d + \sum_{i = 1}^N \rho_i {\mathbbm{1}}(z_i = d, b_i = 0)]$. For example, $\rho_i = v_i / W$. De-escalation is incoherent if $\tilde{p}_d < p_{{\text{T}}}$, and escalation is incoherent if $\tilde{p}_d > p_{{\text{T}}}$. However, the specification of the weight can be arbitrary.
Overdosing Decisions and Incompatible Decisions {#sec:incompatible}
-----------------------------------------------
To better understand the decision rules in TITE and POD designs, we introduce the concept of overdosing decisions and incompatible decisions. A design’s frequency of making such decisions measures the safety of this design.
#### *Overdosing decisions*.
We call a dose-finding decision ${\mathcal{A}}({\mathcal{H}})$ an *overdosing decision* if ${\mathcal{A}}({\mathcal{H}}) > d^*$, where $d^*$ denotes the MTD. Similarly, we call ${\mathcal{A}}({\mathcal{H}}) < d^*$ an underdosing decision. For example, consider a trial with 2 doses, target DLT rate $p_{{\text{T}}} = 0.2$, and true DLT probabilities $(p_1, p_2) = (0.2, 0.5)$. Then, $d^* = 1$ is the MTD, and any decision that allocates a patient to dose 2 is an overdosing decision. Since the true DLT probabilities are unknown in practice, we cannot check whether a decision is an overdosing/underdosing decision in real-world trials.
The decision rules in TITE designs can be viewed as minimizing various loss functions associated with overdosing/underdosing decisions. For example, in the TITE-SPM design (Section \[sec:tite\_new\]), recall that $\gamma$ represents the location of the MTD. Let $\ell(\gamma, \hat{\gamma} )$ denote the loss of allocating the new patient to dose $\hat{\gamma}$ if $\gamma$ is the true MTD, and consider the 0-1 loss $\ell(\gamma, \hat{\gamma} ) = {\mathbbm{1}}(\gamma \neq \hat{\gamma})$. That is, there is a loss for an overdosing/underdosing decision. Then, the decision of allocating the new patient to $\hat{\gamma} = \arg \max_{\gamma} \pi(\gamma \mid {\mathcal{H}})$ minimizes the posterior expected loss.
Overdosing decisions apply to both complete-data and time-to-event designs and are inevitable due to random sampling. As in Figure \[fig:safety\_concern\](a), suppose when the 4th patient is enrolled, patients 1–3 have finished DLT assessment (assuming $W = 28$ days) with no event. Then, the 4th patient might be assigned to dose 2, which is actually overly toxic. One may justify such a decision by sampling error.
-------------------------- ----------------------------
\(a) Overdosing decision \(b) Incompatible decision
-------------------------- ----------------------------
#### *Incompatible decisions.*
We say that a time-to-event dose-finding decision ${\mathcal{A}}[{\mathcal{H}}(\tau)]$ is *incompatible* with a complete-data decision ${\mathcal{A}}^*[{\mathcal{H}}_{N(\tau)}^*]$ if ${\mathcal{A}}[{\mathcal{H}}(\tau)] \neq {\mathcal{A}}^*[{\mathcal{H}}_{N(\tau)}^*]$. Here, ${\mathcal{H}}(\tau) = \{ (\tilde{Y}_i(\tau) , $ $V_i(\tau), $ $Z_i) : i \leq N(\tau) \}$ represents the available time-to-event information at time $\tau$, and ${\mathcal{H}}_{N(\tau)}^* = \{ (Y_i, Z_i) : i \leq N(\tau) \}$ represents the complete toxicity information for the first $N(\tau)$ patients that would have been observed if these patients had completed their follow-up. For example, as in Figure \[fig:safety\_concern\](b), consider a trial with 2 doses, target DLT rate $p_{{\text{T}}} = 0.2$, true DLT probabilities $(p_1, p_2) = (0.1, 0.2)$ and DLT assessment window $W = 28$ days. Suppose when the 4th patient arrives, patient 1 have finished DLT assessment with no event, and patients 2 and 3 are still being followed without definitive outcomes. Using a time-to-event design, the 4th patient might be treated by dose 2. However, it is possible that patient 2 or 3 could eventually experience DLT, making the dose escalation for patient 4 incompatible with a decision that uses complete data of patients 1–3. In practice, we can check whether a time-to-event decision at time $\tau$ is incompatible with a complete-data decision after all patients enrolled before $\tau$ have finished their DLT assessment.
The decision rules in POD designs can be viewed as minimizing a loss function associated with incompatible decisions. For example, in the POD-TPI design (Section \[sec:pod\_example\]), recall that $A$ denotes the random mTPI-2 decision in the presence of pending outcomes. Let $\ell(A, \hat{A} )$ denote the loss of making decision $\hat{A}$ if $A$ is the true complete-data decision, and consider the 0-1 loss $\ell(A, \hat{A} ) = {\mathbbm{1}}(A \neq \hat{A})$. Then, the decision $a^* = \operatorname*{arg\,max}_a \Pr(A = a \mid {\mathcal{H}})$ minimizes the posterior expected loss.
Incompatible decisions only apply to time-to-event designs and can be avoided by following patients for the full length of the assessment window. We note that an incompatible decision is not necessarily an overdosing/underdosing decision. In the Figure \[fig:safety\_concern\](b) example, the incompatible decision actually allocates patient 4 to the MTD. Still, since the true DLT probabilities are unknown in practice, such a decision cannot be justified based on the observed data, and the safety review boards should express concerns regarding the decision. Incompatible and overly aggressive decisions (${\mathcal{A}}[{\mathcal{H}}(\tau)] > {\mathcal{A}}^*[{\mathcal{H}}_{N(\tau)}^*]$) are a major concern for drug companies and regulatory agencies to use and approve time-to-event designs. Nevertheless, we next show that such decisions may be eliminated in POD designs through a suspension rule.
Practical Considerations {#sec:practical}
========================
Safety Rules {#sec:safety_rule}
------------
In addition to statistical modeling, safety rules play an important role in dose-finding designs. For example, when a dose is deemed overly toxic, future dose assignment to this dose or higher doses should be prohibited due to ethical concerns. If the lowest dose is too toxic, the trial should be terminated and redesigned using lower doses. From a Bayesian perspective, toxicity can be quantified using posterior probability. Similar to [@ji2010modified] and [@yuan2016bayesian], we consider the following safety rules.
Safety Rule 1 (Dose Exclusion)
: At any moment in the trial, if $n_z + m_z \geq 3$ and $\Pr(p_z > p_{{\text{T}}} \mid {\text{data}}) > \nu$ for a pre-specified threshold $\nu$ close to 1, suspend dose $z$ and higher doses from the trial;
Safety Rule 2 (Early Termination)
: At any moment in the trial, if $n_1 + m_1 \geq 3$ and $\Pr(p_1 > p_{{\text{T}}} \mid {\text{data}}) > \nu$ for a pre-specified threshold $\nu$ close to 1, terminate the trial due to excessive toxicity.
From a frequentist perspective, a dose $z$ can be considered overly toxic if the lower one-sided $\nu$ confidence interval of $p_z$ does not cover $p_{{\text{T}}}$.
It is possible that a dose is considered overly toxic when some toxicity outcomes at this dose are still pending. In this case, we allow this dose and upper doses to be re-opened if some pending outcomes turn out to be safe and suggest this dose is no longer overly toxic. If the lowest dose is considered overly toxic with some pending outcomes, we temporarily suspend the trial. If later pending outcomes are observed and suggest the lowest dose is no longer overly toxic, we resume the trial; otherwise, the trial is permanently terminated. There is a positive probability that a dose is excluded even if it is actually safe, or the trial is terminated early even when the lowest dose is safe. This is the type I error associated with the safety rules.
Finally, similar to the implementation of the TITE-CRM [@normolle2006designing], we introduce one more safety rule to prevent risky dose escalation until at least one patient at the current dose level has completed DLT assessment without experiencing DLT.
Safety Rule 3 (Restricting Dose Escalation)
: Suppose the current dose is $d$. If $m_d < 1$, dose escalation is not allowed.
In words, if there is no patient with confirmed safety outcome, escalation is not allowed.
Enrollment and Suspension
-------------------------
Many existing designs (e.g., 3+3, mTPI, BOIN, TITE-BOIN and POD-TPI) employ a cohort-based enrollment. The patients in the same cohort are always treated by the same dose. Cohort-based enrollment is especially necessary for complete-data designs, as the trial can take exceedingly long to complete if each patient has to wait for the completion of the DLT assessment for the previous patient. Cohort-based enrollment can also help avoid overly fast dose escalation and can be more convenient for trial administration. For most existing designs, a common cohort size is 2 to 4. Cohort-based enrollment is not necessary for time-to-event designs. The trial speed is no longer a concern, since new patients can be enrolled when some previous outcomes are pending. The potentially fast dose escalation might be a concern, but can be restricted with suspension rules, which we discuss next.
For time-to-event designs, the dose assignment decision for a new patient may be uncertain and risky if the toxicity outcomes of many previous patients are still pending. For example, three patients have been treated at the lowest dose and have been followed up for a while, but none of them have completed the DLT assessment. In this case, it might be too conservative to treat the fourth patient at the lowest dose as it might be subtherapeutic, but it is also too risky to treat the fourth patient at a higher dose as no safe outcome has been observed. Therefore, it may be more sensible to temporarily suspend the trial until at least one outcome has been observed, although trial suspension is not necessary from a purely statistical modeling perspective.
In the existing methods (e.g., [@yuan2018time] and [@guo2019rtpi]), the suspension rule is usually defined based on the number of pending outcomes.
Ad-hoc Suspension Rule
: Suppose the current dose is $d$. If $r_d > C$ for a pre-specified threshold $C$, suspend the enrollment.
Here, $C$ can be a fixed number (e.g. $C = 3$, [@guo2019rtpi]) or a portion of the total number of patients at the current dose (e.g. $C = N_d / 2$, [@yuan2018time]). Alternatively, as in [@zhou2019pod], specifically for the POD designs, the POD can be directly used to calibrate the suspension and risk trade-off.
Probability Suspension Rule
: Let $a^* = \operatorname*{arg\,max}_a \Pr(A = a \mid {\mathcal{H}})$. If $\Pr(A < a^* \mid {\mathcal{H}}) > q_{a^*}$ for a pre-specified threshold $q_{a^*}$, suspend the enrollment.
Here, $a^*$ is the optimal decision under the 0-1 loss, and $\Pr(A < a^* \mid {\mathcal{H}})$ is the posterior probability that a more conservative decision compared to $a^*$ should be made. In other words, if the posterior probability that a more conservative decision than $a^*$ should be made is higher than some threshold $q_{a^*}$, the enrollment is suspended. The threshold $q_{a^*}$ can be different for different $a^*$.
We note that the speed of the trial is solely determined by the suspension rule, i.e., how many times an eligible patient is turned away. If no pending outcome is allowed for making the dose assignment decision ($C = 0$), the time-to-event design reduces to a design using complete outcomes. On the other hand, if a trial never suspends, all eligible patients are enrolled and treated immediately, and the trial achieves its optimal speed. The Probability Suspension Rule allows a meaningful calibration between trial speed and safety of the design. This will be clear later in our numerical examples.
After patient enrollment is terminated and all DLT assessments are finished, the trial completes, and the next step is to recommend an MTD. We summarize several methods for MTD selection in Appendix \[supp:sec:sel\_mtd\].
#### *Coherence principle.*
The coherence principle for dose-finding designs was first introduced in [@cheung2005coherence], including a time-to-event version. It states that dose de-escalation from time $\tau$ to $\tau+v$ is not coherent if no toxicity occurs during $[\tau, \tau+v)$, and dose escalation is not coherent if a toxicity outcome occurs within $[\tau, \tau+v)$ for $v \rightarrow 0$. Later, [@liu2015bayesian] introduced the concept of long-term memory coherence (with complete outcomes). Under the time-to-event setting, the long-term coherence principle may be modified as follows.
Time-to-event coherence principle:
: At any moment in the trial, suppose the current dose is $d$. Let $\hat{p}_d$ be an estimate of $p_d$ under the likelihood . Dose escalation is not coherent if $\hat{p}_d > p_{{\text{T}}}$, and dose de-escalation is not coherent if $\hat{p}_d < p_{{\text{T}}}$.
Simulation Studies {#sec:simulation}
==================
Simulation Set-up
-----------------
We conduct simulation studies to compare the operating characteristics of some TITE and POD designs that we have discussed in the previous sections. We consider 18 dose-toxicity scenarios with target DLT probability $p_{{\text{T}}} = 0.2$ or $0.3$ and $D = 7$ dose levels. The 18 representative scenarios consist of the 16 scenarios reported in [@yuan2018time] and 2 additional scenarios that cover various MTD locations. The scenarios are summarized in Appendix Table \[tbl:simu\_DLT\_prob\].
We assume the DLT assessment window $W = 28$ days and use a maximum sample size of $N^* = 36$ patients. The time-to-toxicity for each patient, $T_i$, is generated from a Weibull distribution with shape $\zeta_{1z}$ and scale $\zeta_{2z}$, given the patient is treated by dose $z$. That is, $$\begin{aligned}
(T_i \mid Z_i = z) \sim {\text{Weibull}}(\zeta_{1z}, \zeta_{2z}).\end{aligned}$$ The parameters $\zeta_{1z}$ and $\zeta_{2z}$ are chosen such that $\Pr(T_i \leq W \mid Z_i = z) = p_z$ and $\Pr(T_i \leq W^* \mid Z_i = z) = (1 - q)p_z$. Here $ W^* \in (0, W)$ and $q \in (0, 1)$ are arbitrary numbers, meaning if a toxic outcome occurs within the assessment window, with probability $q$ it occurs within the interval $(W^*, W]$. The time between the accrual of two consecutive patients is generated from an exponential distribution with rate $\delta$, which means the average wait time between two consecutive patients is $1 / \delta$. We consider the following three settings with different time-to-toxicity and accrual rate profiles:
Setting 1 (default)
: inter-arrival time is 10 days, and 50% of DLTs occur in the second half of the assessment window. This corresponds to $\delta = 0.1$, $W^* = W/2$ and $q = 0.5$;
Setting 2 (more late-onset DLTs)
: inter-arrival time is 10 days, and 80% of DLTs occur in the last quarter of the assessment window. This corresponds to $\delta = 0.1$, $W^* = 3W/4$ and $q = 0.8$;
Setting 3 (faster accrual)
: inter-arrival time is 5 days, and 50% of DLTs occur in the second half of the assessment window. This corresponds to $\delta = 0.2$, $W^* = W/2$ and $q = 0.5$.
We do not consider another setting with a longer DLT assessment window, as it would be equivalent to faster accrual after rescaling the time.
Design Specifications
---------------------
We consider the TITE-CRM, TITE-TPI, TITE-BOIN (Section \[sec:tite\_design\]) and POD-TPI (Section \[sec:pod\_design\]) designs as examples of different types of time-to-event designs. In addition, we include the mTPI-2 and R-TPI in the comparison as examples of complete-data designs and designs that allow for pending outcomes but do not utilize time-to-event information, respectively. By default, TITE-BOIN, POD-TPI and mTPI-2 enroll patients in cohorts of 3, and TITE-CRM and R-TPI enroll patients one by one. For the newly proposed TITE-TPI, we employ a cohort-based enrollment in sizes of 3. For a fair comparison, we impose the same Safety Rules 1, 2 and 3 to all designs with $\nu = 0.95$ (see Section \[sec:safety\_rule\]). For TITE-CRM, TITE-TPI and TITE-BOIN, we suspend the trial if the number of pending patients at the current dose $r_d > N_d / 2$ (Ad-hoc Suspension Rule), as in [@yuan2018time]. For R-TPI, the trial is suspended if $r_d > 3$ by default. For POD-TPI, we suspend the trial according to the Probability Suspension Rule with $q_{a^*} \equiv 0$ for all $a^*$. That is, the trial is suspended if there is a positive probability that the optimal decision is overly aggressive. During trial suspension, the available patients are turned away. More details about the design specifications are reported in Appendix \[supp:sec:design\_spec\].
Performance Metrics
-------------------
The performance of a design is evaluated based on the following metrics.
1. **Selection & Allocation**, including (1.1) percentage of correct selection (PCS) of the MTD; (1.2) percentage of patients treated at the MTD (percentage of correct allocation, PCA); (1.3) percentage of dose selection above the MTD (percentage of overdosing selection, POS); (1.4) percentage of patients treated at doses above the MTD (percentage of overdosing allocation, POA); and (1.5) percentage of patients who have experienced toxicity (POT);
2. **Risk**, which is measured by the percentage of incompatible dose assignment decisions (see Section \[sec:incompatible\]). Recall that an incompatible decision refers to a decision that is different from what would be made if complete outcomes were observed. We are particularly concerned about the decisions that are overly aggressive, including the decisions that (2.1) should be de-escalation based on complete data but are stay (DS), (2.2) should be de-escalation but are escalation (DE), and (2.3) should be stay but are escalation (SE);
3. **Speed**, which is measured by the average trial duration (Dur).
Metrics (1.1)–(1.5) assess the design’s performance in selecting the right dose as the MTD and assigning patients to appropriate doses. Metrics (2.1)–(2.3) are about the risk associated with allowing patient enrollment in the presence of pending outcomes. In particular, the incompatible decisions of the time-to-event designs are obtained by comparing with their complete-data counterparts. For example, TITE-CRM is compared with CRM, and POD-TPI is compared with mTPI-2. Metric (3) is about the speed of the trial.
Simulation Results
------------------
For each dose-toxicity scenario in Appendix Table \[tbl:simu\_DLT\_prob\], we simulate 4,000 trials with each design. Table \[tbl:simu\_result\] summarizes the results by averaging over the 18 scenarios. The scenario-specific results under Setting 1 are reported in Appendix \[supp:sec:ss\_result\]. The performances of the designs are generally similar if averaged over scenarios, although they may have a large variation across different scenarios. The comparison among mTPI-2, R-TPI, TITE-TPI and POD-TPI illustrates the different behaviors of various extensions of the same complete-data design.
[c@c@c@c@c@c@c@c@c@c]{} & & & Speed\
& PCS & PCA & POS & POA & POT & DS & DE & SE & Dur\
\
mTPI-2 & 51.9 & 34.4 & 15.7 & 22.5 & 19.4 & 0.0 & 0.0 & 0.0 & 594\
R-TPI & 48.1 & 32.9 & 13.5 & 20.1 & 18.3 & 33.5 & 0.0 & 0.0 & 521\
TITE-TPI & 51.6 & 32.5 & 16.2 & 21.4 & 18.8 & 13.1 & 3.3 & 21.7 & 435\
POD-TPI & 51.0 & 32.9 & 16.0 & 21.1 & 18.7 & 0.0 & 0.0 & 0.0 & 541\
TITE-CRM & 55.4 & 36.3 & 29.8 & 27.0 & 21.3 & 48.5 & 0.2 & 7.9 & 438\
TITE-BOIN & 54.1 & 32.5 & 21.8 & 21.1 & 18.7 & 10.4 & 3.0 & 18.4 & 435\
\
mTPI-2 & 52.1 & 34.2 & 16.0 & 22.8 & 19.5 & 0.0 & 0.0 & 0.0 & 613\
R-TPI & 50.0 & 33.5 & 14.7 & 22.0 & 19.1 & 51.8 & 0.0 & 0.0 & 543\
TITE-TPI & 51.3 & 31.8 & 17.6 & 23.5 & 19.5 & 21.8 & 7.5 & 33.0 & 444\
POD-TPI & 51.0 & 32.6 & 16.1 & 21.9 & 19.0 & 0.0 & 0.0 & 0.0 & 558\
TITE-CRM & 55.3 & 35.4 & 29.4 & 29.4 & 22.1 & 80.7 & 0.9 & 14.8 & 449\
TITE-BOIN & 53.8 & 32.3 & 23.7 & 23.7 & 19.7 & 22.6 & 7.2 & 31.9 & 444\
\
mTPI-2 & 52.0 & 34.2 & 16.1 & 23.2 & 19.6 & 0.0 & 0.0 & 0.0 & 438\
R-TPI & 48.5 & 32.6 & 14.2 & 20.4 & 18.4 & 45.7 & 0.0 & 0.0 & 387\
TITE-TPI & 51.0 & 31.5 & 15.8 & 21.1 & 18.6 & 15.8 & 3.7 & 25.5 & 290\
POD-TPI & 50.7 & 31.9 & 15.9 & 20.5 & 18.4 & 0.0 & 0.0 & 0.0 & 379\
TITE-CRM & 55.1 & 35.0 & 29.8 & 26.8 & 21.1 & 58.1 & 0.3 & 9.8 & 304\
TITE-BOIN & 53.8 & 31.8 & 21.5 & 20.5 & 18.4 & 12.9 & 3.0 & 21.4 & 290\
On average, the trial duration is shortened by about 150–170 days using TITE designs, and is shortened by about 50–60 days using POD designs. This is a major benefit for drug development. The trial durations under TITE-TPI, TITE-CRM and TITE-BOIN are highly similar, because the same suspension rule is imposed. The trial durations under POD-TPI are longer, because a more conservative suspension rule is used, resulting in more patients being turned away. The PCS of the time-to-event designs is comparable to the complete-data design (mTPI-2). This is not surprising, because we always use complete outcomes to make the final selection of MTD. In the presence of pending outcomes, time-to-event designs may lead to incompatible assignments. For example, in Table \[tbl:simu\_result\], the DS, DE and SE of the TITE designs are non-zero, meaning the TITE designs sometimes make incompatible and aggressive decisions. Nevertheless, through the Probability Suspension Rule, POD designs may completely eliminate the chance of making these decisions. For example, in Table \[tbl:simu\_result\], DS, DE and SE are not observed for POD-TPI. This is a major advantage for POD designs. Although the time-to-event designs make incompatible and aggressive decisions, their PCA and POA are not necessarily higher, as they also make incompatible and conservative decisions (not shown). With more late-onset toxicities (Setting 2) or faster patient accrual (Setting 3), the performances of the time-to-event designs are slightly decreased. In particular, the frequencies of incompatible decisions for TITE designs under Settings 2 and 3 are generally increased compared to the results under Setting 1. Lastly, there is always a trade-off among the different performance metrics. For example, TITE-CRM has the highest PCS and high PCA, but it also has the highest POS and high POA and POT due to the more aggressive decision rules and MTD selection. R-TPI has the lowest POS and low POA and POT, but as a compromise, its PCS is slightly lower due to the more conservative decisions.
#### *Sensitivity of time-to-toxicity model.*
We have listed several possible specifications of the time-to-toxicity model in Section \[sec:model\_tite\] and Appendix \[supp:sec:model\_tite\]. To explore how these specifications can affect the operating characteristics of a design, we conduct additional simulation studies using POD-TPI as an example. We consider five different time-to-toxicity models: (1) uniform distribution; (2) piecewise uniform distribution with 3 sub-intervals (default); (3) piecewise uniform distribution with 9 sub-intervals; (4) discrete hazard model; and (5) piecewise constant hazard model with 3 sub-intervals. More details about the model specifications are reported in Appendix \[supp:sec:t\_model\_spec\]. Recall that the true distribution of $[T \mid Z, Y = 1]$ is a truncated Weibull distribution.
Table \[tbl:simu\_result\_t\_model\] summarizes the simulation results. The performances of POD-TPI under different time-to-toxicity models are generally similar. Importantly, no matter what time-to-toxicity model is used, the Probability Suspension Rule guarantees that no incompatible and aggressive decisions are ever made. The average number of DLTs in the trial is $N^* \times \text{POT} \approx 36 \times 20\% = 7.2$. As a result, there is very limited information for estimating the true time-to-toxicity distribution, and the specification of the time-to-toxicity model matters little. Under the discrete hazard model or the piecewise constant hazard model, the pending patients are weighted less if many DLTs are late-onset, making the design safer in such situations (in terms of lower POA and POT). This is consistent with the results reported in [@yuan2011robust] and [@liu2013bayesian].
[c@c@c@c@c@c@c@c@c@c]{} & & & Speed\
& PCS & PCA & POS & POA & POT & DS & DE & SE & Dur\
\
Uniform & 50.7 & 32.8 & 16.1 & 21.3 & 18.8 & 0.0 & 0.0 & 0.0 & 541\
Piecewise Uniform 3 & 51.0 & 32.9 & 16.0 & 21.1 & 18.7 & 0.0 & 0.0 & 0.0 & 541\
Piecewise Uniform 9 & 51.3 & 33.0 & 15.8 & 21.1 & 18.7 & 0.0 & 0.0 & 0.0 & 541\
Discrete hazard & 48.6 & 28.1 & 15.9 & 18.7 & 17.0 & 0.0 & 0.0 & 0.0 & 529\
Piecewise hazard 3 & 50.2 & 31.2 & 15.8 & 20.0 & 18.1 & 0.0 & 0.0 & 0.0 & 534\
\
Uniform & 51.0 & 32.5 & 16.0 & 21.6 & 18.9 & 0.0 & 0.0 & 0.0 & 556\
Piecewise Uniform 3 & 51.0 & 32.6 & 16.1 & 21.9 & 19.0 & 0.0 & 0.0 & 0.0 & 558\
Piecewise Uniform 9 & 50.6 & 32.7 & 16.0 & 21.9 & 19.1 & 0.0 & 0.0 & 0.0 & 560\
Discrete hazard & 48.3 & 27.3 & 15.6 & 18.7 & 16.9 & 0.0 & 0.0 & 0.0 & 536\
Piecewise hazard 3 & 50.5 & 30.6 & 15.9 & 20.3 & 18.1 & 0.0 & 0.0 & 0.0 & 545\
\
Uniform & 50.1 & 31.8 & 15.8 & 20.4 & 18.4 & 0.0 & 0.0 & 0.0 & 379\
Piecewise Uniform 3 & 50.7 & 31.9 & 15.9 & 20.5 & 18.4 & 0.0 & 0.0 & 0.0 & 379\
Piecewise Uniform 9 & 50.5 & 31.7 & 15.7 & 20.4 & 18.4 & 0.0 & 0.0 & 0.0 & 379\
Discrete hazard & 47.3 & 26.2 & 15.7 & 17.9 & 16.4 & 0.0 & 0.0 & 0.0 & 374\
Piecewise hazard 3 & 50.2 & 30.8 & 15.7 & 19.7 & 18.0 & 0.0 & 0.0 & 0.0 & 377\
#### *Sensitivity of ad-hoc rules.*
As described in Section \[sec:practical\], the dose-finding decisions are always subject to additional ad-hoc rules. To explore how these rules can affect the operating characteristics of a design, we conduct additional simulation studies under Setting 1 using POD-TPI. We consider the following five sets of rules: (1) Probability Suspension Rule + cohort-based enrollment in sizes of 3 (default); (2) Ad-hoc Suspension Rule + cohort-based enrollment in sizes of 3; (3) No Suspension + cohort-based enrollment in sizes of 3; (4) Probability Suspension Rule + rolling (one by one) enrollment; and (5) Ad-hoc Suspension Rule + rolling enrollment. Safety Rules 1, 2 and 3 are imposed under all settings. In particular, we set $\nu = 0.95$ in Safety Rules 1 and 2, $C = N_d / 2$ in the Ad-hoc Suspension Rule and $q_{a^*}\equiv 0$ in the Probability Suspension Rule.
The performance of POD-TPI under each rule set is reported in Table \[tbl:simu\_result\_rule\]. Under rule set 1, the strict thresholds $q_{a^*} = 0$ in the Probability Suspension Rule guarantee that no incompatible and aggressive decision can be made. These are the most conservative choices in practice. Alternatively, one may use less conservative thresholds, e.g., $q_{a^*} = 0.1$, such that certain risks are allowed to achieve faster trials. This may still be acceptable if a benchmark is set as risk tolerance. See [@zhou2019pod] for more details and numerical results. Under rule set 2, POD-TPI has a faster speed (compared to that under rule set 1) at the cost of making some incompatible and aggressive decisions. Specifically, the trial duration under rule set 2 is similar to that of TITE-TPI, TITE-CRM and TITE-BOIN in Table \[tbl:simu\_result\], because the same suspension rule is used. Under rule set 3, POD-TPI abandons the suspension rule and achieves its fastest speed at the cost of lower PCS and PCA. Under rule set 4, the trial needs to be frequently suspended after each patient is treated due to the strict thresholds in the Probability Suspension Rule, and as a result, the trial duration is prolonged. Finally, the results under rule set 5 are similar to those under rule set 2.
[c@c@c@c@c@c@c@c@c@c@c]{} & & & & Speed\
& & PCS & PCA & POS & POA & POT & DS & DE & SE & Dur\
\
Prob & 3 & 51.0 & 32.9 & 16.0 & 21.1 & 18.7 & 0.0 & 0.0 & 0.0 & 541\
Ad-hoc & 3 & 51.7 & 32.6 & 17.9 & 24.1 & 19.8 & 21.4 & 3.8 & 23.0 & 437\
No & 3 & 50.1 & 28.0 & 17.5 & 20.5 & 17.6 & 14.5 & 5.9 & 26.7 & 364\
Prob & 1 & 50.9 & 34.6 & 15.2 & 22.9 & 19.6 & 0.0 & 0.0 & 0.0 & 702\
Ad-hoc & 1 & 50.8 & 33.6 & 17.7 & 25.6 & 20.5 & 23.2 & 6.2 & 14.8 & 431\
Concluding Remarks {#sec:discussion}
==================
We have presented a general statistical framework for time-to-event modeling in dose-finding trials. Two classes of time-to-event designs, TITE designs and POD designs, can be built upon the framework. We have demonstrated that existing time-to-event designs (such as TITE-CRM, TITE-BOIN and TITE-keyboard) fall within this framework, and new time-to-event designs (such as TITE-SPM, TITE-TPI and POD-CRM) can be developed under this framework. The framework opens the way to a deeper study of time-to-event designs.
We have discussed and studied theoretical properties of time-to-event designs with an emphasis on interval-based nonparametric designs. A future direction is to investigate more on these theoretical properties, especially for point-based or parametric designs.
We have evaluated the operating characteristics of several designs through computer simulations. As we have seen, there is no single design that outperforms the other designs in all aspects. To choose one specific design to use in practice, we may run large-scale simulations and consider a loss function of the form $$\begin{gathered}
\ell = -\ell_1 \text{PCS} - \ell_2 \text{PCA} + \ell_3 \text{POS} + \ell_4 \text{POA} +
\ell_5 \text{POT} + \\
\ell_6 \text{DS} + \ell_7 \text{DE} + \ell_8 \text{SE} + \ell_9 \text{Dur}, \end{gathered}$$ where $\ell_1, \ldots, \ell_9 \geq 0$. The design that minimizes the loss function can be selected as the winner. Usually, safety is the major concern. For example, the safety review boards may express concerns regarding incompatible and aggressive decisions. If the probability of such decisions needs to be controlled, we recommend the POD designs together with the Probability Suspension Rule.
Lastly, the proposed framework can be extended to accommodate non-binary endpoints. For example, ordinal endpoints that account for multiple toxicity grades [@van2012dose; @mu2019gboin]. Another direction for further exploration is to apply the framework to drug combination trials [@wages2011dose; @liu2013bayesian2] or phase I/II trials that simultaneously consider toxicity and efficacy [@jin2014using].
[ @text@text]{}
Review of Complete-Data Designs {#supp:sec:review}
===============================
In this section, we provide a brief review of six main-stream complete-data dose-finding designs: CRM, BOIN, mTPI-2, keyboard, SPM and i3+3. The 3+3 design is excluded from the discussion, as it does not allow the specification of a particular DLT target $p_{{\text{T}}}$ and a maximum sample size. It is widely recognized that 3+3 has worse performance than, e.g., mTPI [@ji2013modified]. We denote by $N_z = \sum_{i = 1}^N {\mathbbm{1}}(Z_i = z)$, $n_z = \sum_{i = 1}^N {\mathbbm{1}}(Z_i = z, Y_i = 1)$ and $m_z = \sum_{i = 1}^N {\mathbbm{1}}(Z_i = z, Y_i = 0)$ the total number of patients, number of DLTs and number of non-DLTs at dose $z$, respectively.
Continual Reassessment Method (CRM) {#supp:sec:crm}
-----------------------------------
The CRM design assumes a dose-toxicity response curve $p_z = \phi(z, \alpha)$, such that $\phi$ monotonically increases with $z$, and $\alpha$ is an unknown parameter. For example, a commonly used dose-toxicity curve is $\phi(z, \alpha) = p_{0z}^{\exp(\alpha)}$, where $p_{0z}$’s are pre-specified constants satisfying $p_{01} < \cdots < p_{0D}$. The likelihood becomes $$\begin{aligned}
L( \alpha \mid {\bm{y}}, {\bm{z}}) = \prod_{i = 1}^N \phi(z_i, \alpha)^{y_i} [1 - \phi(z_i, \alpha)]^{1 - y_i}.\end{aligned}$$ Inference on $\alpha$ can be Bayesian [@o1990continual] or be based on maximum likelihood [@o1996continual]. From a Bayesian perspective, a prior distribution $\pi_0(\alpha)$ is specified for $\alpha$ (for example, $\alpha \sim \text{N}(0, 1.34^2)$), leading to the posterior $\pi(\alpha \mid {\bm{y}}, {\bm{z}}) \propto \pi_0(\alpha) L(\alpha \mid {\bm{y}}, {\bm{z}})$. The DLT probabilities can thus be estimated by $\hat{p}_z = \int \phi(z, \alpha) \pi(\alpha \mid {\bm{y}}, {\bm{z}}) d \alpha$. On the other hand, the maximum likelihood estimate (MLE) for $\alpha$ is $\hat{\alpha} = \operatorname*{arg\,max}_{\alpha} L(\alpha \mid {\bm{y}}, {\bm{z}})$, and $p_z$ can be estimated by $\hat{p}_z = \phi(z, \hat{\alpha})$. In both cases, the dose $d^* = \operatorname*{arg\,min}_{z} | \hat{p}_{z} - p_{{\text{T}}} |$ is recommended for the next patient, subject to some practical safety restrictions [@goodman1995some; @cheung2005coherence].
Bayesian Optimal Interval (BOIN) Design {#supp:sec:review_boin}
---------------------------------------
We refer to the local BOIN design [@liu2015bayesian], which considers a hypothesis test of three hypotheses: $$\begin{aligned}
H_0: p_d = p_{{\text{T}}}, \quad
H_1: p_d = p_{{\text{L}}}, \quad
H_2: p_d = p_{{\text{R}}}.
\label{eq:boin_hypothesis}\end{aligned}$$ Here $d$ is the current dose level, $p_{{\text{L}}}$ denotes the highest toxicity probability that is deemed subtherapeutic such that dose escalation should be made, and $p_{{\text{R}}}$ denotes the lowest toxicity probability that is deemed overly toxic such that dose de-escalation is required. The quantities $p_{{\text{L}}}$ and $p_{{\text{R}}}$ need to be pre-specified by physicians. Assuming equal prior weights on the three hypotheses, the optimal decision boundaries $\lambda_{{\text{L}}}(p_{{\text{T}}}, p_{{\text{L}}})$ and $\lambda_{{\text{R}}}(p_{{\text{T}}}, p_{{\text{R}}})$ minimizing the decision error rate are, $$\begin{aligned}
\begin{split}
\lambda_{{\text{L}}} &= \left. \log \left( \frac{1 - p_{{\text{L}}}}{1 - p_{{\text{T}}}} \right)
\middle/
\log \left[ \frac{ p_{{\text{T}}} (1 - p_{{\text{L}}}) }{ p_{{\text{L}}} (1 - p_{{\text{T}}}) } \right] \right., \\
\lambda_{{\text{R}}} &= \left. \log \left( \frac{1 - p_{{\text{T}}}}{1 - p_{{\text{R}}}} \right)
\middle/
\log \left[ \frac{ p_{{\text{R}}} (1 - p_{{\text{T}}}) }{ p_{{\text{T}}} (1 - p_{{\text{R}}}) } \right] \right..
\end{split}\end{aligned}$$ Let $\hat{p}_d = n_d / N_d$ denote the MLE for $p_d$. If $\hat{p}_d \leq \lambda_{{\text{L}}}$, the dose is escalated for the next patient; if $\hat{p}_d \geq \lambda_{{\text{R}}}$, the dose is de-escalated; and otherwise, the same dose level is retained. Note that $\lambda_{{\text{L}}}$ and $\lambda_{{\text{R}}}$ can be pre-specified without the optimization procedure in [@liu2015bayesian]. When they are pre-specified, BOIN uses essentially the same up-and-down rules as the cumulative cohort design [@ivanova2007cumulative].
mTPI-2 Design
-------------
The mTPI-2 design considers a partition of the $[0, 1]$ interval into an equivalence interval $I_{{\text{E}}} = [p_{{\text{T}}} - \epsilon_1, p_{{\text{T}}} + \epsilon_2]$, an underdosing interval $I_{{\text{U}}} = [0, p_{{\text{T}}} - \epsilon_1)$ and an overdosing interval $I_{{\text{O}}} = (p_{{\text{T}}} + \epsilon_2, 1]$. Any dose with toxicity probabilities inside $I_{{\text{E}}}$ is considered a true MTD, and the doses in $I_{{\text{U}}}$ (or $I_{{\text{O}}})$ are considered subtherapeutic (or overly toxic) and are deemed lower (or higher) than the MTD. The values $\epsilon_1$ and $\epsilon_2$ need to be specified by physicians. The intervals $I_{{\text{U}}}$ and $I_{{\text{O}}}$ are further divided into several sub-intervals, $I_{{\text{U}}_0}, \ldots, I_{{\text{U}}_{K_1}}$ and $I_{{\text{O}}_0}, \ldots, I_{{\text{O}}_{K_2}}$, such that all the sub-intervals have the same length $(\epsilon_1 + \epsilon_2)$ except for the two intervals ($I_{{\text{U}}_0}$ and $I_{{\text{O}}_0}$) reaching the boundary of $[0, 1]$. Let $d$ denote the current dose level, and let model $\mathcal{M}_d = j$ represent $\{ p_d \in I_j \}$, $j = {\text{U}}_0, \ldots, {\text{U}}_{K_1}, {\text{E}}, {\text{O}}_0, \ldots, {\text{O}}_{K_2}$. The mTPI-2 design is based on the following hierarchical model, $$\begin{aligned}
\begin{split}
&\Pr(\mathcal{M}_d = j) = 1/(K_1 + K_2 + 3), \; \text{for $j = {\text{U}}_0, \ldots, {\text{U}}_{K_1}, {\text{E}}, {\text{O}}_0, \ldots, {\text{O}}_{K_2}$}; \\
&p_d \mid \mathcal{M}_d \sim {\text{TBeta}}(1, 1; I_{\mathcal{M}_d}),
\end{split}\end{aligned}$$ where ${\text{TBeta}}(\cdot, \cdot; I)$ represents a truncated beta distribution restricted to interval $I$. The dose assignment decision for the next patient is to escalate, stay or de-escalate, if $\arg \max_{j}$ $\Pr(\mathcal{M}_d = j \mid n_d, m_d)$ belongs to $\{ {\text{U}}_0, \ldots, {\text{U}}_{K_1} \}$, equals E or belongs to $\{ {\text{O}}_0, \ldots, {\text{O}}_{K_2} \}$, respectively.
Keyboard Design
---------------
The keyboard design, similar to the mTPI-2 design, considers a partition of the $[0, 1]$ interval into equal-length sub-intervals (except for the two boundary intervals $I_{{\text{U}}_0}$ and $I_{{\text{O}}_0}$). The equal-length sub-intervals $I_{{\text{U}}_1}, \ldots, I_{{\text{U}}_{K_1}}$, $I_{{\text{E}}}$, $I_{{\text{O}}_1}, \ldots, I_{{\text{O}}_{K_2}}$ are referred to as keys, and the equivalence interval $I_{{\text{E}}}$ is referred to as the target key. The two boundary intervals $I_{{\text{U}}_0}$ and $I_{{\text{O}}_0}$ may not be long enough to form a key. Instead of using a hierarchical prior for $p_d$, keyboard considers a simple prior $p_d \sim {\text{Beta}}(1, 1)$, leading to the posterior $p_d \mid n_d, m_d \sim {\text{Beta}}(n_d + 1, m_d + 1)$. The dose assignment decision for the next patient is to escalate, stay or de-escalate, if $\arg \max_{j}$ $\Pr(p_d \in I_j \mid n_d, m_d)$ belongs to $\{ {\text{U}}_1, \ldots, {\text{U}}_{K_1} \}$, equals E or belongs to $\{ {\text{O}}_1, \ldots, {\text{O}}_{K_2} \}$, respectively.
Semiparametric Dose Finding Method (SPM) {#supp:sec:spm}
----------------------------------------
The SPM directly models the the location of the MTD $\gamma$, $1 \leq \gamma \leq D$. Conditional on $\gamma$ being the MTD, the support of $p_z$ is restricted to $$\begin{aligned}
\text{supp}(p_z) =
\begin{cases}
[0, p_{{\text{T}}} - \epsilon_1), & \text{if } z < \gamma; \\
[p_{{\text{T}}} - \epsilon_1, p_{{\text{T}}} + \epsilon_2], & \text{if } z = \gamma; \\
(p_{{\text{T}}} + \epsilon_2, 1], & \text{if } z > \gamma.
\end{cases}\end{aligned}$$ This restriction guarantees the partial ordering of the $p_z$’s. The partition of the $[0, 1]$ interval in the SPM coincides with the mTPI and mTPI-2 designs, while the center interval is interpreted differently as an indifference interval [@cheung2002simple]. The priors on $\gamma$ and $p_z$’s can be specified as follows, $$\begin{aligned}
\begin{split}
&\Pr(\gamma = z^*) = \kappa_{z^*}, \\
&p_z \mid \gamma \sim {\text{TBeta}}(c \theta_z^{\gamma} + 1, c(1 - \theta_z^{\gamma}) + 1; I_{z}^{\gamma}).
\end{split}\end{aligned}$$ Here $\kappa_{z^*}$, $c$ and $\theta_{z}^{\gamma}$ are hyperparameters, and $I_z^{\gamma} = I_{{\text{U}}}, I_{{\text{E}}}$ or $I_{{\text{O}}}$ for $z < \gamma$, $z = \gamma$ or $z > \gamma$, respectively. The posterior $\pi(\gamma, {\bm{p}}\mid {\bm{y}}, {\bm{z}}) \propto \pi_0(\gamma) \pi_0({\bm{p}}\mid \gamma) L( {\bm{p}}\mid {\bm{y}}, {\bm{z}})$, and the dose $\hat{\gamma} = \operatorname*{arg\,max}_{\gamma} \pi(\gamma \mid {\bm{y}}, {\bm{z}})$ is recommended for the next patient, again subject to some restrictions as in the CRM design.
i3+3 Design
-----------
The i3+3 design consists of a set of algorithmic decision rules, that is, model free. Similar to mTPI-2, it considers a partition of the $[0, 1]$ interval into $I_{{\text{E}}}$, $I_{{\text{U}}}$ and $I_{{\text{O}}}$. Suppose the current dose is $d$. If $n_d / N_d \in I_{{\text{U}}}$, the decision is escalation. If $n_d / N_d \in I_{{\text{E}}}$, the decision is stay. If $n_d / N_d \in I_{{\text{O}}}$, the decision is stay when $(n_d - 1) / N_d \in I_{{\text{U}}}$ and is de-escalation otherwise.
Modeling Time-to-Toxicity Data {#supp:sec:model_tite}
==============================
We introduce several examples for the specification of $f_{T \mid Z, Y}(t \mid z, Y = 1, {\bm{\xi}})$. That is, the model for the conditional distribution $[T \mid Z, Y = 1]$. Recall that the event $Y = 1$ is equivalent to $T \leq W$.
Uniform Distribution {#supp:sec:model_tite_unif}
--------------------
The simplest choice for the conditional distribution of $T$ is a uniform distribution, $$\begin{aligned}
T \mid Z, Y = 1 \sim {\text{Unif}}(0, W). \end{aligned}$$ This leads to the conditional pdf, $$\begin{aligned}
f_{T \mid Z, Y}(t \mid z, Y = 1) = 1 / W,\end{aligned}$$ where no additional parameter ${\bm{\xi}}$ is involved. The weight function in this case is $\rho(t \mid z) = t / W$. The uniform distribution, albeit simple, has been shown to be sufficient in many cases [@cheung2000sequential]. It is the default choice in [@cheung2000sequential], [@yuan2018time] and [@lin2018time].
Piecewise Uniform Distribution {#supp:sec:model_tite_pwunif}
------------------------------
A more flexible specification for the conditional distribution of $T$ is a piecewise uniform distribution. The interval $(0, W]$ is first partitioned into $K$ sub-intervals $\{ (h_{k-1}, h_k], k = 1, \ldots, K\}$, where $0 = h_0 < h_1 < \cdots < h_K = W$. Next, each sub-interval is assigned a weight $\omega_k$, $\sum_{k = 1}^K \omega_k = 1$. Conditional on $Y = 1$, $T$ falls into $(h_{k-1}, h_k]$ with probability $\omega_k$ and follows a uniform distribution within this interval. The conditional pdf of $T$ is thus $$\begin{aligned}
f_{T \mid Z, Y} (t \mid z, Y = 1, {\bm{\xi}}) = \omega_k \cdot \frac{1}{h_k - h_{k-1}}, \quad \text{for} \; h_{k-1} < t \leq h_k.
\label{eq:piecewise_uniform}\end{aligned}$$ The weight function is $$\begin{aligned}
\rho(t \mid z, {\bm{\xi}}) = \sum_{k = 1}^K \omega_k \beta(t, k),\end{aligned}$$ where $$\begin{aligned}
\beta(t, k) =
\begin{cases}
1, & \text{if } t > h_k; \\
\frac{t - h_{k-1}}{h_k - h_{k-1}}, & \text{if } t \in (h_{k-1}, h_k], k = 1, \ldots, K; \\
0, & \text{otherwise.}
\end{cases}
\label{eq:beta_fun}\end{aligned}$$ The parameters ${\bm{\xi}}= (K, h_1, \ldots, h_{K-1}, \omega_1, \ldots, \omega_K)$. The number of sub-intervals $K$ and the interval boundaries $h_k$’s are usually fixed. For example, $K = 3$ and $h_k = k W / K$ for $k = 1, \ldots, K$, representing three sub-intervals with equal lengths. Alternatively, let $n$ denote the number of observed DLTs, and let $0 < t_{(1)} \leq \cdots \leq t_{(n)} \leq W$ denote the ordered observed DLT times. One can set $K = n+1$, $h_k = t_{(k)}$ for $k = 1, \ldots, n$ and $h_K = W$. The weights of the sub-intervals $\omega_k$’s can be fixed if prior information is available or can be estimated otherwise. A Dirichlet distribution can be used as the prior for $(\omega_1, \ldots, \omega_K)$. The piecewise uniform distribution with equal-length sub-intervals and fixed sub-interval weights is considered in [@yuan2018time] and [@lin2018time]. The piecewise uniform distribution with empirically calibrated sub-intervals and same sub-interval weights $\omega_k = 1 / K$ is equivalent to the adaptive weighting scheme in [@cheung2000sequential].
Discrete Hazard Model {#supp:sec:model_tite_dhm}
---------------------
The conditional distribution of $T$ can be constructed from a discrete hazard model. Let $0 < t_{(1)} \leq \cdots \leq t_{(n)} \leq W$ denote the ordered observed DLT times, let $h_1 < \cdots < h_{K-1}$ represent the distinct DLT times that are strictly less than $W$, and let $h_K = W$. We assume $T$ can only take discrete values at the $h_k$’s given $Y = 1$. Let $\omega_k = \Pr(T = h_k \mid T \geq h_k, Y = 1)$ represent the discrete hazard at time $h_k$, with $\omega_K = 1$. The conditional probability mass function of $T$ is $$\begin{aligned}
\Pr(T = h_k \mid Z, Y = 1, {\bm{\xi}}) = \omega_k \prod_{j = 1}^{k-1} (1 - \omega_{j}), \quad k = 1, \ldots, K,\end{aligned}$$ and the weight function is $$\begin{aligned}
\rho(t \mid z, {\bm{\xi}}) = \Pr(T \leq t \mid Z, Y = 1, {\bm{\xi}}) = 1 - \prod_{k: h_k \leq t} (1 - \omega_k).\end{aligned}$$ The discrete hazard model is used in [@yuan2011robust].
Piecewise Constant Hazard Model {#supp:sec:model_tite_pchm}
-------------------------------
The conditional distribution of $T$ can also be constructed from a piecewise constant hazard model. Again, consider a partition of $(0, W]$ into $K$ sub-intervals $\{ (h_{k-1}, h_k], k = 1, \ldots, K\}$, where $0 = h_0 < h_1 < \cdots < h_K = W$. Given $Y = 1$, we assume the hazard of toxicity is $\omega_k$ in the interval $(h_{k-1}, h_k]$. This leads to the conditional pdf, $$\begin{aligned}
f_{T \mid Z, Y} (t \mid z, Y = 1, {\bm{\xi}}) = \omega_{k}^{{\mathbbm{1}}(t \in (h_{k-1}, h_k])} \exp \left[ - \sum_{k = 1}^{K} \omega_{k} (h_k - h_{k-1}) \beta(t, k) \right],
\label{eq:simple_hazard}\end{aligned}$$ where $\beta(t, k)$ is the same as in Equation . The weight function is $$\begin{aligned}
\rho(t \mid z, {\bm{\xi}}) = 1 - \exp \left[ - \sum_{k = 1}^{K} \omega_{k} \beta(t, k) \right].\end{aligned}$$ This model specification is used in [@liu2013bayesian] and [@jin2014using]. We note that although this specification can facilitate inference, it has a potential pitfall: $\int_0^W f_{T \mid Z, Y} (t \mid z, Y = 1, {\bm{\xi}}) {\text{d}}t = \rho(W \mid z, {\bm{\xi}}) < 1$, which means is not a proper pdf in $(0, W]$.
To construct a proper piecewise constant hazard model, each dose should have its distinct hazard parameters, and the hazards should depend on ${\bm{p}}$. Let $h_{K+1} = \infty$, leading to the $(K+1)$th interval $(W, \infty)$. For dose $z$, we assume a constant hazard $\omega_{zk}$ in the $k$th interval $(h_{k - 1}, h_k]$. The survival function is thus $$\begin{aligned}
S_{T \mid Z} (t \mid z, {\bm{p}}, {\bm{\xi}}) = \exp \left[ - \sum_{k = 1}^{K+1} \omega_{zk} \beta(t, k) \right],\end{aligned}$$ The conditional pdf of $T$ is $$\begin{gathered}
f_{T \mid Z, Y} (t \mid z, Y = 1, {\bm{p}}, {\bm{\xi}}) = \omega_{zk}^{{\mathbbm{1}}(t \in (h_{k-1}, h_k])} \times \\
\left\{ 1 - \exp \left[ - \sum_{k = 1}^{K} \omega_{zk} (h_k - h_{k-1}) \right] \right\}^{-1} \exp \left[ - \sum_{k = 1}^{K} \omega_{zk} \beta(t, k) \right],\end{gathered}$$ which depends on ${\bm{p}}$. To see this, notice that $S_{T \mid Z} (W \mid z, {\bm{p}}, {\bm{\xi}}) = 1 - p_z$, i.e., $\sum_{k = 1}^K \omega_{zk} (h_k - h_{k-1}) = - \log(1-p_z)$. Intuitively, a dose with a higher toxicity probability has a higher cumulative hazard, thus the hazards must depend on ${\bm{p}}$. Such dependency might cause difficulty in parameter estimation, for example, in the case that a prior distribution is specified for ${\bm{p}}$. Let $h_{K+1} = \infty$, leading to the $(K+1)$th interval $(W, \infty)$. For dose $z$, we assume a constant hazard $\omega_{zk}$ in the $k$th interval $(h_{k - 1}, h_k]$. The survival function is thus $$\begin{aligned}
S_{T \mid Z} (t \mid z, {\bm{p}}, {\bm{\xi}}) = \exp \left[ - \sum_{k = 1}^{K+1} \omega_{zk} \beta(t, k) \right],\end{aligned}$$ where $$\begin{aligned}
\beta(t, k) =
\begin{cases}
h_{k} - h_{k-1}, & \text{if } t > h_k; \\
t - h_{k-1}, & \text{if } t \in (h_{k-1}, h_k], k = 1, \ldots, K; \\
0, & \text{otherwise.}
\end{cases}\end{aligned}$$ The conditional pdf of $T$ is $$\begin{gathered}
f_{T \mid Z, Y} (t \mid z, Y = 1, {\bm{p}}, {\bm{\xi}}) = \omega_{zk}^{{\mathbbm{1}}(t \in (h_{k-1}, h_k])} \times \\
\left\{ 1 - \exp \left[ - \sum_{k = 1}^{K} \omega_{zk} (h_k - h_{k-1}) \right] \right\}^{-1} \exp \left[ - \sum_{k = 1}^{K} \omega_{zk} \beta(t, k) \right],\end{gathered}$$ which depends on ${\bm{p}}$. To see this, notice that $S_{T \mid Z} (W \mid z, {\bm{p}}, {\bm{\xi}}) = 1 - p_z$, i.e., $\sum_{k = 1}^K \omega_{zk} (h_k - h_{k-1}) = - \log(1-p_z)$. Intuitively, a dose with a higher toxicity probability has a higher cumulative hazard, thus the hazards must depend on ${\bm{p}}$. Such dependency might cause difficulty in parameter estimation, for example, in the case that a prior distribution is specified for ${\bm{p}}$.
A simplified piecewise constant hazard model specification is considered in the literature (e.g., @liu2013bayesian and @jin2014using), which assumes $$\begin{aligned}
f_{T \mid Z, Y} (t \mid z, Y = 1, {\bm{\xi}}) = \omega_{k}^{{\mathbbm{1}}(t \in (h_{k-1}, h_k])} \exp \left[ - \sum_{k = 1}^{K} \omega_{k} \beta(t, k) \right],
\label{eq:simple_hazard}\end{aligned}$$ without the constraint on the hazards and also does not depend on ${\bm{p}}$. This can greatly facilitate inference. A potential pitfall is that $$\begin{aligned}
\int_0^W f_{T \mid Z, Y} (t \mid z, Y = 1, {\bm{\xi}}) {\text{d}}t = \exp\left[ - \sum_{k=1}^K \omega_k (h_k - h_{k-1})\right] < 1,\end{aligned}$$ which means is not a proper pdf.
Rescaled Beta Distribution
--------------------------
Another possible specification for the conditional distribution of $T$ is a rescaled beta distribution. Specifically, $$\begin{aligned}
T/W \mid Z, Y = 1, \xi_1, \xi_2 \sim {\text{Beta}}(\xi_1, \xi_2),\end{aligned}$$ and $$\begin{aligned}
f_{T \mid Z, Y}(t \mid z, Y = 1, {\bm{\xi}}) = \frac{1}{\text{B}(\xi_1, \xi_2)} \cdot \frac{t^{\xi_1 - 1} (W - t)^{\xi_2 - 1}}{ W^{\xi_1 + \xi_2 - 1}},\end{aligned}$$ where $\text{B}(\cdot, \cdot)$ is the beta function. The rescaled beta distribution is considered in [@lin2018time]. Gamma distribution priors can be put on $\xi_1$ and $\xi_2$.
In all the examples above, the conditional distribution $f_{T \mid Z, Y}(t \mid z, Y = 1, {\bm{\xi}})$ does not depend on the dose, which implies $T$ and $Z$ are conditionally independent given $Y = 1$. This allows pooling of time-to-event information across doses. If desired, it is easy to let $f_{T \mid Z, Y}(t \mid z, Y = 1, {\bm{\xi}})$ vary across doses. For example, in Equation the parameters $\omega_k$’s can be changed to dose-specific parameters $\omega_{zk}$’s. Since the time-to-event data are usually sparse in dose-finding trials, information borrowing is recommended for estimating dose-specific parameters using, e.g., hierarchical priors.
Inference on the Toxicity Probabilities {#supp:sec:inference_p}
=======================================
Inference Based on the Survival Likelihood {#supp:sec:inference_sur}
------------------------------------------
With the survival likelihood , one can proceed with inference on ${\bm{p}}$. In this section, we propose some general strategies to conduct such inference.
#### *Independent Metropolis-Hastings Algorithm.*
From a Bayesian perspective, prior distributions $\pi_0({\bm{p}})$ and $\pi_0({\bm{\xi}})$ are specified for ${\bm{p}}$ and ${\bm{\xi}}$. Inference on ${\bm{p}}$ is summarized in the posterior distribution, $$\begin{aligned}
\pi({\bm{p}}, {\bm{\xi}}\mid {\mathcal{H}}) \propto \pi_0({\bm{p}}) \pi_0({\bm{\xi}}) L ( {\bm{p}}, {\bm{\xi}}\mid {\mathcal{H}}).\end{aligned}$$ In general, the posterior is not available in closed form, and Monte Carlo simulation is needed to approximate the posterior. When conjugate priors are used, the independent Metropolis-Hastings (IMH) algorithm (@robert2004monte, Section 7.4) can be employed. Let $\tilde{L}({\bm{p}}, {\bm{\xi}}\mid {\mathcal{H}})$ denote the complete case likelihood (i.e. the likelihood for the patients with complete outcomes), $$\begin{aligned}
\tilde{L}({\bm{p}}, {\bm{\xi}}\mid {\mathcal{H}})
= \prod_{i = 1}^N \Big[ p_{z_i}^{{\mathbbm{1}}({\tilde{y}}_i = 1)} (1 - p_{z_{i}})^{{\mathbbm{1}}({\tilde{y}}_i = 0, v_i = W)}
f_{T \mid Z, Y}(v_i \mid z_i, Y = 1, {\bm{\xi}})^{{\mathbbm{1}}({\tilde{y}}_i = 1)} \Big].\end{aligned}$$ The complete case likelihood can be factorized as $\tilde{L} = \tilde{L}_1 ({\bm{p}}\mid {\mathcal{H}}) \tilde{L}_2 ({\bm{\xi}}\mid {\mathcal{H}})$, where $$\begin{aligned}
\tilde{L}_1 ({\bm{p}}\mid {\mathcal{H}}) &= \prod_{i = 1}^N \Big[ p_{z_i}^{{\mathbbm{1}}({\tilde{y}}_i = 1)} (1 - p_{z_{i}})^{{\mathbbm{1}}({\tilde{y}}_i = 0, v_i = W)} \Big], \text{ and} \\
\tilde{L}_2 ({\bm{\xi}}\mid {\mathcal{H}}) &= \prod_{i = 1}^N \Big[ f_{T \mid Z, Y}(v_i \mid z_i, Y = 1, {\bm{\xi}})^{{\mathbbm{1}}({\tilde{y}}_i = 1)} \Big].\end{aligned}$$ To implement the IMH algorithm, we first randomly initialize ${\bm{p}}^{(0)}$ and ${\bm{\xi}}^{(0)}$. At iteration $j$ ($j = 1, 2, \ldots$), we propose $\tilde{{\bm{p}}}$ and $\tilde{{\bm{\xi}}}$ from the complete case posteriors, $$\begin{aligned}
\tilde{\pi}({\bm{p}}\mid {\mathcal{H}}) \propto \pi_0({\bm{p}}) \tilde{L}_1 ({\bm{p}}\mid {\mathcal{H}}), \;\; \text{and} \;\; \tilde{\pi}({\bm{\xi}}\mid {\mathcal{H}}) \propto \pi_0({\bm{\xi}}) \tilde{L}_2 ({\bm{\xi}}\mid {\mathcal{H}}).\end{aligned}$$ If conjugate priors are specified for ${\bm{p}}$ and ${\bm{\xi}}$ (e.g., a beta distribution prior for $p_z$), the complete case posteriors are available in closed form. The proposals are accepted with probability $$\begin{aligned}
q_{\text{acc}} \left( \tilde{{\bm{p}}}, \tilde{{\bm{\xi}}}; {\bm{p}}^{(j-1)}, {\bm{\xi}}^{(j-1)} \right) = 1 \wedge \prod_{i: {\tilde{y}}_i = 0, v_i < W} \frac{ 1 - \rho(v_i \mid z_i, \tilde{{\bm{\xi}}}) \tilde{p}_{z_i} }{ 1 - \rho(v_i \mid z_i, {\bm{\xi}}^{(j-1)}) p_{z_i}^{(j-1)} },\end{aligned}$$ and otherwise, ${\bm{p}}^{(j-1)}$ and ${\bm{\xi}}^{(j-1)}$ are retained. Under standard regularity conditions [@robert2004monte], the sequence $\{ {\bm{p}}^{(j)}, j = 1, 2, \ldots \}$ has a stationary distribution $\pi({\bm{p}}\mid {\mathcal{H}})$.
#### *Partial Derivatives of the Log-Likelihood.*
From a frequentist perspective, the MLE $\hat{{\bm{p}}}$ can be used as an estimate for ${\bm{p}}$. We can calculate $(\hat{{\bm{p}}}, \hat{{\bm{\xi}}}) = \operatorname*{arg\,max}_{{\bm{p}}, {\bm{\xi}}} L( {\bm{p}}, {\bm{\xi}}\mid {\mathcal{H}})$ by taking partial derivatives of the log-likelihood with respect to all the parameters. It suffices to solve the following equations, $$\begin{aligned}
\frac{\partial \log L} {\partial p_z} = \frac{n_z}{p_z} - \frac{m_z}{1 - p_z} - \sum_{i : {\tilde{y}}_i = 0, v_i < W, z_i = z} \frac{\rho(v_i \mid z_i, {\bm{\xi}})}{1 - \rho(v_i \mid z_i, {\bm{\xi}}) p_z } = 0,\end{aligned}$$ and $$\begin{gathered}
\frac{\partial \log L} {\partial \xi_k} = -\sum_{i : {\tilde{y}}_i = 0, v_i < W} \frac{p_{z_i}}{1 - \rho(v_i \mid z_i, {\bm{\xi}}) p_z } \frac{\partial \rho(v_i \mid z_i, {\bm{\xi}})} {\partial \xi_k} + \\
\sum_{i: {\tilde{y}}_i = 1} \frac{1}{f_{T \mid Z, Y}(v_i \mid z_i, Y = 1, {\bm{\xi}})} \frac{\partial f_{T \mid Z, Y}(v_i \mid z_i, Y = 1, {\bm{\xi}})} {\partial \xi_k} = 0.\end{gathered}$$
Equivalence of the Survival Likelihood and the Augmented Likelihood {#supp:sec:eq_sur_mis}
-------------------------------------------------------------------
For notational clarity, let $L_{{\text{mis}}}({\bm{p}}, {\bm{\xi}}\mid {\mathcal{H}})$ denote the derived data likelihood by marginalizing over ${\bm{y}}_{{\text{mis}}}$, and let $L_{\text{sur}}({\bm{p}}, {\bm{\xi}}\mid {\mathcal{H}})$ denote the survival likelihood . We have $$\begin{aligned}
L_{{\text{mis}}} ( {\bm{p}}, {\bm{\xi}}\mid {\mathcal{H}})
= {}&{} \sum_{{\bm{y}}_{{\text{mis}}} \in \{ 0, 1\}^r } L_{{\text{mis}}}( {\bm{p}}, {\bm{\xi}}, {\bm{y}}_{{\text{mis}}} \mid {\mathcal{H}}) \\
= {}&{} \prod_{i : b_i = 1} \Big [ p_{z_i}^{{\mathbbm{1}}(y_i = 1)} (1 - p_{z_i})^{{\mathbbm{1}}(y_i = 0)} f_{T \mid Z, Y}(v_i \mid z_i, Y_i = 1, {\bm{\xi}})^{{\mathbbm{1}}(y_i = 1)} \Big ] \times \\
{}&{} \qquad \prod_{i : b_i = 0} \Big \{ p_{z_i} \left[1 - \rho(v_i \mid z_i, {\bm{\xi}}) \right] + (1 - p_{z_i}) \Big \} \\
= {}&{} \prod_{i = 1}^N \Big \{ p_{z_i}^{{\mathbbm{1}}(y_i = 1, b_i = 1)} (1 - p_{z_i})^{{\mathbbm{1}}(y_i = 0, b_i = 1)} \times \\
{}&{} \qquad f_{T \mid Z, Y}(v_i \mid z_i, Y_i = 1, {\bm{\xi}})^{{\mathbbm{1}}(y_i = 1, b_i = 1)} \left[1 - \rho(v_i \mid z_i, {\bm{\xi}}) p_{z_i} \right]^{{\mathbbm{1}}(b_i = 0)} \Big \} \\
= {}&{} L_{\text{sur}} ( {\bm{p}}, {\bm{\xi}}\mid {\mathcal{H}}).\end{aligned}$$ $$\begin{gathered}
L_{{\text{mis}}} ( {\bm{p}}, {\bm{\xi}}\mid {\mathcal{H}})
= \sum_{{\bm{y}}_{{\text{mis}}} \in \{ 0, 1\}^r } L_{{\text{mis}}}( {\bm{p}}, {\bm{\xi}}, {\bm{y}}_{{\text{mis}}} \mid {\mathcal{H}})
= \prod_{i : b_i = 1} \Big [ p_{z_i}^{{\mathbbm{1}}(y_i = 1)} (1 - p_{z_i})^{{\mathbbm{1}}(y_i = 0)} \times \\
f_{T \mid Z, Y}(v_i \mid z_i, Y_i = 1, {\bm{\xi}})^{{\mathbbm{1}}(y_i = 1)} \Big ] \cdot
\prod_{i : b_i = 0} \Big \{ p_{z_i} \left[1 - \rho(v_i \mid z_i, {\bm{\xi}}) \right] + (1 - p_{z_i}) \Big \} = \\
\prod_{i = 1}^N \Big \{ p_{z_i}^{{\mathbbm{1}}(y_i = 1, b_i = 1)} (1 - p_{z_i})^{{\mathbbm{1}}(y_i = 0, b_i = 1)} f_{T \mid Z, Y}(v_i \mid z_i, Y_i = 1, {\bm{\xi}})^{{\mathbbm{1}}(y_i = 1, b_i = 1)} \times \\
\left[1 - \rho(v_i \mid z_i, {\bm{\xi}}) p_{z_i} \right]^{{\mathbbm{1}}(b_i = 0)} \Big \} = L_{\text{sur}} ( {\bm{p}}, {\bm{\xi}}\mid {\mathcal{H}}).\end{gathered}$$ Here, $r = \sum_{i = 1}^N {\mathbbm{1}}(b_i = 0) = \sum_{z = 1}^D r_z$ is the number of missing outcomes. The last equation holds because $\{ {\tilde{Y}}_i = 1\}$ is equivalent to $\{ Y_i = 1, B_i = 1\}$, and $\{ {\tilde{Y}}_i = 0 \}$ contains $\{ Y_i = 0, B_i = 1 \}$ and $\{ B_i = 0 \}$.
Inference Based on the Augmented Likelihood {#supp:sec:inference_mis}
-------------------------------------------
With the augmented likelihood , one can proceed with inference on ${\bm{p}}$. Specifically, can be factorized into $L( {\bm{p}}, {\bm{\xi}}, {\bm{y}}_{{\text{mis}}} \mid {\mathcal{H}}) = L_1({\bm{p}}, {\bm{y}}_{{\text{mis}}} \mid {\mathcal{H}}) L_2 ({\bm{\xi}}, {\bm{y}}_{{\text{mis}}} \mid {\mathcal{H}})$, where $$\begin{aligned}
L_1({\bm{p}}, {\bm{y}}_{{\text{mis}}} \mid {\mathcal{H}}) &= \prod_{i = 1}^N \Big[ p_{z_i}^{{\mathbbm{1}}(y_i = 1)} (1 - p_{z_{i}})^{{\mathbbm{1}}(y_i = 0)} \Big], \text{ and} \\
L_2 ({\bm{\xi}}, {\bm{y}}_{{\text{mis}}} \mid {\mathcal{H}}) &= \prod_{i = 1}^N \Big \{ f_{T \mid Z, Y}(v_i \mid z_i, Y = 1, {\bm{\xi}})^{{\mathbbm{1}}(y_i = 1, b_i = 1)} \left[1 - \rho(v_i \mid z_i, {\bm{\xi}}) \right]^{{\mathbbm{1}}(y_i = 1, b_i = 0)} \Big \}.\end{aligned}$$ This factorization facilitates inference.
#### *Data Augmentation.*
From a Bayesian perspective, the posterior distribution $\pi({\bm{p}}\mid {\mathcal{H}})$ can be simulated using the data augmentation (DA) method [@tanner1987calculation; @higdon1998auxiliary]. We first randomly initialize ${\bm{p}}^{(0)}$, ${\bm{\xi}}^{(0)}$ and ${\bm{y}}_{{\text{mis}}}^{(0)}$. At iteration $j$ ($j = 1, 2, \ldots$), implement the following procedures until the stationary distribution is reached and the desired number of posterior samples is obtained.
\(1) Imputation step. Sample ${\bm{y}}_{{\text{mis}}}^{(j)} \mid {\mathcal{H}}, {\bm{p}}^{(j-1)}, {\bm{\xi}}^{(j-1)}$ from ;
\(2) Posterior step. Sample ${\bm{p}}^{(j)} \mid {\mathcal{H}}, {\bm{y}}_{{\text{mis}}}^{(j)}$ and ${\bm{\xi}}^{(j)} \mid {\mathcal{H}}, {\bm{y}}_{{\text{mis}}}^{(j)}$ from the corresponding posteriors, $$\begin{aligned}
&\pi({\bm{p}}\mid {\mathcal{H}}, {\bm{y}}_{{\text{mis}}}^{(j)}) \propto \pi_0({\bm{p}}) L_1({\bm{p}}, {\bm{y}}_{{\text{mis}}}^{(j)} \mid {\mathcal{H}}), \quad \text{and} \\
&\pi({\bm{\xi}}\mid {\mathcal{H}}, {\bm{y}}_{{\text{mis}}}^{(j)}) \propto \pi_0({\bm{\xi}}) L_2 ({\bm{\xi}}, {\bm{y}}_{{\text{mis}}}^{(j)} \mid {\mathcal{H}}).\end{aligned}$$ If conjugate priors are specified for ${\bm{p}}$ and ${\bm{\xi}}$ (e.g., a beta distribution prior for $p_z$), the above posteriors are available in closed form. Through this procedure, we obtain a Markov chain $\{{\bm{p}}^{(j)},{\bm{\xi}}^{(j)}, {\bm{y}}_{{\text{mis}}}^{(j)}, j = 1, 2, \ldots \}$, whose stationary distribution is $\pi({\bm{p}}, {\bm{\xi}}, {\bm{y}}_{{\text{mis}}} \mid {\mathcal{H}})$ under standard regularity conditions. The sequence $\{ {\bm{p}}^{(j)}, j = 1, 2, \ldots \}$ has a marginal stationary distribution $\pi({\bm{p}}\mid {\mathcal{H}})$.
#### *Expectation Maximization.*
From a frequentist perspective, the MLE of ${\bm{p}}$ can be calculated through the expectation maximization (EM) algorithm [@dempster1977maximum]. We first randomly initialize ${\bm{p}}^{(0)}$, ${\bm{\xi}}^{(0)}$ and ${\bm{y}}_{{\text{mis}}}^{(0)}$. At iteration $j$ ($j = 1, 2, \ldots$), implement the following procedures until the desired convergence criteria is met.
\(1) Expectation step. Set ${\bm{y}}_{{\text{mis}}}^{(j)}$ at the expected values as in given the current parameter estimates ${\bm{p}}^{(j-1)}$ and ${\bm{\xi}}^{(j-1)}$;
\(2) Maximization step. Set ${\bm{p}}^{(j)}$ and ${\bm{\xi}}^{(j)}$ at the corresponding MLEs of , using $({\bm{y}}_{{\text{obs}}}, {\bm{y}}_{{\text{mis}}}^{(j)})$ as the full data. That is, $$\begin{aligned}
&{\bm{p}}^{(j)} = \operatorname*{arg\,max}_{{\bm{p}}} L_1({\bm{p}}, {\bm{y}}_{{\text{mis}}}^{(j)} \mid {\mathcal{H}}), \quad \text{and} \\
&{\bm{\xi}}^{(j)} = \operatorname*{arg\,max}_{{\bm{\xi}}} L_2 ({\bm{\xi}}, {\bm{y}}_{{\text{mis}}}^{(j)} \mid {\mathcal{H}}).\end{aligned}$$ Here, ${\bm{y}}_{{\text{mis}}}^{(j)}$ can be a fraction. Under standard regularity conditions, the sequence $\{{\bm{p}}^{(j)},$ ${\bm{\xi}}^{(j)},$ $j = 1, 2, \ldots \}$ converges to $\operatorname*{arg\,max}_{{\bm{p}}, {\bm{\xi}}} L({\bm{p}}, {\bm{\xi}}\mid {\mathcal{H}})$.
Imputation
----------
Imputation is an alternative way to deal with the unknown latent variables ${\bm{Y}}_{{\text{mis}}}$ in . After imputation, inference can be conducted as if the full data are available.
#### *Single imputation.*
Using single imputation, each $Y_{{\text{mis}}, i} \in {\bm{Y}}_{{\text{mis}}}$ is substituted with a reasonable estimate, for example, the expected value $\text{E}(Y_{{\text{mis}}, i} = 1 \mid Z_i = z_i, T_i > v_i, {\bm{p}}, {\bm{\xi}}) = \Pr(Y_{{\text{mis}}, i} = 1 \mid Z_i = z_i, T_i > v_i, {\bm{p}}, {\bm{\xi}})$ as in . This involves the unknown parameters ${\bm{p}}$ and ${\bm{\xi}}$. In TITE-BOIN [@yuan2018time], estimates of ${\bm{p}}$ and ${\bm{\xi}}$ based on the observed DLT outcomes are plugged in.
#### *Multiple imputation.*
Using multiple imputation [@rubin1996multiple], the imputed values are drawn multiple times from a distribution, for example, the posterior predictive $\Pr(Y_{{\text{mis}}} = {\bm{y}}_{{\text{mis}}} \mid {\bm{y}}_{{\text{obs}}}, {\bm{b}}, {\bm{v}}, {\bm{z}})$. Inferences are performed on all imputed datasets and are combined. Multiple imputation better accounts for the variability associated with the inference compared to single imputation.
Design Properties {#supp:sec:property}
=================
Large-Sample Convergence Properties {#supp:sec:convergence}
-----------------------------------
\(1) In the likelihood , we first treat ${\bm{\xi}}$ as a fixed parameter. Conditional on ${\bm{\xi}}$, $p_z$ is independent of the information at the other doses, which simplifies the problem. In the end, we will integrate out ${\bm{\xi}}$ and show the lemma also holds unconditional on ${\bm{\xi}}$. For notation simplicity, we suppress the subscript $z$ in the following proof and only consider the patients at dose $z$. That is, $N_z$, $n_z$, $m_z$, $r_z$, $p_z$ and $p_{0z}$ are simplified as $N$, $n$, $m$, $r$, $p$ and $p_0$. Also, $y_i$, ${\tilde{y}}_i$, $v_i$ and $b_i$ now only refer to the DLT outcome, current toxicity status, follow-up time and observed outcome indicator for a patient $i$ at dose $z$, $i = 1, \ldots, N_z$. Without loss of generality, assume $0 < p_0 - \epsilon < p_0 < p_0 + \epsilon < 1$, thus $\log (p_0 - \epsilon)$, $\log p_0$ and $\log (1 - p_0 - \epsilon)$ are finite. The likelihood of $p$ is $$\begin{aligned}
L ( p \mid {\mathcal{H}}, {\bm{\xi}}) &= \prod_{i = 1}^N \Big\{ p^{{\mathbbm{1}}({\tilde{y}}_i = 1)} (1 - p)^{{\mathbbm{1}}({\tilde{y}}_i = 0, v_i = W)}
\left[ 1 - \rho(v_i \mid z, {\bm{\xi}}) p \right]^{{\mathbbm{1}}({\tilde{y}}_i = 0, v_i < W)}
\Big\}\\
&= \prod_{i = 1}^N \Big\{ p^{{\mathbbm{1}}(y_i = 1, b_i = 0)} (1 - p)^{{\mathbbm{1}}(y_i = 0, b_i = 0)}
\left[ 1 - \rho(v_i \mid z, {\bm{\xi}}) p \right]^{{\mathbbm{1}}(b_i = 1)}
\Big\}.\end{aligned}$$ Define $$\begin{aligned}
\eta_N (p; {\mathcal{H}}, {\bm{\xi}})
= \frac{1}{N} \log \frac{L ( p_0 \mid {\mathcal{H}}, {\bm{\xi}})}{L ( p \mid {\mathcal{H}}, {\bm{\xi}})}.
\label{supp:eq_eta_fn}\end{aligned}$$ We have $$\begin{gathered}
\eta_N (p; {\mathcal{H}}, {\bm{\xi}}) = \frac{1}{N} \Big \{ n (\log p_0 - \log p) + m [\log (1 - p_0) - \log (1 - p) ] + \\
\sum_{i: b_i = 1} [ \log (1 - \rho_i p_0) - \log (1 - \rho_i p) ] \Big \},\end{gathered}$$ where $\rho_i = \rho(v_i \mid z, {\bm{\xi}})$, and $0 \leq \rho_i \leq 1$. Thus, $$\begin{aligned}
\underline{\eta}_N (p; {\mathcal{H}}) \leq \eta_N (p; {\mathcal{H}}, {\bm{\xi}}) \leq \overline{\eta}_N (p; {\mathcal{H}}),\end{aligned}$$ where $$\begin{aligned}
\underline{\eta}_N (p; {\mathcal{H}}) &= \frac{1}{N} \left[ n (\log p_0 - \log p) + (m + r) \log (1 - p_0) - m \log(1 - p) \right], \\
\overline{\eta}_N (p; {\mathcal{H}}) &= \frac{1}{N} \left[ n (\log p_0 - \log p) + m \log (1 - p_0) - (m + r) \log(1 - p) \right].\end{aligned}$$ By taking derivatives, we know $\underline{\eta}_N (p; {\mathcal{H}})$ monotonically decreases on $[0, n/(n+m))$, reaches the minimum at $n/(n+m)$ and monotonically increases on $(n/(n+m), 1]$; $\overline{\eta}_N (p; {\mathcal{H}})$ monotonically decreases on $[0, n/N)$, reaches the minimum at $n/N$ and monotonically increases on $(n/N, 1]$
Furthermore, let $$\begin{aligned}
\eta^* (p) = p_0 (\log p_0 - \log p) + (1 - p_0) [\log (1 - p_0) - \log (1 - p)].\end{aligned}$$ Similarly, $\eta^* (p)$ monotonically decreases on $[0, p_0)$, reaches the minimum at $p_0$ and monotonically increases on $(p_0, 1]$. We have $\eta^* (p_0) = 0$. Since $\eta^* (p)$ is continuous, $\exists \delta > 0$ and $0 < \epsilon_0 < \epsilon$, such that $\min \{ \eta^* (p_0 - \epsilon), \eta^* (p_0 + \epsilon) \} > 5 \delta$ and $\max \{ \eta^* (p_0 - \epsilon_0), \eta^* (p_0 + \epsilon_0) \} < \delta$.
Let $\bar{{\mathcal{C}}}_{\varepsilon}$ denote the complement of ${\mathcal{C}}_{\varepsilon}$. Recall that ${\mathcal{C}}_{\varepsilon} = \{ p: | p - p_{0} | < \varepsilon \}$. Consider the posterior odds, $$\begin{aligned}
\frac{\int_{{\mathcal{C}}_{\varepsilon}} \pi(p \mid {\mathcal{H}}, {\bm{\xi}}) {\text{d}}p}{\int_{\bar{{\mathcal{C}}}_{\varepsilon}} \pi(p \mid {\mathcal{H}}, {\bm{\xi}}) {\text{d}}p} =
&\frac{\int_{{\mathcal{C}}_{\varepsilon}} \pi_0(p) L(p \mid {\mathcal{H}}, {\bm{\xi}}) {\text{d}}p}{\int_{\bar{{\mathcal{C}}}_{\varepsilon}} \pi_0(p) L(p \mid {\mathcal{H}}, {\bm{\xi}}) {\text{d}}p} \nonumber \\
= &\frac{\int_{{\mathcal{C}}_{\varepsilon}} \pi_0(p) \exp [-N \eta_N (p; {\mathcal{H}}, {\bm{\xi}})] {\text{d}}p}{\int_{\bar{{\mathcal{C}}}_{\varepsilon}} \pi_0(p) \exp [-N \eta_N (p; {\mathcal{H}}, {\bm{\xi}})] {\text{d}}p}.
\label{supp:eq:post_odd}\end{aligned}$$
According to the strong law of large numbers, and since $r = o(N)$, $\exists N_0$ (note that $N_0$ does not depend on ${\bm{\xi}}$), $\forall N > N_0$, $$\begin{gathered}
\max \left\{ \left| \frac{n}{n+m} - p_0 \right|, \left| \frac{n}{N} - p_0 \right|, \left| \frac{n+r}{N} - p_0 \right| \right\} < \\
\min \left\{ \epsilon_0, \frac{1}{4} \left| \frac{\delta}{\log(p_0 - \epsilon)}\right|, \frac{1}{4} \left| \frac{\delta}{\log(1 - p_0 - \epsilon)}\right| \right\},\end{gathered}$$ almost surely (a.s.). We have $$\begin{gathered}
| \underline{\eta}_N (p_0 - \epsilon; {\mathcal{H}}) - \eta^* (p_0 - \epsilon) | \leq \left| \frac{n}{N} - p_0 \right| \cdot \left| \log p_0 \right| + \left| \frac{n}{N} - p_0 \right| \cdot \left | \log (p_0 - \epsilon) \right | + \\
\left| \frac{n}{N} - p_0 \right| \cdot \left| \log (1 - p_0) \right| + \left| \frac{n+r}{N} - p_0 \right| \cdot \left| \log (1 - p_0 + \epsilon) \right| \leq \delta.\end{gathered}$$
Thus, for all $p \leq p_0 - \epsilon$, $$\begin{aligned}
\eta_N (p; {\mathcal{H}}, {\bm{\xi}}) &\geq \underline{\eta}_N (p; {\mathcal{H}}) \geq \underline{\eta}_N (p_0 - \epsilon; {\mathcal{H}}) \\
&\geq \eta^* (p_0 - \epsilon) - | \underline{\eta}_N (p_0 - \epsilon; {\mathcal{H}}) - \eta^* (p_0 - \epsilon) | > 4 \delta.\end{aligned}$$ Similarly, for all $p \geq p_0 + \epsilon$, we have $\eta_N (p; {\mathcal{H}}, {\bm{\xi}}) > 4 \delta$.
Next, consider $p_0 - \epsilon_0 \leq p \leq p_0 + \epsilon_0$. We have $$\begin{gathered}
| \overline{\eta}_N (p_0 - \epsilon_0; {\mathcal{H}}) - \eta^* (p_0 - \epsilon_0) | \leq \left| \frac{n}{N} - p_0 \right| \cdot \left| \log p_0 \right| + \left| \frac{n}{N} - p_0 \right| \cdot \left | \log (p_0 - \epsilon_0) \right | + \\
\left| \frac{n+r}{N} - p_0 \right| \cdot \left| \log (1 - p_0) \right| + \left| \frac{n}{N} - p_0 \right| \cdot \left| \log (1 - p_0 + \epsilon_0) \right| \leq \delta.\end{gathered}$$ Thus, for all $p_0 - \epsilon_0 \leq p \leq n/N$, $$\begin{aligned}
\eta_N (p; {\mathcal{H}}, {\bm{\xi}}) &\leq \overline{\eta}_N (p; {\mathcal{H}}) \leq \underline{\eta}_N (p_0 - \epsilon_0; {\mathcal{H}}) \\
&\leq \eta^* (p_0 - \epsilon_0) + | \overline{\eta}_N (p_0 - \epsilon_0; {\mathcal{H}}) - \eta^* (p_0 - \epsilon_0) | < 2 \delta.\end{aligned}$$ Similarly, for all $n/N \leq p \leq p_0 + \epsilon_0$, we have $\eta_N (p; {\mathcal{H}}, {\bm{\xi}}) < 2 \delta$.
As a result, in Equation , the numerator and the denominator satisfy $$\begin{aligned}
&\int_{{\mathcal{C}}_{\varepsilon}} \pi_0(p) \exp [-N \eta_N (p; {\mathcal{H}}, {\bm{\xi}})] {\text{d}}p \geq \exp (- 2 N \delta) \pi_0(p \in {\mathcal{C}}_{\epsilon_0}), \\
&\int_{\bar{{\mathcal{C}}}_{\varepsilon}} \pi_0(p) \exp [-N \eta_N (p; {\mathcal{H}}, {\bm{\xi}})] {\text{d}}p \leq \exp (- 4 N \delta),\end{aligned}$$ respectively, for all $N > N_0$ a.s. That is, for some $\delta > 0$ and $0 < \epsilon_0 < \epsilon$, $\exists N_0 > 0$, $\forall N > N_0$, $\eqref{supp:eq:post_odd} \geq \exp (2 N \delta) \pi_0(p \in {\mathcal{C}}_{\epsilon_0})$ a.s.
Since the numerator and the denominator of add up to 1, through simple algebra we have $\pi(p \in {\mathcal{C}}_{\varepsilon} \mid {\mathcal{H}}, {\bm{\xi}}) \geq 1 - [1 + \exp (2 N \delta) \pi_0(p \in {\mathcal{C}}_{\epsilon_0}) ]^{-1}$. Notice that none of the terms on the right hand side depend on ${\bm{\xi}}$. Thus, $$\begin{aligned}
\pi(p \in {\mathcal{C}}_{\varepsilon} \mid {\mathcal{H}}) &= \int_{{\bm{\xi}}} \pi(p \in {\mathcal{C}}_{\varepsilon} \mid {\mathcal{H}}, {\bm{\xi}}) \pi({\bm{\xi}}\mid {\mathcal{H}}) {\text{d}}{\bm{\xi}}\\
&\geq \int_{{\bm{\xi}}} \left\{ 1 - [1 + \exp (2 N \delta) \pi_0(p \in {\mathcal{C}}_{\epsilon_0}) ]^{-1} \right\} \pi({\bm{\xi}}\mid {\mathcal{H}}) {\text{d}}{\bm{\xi}}\\
&= 1 - [1 + \exp (2 N \delta) \pi_0(p \in {\mathcal{C}}_{\epsilon_0}) ]^{-1}\end{aligned}$$ and goes to 1 a.s. as $N \rightarrow \infty$.
\(2) In (1) we have already proved $\forall \epsilon > 0$, $\exists N_0$, $\forall N > N_0$, $\exists \delta > 0$ such that $\eta_N (p; {\mathcal{H}}, {\bm{\xi}}) > 4 \delta$ for all $p \in \bar{{\mathcal{C}}}_{\varepsilon}$ and for all ${\bm{\xi}}$ a.s. Since the log-likelihood at the MLE $\hat{p}$ must be greater than or equal to the log-likelihood at any other place (including $p_0$), we have $\eta_N (\hat{p}; {\mathcal{H}}, \hat{{\bm{\xi}}}) \leq 0$. As a result, $\hat{p} \in \bar{{\mathcal{C}}}_{\varepsilon}$ would cause a contradiction. Therefore, $\forall \epsilon > 0$, $\exists N_0$, $\forall N > N_0$, $\hat{p} \in {\mathcal{C}}_{\varepsilon}$ a.s., which means $\hat{p} \rightarrow p_0$.
The proof follows [@oron2011dose] as a consequence of Lemma \[lem:consistency\]. Define the random set $$\begin{aligned}
{\mathcal{Z}}= \{ z \in \{ 1, \ldots, D\}: N_z \rightarrow \infty \text{ as } N \rightarrow \infty \},\end{aligned}$$ where $N_z$ is the number of patients assigned to dose $z$. Obviously, ${\mathcal{Z}}$ is nonempty and is composed of consecutive dose levels, ${\mathcal{Z}}= \{ Z_1, \ldots, Z_2 \}$. Here $Z_1, \ldots, Z_2$ are consecutively ordered integers ($Z_1 \leq Z_2$). For a particular dose $d^*$, the possible configurations of ${\mathcal{Z}}$ can be partitioned into three subspaces: $A = \{ Z_1 = Z_2 = d^* \}$, $B = \{ Z_1 < Z_2 \text{ and } d^* \in {\mathcal{Z}}\}$ and $C = \{ d^* \notin {\mathcal{Z}}\}$. Almost sure convergence to $d^*$ is equivalent to $\Pr(A) = 1$.
Suppose the dose $d^*$ satisfies $p_{0d^*} \in (p_{{\text{T}}} - \epsilon_1, p_{{\text{T}}} + \epsilon_2)$, and $d^*$ is also the only dose such that $p_{0d^*} \in [p_{{\text{T}}} - \epsilon_1, p_{{\text{T}}} + \epsilon_2]$. We first prove $\Pr(C) = 0$ by contradiction. With out loss of generality, we assume there is some specific dose $z_1 > d^*$ for which $\Pr(Z_1 = z_1) > 0$. From the theorem’s condition, $p_{0 z_1} > p_{{\text{T}}} + \epsilon_2$. When the event $Z_1 = z_1$ happens, $N_{z_1} \rightarrow \infty$ as $N \rightarrow \infty$. Thus $\forall \epsilon > 0$, for $N$ large enough, $\pi[p_{z_1} \in (p_{0 z_1} - \epsilon, p_{0 z_1} + \epsilon) \mid {\mathcal{H}}] \rightarrow 1$, meaning $\arg \max_{j} \Pr(p_{z_1} \in I_j \mid {\mathcal{H}}) \in \{ {\text{O}}_1, \ldots, {\text{O}}_{K_2} \}$. Similarly, for $N$ large enough, the MLE $\hat{p}_{z_1} > p_{{\text{T}}} + \epsilon_2$. According to the transition rule of interval-based designs, the next lower dose level $z_1 - 1$ will be assigned a.s. following each allocation to $z_1$. Thus $z_1 - 1 \in {\mathcal{Z}}$, reaching a contradiction. As a result, for all $z_1 > d^*$, $\Pr(Z_1 = z_1) = 0$. Based on similar reasoning, for all $z_2 < d^*$, $\Pr(Z_2 = z_2) = 0$. This means with probability 1, $Z_1 \leq d^*$ and $Z_2 \geq d^*$, i.e., $d^* \in {\mathcal{Z}}$.
Since $d^* \in {\mathcal{Z}}$, $N_{d^*} \rightarrow \infty$ as $N \rightarrow \infty$. Thus $\forall \epsilon > 0$, for $N$ large enough, $\pi[p_{d^*} \in (p_{0 d^*} - \epsilon, p_{0 d^*} + \epsilon) \mid {\mathcal{H}}] \rightarrow 1$, meaning $\Pr(p_{d^*} \in I_{{\text{E}}} \mid {\mathcal{H}}) \rightarrow 1$. Similarly, for $N$ large enough, the MLE $\hat{p}_{d^*} \in I_{{\text{E}}}$. According to the transition rule of interval-based designs, the same dose level $d^*$ will be retained a.s. following each assignment to $d^*$. Thus, there is no other level in ${\mathcal{Z}}$ with probability 1, and the dose allocation converges a.s. to $d^*$.
For notation simplicity, we suppress the subscript $d$ in the following proof and only consider the patients with $Z_i = d$, as interval-based nonparametric designs (defined in Section \[sec:complete\_data\_design\]) only use information at the current dose. Suppose ${\mathcal{A}}^*({\bm{y}})$ is the dose decision function of an interval-based nonparametric complete-data design, where ${\bm{y}}$ denotes the vector of outcomes at the current dose only. We have $$\begin{aligned}
\Pr(A = a \mid {\mathcal{H}}) = \sum_{{\bm{y}}_{{\text{mis}}}: {\mathcal{A}}^*({\bm{y}}_{{\text{obs}}}, {\bm{y}}_{{\text{mis}}}) = a} \Pr({\bm{Y}}_{{\text{mis}}} = {\bm{y}}_{{\text{mis}}} \mid {\mathcal{H}}).\end{aligned}$$ Again, ${\bm{y}}_{{\text{obs}}}$ and ${\bm{Y}}_{{\text{mis}}}$ now only refer to the observed outcomes and unknown outcomes for patients at the current dose. Let $\mathcal{Y}_{{\text{mis}}} = \{ 0, 1 \}^r$ denote the support of ${\bm{Y}}_{{\text{mis}}}$, where $r$ is the number of pending patients at the current dose.
\(1) Suppose $p_0 \in (p_{{\text{T}}} - \epsilon_1, p_{{\text{T}}} + \epsilon_2)$. It suffices to show $\forall {\bm{y}}_{{\text{mis}}} \in \mathcal{Y}_{{\text{mis}}}$, $\exists N_0 > 0$, when $N > N_0$, ${\mathcal{A}}^*({\bm{y}}_{{\text{obs}}}, {\bm{y}}_{{\text{mis}}}) = d$ a.s. For a complete-data design with outcomes $({\bm{y}}_{{\text{obs}}}, {\bm{y}}_{{\text{mis}}})$, the likelihood of $p$ is $$\begin{aligned}
L( p \mid {\bm{y}}_{{\text{obs}}}, {\bm{y}}_{{\text{mis}}}) = \prod_{i = 1}^N p^{y_i} (1-p)^{1 - y_i} = p^{n + s} (1-p)^{m+r-s},\end{aligned}$$ where $s = \sum_{l = 1}^r {\mathbbm{1}}(y_{{\text{mis}}, l} = 1)$ counts the number of DLTs in ${\bm{y}}_{{\text{mis}}}$, $0 \leq s \leq r$. Similar to Equation , consider $$\begin{gathered}
\eta_N (p; {\bm{y}}_{{\text{obs}}}, {\bm{y}}_{{\text{mis}}})
= \frac{1}{N} \log \frac{L(p_0 \mid {\bm{y}}_{{\text{obs}}}, {\bm{y}}_{{\text{mis}}})}{L(p \mid {\bm{y}}_{{\text{obs}}}, {\bm{y}}_{{\text{mis}}})} \\
= \frac{1}{N} \Big\{ (n+s) (\log p_0 - \log p) + (m+r+s) [\log(1 - p_0) - \log (1 - p)] \Big\}.\end{gathered}$$ The function $\eta_N (p; {\bm{y}}_{{\text{obs}}}, {\bm{y}}_{{\text{mis}}})$ monotonically decreases on $[0, (n+s) / N)$, reaches the minimum at $(n+s) / N$ and monotonically increases on $((n+s) / N, 1]$.
Let $\epsilon_0 = \min \{ \epsilon_1, \epsilon_2 \}$. Similar to the proof of Lemma \[lem:consistency\], by the strong law of large numbers, and since $r = o(N)$, we can show $\exists N_0 > 0$, when $N > N_0$, $\pi(p \in {\mathcal{C}}_{\epsilon_0} \mid {\bm{y}}_{{\text{obs}}}, {\bm{y}}_{{\text{mis}}}) \geq 0.99$ a.s., for any $0 \leq s \leq r$. Also, when $N > N_0$, $| \hat{p} - p_0 | < \epsilon_0 $ a.s. If $p_{0} \in (p_{{\text{T}}} - \epsilon_1, p_{{\text{T}}} + \epsilon_2)$, then for $N > N_0$ and for any ${\bm{y}}_{{\text{mis}}}$, $\arg \max_{j} \Pr(p \in I_j \mid {\bm{y}}_{{\text{obs}}}, {\bm{y}}_{{\text{mis}}}) = {\text{E}}$ and $\hat{p} \in I_{{\text{E}}}$ a.s. That is, ${\mathcal{A}}^*({\bm{y}}_{{\text{obs}}}, {\bm{y}}_{{\text{mis}}}) = d$ for any ${\bm{y}}_{{\text{mis}}}$. As a result, $$\begin{aligned}
\Pr(A = d \mid {\mathcal{H}}) = \sum_{{\bm{y}}_{{\text{mis}}} \in \mathcal{Y}_{{\text{mis}}}} \Pr({\bm{Y}}_{{\text{mis}}} = {\bm{y}}_{{\text{mis}}} \mid {\mathcal{H}}) = 1,\end{aligned}$$ a.s. for $N > N_0$.
(2, 3) For the same reason, if $p_{0} < p_{{\text{T}}} - \epsilon_1$, then $\exists N_0$, when $N > N_0$, $\forall {\bm{y}}_{{\text{mis}}} \in \mathcal{Y}_{{\text{mis}}}$, ${\mathcal{A}}^*({\bm{y}}_{{\text{obs}}}, {\bm{y}}_{{\text{mis}}}) = d + 1$ a.s. Thus $\Pr(A = d + 1 \mid {\mathcal{H}}) = 1$ a.s. If $p_{0} > p_{{\text{T}}} + \epsilon_2$, then $\exists N_0$, when $N > N_0$, $\forall {\bm{y}}_{{\text{mis}}} \in \mathcal{Y}_{{\text{mis}}}$, ${\mathcal{A}}^*({\bm{y}}_{{\text{obs}}}, {\bm{y}}_{{\text{mis}}}) = d - 1$ a.s. Thus $\Pr(A = d-1 \mid {\mathcal{H}}) = 1$ a.s.
Coherence Principles {#supp:sec:coherence}
--------------------
We prove below for a simple case that an interval-based TITE design is interval coherent in the sense of Definition \[supp:def:coherence\_interval\].
Consider an interval-based TITE design that makes dose-finding decisions based on the MLE (see Section \[sec:complete\_data\_design\]). We establish its interval coherence in de-escalation. First, consider a function $$\begin{aligned}
\ell(p, \rho_1, \ldots, \rho_r; n, m) \triangleq \frac{n}{p} - \frac{m}{1 - p} - \sum_{i = 1}^r \frac{\rho_i}{1 - \rho_i p },\end{aligned}$$ where $n$, $m$ and $r$ are some non-negative integers. We have $$\begin{aligned}
\frac{\partial \ell}{\partial p} &= - \frac{n}{p^2} - \frac{m}{(1-p)^2} - \sum_{i = 1}^r \frac{\rho_i^2}{(1 - \rho_i p)^2} < 0, \quad \text{and} \\
\frac{\partial \ell}{\partial \rho_i} &= - \frac{1}{(1 - \rho_i p)^2} < 0,\end{aligned}$$ for all $p$ and $\rho_i$, which mean that $\ell$ monotonically decreases with $p$ and $\rho_i$.
Next, suppose the currently-administrated doses just prior to $\tau$ and $\tau + \tau'$ are both $d$. At dose $d$, let $n_d$ and $m_d$ denote the numbers patients that have or do not have DLTs just prior to $\tau$, let $i = 1, \ldots, r_1$ index the patients that are still being followed just prior to $\tau$, and let $i = r_1 + 1, \ldots, r_2$ index the patients that are enrolled between $[\tau, \tau + \tau')$. The MLE of $p_d$ just prior to $\tau$, denoted by $\hat{p}_d(\tau)$, satisfies $$\begin{aligned}
\ell[\hat{p}_d(\tau), \rho_1(\tau), \ldots, \rho_{r_1 + r_2} (\tau); n_d, m_d] = 0,\end{aligned}$$ where $\rho_i(\tau) = \rho[v_i(\tau) \mid d, \hat{{\bm{\xi}}}]$ for some $\hat{{\bm{\xi}}}$ for $i = 1, \ldots, r_1$, and $\rho_i(\tau) = 0$ for $i = r_1 + 1, \ldots, r_2$. On the other hand, suppose no DLT occurs at dose $d$ during $[\tau, \tau + \tau')$. Then, the MLE of $p_d$ just prior to $\tau + \tau'$, denoted by $\hat{p}_d(\tau + \tau')$, satisfies $$\begin{aligned}
\ell[\hat{p}_d(\tau + \tau'), \rho_1(\tau + \tau'), \ldots, \rho_{r_1 + r_2} (\tau + \tau'); n_d, m_d] = 0,\end{aligned}$$ where $\rho_i(\tau + \tau') = \rho[v_i(\tau + \tau') \mid d, \hat{{\bm{\xi}}}]$ if patient $i$ is still being followed just prior to $\tau + \tau'$, and $\rho_i(\tau + \tau') = 1$ if patient $i$ has finished followup just prior to $\tau + \tau'$ with no DLT.
By definition, $\rho_i(\tau + \tau') \geq \rho_i(\tau)$. Therefore, $$\begin{gathered}
\ell[\hat{p}_d(\tau + \tau'), \rho_1(\tau), \ldots, \rho_{r_1 + r_2} (\tau); n_d, m_d] \geq \\
\ell[\hat{p}_d(\tau + \tau'), \rho_1(\tau + \tau'), \ldots, \rho_{r_1 + r_2} (\tau + \tau'); n_d, m_d] = 0 = \\
\ell[\hat{p}_d(\tau), \rho_1(\tau), \ldots, \rho_{r_1 + r_2} (\tau); n_d, m_d].\end{gathered}$$ Since $\ell$ monotonically decreases with $p$, we have $\hat{p}_d(\tau + \tau') \leq \hat{p}_d(\tau)$, thus ${\mathcal{A}}[{\mathcal{H}}(\tau + \tau')] \geq {\mathcal{A}}[{\mathcal{H}}(\tau)]$. Coherence in escalation can be proved in a similar way.
Selection of the MTD {#supp:sec:sel_mtd}
====================
Except for the 3+3 and R6 designs, the patient enrollment is terminated if the number of enrolled patients reaches the pre-specified maximum sample size $N^*$ or an early stopping rule (e.g. Safety rule 1) is triggered. After all patients have finished their DLT assessment, the trial completes, and the next step is to recommend an MTD. The selection of MTD does not involve any pending outcomes and is simply a problem of statistical inference about ${\bm{p}}$ under the likelihood and the order constraint $p_1 \leq p_2 \leq \cdots \leq p_D$. Usually, the doses with $\Pr(p_z > p_{{\text{T}}} \mid {\text{data}}) > \nu$ for a $\nu$ close to 1 and the doses that have never been tried are excluded from the MTD candidates. If the trial is stopped early because the lowest dose is overly toxic, no MTD will be selected.
The MTD selection rules for the CRM and SPM are consistent with their dose assignment rules. For CRM, the dose $d^* = \operatorname*{arg\,min}_{z} | \hat{p}_{z} - p_{{\text{T}}} |$ is selected as the MTD. For SPM, the dose $\hat{\gamma} = \operatorname*{arg\,max}_{\gamma} \pi(\gamma \mid {\bm{y}}, {\bm{z}})$ is selected as the MTD. See more details in Sections \[supp:sec:crm\] and \[supp:sec:spm\]. On the other hand, the MTD selection rules for the BOIN, mTPI-2, keyboard and i3+3 are different from their dose assignment rules, as their dose assignments only depend on outcomes at the current dose. To impose the order constraint, an isotonic regression is performed using the pooled adjacent violators algorithm [@ji2007dose], resulting in estimates $\hat{{\bm{p}}}$ satisfying $\hat{p}_1 \leq \hat{p}_2 \leq \cdots \leq \hat{p}_D$. For BOIN and keyboard, the dose $d^* = \operatorname*{arg\,min}_{z} | \hat{p}_{z} - p_{{\text{T}}} |$ is selected as the MTD. For mTPI-2 and i3+3, the dose $d^*$ with the smallest distance will be selected only if $\hat{p}_{d^*} \leq p_{{\text{T}}} + \epsilon_2$, and otherwise, the highest dose with DLT probability lower than $p_{{\text{T}}} + \epsilon_2$ is selected, which is more conservative. For the time-to-event designs, we can simply apply the MTD selection rules of their complete-data counterparts.
Simulation Details
==================
Dose-Toxicity Scenarios
-----------------------
We summarize the 18 dose-toxicity scenarios in Table \[tbl:simu\_DLT\_prob\]. We follow [@guo2017bayesian] to define the MTD as the highest dose whose probability of DLT is close to or lower than $p_{\text{T}}$. Specifically, the doses $z$ with $p_z \in [p_{\text{T}}- 0.05, p_{\text{T}}+ 0.05]$ are considered MTDs, and if such doses do not exist, the highest dose $z$ with $p_z < p_{{\text{T}}}$ is considered as the MTD. We note the definition of MTD may be slightly different in other articles.
[c@ccccccc]{} &\
& 1 & 2 & 3 & 4 & 5 & 6 & 7\
\
1 & 0.28 & 0.36 & 0.44 & 0.52 & 0.60 & 0.68 & 0.76\
2 & 0.05 & **0.20** & 0.46 & 0.50 & 0.60 & 0.70 & 0.80\
3 & 0.02 & 0.05 & **0.20** & 0.28 & 0.34 & 0.40 & 0.44\
4 & 0.01 & 0.05 & 0.10 & **0.20** & 0.32 & 0.50 & 0.70\
5 & 0.01 & 0.04 & 0.07 & **0.10** & 0.50 & 0.70 & 0.90\
6 & 0.01 & 0.05 & 0.10 & 0.14 & **0.20** & 0.26 & 0.34\
7 & 0.01 & 0.02 & 0.03 & 0.05 & **0.20** & 0.40 & 0.50\
8 & 0.01 & 0.04 & 0.07 & 0.10 & **0.15** & **0.20** & **0.25**\
9 & 0.01 & 0.02 & 0.03 & 0.04 & 0.05 & **0.20** & 0.45\
\
10 & 0.40 & 0.45 & 0.50 & 0.55 & 0.60 & 0.65 & 0.70\
11 & **0.30** & 0.40 & 0.50 & 0.60 & 0.70 & 0.80 & 0.90\
12 & 0.14 & **0.30** & 0.39 & 0.48 & 0.56 & 0.64 & 0.70\
13 & 0.07 & **0.23** & 0.41 & 0.49 & 0.62 & 0.68 & 0.73\
14 & 0.05 & 0.15 & **0.30** & 0.40 & 0.50 & 0.60 & 0.70\
15 & 0.05 & 0.12 & 0.20 & **0.30** & 0.38 & 0.49 & 0.56\
16 & 0.01 & 0.04 & 0.08 & 0.15 & **0.30** & 0.36 & 0.43\
17 & 0.02 & 0.04 & 0.08 & 0.10 & 0.20 & **0.30** & 0.40\
18 & 0.01 & 0.03 & 0.05 & 0.07 & 0.09 & **0.30** & 0.50\
Design Specifications {#supp:sec:design_spec}
---------------------
We first provide a brief review of the R-TPI design [@guo2019rtpi]. R-TPI is an extension to the mTPI-2 design. Suppose the current dose is $d$. R-TPI allows dose escalation only in the case that even all the $r_d$ pending outcomes are toxic, the mTPI-2 decision is still escalation. In other cases, the R-TPI decision is stay or de-escalation, based on conservative guess of ${\bm{Y}}_{{\text{mis}}}$. R-TPI does not utilize time-to-event information, although it is among the safest designs that allow pending outcomes.
For all the designs, we start from the lowest dose. The specification of each design is as follows. For TITE-CRM, we use the power model, $p_z = \phi(z, \alpha) = p_{0z}^{\exp(\alpha)}$, with $\alpha \sim \text{N}(0, 1.34^2)$. The skeleton $(p_{01}, \ldots, p_{0D})$ is calibrated based on [@lee2009model], with prior guess of MTD being the middle dose 4 and halfwidth of the indifference interval being 0.05. A uniform distribution is assumed for $[T \mid Z, Y = 1]$. For TITE-TPI, POD-TPI, mTPI-2 and R-TPI, the equivalence interval is chosen with $\epsilon_1 = \epsilon_2 = 0.05$. For TITE-TPI and POD-TPI, a piecewise uniform distribution is assumed for $[T \mid Z, Y = 1]$ with 3 equal-length sub-intervals. A ${\text{Dir}}(1, 1, 1)$ prior is assumed for the sub-interval weights $(\omega_1, \omega_2, \omega_3)$. For TITE-BOIN, the decision boundaries are calculated based on $p_{{\text{L}}} = 0.6 p_{{\text{T}}}$ and $p_{{\text{R}}} = 1.4 p_{{\text{T}}}$. A uniform distribution is assumed for $[T \mid Z, Y = 1]$.
The MTD selection rules of the time-to-event designs follow their complete-data counterparts. The doses with $\Pr(p_z > p_{{\text{T}}} \mid {\text{data}}) > 0.95$ are excluded from the MTD candidates for all designs.
Scenario-specific Results {#supp:sec:ss_result}
-------------------------
Figures \[supp:fig:pcs\], \[supp:fig:poa\] and \[supp:fig:ds\] show the scenario-specific operating characteristics for mTPI-2, R-TPI, TITE-TPI, POD-TPI, TITE-CRM and TITE-BOIN under Setting 1.
![Scenario-specific PCS, PCA and POS for mTPI-2, R-TPI, TITE-TPI, POD-TPI, TITE-CRM and TITE-BOIN under Setting 1. []{data-label="supp:fig:pcs"}](./figs/pcs.pdf "fig:"){width="\textwidth"} ![Scenario-specific PCS, PCA and POS for mTPI-2, R-TPI, TITE-TPI, POD-TPI, TITE-CRM and TITE-BOIN under Setting 1. []{data-label="supp:fig:pcs"}](./figs/pca.pdf "fig:"){width="\textwidth"} ![Scenario-specific PCS, PCA and POS for mTPI-2, R-TPI, TITE-TPI, POD-TPI, TITE-CRM and TITE-BOIN under Setting 1. []{data-label="supp:fig:pcs"}](./figs/pos.pdf "fig:"){width="\textwidth"}
![Scenario-specific POA, POT and Dur for mTPI-2, R-TPI, TITE-TPI, POD-TPI, TITE-CRM and TITE-BOIN under Setting 1. []{data-label="supp:fig:poa"}](./figs/poa.pdf "fig:"){width="\textwidth"} ![Scenario-specific POA, POT and Dur for mTPI-2, R-TPI, TITE-TPI, POD-TPI, TITE-CRM and TITE-BOIN under Setting 1. []{data-label="supp:fig:poa"}](./figs/pot.pdf "fig:"){width="\textwidth"} ![Scenario-specific POA, POT and Dur for mTPI-2, R-TPI, TITE-TPI, POD-TPI, TITE-CRM and TITE-BOIN under Setting 1. []{data-label="supp:fig:poa"}](./figs/dur.pdf "fig:"){width="\textwidth"}
![Scenario-specific frequencies of incompatible and risky decisions (DS, DE and SE) for mTPI-2, R-TPI, TITE-TPI, POD-TPI, TITE-CRM and TITE-BOIN under Setting 1. []{data-label="supp:fig:ds"}](./figs/ds.pdf "fig:"){width="\textwidth"} ![Scenario-specific frequencies of incompatible and risky decisions (DS, DE and SE) for mTPI-2, R-TPI, TITE-TPI, POD-TPI, TITE-CRM and TITE-BOIN under Setting 1. []{data-label="supp:fig:ds"}](./figs/de.pdf "fig:"){width="\textwidth"} ![Scenario-specific frequencies of incompatible and risky decisions (DS, DE and SE) for mTPI-2, R-TPI, TITE-TPI, POD-TPI, TITE-CRM and TITE-BOIN under Setting 1. []{data-label="supp:fig:ds"}](./figs/se.pdf "fig:"){width="\textwidth"}
Time-to-Toxicity Model Specifications {#supp:sec:t_model_spec}
-------------------------------------
To explore the role of the time-to-toxicity model, we run additional simulations with the following five models: (1) uniform distribution; (2) piecewise uniform distribution with 3 sub-intervals; (3) piecewise uniform distribution with 9 sub-intervals; (4) discrete hazard model; and (5) piecewise constant hazard model with 3 sub-intervals. For model (2), we consider 3 equal-length sub-intervals, $h_k = kW/K$ for $K = 3$. A ${\text{Dir}}(1, 1, 1)$ prior is assumed for the sub-interval weights $(\omega_1, \omega_2, \omega_3)$ (see Section \[supp:sec:model\_tite\_pwunif\]). For model (3), we consider 9 equal-length sub-intervals, $h_k = kW/K$ for $K = 9$. A ${\text{Dir}}(1, \ldots, 1)$ prior is assumed for the sub-interval weights $(\omega_1, \ldots, \omega_9)$. For model (4), we assume a ${\text{Beta}}(0.5, 0.5)$ prior for $\omega_k$, which is the discrete hazard at time $h_k$ (see Section \[supp:sec:model\_tite\_dhm\]). For model (5), we consider 3 equal-length sub-intervals, $h_k = kW/K$ for $K = 3$. We follow [@liu2013bayesian] and assume a $\text{Gamma}(K / [2W(K - k + 0.5)], 1/2)$ prior for $\omega_k$, which is the hazard in the $k$th sub-interval (see Section \[supp:sec:model\_tite\_pchm\]).
|
---
address:
- |
Institut für Theoretische Physik der Universität zu Köln\
D-50937 Köln, Germany
-
author:
- Rochus Klesse and Marcus Metzler
-
title: |
Spectral Compressibility at the Metal-Insulator Transition\
of the Quantum Hall Effect [^1]
---
Since the development of the classical Random Matrix Theory (RMT) by Wigner, Dyson, Mehta and others the statistics of energy levels of complex quantum systems have become the subject of research in many areas of physics, from the study of atomic nuclei to the investigation of disordered metals and quantum chaos [@rmt]. The remarkable property the level statistics of these systems have in common is their universality, regardless of the microscopic details of the particular system. In the following we will focus on the spectral properties of systems undergoing a metal-insulator transition. In this case at least two entirely different types of level statistics are involved. In the insulating phase the probability density of an eigenstate is almost completely localized within a comparatively small volume. As long as the spatial extension of any two states is small compared to their distance, they are independent in the same way as two states of two separate systems. Consequently, in the localized regime the energy levels are uncorrelated and therefore governed by Poisson statistics. For example, the probability to find two consecutive levels separated by an energy $\epsilon = s \Delta$, $\Delta$ being the average level spacing, is given by the Poisson-distribution $P(s) = \exp(-s)$. The number variance $\Sigma_2(N) \equiv \langle (n - \langle n \rangle )^2 \rangle $ of an energy interval which on average contains $N=\langle n \rangle$ levels is $\Sigma_2(N)=N$, according to the central limit theorem.
In the metallic or delocalized phase, on the other hand, the eigenstates are extended over the entire system. In this case the disorder potential causes the levels to repel each other. This level repulsion leads to a considerable rigidity of the spectrum with respect to fluctuations in the level density: the number variance increases only logarithmically with the number of levels, $\Sigma_2(N) \propto
\ln(N)$ and the spectral compressibility defined by $$\label{chi}
\chi=\lim_{N\rightarrow\infty}\lim_{L\rightarrow\infty}{d\Sigma_2(N)\over
dN},$$ vanishes; in the delocalized regime the spectra are incompressible. The level spacing distribution is described by the so called Wigner surmise, $P(s) \propto s^\beta \exp(-c_\beta s^2)$, where $\beta$ is a number of order one which depends on the symmetry of the system’s Hamiltonian [@rmt]. The factor $s^\beta$, missing in the Poisson statistics, reflects the strong level-repulsion.
At the mobility edge, where the two phases with their different kinds of spectra meet, things become more complicated and have given rise to extended investigations and controversial discussions concerning the critical level statistics [@shklovskii; @altshuler; @kravtsov_1994; @aronov; @kravtsov_1995; @chalker_condmat; @braun]. In the vicinity of the transition energy the probability density of a state is neither localized on a small confined area nor smeared out almost homogeneously over the whole system, but forms a self-similar measure which fluctuates very strongly on all length scales. It is best described in terms of multifractality [@multifraktales]. Since the repulsion between levels is strongly influenced by the spatial correlations of the corresponding eigenstates, their multifractal structure enters the level distribution.
In recent publications Chalker, Lerner and Smith presented a treatment on level distributions in disordered systems, which in contrast to earlier works takes care of a possible non-trivial structure of the eigenstates [@chalker_lerner]. A central result of their work is a relation between the spectral form factor $K(t)$ (i.e. the Fourier transformed of the two-level-correlation function $R(s)$ ) and the ensemble averaged quantum return probability $p(t)$ of a state initially confined to a small volume. Using this result and from scaling theory that $p(t) \propto t^{-D_2/d} $, $D_2< d$ being the fractal (correlation) dimension [@scaling], Chalker et al. [@chalker_condmat] derived the critical spectral compressibility $$\label{result}
\chi = {\eta \over 2d} \:\: (<1),$$ where the anomalous diffusion exponent $\eta$ is related to $D_2$ by $\eta=d-D_2$ [@janssen_thesis].
In spite of the Poisson-like behavior of the number variance $\Sigma_2(N)$, there still is a strong level repulsion for small level separations. This is obvious in the spacing distribution function $P(s)$, which deviates only in the tail from the Wigner surmise [@shklovskii; @kravtsov_1994; @braun; @critical_pvons] (Fig. \[fig-svonn\]). Hence, the non-vanishing compressibility must be due to level density fluctuations on larger level separations.
Up to now, Eq. (\[result\]) could not be confirmed directly by numerical calculations. The numerical results achieved for the Anderson transition in $d=3$ dimensions [@altshuler; @braun] are compatible with a linear increase of the number variance with $N$ but their rather large numerical uncertainties do not allow to strictly exclude other dependencies. In particular, the slope of $\Sigma_2(N)$ for large $N$ could not be determined precisely. Besides, the value of the multifractal exponent $D_2$ for the 3d-MIT is not known very accurately [@3dd2]. In case of the MIT in two-dimensional quantum Hall systems the only numerical simulation [@feingold] presented so far shows no significant linear contribution to $\Sigma_2(N)$.
The difficulties in determining the compressibility are due to the necessity to investigate level separations that are large compared to the average level spacing. Since only the critical region (which is usually a small portion of the entire spectrum) can be used for the statistics, one has to go to very large system sizes to produce a sufficient number of critical levels. Unfortunately, the widths of the critical region achievable with present computer capacities extends over no more than a few hundred levels, so that the results are strongly affected by incalculable finite size effects. This can of course not be compensated by a large number of disorder realizations. Consequently, to study critical level statistics numerically another way has to be found to produce larger critical spectra.
In this communication we show that by using the Chalker-Coddington network model [@chalker_coddington] for the Quantum Hall Effect the problems we just mentioned can be avoided. As pointed out by Fertig in a detailed semiclassical analysis leading to a description very similar to that in [@chalker_coddington], an energy dependent unitary operator $U(E)$ associated to the network model offers an alternative method for the numerical calculations of energy spectra and eigenstates [@fertig]. However, here we do not use $U(E)$ for determining the real energy spectrum $E_n$, but calculate the eigenvalues $e^{i\omega_l}$ of $U(E)$ itself, whereby the energy $E$ is fixed to the critical value $E_c$. The obtained quasi-spectrum $\omega_l$ is governed by the same statistics as the energy spectrum at $E_c$ and is therefore suitable for our purposes. By using this new method it is possible to improve the statistics considerably and to confirm that $\chi= \eta/2 d$ to a high degree of accuracy.
Our numerical approach is closely related to that of Edrei et al. [@edrei], where the concept of a network model has been used for calculating wave propagation through random media. Actually, the definition of network states and operator used here are in principle identical to those in [@edrei]. However, the main difference are the boundary conditions. Whereas in [@edrei] open systems are treated, since they focused on transmission amplitudes, here we use closed systems in order to get information about the energy level distribution.
The network-model for the Quantum Hall Effect [@chalker_coddington] is based on ideas developed in the early eighties [@anderson_shapiro] for the description of the Anderson transition in terms of scattering theory. It provides a semiclassical description of a 2d electron in a quantizing magnetic field and a smooth disorder potential with correlation length $\lambda$ large compared to the magnetic length $l_c$. The electron executes a fast cyclotron motion on a circle of radius $l_c$ around a guiding center, which drifts slowly along a contour $r$ of constant energy, $V(r)= E \equiv E' -
\hbar \omega_c/2$. At saddle-points of the potential with energies close to $E$ the electron tunnels with an appreciable probability between different contours.
The motion of electrons along the contours is depicted by one-dimensional, unidirectional channels, called links. The electron tunneling between them is represented by $ 2\times 2$ scattering matrices $S = \{t_{ml}\}$, which connect the complex current amplitudes $\psi_k, \psi_l$ of incoming links to those in outgoing links $\psi_m, \psi_n$ (Fig. \[fig-u\]). Due to the random length of the links between neighboring saddle-points an electron acquires random phases $\phi_j$, which we absorb into the scattering coefficients $t_{ml}$.
The coefficients $t_{ml}$ depend on the electron energy $E$ and can, in principle, be determined by semiclassical methods for a given disorder potential [@fertig]. For a saddle-point at zero energy and tunneling energy $E_t$ the tunneling amplitudes are $T = |t_{mk}|^2 = |t_{nl}|^2 = (1+
\exp(-E/ E_t))^{-1}$ and $R = |t_{ml}|^2 = |t_{nk}|^2 =
1-T $. Moreover, the random phases $\phi_{ml} \equiv
\arg(t_{ml})$ depend in a rather complicated manner on the energy $E$ [@fertig].
A state $\Psi$ of the network is given by its complex amplitudes $\psi_j$ on the links, $ \Psi=\{ \psi_j \}_j$. It is stationary at energy $E$ if the scattering condition $$\psi_m = t_{mk}(E)\psi_k + t_{ml}(E)\psi_{l} \label{scattering_condition}$$ is satisfied at each saddle point.
Defining an unitary operator $U(E)$ by $$\label{weq}
U(E)e_l = t_{ml}(E)e_m + t_{nl}(E)e_n,$$ where $e_i = \{ \delta_{ji} \}_j $, this condition can be written as $$\label{wpsi}
U(E)\Psi = \Psi.$$ The energy enters parametrically via the coefficients $t_{ml}(E)$. This equation has non-trivial solutions only for discrete energies $E_n$. According to Fertig [@fertig], these energies $E_n$ are eigenenergies of the system and their eigenvectors $\Psi_n$ determine the amplitudes of the corresponding eigenstates on the different equipotential contours (links). Hence, under certain circumstances the scattering condition (\[wpsi\]) offers an alternative method of determining numerically eigenenergies and states of an electron in an disordered system, as it was first — to the best of our knowledge — pointed out by Fertig. This method was utilized in [@klesse_metzler] for the numerical calculation of critical eigenstates in quantum Hall systems. (Like in [@edrei], Eq. (\[wpsi\]) can also be taken as a definition of a time evolution for states at energy $E$, $\Psi(t+\tau) \equiv
U(E) \Psi(t)$, whereby the energy dispersion is neglected [@rk_thesis; @bh_rk].)
Before we proceed it is necessary to consider the length and energy scales determining the critical region. The averaged level spacing $\Delta$ is related to the system size $L$ and magnetic length $l_c$ via $\Delta \sim \Gamma (l_c/ L)^2$, $\Gamma$ being the width of the disorder broadened Landau band. Close to the critical energy $E_c=0$ the localization length $\xi$ is $\xi \sim \lambda
|E/\Gamma|^{-\nu}$. Therefore, the width of the critical energy region is $\Delta_c \sim \Gamma (\lambda / L ) ^{1/\nu}$. Consequently, the number $N_c$ of critical levels behaves like $N_c = \Delta_c /\Delta
\sim (\lambda/l_c)^2 (L/\lambda)^{2-1/\nu}$.
Note that even for a fixed ratio $L/\lambda$ the number $N_c$ can be enhanced by increasing the ratio $\lambda/l_c$. We emphasize that in the Chalker-Coddington model this ratio is arbitrarily large, so that the number of critical levels is not restricted, which makes the model in principle very convenient for investigations of critical level statistics. However, the calculation of the energies $E_n$ by solving the non-linear Eq. (\[wpsi\]) is a difficult numerical task and not suitable for practical purposes.
Therefore, in order to determine spectral statistics let us discuss instead of Eq. (\[wpsi\]) the eigenvalue problem $$\label{eve}
U(E) \Psi_l(E) = e^{i \omega_l(E)} \Psi_l(E).$$ For a given energy $E$ the unimodular eigenvalues $e^{i\omega_l(E)}$ define quasi-energies $\omega_l(E)$, $l=1,\dots,M=\dim U(E)$. They are smooth functions of the energy and do not cross each other, following a theorem by von Neumann and Wigner [@neumann]. According to Eq. (\[wpsi\]), the intersection points of the curves $\omega_l(E)$ with the lines $\omega = 0, \pm 2\pi, \pm 4\pi, \dots$ determine the energy levels $E_n$.
The flow of the levels $\omega_l(E)$ obeys two symmetries: First, the intersection points $E_n'$ with shifted lines $\omega'=\Omega,\Omega\pm 2\pi, \dots $ must exhibit the same statistics as the original spectrum $E_n$, since the corresponding transformed operator $U'(E)=e^{-i\Omega}U(E)$ belongs to the same universality class as $U(E)$ ($U'$ deviates from $U$ only by a global phase shift, which has no influence on the statistical properties). Second, as long as the critical regime is not left, $|E|<\Delta_c$, the statistical properties of the flow $\omega_l(E)$ can not change significantly with energy, because such a change would be accompanied by a new energy scale inbetween $\Delta$ and $\Delta_c$ ($ \ll E_t \ll \Gamma$), which makes no physical sense.
Due to this homogeneity in both directions and due to the strong repulsion of the $\omega_l(E)$ they must behave as depicted in Fig. \[fig-flow\]: The average slope of the curves varies neither strongly with the level number $l$ nor with the energy $E$. Further, this homogeneity implies that the intersection points $\omega_l^c$ with a cut $c$ crossing the band of curves show essentially the same statistics, independent of the precise position of $c$. For this reason, instead of the real energy spectrum $E_n$ one can also use a quasi-spectrum $\omega_l(E)$ with $E$ within the critical regime for an analysis of the critical level statistics [@jalabert]. A big advantage of this method is the simple fact that the $\omega_l(E)$ are far better numerically accessible than the real energies $E_n$. They can be calculated by solving the linear eigenvalue problem (\[wpsi\]) with standard numerical methods. [@erlaeuterung]
For our calculations we used closed networks of $50 \times 50 $ saddle-points with periodic boundaries in one and reflecting in the other direction. The transmission amplitudes were set to the critical value $ T = 1/2$ for models describing the transition point and to $ T_\pm= (1 + \exp(\pm E/E_t))^{-1}$ with $E/E_t=10$ for non-critical systems with strongly localized states. The disorder is represented by random scattering phases $\phi_{ml} = \arg(t_{ml})$. Note that the calculations are done at constant energies $E=E_c$ and $E=10E_t$, respectively, hence we did not have to know the energy dependence of the phases $\phi_{lm}(E)$. From this settings we obtained random network operators at the critical point, $U(E=E_c)$, and deep in the localized regime, $U(E=10E_t)$, of dimension $M=2\times 50\times 50$. Diagonalizing them by standard numerical methods yields critical ($E=E_c$) and non-critical ($E=10E_t$) quasi-spectra of $M=5000$ levels each.
As explained in the considerations given above, the critical level statistics can be determined by analyzing the critical quasi-spectra. Although the statistics do not change within the quasi-spectra one has to take into account that the total number of quasi-levels per spectrum is fixed to $M$. So, when calculating the number variance $\Sigma_2(N)$ one has to confine oneself to interval sizes $\Delta\omega$ with averaged level number $N=\langle n
\rangle_{\Delta\omega}$ small compared to $M$. We checked by numerical simulations with Poisson distributed levels that at a total number of $M=5000$ levels deviations from the expected number variance $\Sigma_2(N)=N$ are negligible for $N < 300$.
For the determination of the critical number variance $\Sigma_2(N)$ we divided the quasi-spectra of 40 different disordered critical network operators $U(E_c)$ into non-overlapping intervals of length $\Delta\omega=(2\pi/5000)N$, $N$ ranging from 1 to 300. For each $N$ this results in an ensemble of $m_N = 40 \times M /N$ intervals with level numbers $n_i$ and $\langle n_i \rangle = N$, from which we calculate the level number variance $\Sigma_2(N) = m_N^{-1}\sum_{i=1}^{m_N} (n_i-N)^2$.
The results plotted in Fig. \[fig-svonn\] show a clearly linear behavior of the number variance $\Sigma(N)$ for a wide range of $N$. Eq. (\[result\]) predicts a slope of $\eta /2d = (2-D_2)/2d = 0.125 \pm 0.01$, where we have taken $D_2=1.5 \pm 0.05$ from independent numerical calculations of critical states [@dezwei; @mocm_thesis; @rk_thesis]. A least square fit of our data yields a slope $0.124 \pm 0.006$, which agrees with the prediction very well.
The dashed lines mark the range of the expected fluctuations of $\Sigma_2(n)$ due to the finite number $m_N$ of intervals, calculated via the $\chi^2_\alpha$-distribution for $\alpha = 0.8, 0.2$. This clearly indicates that the deviations of the data from the straight line are not systematic but due to statistical fluctuations.
The spacing distribution $P(s)$ plotted in the inset of Fig. \[fig-svonn\] shows for small spacings $s$ a WD-type behavior for a GUE ensemble (dashed line), $P(s) \propto s^2$, indicating strong level repulsion for small level spacing.
We use the same procedure as for the critical quasi-spectra for five non-critical quasi-spectra at $E= 10E_t$ in the strongly localized regime. Here $\Sigma_2(N)$ follows a straight line of slope 1, as it should be, since in this region the levels are Poisson-distributed.
To summarize, the recently derived relation between the multifractal exponent $\eta= d-D_2$ of eigenstates and the spectral compressibility at the mobility edge, $\chi=\eta/2d$, has been confirmed numerically for the integer quantum Hall delocalization transitions. This has been done by introducing a new method to investigate spectral properties of disordered systems.
We would like to thank János Hajdu, Bodo Huckestein and Martin Janssen for valuable discussions and the research program Sonderforschungsbereich 341, Köln-Aachen-Jülich for their support.
[10]{}
\#1\#2\#3\#4[ \#1, Phys. Rev. A [**\#2**]{}, \#3 (\#4).]{} \#1\#2\#3\#4[ \#1, Phys. Rev. B [**\#2**]{}, \#3 (\#4).]{} \#1\#2\#3\#4[ \#1, Phys. Rev. Lett. [**\#2**]{}, \#3 (\#4).]{}
see e.g. M. L. Mehta, [*Random Matrices*]{}, 2nd ed. (Academic Press, New York, 1991), and references therein.
B. L. Altshuler, I. Zharekeshev, S. Kotochigova, B. Shklovskii, JETP [**67**]{}, 625 (1988).
J. T. Chalker, V. E. Kravtsov, I. V. Lerner, JETP Lett. [**64**]{}, 386 (1996).
M. Janssen, Int. J. Mod. Phys. [**8**]{}, 943 (1994).
J. T. Chalker, I. V. Lerner, R. S. Smith, Phys. Rev. Lett [**77**]{}, 554 (1996); J. Math. Phys. [**37**]{}, 5061 (1996).
J. T. Chalker, G. J. Daniell, Phys. Rev. Lett. [**61**]{}, 593 (1988); B. Huckestein, L. Schweitzer, Phys. Rev. Lett. [**72**]{}, 713 (1994); L. T. Brandes, B. Huckestein, L. Schweitzer Ann. Physik [**5**]{}, 633 (1996).
M. Janssen, Ph.D. Thesis, Universität zu Köln, 1990; J. Hajdu, M. Janssen, in: G. Györgyi, I. Kondor, L. Sasvàri, T. Tel (eds.), [*From Phase Transitions to Chaos*]{}, World Scientific, Singapore, 1992.
S. N. Evangelou, Phys. Rev. B [**49**]{}, 16805 (1994); Y. Ono, T. Ohtsuki, B. Kramer, J. Phys. Soc. Jpn [**65**]{} 6 (1996).
The values for $D_2$ ranges from 1.3 up to 1.9,
J. T. Chalker, P. D. Coddington, J. Phys. C [**21**]{}, 2665 (1988).
P. W. Anderson, Phys. Rev. B [**23**]{}, 4828 (1981);
B. Huckestein, R. Klesse, Phys. Rev. B [**55**]{}, R7303 (1997).
R. Klesse, Ph.D. Thesis, Universität zu Köln, 1996. M. Metzler, Ph.D. Thesis, Universität zu Köln, 1996.
J. v. Neumann, E. Wigner, Physikalische Zeitschrift [**30**]{}, 467 (1929).
A similar approach taking advantage of a relation between scattering phases and energy-levels was used by Jalabert and Pichard in another context , R. A. Jalabert, J.-L. Pichard, J. Phys. I France [**5**]{}, 287 (1995)
In order to get a better understanding of the quasi-energies $\omega_l$ and their eigenfunctions $\Psi_l$ we refer to recent numerical works [@rk_thesis; @mocm_thesis; @bh_rk]. In dynamical simulations [@rk_thesis], where $U(E_c)$ is interpreted as evolution operator for microscopic time steps — this means that the $\omega_l(E_c)$ are thought of as actual eigenenergies $E_l$ with eigenstates $\Psi_l(E_c)$ —, the well known critical anomalous diffusion behavior shows up in good agreement with the predictions of scaling theory [@scaling]. Moreover, the correlations in the local amplitudes of eigenfunctions $\Psi_l(E_c)$, $\Psi_{l'}(E_c)$ were found to obey the same scaling behavior in the spatial and quasi-energy difference $\Delta \omega =
|\omega_l-\omega_{l'}|$ as those of real critical eigenfunctions in real energy differences [@mocm_thesis; @bh_rk]. This suggests to interpret the quasi-spectrum $\omega_l(E)$ of $U(E)$ as a spectrum, which is statistically equivalent to the excitation spectrum of the real system in the vicinity of the energy $E$.
R. Klesse, M. Metzler, Euro Phys. Lett., [**32**]{}, 229 (1995).
W. Pook, M. Janssen, Z. Phys. [**82**]{}, 295 (1991).
[^1]: Work performed within the research program of the Sonderforschungsbereich 341, Köln-Aachen-Jülich
|
---
abstract: 'We study analytically the full counting statistics of charge transport through single molecules, strongly coupled to a weakly damped vibrational mode. The specifics of transport in this regime – a hierarchical sequence of avalanches of transferred charges, interrupted by “quiet" periods – make the counting statistics strongly non-Gaussian. We support our findings for the counting statistics as well as for the frequency-dependent noise power by numerical simulations, finding excellent agreement.'
author:
- Jens Koch
- 'M.E. Raikh'
- Felix von Oppen
date: 'August 4, 2005'
title: 'Full counting statistics of strongly non-Ohmic transport through single molecules'
---
[*Introduction.*]{}—A prime qualitative difference of transport through single molecules as compared to artificial nanostructures lies in the role of the vibrational motion of the nuclei. This aspect is at the focus of current experiments [@Parks; @Ruitenbeek; @Ralph], and is also being studied theoretically within a number of approaches [@Glazman; @Schoeller; @Flensberg; @Aleiner; @Nitzan; @Takei; @Koch]. The incorporation of molecular vibrations (phonons) into the theoretical description is mostly done within simplified (phenomenological) models, as opposed to purely electronic first-principles studies [@Ratner; @Baranger].
Even within minimal models involving one molecular orbital coupled to a single vibrational mode, “unidirectional” transport (i.e., for voltages large compared to temperature) depends radically on the strength of the electron-phonon coupling, already at the qualitative level [@Flensberg; @Aleiner; @Koch]. For weak and intermediate coupling [@terminology], transport is adequately described in terms of individual electron transitions. By contrast when vibrational relaxation is slow, transport in the regime of strong electron-phonon coupling is appropriately captured within a scenario of [*avalanches*]{} of transferred electrons, with exponential spreads of height and duration [@Koch].
More quantitatively, the time dependence of the current in the strong-coupling regime can be presented as $$\label{current}
I(T) = f_1^{(0)}(T-t_1) + f_2^{(0)}(T-t_1-t_2) + \ldots,$$ where $t_i$ are the time intervals between avalanches (quiet periods). These intervals are much longer than the typical duration $\tau^{(0)}$ of an avalanche which occurs during the sparse periods when the vibrations are excited. The random function $f_i^{(0)}(\tau)$ (which is nonzero only for times $|\tau|\lesssim \tau^{(0)}$) describes the passage of a [*large number*]{} $\int d\tau f_i^{(0)}(\tau)=N_i \gg 1$ of electrons through the molecule during the $i$th avalanche. Moreover, a numerical study of the avalanches [@Koch] revealed their self-similar hierarchical structure, see Fig. 1. Quantitatively, this structure manifests itself in the fact that, during the time of an avalanche $\sim \tau^{(0)}$, each function $f_i^{(0)}$ itself takes the form of Eq. (\[current\]), with $f_i^{(0)}$ replaced by random functions $f_i^{(1)}$, which describe avalanches of the [*first generation*]{} interrupted by quiet periods. Again, these quiet periods are much longer than the characteristic time scale $\tau^{(1)}$ of the functions $f_i^{(1)}$. For times shorter than $\tau^{(1)}$, the functions $f_i^{(1)}$ have the form of Eq. (\[current\]) with corresponding second-generation avalanches, $f_i^{(2)}$, having even shorter time-scale, $\tau^{(2)}$, and so on [@cutoff]. Numerical results supporting this scenario are shown in Fig. 1.
The above discussion implies that the statistical properties of charge transport through a molecule in the regime of strong electron-phonon coupling and through a conventional nanostructure are drastically different. For a nanostructure, all $f_i^{(0)}$ are $\delta$ functions, so that $N_i=1$. Hence, the distribution function $P_T(Q)$ of the net transmitted charge $Q$ during time $T$ (full counting statistics [@Lesovik]) is completely encoded in the distribution of the [*waiting times*]{} $t_i$ for single-electron transitions. This distribution reflects the details of the transport mechanism, and might be quite nontrivial [@Blanter]. Nevertheless, with all $t_i$ being of the same order, the full counting statistics differs only weakly from a Gaussian distribution. Small deviations are caused by correlations [@Levitov; @Imry], interactions [@Kindermann], or the influence of the environment [@Beenakker], and have been extensively studied theoretically.
By contrast, the counting statistics of avalanche-type transport is [*insensitive*]{} to the details of the passage of a single electron through the molecule, since the number of electrons involved in each avalanche is large. Instead, the counting statistics is governed [*exclusively*]{} by the transition rates between different vibrational states. These rates have a simple structure in the limit of strong coupling which allows us to develop a complete analytical theory for the regime of avalanche-type transport. In particular, we demonstrate in this paper that the full counting statistics $P_T(Q)$ is given by a concise analytical expression, which is strongly skewed at “short” times ($\sim$ zero-order quiet period) and evolves into a Gaussian only for very large $T$. Along with the counting statistics, we also study analytically how the hierarchy of avalanches manifests itself in the frequency dependence of the noise power $S(\omega)$. Our analytical results are in excellent agreement with numerical Monte-Carlo (MC) simulations.
![Hierarchical character of transport. Left: Three generations of self-similar MC plots for $\lambda=4$ and $eV=3\hbar\omega_v$, showing the net-transferred charge $N$ and phonon state $q$ as functions of time. ($\omega_v$: phonon frequency; $\rho$: density of states of the leads; $t_0$: molecule-lead coupling). Right: Comparison of the fixed-point distribution for the transferred charge per generation-$q$ avalanche to numerical simulations for $q=0,1,2$ (mean values $\overline{N}^{(0)}=91.2$, $\overline{N}^{(1)}=11.1$, and $\overline{N}^{(2)}=2.9$). []{data-label="montecarlo"}](fig1.ps){width="\columnwidth"}
[*Full counting statistics.*]{}—Since different zeroth-generation avalanches are statistically independent, it is easy to derive a relation between the counting statistics $P_T(Q)$ of the [*net charge*]{} $Q$ and the conventional counting statistics ${\varphi}_T(n)$ [@Lesovik] of the [*number*]{} of zeroth-generation avalanches $n$ during the time interval $T$. Indeed, using the definition $P_T(Q)=\langle \delta(Q - \sum_{j=1}^n N_j) \rangle_{N_j,n}$, and a Fourier representation of the RHS, one obtains $$\label{general}
P_T(Q)=\int \frac{d\alpha}{2\pi}e^{i\alpha Q}\sum_n \left[\tilde{\cal P}_0(\alpha)\right]^n \varphi_T(n),$$ where $\tilde{\cal P}_0(\alpha)=\langle\exp\left(-i\alpha N_j\right)\rangle_{N_j}$ denotes the Fourier transform of the distribution function ${\cal P}_0(N)$ of the total charge passing per zeroth-generation avalanche. The durations of the quiet periods obey Poisson statistics so that $\varphi_T(n) = \exp(-{\overline n}_T){\overline n}_T^n/n!$. Here, ${\overline n}_T$ denotes the average number of zeroth-generation avalanches within time $T$. Substituting this form into Eq. (\[general\]) and performing the summation over $n$ yields an expression for the counting statistics similar to the Holtsmark distribution [@Holtsmark], $$\label{similar}
P_T(Q) = \int \frac{d\alpha}{2\pi} \exp\left\{ i\alpha Q + {\overline n}_T \left[\tilde{\cal P}_0(\alpha)
- 1 \right]\right\}.$$ Thus, the problem of the counting statistics is reduced to finding the distribution ${\cal P}_0(N)$. Two facts allow us to find ${\cal P}_0(N)$, namely ([*i*]{}) the self-similar structure of avalanches and ([*ii*]{}) the large number $n_q$ of generation-$(q+1)$ avalanches within a given generation-$q$ avalanche.
Our basic observation is that we can derive a recursion relation, relating the distribution functions ${\cal P}_q(N)$ and ${\cal P}_{q+1}(N)$ of the total passing charge ($N^{(q)}$ and $N^{(q+1)}$, respectively) per avalanche for subsequent generations. This recursion follows from the obvious facts that $N^{(q)}=\sum_{j=1}^{n_q} N^{(q+1)}_j$ and that different avalanches of a given generation are statistically independent. By analogy with the derivation of Eq. (\[general\]), we thus obtain $$\label{recursive}
{\cal P}_q(N) = \int \frac{d\alpha}{2\pi} e^{i\alpha N} \sum_{n}
\left[\tilde {\cal P}_{q+1}(\alpha)\right]^{n} p_q(n),$$ where $p_q(n)$ denotes the distribution function of $n_q$. To proceed further one has to specify the form of the distribution $p_q(n)$. This distribution is governed by the [*microscopic*]{} characteristics of the Franck-Condon transitions. We demonstrate below that $p_q(n)=(1/{\overline n}_q) \exp(-n/{\overline n}_q)$. Upon substituting this form into Eq. (\[recursive\]), the summation over $n$ on the RHS can be easily performed and we obtain, after a Fourier transform of both sides, $$\tilde {\cal P}_q(\alpha) = \frac{1}{\overline n_q}\frac{1}{1- \tilde {\cal P}_{q+1}(\alpha) \exp(-1/\overline n_q ) }.
\label{con}$$ The distribution ${\cal P}_q$ can now be obtained from this equation by writing its general solution as $\tilde {\cal P}_q(\alpha) = [1 + i\alpha \overline N^{(q)} + c_q (\alpha \overline
N^{(q)})^2 +\ldots]^{-1}$. Inserting this into Eq. (\[con\]), we find that the numerical coefficients $c_q$ flow to zero with $q$ by virtue of the small parameter $1/\overline
n_q$. Thus, the solution $\tilde {\cal P}_q(\alpha) = [1 + i\alpha \overline N^{(q)}]^{-1}$ with Fourier transform ${\cal P}_q(N)=\theta(N) \exp(-N/\overline N^{(q)})/\overline N^{(q)}$ can be viewed as a fixed point of the recursion equation Eq. (\[recursive\]) and since $\overline N^{(q)} = \overline n_q\overline N^{(q+1)}$, self-similarity is obeyed asymptotically. The existence of this fixed-point solution can be viewed as a consequence of remark ([*i*]{}) which implies that up to rescalings, the distribution functions ${\cal P}_q(N)$ have the same functional form for [*all*]{} $q$. Fig. \[montecarlo\] numerically confirms this result for three different generations.
With ${\cal P}_q(N)$ established, we obtain the counting statistics by substituting $\tilde{\cal P}_0(\alpha)=
(1+i\alpha\overline N^{(0)})^{-1}$ into Eq. (\[similar\]) and performing the integral. This yields $$\label{main}
P_T(Q) = e^{-{\overline n}_T} \delta(Q) +
e^{-\frac{Q}{ {\overline N^{(0)}}}-{\overline n}_T}
\sqrt{
\frac{{\overline n}_T } {{\overline N}^{(0)} Q}
}\,
I_1\!\left(\sqrt{\frac{4{\overline n}_T Q}{{\overline N}^{(0)}}} \right).$$ Here $I_1(z)$ denotes a modified Bessel function. Eq. (\[main\]) is our central result. It is nicely confirmed by our MC results, as shown in Fig. \[fullcount\], and describes the evolution of the counting statistics between the following two transparent limits. ([*i*]{}) Short times, ${\overline n}_T=T/\langle t_i \rangle \ll 1$: Using the expansion $I_1(z)\approx z/2$ for $z\ll 1$ we obtain from Eq. (\[main\]) $$\label{short}
P_T(Q)\simeq e^{-{\overline n}_T}\left[\delta(Q)+({\overline n}_T/\overline N^{(0)})
e^{-{Q/ {\overline N}^{(0)}}}\right],$$ Typically only a few electrons are transmitted through a molecule. The long tail described by the second term in Eq. (\[short\]) arises from realizations where an avalanche occurs within the time $T$ and reflects the spread of charge within a single avalanche. ([*ii*]{}) Long times, ${\overline n}_T \gg 1$: Upon substituting the large-$z$ asymptote of $I_1(z)$ into Eq. (\[main\]), it is easy to see that the second term has a sharp maximum centered at $Q={\overline n}_T\overline N^{(0)}$, which is the average charge passed through the molecule after a large number of avalanches. Expansion of the exponent around the maximum yields the Gaussian $$\label{long}
P_T(Q) \simeq (\sqrt{2\pi}\sigma_Q)^{-1}\exp\left[-{\left(Q-{\overline n}_T
\overline N^{(0)}\right)^2}/{2\sigma_Q^2}\right]$$ with a width $\sigma_Q=\left(2{\overline n}_T\right)^{1/2}\overline N^{(0)}$. This width is [*twice*]{} the width expected from the fluctuations of the waiting times. This enhanced broadening is due to fluctuations of the charge passed per avalanche. These additional fluctuations also manifest themselves in the noise characteristics of transport as analyzed below.
![(Color online) Evolution of full counting statistics $P_T(Q)$ for four different time intervals $T$ with $\lambda=4.0$ and $eV=3\hbar\omega_v$. The MC data (solid lines) are in excellent agreement with the analytical full counting statistics, Eq. (\[main\]), (dashed lines), with $\overline{N}^{(0)}=91.2$ and $\overline{n}_T=2.4\cdot10^{-6}T/T_0$ (no fit), as well as $T_0=(2\pi\rho t_0^2/\hbar)^{-1}$.[]{data-label="fullcount"}](fig2.ps){width="0.8\columnwidth"}
[*Microscopic derivation of $p_q(n)$.*]{}—The distribution $p_q(n)$ is obtained by averaging the Poisson distribution of $n$ for a [*given*]{} avalanche duration over the distribution of [*durations*]{}. On microscopic grounds, the latter distribution is a simple exponent, which immediately transforms into $p_q(n)=(1/{\overline n}_q) \exp(-n/{\overline n}_q)$ since $\overline n_q$ is large.
To see this, we note that the duration $\tau^{(q)}$ of a generation-$q$ avalanche is determined by the waiting times in the vibrational state $q+1$ since the durations of intermittent higher-generation avalanches can be neglected. Two processes terminate a generation-$q$ avalanche: a direct transition from $q+1$ to $q$ or a transition back to $q$ during a generation-$(q+1)$ avalanche. Denoting the total rate for both processes by $\Gamma_q$, we obtain an exponential distribution of durations $\Gamma_q \exp(-\Gamma_q\tau^{(q)})$.
[*Noise spectrum $S(\omega)$ of avalanche-type transport.*]{}—We first derive a general expression for $S(\omega)$ assuming arbitrary distributions ${\cal P}_0(N)$ of the avalanche magnitudes and $W(t)$ of the waiting times. For frequencies smaller than $1/\tau^{(0)}$, we have $f_i^{(0)}(t)\simeq N_i\delta(t)$ in Eq. (\[current\]). Using Fourier representations of the $\delta$ functions and averaging over the $t_i$ and $N_i$, the average current becomes $$\langle I(T)\rangle = \langle N_i\rangle \int\frac{d\alpha}{2\pi} e^{i\alpha T} \frac{\tilde W(\alpha)}{
1 - \tilde W(\alpha)} ,
\label{avcur}$$ where $\tilde W(\alpha)=\langle \exp(-i\alpha t_i)\rangle_{t_i}$ denotes the Fourier transform of $W(t)$. In the long-time limit, only small values of $\alpha$ contribute to the integral in Eq. (\[avcur\]) so that we can use the expansion $\tilde W(\alpha) = 1 -i\alpha \langle t_i\rangle - (1/2) \alpha^2
\langle t_i^2 \rangle + \ldots$. Inserting this expansion into Eq. (\[avcur\]), keeping only the leading order in $\alpha$ and performing the contour integration over $\alpha$, we recover the obvious result $\langle I(T)\rangle = \langle N_i\rangle /\langle t_i\rangle$.
Similarly, we express the current-current correlator as
$$\begin{aligned}
&&\langle I(T_1) I(T_2)\rangle = \int \frac{d\alpha}{2\pi}\frac{d\beta}{2\pi} e^{-i\alpha T_1 - i\beta T_2}
\left\langle [N_1 e^{i\alpha t_1}+N_2 e^{i\alpha (t_1+t_2)}+\ldots ][N_1 e^{i\beta t_1}+N_2 e^{i\beta
(t_1+t_2)}+\ldots ]\right\rangle \nonumber\\
&&\,\,\,\,= \int \frac{d\alpha}{ 2\pi}\frac{d\beta}{ 2\pi} e^{-i\alpha T_1 -i\beta T_2}
\left\{ \frac{\langle N_i\rangle^2 \tilde W(\alpha+\beta)}{
1 - \tilde W(\alpha+\beta)}\left( \frac{1}{
1 - \tilde W(\alpha)} + \frac{1}{
1 - \tilde W(\beta)} -1 \right)\right.
+ \left. \frac{(\langle N_i^2\rangle -\langle N_i\rangle^2) \tilde W(\alpha+\beta)}{
1 - \tilde W(\alpha+\beta)}\right\}.
\label{corcur}\end{aligned}$$
The last equality follows upon term-by-term averaging and resummation of the series. To access the limit of long times $T=(T_1+T_2)/2$, we introduce $\omega = (\alpha -\beta)/2$ and $\Omega = \alpha +\beta$. Then, the exponent in the integrand in Eq. (\[corcur\]) assumes the form $\exp(i\omega\tau - i\Omega T)$ with $\tau=T_2-T_1$. The limit $T\to\infty$ can now be taken in analogy with the derivation of $\langle I(T)\rangle$ above. The integrand can be directly identified with the noise spectrum $S(\omega)$, so that $$\begin{aligned}
\nonumber
S(\omega)&= \frac{2}{ \langle t_i\rangle} \bigg\{ \langle N_i\rangle^2 \left[ \frac{1}{
1 - \tilde W(\omega)} + \frac{1}{
1 - \tilde W(-\omega)} -1 \right]\\
&\qquad +(\langle N_i^2\rangle -\langle N_i\rangle^2) \bigg\}.
\label{noisespectrum}\end{aligned}$$ Taking the zero-frequency limit requires one to keep terms of order $\omega^2$ in the expansion of $\tilde W(\pm \omega)$. In this way, the Fano factor $F=S(\omega=0)/2e\langle I\rangle$ becomes $$F=\langle N_i\rangle \frac{\langle t_i^2\rangle-\langle t_i\rangle^2}{\langle t_i\rangle^2}
+\frac{\langle N_i^2\rangle-\langle N_i\rangle^2}{ \langle N_i\rangle}.
\label{fanogen}$$ This equation allows for a transparent interpretation: Noise originates from two sources, namely the fluctuations in the intervals between avalanches and the fluctuations in the transmitted charge per avalanche. In the conventional situation where $N_i=1$ for all $i$, the Fano factor is given by the fluctuations of the waiting times $t_i$ for a transition in which an electron passes either directly or sequentially from the left to the right lead. For example, for transport through a symmetric junction in the Coulomb-blockade regime, one immediately recovers $F=5/9$ [@Nazarov] when taking into account that the rates of entering and leaving the dot are related as 2:1 due to spin. For the specific distributions adopted in our model, both terms in Eq. (\[fanogen\]) contribute equally, and the Fano factor reduces to $F=2{\overline N}^{(0)}$, which, in agreement with Eq. (\[long\]), is twice the value expected for a fixed magnitude of avalanches. This is confirmed by numerical results.
For frequencies larger than $1/\tau^{(0)}$, the “fine structure" of the avalanches described by the functions $f_i^{(0)}$ in Eq. (\[current\]) must be taken into account. This fine structure can be incorporated into the noise spectrum Eq. (\[noisespectrum\]) by replacing $\langle N_i\rangle^2$ by $\langle \tilde f(\alpha)\rangle\langle \tilde f(\beta)\rangle$ and $\langle N_i^2\rangle -\langle N_i\rangle^2$ by $\langle \tilde f(\alpha)
\tilde f(\beta)\rangle - \langle \tilde f(\alpha)\rangle\langle \tilde f(\beta)\rangle$, where $\tilde f(\alpha)$ denotes the Fourier transform. Explicitly employing the Poisson distribution of the waiting times leads to the remarkable simplification $[1-\tilde W(\omega)]^{-1}+[1-\tilde W(-\omega)]^{-1} -1 = 1$. In this way, we obtain $$S(\omega) = \frac{2}{ \langle t_i\rangle} \langle \tilde f(\omega)\tilde f(-\omega)\rangle.
\label{Somega}$$ For frequencies of order $\omega\simeq 1/\tau_0$ (where $\tau_q$ denotes the average waiting time $\langle t_i\rangle$ at level $q$ of the hierarchy), we can ignore the fine structure of the avalanche and replace $\tilde f(\omega) = N_i^{(0)}$. Thus, we find $S(\omega) = 2 \langle [N_i^{(0)}]^2\rangle /
\tau_0$. At higher frequencies $\omega \simeq 1/\tau_1$, the function $f(t)$ is resolved into avalanches of generation $q=1$. Then, we can write $\langle\tilde f(\omega)\tilde f (-\omega)\rangle
= \int dT\int d\tau e^{i\omega\tau} \langle f(T+\tau/2)f(T-\tau/2)\rangle$. Up to the integral over $T$, this expression is analogous to $S(\omega)$ itself, with zeroth-generation quantities replaced by corresponding $q=1$ quantities. For the frequencies of interest, we therefore find $S(\omega) = (2/\tau_0)(\tau^{(0)}\langle
[N_i^{(1)}]^2\rangle/\tau_1)$. Using the obvious relations $\tau^{(0)}=\tau_1 {\overline n}_0$ and $\overline N^{(0)}=\overline n_0 \overline N^{(1)}$ and generalizing to arbitrary $q$, we find $$S_{q+1} = \frac{{\overline N}^{(q+1)}}{ {\overline N}^{(q)}} S_q.
\label{Shierarchy}$$ Here, we define $S_q = S(\omega \simeq 1/\tau_q)$ so that Eq. (\[Shierarchy\]) provides a rule for extending the noise spectrum to progressively higher frequencies.
The essential [*microscopic*]{} inputs are the ratios $\tau_{q+1}/\tau_q$ and ${\overline N}^{(q+1)}/
{\overline N}^{(q)}$. Both ratios are determined by overlaps of displaced vibrational wavefunctions [@Koch]. The rate $1/\tau_q$ is dominated by the transition $q\to q+1$. Thus, it involves the overlap of [*neighboring*]{} harmonic oscillator states. By contrast, ${\overline N}^{(q)}$ is inversely proportional to the transition rate from a highly excited phonon levels to the $q$th vibrational level. The difference between these two rates is thus that the first involves four wavefunctions with index of order $q$, while the second involves only two. As a result, we can immediately establish from a quasiclassical evaluation of the matrix elements that $\tau_q/({\overline N}^{(q)})^2$ is essentially independent of $q$. With this input, we conclude that $S(\omega) \sim \omega^{-\alpha}$ with exponent $\alpha=1/2$. Since the noise power does not depend sensitively on $\omega$ in finite intervals around $ 1/\tau_q$, this power law should be superimposed with steplike features in $S(\omega)$. These conclusions agree with numerical simulations (see Ref. [@Koch]) over several orders of magnitude in frequency.
[*Conclusions.*]{}—Our complete analytical description for the full-counting statistics and the frequency-dependent noise power of self-similar avalanche-type transport was made possible by the fact that current flow is essentially unidirectional. We emphasize that our arguments are quite general, with rather limited microscopic input, making our results potentially applicable far beyond the particular realization [@Koch] of avalanche-type transport considered in the numerical simulations. Finally, we remark that direct vibrational relaxation (with rate $\gamma_\text{rel}$), neglected so far, only gradually suppresses avalanche-type transport. Indeed, one readily argues that $\overline N^{(0)}\sim 1/\gamma_\text{rel}$ over a wide range of relaxation times, leading to $S(0)\sim 1/\gamma_\text{rel}^2$.
This work was supported by NSF-DAAD Grant INT-0231010, Sfb 658, and Studienstiftung des deutschen Volkes.
[50]{}
H. Park *et al.*, Nature [**407**]{}, 57 (2000).
R.H.M. Smit *et al.*, Nature [**419**]{}, 906 (2002).
A.N. Pasupathy *et al.*, cond-mat/0311150 (2003).
L.I. Glazman and R.I. Shekhter, Zh. Eksp. Teor. Fiz. [**94**]{}, 292 (1987) \[Sov. Phys. JETP [**67**]{}, 163 (1988)\].
D. Boese and H. Schoeller, Europhys. Lett. [**54**]{}, 668 (2001).
S. Braig and K. Flensberg, Phys. Rev. B [**70**]{}, 085317 (2004).
A. Mitra, I. Aleiner, and A.J. Millis, Phys. Rev. [**69**]{}, 245302 (2004).
M. Galperin, M.A. Ratner, and A. Nitzan, J. Chem. Phys. [**121**]{}, 11965 (2004).
S. Takei, Y.-B. Kim, and A. Mitra, cond-mat/0412710.
J. Koch and F. von Oppen, Phys. Rev. Lett. [**94**]{}, 206804 (2005).
Y. Xue and M. Ratner, Int. J. Quantum Chem. [**102**]{}, 911 (2005).
R.U.I. Liu *et al.*, J. Chem. Phys. [**122**]{}, 044703 (2005).
Strong electron-phonon coupling means $\lambda\gg 1$, where $\lambda$ is the shift of the equilibrium nuclear normal mode between neutral molecule and ion, in units of the spatial extent of the vibrational ground-state wavefunction.
The hierarchy breaks off once the transferred charge per avalanche is of order 1.
L.S. Levitov and G.B. Lesovik, JETP Lett. [**58**]{}, 230 (1993).
Ya.M. Blanter and M. Büttiker, Phys. Rep. [**336**]{}, 1 (2000).
L.S. Levitov and M. Reznikov, Phys. Rev. B [**70**]{}, 115305 (2004), and references therein.
U. Gavish, Y. Imry, Y. Levinson, and B. Yurke, in [*Quantum Noise in Mesoscopic Physics*]{}, ed. by Yu.V. Nazarov (Kluwer, Dordrecht, 2003).
M. Kindermann and Yu.V. Nazarov Phys. Rev. Lett. [**91**]{}, 136802 (2003).
M. Kindermann, Yu.V. Nazarov, and C.W.J. Beenakker, Phys. Rev.B [**69**]{}, 035336 (2004).
J. Holtsmark, Ann. Physik [**58**]{}, 577 (1919).
Yu.V. Nazarov and J.J.R. Struben, Phys. Rev. B [**53**]{}, 15466 (1996).
|
---
abstract: 'Kazhdan-Lusztig polynomials $P_{x,w}(q)$ play an important role in the study of Schubert varieties as well as the representation theory of semisimple Lie algebras. We give a lower bound for the values $P_{x,w}(1)$ in terms of “patterns”. A pattern for an element of a Weyl group is its image under a combinatorially defined map to a subgroup generated by reflections. This generalizes the classical definition of patterns in symmetric groups. This map corresponds geometrically to restriction to the fixed point set of an action of a one-dimensional torus on the flag variety of a semisimple group $G$. Our lower bound comes from applying a decomposition theorem for “hyperbolic localization” [@Br] to this torus action. This gives a geometric explanation for the appearance of pattern avoidance in the study of singularities of Schubert varieties.'
address:
- |
Dept. of Mathematics, 2-334\
Massachusetts Institute of Technology\
Cambridge, MA 02139
- |
Dept. of Mathematics and Statistics\
University of Massachusetts\
Amherst, MA 01003
author:
- 'Sara C. Billey'
- Tom Braden
title: 'Lower bounds for Kazhdan-Lusztig polynomials from patterns'
---
[^1]
Introduction
============
Many recent results on the singularities of Schubert varieties $X_w$ in the variety ${{\mathcal}F}_n$ of flags in ${{\mathbb C}}^n$ are expressed by the existence of certain patterns in the indexing permutation $w \in {{\mathfrak S}}_n$. For example, Lakshmibai and Sandhya [@LS] proved that $X_w$ is singular if and only if $w$ contains either of the patterns $4231$ or $3412$ (see also [@R], [@W]). A permutation $w\in {{\mathfrak S}}_n$ is said to contain the pattern $\tilde{w} \in {{\mathfrak S}}_k$ for $k < n$ if the permutation matrix of $w$ has the permutation matrix of $\tilde{w}$ as a submatrix.
This implies that if $\tilde{w} \in {{\mathfrak S}}_k$ is any pattern for $w$ and $X_{\tilde{w}} \subset {{\mathcal}F}_k$ is singular, then $X_{w}$ is singular as well. In this paper, we give a general geometric explanation of this phenomenon which works for the flag variety ${{\mathcal}F}$ and Weyl group $W$ of any semisimple algebraic group $G$.
Our result concerns the Kazhdan-Lusztig polynomials $P_{x,w}(q) \in
{{\mathbb Z}}_{\ge 0}[q]$, $x,w \in W$. Although defined purely combinatorially, they carry important information about representation theory of Hecke algebras and Lie algebras (see [@KL1; @BeBe; @BryK; @BGS] among many others), as well as geometric information about the singularities of Schubert varieties $X_w$ in ${{{\mathcal}F}}$.
More precisely, $P_{x,w}(q)$ is the Poincaré polynomial (in $q^{1/2}$) of the local intersection cohomology of $X_w$ at a generic point of $X_x$, and $P_{{1},w}(1)=1$ if and only if $X_{w}$ is rationally smooth [@KL2]. If $G$ is of type A,D, or E, then $X_w$ is singular if and only if $P_{{1},w}(1) > 1$ (Deodhar [@De] proved this for type A, while Peterson (unpublished) proved it for all simply laced groups. See [@CK]).
Our main result (Theorem \[main theorem\]) is a lower bound for $P_{x,w}(1)$ in terms of Kazhdan-Lusztig polynomials of patterns appearing in $x$ and other elements of $W$ determined by $x$ and $w$. Here a pattern of an element of $W$ is its image under a function $\phi\colon W \to W'$, which we define for any finite Coxeter group and any (not necessarily standard) parabolic subgroup $W' \subset W$. It agrees with the standard definition of patterns in type A, but is more general than the one using signed permutations used in [@Bi] for types B and D.
One consequence of our result is the following:
\[pattern monotonicity\] For any parabolic $W' \subset W$, we have $P_{{1},w}(1) \ge P_{{1},\phi(w)}(1)$.
In particular, this gives another proof that $X_{\tilde{w}}$ singular implies $X_w$ singular in type A. See also the remark after Theorem \[patterns and geometry\].
The definition of the pattern map $\phi$ is combinatorial, but it is motivated by the geometry of the action of the torus $T$ on ${{\mathcal}F}$, and the proof of Theorem \[main theorem\] is entirely geometrical. For $W'\subset W$ parabolic, there is a cocharacter $\rho\colon {{\mathbb C}}^*\to T$ whose fixed point set in ${{\mathcal}F}$ is a disjoint union of copies of the flag variety ${{{\mathcal}F}}'$ of a group $G'$ with Weyl group $W'$. The action of $\rho$ gives rise to a “hyperbolic localization” functor which takes sheaves on ${{\mathcal}F}$ to sheaves on ${{{\mathcal}F}}'$. Theorem \[main theorem\] then follows from a “decomposition theorem” for this functor, proved in [@Br], together with the fact that hyperbolic localization preserves local Euler characteristics.
If the action is totally attracting or repelling near a fixed point, hyperbolic localization is just ordinary restriction or its Verdier dual. This gives stronger coefficient-by-coefficient inequalities in some special cases (see Theorem \[parabolic main\]). The attracting/repelling case of [@Br] has been known for some time; it was used in [@BrM] to prove a conjecture of Kalai on toric $g$-numbers of rational convex polytopes.
Matthew Dyer has recently given us a preprint [@Dy] containing an inequality equivalent to Theorem \[main theorem\], which he proves using his theory of abstract highest weight categories. It seems likely that his approach is dual to ours under some version of Koszul duality [@BGS].
This work was originally motivated by the following question asked by Francesco Brenti: How can we describe the Weyl group elements $w$ such that $P_{\id, w}(1)=2$? In type A, we can show that if $P_{\id, w}(1)=2$ then the singular locus of the Schubert variety $X_w$ has only one irreducible component and $w$ must avoid the patterns: $$\begin{array}{ccc}
(5 2 6 4 1 3)& (5 4 6 2 1 3)& (4 6 3 1 5 2)\\ (4 6 5 1 3 2) &
(6 3 2 5 4 1) & (6 5 3 4 2 1)
\end{array}$$ We conjecture the converse holds as well.
We outline the sections of this paper. In §\[classical.patterns\], we discuss pattern avoidance on permutations and some applications from the literature. In §\[patterns\] we describe the pattern map for arbitrary finite Coxeter groups. §\[classical and Coxeter\] explains why the two notions agree for permutations. The main result of §\[patterns\] is proved in §\[coset proof\]. In §\[main result\] we state our main theorem. In §\[applications\] we highlight two particularly interesting special cases, including Theorem \[pattern monotonicity\]. Our geometric arguments are in §\[geometry\].
Pattern avoidance
=================
Classical pattern avoidance {#classical.patterns}
---------------------------
We can write an element $w$ of the permutation group ${{{\mathfrak S}}}_{n}$ on $n$ letters in one-line notation as $w=w_{1}w_{2}\dotsb w_{n}$, i.e. $w$ maps $i$ to $w_{i}$. We say a permutation $w$ *contains* a pattern $v \in {{\mathfrak S}}_{k}$ if there exists a subsequence $w_{i_{1}}w_{i_{2}}\dotsb w_{i_{k}}$, with the same relative order as $v=v_{1}\dotsb v_{k}$. If no such subsequence exists we say $w$ *avoids* the pattern $v$.
More formally, let $a_{1}\dotsb a_{k}$ be any list of distinct positive integers. Define the *flattening function* ${\mbox{\rm fl}}(a_{1}\dotsb a_{k})$ to be the unique permutation $v\in {{\mathfrak S}}_{k}$ such that $v_{i}>v_{j} \ \iff \ a_{i}>a_{j}$. Then it is equivalent to say that $w$ *avoids* $v$ if no ${\mbox{\rm fl}}(w_{i_{1}}w_{i_{2}}\dotsb
w_{i_{k}})=v$. For example, $w=4 5 3 6 1 7 2$ contains the pattern $3412$, since ${\mbox{\rm fl}}(w_1 w_{4} w_{5} w_{7}) = {\mbox{\rm fl}}(4612) = 3412$, but it avoids $4321$.
Several properties of permutations have been characterized by pattern avoidance and containment. For example, as mentioned in the introduction, for the Schubert variety $X_w$ we have Schubert variety $X_{w}$ is nonsingular if and only if $P_{{1},w}=1$ if and only if $w$ avoids $3412$ and $4231$ [@LS; @Car; @De; @KL2]. The element $C_{w}'$ of the Kazhdan-Lusztig basis of the Hecke algebra of $W$ equals the product $C'_{s_{a_{1}}}
C'_{s_{a_{2}}}\dotsb C'_{s_{a_{p}}}$ for any reduced expression $w = s_{a_{1}}s_{a_{2}}\dotsb s_{a_{p}}$ if and only if $w$ is *321-hexagon-avoiding* [@BiW]. Here 321-hexagon-avoiding means $w$ avoids the five patterns $321,$ $56781234,$ $46781235,$ $56718234,$ $46718235$.
The notion of pattern avoidance easily generalizes to the Weyl groups of types B,C,D since elements can be represented in one-line notation as permutations with $\pm$ signs on the entries. Once again, the properties $P_{{1},w}=1$ and $C_{w}' = C'_{s_{a_{1}}}
C'_{s_{a_{2}}}\dotsb C'_{s_{a_{p}}}$ can be characterized by pattern avoidance [@Bi; @BiW], though the list of patterns can be rather long. More examples of pattern avoidance appear in [@LasSc; @Stem; @BiP; @BiW2; @Manivel; @KLR; @Co; @Co2].
Patterns in Coxeter groups {#patterns}
--------------------------
In this section, we generalize the flattening function for permutations to an arbitrary finite Coxeter group $W$.
Let $S$ be the set of simple reflections generating $W$. The set $R$ of all reflections is $R = \bigcup_{w\in W} wSw^{-1}$. Given $w\in W$, its length $l(w)$ is the length of the shortest expression for $w$ in terms of elements of $S$. The Bruhat-Chevalley order is the partial order $\le$ on $W$ generated by the relation $$x < y \text{ if } l(x) < l(y)\;\text{and}\; xy^{-1} \in R.$$
Each subset $I \subset S$ generates a subgroup $W_I$; a subgroup $W'\subset W$ which is conjugate to $W_I$ for some $I$ is called a [*parabolic*]{} subgroup. The $W_I$’s themselves are known as [*standard*]{} parabolic subgroups.
A parabolic subgroup $W' = xW_Ix^{-1}$ of $W$ is again a Coxeter group, with simple reflections $S' = xIx^{-1}$ and reflections $R' = R \cap
W'$. Note that $S' \not\subset S$ unless $W'$ is standard.
We denote the length function and the Bruhat-Chevalley order for $(W',S')$ by $l'$ and $\le'$, respectively. If $W' = W_I$ then $$l' = l|_{W'}\;\text{and}\; \mathord{\le'} = \mathord{\le}|_{W'\times W'},$$ but in general we only have $l'(w) \le l(w)$ and $x \le' y \implies x \le y$. For instance, if $W' \subset {{\mathfrak S}}_4$ is generated by the reflections $r_{23} = 1324$ and $r_{14} = 4231$, then $r_{23} \le r_{14}$ although they are not comparable for $\le'$.
The following theorem/definition generalizes the flattening function for permutations.
\[coset theorem\] Let $W' \subset W$ be a parabolic subgroup. There is a unique function $\phi\colon W\to W'$, the [*pattern map*]{} for $W'$, satisfying:
1. $\phi$ is $W'$-equivariant: $\phi(wx) = w\phi(x)$ for all $w\in W'$, $x\in W$,
2. If $\phi(x) \le' \phi(wx)$ for some $w\in W'$, then $x \le wx$.
In particular, $\phi$ restricts to the identity map on $W'$.
If $W' = W_I$ is a standard parabolic, then (b) can be strengthened to “if and only if”. In this case the result is well-known.
To show uniqueness, note that (a) implies that $\phi$ is determined by the set $\phi^{-1}(1)$, and (b) implies that $\phi^{-1}(1) \cap W'x$ is the unique minimal element in $W'x$. Existence is more subtle; it is not immediately obvious that the function so defined satisfies (b). We give a construction of a function $\phi$ that satisfies (a) and (b) in Section \[coset proof\].
Relation with classical patterns {#classical and Coxeter}
--------------------------------
Take integers $1 \le a_1 < \dots < a_k \le n$, and let $\Sigma = \{a_1, a_2,\ldots, a_k\}$. Define a generalized flattening function ${\mbox{\rm fl}}_\Sigma\colon {{\mathfrak S}}_n \to {{\mathfrak S}}_k$ by ${\mbox{\rm fl}}_\Sigma(w)={\mbox{\rm fl}}(w_{i_1}w_{i_2}\ldots w_{i_k})$, where $w_{i_j} \in \Sigma$ for all $1 \leq j \leq k$ and $1 \leq i_1 < i_2 < \ldots < i_k \leq n$.
Let $W'\subset{{\mathfrak S}}_n$ be the subgroup generated by the transpositions $r_{a_i, a_j}$ for all $i<j$. It is parabolic; conjugating by any permutation $z$ with $z_i = a_i$ for $1 \le i \le k$ gives an isomorphism $\iota\colon {{\mathfrak S}}_k \to W'$, where ${{\mathfrak S}}_k \subset {{\mathfrak S}}_n$ consists of permutations fixing the elements $k+1,\dots, n$.
The function $\iota\circ {\mbox{\rm fl}}_\Sigma$ satisfies the properties of Theorem \[coset theorem\], and so $\iota\circ {\mbox{\rm fl}}_\Sigma(w) = \phi(w)$. Property (a) follows since left multiplication by a permutation $w \in W'$ acts only on the values in the set $\{a_1,
a_2,\ldots, a_k\}$. To prove (b), note that if $v_i = w_i$ for two permutations $v,w \in {{\mathfrak S}}_n$ then $v \leq w$ if and only ${\mbox{\rm fl}}(\hat v)\leq {\mbox{\rm fl}}(\hat w)$ where $\hat v,\hat w$ are the sequences obtained by removing the $i$th entry from each. This implies that $\iota\circ{\mbox{\rm fl}}_\Sigma(x) \leq' \iota\circ {\mbox{\rm fl}}_\Sigma(wx)$ if and only if $x \leq wx$.
For example, take $\Sigma = \{1,4,6,7\}$; the associated subgroup $W' \subset {{\mathfrak S}}_7$ is generated by $\{r_{14}, r_{46}, r_{67}\}$. If $x=6
2 1 3 4 7 5$ then $y=1 2 4 3 6 7 5$ is the unique minimal element in $W'x$ and $x= r_{46}r_{14} y$, so $\phi(x)=r_{46}r_{14}$. This agrees with the classical flattening using the isomorphism $W' \cong {{\mathfrak S}}_4$ given by $r_{14} \mapsto s_1$, $r_{46} \mapsto s_2$, $r_{67} \mapsto s_3$: in fact, $${\mbox{\rm fl}}_{\{1,4,6,7\}}(6 2 1 3 4 7 5)={\mbox{\rm fl}}(6147) = 3124=s_{2}s_{1}.$$
To obtain the most general parabolic subgroup of ${{\mathfrak S}}_n$, let $\Sigma_1,\dots,\Sigma_l$ be disjoint subsets of $1\dots n$. To each $\Sigma_j$ is associated a parabolic subgroup $W'_j$ as before, and then $$W' = W'_1W'_2\dots W'_l \cong {{\mathfrak S}}_{|\Sigma_1|} \times \dots \times
{{\mathfrak S}}_{|\Sigma_l|}$$ is a parabolic subgroup. The corresponding flattening function is $$w \mapsto ({\mbox{\rm fl}}_{\Sigma_1}(w),\dots,{\mbox{\rm fl}}_{\Sigma_l}(w)).$$
In types B and D, the flattening function of [@Bi] given in terms of signed permutations can also be viewed as an instance of our pattern map. The group $W'$ of signed permutations which fix every element except possibly the $\pm a_i$, $1 \le i \le k$ is parabolic. Multiplication on the left by $w \in W'$ acts only on the values in the set $\{\pm a_1, \pm a_2,\ldots, \pm a_k\}$ and if $v_i = w_i$ for two signed permutations $v,w$ then $v \leq w$ if and only ${\mbox{\rm fl}}(\hat v)\leq
{\mbox{\rm fl}}(\hat w)$ where $\hat v,\hat w$ are the sequences obtained by removing the $i$th entry from each. It follows that $v \mapsto {\mbox{\rm fl}}(\hat v)$ satisfies the conditions of Theorem \[coset theorem\]
There are other types of parabolic subgroups in types B and D which give rise to other pattern maps. For instance, the group $W'$ of all unsigned permutations is a parabolic subgroup of either B${}_n$ or D${}_n$. In this case the pattern map “flattens” the signed permutation to an unsigned one (e.g. $-4,2,1,-3 \mapsto 1432$). Other cases of pattern maps for classical groups are more difficult to describe combinatorially.
The first author and Postnikov [@BiP] have used these more general pattern maps to reduce significantly the number of patterns needed to recognize smoothness and rational smoothness of Schubert varieties. They reduce the list even further by generalizing pattern maps to the case of “root system embeddings” which do not necessarily preserve the inner products of the roots; for instance, there is a root system embedding of A${}_3$ into B${}_3$. We do not know of a geometric interpretation of these more general pattern maps.
Spanning subgroups and the reflection representation {#coset proof}
----------------------------------------------------
To prove Theorem \[coset theorem\] we use the action of $W$ on its root system. See [@H Section 1] for proofs of the following facts.
We have the following data: a representation of $W$ on a finite-dimensional real vector space $V$, a $W$-invariant subset $\Phi \subset V$ (the roots), a subset $\Pi \subset \Phi$ (the positive roots), and a bijection $r \mapsto \alpha_r$ between $R$ and $\Pi$.
These data satisfy the following properties: $\Phi$ is the disjoint union of $\Pi$ and $-\Pi$. The vectors $\{\alpha_s\}_{s\in S}$ form a basis for $V$; a root $\alpha \in \Phi$ is positive if and only if it can be expressed in this basis with nonnegative coefficients. For any $r\in R$ and $w \in W$, we have $$\label{order and roots}
rw > w \iff \alpha_r \in w\Pi.$$
Given a linear function $H\colon V\to {{\mathbb R}}$, define $$\Pi_H = \{\alpha \in \Phi\mid H(\alpha)>0\}.$$ Call $H$ [*generic*]{} if $\Phi \cap\ker H = \emptyset$. If we take $H_1(\alpha_s) = 1$ for all $s\in S$, then $H_1$ is generic and $\Pi = \Pi_{H_1}$. If we put $H_w = {H_1 \circ w^{-1}}$, then $\Pi_{H_w} = w\Pi$. Conversely, if $H$ is generic, then $\Pi_H = w\Pi$ for a unique $w\in W$.
\[geometric spanning\] Let $W'\subset W$ be a subgroup generated by reflections. Then $W'$ is parabolic if and only if there is a subspace $V'\subset V$ so that $W'$ is generated by $R' = \{r\in R\mid \alpha_r \in V'\}$. If so, then $V'$ is $W'$-stable, and putting $\Phi' = \Phi\cap V'$, $\Pi' = \Pi \cap V'$, and $\alpha'_r = \alpha_r$ for $r \in R'$ gives the reflection representation of $W'$.
See [@H §1.12].
In type A, all subgroups generated by reflections are parabolic. In other types this is no longer the case – for instance, the subgroup $W' \cong ({{\mathbb Z}}_2)^n$ of B${}_n$ generated by reflections in the roots $\{\pm e_j\}$ is not parabolic for any $n \ge 2$, since these roots span $V$.
We now prove the existence of the function $\phi$ from Theorem \[coset theorem\]. Let $V' \subset V$ be as in Proposition \[geometric spanning\]. Given $w \in W$, we have $w\Pi = \Pi_{H_w}$, and so $w\Pi\cap V' = \Pi'_{H'}$, where $\Pi' = \Pi\cap V$ and $H' = H_w|_{V'}$. It follows that there is a unique $\phi(w) \in W'$ so that $$\phi(w)\Pi' = w\Pi\cap V'.$$ We show that the function $\phi$ defined this way satisfies (a) and (b) from Theorem \[coset theorem\]. Any $w\in W'$ fixes $V'$, so if $x \in W$ then $$\phi(wx) \Pi' =
(wx\Pi) \cap V' = w(x\Pi \cap V') = w\phi(x)\Pi',$$ giving (a).
To prove (b), it will be enough to show that $\phi(x) \le' \phi(rx)$ implies $x \le rx$ for any $x\in W$, $r\in R'$, since these relations generate the Bruhat-Chevalley orders on $W$ and $W'$. We have $$\begin{aligned}
\phi (x)<'\phi (rx)=r\phi(x) &\iff& \alpha_{r}
\in \phi (x)\Pi'=x\Pi \cap V'\\ & \implies & \alpha_{r} \in x\Pi \\
&\implies& x <rx.\end{aligned}$$
The main result {#main result}
===============
Suppose now that $W$ is the Weyl group of a semisimple complex algebraic group $G$. Let $W' \subset W$ be parabolic, and let $\phi\colon
W\to W'$ be the pattern map of Theorem \[coset theorem\]. For any $x\in W$, define a partial order on $W'x$ by “pulling back” the Bruhat order from $W'$: if $w,w' \in W'$, say $wx \le_x w'x$ if and only if $\phi(wx) \le' \phi(w'x)$. By Theorem \[coset theorem\], this is weaker than the Bruhat order on $W'x$.
Our main result is the following.
\[main theorem\] If $x,w \in W$, then $$P_{x,w}(1) \ge \sum_{y\in M(x,w;W')} P_{y,w}(1)P'_{\phi(x), \phi(y)}(1),$$ where $M(x,w;W')$ is the set of maximal elements with respect to $\le_x$ in $[1, w] \cap W'x$, and $P'$ denotes the Kazhdan-Lusztig polynomial for the Coxeter system $(W',S')$.
Conjecturally this should hold for any finite Coxeter group $W$. There is a stronger formulation when $W'$ is a standard parabolic subgroup of $W$; see the next section.
Take $W = {{\mathfrak S}}_4$, $w = 4231$, $x = 2143$. Let $W' \cong {{\mathfrak S}}_2 \times {{\mathfrak S}}_2$ be the group generated by reflections $r_{13} = 3214$, $r_{24} = 1432$. Then $W'x = \{2143, 4123, 2341, 4321\}$. All but $4321$ are in the interval $[1,w]$, so the maximal elements of $[1,w] \cap W'x$ are $4123 = r_{24}x$ and $2341 = r_{13}x$. Theorem \[main theorem\] gives $$P_{2143,4231}(1) \ge P_{4123,4231}(1)P'_{{1},r_{24}}(1) + P_{2341,4231}(1)P'_{{1},r_{13}}(1)$$ $$= 1\cdot 1 + 1 \cdot 1 = 2,$$ which holds since $P_{2143,4231}(q) = 1 + q$.
Note that this shows $X_{4231}$ is singular, even though all the Schubert varieties corresponding to terms on the right hand side are smooth.
One can calculate $P_{1234567,6734512}(1)=44$ in type $A$. This is the maximum value of $P_{x,w}(1)$ for any $x,w \in {{\mathfrak S}}_{7}$. Let $W' \subset {{\mathfrak S}}_9$ be the subgroup generated by the reflections $\{r_{13},r_{34},r_{45},r_{57},r_{78},r_{89}\}$; it is a parabolic subgroup isomorphic to ${{\mathfrak S}}_{7}$. If $w=869457213$ and $x=163457289$, then $W'x=W'w$ so $M(x,w;W')=\{w \}$, giving $\phi(x)=1234567$ and $\phi(w)=6734512$. Hence $$P_{x,w}(1)\geq
P'_{1234567,6734512}(1)P_{w,w}(1)=44.$$
Special cases/applications {#applications}
--------------------------
The complicated interaction of the multiplicative structure of $W$ and the Bruhat-Chevalley order makes computing the set $M(x,w;W')$ difficult. We mention two cases in which the answer is nice:
**(a)** If $w$ and $x$ lie in the same $W'$-coset then $M(x,w;W') = \{w\}$. In this case Theorem \[main theorem\] says $$P_{x,w}(1) \ge P'_{\phi(x), \phi(w)}(1).$$
This allows us to prove Theorem \[pattern monotonicity\] from the introduction’: given $w\in W$, let $x \in W'w$ satisfy $\phi(x)= 1$. Then $$P_{{1},w}(1) \ge P_{x,w}(1) \ge P'_{{1},\phi(w)}(1).$$ The first inequality comes from the monotonicity of Kazhdan-Lusztig polynomials [@I],[@BrM2 Corollary 3.7].
**(b)** If either $W'$ or $x^{-1}W'x$ is a standard parabolic subgroup of $W$, then $M(x,w;W')$ has only one element. The case where $x = 1$ was studied by Billey, Fan, and Losonczy [@BiFL].
In this case the inequality will hold coefficient by coefficient rather than just at $q = 1$:
\[parabolic main\] If $W'$ or $x^{-1}W'x$ is a standard parabolic subgroup, then $$[q^k]P_{x,w} \ge \sum_{i+j = k} [q^i]P_{y,w}[q^j]P'_{\phi(x),\phi(y)},$$ where $M(x,w;W') = \{y\}$. Here the notation $[q^k]P$ means the coefficient of $q^k$ in the polynomial $P$.
If both (a) and (b) hold, then Theorem \[parabolic main\] is implied by a well-known equality (see [@P Lemma 2.6]):
\[t:parabolic\] If $W'$ or $x^{-1}W'x$ is a standard parabolic subgroup of $W$ and $w \in W'x$, then $$P_{x,w}(q) = P'_{\phi(x),\phi(w)}(q).$$
Theorem \[t:parabolic\] can be thought of as a generalization of a theorem due to Brenti and Simion:
[@BreS] \[t:BS\] Let $u,v \in {{\mathfrak S}}_{n}$. For any $1 \leq i\leq n$ such that $\{1,2,\dots,i \}$ appear in the same set of positions (though not necessarily in the same order) in both $u$ and $v$, then $$P_{u,v}(q)= P_{u[1,i],v[1,i]}(q) \cdot P_{{\mbox{\rm fl}}(u[i+1,n]),{\mbox{\rm fl}}(v[i+1,n]) }(q),$$ where $u[j,k]$ is obtained from $u$ by only keeping the numbers $j,j+1,\dots, k$ in the order they appear in $u$.
We demonstrate the relationship between the two theorems on an example. Let $I_1 = \{s_{1},s_{2},s_{3}\}$, $I_2 = \{s_{5},s_{6},s_{7}\}$, $I = I_1 \cup I_2$. Let $W' = W_I \cong W_{I_1} \times W_{I_2}$. Any pair $x,w$ in the same coset of $W'\backslash W$ satisfies the conditions of Theorem \[t:BS\] and Theorem \[t:parabolic\]. Take $x=25174683$ and $w=48273561$. Then Theorem \[t:parabolic\] gives $$\begin{aligned}
\label{e:BS.1}
P_{25174683,48273561}(q) &= P'_{\phi(25174683),\phi(48273561)}(q) \\
&= P'_{21435768,42318756}(q) = P_{2143,4231}(q) P_{1324,4312}(q) \end{aligned}$$ agreeing with Theorem \[t:BS\]. The last equality results because we have $P_{x_1\times x_2, w_1 \times w_2}(q) = P_{x_1,w_1}(q)P_{x_2,w_2}(q)$ for any $x_1 \times x_2$, $w_1 \times w_2$ in the reducible Coxeter group $W_{I_1}\times W_{I_2}$.
Geometry of flag varieties {#geometry}
==========================
Let $G$ be a connected semisimple linear algebraic group over ${{\mathbb C}}$. It acts transitively on the flag variety ${{\mathcal}F}$ of Borel subgroups of $G$ by conjugation: $g\cdot B = gBg^{-1}$. For any $g \in G$, the point $B\in {{{\mathcal}F}}$ is fixed by $g$ if and only if $g \in B$. Fix a Borel subgroup and a maximal torus $T \subset B \subset G$. The Weyl group $W = N_G(T)/T$ is a finite Coxeter group. The point $g\cdot B \in {{{\mathcal}F}}$ is fixed by $T$ if and only if $g \in N_G(T)B$, and so $g\mapsto g\cdot B$ induces a bijection between $W$ and ${{\mathcal}F}^T$. We abuse notation and refer to $w \in W$ and the corresponding point of ${{\mathcal}F}$ by the same symbol.
Every $B$-orbit on ${{{\mathcal}F}}$ contains a unique $T$-fixed point; for $w \in W$, the Bruhat cell $C_w$ is the $B$-orbit $B\cdot w$. The Schubert variety $X_w$ is the closure of $C_w$; we have $X_w = \bigcup_{x\le w} C_x$ and so $X_x \subset X_w \iff x \le w$.
Torus actions {#T actions}
-------------
Let $\rho\colon {{\mathbb C}}^* \to T$ be a cocharacter of $T$, and let $G'$ be the centralizer of $T_0 = \rho({{\mathbb C}}^*)$.
[@Sp Theorem 6.4.7]\[restricting Borels\] $G'$ is connected and reductive; $T$ is a maximal torus in $G'$. If $T_0$ fixes a point $B_0 \in {{{\mathcal}F}}$, so that $T_0 \subset B_0$, then $B_0 \cap G'$ is a Borel subgroup of $G'$.
Let ${{\mathcal}F}'\cong G'/B'$ be the flag variety of $G'$, and put ${{\mathcal}F}^\rho = {{\mathcal}F}^{T_0}$. Using Theorem \[restricting Borels\], we can define a $G'$-equivariant algebraic map $\psi\colon {{\mathcal}F}^{\rho}\to {{\mathcal}F}'$ by $\psi(B_0) = (B_0) \cap G'$.
Fix a maximal torus and Borel subgroup of $G'$ by setting $B' = B\cap G'$, $T' = T$. The Weyl group of $G'$ is $W' = N_{G'}(T')/T' = W \cap (G'/B')$. The Schubert varieties of ${{{\mathcal}F}}'$ defined by the action of $B'$ are indexed by elements of $W'$; denote them by $X'_w$, $w\in W'$.
\[tori exist\] $W'$ is a parabolic subgroup of $W$, and all parabolic subgroups arise in this way for some choice of $\rho$.
This is well-known; the groups $G'$ which arise this way are Levi subgroups of parabolic subgroups of $G$. The second half of the statement (which is the only part we need) can be deduced from [@Sp 6.4.3 and 8.4.1], for instance.
Now we can connect the pattern map $\phi$ defined by Theorem \[coset theorem\] to geometry.
\[patterns and geometry\] The map $\psi$ restricts to an isomorphism on each connected component of ${{\mathcal}F}^{\rho}$. The restriction $\psi|_{{{{\mathcal}F}}^T} \colon {{{\mathcal}F}}^T \to ({{{\mathcal}F}}')^T$ is the pattern map $\phi$, using the identifications ${{{\mathcal}F}}^T = W$, $({{{\mathcal}F}}')^T = W'$. In particular, the components of ${{{\mathcal}F}}^\rho$ are in bijection with $W'\backslash W$.
To show the first assertion, it is enough to show that $\psi$ is a finite map, since it is $G'$-equivariant and its image ${{{\mathcal}F}}'$ is maximal among the compact homogeneous spaces for $G'$. But $\psi(g\cdot B) \in ({{{\mathcal}F}}')^T \implies T\subset g\cdot B
\implies g\cdot B \in {{{\mathcal}F}}^T$, a finite set.
Certainly $\psi$ takes $T$-fixed points to $T$-fixed points, so it induces a function $W \to W'$ by restriction. We need to show that it satisfies the properties of Theorem \[coset theorem\]. The $W'$-equivariance (a) follows immediately from the $G'$-equivariance of $\psi$.
To see property (b), take $x \in W$ and $w \in W'$, and suppose that $\psi(x) \le' \psi(wx)$. This implies that $\psi(x) \in {\overline}{B'\cdot \psi(wx)}$, and since $x$ and $wx$ lie in the same component of ${{\mathcal}F}^{\rho}$, we must have $x \in {\overline}{B'\cdot wx} \subset {\overline}{B \cdot wx}$. Thus $x \le wx$.
\[r:t-fixed-pts\] Given $w\in W$, let $Y \cong {{\mathcal}F}'$ be the component of ${{\mathcal}F}^{\rho}$ which contains $w$. Then one can show that $X_w \cap Y \cong X'_{\phi(w)}$. Therefore, $X'_{\phi(w)}$ singular implies that $X_w$ is singular, using the result of Fogarty and Norman [@FN]: a linearly algebraic group $G$ is linearly reductive (this class includes all tori) if and only if for all smooth algebraic $G$-schemes $X$ the fixed point scheme $X^{G}$ is smooth.
Hyperbolic localization {#T localization}
-----------------------
Let $X$ be a normal complex variety with an action of ${{\mathbb C}}^*$. Let $X^\circ = X^{{{\mathbb C}}^*}$, and let $X^\circ_1 \dots X^\circ_r$ be the connected components of $X^\circ$. For $1 \le k \le r$, define a variety $$X^+_k = \{x \in X \mid \lim_{t\to 0} t\cdot x \in X^\circ_k\},$$ and let $X^+$ be the disjoint (disconnected) union of all the $X^+_k$. The inclusions $X^\circ_k \subset X^+_k \subset X$ induce maps $$X^\circ {\mathop{\rightarrow}\limits}^f X^+ {\mathop{\rightarrow}\limits}^g X.$$
Let $D^b(X)$ denote the constructible derived category of ${{\mathbb Q}}$-sheaves on $X$.
Given $S\in D^b(X)$, define its [*hyperbolic localization*]{} $$S^{!*} = f^!g^*S \in D^b(X^\circ).$$
Hyperbolic localization is better adapted to ${{\mathbb C}}^*$-equivariant geometry than ordinary restriction. It was first studied by Kirwan [@Ki], who showed that if $S$ is the intersection cohomology sheaf of a projective variety with a linear ${{\mathbb C}}^*$-action, then $S$ and $S^{!*}$ have isomorphic hypercohomology groups. We will need two properties of hyperbolic localization from [@Br]. For any $S \in D^b(X)$ and $p \in X$, we let $\chi_p(X)$ denote the Euler characteristic of the stalk cohomology at $p$.
[@Br Proposition 3] \[localization chi\] If $p \in X^\circ$, then $$\chi_p(S) = \chi_p(S^{!*}).$$
Second, hyperbolic localization satisfies a decomposition theorem [@Br Theorem 2]. When applied to $X = {{\mathcal}F}$ and the action given by $\rho$, this gives the following.
\[localization purity\] Let $L_w$, and $L'_v$ be the intersection cohomology sheaves of the Schubert varieties $X_w$ and $X'_v$, respectively. For any $w\in W$ and $1 \le k \le r$, there is an isomorphism $$\psi_*((L_w)^{!*}|_{{{{\mathcal}F}}^\circ_k}) \cong \bigoplus_{j=1}^{m} L'_{v_j}[d_j],$$ for some $v_j \in W'$ (not necessarily distinct) and $d_j \in 2{{\mathbb Z}}$.
Here we use the fact that hyperbolic localization preserves $B'$-equivariance. The fact that $d_j \in 2{{\mathbb Z}}$ follows from the purity of the stalks of simple mixed Hodge modules of Schubert varieties.
Proof of Theorem \[main theorem\]
---------------------------------
The description of Kazhdan-Lusztig polynomials as the local intersection cohomology Poincaré polynomials of Schubert varieties [@KL2] implies that for any $u, v \in W$, we have $$P_{u,v}(1) = \chi_u(L_v) = \sum_i \dim_{{\mathbb Q}}{{\mathbb H}}^{2i}((L_v)_u).$$
Now, given $x,w\in W$, let ${{{\mathcal}F}}^\circ_k$ be the component of ${{{\mathcal}F}}^{\rho}$ which contains $x$, and thus all of $W'x$. For every $y\in W'$, let $a_y$ be the number of $j$ for which $v_j = y$ in Theorem \[localization purity\].
For any $z \in W'x$ we have, using Theorem \[localization purity\] and Proposition \[localization chi\], $$\begin{aligned}
\nonumber
P_{z,w}(1)&= \chi_z(L_w)= \chi_{\phi(z)}
\left(\psi_*((L_w)^{!*}|_{{{{\mathcal}F}}^\circ_k}) \right)\\
\label{loc sum} &=\sum_{j=1}^{m} \chi_{\phi(z)}\left(L'_{v_j}[d_j] \right)\\
\nonumber &=\sum_{y\in W'z} a_y P'_{\phi(z),\phi(y)}(1)\end{aligned}$$ (note that the shift $[d_j]$ does not change the Euler characteristic, since $d_j \in 2{{\mathbb Z}}$).
If $z \notin [1,w]$ then equation (\[loc sum\]) implies $a_z = 0$, since $P_{z,w} = 0$, $P'_{z,z} = 1$, and all the terms in the sum are nonnegative. Using (\[loc sum\]) again shows that if $y \in M(x;w;W')$, i.e.$y$ is maximal in $[1,w]\cap W'x$, then $a_y = P_{y,w}(1)$. Finally, evaluating at $x$ and keeping only the terms with $y \in M(x,w;W')$ proves Theorem \[main theorem\].
Proof of Theorem \[parabolic main\]
-----------------------------------
Suppose first that $x^{-1}W'x = W_I$ is a standard parabolic subgroup. Take $\mu$ to be any dominant integral cocharacter which annihilates a root $\alpha_r$ if and only if $r \in W'$, and let $\rho = Ad(x)\mu$. Then the action of $\rho$ is [*completely repelling*]{} near the component ${{\mathcal}F}^\circ_k$ of ${{\mathcal}F}^{\rho}$ which contains $W'x = xW_I$, meaning that ${{\mathcal}F}_k^+ = {{\mathcal}F}^\circ_k$, in the notation of §\[T localization\].
This implies that hyperbolic localization to ${{{\mathcal}F}}^\circ_k$ is just ordinary restriction: setting $h\colon {{{\mathcal}F}}^\circ_k \to {{{\mathcal}F}}^{\rho}$ for the inclusion, we have $$(S^{!*})|_{{{{\mathcal}F}}^\circ_k} = h^!f^!g^*S = (fh)^!g^*S = (fh)^*g^*S =
S|_{{{{\mathcal}F}}^\circ_k},$$ since both $h$ and $fh$ are open immersions. The same argument given for Theorem \[main theorem\] now proves Theorem \[parabolic main\], using local Poincaré polynomials instead of local Euler characteristics.
If instead $W' = W_I$, we can use the anti-involution $g \mapsto g^{-1}$ to replace left cosets by right cosets, since $P_{x^{-1}, w^{-1}} = P_{x,w}$ for all $x,w \in W$.
Acknowledgments {#acknowledgments .unnumbered}
---------------
We have benefitted greatly from conversations with Francesco Brenti, Victor Guillemin, Victor Ginzburg, Bert Kostant, Sue Tolman, David Vogan and Greg Warrington. We are grateful to Patrick Polo for spotting an error in Theorem 5, and to the referees for thoughtful comments.
[99]{}
A. Beilinson and J. Bernstein, [ *Localisation de $\mathfrak g$-modules*]{}, C. R. Acad. Sci. Paris Ser. I Math. [**292**]{} (1981), no. 1, 15–18.
A. Beilinson, V. Ginzburg, and W. Soergel [*Koszul duality patterns in representation theory*]{}, J. Amer. Math. Soc. [**9**]{} (1996), no. 2, 473–527.
S. Billey, [*Pattern avoidance and rational smoothness of Schubert varieties*]{} Adv. Math. [**139**]{} (1998), no. 1, 141–156.
S. Billey, C.K. Fan, and J. Losonczy, [*The parabolic map*]{}. J. Algebra, [**214**]{} (1999), no. 1, 1–7.
S. Billey and A. Postnikov, [*A root system description of pattern avoidance with applications to Schubert varieties*]{}, preprint [math.CO/0205179]{} (2002), 17 pp.
S. Billey and G. Warrington, [*Kazhdan–Lusztig Polynomials for 321-hexagon-avoiding permutations*]{}, J. Alg. Comb. **13** (2000), 111–136.
, [*Maximal Singular Loci of Schubert Varieties in $SL_{n}/B$*]{}, Trans. Amer. Math. Soc. [**355**]{} (2003), no. 10, 3915–3945.
T. Braden, [*Hyperbolic localization of intersection cohomology*]{}, preprint [math.AG/0202251]{} (2002), 7 pp., submitted.
T. Braden and R. MacPherson, [*Intersection homology of toric varieties and a conjecture of Kalai*]{}, Comment. Math. Helv. [**74**]{} (1999), no. 3, 442–455.
, [*From moment graphs to intersection cohomology*]{}, Math. Ann. [**321**]{} (2001) no. 3, 533–551.
, [*Explicit formulae for some Kazhdan-Lusztig polynomials*]{}, J. Algebraic Combin. [**11**]{} (2000), no. 3, 187–196.
J.-L. Brylinski and M. Kashiwara, [*Kazhdan-Lusztig conjecture and holonomic systems*]{}, Invent. Math. [**64**]{} (1981), no. 3, 387–410.
J. Carrell, [*The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and rational smoothness of Schubert varieties*]{}, Proceedings of Symposia in Pure Math., **56** (1994), 53–61.
J. Carrell and J. Kuttler, [*Smooth points of $T$-stable varieties in $G/B$ and the Peterson map*]{}, to appear in Invent. Math.
A. Cortez, [*Singularités génériques des variétés de Schubert covexillaires*]{}, Ann. Inst. Fourier (Grenoble) [**51**]{} (2001), no. 2, 375-393.
A. Cortez, [*Singularités génériques et quasi-resolutions des variétés de Schubert pour le groupe linéaire*]{}, C. R. Acad. Sci. Paris Sér. I Math. [**333**]{} (2001), no. 6, 561–566.
V. Deodhar, [*Local Poincaré duality and nonsingularity of Schubert varieties*]{}, Comm. Algebra [**13**]{} (1985), no. 6, 1379–1388.
M. Dyer, [*Rank two detection of singularities of Schubert varieties*]{}, preprint (2001).
, [*A fixed-point characterization of linearly reductive groups*]{}, in Contributions to algebra (collection of papers dedicated to [Ellis Kolchin]{}), Academic Press, New York, 1977, pp. 151–155.
J. Humphreys, Reflection groups and Coxeter groups, Cambridge studies in advanced mathematics [**29**]{}.
R. Irving, [*The socle filtration of a Verma module*]{}, Ann. Sci. École Norm. Sup. series 4 [**21**]{} (1988), no. 1, 47–65.
D. Kazhdan and G. Lusztig, [*Representations of Coxeter groups and Hecke algebras*]{}, Invent. Math., **53** (1979), 165–184.
, [*Schubert varieties and Poincaré duality*]{}, Proc. Symp. Pure. Math., AMS, **36** (1980), 185–203.
F. Kirwan, [*Intersection homology and torus actions*]{}, J. Amer. Math. Soc. [**1**]{} (1988), no. 2, 385–400.
V. Lakshmibai and B. Sandhya, [*Criterion for smoothness of Schubert varieties in ${\rm Sl}(n)/B$*]{} Proc. Indian Acad. Sci. Math. Sci. [**100**]{} (1990), no. 1, 45–52.
A. Lascoux and M.P. Schützenberger, [*Polynômes de Kazhdan and Lusztig pour les grassmanniennes.*]{} (French) \[Kazhdan–Lusztig polynomials for Grassmannians\], Astérisque, **87–88** (1981), 249–266, Young tableaux and Schur functions in algebra and geometry (Toruń, 1980).
C. Kassel, A. Lascoux and C. Reutenauer, *The singular locus of a [Schubert]{} variety*, Prépublication de l’[Institut de Recherche Mathématique Avancée]{} (2001). To appear J. of Alg.
L. Manivel, *Le lieu singulier des variétés de [Schubert]{}*, Internat. Math. Res. Notices (2001), no. 16, 849–871.
P. Polo, [*Construction of arbitrary Kazhdan-Lusztig polynomials in symmetric groups*]{} Represent. Theory [**3**]{} (1999), 90-104 (electronic).
K. M. Ryan, [*On Schubert varieties in the flag manifold of ${\rm Sl}(n,{\bf C})$*]{}, Math. Ann. [**276**]{} (1987), no. 2, 205–224.
T. Springer, Linear algebraic groups, Second edition. Birkhäuser, Boston, MA, 1998.
J. Stembridge, [*On the fully commutative elements of [Coxeter]{} groups*]{}, J. Algebraic Combin. **5** (1996), no. 4, 353–385.
J. S. Wolper, [*A combinatorial approach to the singularities of Schubert varieties*]{}, Adv. Math. [**76**]{} (1989), no. 2, 184–193.
[^1]: The first author was supported by National Science Foundation grant DMS-9983797; the second author was supported in part by grant DMS-0201823
|
---
abstract: 'The Göppert-Mayer (GM) gauge transformation, of central importance in atomic, molecular, and optical physics since it connects the length gauge and the velocity gauge, becomes unphysical as the field frequency declines towards zero. This is not consequential for theories of transverse fields, but it is the underlying reason for the failure of gauge invariance in the dipole-approximation version of the Strong-Field Approximation (SFA). This failure of the GM gauge transformation explains why the length gauge is preferred in analytical approximation methods for fields that possess a constant electric field as a zero-frequency limit.'
author:
- 'H. R. Reiss'
title: 'Low-frequency failure of the Göppert-Mayer gauge transformation and consequences for the Strong-Field Approximation'
---
The Strong-Field Approximation (SFA) is the basic analytical method for the treatment of the interaction of nonperturbatively strong laser fields with atoms and molecules, but it is known to be gauge-dependent. It is important to note from the outset that results obtained herein apply only to theories that make *a priori* use of the dipole approximation. Such theories are identifiable by the fact that a zero-frequency limit exists, and that this limit corresponds to a constant electric field. In other words, this work applies only to longitudinal fields. Thus theories that are derived from propagating-wave formalisms are excluded, since such transverse-field theories have extremely low-frequency radio waves as a low frequency limit.
Gauge dependence of the SFA is probably shown most clearly in Ref. [@bmb], where the length gauge (LG) results are plausible, but the velocity gauge (VG) results are not. This long-known lack of gauge invariance has led to statements of alarm, such as *... the SFA is not gauge invariant, which is really bad news for a theory.*. [@mishajmo] (emphasis from the original.) Another expression of concern is: ...how can a noninvariant theory be used for the calculation of observables? [@poprtun].
The approach taken here for examination of the gauge problem is entirely general, with no dependence on the particular properties of any problem or class of problems beyond the statement that the field is treated as a longitudinal field (or the equivalent statement that the field has a zero-frequency limit corresponding to a static electric field). We start with a re-derivation of the known result [@hrjmo; @hrtun] that the static electric field can be described only within a unique gauge if all physical constraints are to be satisfied. A nominally alternative gauge is discarded on the grounds that it violates the physical condition that a charged particle in a static electric field represents a system for which total energy is conserved. It is then shown that this unphysical gauge arises from a Göppert-Mayer (GM) gauge transformation from the length gauge to the velocity gauge as applied to an oscillatory electric field in the zero frequency limit. This establishes the unphysical nature of the GM gauge transformation when zero frequency is a possibility. This is consequential in that photoelectron spectra that extend to zero frequency are a necessary part of any strong-field, nonperturbative problem. The GM gauge transformation from the LG to the VG is thus shown to be unphysical in the zero-frequency limit, leaving the LG as the only physical alternative. The two constraints of strong fields and the accessibility of a zero-frequency limit are all that is necessary to confirm the LG as the only physical gauge for the SFA in the form appropriate to oscillatory electric fields.
Consider a static electric field with the amplitude $E_{0}$. It is known from electrostatics that this field can be specified by the scalar and vector potentials$$\phi=-\mathbf{r\cdot E}_{0},\quad\mathbf{A}=0. \label{a}$$ A gauge transformation can be accomplished by a scalar generating function $\Lambda$ subject only to the constraint that the generating function satisfy the homogeneous wave equation$$\partial^{\mu}\partial_{\mu}\Lambda=0. \label{b}$$ The 4-vector potential following from a gauge transformation is$$\widetilde{A}^{\mu}=A^{\mu}+\partial^{\mu}\Lambda, \label{c}$$ which is equivalent to the transformed scalar and 3-vector potentials$$\begin{aligned}
\widetilde{\phi} & =\phi+\frac{1}{c}\partial_{t}\Lambda,\label{d}\\
\widetilde{\mathbf{A}} & =\mathbf{A}-\mathbf{\nabla}\Lambda. \label{e}$$ It is well-known that the representation of a static electric field by a scalar potential alone, as in Eq.(\[a\]) can be gauge-transformed so that the field can be described by a vector potential alone by using the generating function$$\Lambda=ct\mathbf{r\cdot E}_{0}, \label{f}$$ which leads to the new potentials$$\widetilde{\phi}=0,\quad\widetilde{\mathbf{A}}=-ct\mathbf{E}_{0}. \label{g}$$ The potentials in Eq. (\[g\]) are unphysical in the sense that a charged particle subject to those potentials is described by Lagrangian and Hamiltonian functions that possess explicit time dependence; and explicit time dependence of these system functions is a clear indicator that total energy is not conserved. This contrasts with the time independence of the potentials in Eq.(\[a\]), signifying energy conservation.
The formal foundations for Noether’s Theorem connecting symmetries with physical conservation laws are expressed in terms of the Lagrangian function. (See, for example, Ref. [@goldstein].) The potentials (\[a\]) lead to a Lagrangian that has no explicit dependence on time, and thereby demonstrates energy conservation, whereas the potentials (\[g\]) signify a Lagrangian that depends explicitly on the time, and is thus unphysical.
The GM gauge transformation is usually expressed in terms of the vector potential that arises after the transformation. That is, the generator of the GM gauge transformation is usually written as$$\Lambda^{GM}=-\mathbf{r\cdot}\widetilde{\mathbf{A}}.\label{g1}$$ This is exactly what follows from Eqs. (\[f\]) and (\[g\]), so the above discussion amounts to concluding that the GM gauge transformation is unphysical when $\omega=0$.
Problems described by nonperturbative methods such as tunneling methods [@kel; @nr; @ppt; @adk], have spectra that are always inclusive of zero frequency. This is straightforward to describe within the LG, but the extension to $\omega\rightarrow0$ defies treatment within the VG.
The failure of the GM gauge transformation has no significance for transverse fields, such as laser fields. Such fields are propagating fields that do not have a zero frequency limit in the same sense as longitudinal fields. Propagating fields have extremely low-frequency radio fields as the limit when $\omega\rightarrow0$ [@hr101; @hrtun]. The limit point of $\omega=0$ cannot be achieved for a variety of (inter-related) reasons: propagation is not defined when $\omega=0$; the magnetic field must always have the same magnitude as the electric field (in Gaussian units), so it can never be set to zero when the electric field is nonzero; $\omega\rightarrow0$ implies wavelength $\lambda\rightarrow\infty$; the ponderomotive energy $U_{p}$ for a transverse field is proportional to $1/\omega^{2}$, so infinite energy must be supplied; there is no gauge freedom at all for propagating fields [@hrjmo; @hrtun]; and so on.
The overall conclusion is that the LG is the sole physical gauge for oscillatory electric fields when the zero-frequency limit must be considered.
I thank Prof. D. Bauer of Rostock University for useful discussions.
[99]{}
D. Bauer, D. B. Milosevic, and W. Becker, Phys. Rev. A **72**, 023415 (2005).
M. Yu. Ivanov, M. Spanner, and O. Smirnova, J. Mod. Opt. **52**, 165 (2005).
S. V. Popruzhenko, J. Phys. B **47**, 204001 (2014).
H. R. Reiss, J. Mod. Opt. **59**, 1371 (2012); **60**, 687 (2013).
H. R. Reiss, J. Phys. B **47**, 204006 (2014).
H. Goldstein, C. Poole, and J. Safko, *Classical Mechanics*, third ed. (Addison-Wesley, San Francisco, 2002).
L. V. Keldysh, Zh. Eksp. Teor. Fiz. **47**, 1945 (1964) \[Sov. Phys.-JETP **20**, 1307 (1965)\].
A. I. Nikishov and V. I. Ritus, Zh. Eksp. Teor. Fiz. **50**, 255 (1966) \[Sov. Phys.-JETP **23**, 168 (1966)\].
A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, Zh. Eksp. Teor. Fiz. **50**, 1393 (1966) \[Sov. Phys.-JETP **23**, 924 (1966)\].
M. V. Ammosov, N. B. Delone, and V. P. Krainov, Zh. Eksp. Teor. Fiz. **91**, 2008 (1986) \[Sov. Phys.-JETP **64**, 1191 (1986)\].
H. R. Reiss, Phys. Rev. Lett. **101**, 043002 (2008); **101**, 159901(E) (2008).
|
---
abstract: |
Recent work on single bubble sonoluminescence (SBSL) has shown that many features of this phenomenon, especially the dependence of SBSL intensity and stability on experimental parameters, can be explained within a [*hydrodynamic approach*]{}. More specifically, many important properties can already be derived from an analysis of [*bubble wall dynamics*]{}. This dynamics is conveniently described by the Rayleigh–Plesset (RP) equation. In this work we derive analytical approximations for RP dynamics and subsequent analytical laws for parameter dependences. These results include (i) an expression for the onset threshold of SL, (ii) an analytical explanation of the transition from diffusively unstable to stable equilibria for the bubble ambient radius (unstable and stable sonoluminescence), and (iii) a detailed understanding of the resonance structure of the RP equation. It is found that the threshold for SL emission is shifted to larger bubble radii and larger driving pressures if surface tension is enlarged, whereas even a considerable change in liquid viscosity leaves this threshold virtually unaltered. As an enhanced viscosity stabilizes the bubbles against surface oscillations, we conclude that the ideal liquid for violently collapsing, surface stable SL bubbles should have small surface tension and large viscosity, although too large viscosity ($\eta_l{\mbox{\
\raisebox{-.9ex}{$\stackrel{\textstyle >}{\sim}$}\ }}40\eta_{\em water}$) will again preclude collapses.
author:
- |
\
SASCHAHILGENFELDT$^1$,MICHAELP.BRENNER$^2$,\
SIEGFRIEDGROSSMANN$^1$,
- DETLEFLOHSE$^1$
date: 22 February 1997 and in revised form 21 January 1998
title: ' Analysis of Rayleigh–Plesset dynamics for sonoluminescing bubbles'
---
Introduction {#secintro}
============
Sonoluminescence {#secsl}
----------------
The analysis of the dynamics of a small bubble or cavity in a fluid dates back to the work of Lord [@ray17] at the beginning of this century. A large number of publications followed in subsequent decades, including the studies of oscillating bubbles by Plesset (1949,1954), [@ell70], Flynn (1975a,1975b), [@lau76; @pro77; @ple77], and others. In recent years, a renascence of bubble dynamics has occurred initiated by the discovery of single bubble sonoluminescence (SBSL) by [@gai90], see also [@gai92].
SBSL is an intriguing phenomenon: A single gas bubble of only a few $\mu m$ size, levitated in water by an acoustic standing wave, emits light pulses so intense as to be visible to the naked eye. The standing ultrasound wave of the driving keeps the bubble in position at a pressure antinode and, at the same time, drives its oscillations. The experiments of Putterman’s group (Barber & Putterman 1991; Barber (1994,1995); Hiller 1994; Löfstedt, Barber & Putterman 1993; Löfstedt 1995; Weninger, Putterman & Barber 1996) and others have revealed a multitude of interesting facts about SBSL: the width of the light pulse is small (Barber & Putterman 1991 give $50\,ps$ as upper threshold, Moran 1995 $10\,ps$ – recent measurements by Gompf 1997 report 100-300$ps$, depending on the forcing pressure and gas concentration in the liquid), the spectrum shows no features such as lines (Hiller, Putterman & Barber 1992; Matula 1995). While the exact mechanism of light emission is still an open issue, almost all suggested theories – see e.g. Löfstedt (1993), Hiller (1992), [@fli89], [@wu93], [@fro94], Moss (1994), Bernstein & Zakin (1995), [@mos97] – agree that temperatures of at least $10^4$-$10^5\,K$ are reached during bubble collapse. This, together with the light intensity, clearly shows that SBSL relies on an extraordinarily powerful energy focusing process.
In our previous publications [@bre95], Brenner (1996a, 1996b), [@hil96], [@bre96b],[@loh97], and [@loh97b] we calculated phase diagrams for bubbles and have focused on the identification of parameter regimes where SBSL occurs. As a scan of the whole multi-dimensional parameter space is by far too expensive for full numerical simulations of the underlying fundamental equations (i.e., Navier–Stokes and advection-diffusion PDEs), it is necessary to introduce approximations. The necessary conditions for SL to occur could be calculated from the dynamics $R(t)$ of the bubble wall, which is – apart from a tiny interval around the bubble collapse – very well described by the Rayleigh–Plesset (RP) equation. We call this approach the [*RP-SL bubble approach*]{}.
The key parameters in an SL experiment are the ambient bubble radius $R_0$ (radius under normal conditions of $1.013\times 10^5\,{\mbox{\it Pa}}=1\,atm$ and $20^\circ$C), the driving pressure amplitude $P_a$, and the gas concentration in the water surrounding the bubble $p_\infty/P_0$, measured by its partial pressure divided by the ambient pressure. Note that $R_0$ is not at the experimenter’s disposal, but adjusts itself by gas diffusion on a slow time scale of seconds. Its size can, however, be measured in experiment, e.g., by Mie scattering techniques as in [@bar95] or by direct microscopic imaging, see [@tia96; @hol96]. On time scales much smaller than those of diffusive processes, e.g. for one period of driving, $R_0$ may be regarded as a constant to high accuracy.
In Hilgenfeldt (1996) we found that the $P_a/P_0 - p_\infty/P_0$ state space is divided into regions where (diffusively) [*stable SL*]{}, [*unstable SL*]{} or [*no SL*]{} are to be expected, in excellent agreement with experimental findings. These results will now be briefly presented in the following subsection.
Stability requirements {#secstab}
----------------------
Stable sonoluminescence is characterized by light emission in each period of driving at precisely the same oscillation phase and precisely the same brightness for millions (and sometimes billions) of cycles. We found that it occurs in a tiny section of the whole parameter space only, and that the calculated domain agrees very well with experimental findings, cf. Hilgenfeldt (1996), [@loh97]. Its boundaries are set by certain dynamical and stability conditions imposed upon the oscillating bubble (Brenner 1995, Brenner 1996a, Hilgenfeldt 1996): (i) The bubble wall velocity during collapse must reach the speed of sound in the gas $c_{g}$ to ensure sufficient energy transfer from the liquid to the gas. (ii) The bubble must be stable towards non-spherical oscillations of its surface which lead to fragmentation. Bubble fragments have meanwhile been experimentally observed by J. Holzfuss (private communication, 1997). (iii) The bubble must be stable towards diffusive processes, i.e., it must not dissolve or grow by rectified diffusion; diffusively growing bubbles show [*unstable SL*]{}. A further requirement of (iv) chemical stability becomes important when the bubble contains molecular gases which are able to dissociate and recombine with liquid molecules (Brenner 1996, Lohse 1997). E.g., the differences in the parameter regimes of SL in air bubbles vs.SL in noble gas bubbles can consistently be accounted for by dissociation of N$_2$ and O$_2$ in an air bubble; these molecular constituents of air are burned, leaving only inert gases in the bubble (the experimental work of Holt & Gaitan 1996 supports this model). We therefore restrict ourselves for simplicity to the case of a bubble filled with argon. An extension to reactive gas mixtures as analysed in [@loh97] is straightforward. Also, we specify the liquid in which the bubble oscillates to be water, as in most SBSL experiments.
(12,11) (0.5,0.5)
Figure \[total\] illustrates how the conditions (i)–(iii) determine domain boundaries in the $P_a$–$R_0$ state space. Criterion (i) means that the Mach number with respect to $c_{g}$ is larger than 1, i.e., |M\_[g]{}|= [|R|c\_[g]{} ]{} 1, \[mach\] and it is fulfilled for bubbles with large enough ambient radii $R_0$ and at large enough forcing amplitude $P_a$, i.e., right of the dashed line in figure \[total\]. The shape stability condition (ii) – see [@ple54; @bir54; @ell70; @str71; @pro77], Brenner (1995) or Hilgenfeldt (1996) for detailed studies on the other hand, limits the parameter domain in which bubbles can stably oscillate to small $R_0{\mbox{\
\raisebox{-.9ex}{$\stackrel{\textstyle<}{\sim}$}\ }}4-5\,\mu m$, within our boundary layer approximation. As the RP SL approach neglects effects of thermal conduction, which has a damping influence on surface oscillations, this upper limit on $R_0$ may be somewhat higher in reality. Holt & Gaitan’s (1996) experimental results seem to give a threshold around $7\,\mu m$. (i) and (ii) together determine the shaded area of potentially sonoluminescing bubbles in figure \[total\]. The actual position of a stable SL bubble in $P_a$–$R_0$ parameter space is determined by condition (iii) for stable diffusive equilibria of the gas inside the bubble and the dissolved gas in the liquid (thick lines in figure \[total\]). These equilibrium lines show negative slope whenever the equilibria are unstable, i.e., bubbles below the line shrink and dissolve, bubbles above the line grow. At large gas concentrations in the liquid (e.g. $p_\infty/P_0\sim 0.2$, left curve), only unstable equilibria are possible in the parameter range of interest. Tiny ratios $p_\infty/P_0\sim 0.002$ (right curve) are necessary for diffusive stability (i.e., the fluid must be strongly degassed). The positive slope of the upper branch of the curve characterizes these bubbles as stable. The computation of diffusive equilibria is explained in §\[subsecdiff\].
Summary of results of the present work {#secsummary}
--------------------------------------
Having identified the parameter regions for SBSL through numerically solving the RP equation, the question arises if one can understand the shape and size of these regions [*analytically*]{}, i.e., by analysing the bubble dynamics equations. In principle, all of the conditions that determine the occurrence of stable/unstable/no SL depend only on properties of bubble dynamics. Therefore, we set out in this work to derive analytical approximations for RP dynamics and subsequently find scaling laws or approximate analytical expressions for our numerical curves presented above, in order to give a clearer insight into the role of different physical processes governing the dynamical equations. Moreover, more practical reasons make analytical expressions highly desirable, as the multi-dimensional parameter space of SBSL experiments cannot be scanned in detail just by numerical solution of the RP equation. Our analytical efforts strongly build on previous work, most notably that of Löfstedt (1993). We present the most important results in this subsection, written such that experimental parameters can be directly inserted to yield numerical values. Here we have used fixed $\om=2\pi\times 26.5\,{\mbox{\it kHz}}$ and $P_0=1\,atm$. More detailed results and the complete derivations for general $\om, P_0$ will be given in the corresponding Sections. All the presented approximations naturally have limited parameter regimes of validity, which include the region of sonoluminescing bubbles in all cases.
We will demonstrate in §\[secrp\] that, in order to understand the location of diffusive equilibria, it is sufficient to analyse the parameter dependence of the ratio of the maximum bubble radius to its ambient radius $(R_{max}/R_0)$, see Löfstedt (1993). We show in §\[sectwo\] that two clearly distinct kinds of bubble dynamical behaviour exist depending on $P_a$ and $R_0$: [*weakly oscillating*]{} and [*strongly collapsing*]{} bubbles. The transition between these two states is rather abrupt and occurs for given $P_a$ at an ambient radius $$R_0^{tr}={4\over 9}\sqrt{3}{\sigma\over P_a-P_0}
\approx {0.562\,\mu m \over P_a/P_0-1} \, .
\label{r0trintro}$$ This transition is controlled by the surface tension $\sigma$, i.e., strong collapses are easier to achieve for small $\sigma$.
In §\[sectour\], we derive analytical approximations to RP dynamics for all phases of the oscillation cycle of a strongly collapsing bubble. We find that in this regime the bubble essentially collapses like an empty cavity (see [@ray17]) according to R(t)14.3m (R\_[max]{}m)\^[3/5]{} ([t\^\*-tT]{})\^[2/5]{} , \[rayintro\] with the time of maximum bubble compression $t^*$ and the driving period $T=2\pi/\om$.
Following the collapse, a series of characteristic afterbounces of the bubble radius occurs. We show in §\[secafter\] that they are the cause for the wiggly structure of the diffusive equilibrium curves and the $|M_{g}|=1$ line in figure \[total\]. The location of the wiggles can be understood as a parametric resonance phenomenon. A Mathieu approximation yields the ambient radius of the $k^{th}$ wiggle as R\_0\^[(k)]{}37.0m + 0.487m ([q\^[5/3]{}-2q+1q\^[5/3]{}-1]{}+2[2-q\^[2/3]{}q\^[2/3]{}]{}) \[resintro\] with the abbreviation $q=(1+P_a/P_0)$.
Section \[secexp\] deals with the bubble expansion. In the regime of strong bubble collapses, an approximate result for the dependence of the maximum radius on $P_a$ and $R_0$ is 67.2+0.112(R\_0m)\^2+99.5(P\_a/P\_0-/2) . \[practrmaxintro\] With $R_{max}$, the location of diffusive equilibria in ($P_a,R_0$) parameter space can be calculated.
A closer discussion of the role of surface tension and viscosity of the liquid $\eta_l$ is presented in §\[secsurfvis\]. In particular, the viscosity of water is so small that it has no significant influence on bubble dynamics. Oscillations are only viscosity-dominated if \_l\^c(1+[0.487mR\_0]{})8.72 ([R\_0m]{})\_[*water*]{} , \[etacintro\] which corresponds to $\eta_l{\mbox{\
\raisebox{-.9ex}{$\stackrel{\textstyle >}{\sim}$}\ }}40\eta_{\em water}$ for typical $R_0$.
Note that these equations are [*not*]{} fit formulas, but are analytically [*derived*]{} from the RP dynamics. They are all verified by comparison to full numerical solutions in the appropriate domains of validity. With these formulas, we are able to understand most of the parameter dependences of SL analytically. Section \[secconcl\] presents conclusions.
Rayleigh–Plesset bubble dynamics {#secrp}
================================
Notation and parameters {#subsecnot}
-----------------------
Since Lord Rayleigh (1917, see Lamb 1932 for earlier references) treated the collapse of an empty cavity in a liquid, a lot of refinement has been done in the modelling of the dynamics of spherical domain walls in liquids. The main step towards bubble dynamics was the introduction of a variable external driving pressure and of the influence of surface tension by [@ple49].
An ODE for the bubble radius can be derived from the Navier–Stokes equations from an approximation valid to the order of $\dot{R}/c_l$, where $\dot{R}$ is the speed of the bubble wall and $c_l$ is the sound speed in the liquid. Following [@pro86], Löfstedt (1993) and many others, we will henceforth denote the following ODE as Rayleigh–Plesset (RP) equation: $$\begin{aligned}
\rho_l \left( R \ddot R + {3\over 2} \dot R^2 \right) &=&
p_{gas}(R,t) - P(t) - P_0
\nonumber \\
&+& {R\over c_l} {{\mbox d}\over {\mbox d}t} p_{gas}(R,t) - 4 \eta_l
{\dot R \over R} - {2\sigma \over R}.
\label{rp}\end{aligned}$$ The left-hand side of this ODE for the bubble radius $R$ consists of dynamical pressure terms already known to Rayleigh ($\rho_l= 1000\, kgm^{-3}$ is the density of water). $P_0=1\,atm$ is the constant ambient pressure, $P(t)$ the ultrasound driving, modelled as a spatially homogeneous, standing sound wave, i.e., P(t) = - P\_a t = - P\_0p t \[eq1\] with the dimensionless forcing pressure amplitude $p\equiv P_a/P_0$ and a fixed frequency of $\om=2\pi\times 26.5\,{\mbox{\it kHz}}$ (period $T\approx 38\,\mu s$), which is a common value in many experiments like those of [@bar94] and Hiller (1992). The wavelength of this sound in water is about $5\,cm$, while the bubble radii treated in this work never exceed $200\,\mu m$. Because of this separation of scales, it is common to assume spatial homogeneity, as stated above. We will refer to the sum of experimentally controllable pressures as the [*external pressure*]{} $p_{ext}=P_0+P(t)$. By definition, the external pressure exerts maximally outward directed forces ($p_{ext}=P_0(1-p)<0$) on the bubble at $t=0$.
The other terms on the right-hand side of equation (\[rp\]) model the influence of the surface tension at the bubble-water interface ($\sigma = 0.073\,kg\,s^{-2}$), the water viscosity ($\eta_l = 1.00\times 10^{-3}\,{\mbox{\it Pa}}\, s$), and of emitted sound waves from the bubble (cf. Keller & Miksis 1980, this term contains the speed of sound in water $c_l=1481\,m\, s^{-1}$).
(12,15.2) (-0.,5.4) (0,-4.1)
The gas pressure $p_{gas}(R,t)$ inside the bubble is assumed to obey a van der Waals type process equation p\_[gas]{}(R,t) = p\_[gas]{}(R(t)) = (P\_0 + [2R\_0]{}) ( [R\_0\^3 - h\^3R\^3(t) - h\^3 ]{})\^, \[vdw\] $R_0$ being the ambient bubble radius and $h$ the (collective) van der Waals hard core radius $h= R_0/8.86$ (for argon) (Lide 1991). The pressure exerted by surface tension was included explicity in (\[rp\]). The $\sigma$ dependence of the prefactor of the polytropic expression ensures that $R_0$ is the radius of a static (unforced) bubble, neglecting effects of gas diffusion. Note that (\[vdw\]) presupposes homogeneity of the pressure inside the bubble. This is of course not satisfied in the final stages of bubble collapse, as a more detailed investigation of the gas dynamics inside the bubble reveals, cf. [@wu93; @mos94; @vuo96; @eva96; @bre96d], Moss (1997), but the violent collapse phase lasts only $\sim 1\,ns$ out of the $T\approx 38\,\mu s$ of the oscillation cycle. Therefore, this approximation does not severely affect our analysis of bubble wall dynamics. We furthermore set the effective polytropic exponent $\kappa \approx 1$ as for this frequency and bubble ambient radii below $\sim20\,\mu m$ the bubbles can be considered to be isothermally coupled to the surrounding liquid (Plesset & Prosperetti 1977), except during the small time interval around the bubble collapse, where the extremely rapid bubble dynamics requires adiabatic treatment of the gas. This will be taken into account in §$\!$§\[seccollapse\] and \[turn\]. The solid line of figure \[roft\]($a$) shows a time series $R(t)$ from (\[rp\]) for relatively strong driving $P_a=1.4\,atm$ and moderate ambient radius $R_0=4.0\,\mu m$. The typical feature of the oscillations of $R(t)$ is a slow expansion for approximately half a cycle of driving, followed by a rapid and violent collapse and a series of afterbounces corresponding to an almost free oscillation of the bubble. The time scale of the afterbounces is thus set by the period of the bubble’s (small amplitude) eigenoscillations, whose frequency $\om_e\sim 1\,{\mbox
{\it MHz}}$ can be easily obtained from a linearization of (\[rp\]): +\_e\^2 (R-R\_0)=[P\_a t\_l R\_0]{} \[fulllin\] \_e\^2=[3P\_0\_l R\_0\^2]{} , \[ome\] where we have set $\kappa=1$ and neglected surface tension and viscosity effects. Including surface tension yields $$\om_s^2={3P_0\over\rho_l R_0^2}+{4\sigma\over \rho_l R_0^3} =\left(1+{2\over
3}\alpha_s\right) \om_e^2 \, ,
\label{omes}$$ where $\alpha_s=2\sigma/(P_0 R_0)$ is the ratio of surface tension pressure to $P_0$ at $R=R_0$. $\alpha_s\approx1$ for $R_0\approx 1.5\mu m$, while for larger $R_0$ it becomes very small.
The [*resonance radius*]{}, on the other hand, is defined as the ambient radius of a bubble with $\om_e=\om$, i.e., R\_[res]{}=([3P\_0\_l \^2]{})\^[1/2]{}105m . \[rres\]
------------------- --------------------------------------------------------------------
[pressure term]{} [definition]{}
\[3pt\] $p_{acc}$ $\rho_l R\ddot{R}$
$p_{vel}$ ${3\over 2}\rho_l\dot{R}^2$
$p_{gas}$ $\left(P_0+{2\sigma\over R_0}\right)\left({R_0^3-h^3\over R^3-h^3}
\right)^\kappa$
$p_{sur}$ ${2\sigma\over R}$
$p_{vis}$ $4\eta_l{\dot{R}\over R}$
$p_{snd}$ ${R\over c_l} \left(P_0+{2\sigma\over R_0}\right)
{{\mbox d}\over {\mbox d}t}
\left({R_0^3-h^3\over R^3-h^3}\right)^\kappa$
$p_{ext}$ $P_0-P_a\cos\om t = P_0(1-p\cos\om t)$
------------------- --------------------------------------------------------------------
: Definition of the pressure terms in the RP equation (\[rp\]) used in this work.[]{data-label="table1"}
For convenience, we list in table \[table1\] the definition of the different pressure terms of (\[rp\]) which will appear throughout this paper.
Besides the solution of the RP equation (\[rp\]), figure \[roft\]($a$) shows time series obtained from other commonly used bubble dynamical equations, namely Flynn’s and Gilmore’s equation, which are discussed in detail in Appendix A. It is obvious that, for bubbles in the SBSL regime, all equations yield very similar $R(t)$ dynamics. It is only upon magnification of the small time interval around the collapse (figure \[roft\]$b$) that the differences between these descriptions of bubble dynamics becomes apparent.
The deviations of the RP, Flynn, and Gilmore equations from each other may become pronounced when the bubble is driven at very high pressure amplitudes such as $P_a=5\,atm$ (cf. Lastman & Wentzell 1981). These pressures are common in cavitation fields, but they are far too high to allow for stable bubbles in SBSL experiments (with the possible exception of SBSL in high magnetic fields described by Young, Schmiedel & Kang 1996). Sonoluminescent bubbles require a driving pressure amplitude in a narrow window $1.1\,atm$${\mbox{\
\raisebox{-.9ex}{$\stackrel{\textstyle<}{\sim}$}\ }}P_a{\mbox{\
\raisebox{-.9ex}{$\stackrel{\textstyle<}{\sim}$}\ }}1.5\,atm$. It is this range of $P_a$ that we will mainly focus on in this work. Only in §\[secexp\] results in the range of cavitation field pressures will briefly be displayed. Direct and indirect measurements of the size of SL bubbles e.g. in [@bar92], Tian (1996), or [@hol96] indicate that typical $R_0$ lie around $5\,\mu m$.
Calculating diffusive equilibria from RP dynamics {#subsecdiff}
-------------------------------------------------
A computation of points of diffusive equilibrium in the $P_a$–$R_0$ plane from first principles requires solution of an advection diffusion PDE with appropriate boundary conditions, coupled to the RP equation. This is numerically far too expensive to allow for a scan of the whole $P_a$–$R_0$ parameter space. In [@bre96] and Hilgenfeldt (1996), we therefore employed the method introduced by [@fyr94] and [@loe95], which is based on the separation of the driving time scale $T$ and the diffusive time scale $\tau_{\em diff}\gg T$. Within this approximation, the task is massively reduced to the solution of the RP equation and the computation of weighted averages of the form \_i = \_0\^T f(t) R\^i(t) [d]{}t / \_0\^T R\^i(t) [d]{}t . . \[av\] The mass flux into or out of the bubble is then proportional to $p_\infty - \left< p_{gas} \right>_{4}$ (see Fyrillas & Szeri 1994). An equilibrium point is characterized by the simple condition p\_= \_[4]{} \[diffeq\] and it is stable if = [d\_[4]{}dR\_0]{} \[diffstab\] is positive.
(12,11) (-0.5,0.5)
Figure \[p4\]$a$ displays $\left< p_{gas} \right>_{4}$ for different $P_a$. The graphs show characteristic wiggles for larger $R_0$ (which can be explained from resonance effects, see §\[secafter\]) and, for large enough $P_a$, a global [*minimum*]{} at some critical $R_0=R_0^c$. If $R_0>R_0^c$, even with no wiggles present, the bubbles are diffusively stable according to the sign of the slope $\beta$. For small $R_0<R_0^c$, all equilibria are unstable, i.e., the bubble either dissolves or grows by rectified diffusion, see [@bla49; @ell64] (the latter case can lead to unstable SBSL, cf. Hilgenfeldt 1996). The possibility of multiple stable equilibria because of the resonance structure was recognized earlier by [@chu88] and [@kam93]. Here we analyse the formal and physical origin of the positive [*overall*]{} slope of $\left< p_{gas}\right>_{4}(R_0)$ for large $R_0$, which is an essential property of stable SBSL bubbles.
In the average $\left< p_{gas} \right>_{4}$ the pressure is weighted with $R^4(t)$ and will therefore be dominated by the value of $p_{gas}$ at $R_{max}$. For large radii, we can neglect the excluded volume $h^3$ in the van der Waals formula and replace (\[vdw\]) by an ideal gas law under isothermal conditions, (1+\_s) [\_0\^T R\_0\^3R(t) [d]{}t \_0\^T R\^4(t) [d]{}t ]{}(1+\_s ) ([R\_0R\_[max]{}]{})\^3 . \[diffp4rmax\] $\xi$ is a prefactor that is due to the different shape of the integrands $R(t)$ and $R^4(t)$. A crude estimate of $\xi$ can be obtained by approximating $R(t)$ by a parabola $\widetilde{R}(t)\sim
R_{max}(1-16t^2/T^2)$ and integrating $\widetilde{R}$ and $\widetilde{R}^4$ over one half cycle from $-T/4$ to $T/4$. This gives $\xi=105/64\approx 1.64$, which is quite accurate in reproducing numerical results. With this saddle point approximation introduced by Löfstedt (1993), the key parameter for diffusive equilibria is the expansion ratio $R_{max}/R_0$. Figure \[p4\] demonstrates the close relation between $\left< p_{gas} \right>_{4}$ and $R_{max}/R_0$ as functions of $R_0$. The expansion ratio displays a maximum at $R_0^c$, corresponding to the minimum of $\left< p_{gas}\right>_{4}$. In order to determine diffusive equilibrium points, one has to look for the intersections of the $\left< p_{gas}\right>_{4}/P_0$ curves in figure \[p4\] with a horizontal line given by $p_\infty/P_0$ (cf. equation (\[diffeq\])). Note that degassing to tiny partial pressures is necessary to achieve equilibria in the $R_0$ range of pure argon SL bubbles; this fact was first realized by [@loe95].
For high enough $P_a$, there are two equilibrium values for $R_0$, the larger one being a stable equilibrium, the smaller one being unstable. If $P_a$ is decreased, $\left< p_{gas}\right>_{4}/P_0$ increases and the equilibria come closer together. This can also be seen in figure \[total\]: for decreasing $P_a$, the $R_0$ values given by the $p_\infty/P_0=0.002$ equilibrium curve approach each other. Eventually, at a certain $P_a$ the stable und the unstable equilibrium coalesce and for smaller $P_a$ no equilibrium is possible. This is reflected in figure \[p4\] by the fact that the whole $\left< p_{gas}\right>_{4}/P_0$ curve lies above $p_\infty/P_0$.
For relatively high gas concentrations such as $p_{\infty}/P_0=0.2$, stable equilibria can only exist for very large $R_0$, where the bubbles are shape unstable. But if the concentration is lowered, e.g.to $p_{\infty}/P_0=0.002$, the stable branch (positive slope in figure \[total\]) enters the region of sonoluminescent bubbles, whereupon stable SL can set in. The occurrence of stable and unstable branches depends on the existence of a minimum in $\left< p_{gas}\right>_{4}$, which in turn necessitates a maximum in $R_{max}/R_0$ (figure \[p4\]$a$ and $b$). Therefore, to analyse the lines of diffusive equilibria in figure \[total\], it is sufficient to explain the maximum of the expansion ratio figure \[p4\]($b$) and its dependence on $R_0$ and $p$; this question will be addressed in §\[secexp\].
Quasistatic Blake threshold {#sectwo}
===========================
The transition from sharply increasing $R_{max}/R_0$ for small $R_0$ to decreasing expansion ratios for large $R_0$ (figure \[p4\]$b$) marks an important boundary between two very different types of bubble dynamics. Consider figure \[strongweak\] where two examples of bubble dynamics for the same $P_a=1.5\,atm$ and only minutely different ambient radii are displayed. The smaller bubble exhibits a weak (although obviously not sinusoidal) oscillation with a maximum expansion ratio $R_{max}/R_0\approx 2$; no collapse is visible. The time series of the larger bubble is almost indistinguishable from the other until $t\approx 0$. But then, a rapid expansion to $R_{max}/R_0\approx 10$ occurs, followed by a strong collapse, the typical dynamics of a sonoluminescing bubble, cf.figure \[roft\]($a$).
(12,11) (-0.5,0.5)
(12,8) (-6.2,-11.2)
Figure \[rminr0\] shows the compression ratio $R_{min}/R_0$ of the minimum radius achieved during bubble oscillation to the ambient radius as a function of $P_a$ and $R_0$. A sharp transition, like in the expansion ratio, is obvious in this graph and it occurs at the same $R_0$. For small $P_a$ and small $R_0$, $R_{min}/R_0$ is near one; we denote such bubbles as [*weakly oscillating*]{}. For large $P_a$ and $R_0$, a horizontal plane at $R_{min}/R_0\approx h/R_0$ indicates collapse to a radius very near the hard core radius. We say that these latter bubbles exhibit [*strong collapses*]{}.
The key to understanding this transition from weakly oscillating to strongly collapsing bubbles lies in the existence of a threshold for spontaneous bubble expansion known as the Blake threshold (Blake 1949, Atchley 1989). It is normally considered for bubbles under static conditions: let us first set $P_a$ (and thus also $p_{ext}$) constant in time, and correspondingly take $R(t)$ to be time-independent. Then the RP equation reduces to $$0 = (P_0 + {2\sigma \over R_0})
\left( {R_0\over R}\right)^3
- p_{ext} - {2\sigma \over R}\, ,
\label{blake}$$ where for $p_{gas}$ again the isothermal ideal gas law was used, which is certainly an excellent approximation for the static situation. For $p_{ext}>0$, equation (\[blake\]) has exactly one solution for positive $R$, and it corresponds to a stable equilibrium. If $p_{ext}<0$ but small in absolute magnitude, two equilibria exist, the one at larger $R$ being unstable, i.e., a bubble with larger radius would grow indefinitely. Finally, at a critical $p_{ext}^B<0$ (Blake threshold pressure, cf.Prosperetti 1984) the two equilibrium points merge and disappear in an inverse tangent bifurcation. In this situation, $p_{gas}$ is always larger than $p_{ext}+p_{sur}$ and (\[blake\]) cannot be fulfilled for any radius. Thus, the assumption of a time-independent $R(t)$ has to be dropped. A dynamical expansion ensues with significant contributions from the dynamical pressure terms on the left-hand side of (\[rp\]).
Returning to the oscillatory driving $p_{ext}=P_0(1-p\cos\om t)$, we notice that the driving period $T=2\pi/\om\approx 40\,\mu s$ is long compared to the time scale of the bubble’s eigenoscillations $2\pi/\om_e\sim 1\,\mu s$. Thus, we can consider the external pressure oscillations as quasistatic and follow Blake’s argument as above. As $p_{ext}<0$ is necessary to cross the Blake threshold, we must require $p>1$ here. Obviously, the most sensitive point in the cycle is $t=0$, where $p_{ext}$ is negative and of magnitude $(p-1)P_0$.
The quasistatic approximation (\[blake\]) describes the [*complete*]{} time series of a weakly oscillating bubble with good accuracy. Rewriting (\[blake\]), we obtain the cubic equation (pt-1)R\^3-[2P\_0]{}R\^2+ (1+[2R\_0P\_0]{})R\_0\^3 = 0 . \[cubicr\] Given a time $t$ for which $p_{ext}<0$, there is a critical $R_0=R_0^{tr}$ above which the two positive real solutions of (\[cubicr\]) become complex. When this happens, the weak oscillation dynamics is no longer a valid description and the transition to strong collapses occurs. For given $p$, the smallest transition radius $R_0^{tr}$ is required for $t=0$. For $R_0^{tr}$, therefore, the discriminant of (\[cubicr\]) at $t=0$ must vanish, i.e., R\_0\^3+[2P\_0]{}R\_0\^2-[3227]{}[\^3P\_0\^3(p-1)\^2]{} = 0. \[discriminant\] After a lengthy but straightforward calculation, the transition ambient radius $R_0^{tr}$ at given $p=P_a/P_0$ is $$\begin{aligned}
R_0^{tr} &=& {2\over 3}{\sigma\over P_0}\left\{
\left(
{2\over (p-1)^2}-1+{2\over (p-1)}\sqrt{{1\over (p-1)^2}-1} \;\,
\right)^{1/3} \right. \nonumber \\
& + & \left.\left(
{2\over (p-1)^2}-1+{2\over (p-1)}\sqrt{{1\over (p-1)^2}-1}\;\,
\right)^{-1/3}
-1\right\} .
\label{r0trans}\end{aligned}$$ Note that $R_0^{tr}$ is a real number for all $p$. In figure \[transition\] the calculated $R_0^{tr}$ from (\[r0trans\]) is compared to the numerical values (identified by the condition $R_{min}/R_0=0.5$). The agreement is very good, the errors at higher $P_a$ being only about $0.01\,\mu m$.
When $R_0$ exceeds $R_0^{tr}$, there is a period of time around $t=0$ where the right-hand side of (\[blake\]) cannot be zero, but must be positive. Then, the dynamical terms neglected so far must become noticeable and a dynamical expansion follows which can only be stopped when $p_{ext}$ has again become large enough to allow for a stable radius equilibrium. When the bubble growth is stopped, the expanded bubble does not experience significant outward directed forces and, consequently, undergoes a violent collapse. If $R_0$ is only slightly larger than $R_0^{tr}$, the time scale separation still holds for a large portion of the cycle, cf. figure \[strongweak\].
(12,11) (0.5,0.5)
It is immediately obvious from (\[discriminant\]) and (\[r0trans\]) that surface tension plays a key role in this transition mechanism from weak oscillations to strong collapses. If $p>1$, weak oscillations at small $R_0$ are dominated by the influence of $\sigma$, whereas strongly collapsing (larger) bubbles are controlled by the properties of dynamical expansion and collapse (cf. §$\!$§\[seccollapse\],\[secexp\]). Note that in a fluid with very small $\sigma$, already bubbles of very small size will show collapses (see also Löfstedt 1995 and Akhatov 1997). It should also be emphasized here that the crucial driving parameter for the transition is $(p-1)$, i.e., the [*difference*]{} of driving pressure amplitude $P_a$ and ambient pressure $P_0$, rather than $P_a$ itself.
In the limit of large forcing $p\gg 1$, (\[discriminant\]) yields the much simpler formula $$R_0^{tr}={4\over 9}\sqrt{3}{\sigma\over P_0}{1\over p-1} \, .
\label{r0tlargep}$$ It can be seen from figure \[p4\] that in this limit the difference between $R_0^{tr}$ (onset of transition) and $R_0^c$ (extremum of expansion and compression ratio) becomes negligibly small. Thus, (\[r0tlargep\]) is also a good approximation to the critical $R_0^c$ we were trying to identify. This is confirmed by figure \[r0c\], from which also the (small) errors of the saddle point approximation (determining $R_0^c$ from $R_{max}/R_0$ instead of $\la p_{gas}\ra_{4}$) can be read off.
(12,11) (0.,0.5)
What is the maximum radius of a bubble weakly oscillating at $R_0^{tr}(p)$? Inserting (\[r0tlargep\]) into (\[cubicr\]) with $t=0$ and expanding to the same order in $1/(p-1)$ gives $$R_{max}={4\over 3}{\sigma\over P_0}{1\over p-1}
\label{rmaxt}$$ in the large $p$ limit. This yields a minimum expansion ratio of $R_{max}/R_0^{tr}=\sqrt{3}$ for the onset of bubble collapse, which is an analytical justification of Flynn’s (1975b) definition of a [*transient cavity*]{}. In that work, a strongly collapsing bubble was characterized by an expansion ratio ${\mbox{\
\raisebox{-.9ex}{$\stackrel{\textstyle >}{\sim}$}\ }}2$.
As the collapse sets in rather abruptly when $R_0$ is enlarged, we expect that $R_0^{tr}$ also marks the transition to bubbles which fulfill the Mach criterion (\[mach\]). Figure \[transition\] shows the $|M_{g}|=1$ line of figure \[total\] together with the $R_0^{tr}(p)$ line according to (\[r0trans\]). Both curves display the same trend, approaching each other at large $p$. The Blake transition occurs for smaller $P_a$ and $R_0$ than those necessary for $|M_{g}|{\mbox{\
\raisebox{-.9ex}{$\stackrel{\textstyle >}{\sim}$}\ }}1$, i.e., for possible light emission. The physical consequence of this is that, upon increasing the driving force, the bubble first emits cavitation noise due to collapses and only afterwards starts to emit light. Indeed, such a sequence of events has been reported by W. Eisenmenger & B. Gompf (private communication, 1996).
The transition line $R_0^{tr}(p)$ is shifted towards smaller $R_0$ for smaller $\sigma$. This means that collapses of the same violence can be achieved (for a given $R_0$ range) with smaller driving pressures in a liquid with less surface tension. Note however that such bubbles will also be stronger affected by surface instabilities, whereas in a liquid with high $\sigma$, bubbles are more surface stable. It is therefore possible to obtain violent collapses at larger $R_0$ in liquids with larger surface tension using larger driving pressures.
A guided tour of RP dynamics {#sectour}
============================
Let us now explain in detail the dynamics of strongly collapsing bubbles (as shown e.g.in figure \[roft\]$a$). To this end, we divide the oscillation cycle of the bubble into several time intervals indicated in figure \[roft\]($b$), where $t_m$ is the time of maximum bubble radius, $t^*$ the time of minimum bubble radius (after collapse), and $t_+=-t_-=\arccos(1/p)/\omega$ the time when $p_{ext}$ changes its sign from positive (contracting) to negative (expanding) values. With this interval division scheme we extend an approach presented in the pioneering paper by Löfstedt (1993). In particular, we will treat the [*bubble collapse phase*]{} denoted by C in figure \[roft\]($b$) in the interval $t_m\leq t\leq t^*$, the [*reexpansion interval*]{} (R) very close to the time of maximum compression ($t\approx t^*$), the [*afterbounces*]{} (AB) for $t^*\leq t\leq t_-$ and the [*bubble expansion*]{} in two stages for $t_-\leq t \leq t_+$ (E$_1$) and $t_+\leq t \leq t_m$ (E$_2$).
Within each of these intervals, certain pressure terms in (\[rp\]) are dominant, whereas others are negligible. Thus, simplified equations with analytical solutions can be derived, which enable us to characterize the complex bubble behaviour analytically and quantitatively. Our approximate formulas hold in the regime of strongly collapsing bubbles, i.e., for $R_0>R_0^{tr}(P_a)$; in the weakly oscillating regime, the bubble dynamics becomes of course trivial.
Rayleigh collapse (region C) {#seccollapse}
----------------------------
We first take a closer look at the main collapse (after $R_{max}$ has been reached, interval C in figure \[roft\]$b$). Figure \[rpcolbefore\] shows the behaviour of the most important terms in the RP equation (defined in table \[table1\]) just prior to the main collapse. The abscissa displays the logarithm of the time interval before the collapse time $t^*$ which is identified by the condition $\dot{R}(t^*)=0$, i.e., the bubble reaches its minimum radius at $t^*$. The ordinate gives the logarithms of the absolute values of the various pressure contributions. As the whole time interval treated in this subsection only comprises $\approx 0.1\,\mu s$, and we want to discuss processes as fast as $1\,ns$, we choose the polytropic exponent in (\[vdw\]) to be $\kappa=5/3$, the adiabatic value for argon. Note that the portions of the graphs for $|t^*-t|{\mbox{\
\raisebox{-.9ex}{$\stackrel{\textstyle<}{\sim}$}\ }}10^{-7}T$ in figure \[rpcolbefore\], as well as in figures \[roftcol\] – \[expsnd\] below, represent time scales on or below the picosecond scale. As hydrodynamics breaks down here, this part of the computation will only be able to give a reasonable effective dynamics. We will take care not to draw physical conclusions from data in this range.
(12,11) (0.,0.5)
In a large part of the collapse phase (figure \[rpcolbefore\]) the dynamical terms $p_{acc}$ and $p_{vel}$ give the dominant contribution; they compensate each other, so that the dynamics is well described by the classical Rayleigh collapse R R + [32]{} R\^2 = 0. \[raycol\] This formula complements the quasistatic approximation (\[blake\]) above. Equation \[raycol\] implies a scaling law for $R(t)$: R(t)= R\_R ([t\^\*-tT]{})\^[2/5]{} . \[scalecol\] Here, the oscillation period $T$ is used for non-dimensionalization of the time coordinate. The characteristic radius $R_R$ can be estimated from an energy argument: at $R=R_{max}$, the potential energy of the bubble is approximately $E_{pot}\sim 4\pi P_0 R_{max}^3/3$, see e.g. [@sme87]. Converting this into kinetic energy of the fluid at $R=R_0$, we get as an estimate for the bubble wall speed at $R=R_0$ |\_[R=R\_0]{}.= -([2P\_03\_l]{})\^[1/2]{} ([R\_[max]{}R\_0]{})\^[3/2]{} . \[r0dot\] Using the time derivative of (\[scalecol\]), we find R\_R=R\_0 ([5T||\_[R=R\_0]{}.|2R\_0]{} )\^[2/5]{} =([25P\_0T\^26\_l]{})\^[1/5]{} R\_[max]{}\^[3/5]{} 14.3m(R\_[max]{}m)\^[3/5]{} . \[rr\] With this $R_R$, (\[scalecol\]) is compared to the numerical result of the RP ODE in figure \[roftcol\]. Both slope and prefactor are reproduced excellently, despite the rather crude approximations leading to (\[rr\]). The only characteristic value for the Rayleigh collapse is $R_{max}$, which depends on $P_a$ and (although weakly) on $R_0$. Analytical expressions for these dependences will be given in Section \[secexp\].
(12,11) (0.,0.2)
We now examine the range of validity of (\[raycol\]); one could worry whether it is justified to neglect $p_{gas}$ and $p_{snd}$ during collapse. For the solution (\[scalecol\]), we have $p_{vel}=-p_{acc}\propto (t^*-t)^{-6/5}$, whereas (as long as $R(t)^3\gg h^3$) $p_{gas}\propto (t^*-t)^{-2}$ and $p_{snd}\propto(t^*-t)^{-13/5}$ for $\kappa=5/3$, i.e., the latter two pressure contributions grow stronger than the dynamical terms as $t\to t^*$. This can also be observed in figure \[rpcolbefore\], but the absolute value of $p_{gas}$ and $p_{snd}$ is negligible compared to $p_{vel}, p_{acc}$ except for times very close to $t^*$. We can compute the range of validity of (\[raycol\]) by equating $p_{gas}=p_{vel}$ and $p_{snd}=p_{vel}$, respectively, using (\[scalecol\]), (\[rr\]). It turns out that the sound pressure contribution is the first to violate (\[raycol\]). This happens at $t_{snd}$ with $$\begin{aligned}
(t^*-t_{snd})/T&=&\left({192\rho_l c_l^2\over 25P_0}\right)^{1/7}
{R_0\over c_lT}
\left(1+\alpha_s\right)^{5/7}
\left({R_0\over R_{max}}\right)^{18/7} \nonumber \\
&\approx& 1.0\times 10^{-4} \left(1+\alpha_s\right)^{5/7}
\left({R_0\over R_{max}}\right)^{18/7}
\left({R_0\over \mu m}\right)\, ,
\label{tsnd}\end{aligned}$$ which agrees with the numerical result e.g. in figure \[rpcolbefore\] (where $R_0=4\,\mu m$ and $R_{max}\approx 47\,\mu m$). For the approximation (\[tsnd\]), $R^3(t_{snd})\gg h^3$ was assumed; $\alpha_s$ is the surface tension parameter introduced in (\[omes\]). The collapse behaviour changes due to $p_{snd}$ shortly before another assumption for (\[raycol\]) breaks down: obviously, $R(t)$ cannot be smaller than the van der Waals hard core $h$. Equating $R(t)=h$ using (\[scalecol\]) with $h=R_0/8.86$, we obtain the “hard core time” (t\^\*-t\_[vdw]{})/T=([6\_l25P\_0]{})\^[1/2]{} [1T]{}[h\^[5/2]{}R\_[max]{}\^[3/2]{}]{} 5.510\^[-6]{} ([R\_0R\_[max]{}]{})\^[3/2]{} ([R\_0m]{}) . \[tvdw\] At $t\approx t_{vdw}$, the van der Waals hard core cuts off the scaling behaviour abruptly. However, for typical values of $R_0\approx 4\,\mu m$, the bubble collapses like an empty cavity for a time interval from $(t^*-t)\sim 1\,\mu s$ down to $(t^*-t)\sim 100\,ps$ ($t^*-t\sim 0.03T\dots 3\times 10^{-6}T$).
Turnaround and delayed reexpansion (region R) {#turn}
---------------------------------------------
As the gas is compressed to the hard core radius, the collapse is halted abruptly. Löfstedt (1993) have shown that – in the Hamiltonian limit neglecting $p_{sur}, p_{vis}, p_{snd}$ and the temporal variation of $p_{ext}$ – the turnaround time interval of the bubble is approximately $$\tau_{turn}\equiv\left({R(t^*)\over\ddot{R}(t^*)}\right)^{1/2}
\approx\left[3^\kappa{\rho_l h^2\over P_0(1+\alpha_s)}
\left({h\over R_0}\right)^{3\kappa}
\left({R_{min}-h\over h}\right)^\kappa\right]^{1/2}\, .
\label{loeturn}$$ This equation also follows from approximating the RP equation (\[rp\]) by keeping only the dominant terms in the immediate vicinity of the collapse, i.e., $p_{acc}$ and $p_{gas}$ (cf.figures \[rpcolbefore\] and \[rpcolafter\]): \_l R R = P\_0(1 + \_s) ( [R\_0\^3 - h\^3R\^3(t) - h\^3 ]{})\^ . \[rpgasacc\] (\[rpgasacc\]) is a good description of bubble dynamics for a time interval around the collapse of length $\sim\tau_{turn}$.
(12,11) (0.,0.5)
Figure \[rpcolafter\] shows the pressure contributions [*after*]{} $t^*$. From $(t-t^*)/T\sim 10^{-6}$ to $(t-t^*)/T\sim 10^{-4}$ (i.e., from $(t-t^*)\sim 30\,ps$ to $\sim 3\,ns$) the dominant terms in (\[rp\]) are $p_{gas}$ and $p_{snd}$, which compensate each other. This means that the energy stored in the compressed gas is released almost exclusively through emission of sound waves (cf. Church 1989) – it is not converted back to kinetic energy of the liquid surrounding the bubble. The corresponding dynamics shows a relatively low expansion velocity and small acceleration, keeping a very small bubble radius for a few $ns$ (figure \[expsnd\]). This time interval of [*delayed reexpansion*]{} (denoted by R in figure \[roft\]$b$) is described by 0 = p\_[gas]{}(R(t)) + [R(t)c\_l]{} [[d]{}t]{} p\_[gas]{}(R(t)) \[gassound\] with $p_{gas}$ given by (\[vdw\]). This ODE has an analytical solution : $${c_l\over 3\kappa} (t-t^*) =
\left[R+{h\over 6} \ln{(R-h)^2\over R^2+h^2+Rh} - {h\over\sqrt{3}}
\arctan {2R+h\over\sqrt{3}h}\right]_{R_{min}}^{R(t)} \, .
\label{soundsol}$$ For $(R(t)-R_{min})\ll R_{min}$, i.e., just after the collapse, this implicit equation can be simplified to yield $$R(t)\approx R_{min} + {c_l\over 3\kappa} {R_{min}^3-h^3\over R_{min}^3}
(t-t^*) \, .
\label{soundapp}$$ This linear expansion law holds for a longer time interval if $R_{min}$ is larger, i.e., for smaller $P_a$. Its validity is demonstrated in figure \[expsnd\]. Note that although the turnaround time $\tau_{turn}$ becomes smaller for decreasing $R_{min}-h$, the velocity of the bubble immediately after collapse is actually [*smaller*]{} because of the larger energy losses through acoustic radiation. The strongly asymmetric shape of $R(t)$ around $t^*$ has also been observed in experimental measurements of bubble dynamics, e.g. by [@bar92], Tian (1996), [@wen97], and [@mat97].
(12,10.7) (0.,0.5)
After the delayed reexpansion phase, the bubble wall gains speed and enters another short time interval around $(t-t^*)\approx 10^{-3}T$ well described by Rayleigh’s equation (\[raycol\]) with $R(t)\propto (t-t^*)^{2/5}$ as the bubble expands. At $(t-t^*)\approx 10^{-2}T$ it enters the phase of subsequent afterbounces.
Afterbounces: a parametric resonance (region AB) {#secafter}
------------------------------------------------
The discussion of the afterbounce interval (AB in figure \[roft\]$b$) is intimately connected to the explanation of the wiggly structure of various dynamically computed terms, like the expansion ratio (figure \[p4\]$b$) or the diffusive equilibrium lines in Fig \[total\]. Obviously, as the RP equation (\[rp\]) describes a driven oscillator, the maxima in the expansion ratio represent parameter values of resonant driving. Figure \[minmax\] clarifies the character of these resonances. It shows two time series of the bubble radius $R(t)$ at values of $R_0$ corresponding to a relative maximum and a relative minimum of $R_{max}/R_0$, respectively. A large or small expansion ratio results from the phase of the afterbounces at the time when $p_{ext}$ becomes negative, i.e., when the external forces start the rapid expansion: for the bubble with the large expansion ratio, the last afterbounce “fits” into the expansion, which is enhanced. For the other bubble, growth is inhibited as the last afterbounce collapse interferes with the expanding external force.
(12,11.3) (0.,0.2)
The afterbounce oscillations show relatively small amplitude, and it is therefore possible to linearize the RP equation in this region of the driving cycle. Moreover, sound radiation and viscosity contributions are negligible. For simplicity, we also neglect $p_{sur}$ for the moment. In order to separate the time scale $1/\om$ of the driving from the much shorter time scale of the afterbounces, which is $\sim 1/\om_e$, we use the ansatz (cf. e.g. Hinch 1991) R(t)=\_0()(1+y(t)) \[separation\] with small $y(t)$ and a slowly varying function $\widetilde{R}_0(\tau)$ which is to be determined; $\tau=\epsilon t, \epsilon=\om/\om_{e}\ll 1$. To leading orders in $y$ and $\epsilon$, equation (\[rp\]) is transformed into \_0\^2 = -\_e\^2[R\_0\^5\_0\^3]{}y + [P\_0\_l]{}([R\_0\^3\_0\^3]{}-(1-p\_[e]{} ) ) \[linrp\] Requiring the slowly varying (secular) term on the right-hand side to vanish, we have to choose \_0()=R\_0/ (1-p (\_[e]{} ))\^[1/3]{} =R\_0/ (1-p (t))\^[1/3]{}. \[r0tilde\] With this definition, (\[linrp\]) results in a Hill equation: +\_e\^2(1-pt)\^[5/3]{} y = 0. \[hill\] Because of the separation of time scales $\om_e\gg\om$ this equation represents a harmonic oscillator with slowly varying eigenfrequency, i.e., the afterbounce frequency $\om_{ab}=\om_e (1-p\cos \om t)^{5/6}$. For this system, $E(\om_{ab})/\om_{ab}$ (with $E=\la y^2\ra \om_{ab}^2/2$ being the oscillator energy) is an adiabatic invariant (see Hinch 1991), i.e., y\^2(1-pt)\^[5/6]{}=const. , \[adinv\] where the mean $\la\cdot\ra$ is an average over the fast time scale. Note that in the time interval $\pi/2{\mbox{\
\raisebox{-.9ex}{$\stackrel{\textstyle<}{\sim}$}\ }}\om t{\mbox{\
\raisebox{-.9ex}{$\stackrel{\textstyle<}{\sim}$}\ }}3\pi/2$ of afterbounces $(1-p\cos\om t)>0$. It follows that the amplitude of the afterbounces changes as $\tilde{R}_0y \propto (1-p\cos \om t)^{-3/4}$.
The resonance structure of (\[hill\]) still cannot be evaluated analytically. Yet the parametric driving of (\[hill\]) has a very similar shape to the cosine driving of a Mathieu equation. We can therefore further approximate (\[hill\]) by choosing suitable constants $Q_1,Q_2$, where we require Q\_1-Q\_2 (t) = (1-p (t))\^[5/3]{} t=[2]{}, , \[equal\] i.e., $Q_1=1$, $Q_2=(1+p)^{5/3}-1$. The errors in this approximation are only a few percent in the time interval $\sim[\pi/2,3\pi/2]$ of afterbounces we focus on. As an analytically accessible approximation to the afterbounce dynamics of (\[rp\]) we have thus the Mathieu equation y”+4[\_e\^2\^2]{}(1-2) y = 0 \[mathieu\] with dimensionless time $\hat{x}=\om t/2$; the primes denote derivatives with respect to $\hat{x}$.
The contribution of surface tension may be included if $\alpha_s\ll 1$ to yield a refinement of (\[mathieu\]): y”+4[\_e\^2\^2]{} ((1+[23]{}\_s) -2) y = 0 , \[mathieusurf\] with the factor $\alpha_s$ from (\[omes\]). Note that a simple substitution $\om_e\to\om_s$ does not cover all first-order effects of $\alpha_s$.
For certain parameter combinations, equation (\[mathieusurf\]) shows parametrically stable or unstable solutions. Because $\om_e/\om\gg 1$, the best analytical approximation to these characteristic values is given by the asymptotic series (Abramowitz & Stegun 1972) b = - [\^2+18]{}-[\^3+32\^6]{} …\[asyseries\] with $b = 4{\om_e^2\over\om^2}[(1-2\alpha_s/3)(1+p)^{5/3}+
4\alpha_s/3\cdot(1+p)],\;
s=8{\om_e^2\over\om^2}[(1-2\alpha_s/3)(1+p)^{5/3}
+4\alpha_s/3\cdot p-1+2\alpha_s/3]$, and $\nu=2k_M\pm1$, where the sign distinguishes even from odd Mathieu solutions. $k_M$ is the order of the Mathieu resonance, corresponding to the number of afterbounces in the RP equation (see below). We take here only the leading term on the right-hand side of (\[asyseries\]) and treat the case $k_M\gg1$, so that $\nu\approx2k_M$; moreover, we only keep terms up to first order in $\alpha_s$. This yields ambient radii $R_0^{(k_M)}$ for which the oscillation shows maximum stability against parametric excitation: $$\begin{aligned}
R_0^{(k_M)}&=&\left({3P_0\over 2\rho_l\om^2}\right)^{1/2}
{q^{5/3}\over\sqrt{q^{5/3}-1}}
{1\over k_M} +{2\sigma\over 3P_0}\left({q^{5/3}-2q+1\over q^{5/3}-1}+
2{2-q^{2/3}\over q^{2/3}}\right) \nonumber \\
&\approx& 74.0\,\mu m \cdot {q^{5/3}\over\sqrt{q^{5/3}-1}}{1\over k_M} +
0.487\,\mu m \cdot \left({q^{5/3}-2q+1\over q^{5/3}-1}+2{2-q^{2/3}\over
q^{2/3}}\right) \, .
\label{resradii}\end{aligned}$$ Here we have abbreviated $q=(p+1)$. Note that the correction term due to surface tension does not depend on $k_M$.
(12,11.3) (0.,0.2)
Although the behaviour of the RP oscillator is fairly well described by Mathieu oscillations in the afterbounce phase (see e.g. figure \[dynmathrp\]), it is of course entirely different during the expansion interval of the cycle. Therefore, some information about the overall shape of the oscillation must enter into our analysis. Especially, Mathieu solutions can be $T$- or $2T$-periodic. RP dynamics in the SL regime, however, only allows for $T$-periodic solutions, as the $2T$-periodic Mathieu solutions would require large negative values for $y$. Therefore, every second resonance of (\[mathieusurf\]) must be dropped, i.e., the resonance of order $k_M$ of (\[mathieusurf\]) corresponds to resonance number $k=k_M/2$ of (\[rp\]), so that the $k^{th}$ resonance radius $R_0^{(k)}$ of (\[mathieusurf\]) for a dynamics with $k$ afterbounces is obtained by replacing $k_M$ by $2 k$.
We must also provide additional information about the length of the afterbounce interval. Figure \[dynmathrp\] shows that this length, which is almost independent of $p$ for our Mathieu approximation, is significantly reduced for increasing $p$ in the case of the RP equation, as the expansion interval lasts longer. This is a property of the nonlinear part of the RP cycle (cf. the next subsection), which cannot be modelled within the Mathieu approximation. In the Appendix it is shown that the length of the afterbounce phase (and therefore the resonance number $k$) has to be rescaled according to $k\to C(p)\times k$ with $C(p)$ approximately given by the expansion $C(p)\approx 0.688-0.548(p-\pi/2)+0.418(p-\pi/2)^2$ (cf. (\[ckexp\])).
(12,11.3) (-0.3,0.5)
Figure \[comprpmath\] presents a comparison of the computed resonance radii of order (number of afterbounces) $k$ from numerical solutions of (\[rp\]) – both for relative maxima and minima of $R_{max} (R_0)$ – and from (\[mathieusurf\]) for driving pressure amplitudes $p=1.2$ and 1.5, corrected with $C(p)$. The resonance locations are in good agreement for both pressures, considering the multitude of approximations they were calculated with.
In the stability maxima marked by $R_0^{(k)}$ the bubble is less excitable by the driving than bubbles with neighboring $R_0$. Therefore the expansion ratio has a local minimum and the average pressure $\la p_{gas}\ra_{4} \propto
1/R_{max}^3$ experiences a local maximum. The existence of such wiggles in $R_{max}/R_0$ (and therefore in $\left< p_{gas}\right>_{4}$) leads to the possibility of multiple equilibria for given experimental parameters $P_a$ and $p_\infty$: the ambient radius can adjust itself diffusively to different stable equilibrium values, depending on initial conditions and/or perturbations. For an analysis of the physical consequences of multiple equilibria we refer the reader to the work of [@bre96], Hilgenfeldt (1996) and [@cru94b].
Bubble Expansion (regions E$_1$, E$_2$) {#secexp}
---------------------------------------
We wish to be more quantitative about the properties of the expansion phase now. Despite the small portion of parameter space for SL bubbles, there are different types of expansion behaviour to be identified depending on $p$ and $R_0$. For $p{\mbox{\
\raisebox{-.9ex}{$\stackrel{\textstyle >}{\sim}$}\ }}1$ and large $R_0{\mbox{\
\raisebox{-.9ex}{$\stackrel{\textstyle >}{\sim}$}\ }}10\,\mu m$, the gas pressure plays an important role and balances the dynamical pressure, which is dominated by $p_{vel}$ for most of the cycle, so that a first approximation to the dynamics is: \_l [32]{} R\^2 = p\_[gas]{}(R,t) \[exp2\] With $p_{gas}(R,t)\approx P_0 R_0^3/R^3$ for large $R_0$, this equation yields a solution for R(t): R(t) = \^[2/5]{} \[sol35\] with the starting time of expansion $t_-$ and starting radius $R_-=R(t_-)$. For radii $R\gg R_-$, (\[sol35\]) reduces to a Rayleigh-type expansion law, which gives a scaling relation for the maximum radius: R\_[max]{} R\_0\^[3/5]{} \[scale35\] if we assume that the length of the expansion interval is independent of $R_0$, which is a good assumption except for small $R_0$ below the Blake threshold (\[r0trans\]). The law (\[scale35\]) can numerically be confirmed for large $R_0$, see Hilgenfeldt (1996). Together with (\[diffp4rmax\]) this yields $\la p_{gas} \ra_{4}\propto R_0^{6/5}$.
For higher driving pressure amplitudes or smaller $R_0$, i.e., in most of the SL parameter region, the approximation (\[exp2\]) is too crude. Instead, one has to take into account the balance of the dynamical pressures $p_{acc}, p_{vel}$ and the external pressure $p_{ext}$ \_l ( R R + [32]{} R\^2 ) = P\_0 (pt - 1) , \[exp1\] as can be seen e.g. in figure \[exp1440\].
(12,11) (-0.2,0.)
In the work of Löfstedt (1993), the left-hand side of (\[exp1\]) has been approximated using a power series for $R(t)$. This leads to a bubble expansion which is linear in time, with a velocity proportional to $\sqrt{p-1}$. The first nonlinear corrections are of fourth order in $t$.
(12,11) (-0.2,0.)
However, an expansion like this turns out to be a series which is not well-controlled. Especially, no reliable values for the time $t_{max}$ and value $R_{max}$ of the maximal radius can be derived (cf.figure \[expana\]). We therefore follow a different ansatz: consider the dynamical terms $p_{acc}$ and $p_{vel}$ for a typical bubble expansion (figure \[exp1440\]). During the first, almost linear part $|p_{vel}|\gg |p_{acc}|$, whereas when the maximum is approached and the bubble decelerates, $|p_{acc}|\gg |p_{vel}|$. This suggests a division of the expansion interval into two parts (denoted by E$_1$ and E$_2$ in figures \[roft\]($b$),\[exp1440\], and \[expana\]). Observing that the combination $R\ddot{R}+\dot{R}^2$ is just the second derivative of $R^2/2$, (\[exp1\]) can be approximated with good accuracy by
[d\^2dx\^2]{}R\^2 &=& [49]{}R\_[res]{}\^2(px -1) + [O]{}(R [d\^2dx\^2]{} R) x\_-xx\_+ , \[expleft\]\
[d\^2dx\^2]{}R\^2 &=& [23]{}R\_[res]{}\^2(px -1) + [O]{}(([ddx]{} R)\^2) x\_+ xx\_m . \[expright\]
Here we have introduced the dimensionless time $x\equiv \om t$ and the linear bubble resonance radius $R_{res}$ from (\[rres\]); $x_m$ is given by $R(x_{m})=R_{max}$. The rational prefactors on the right-hand side make sure that the dominant terms $p_{vel}$ in (\[expleft\]$a$) and $p_{acc}$ in (\[expright\]$b$) are correctly represented, while the other terms gives contributions of the indicated order in each case. The starting time $x_-=x_-(p)$ and the transition time $x_+=x_+(p)$ between both solutions are given by the zeros of $p_{ext}$, i.e., $x_+(p)=-x_-(p)=\arccos{1/p}$. Equations (\[expleft\]$a,b$) can be integrated analytically requiring continuity and differentiability at $x_+$ for the overall solution. To complete the problem, initial conditions at $x_-$ have to be imposed: we set $R_-=R(x_-)=\zeta R_0$ with a parameter $\zeta\sim 1$ whose value is not crucial for the shape of the solution. An estimate of $\zeta$ can be computed from algebraic equations, but not in an explicit form. $\zeta$ lies between $1.2$ and $2.0$ for typical $R_0$ of SL bubbles; for simplicity, we choose $\zeta=1.6$ in all calculations. For the initial velocity, we observe that $x_-$ marks the transition from the afterbounce regime, where the bubble essentially oscillates with its eigenfrequency, to the expansion regime, where the governing time scale is the driving period $T$. Therefore, we set $R'_-=(dR/dx)(x_-)=R_-$, corresponding to $\dot{R}_-=\om R_-$ in dimensional terms.
Figure \[expana\] shows that the shape of the expansion as well as time and value of the maximum are reproduced satisfactorily. From the solutions of (\[expleft\]$a$) in E$_1$ and (\[expright\]$b$) in $E_2$ one obtains a system of equations for $R_{max}$ and $x_{m}$: $$\begin{aligned}
R_{max}^2 &=& R_-^2 \left(1+2(x_m+x_+)\right) \nonumber \\
&+& {2\over 3}R_{res}^2\left[1-p\cos x_m -{1\over 2}(x_m^2-x_+^2)+
{1\over 3}(p\sin x_+-x_+)(x_m+3x_+)\right],
\label{rmaxana}\end{aligned}$$ $$p\sin x_m-x_m +{1\over 3}(p\sin x_+-x_+)+{3R_-^2\over R_{res}^2} = 0 \, .
\label{xmaxana}$$ Note that (\[rmaxana\]), (\[xmaxana\]) give the position and height of the radius maximum [*without*]{} any freely adjustable parameters. The inset of figure \[rmaxtheo\] shows the maximum radii obtained with these formulas for $R_0=5\,\mu m $ and $9\,\mu m$ and $p=1-5$ together with results from a complete RP computation. Apart from the resonance wiggles (cf. Section \[secafter\]), the curves are very well reproduced both within the $p$ regime for SL bubbles and for the much higher pressure amplitudes of cavitation field experiments.
(12,11) (-0.3,0.2)
Equation (\[xmaxana\]) is transcendental, and $R_{max}$ and $x_m$ do not have simple analytical representations. One can, however, derive simple expressions in several limiting cases: if $p\gg 1$, we obtain after a lengthy calculation $$R_{max}\approx\sqrt{2/3}R_{res}(Fp-G)^{1/2}
\label{rmaxlargep}$$ with constants $F=1+5\pi/6=3.618\dots,\, G=19\pi^2/24-1=6.813\dots$ . This formula is a good approximation only if $p{\mbox{\
\raisebox{-.9ex}{$\stackrel{\textstyle >}{\sim}$}\ }}5$. For $p{\mbox{\
\raisebox{-.9ex}{$\stackrel{\textstyle >}{\sim}$}\ }}1$, i.e., the case of interest for sonoluminescent bubbles, we can expand $x_m$ around $x=\pi/2$. Moreover, as $R_{res}\gg R_0$ for SL driving frequencies, we can also neglect the last term on the right-hand side of (\[xmaxana\]). To leading order in ($x_m-\pi/2$), (\[xmaxana\]) then becomes $$x_m = p + {1\over 3}(p\sin x_+-x_+) \, ;
\label{xmapprox}$$ remember that $x_+=x_+(p)=\arccos(1/p)$. For $p\in[1.0,1.5]$, the second term of the right-hand side of this equation is never greater than 0.185$p$ in absolute magnitude, so that $x_m = p$ is a good approximation. Inserting into (\[rmaxana\]) gives
R\_[max]{}\^2 &=& f(p) R\_0\^2 + g(p) R\_[res]{}\^2\
f(p) &=& \^2 (1+2(p+x\_+)) , \[rmaxapprox\]\
g(p) &=& [23]{} .
The second term in (\[rmaxapprox\]a) is much greater than the first; thus, it is [*not*]{} primarily $R_0$ which determines $R_{max}$, but the resonance radius $R_{res}$. With $R_{res}\propto1/\om$, we see that the expansion ratio is (at constant $p$ and $R_0$) roughly inversely proportional to the driving frequency, i.e., upscaling of SL collapses can be achieved by lowering $\om$, which was also seen in experiment by R. E. Apfel (private communication, 1996). In the same way, a higher ambient pressure $P_0$ (while keeping $p=P_a/P_0$ constant) will lead to higher expansion ratios because of the dependence of $R_{res}$ on $P_0$ (see (\[rres\])). A further simplification of (\[rmaxapprox\]) can be obtained from a stringent expansion in $(p-\pi/2)$ and $R_0$, which yields to leading order the simple result =67.2+0.112(R\_0m)\^2+99.5(p-/2) +[O]{}((p-/2)\^2) . \[practicalrmax\] Figure \[rmaxtheo\] shows the very good agreement of (\[rmaxapprox\]) and (\[practicalrmax\]) with full RP dynamics for several $R_0$ over the whole range of pressures in SL experiments. The approximation breaks down only at $p\sim 3$, where $x_m\approx p$ is no longer valid, see inset of figure \[rmaxtheo\]. The expansion ratio is also accurately reproduced for moderate or large $R_0$ by this formula, as seen in figure \[rmaxr0theo\]. The deviations for small $R_0$ are due to neglecting $p_{sur}$, which becomes the dominant influence as $R_0$ approaches $R_0^{tr}$.
(12,10.5) (0.,0.5)
One would therefore like to include the effects of surface tension into (\[rmaxapprox\]). We make the following ansatz: instead of (\[expleft\]$a,b$), we write
[d\^2dx\^2]{}R\^2 &=& [49]{}R\_[res]{}\^2(px - (1+[\_sK(p)]{})) x\_-xx\_+ , \[expleftsurf\]\
[d\^2dx\^2]{}R\^2 &=& [23]{}R\_[res]{}\^2(px - (1+[\_sK(p)]{})) x\_+ xx\_m . \[exprightsurf\]
This models the influence of $p_{sur}$ by an average pressure contribution of $P_0\alpha_s/K(p)$, where $K$ is taken to be independent of $R_0$. Expanding $x_m$ again around $\pi/2$, we get R\_[max]{}\^2 = f(p) R\_0\^2 + R\_[res]{}\^2 \[rmaxapproxsurf\] with $f(p)$, $g(p)$ from (\[rmaxapprox\]$b,c$). With this expression, $(R_{max}/R_0)(R_0)$ shows a global maximum at R\_0\^c(p)=[3P\_0K(p)]{}[[12]{}(p\^2+x\_+\^2)+ [13]{}px\_+g(p)]{} \[r0csurf\] For large enough $p$, we can equate (\[r0csurf\]) and (\[r0tlargep\]), because $R_0^{tr}$ and $R_0^c$ are very close then. This gives an estimate for $K(p)$: K(p)=[94]{}(p-1)[[12]{}(p\^2+x\_+\^2)+ [13]{}px\_+g(p)]{} \[ksurf\] In the regime of SL driving pressures ($p=1.2\dots 1.5$) $K(p)$ depends only weakly on $p$; its value lies between $7.5$ and $9.4$. The ansatz (\[expleftsurf\]$a,b$) proves very successful for the description of $(R_{max}/R_0)(R_0)$ over the whole range of relevant $R_0$, as can be seen from figure \[rmaxr0theo\].
From (\[rmaxapproxsurf\]), we obtain expansion ratios and, using (\[diffeq\]) and (\[diffp4rmax\]) for given gas concentration $p_\infty/P_0$, an approximation for the location $R_0(p)$ of diffusive equilibria can be computed. Figure \[diffeqsurf\] shows that both the stable and the unstable branches of the equilibrium curves (taken from figure \[total\]) are reproduced satisfactorily for both high and low gas concentrations.
(12,11) (0.5,0.2)
When, starting on the stable branch, $p$ is lowered, $R_{max}/R_0$ becomes smaller and, by (\[diffp4rmax\]), $\la p_{gas}\ra_{4}$ becomes larger. The corresponding equilibrium ambient radii $R_0$ shrink. Eventually, the minimum of $\la p_{gas}\ra_{4}/P_0$ becomes larger than $p_\infty/P_0$ (see figure \[p4\]) and no $R_0$ can fulfill the equilibrium condition. This situation corresponds to the turning point of the diffusive equilibria in figure \[total\] and figure \[diffeqsurf\].
Role of surface tension and liquid viscosity {#secsurfvis}
============================================
The previous sections have provided a detailed analysis of the dynamics of SL bubbles. How will these results change if we introduce a different fluid with different surface tension $\sigma$ and/or fluid viscosity $\eta_l$?
Surface tension is the crucial parameter for the location of the Blake threshold in parameter space (cf. also Löfstedt 1995 or Akhatov 1997). The transition from weakly oscillating to strongly collapsing bubbles and therefore the boundary of the SL region determined by (\[mach\]) is entirely controlled by $\sigma$. If we had $\sigma\to 0$, bubbles with any $R_0$ would be strongly collapsing, i.e., liquids with small surface tension should allow for violent collapses at smaller $P_a$. On the other hand, in liquids with high $\sigma$ larger $P_a$ and $R_0$ are required for collapses. Although a larger $\sigma$ has a stabilizing effect on the bubble surface, (\[r0tlargep\]) and (\[r0csurf\]) show that the $|M_{g}|=1$ line overtakes the shape instability threshold in $P_a$–$R_0$ parameter space (cf. Hilgenfeldt 1996), so that no stable SL should be possible if $\sigma$ is e.g. five times higher than in water. This is easily confirmed by the numerical solution of the RP equation, see figure \[nusig\]($a$).
(12,11) (-0.2,0.5)
At first sight it seems that fluid viscosity could have been neglected in (\[rp\]) right from the start. Apart from a slight damping effect during the afterbounce phase, the influence of $\eta_l$ for water on bubble dynamics is hardly noticeable, even a tenfold increase of $\eta_l$ only reduces the maximum radius by $\approx 10\%$ (figure \[etac\]$a$,$b$). We can, however, estimate by how much the viscosity would have to be enhanced to have a significant effect: the damping of a high viscosity liquid should ultimately prevent the bubble from collapsing violently and therefore it will never fulfill the energy transfer condition (\[mach\]). As the collapse is in fact the first afterbounce minimum, an estimate for this critical $\eta_l^c$ can be obtained if we demand the afterbounces to be overdamped. The viscosity $\eta_l$ introduces a damping term in the linearization of the RP equation (\[fulllin\]). It is easy to see that overdamped motion requires \_e(1+[\_s3]{}) . \[etacrit1\] With the definition of $\om_e$, it follows \_l\^c(1+[\_s3]{}) ([34]{}\_l P\_0 R\_0\^2)\^[1/2]{} . \[etacrit2\] With a typical value $R_0=4\mu m$ and keeping $\sigma$, $\rho_l$ at the values for water, we obtain that $\eta_l^c{\mbox{\
\raisebox{-.9ex}{$\stackrel{\textstyle >}{\sim}$}\ }}40\eta_{\em water}$. This is confirmed by direct computation of (\[rp\]) using $\eta_l^c$ and strong driving pressure $P_a=1.4\,atm$, see figure \[etac\]($c$)-($e$). Viscosities in this range can be easily achieved in mixtures of water and glycerine. For moderate glycerine percentage, the viscosity is not very different from pure water, but for high glycerine contents it rises dramatically. The required factor of 40 is (at $10^\circ$C) reached for $\approx 70\%$ glycerine (weight percentage). Above this percentage, it would be extremely difficult to obtain collapses strong enough to ensure energy transfer and the ignition of SL. Moreover, chemical dissociation reactions in air cannot take place, which seem to be necessary for SL stability using air at moderate degassing levels (Lohse 1997). This may be the reason why [@gai90] was not able to observe stable bubbles above a glycerine percentage of $\approx 60\%$. The actual threshold for SL should occur at slightly smaller $\eta_l$ than predicted by (\[etacrit2\]), because even if the collapse minimum is not completely damped out, the collapse is already considerably weakened (figure \[etac\]$c$). Also, the threshold should be higher because of the additional damping effect of thermal dissipation (see Vuong & Szeri 1996, Yasui 1995) which is not included in our approach.
(12,11) (-0.2,0.5)
Even for smaller $\eta_l\sim\eta_{\em water}$, fluid viscosity is an important contribution to the damping of bubble surface oscillations, as was shown in Hilgenfeldt (1996). Therefore, a moderate increase in $\eta_l$ does not lead to significant changes in the $R(t)$ dynamics itself (and therefore in the $|M_{g}|=1$ curve), but it helps to stabilize bubbles at larger $R_0$. This change affects only the parametric and afterbounce instabilities (see Hilgenfeldt 1996), which can show accumulation effects over several driving periods, but not the Rayleigh-Taylor instability, which is directly dependent on the acceleration of the bubble wall and cannot change significantly when the $R(t)$ dynamics does not. Therefore, the Rayleigh-Taylor process still cuts off the bubble stability region (cf. figure 6 of Hilgenfeldt 1996) at $P_a\approx 1.45atm$ almost independent of viscosity, whereas at smaller $P_a$, much larger bubbles can be stable if $\eta_l$ is enhanced.
In figure \[nusig\]($b$), we show surface stability curves for different fluid viscosities. For high $\eta_l$, the region of stably sonoluminescing bubbles between the almost unaffected $|M_{g}|=1$ curve and the increased stability threshold is considerably enlarged. This would probably correspond to a substantial upscaling of SL intensity. Gaitan’s (1990) experimental observation that a moderate percentage of glycerine helps to establish stable SL bubbles supports this conjecture (see also the experimental results of Gaitan 1996 for fluids of different viscosity and surface tension). Combining our results for $\sigma$ and $\eta_l$, we conclude that the ideal fluid for violently collapsing, but surface stable bubbles should have small surface tension and high viscosity.
Conclusions {#secconcl}
===========
The analysis of the RP equation presented here has explained quantitatively quite a lot of features seen in numerical computations of RP dynamics. One cycle of oscillation of this highly nonlinear system can be completely divided into subsections in which its behaviour can be accurately approximated by analytically integrable equations. While being in the spirit of Löfstedt ’s (1993) previous analysis, the present work presents more complete and more detailed results. We emphasize that there are no freely adjustable parameters in our approach.
We made use of these approximations to calculate analytical laws for the bubble’s collapse, afterbounce behaviour, expansion dynamics, maximum radius, and expansion ratio. With these results we could clarify parameter dependences of numerically calculated curves of diffusive equilibria in $P_a$–$R_0$ parameter space as those in figure \[total\]. A summary of relevant analytical relations and predictions for experimental verification was already given in the Introduction.
An approximation of the RP equation by a Mathieu equation has explained the wiggly resonance structure characteristic for many quantities derived from RP dynamics. The concept of a quasistatic Blake threshold between regimes of weakly oscillating and strongly collapsing bubbles was able to shed light on the existence of [*stable*]{} diffusive equilibria in the SL regime for high driving pressures. The change of sign in the slope of $\left< p_{gas}\right>_{4}(R_0)$ is a generic feature of RP dynamics, resulting from the dominance of surface tension pressure at small $R_0$. This allows the bubble to reach a stable diffusive equilibrium.
In all approximations of the RP equation, the fluid viscosity term for water or similar liquids could be neglected without causing large errors. Both numerical computations and analytical estimates of the magnitude of $p_{vis}$ show that $\eta_l$ has to be as high as $\sim 40\,\eta_{\em water}$ to become a dominant contribution to bubble dynamics. Viscosity does, however, have a strong influence on the dynamics of surface oscillations; parametric instabilities are weakened for larger $\eta_l$.
Surface tension is the underlying cause for the change from unstable to stable diffusive equilibria, which stabilizes small bubbles to an extent that they can only show weak oscillations. For fluids with low $\sigma$, collapses of bubbles of a given size are more violent. This is especially interesting if this effect is combined with higher fluid viscosity to establish bubbles which show a similarly violent collapse dynamics as bubbles in water while maintaining larger radii.
Other possibilities for an upscaling of the collapse intensity are the use of lower driving frequencies $\om$ or of larger ambient pressures $P_0$ at the same $P_a/P_0$. These predictions offer a useful guideline to experimenters in search of upscaled single bubble sonoluminescence.
This work has been supported by the DFG uder grant SFB185-D8 and by the joint DAAD/NSF Program for International Scientific Exchange.
Modifications of the RP ODE {#subsecmod}
===========================
As stated in the Introduction, a lot of variations to the RP equation (\[rp\]) are known from literature, see Lastman & Wentzell (1981,1982) for an overview. We mention here the form derived by Flynn (1975a,1975b) $$\begin{aligned}
\rho_l \left[ \left(1-{\dot{R}\over c_l}\right) R \ddot R + {3\over 2}
\left(1-{\dot{R}\over 3c_l}\right) \dot R^2 \right] &=&
\left(1+{\dot{R}\over c_l}\right)\left[p_{gas}(R,t) - P(t) - P_0 - 4 \eta_l
{\dot R \over R} - {2\sigma \over R}\right]
\nonumber \\
&+& \left(1-{\dot{R}\over c_l}\right){R\over c_l} {{\mbox d}\over
{\mbox d}t}
\left[p_{gas}(R,t) - 4 \eta_l {\dot R \over R} - {2\sigma \over R}
\right],
\label{flynn}\end{aligned}$$ which contains correction terms of higher order in $\dot{R}/c_l$. It also includes time derivatives of the surface tension and viscosity terms.
Gilmore’s equation (see e.g. Hickling 1963) differs from the other RP variations in that its key variable is not pressure, but the enthalpy $H$ of the gas at the bubble wall: \_l = (1+[C\_l]{}) H + (1-[C\_l]{})[RC\_l]{} [[d]{}t]{} H. \[gilmore\] Here, the sound speed $C_l$ is not a constant, but depends on $H$. The exact form of this dependence has to be specified by an equation of state for water, e.g. of modified Tait form (Prosperetti & Lezzi 1986, Cramer 1980). Gilmore’s equation was shown in [@hic63] to be an accurate description of a collapsing [*cavity*]{} up to Mach numbers $|M_l|=|\dot{R}|/C_l$ as high as 5. Its validity for the present problem of collapsing gas bubbles is, however, not well established (Prosperetti 1984).
Figure \[roft\]($a$) compares the bubble radius dynamics $R(t)$ computed from Eqs. (\[flynn\]) and (\[gilmore\]) to the solution of (\[rp\]). On this scale, the curves are almost indistinguishable. Only a blow-up of the region around the radius minimum reveals deviations.
As all equations (\[rp\]),(\[flynn\]),(\[gilmore\]) have a common limit for small $M_l$, significant deviations are only to be expected during collapses, when the bubble wall velocity becomes of order of the sound speed. Figure \[roft\]($c$) shows $M_l$ for the different $R(t)$ dynamics. Obviously, large differences occur only around the main collapse, and they are not important for the overall dynamics of the bubble, the collapse time interval being exceedingly small. Thus, the RP equation provides a relatively simple and very accurate description of bubble wall motion, which is confirmed by comparison with recent experimental measurements of $R(t)$ in Tian (1996). Note also that our criterion (\[mach\]) for energy transfer uses $M_{g}$, the Mach number with respect to $c_{g}$, which (for argon) is almost 5 times smaller than $c_l$ (in water). Therefore, the energy transfer threshold can be well computed within the RP-SL approach.
Length of afterbounce interval {#secablength}
==============================
The Mathieu model equation (\[mathieusurf\]) can only be expected to give an accurate description of RP dynamics in the time interval $[\pi/2,3\pi/2]$, for which it was matched to the Hill equation (\[hill\]) via (\[equal\]). Therefore, we have to compare the afterbounce interval of RP dynamics to this constant interval of length $\pi$. Figure \[dynmathrp\] shows that for large driving $p$ the Mathieu dynamics gives a good approximation for the end point of the afterbounce interval (starting point of expansion phase), but fails to model the onset time of afterbounces, i.e., the dimensionless collapse time $x^*$. Thus, the length of the afterbounce interval is smaller than $\pi$ by a factor of $C(p)=(3\pi/2-x^*(p))/\pi$.
The collapse time $x^*$ is relatively close to the maximum time $x_m$. It is therefore convenient to compute it from an expansion of (\[expright\]$b$) in powers of $(x-\pi/2)$ and $(p-\pi/2)$. This yields x\^\*2.55+1.72 (p-/2) - 1.31 (p-/2)\^2 . \[xstar\] The coefficients can be computed analytically, but are of very complicated form. This expression corresponds to a correction factor for the length of the afterbounce interval and, equivalently, for the resonance order $k$, of C0.688 -0.548 (p-/2) + 0.418 (p-/2)\^2 . \[ckexp\] This is the correction introduced in Section \[secafter\] which is important for a satisfactory description of the resonances of (\[rp\]) by (\[mathieusurf\]), see figure \[comprpmath\].
[10]{} 1965 Handbook of Mathematical Functions. Dover.
1997, [*Phys. Rev. Lett.*]{} [**78**]{}, 227.
1989, [*J. Acoust. Soc. Am.*]{} [**85**]{}, 152.
1991, [*Nature*]{} [**352**]{}, 318.
1992, [*Phys. Rev. Lett.*]{} [**69**]{}, 3839.
1995, [*Phys. Rev. Lett.*]{} [**74**]{}, 5276.
1994, [*Phys. Rev. Lett.*]{} [**72**]{}, 1380.
1995, [*J. Phys. Chem.*]{} [**99**]{}, 14619.
1949, Harvard University Acoustic Research Laboratory Tech. Memo. [**12**]{}, 1.
1996 Why air bubbles in water glow so easily. In [*Nonlinear Physics of Complex Systems – Current Status and Future Trends*]{} (ed. J. Parisi, S. C. Müller, & W. Zimmermann). Lecture Notes in Physics, vol. 476, p. 79. Springer.
1996b, [*Phys. Rev. Lett.*]{} [**77**]{}, 3467.
1995, [*Phys. Rev. Lett.*]{} [**75**]{}, 954.
1996a, [*Phys. Rev. Lett.*]{} [**76**]{}, 1158.
1954, [*Quart. Appl. Math.*]{} [**12**]{}, 306.
1988, [*J. Acoust. Soc. Am.*]{} [**83**]{}, 2210.
1989, [*J. Acoust. Soc. Am.*]{} [**86**]{}, 215.
1980\. In [*Cavitation and Inhomogeneities in Underwater Acoustics*]{}, p. 54\
(ed. W. Lauterborn). Springer.
1994\. In [*Bubble dynamics and interface phenomena*]{}, p. 287\
(ed. J. Blake [*et al.*]{}). Kluwer Academic Publishers.
1970, [*J. Acoust. Soc. Am. Suppl.*]{} [**47**]{}, 762.
1964, [*J. Acoust. Soc. Am.*]{} [**37**]{}, 493.
1996, [*Phys. Rev.*]{} E [**54**]{}, 5004.
1989, [*J. Am. Chem. Soc.*]{} [**111**]{}, 6987.
1975a, [*J. Acoust. Soc. Am.*]{} [**57**]{}, 1379.
1975b, [*J. Acoust. Soc. Am.*]{} [**58**]{}, 1160.
1994, [*Phys. Rev. Lett.*]{} [**74**]{}, 2883.
1994, [*J. Fluid Mech.*]{} [**277**]{}, 381.
1990 An experimental investigation of acoustic cavitation in gaseous liquids. Ph.D. thesis, The University of Mississippi.
1992, [*J. Acoust. Soc. Am.*]{} [**91**]{}, 3166.
1996, [*Phys. Rev. E*]{} [**54**]{}, 525.
1997, [*Phys. Rev. Lett.*]{} [**79**]{}, 1405.
1963, [*J. Acoust. Soc. Am.*]{} [**35**]{}, 967.
1996, [*Phys. Fluids*]{} [**8**]{}, 2808.
1992, [*Phys. Rev. Lett.*]{} [**69**]{}, 1182.
1994, [*Science*]{} [**266**]{}, 248.
1991 Perturbation methods. Cambridge University Press.
1994, [*Phys. Rev. Lett.*]{} [**72**]{}, 1376.
1996, [*Phys. Rev. Lett.*]{} [**77**]{}, 3791.
1993, [*J. Acoust. Soc. Am.*]{} [**94**]{}, 248.
1980, [*J. Acoust. Soc. Am.*]{} [**68**]{}, 628.
1932 Hydrodynamics. Cambridge University Press.
1981, [*J. Acoust. Soc. Am.*]{} [**69**]{}, 638.
1982, [*J. Acoust. Soc. Am.*]{} [**71**]{}, 835.
1976, [*J. Acoust. Soc. Am.*]{} [**59**]{}, 283.
1991 CRC Handbook of Chemistry and Physics. CRC.
1997, [*Phys. Rev. Lett.*]{} [**78**]{}, 1359.
1997, [*J. Chem. Phys.*]{}, [**107**]{}, 6986.
1993, [*Phys. Fluids*]{} A [**5**]{}, 2911.
1995, [*Phys. Rev.*]{} E [**51**]{}, 4400.
1995, [*Phys. Rev. Lett.*]{} [**75**]{}, 2602.
1997 The acoustic emissions from single-bubble sonoluminescence. Preprint.
1995, [*Nucl. Instr. and Meth. in Phys. Res.*]{} B [**96**]{}, 651.
1994, [*Phys. Fluids*]{} [**6**]{}, 2979.
1997, [*Science*]{} [**276**]{}, 1398.
1949, [*J. Appl. Mech.*]{} [**16**]{}, 277.
1954, [*J. Appl. Phys.*]{} [**25**]{}, 96.
1977, [*Ann. Rev. Fluid Mech.*]{} [**9**]{}, 145.
1977, [*Quart. Appl. Math.*]{} [**34**]{}, 339.
1984, [*Ultrasonics*]{} [**22**]{}, 69.
1986, [*J. Fluid Mech.*]{} [**168**]{}, 457.
1917, [*Philos. Mag.*]{} [**34**]{}, 94.
1987, [*Phys. Fluids*]{} [**30**]{}, 3342.
1971, [*Acustica*]{} [**25**]{}, 289.
1996, [*J. Acoust. Soc. Am.*]{} [**100**]{}, 3976.
1996, [*Phys. Fluids*]{} [**8**]{}, 2354.
1997, [*Phys. Rev. Lett.*]{} [**78**]{}, 1799.
1996, [*Phys. Rev.*]{} E [**54**]{}, R2205.
1993, [*Phys. Rev. Lett.*]{} [**70**]{}, 3424.
1995, [*J. Acoust. Soc. Am.*]{} [**98**]{}, 2772.
1996, [*Phys. Rev. Lett.*]{} [**77**]{}, 4816.
|
---
abstract: 'The optimization of large portfolios displays an inherent instability to estimation error. This poses a fundamental problem, because solutions that are not stable under sample fluctuations may look optimal for a given sample, but are, in effect, very far from optimal with respect to the average risk. In this paper, we approach the problem from the point of view of statistical learning theory. The occurrence of the instability is intimately related to over-fitting which can be avoided using known regularization methods. We show how regularized portfolio optimization with the expected shortfall as a risk measure is related to support vector regression. The budget constraint dictates a modification. We present the resulting optimization problem and discuss the solution. The L2 norm of the weight vector is used as a regularizer, which corresponds to a diversification “pressure". This means that diversification, besides counteracting downward fluctuations in some assets by upward fluctuations in others, is also crucial because it improves the stability of the solution. The approach we provide here allows for the simultaneous treatment of optimization and diversification in one framework that enables the investor to trade-off between the two, depending on the size of the available data set.'
address:
- 'Information and Computer Sciences, University of Hawaii at M$\bar{\rm a}$noa, Honolulu, Hawaii, USA'
- 'Collegium Budapest–Institute for Advanced Study and Department of Physics of Complex Systems, E[ö]{}tv[ö]{}s University, Budapest, Hungary'
author:
- Susanne Still
- Imre Kondor
bibliography:
- 'RPOsubmit.bib'
title: 'Regularizing Portfolio Optimization.'
---
Introduction
============
Markowitz’ portfolio selection theory [@Markowitz52; @Markowitz59] is one of the pillars of theoretical finance. It has greatly influenced the thinking and practice in investment, capital allocation, index tracking, and a number of other fields. Its two major ingredients are (i) seeking a trade-off between risk and reward, and (ii) exploiting the cancellation between fluctuations of (anti-)correlated assets. In the original formulation of the theory, the underlying process was assumed to be multivariate normal. Accordingly, reward was measured in terms of the expected return, risk in terms of the variance of the portfolio.
The fundamental problem of this scheme (shared by all the other variants that have been introduced since) is that the characteristics of the underlying process generating the distribution of asset prices are not known in practice, and therefore averages are replaced by sums over the available sample. This procedure is well justified as long as the sample size, $T$ (i.e. the length of the available time series for each item), is sufficiently large compared to the size of the portfolio, $N$ (i.e. the number of items). In that limit, sample averages asymptotically converge to the true average due to the central limit theorem.
Unfortunately, the nature of portfolio selection is not compatible with this limit. Institutional portfolios are large, with $N$’s in the range of hundreds or thousands, while considerations of transaction costs and non-stationarity limit the number of available data points to a couple of hundreds at most. Therefore, portfolio selection works in a region, where $N$ and $T$ are, at best, of the same order of magnitude. This, however, is not the realm of classical statistical methods. Portfolio optimization is rather closer to a situation which, by borrowing a term from statistical physics, might be termed the “thermodynamic limit", where $N$ and $T$ tend to infinity such that their ratio remains fixed.
It is evident that portfolio theory struggles with the same fundamental difficulty that is underlying basically every complex modeling and optimization task: the high number of dimensions and the insufficient amount of information available about the system. This difficulty has been around in portfolio selection from the early days and a plethora of methods have been proposed to cope with it, e.g. single and multi-factor models [@eltongruber], Bayesian estimators [@jobson1979; @jorion1986; @frost_savarino; @safePO; @Jagannathan2003; @LedoitWolf2003; @LedoitWolf2004; @LedoitWolfHoney; @DeMiguel2007; @Garlappi2007; @Golosnoy2007; @Kan2007; @frahm_memmel; @DeMiguel2009], or, more recently, tools borrowed from random matrix theory [@Laloux1999; @Plerou1999; @Laloux2000; @Plerou2002; @burda; @Potters2005]. In the thermodynamic regime, estimation errors are large, sample to sample fluctuations are huge, results obtained from one sample do not generalize well and can be quite misleading concerning the true process.
The same problem has received considerable attention in the area of machine learning. We discuss how the observed instabilities in portfolio optimization (elaborated in Section \[instab\]) can be understood and remedied by looking at portfolio theory from the point of view of machine learning.
Portfolio optimization is a special case of regression, and therefore can be understood as a machine learning problem (see Section \[reasons\]). In machine learning, as well as in portfolio optimization, one wishes to minimize the [*actual risk*]{}, which is the risk (or error) evaluated by taking the ensemble average. This quantity, however, can not be computed from the data, only the [*empirical risk*]{} can. The difference between the two is not necessarily small in the thermodynamic limit, so that a small empirical risk does not automatically guarantee small actual risk [@VapnikCh71].
Statistical learning theory [@VapnikCh71; @Vapnik95; @Vapnik98] finds upper bounds on the generalization error that hold with a certain accuracy. These error bounds quantify the expected generalization performance of a model, and they decrease with decreasing [*capacity*]{} of the function class that is being fitted to the data. Lowering the capacity therefore lowers the error bound and thereby improves generalization. The resulting procedure is often referred to as regularization and essentially prevents over-fitting (see Section \[RPO\]).
In the thermodynamic limit, portfolio optimization needs to be regularized. We show in Section \[RPO-ES\] how the above mentioned concepts, which find their practical application in support vector machines [@Boser92; @CortesVapnik95], can be used for portfolio optimization. Support vector machines constitute an extremely powerful class of learning algorithms which have met with considerable success. We show that regularized portfolio optimization, using the expected shortfall as a risk measure, is almost identical to support vector regression, apart from the budget constraint. We provide the modified optimization problem which can be solved by linear programming.
In Section \[diversification\], we discuss the financial meaning of the regularizer: minimizing the L2 norm of the weight vector corresponds to a diversification pressure. We also discuss alternative constraints that could serve as regularizers in the context of portfolio optimization.
Taking this machine learning angle allows one to organize a variety of ideas in the existing literature on portfolio optimization filtering methods into one systematic and well developed framework. There are basically two choices to be made: (i) which risk measure to use, and (ii) which regularizer. These choices result in different methods, because different optimization problems are being solved.
While we focus here on the popular expected shortfall risk measure (in Section \[RPO-ES\]), the variance has a long history as an important risk measure in finance. Several existing filtering methods that use the variance risk measure essentially implement regularization, without necessarily stating so explicitly. The only work we found in this context [@safePO] that mentiones regularization in the context of portfolio optimization has not been noticed by the ensuing, closely related, literature. It is easy to show that when the L2 norm is used as a regularizer, then the resulting method is closely related to Bayesian ridge regression, which uses a Gaussian prior on the weights (with the difference of the additional budget constraint). The work on covariance shrinkage, such as [@Jagannathan2003; @LedoitWolf2003; @LedoitWolf2004; @LedoitWolfHoney], falls into the same category. Other priors can be used [@DeMiguel2009], which can be expected to lead to different results (for an insightful comparison see e.g. [@Tibshirani96]). Using the L1 norm has been popularized in statistics as the “LASSO" (least absolute shrinkage and selection operator) [@Tibshirani96], and methods that use any Lp norm are also known as the “bridge" [@Frank93].
Preliminaries – Instability of classical portfolio optimization. {#instab}
================================================================
Portfolio optimization in large institutions operates in what we called the thermodynamic limit, where both the number of assets and the number of data points are large, with their ratio a certain, typically not very small, number. The estimation problem for the mean is so serious [@chopraziemba1993; @merton1980] as to make the trade-off between risk and return largely illusory. Therefore, following a number of authors [@Jagannathan2003; @LedoitWolf2003; @okhrin2006; @kempf_memmel2006; @Frahm2008], we focus on the minimum variance portfolio and drop the usual constraint on the expected return. This is also in line with previous work (see [@kondor2007] and references therein), and makes the treatment simpler without compromising the main conclusions. An extension of the results to the more general case is straightforward.
Nevertheless, even if we forget about the expected return constraint, the problem still remains that covariances have to be estimated from finite samples. It is an elementary fact from linear algebra that the rank of the empirical $N\times N$ covariance matrix is the smaller of $N$ and $T$. Therefore, if $T < N$, the covariance matrix is singular and the portfolio selection task becomes meaningless. The point $T = N$ thus separates two regions: for $T > N$ the portfolio problem has a solution, whereas for $T < N$, it does not.
Even if $T$ is larger than $N$, but not [*much*]{} larger, the solution to the minimum variance problem is unstable under sample fluctuations, which means that it is not possible to find the optimal portfolio in this way. This instability of the estimated covariances, and hence of the optimal solutions, has been generally known in the community, however, the full depth of the problem has only been recognized recently, when it was pointed out that the average estimation error diverges at the critical point $N = T$ [@pafka2002; @pafka2003; @pafka2004].
In order to characterize the estimation error, Kondor and co-workers used the ratio $q_0^2$ between (i) the risk, evaluated at the optimal solution obtained by portfolio optimization using finite data and (ii) the true minimal risk. This quantity is a measure of generalization performance, with perfect performance when $q_0^2 = 1$, and increasingly bad performance as $q_0^2$ increases. As found numerically in [@pafka2003] and demonstrated analytically by random matrix theory techniques in [@burda2003], the quantity $q_0$ is proportional to $(1 - N/T)^{-1/2}$ and diverges when $T$ goes to $N$ from above.
The identification of the point $N = T$ as a phase transition [@kondor2007; @ciliberti_1] allowed for the establishment of a link between portfolio optimization and the theory of phase transitions, which helped to organize a number of seemingly disparate phenomena into a single coherent picture with a rich conceptual content. For example, it has been shown that the divergence is not a special feature of the variance, but persists under all the other alternative risk measures that have been investigated so far: historical expected shortfall, maximal loss, mean absolute deviation, parametric VaR, expected shortfall, and semivariance [@kondor2007; @ciliberti_1; @ciliberti_2; @hasszan2008]. The critical value of the $N/T$ ratio, at which the divergence occurs, depends on the particular risk measure and on any parameter that the risk measure may depend on (such as the confidence level in expected shortfall). However, as a manifestation of universality, the power law governing the divergence of the estimation error is independent of the risk measure [@kondor2007; @ciliberti_1; @ciliberti_2], the covariance structure of the market [@pafka2004], and the statistical nature of the underlying process [@hasszan2007]. Ultimately, this line of thought led to the discovery of the instability of coherent risk measures [@kondor2008].
Statistical reasons for the observed instability in portfolio optimization {#reasons}
==========================================================================
As mentioned above, for simplicity and clarity of the treatment we do not impose a constraint on the expected return, and only look for the global minimum risk portfolio. This task can be formalized as follows: Given a fixed budget, customarily taken to be unity, given $T$ past measurements of the returns of $N$ assets: $x_i^k$, $i=1, \dots,N$, $k = 1, \dots, T$, and given the risk functional $F({{\bf w} \cdot {\bf x}})$, find a weighted sum (the portfolio), ${{\bf w} \cdot {\bf x}}$,[^1] such that it minimizes the [*actual*]{} risk $$R ({{\bf w}}) = \langle F({{\bf w} \cdot {\bf x}}) \rangle_{p({{\bf x}})},$$ under the constraint that $\sum_i w_i = 1$. The central problem is that one does not know the distribution $p({{\bf x}})$, which is assumed to underly the generation of the data. In practice, one then minimizes the [*empirical*]{} risk, replacing ensemble averages by sample averages: $$R_{\rm emp} ({{\bf w}}) = {1 \over T} \sum_{k=1}^T F({{\bf w} \cdot {{\bf x}^{(k)}}})
\label{emprisk}$$ Now, let us interpret the weight vector as a linear model. The model class given by the linear functions has a [*capacity*]{} $h$, which is a concept that has been introduced by Vapnik and Chervonenkis in order to measure how powerful a learning machine is [@VapnikCh71; @Vapnik95; @Vapnik98]. (In the statistical learning literature, a learning machine is thought of as having a function class at its disposal, together with an induction principle and an algorithmic procedure for the implementation thereof [@Bernhard_thesis]). The capacity measures how powerful a function class is, and thereby also how easy it is to learn a model of that class. The rough idea is this: a learning machine has larger capacity if it can potentially fit more different types of data sets. Higher capacity comes, however, at the cost of potentially over-fitting the data. Capacity can be measured, for example, by the Vapnik-Chervonenkis (VC-) dimension [@VapnikCh71], which is a combinatoric measure that counts how many data points can be separated in all possible ways by any function of a given class.
To make the idea tangible for linear models, focus on two dimensions ($N=2$). For each number of points, $n$, one can choose the geometrical arrangement of the points in the plane freely. Once it is chosen, points are labeled by one of two labels, say “red" and “blue". Can a line separate the red points from the blue points for [*any*]{} of the $2^n$ different ways in which the points could be colored? The VC-dimension is the largest number of points for which this can be done. Two points can trivially be separated by a line. Three points that are not arranged collinear can still be separate for any of the 8 possible labelings. However, for four points this is no longer the case, since there is no geometrical arrangement for which one could not find a labeling that can not be separated by a line. The VC-dimension is 3, and in general, for linear models in $N$ dimensions, it is $N+1$ [@Bernhard_thesis; @Bernhard_Book1].
In the regime in which the number of data points are much larger than the capacity of the learning machine, $h/T << 1$, a small empirical risk guarantees small actual risk [@VapnikCh71]. For linear functions through the origin that are otherwise unconstrained, the VC-dimension grows with $N$. In the thermodynamic regime, where $N/T$ is not very small, minimizing the empirical risk does not necessarily guarantee a small actual risk [@VapnikCh71]. Therefore it is not guaranteed to produce a solution that generalizes well to other data drawn from the same underlying distribution.
In solving the optimizing problem that minimizes the [*empirical*]{} risk, Eq. (\[emprisk\]) in the regime in which $N/T$ is not very small, portfolio optimization [*over-fits*]{} the observed data. It thereby finds a solution that essentially pays attention to the seeming correlations in the data which come from estimation noise due to finite sample effects, rather than from real structure. The solution is thus different for different realizations of the data, and does not necessarily come close to the actual optimal portfolio.
Overcoming the instability {#RPO}
==========================
The generalization error can be bounded from above (with a certain probability) by the empirical error plus a confidence term that is monotonically increasing with some measure of the capacity, and depends on the probability with which the bound holds [@Vapnik79]. Several different bounds have been established, connected with different measures of capacity, see e.g. [@Bernhard_Book1].
Poor generalization and over-fitting can be improved upon by decreasing the capacity of the model [@Vapnik95; @Vapnik98], which helps to lower the generalization error. Support vector machines are a powerful class of algorithms that implement this idea.
We suggest that if one wants to find a solution to the portfolio optimization problem in the thermodynamic regime, then one should not minimize the empirical risk alone, but also constrain the capacity of the portfolio optimizer (the linear model).
How can portfolio optimization be regularized? Portfolio optimization is essentially a regression problem, and therefore we can apply statistical learning theory, in particular the work on support vector regression.
Note first that the capacity of a linear model class for which the length of the weight vector is restricted to $\|w\|^2 \leq A$ has an upper bound which is smaller than the capacity of unconstrained linear models [@Vapnik95; @Vapnik98]. The capacity is minimized when the length of the weight vector is minimized [@Vapnik95; @Vapnik98]. Vapnik’s concept of [*structural risk minimization*]{} [@Vapnik79] results in the support vector algorithm [@Boser92; @CortesVapnik95] which finds the model with the smallest capacity that is consistent with the data, that is the model with smallest $\|w\|^2$. This leads to a convex constrained optimization problem [@Boser92; @CortesVapnik95] which can be solved using linear programming.
Regularized portfolio optimization with the expected shortfall risk measure. {#RPO-ES}
============================================================================
While the original Markowitz’ formulation [@Markowitz52] measures risk by the variance, many other risk measures have been proposed since. Today, the most widely used risk measure, both in practice and in regulation, is Value at Risk (VaR) [@jorion; @riskmetrics]. VaR has, however, been criticized for its lack of convexity, see e.g. [@Artzner99; @embrechts; @acerbi1], and an axiomatic approach, leading to the introduction of the class of coherent risk measures, was put forward [@Artzner99]. Expected shortfall, essentially a conditional average measuring the average loss above a high threshold, has been demonstrated to belong to this class [@acerbi2; @acerbi3; @acerbi4].
Expected shortfall has been steadily gaining popularity in recent years. The regularization we propose here is intended to cure its weak point, the sensitivity to sample fluctuations, at least for reasonable values of the ratio $N/T$.
Choose the risk functional $F(z) = z \theta(z-\alpha_\beta)$, where $\alpha_\beta$ is a threshold, such that a given fraction $\beta$ of the (empirical) loss-distribution over $z$ lies above $\alpha_\beta$. One now wishes to minimize the average over the remaining tail distribution, containing the fraction $\nu :=1 - \beta$, and defines the expected shortfall as $$ES = \min_{\epsilon}\left[\epsilon + \frac{1}{\nu T} \sum_{k=1}^T {1
\over 2} \left(-\epsilon -{{\bf w} \cdot {{\bf x}^{(k)}}}+ |-\epsilon -{{\bf w} \cdot {{\bf x}^{(k)}}}|\right)\right].
\label{ExpS}$$ The term in the sum implements the $\theta$-function, while $\nu$ in the denominator ensures normalization of the tail distribution. It has been pointed out [@Rockafellar] that this optimization problem maps onto solving the linear program: $$\begin{aligned}
&&\min_{{{\bf w}}, {\bf \xi}, \epsilon} \left[ {1 \over T} \sum_{k=1}^{T}
\xi_k + \nu \epsilon \right] \label{CVaR} \\
&{\rm s.t.}\;\;\; & {{\bf w} \cdot {{\bf x}^{(k)}}}+ \epsilon + \xi_k \geq 0; \;\;\; \xi_k; \geq
0 \label{ES-constr} \\
&& \sum_i w_i = 1.\end{aligned}$$ We propose to implement regularization by including the minimization of $\|{{\bf w}}\|^2 $. This can be done using a Lagrange multiplier, $C$, to control the trade-off – as we relax the constraint on the length of the weight vector, we can, of course, make the empirical error go to zero and retrieve the solution to the minimal expected shortfall problem. The new optimization problem reads:
$$\begin{aligned}
&&\min_{{{\bf w}}, {\bf \xi}, \epsilon} \left[ {1 \over 2} \|{{\bf w}}\|^2 + C
\left({1 \over T} \sum_{k=1}^{T} \xi_k + \nu \epsilon \right) \right]
\label{newPO} \\
&{\rm s.t.}\;\;\; & - {{\bf w} \cdot {{\bf x}^{(k)}}}\leq \epsilon + \xi_k; \label{con1}\\
&& \xi_k \geq 0; \;\;\; \epsilon \geq 0;\\
&& \sum_i w_i = 1. \label{b}\end{aligned}$$
The problem is mathematically almost identical to a support vector regression (SVR) algorithm called $\nu$-SVR. There are two differences: (i) the budget constraint is added, and (ii) the loss function is asymmetric. Expected shortfall is an asymmetric version of the $\epsilon$-intensive loss, used in support vector regression, defined as the maximum of $\{0;| f({{\bf x}}) - y| - \epsilon \}$, where $f({{\bf x}})$ is the interpolant, and $y$ the measured value (response). In that sense $\epsilon$ measures an allowable error below which deviations are discarded.[^2]
The use of asymmetric risk measures in finance is motivated by the consideration that investors are not afraid of upside fluctuations. However, to make the relationship to support vector regression as clear as possible, we will first solve the more general symmetrized problem, before restricting our treatment to the completely asymmetric case, corresponding to expected shortfall. In addition, one may argue that focusing exclusively on large negative fluctuations might not be advisable even from a financial point of view, especially when one does not have sufficiently large samples. In a relatively small sample it may happen that a particular item, or a certain combination of items, dominates the rest, i.e. produces a larger return than any other item in the portfolio at each time point, even though no such dominance exists on longer time scales. The probability of such an apparent arbitrage increases with the ratio $N/T$, and when it occurs it may encourage an investor acting on a lopsided risk measure to take up very large long positions in the dominating item(s), which may turn out to be detrimental on the long run. This is the essence of the argument that has led to the discovery of the instability of coherent and downside risk measures [@hasszan2008; @kondor2008].
According to the above, let us consider the general case where positive deviations are also penalized. The objective function, Eq. (\[newPO\]), then becomes $$\min_{{{\bf w}}, {\bf \xi}, \epsilon} \left[ {1 \over 2} \|{{\bf w}}\|^2 + C
\left({1 \over T} \sum_{k=1}^{T} \left( \xi_k + \xi_k^*\right) + \nu
\epsilon \right) \right] \label{gen-RES},$$ and additional constraints have to be added to Eqs. (\[con1\]) to (\[b\]): $$\begin{aligned}
{{\bf w} \cdot {{\bf x}^{(k)}}}\leq \epsilon + \xi_k^*; \;\;\; \xi_k^* \geq 0.\label{con1sym}\end{aligned}$$ This problem corresponds to $\nu$-SVR, a well understood regression method [@Nu-SVM], with the only difference that the budget constraint, Eq. (\[b\]) is added here. In the finance context the associated loss might be called [*symmetric tail average*]{} (STA). Solving the regularized expected shortfall minimization problem, Eqs. (\[newPO\])–(\[b\]) is a special case of solving the regularized STA minimization problem, Eq. (\[gen-RES\]) with the constraints Eqs. (\[con1\])–(\[b\]) and (\[con1sym\]). Therefore, we solve the more general problem first (Section \[RSTA\]), before providing, in Section \[RES-par\], the solution to the regularized expected shortfall, Eqs. (\[newPO\])–(\[b\]).
Regularized Symmetric Tail Average Minimization {#RSTA}
-----------------------------------------------
The solution to the regularized symmetric tail average problem, Eq. (\[gen-RES\]) with the constraints Eqs. (\[con1\])–(\[b\]) and (\[con1sym\]), is found in analogy to support vector regression, following [@Nu-SVM], by writing down the Lagrangean, using Lagrange multipliers, $\{ {\bf \alpha}, {\bf \alpha^*}, \gamma, \lambda, {\bf \eta}, {\bf \eta^*} \}$, for the constraints. The solution is then a saddle point, i.e. minimum over primal and maximum over dual variables. The Lagrangean is different from the one that arises in $\nu$-SVR in that it is modified by the budget constraint: $$\begin{aligned}
\!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! L[{{\bf w}}, {\bf \xi}, {\bf \xi^*}, \epsilon, {\bf \alpha}, {\bf \alpha^*}, \gamma, \lambda, {\bf \eta}, {\bf \eta^*}]
&=&{1 \over 2} \|{{\bf w}}\|^2 + {C \over T} \sum_{k=1}^{T} (\xi_k +
\xi_k^*) + C \nu \epsilon - \lambda \epsilon + \gamma \left( \sum_i
w_i -1 \right)\nonumber \\
&& + \sum_{k=1}^{T} \alpha_k^* ({{\bf w} \cdot {{\bf x}^{(k)}}}- \epsilon - \xi_k^*) -
\sum_{k=1}^{T} \alpha_k ({{\bf w} \cdot {{\bf x}^{(k)}}}+ \epsilon + \xi_k)
\nonumber \\
&& - \sum_{k=1}^{T} (\eta_k \xi_k + \eta_k^* \xi_k^*) \label{L-sym} \\
&=& F[{{\bf w}}] + \epsilon \left( C \nu - \lambda - \sum_{k=1}^{T}
(\alpha_k + \alpha_k^*) \right) -\gamma \label{Lagr-w-constr} \\ &&+
\sum_{k=1}^{T} \left[ \xi_k \left( {C \over T} -\alpha_k -
\eta_k\right) + \xi_k^* \left( {C \over T} - \alpha_k^* - \eta_k^*
\right) \right] \nonumber
\label{L1}\end{aligned}$$ with $$\begin{aligned}
F[{{\bf w}}] &=& {{\bf w}}\cdot \left({1 \over 2} {{\bf w}}- \left(\sum_{k=1}^{T}
(\alpha_k - \alpha_k^*) {{\bf x}^{(k)}}- \gamma {\bf 1}\right) \right),\end{aligned}$$ where ${\bf 1}$ denotes the unit vector of length $N$. Setting the derivative of the Lagrangian w.r.t. ${{\bf w}}$ to zero gives: $$\begin{aligned}
{{\bf w}}_{\rm opt} = \sum_{k=1}^{T} (\alpha_k - \alpha_k^*) {{\bf x}^{(k)}}- \gamma
{\bf 1} \label{w-sym}\end{aligned}$$ This solution for the optimal portfolio is sparse in the sense that, due to the Karush-Kuhn-Tucker conditions (see e.g. [@Bertsekas95]), only those points contribute to the optimal portfolio weights, for which the inequality constraints in (\[con1\]), and the corresponding constraints in Eq. (\[con1sym\]), are met exactly. The solution of ${{\bf w}}_{\rm opt} $ contains only those points, and effectively ignores the rest. This sparsity contributes to the stability of the solution. Regularized portfolio optimization (RPO) operates, in contrast to general regression, with a fixed budget. As a consequence, the Lagrange multiplier $\gamma$ now appears in the optimal solution, Eq. (\[w-sym\]). Compared to the optimal solution in support vector (SV) regression, ${{\bf w}}_{\rm SV}$, the solution vector under the budget constraint, ${{\bf w}}_{\rm RPO}$, is shifted by $\gamma$: $${{\bf w}}_{\rm RPO} = {{\bf w}}_{\rm SV} - \gamma {\bf 1}.$$ Let us now consider the dual problem. The dual is, in general, a function of the dual variables, which are here $\{ {\bf \alpha}, {\bf
\alpha^*}, \gamma, \lambda, {\bf \eta}, {\bf \eta^*} \}$, although we will see in the following that some of these variables drop out. The dual is defined as , and the dual problem is then to maximize $D$ over the dual variables. We can replace the minimization over ${{\bf w}}$ by evaluating the Lagrangian at ${{\bf w}}_{\rm opt}$. For that we have to evaluate $$\begin{aligned}
F[{{\bf w}}_{\rm opt}] &=& - {1 \over 2} \| {{\bf w}}_{\rm opt} \|^2 \\
&=& \left[- {1 \over 2}
\left(\sum_{k=1}^{T} (\alpha_k - \alpha_k^*) {{\bf x}^{(k)}}- \gamma {\bf
1}\right)^2 \right].
$$ For the other terms in the Lagrangian, we have to consider different cases:
1. If $\left( C \nu - \lambda - \sum_{k=1}^{T} (\alpha_k +
\alpha_k^*) \right) < 0$, then $L$ can be minimized by letting $\epsilon \rightarrow \infty$, which means that $D = -\infty$.
2. If $\left( C \nu - \lambda - \sum_{k=1}^{T} (\alpha_k +
\alpha_k^*) \right) \geq 0$: The term $\epsilon \left( C \nu - \lambda
- \sum_{k=1}^{T} (\alpha_k + \alpha_k^*) \right)$ vanishes. Reason: if equality holds, this is trivially true, and if the inequality holds strictly then $L$ can be minimized by setting $\epsilon =0$.
Similarly, for the other constraints (the notation $(*)$ means that this is true for variables with and without the asterisk):
1. If $\left( {C \over T} -\alpha_k^{(*)} - \eta_k^{(*)} \right) <
0$, then $L$ can be minimized by letting $\xi_k^{(*)} \rightarrow
\infty$, which means that $D = -\infty$.
2. If $\left( {C \over T} -\alpha_k^{(*)} - \eta_k^{(*)} \right)
\geq 0$, then $\xi_k \left( {C \over T} -\alpha_k^{(*)} - \eta_k^{(*)}
\right) = 0$. Reason: If the inequality holds strictly then $L$ can be minimized by $\xi_k^{(*)} = 0$. If equality holds then it is trivially true.
By a similar argument, the term $\gamma$ in Eq. (\[Lagr-w-constr\]) disappears in the Dual. Altogether we have that either $D = - \infty$, or $$\begin{aligned}
&& D({\bf \alpha}, {\bf \alpha^*}, \gamma) = \min_{{\bf \xi}, {\bf
\xi^*}, \epsilon} F[{{\bf w}}_{\rm opt}({\bf \alpha}, {\bf \alpha^*},
\gamma)] = - {1 \over 2} \| {{\bf w}}_{\rm opt} \|^2 \\
{\rm and} &\;\;\;& \sum_{k=1}^{T} (\alpha_k^* + \alpha_k) \leq C \nu -
\lambda \\
{\rm and} &\;\;\;& \alpha_k^{(*)} + \eta_k^{(*)} \leq {C \over T}.\end{aligned}$$ Note that the variables $\xi_k^{(*)}, \eta_k^{(*)}, \epsilon, \lambda$ do not appear in $F[{{\bf w}}_{\rm opt}({\bf \alpha}, {\bf \alpha^*},
\gamma)]$. The dual problem is therefore given by $$\begin{aligned}
\max_{{\bf \alpha}, {\bf \alpha^*}, \gamma} && \left[- {1 \over 2}
\left(\sum_{k=1}^{T} (\alpha_k - \alpha_k^*) {{\bf x}^{(k)}}- \gamma {\bf
1}\right)^2 \right]. \\
{\rm s.t.}&& \{ \alpha_k, \alpha_k^* \} \in \left[0,{C \over T}\right] \\
&& \sum_{k=1}^{T} (\alpha_k^* + \alpha_k) \leq C\nu.\end{aligned}$$ We can analytically maximize over $\gamma$ and obtain for the optimal value $$\gamma = {1 \over N} \left(\sum_{k=1}^{T} (\alpha_k - \alpha_k^*)
\sum_{i=1}^N {x^{(k)}_i}- 1 \right) \label{gamma}$$ The optimal projection (= optimal portfolio) is given by $$\!\!\!\!\!\!\!\! \!\!\!\! \!\!\!\! \!\!\!\! \!\!\!\! {{\bf w}}_{\rm opt} \cdot {{\bf x}}= \sum_{k=1}^{T} (\alpha_k - \alpha_k^*) {{\bf x}^{(k)}}\cdot {{\bf x}}- {1 \over N}
\left(\sum_{k=1}^{T} (\alpha_k - \alpha_k^*) \sum_{i=1}^N {x^{(k)}_i}- 1
\right) {\bf1} \cdot {{\bf x}}.
\label{RPO-sol-w}$$ For $N \rightarrow \infty$ the second term vanishes and the solution is the same as the the solution in support vector regression. Note that the kernel-trick (see e.g. [@Bernhard_Book1]), which is used in support vector machines to find nonlinear models hinges on the fact that only dot products of input vectors appear in the support vector expansion of the solution. As a consequence of the budget constraint, one can no longer use the kernel-trick (compare Eq. (\[RPO-sol-w\])). As long as we disregard derivatives, this is not a problem for portfolio optimization. Keep in mind, however, that the budget constraint introduces this otherwise undesirable property.
Support vector algorithms typically solve the dual form of the problem (for a recent survey see [@leon2006]), which is in our case given by $$\begin{aligned}
\!\!\!\!\!\!\!\! \max_{{\bf \alpha}, {\bf \alpha^*}, \gamma} && - {1 \over 2} \left[
\sum_{k=1}^{T} \sum_{l=1}^{T} (\alpha_k - \alpha_k^*)(\alpha_l -
\alpha_l^*) \left( {{\bf x}^{(k)}}{{\bf x}^{(l)}}- {1\over N} \sum_{i=1}^{N} x^{(k)}_i
\sum_{i=1}^{N} x^{(l)}_i \right)
\right] \label{dual-sym-rpo} \\
\!\!\!\!\!\! {\rm s.t.}&& \{ \alpha_k, \alpha_k^* \} \in \left[0,{C \over T}\right]; \nonumber \\
&& \sum_{k=1}^{T} (\alpha_k^* + \alpha_k) \leq C\nu. \nonumber\end{aligned}$$ For $N \rightarrow \infty$ the problem becomes [*identical*]{} to $\nu$-SVR, which can be solved by linear programming, for which software packages are available [@LOQO]. For finite $N$, it can still be solved with existing methods, because it is quadratic in the $\alpha_k$’s. Solvers such as the ones discussed in [@bordes-ertekin-weston-bottou-2005] and [@leon2006] can be used, but have to be adapted to this specific problem.
The regularized symmetric tail average minimization problem (Eq. (\[gen-RES\]) with the constraints Eqs. (\[con1\])–(\[b\]) and (\[con1sym\])) is, as we have shown here, directly related to support vector regression which uses the $\epsilon$-insensitive loss function. The $\epsilon$-insensitive loss is stable to local changes for data points that fall outside the range specified by $\epsilon$. This point is elaborated in Section 3 in [@Nu-SVM], and relates this method to robust estimation of the mean. It can also be extended to robust estimation of quantiles [@Nu-SVM] by scaling of the slack variables $\xi_k$ by $\mu$ and $\xi_k^*$ by $1-\mu$, respectively.
This scaling translates directly to the portfolio optimization problem, which is an extreme case: downside risk measures penalize only loss, not gain. The asymmetry in the loss function corresponds to $\mu =1$.
Regularized expected shortfall. {#RES-par}
-------------------------------
By this final change we arrive at the regularized portfolio optimization problem, Eqs. (\[newPO\])–(\[b\]), which we originally set out to solve. This is now easily solved in analogy to the previous paragraphs: the slack variables $\xi_k^*$ disappear, together with the respective Lagrange multipliers which enforce constraints, including $\alpha_k^*$. The optimal solution is now $$\begin{aligned}
&{{\bf w}}_{\rm opt} = \sum_{k=1}^{T} \alpha_k {{\bf x}^{(k)}}- \gamma {\bf1},\end{aligned}$$ with $$\begin{aligned}
\gamma &=& {1 \over N} \left(\sum_{k=1}^{T} \alpha_k \sum_{i=1}^{N}
{x^{(k)}_i}- 1 \right).\end{aligned}$$ The dual problem is given by $$\begin{aligned}
\max_{\alpha_k}&& - {1 \over 2} \left[ \sum_{k=1}^{T} \sum_{l=1}^{T}
\alpha_k \alpha_l \left( {{\bf x}^{(k)}}{{\bf x}^{(l)}}- {1\over N} \sum_{i=1}^{N} x^{(k)}_i
\sum_{i=1}^{N} x^{(l)}_i \right)
\right] \nonumber \\
{\rm s.t.}&& \alpha_k \in \left[0,{C \over T}\right]; \;\;
\sum_{k=1}^{T} \alpha_k \leq C\nu.\end{aligned}$$ which, like its symmetric counterpart, Eq. (\[dual-sym-rpo\]), can be solved by adjusting existing algorithms.
The formalism provides a free parameter, $C$, to set the balance between the original risk function and the regularizer. Its choice may depend on a number of factors, such as the investors time horizon, the nature of the underlying data, and, crucially, on the ratio $N/T$. Intuitively, there must be a maximum allowable value $C_{\rm max}(N/T)$ for $C$, such that when one puts more emphasis on the data, $C > C_{\rm max}(N/T)$, then over fitting will occur with high probability. It would be desirable to know an analytic expression for (a bound on) $C_{\rm max}(N/T)$. In practice, cross-validation methods are often employed in machine learning to set the value of $C$. Those methods are not free of problems (see, for example, the treatment in [@bengio]), and the optimal choice of this parameter remains an open problem.
Regularization corresponds to portfolio diversification. {#diversification}
========================================================
Above, we have controlled the capacity of the linear model by minimizing the L2 norm of the portfolio weight vector. In the finance context, minimizing $$\| {{\bf w}}\|^2 = \sum_i w_i^2 \simeq {1 \over N_{\rm eff}}$$ corresponds roughly to maximizing the effective number of assets, $N_{\rm eff}$, i.e. to exerting a [*pressure*]{} towards portfolio diversification [@Bouchaudpotters]. We conclude that diversification of the portfolio is crucial, because it serves to counteract the observed instability by acting as a regularizer.
Other constraints that penalizes the length of the weight vector could alternatively be considered as a regularizer, in particular any Lp norm. The budget constraint [*alone*]{}, however, does not suffice as a regularizer, since it does not constrain the length of the weight vector. Adding a ban on short selling, $w_i \geq 0$, to the budget constraint, $\sum_i w_i = 1$, limits the allowable solutions to a finite volume in the space of weights and is equivalent to requiring that $\sum_i | w_i | \leq 1$.[^3] It thereby imposes a limit on the L1 norm, that is on the sum of the absolute amplitudes of long and short positions.
One may argue that it may be a good idea to use the L1 norm instead of the L2 norm, because that may make the solution sparser. However, the L1 norm has a tendency to make some of the weights vanish. Indeed, it has been shown that in the orthonormal design case (using the variance as the risk measure) an L1 regularizer will set some of the weights to zero, while an L2 regularizer will scale all the weights [@Tibshirani96]. The spontaneous reduction of portfolio size has also been demonstrated in numerical simulations [@Kondor4]: as one goes deeper and deeper into the regime where $T$ is significantly smaller than $N$, under a ban on short selling, more and more of the weights will become zero. The same “freezing out" of the weights has been observed in portfolio optimization [@scherermartin] as an empirical fact.
It is important to stress that the vanishing of some of the weights does not reflect any structural property of the objective function, it is just a random effect: as clearly demonstrated by simulations [@Kondor4], for a different sample a different set of weights vanishes. The angle of the weight vector fluctuates wildly from sample to sample. (The behavior of the solutions is similar for other limit systems as well.) This means that the solutions will be determined by the limit system and the random sample, rather than by the structure of the market. So the underlying instability is merely “masked", in that the solutions do not run away to infinity, but they are still unstable under sample fluctuations when $T$ is too small. As it is certainly not in the interest of the investor to obtain a portfolio solution which sets weights to zero on the basis of unreliable information from small samples, the above observations speak strongly in favor of using the L2 norm over the L1 norm.
Conclusion
==========
We have made the observation that the optimization of large portfolios minimizes the empirical risk in a regime where the data set size is similar to the size of the portfolio. In that regime, a small empirical risk does not necessarily guarantee a small actual risk [@VapnikCh71]. In this sense naive portfolio optimization over-fits the data. Regularization can overcome this problem by reducing the capacity of the considered model class.
Regularized portfolio optimization has choices to make, not only about the risk function, but also about the regularizer. Here, we have focussed on the increasingly popular expected shortfall risk measure. Using the L2 norm as a regularizer leads to a convex optimization problem which can be solved with linear programming. We have shown that regularized portfolio optimization is then a variant of support vector regression. The differences are an asymmetry, due to the tolerance to large positive deviations, and the budget constraint, which is not present in regression.
Our treatment provides a novel insight into why diversification is so important. The L2 regularizer implements a pressure towards portfolio diversification. Therefore, from a statistical point of view, diversification is important as it is one way to control the capacity of the portfolio optimizer and thereby to find a solution which is more stable, and hence meaningful.
In summary, the method we have outlined in this paper allows for the unified treatment of optimization and diversification in one principled formalism. It shows how known methods from modern statistics can be used to improve the practice of portfolio optimization.
Acknowledgements
================
We thank Leon Bottou for helpful discussions and comments on the manuscript. This work has been supported by the “Cooperative Center for Communication Networks Data Analysis", a NAP project sponsored by the National Office of Research and Technology under grant No. KCKHA005. SS thanks the Collegium Budapest for hosting her during this collaboration, and the community at the Collegium for providing a creative and inspiring atmosphere.
[^1]: Notation: bold face symbols are understood to denote vectors.
[^2]: The mathematical similarity between minimum expected shortfall [*without*]{} regularization and the E$\nu$-SVM algorithm [@EnuSVM] was pointed out, but incorrectly, in [@Takeda2008]. There is an important difference between the two optimization problems. In E$\nu$-SVM, the length of the weight vector, $\| {{\bf w}}\|$, is constrained, which implements capacity control. In the pure expected shortfall minimization, Eq. (\[CVaR\]), this is not done. Instead, the total budget $\sum_i w_i$ is fixed. This difference is not correctly identified in the proof of the central theorem (Theorem 1) in [@Takeda2008].
[^3]: This point has been made independently by [@DeMiguel2009].
|
---
abstract: 'We demonstrate that a dipolar condensate can be prepared into a three-dimensional wavepacket that remains localized when released in free-space. Such self-bound states arise from the interplay of the two-body interactions and quantum fluctuations. We develop a phase diagram for the parameter regimes where these self-bound states are stable, examine their properties, and demonstrate how they can be produced in current experiments.'
author:
- 'D. Baillie'
- 'R. M. Wilson'
- 'R. N. Bisset'
- 'P. B. Blakie'
title: 'Self-bound dipolar droplet: a localized matter-wave in free space'
---
Localized structures such as solitons are of interest to a wide range of fields from photonics to many-body physics. Three-dimensionally localized light pulses, so-called light bullets, have been realized using fabricated waveguides [@Minardi2010a]. The matter-wave equivalent has been the subject of numerous proposals, including using light-induced gravitational forces [@Giovanazzi2002a; @Giovanazz2001b], off-resonant Rydberg dressing [@Maucher2011a], cold atomic gases with three-body interactions [@Bulgac2002a], and spin-orbit coupled binary condensates [@Zhang2015a]. However, to date none of these schemes have been realized in experiments.
Here we show that it is possible to realize a localized matter wave state in current experiments with dipolar condensates (see Fig. \[figiso\]). Such condensates consist of atoms with appreciable magnetic dipole moments and have been experimentally realized with chromium [@Griesmaier2005a; @Beaufils2008], dysprosium [@Mingwu2011a] and erbium [@Aikawa2012a]. The two-body interaction in this system includes a long-ranged and anisotropic dipole-dipole interaction (DDI) in addition to a short-ranged $s$-wave interaction [@Lahaye_RepProgPhys_2009]. For sufficiently strong dipoles the two-body interaction is partially attractive and the system is susceptible to local collapse instabilities [@Koch2008a; @Wilson2009a; @Lahaye2009a]. However, recent experiments exploring this regime with trapped dipolar condensates have observed the formation of droplet arrays, i.e. the atoms coalesce into a set of small and dense droplets that have long life-times ($\gtrsim\!100\,$ms) [@Kadau2016a; @Ferrier-Barbut2016a; @Bisset2016a]. Recent works demonstrated that quantum fluctuations are most likely responsible for stabilizing these droplets [@Ferrier-Barbut2016a; @Wachtler2016a; @Saito2016a], as they contribute a local energy proportional to $n^{5/2}$, where $n$ is the density, that arrests the two-body driven collapse (proportional to $n^2$) at sufficiently high $n$.
In this work we develop a general theory of self-bound dipolar condensates based on the generalized non-local Gross-Pitaevskii equation (GPE) that includes corrections due to quantum fluctuations. We obtain self-bound states directly using numerical methods [@Ronen2006a] and by an approximate variational approach. This allows us to construct a phase diagram for the regime of interaction parameters and atom number $N$ where self-bound states exist, and to explore the typical properties of these states. Finally, we discuss how these states can be produced in experiments beginning from a trapped dipolar condensate by dynamically adjusting the trapping potential and $s$-wave scattering length. These results show that the lifetimes of the self-bound states are ultimately set by the three-body loss rate, which eventually reduces the atom number to the point when the wavepacket is no longer self-bound.
![(color online) Density isosurfaces illustrating the dynamical production of a self-bound droplet starting from a $^{164}$Dy condensate with $a_s=130a_0$ and $10^4$ atoms. In the dynamics, $a_s$ is quenched to $80a_0$ over $10$ms, and then the trapping potential is turned off over $10$ms. Contours are for a density slice in the $y=0$ plane. Each adjacent contour has a density differing by a factor of 10. See Fig. \[figtime\] for other simulation parameters and details.[]{data-label="figiso"}](fig1.pdf){width="3.4in"}
*Formalism*– The meanfield theory for the dynamics of a dipolar condensate is given by a generalized non-local GPE $$\begin{aligned}
i\hbar\dot{\psi}=\left[-\frac{\hbar^2\nabla^2}{2m}+g|\psi|^2+\Phi(\mathbf{r})+\gamma_{\mathrm{QF}}|\psi|^3\right]\psi(\mathbf{r}),\label{GGPE}\end{aligned}$$ where $\psi$ is the condensate wavefunction. Here $g=4\pi a_s\hbar^2/m$ is the $s$-wave coupling constant with $a_s$ being the $s$-wave scattering length. The DDIs are described by the term $\Phi(\mathbf{r})=\int d\mathbf{r}'U_{\mathrm{dd}}(\mathbf{r}-\mathbf{r}')|\psi(\mathbf{r}')|^2$, where $U_{\mathrm{dd}}(\mathbf{r})=\frac{\mu_0\mu^2}{4\pi r^3}(1-3\cos^2\theta)$ and $\theta$ is the angle between $\mathbf{r}$ and the polarization axis of the dipoles, which we take to be the $z$ direction. To leading order the quantum fluctuation correction to the meanfield energy for a uniform dipolar condensate is $\Delta E=\frac{2}{5}\gamma_{\mathrm{QF}}n^{5/2}$ [@Lima2011a], with coefficient [@Ferrier-Barbut2016a; @Bisset2016a] $$\gamma_{\mathrm{QF}}=\frac{32}{3}g\sqrt{\frac{a_s^3}{\pi}}\left(1+\frac{3}{2}\epsilon_{\mathrm{dd}}^2\right).$$ Here $\epsilon_{\mathrm{dd}}=a_{\mathrm{dd}}/a_s$ is the ratio of DDI to $s$-wave interaction strengths and $a_{\mathrm{dd}}=m\mu_0\mu^2/12\pi\hbar^2$ is the [dipole length]{} [@Lahaye_RepProgPhys_2009]. In Eq. (\[GGPE\]) these fluctuations are included via the associated chemical potential shift $\Delta\mu=\gamma_{\mathrm{QF}}n^{3/2}$ making the local density approximation $n^{3/2}\to|\psi|^3$. The applicability of generalized GPE (\[GGPE\]) to dipolar condensates in the regime we consider here has been discussed in Refs. [@Wachtler2016a; @Saito2016a; @Bisset2016a].
![(color online) Phase diagram of self-bound solutions as a function of $1/\epsilon_{\mathrm{dd}}$ and $N$ calculated using the variational approach. The colors show the energy of the solutions. The thick black line corresponds to $E=0$. Colored regions above this line are metastable (i.e. $E>0$) and in the white region only the trivial dispersed solution exists. Stability thresholds from GPE calculations are indicated by circles (also see Fig. \[figDropProps\]). The inset shows isodensity contours of a self-bound GPE (parameters indicated by arrow). The contours have the same scale as Fig. \[figiso\]. []{data-label="figPD"}](fig2){width="\columnwidth"}
Equation (\[GGPE\]) possesses a continuous translational symmetry (due to the absence of an external trapping potential) and has a trivial uniform stationary state $\psi=\sqrt{n}$ with zero energy (noting that $n\to0$ for fixed $N$). Here our interest is in localized (self-bound) stationary solutions to Eq. (\[GGPE\]). If these localized solutions have negative energy then they are thermodynamically stable with respect to the trivial uniform solution.
A useful description of the system is furnished by a Gaussian variational ansatz for the condensate wavefunction $$\psi_{\mathrm{v}}(\mathbf{r})=\sqrt{\tfrac{8N}{\pi^{3/2}\sigma_\rho^2\sigma_z}}e^{-2(\rho^2/\sigma_\rho^2+z^2/\sigma_z^2)},$$ where $\sigma_\rho$ and $\sigma_z$ are the variational width parameters and we have utilized the cylindrical symmetry of the system around the $z$ axis. The equilibrium width parameters can be determined by finding minima of the energy functional associated with Eq. (\[GGPE\]), which has the form [@Bisset2016a] $$\begin{aligned}
\frac{E(\sigma_\rho,\sigma_z)}{E_0/N}&=\frac{2}{\bar{\sigma}^2_\rho}+\frac{1}{\bar{\sigma}^2_z}+\frac{8}{\sqrt{2\pi}\bar{\sigma}_\rho^2\bar{\sigma}_z}\left[\frac{1}{\epsilon_{\mathrm{dd}}}-f\left(\frac{\bar{\sigma}_\rho}{\bar{\sigma}_z}\right)\right]\nonumber \\
&+c\frac{1+\frac{3}{2}\epsilon_{\mathrm{dd}}^2}{\bar{\sigma}_\rho^3\bar{\sigma}_z^{3/2}N\epsilon_{\mathrm{dd}}^{5/2}},\label{Evar}\end{aligned}$$ where $c=2^{14}/75\sqrt{5}\pi^{7/4}\approx13.18$ and $
f(x)=\frac{1+2x^2}{1-x^2}-\frac{3x^2\mathrm{arctanh}\sqrt{1-x^2}}{(1-x^2)^{3/2}}$. Here $E_0=\hbar^2/ma_{\mathrm{dd}}^2$ and $L_0=Na_{\mathrm{dd}}$ are convenient units of energy and length, and we define $\bar{\sigma}_\nu\equiv\sigma_\nu/L_0$.
*Equilibrium results*– The form of Eq. (\[Evar\]) reveals that solution properties only depend on the parameters $N$ and $1/\epsilon_{\mathrm{dd}}$. In terms of these parameters the energy of non-trivial solutions \[i.e. local minima to (\[Evar\])\] are shown in Fig. \[figPD\] as a “phase diagram” for the existence of these solutions. We mark the phase boundary (the binodal line, where $E=0$) with a thick black line, below which these solutions are the stable ground state. We note that localized solutions persist slightly beyond this region as meta-stable states, until they reach the spinodal line.
![(color online) Properties of self-bound solutions. Widths $\{\sigma_\rho,\sigma_z\}$ as a function of (a) $1/\epsilon_{\mathrm{dd}}$ for $N=10^3$ (black), $10^4$ (gray), and $10^5$ (light gray) and (b) $N$ for $1/\epsilon_{\mathrm{dd}} = 0.535$ (black) and $0.765$ (gray). Peak number density as a function of (c) $1/\epsilon_{\mathrm{dd}}$ and (d) $N$. Variational predictions are shown as solid and dashed lines, and GPE results are indicated as circles. The inset to (d) shows the tip of the $\epsilon_{\mathrm{dd}}^{-1}=0.765$ results with a linear horizontal axis to better reveal the difference in variational and GPE predictions for the minimum atom number. The two choices of $\epsilon_{\mathrm{dd}}$ in (b) and (d) correspond to $a_s = 70.0$ and $100 a_0$ for Dy, and $a_s = 35.6$ and $50.9 a_0$ for Er. []{data-label="figDropProps"}](fig3){width="\columnwidth"}
The maximum value of $\epsilon_{\mathrm{dd}}^{-1}$ at which the localized state becomes unstable increases with $N$, approaching unity as $N$ gets large. Equivalently, for any given value of $\epsilon_{\mathrm{dd}}^{-1}<1$ there will be a minimum value of $N$ below which the localized solution becomes unstable. We show later that this minimum number has important implications when atomic loss is accounted for and can limit the lifetime of the self-bound state.
Other properties of the droplet solutions are considered in Fig. \[figDropProps\]. Here we show results from both variational solutions and numerically determined stationary solutions of the generalized GPE (\[GGPE\]). Figure \[figDropProps\](a) shows that the self-bound solutions are elongated along $z$ (i.e. $\sigma_z\gg\sigma_\rho$) since in this configuration the DDI-term \[i.e. $-f$ term in Eq. (\[Evar\])\] becomes negative, arising from the dominant attractive head-to-tail interaction between dipoles. As $\epsilon_{\mathrm{dd}}^{-1}$ increases the widths monotonically increase until the spinodal stability threshold is reached, where they diverge. Considering the widths as a function of $N$ in Fig. \[figDropProps\](b) reveals that the widths monotonically increase with $N$ for sufficiently large $N$. However, as $N$ approaches the minimum value (where the lines terminate), the widths start increasing with decreasing $N$. This occurs because at low $N$ the kinetic energy (quantum pressure) becomes important and can destabilise the self-bound state.
Figure \[figDropProps\](c) shows that the maximum density $n_{\max}$ decreases with increasing $\epsilon_{\mathrm{dd}}^{-1}$. As $\epsilon_{\mathrm{dd}}^{-1}$ increases, the two-body interactions become less attractive and the quantum fluctuation term is able to balance their effect at low $n$. This result also shows that the value of the diluteness parameter $na_{\mathrm{dd}}^3$ remains $\lesssim10^{-2}$ over the parameters considered here. Considering $n_{\max}$ as a function of $N$ in Fig. \[figDropProps\](d) reveals that except near the minimum number, the droplet density saturates, i.e. adding more particles to the system barely changes $n_{\max}$, but instead causes the droplet to expand. This behaviour is reminiscent of liquid states of matter.
![(color online) Droplet density profiles for $N=10^4$. (a) Self-bound solution with $E=-5.2\times10^{-3}E_0$. (b) Solution near the instability threshold with $E=0$. (c) Axial density slices of the self-bound solution shown in (a) for $\rho=0$ (black), $\rho=20a_\mathrm{dd}$ (gray) and $\rho=30a_\mathrm{dd}$ (light gray). (d) Axial density slices of the solution shown in (b) for $\rho=0$ (black), $\rho=50a_\mathrm{dd}$ (gray) and $\rho=100a_\mathrm{dd}$ (light gray). Note the different scales on the axes for the two results. The values of $1/\epsilon_{\mathrm{dd}}$ are chosen to correspond to $a_s = 50.0$ and $98.0 a_0$ for Dy, and $a_s = 25.4$ and $49.9 a_0$ for Er. []{data-label="figGpeWfns"}](fig4){width="\columnwidth"}
Examples of GPE solutions are shown in Fig. \[figGpeWfns\]. Notably, the self-bound solution shown in Fig. \[figGpeWfns\](a) is more deeply bound, i.e. is in a regime where the maximum density has saturated as a function of $N$. As a result this droplet has a flat density profile along the $z$ axis \[see Fig. \[figGpeWfns\](c)\]. In contrast Fig. \[figGpeWfns\](b) shows a self-bound solution with $E=0$, at the threshold of metastability. This state is much larger, has a significantly lower peak density, and does not exhibit density saturation effects \[see Fig. \[figGpeWfns\](d)\].
![(color online) Droplet formation for $^{164}$Dy starting with $N_0=10^4$ atoms (plus $T\!=\!20$nK initial noise) in a harmonic trapping potential with $\omega_\rho=2\pi\times 70$Hz and $\lambda = 0.75$. Different line colors indicate the final $a_s$ value \[see inset to (b)\]. (a) peak density, colored ticks by the vertical axis indicate the equilibrium critical peak density, (b) total number (solid) and number within the cylinder of diameter $3\:\mu$m and height $10\:\mu$m centered at the location of $n_{\mathrm{peak}}$ (dashed), colored ticks indicate the equilibrium critical droplet number from Fig. \[figPD\], (c) oscillation of the RMS widths $\sqrt{\langle r_i^2\rangle -\langle r_i \rangle^2}$ for $r_i=z$ (solid), $r_i=x$ (dashed) and $r_i=y$ (dash-dotted) evaluated using the field within the cylinder. The inset to (b) shows the time sequence used: linear $a_s$ quench to $[60,70,80,100,130]a_0$ over $\tau_{s}=10$ms, followed by a $\tau_{\mathrm{trap}}=10$ms linear ramp of the trap frequencies to zero. For $t\ge20$ms the system evolves in free space.[]{data-label="figtime"}](fig5){width="\columnwidth"}
*Dynamical Results*– We now turn to considering how a self-bound droplet can be obtained from the type of trapped condensate typically prepared in experiments. From this initial state we propose using a sequence of two ramps \[see inset to Fig. \[figtime\](b)\] to produce the desired state in free space: (i) the $s$-wave scattering length is reduced over a time scale $\tau_{s}$ using a Feshbach resonance until $\epsilon_{\mathrm{dd}}^{-1}$ reaches a value necessary for a self-bound droplet; (ii) the trapping potential is then ramped off over a time scale $\tau_{\mathrm{trap}}$ leaving the droplet in a self-bound state.
To accurately model the dynamics of formation it is necessary to augment the generalized GPE (\[GGPE\]) with the additional terms $\mathcal{V}\psi$ where $$\mathcal{V} \equiv\tfrac{1}{2}m\omega_\rho(t)^2(\rho^2+\lambda^2z^2)-\tfrac{i}{2}{\hbar L_3}|\psi|^4,$$ that describe the harmonic trapping potential and three-body loss processes that will occur at high atomic density. Here $\omega_\rho(t)$ is the radial trap frequency at time $t$, $\lambda=\omega_z/\omega_\rho$ is the trap aspect ratio, and $L_3$ is the loss coefficient. We simulate the GPE dynamics using a 3D Fourier method on a grid of $512\times512\times256$ points evolved using a fourth-order Runge-Kutta algorithm. The initial condition for the dynamics, $\psi_0$, is a stationary solution of the GPE subject to the trapping potential with $\epsilon_{\mathrm{dd}}\approx1$ and noise added to mimic quantum and thermal fluctuations. The procedure for adding this noise is as described in Refs. [@Bisset2015a; @Blakie2016a].
We choose to use a prolate trap ($\lambda<1$) since the condensate continuously evolves into a single droplet state as $a_s$ (i.e. $\epsilon_{\mathrm{dd}}^{-1}$) is reduced [@Bisset2016a]. In contrast, for oblate traps there is a first order phase transition between the stable condensate and droplet state, and the heating that occurs when crossing this transition leads to multiple droplets forming (also see [@Blakie2016a; @Wachtler2016a]). For these droplets to remain self-bound as the trap is reduced each must exceed the minimum atom number for stability. Their mutual interaction will, however, cause them to repel and move away from each other.
The results of selected dynamical simulations are summarized in Fig. \[figtime\] for parameters relevant to $^{164}$Dy with $L_3 = 1.2\times 10^{-41}\:\mathrm{m}^6\mathrm{s}^{-1}$ [@Wachtler2016a]. The initial state $\psi_0$ uses $a_s=130a_0$. The peak density is seen to rapidly increase as $a_s$ is reduced, signaling the droplet formation. Once fully formed, the peak density of the self-bound solution is higher if $a_s$ is quenched to a lower value \[also see Fig. \[figDropProps\](c)\], and as a result the atom number decreases through three-body loss most rapidly for smaller $a_s$. In order to distinguish non-self-bound atoms (which are expelled as the trap is turned off) from those in the droplet we evaluate both the total atom number and the number within a cylindrical region centered on the droplet in Fig. \[figtime\](b). The similar behavior of the decay in total number and droplet number shows that most of the atom loss occurs in the dense droplet, and shows that the droplet atoms remain localized in the cylindrical region. We also observe in the two lowest ($a_s$) quenches considered that the steady decay in droplet number is suddenly interrupted by a more rapid decay at later times. This occurs because when the atom number falls below the minimum number for a stable droplet \[which depends on $a_s$, indicated by the colored ticks by the vertical axis in Fig. \[figtime\]\], the droplet suddenly becomes unbounded and disperses. For the quench to $a_s=80a_0$ this does not occur within the time range we simulate (also see Fig. \[figiso\]). For the quench to $a_s=100a_0$ the droplet does not form at all, as can be seen by the small peak density. This is because the initial condensate number ($N_0=10^4$) is lower than the stability threshold ($1.1\times10^4$) for $a_s = 100 a_0$.
We have also investigated droplet formation sensitivity to the three-body loss rate. For $a_s=60a_0$ ($a_s=70a_0$) the droplet lifetimes are $\sim 35,20,15\:$ms ($\sim 80,45,25\:$ms) if we scale the three-body loss parameter by the factors $\tfrac12,1,2$, respectively.
Because the quenches are reasonably fast they excite collective modes of the droplet. These excitations give rise to width oscillations that are seen to persist when the droplet is in free space, providing an additional signature of the self-bound character of the droplet.
*Conclusions*– In this paper, we have shown that dipolar condensates provide an ideal system for realizing a self-bound matterwave. We have presented a universal description of the self-bound states parameterized in terms of the interaction parameter ratio and atom number, thus our results are relevant for, and can be easily extended to describe experiments with Er. Importantly, we observe that in the strongly dipolar regime droplets require a minimum atom number to be stable. We have proposed and simulated a scheme for producing a self-bound droplet in free space in a parameter regime accessible to current experiments. We show that three-body loss plays an important role and will ultimately limit self-bound droplet lifetime. However, our results show that quenching to larger values of $\epsilon_{\mathrm{dd}}^{-1}$ are favourable for producing a long lived droplet because of the slower loss rates.\
*Acknowledgments*– The authors acknowledge valuable conversations with F. Ferlaino. DB and PBB acknowledge the contribution of NZ eScience Infrastructure (NeSI) high-performance computing facilities, and support from the Marsden Fund of the Royal Society of New Zealand. RMW acknowledges partial support from the National Science Foundation under Grant No. PHYS-1516421. RNB acknowledges support by the QUIC grant of the Horizon2020 FET program and by Provincia Autonoma di Trento.\
*Note added*– In the final stages of preparing of this manuscript we became aware of Ref. [@Wachtler2016c], which discusses equilibrium properties of trapped and self-bound droplets.
[24]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\
12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****, ()](\doibase %
10.1103/PhysRevLett.105.263901) [****, ()](\doibase 10.1103/PhysRevLett.88.130402) [****, ()](\doibase 10.1103/PhysRevA.63.031603) [****, ()](\doibase %
10.1103/PhysRevLett.106.170401) [****, ()](\doibase 10.1103/PhysRevLett.89.050402) [****, ()](\doibase %
10.1103/PhysRevLett.115.253902) [****, ()](\doibase %
10.1103/PhysRevLett.94.160401) [****, ()](\doibase %
10.1103/PhysRevA.77.061601) [****, ()](\doibase 10.1103/PhysRevLett.107.190401) [****, ()](\doibase %
10.1103/PhysRevLett.108.210401) [****, ()](http://stacks.iop.org/0034-4885/72/126401) [****, ()](\doibase 10.1038/nphys887) [****, ()](\doibase 10.1103/PhysRevA.80.023614) [****, ()](\doibase 10.1103/PhysRevLett.101.080401) [****, ()](http://dx.doi.org/10.1038/nature16485) [****, ()](\doibase %
10.1103/PhysRevLett.116.215301) @noop [ ]{} [****, ()](\doibase 10.1103/PhysRevA.93.061603) [****, ()](\doibase 10.7566/JPSJ.85.053001) [****, ()](\doibase 10.1103/PhysRevA.74.013623) [****, ()](\doibase 10.1103/PhysRevA.84.041604) [****, ()](\doibase 10.1103/PhysRevA.92.061603) [****, ()](\doibase 10.1103/PhysRevA.93.033644) @noop [ ]{}
|
---
abstract: |
In this article, we study the family of elliptic curves $E/\Q$, having good reduction at $2$ and $3$, and whose $j$-invariants are small. Within this set of elliptic curves, we consider the following two subfamilies: first, the set of elliptic curves $E$ such that the ratio $\Delta(E)/C(E)$ is squarefree; and second, the set of elliptic curves $E$ such that $\Delta(E)/C(E)$ is bounded by a small power $(<3/4)$ of $C(E)$. Both these families are conjectured to contain a positive proportion of elliptic curves, when ordered by conductor.
Our main results determine asymptotics for both these families, when ordered by conductor. Moreover, we prove that the average size of the $2$-Selmer groups of elliptic curves in the first family, again when these curves are ordered by their conductors, is $3$. This implies that the average rank of these elliptic curves is finite, and bounded by $1.5$.
author:
- 'Ananth N. Shankar and Arul Shankar and Xiaoheng Wang'
bibliography:
- 'Arithbib.bib'
title: Families of elliptic curves ordered by conductor
---
Introduction
============
Every elliptic curve over $\Q$ can be uniquely represented as $E_{AB}:y^2=x^3+Ax+B$, where $A$ and $B$ are integers such that there is no prime $p$ with $p^4\mid A$ and $p^6\mid B$, and such that $\Delta(A,B):=-4A^3-27B^2\neq 0$. Given an elliptic curve $E$ over $\Q$, we denote its algebraic rank by $r(E)$ and its analytic rank by $r_\an(E)$. The Birch and Swinnerton-Dyer conjecture asserts that these two quantities are equal, i.e., we have $r(E)=r_\an(E)$.
Foundational conjectures of Goldfeld [@goldfeld-quadtwistconj] (in the case of families of quadratic twists of elliptic curves) and Katz–Sarnak [@katzsarnak] (for the full family of elliptic curves) assert that a density of $50\%$ of elliptic curves have rank $0$, and that $50\%$ have rank $1$, and that the average rank of elliptic curves is $1/2$. Both these conjectures are formulated through a study of the associated family of the $L$-functions $L_E(s)$ attached to the elliptic curves $E$. The behaviour of $L_E(s)$ at and near the critical point is used to control the distribution of analytic ranks, which, assuming the BSD conjecture, can be used to give heuristics for the distribution of the algebraic ranks.
The most natural way to order a family of $L$-functions is by their [*conductors*]{}, which, in this case of $L$-functions of elliptic curves, is equal to the levels of the associated modular forms. Thus in the conjectures of Goldfeld and Katz–Sarnak, it is implicitly assumed that elliptic curves are ordered by their conductors. However, when studying two-parameter families of elliptic curves, the curves $E_{AB}$ are usually ordered by their (naive) [*height*]{} $H(E_{AB})=\max\{4|A|^3,27B^2\}$.[^1] Assuming the generalized Riemann hypothesis, Brumer [@brumer], Heath-Brown [@heathbrown-avgrank], and Young [@young-avgrank], proved the successively better bounds of 2.3, 2, and $25/14$, on the average analytic ranks of elliptic curves when ordered by height. On the algebraic side, Bhargava and the second named author [@bs5sel] proved that the average rank of elliptic curves, when ordered by height, is bounded by $0.885$.
If elliptic curves are instead ordered by conductor, even asymptotics for the number of curves are not known. The [*discriminant*]{} $\Delta(E_{AB})$ of $E_{AB}$ is (up to absolutely bounded factors of $2$ and $3$) $-4A^3-27B^2$. The [*conductor*]{} $C(E_{AB})$ of $E_{AB}$ is (again, up to bounded factors of $2$ and $3$) the product over all primes $p$ dividing $\Delta(E_{AB})$ of either $p$ or $p^2$ depending on if $E_{AB}$ has multiplicative or additive reduction at $p$. Building on the work of Brumer–McGuinnes [@brumermcguinness] on the family of elliptic curves ordered by discriminant, Watkins [@watkins-heuristics] gives heuristics suggesting that the number of elliptic curves with conductor bounded by $X$ grows as $\sim
cX^{5/6}$ for an explicit constant $c$. Lower bounds of this magnitude are easy to obtain, but the best known upper bound is $O(X^{1+\epsilon})$ due to work of Duke–Kowalski [@DukeKowalski].
The difficulties in determining precise upper bounds are twofold. First, it is difficult to rule out the possibility of many elliptic curves with large height but small discriminant. Second, it is difficult to rule out the possibility of many elliptic curves with large discriminant but small conductor. It is interesting to note here that the second difficulty is exactly a nonarchimedean version of the first. Indeed, curves $E_{AB}$ with large height and small discriminant correspond to pairs $(A,B)$ of integers, where $4A^3$ and $-27B^2$ are unusually close as real numbers. On the other hand, curves $E_{AB}$ with large discriminant and small conductor correspond to pairs of integers $(A,B)$ such that $4A^3$ and $-27B^2$ are unusually close as $p$-adic numbers.
In this article, we focus on studying the second difficulty while entirely sidestepping the first. To this end, we let $\E$ denote the set of elliptic curves $E$ over $\Q$ that satisfy the following properties.
1. The $j$-invariant $j(E)$ of $E$ satisfies $j(E)<\log\Delta(E)$.
2. $E$ has good reduction at $2$ and $3$.
The first of the above three properties excludes all elliptic curves $E$ with $\Delta(E)\ll H(E)^{1-\epsilon}$ and is absolutely critical for our results. According to the Brumer–Mcguinnes heuristics [@brumermcguinness], only a negligible number of elliptic curves are being excluded by the assumption of this property, but this is unproven. The second property is a technical assumptions made to simplify local computations at the $2$-adic and $3$-adic places. We will in fact have to further restrict our families of elliptic curves. Define the families $$\begin{array}{rcl}
\E_\sf&:=&
\displaystyle \Bigl\{E\in \E:\frac{\Delta(E)}{C(E)}\;
\mbox{is squarefree} \Bigr\},\\[.2in]
\E_\kappa&:=&
\displaystyle \bigl\{ E\in\E:\Delta(E)<C(E)^{\kappa}\bigr\},
\end{array}$$ for every $\kappa>1$. When ordered by conductor, the family $\E_\kappa$ conjecturally contains $100\%$ of elliptic curves with good reduction at $2$ and $3$, and $\E_\sf$ conjecturally contains a positive proportion of elliptic curves. We prove the following result determining asymptotics for these families of elliptic curves, ordered by their conductors.
\[thmmain\] Let $1<\kappa<7/4$ be a positive constant. Then we have $$\label{eqEF}
\begin{array}{rcl}
\displaystyle\#\{E\in \E_\sf:\; C(E)<X\}&\sim&
\displaystyle \frac{1+\sqrt{3}}{60\sqrt{3}}
\frac{\Gamma(1/2)\Gamma(1/6)}{\Gamma(2/3)}
\cdot\prod_{p\geq 5}\Bigl(1+\frac{1}{p^{7/6}}-\frac{1}{p^2}-\frac{1}{p^{13/6}}\Bigr)
\cdot X^{5/6};\\[.2in]
\displaystyle\#\{E\in \E_\kappa:\; C(E)<X\}&\sim&
\displaystyle\frac{1+\sqrt{3}}{60\sqrt{3}}
\frac{\Gamma(1/2)\Gamma(1/6)}{\Gamma(2/3)}
\cdot\prod_{p\geq 5}
\Bigl[\Bigl(1-\frac{1}{p}\Bigr)\Bigl(1+\frac{1}{p^{5/3}}+\frac{1}{p^{11/6}}+\frac{1}{p^{17/6}}\Bigr)\\[.2in]
&&
+\displaystyle\frac{1}{p}\Bigl(1-\frac{1}{p}\Bigr)
\Bigl(1-\frac{1}{p^{1/6}}\Bigr)^{-1}\Bigl(1+\frac{2}{p}-\frac{2}{p^{3/2}}\Bigr)\Bigr]
\cdot X^{5/6}.
\end{array}$$
We expect Theorem \[thmmain\] to hold for all $\kappa$. Furthermore, since the abc conjecture implies that for $\kappa>6$, we have $\E_\kappa=\E$, we expect these asymptotics to also hold for the family $\E$. We note that the Euler factors appearing in Theorem \[thmmain\] arise naturally from the densities of elliptic curves over $\Q_p$ with fixed Kodaira symbol. These densities are computed in Theorem \[propcasessp\].
Our next main result is on the distribution of ranks of elliptic curves in $\E_\sf$. As in [@bs2sel], we study the ranks of these elliptic curves via their $2$-Selmer groups. Recall that the $2$-Selmer group $\Sel_2(E)$ of an elliptic curve $E$ over $\Q$ is a finite $2$-torsion group which fits into the exact sequence $$\label{eqexseq}
0\to E(\Q)/2E(\Q)\to\Sel_2(E)\to\SH_E[2]\to 0,$$ where $\SH_E$ denotes the Tate–Shafarevich group of $E$. Our result regarding the $2$-Selmer groups of elliptic curves in $\E_\sf$ is as follows.
\[thmsel\] When elliptic curves in $\E_\sf$ are ordered by their conductors, the average size of their $2$-Selmer groups is $3$.
Theorem \[thmsel\] has the following immediate corollary.
\[corrank\] When elliptic curves in $E\in \E_\sf$ are ordered by their conductors, their average $2$-Selmer rank is at most $1.5$; thus, their average rank is at most $1.5$ and the average rank of $\SH_E[2]$ is also at most $1.5$.
Corollary \[corrank\] provides evidence for the widely held belief that the distribution of the ranks of elliptic curves are the same regardless of whether the curves are ordered by height or conductor. Moreover, as expected, the average size of the $2$-Selmer groups of curves in $\E_\sf$ are the same as the average over all elliptic curve ordered by height obtained in [@bs2sel Theorem 1.1]. We remark that our methods are flexible enough to recover verions of Theorems \[thmmain\] and \[thmsel\] where the families $\E_\sf$ and $\E_\kappa$ are restricted so that the curves in them satisfy any finite set of local conditions. This result is stated as Theorem \[thmmainlarge\].
The key ingredient for proving the main results are “uniformity estimates” or “tail estimates”. These are upper bounds on the number of elliptic curves in our families whose discriminants are large compared to their conductors. For the proof of Theorem \[thmsel\], we additionally need bounds on the sum of the sizes of the $2$-Selmer groups of elliptic curves in $\E_\sf$ with large discriminant and small conductor. To this end, we prove the following result for the family $\E_\sf$.
\[thunifsqi\] For positive real numbers $X$ and $M$, we have $$\#\Bigr\{(E,\sigma):E\in\E_\sf,\;C(E)<X,\;\Delta(E)>MC(E),\;
\sigma\in\Sel_2(E)\Bigl\}\,\ll_\epsilon
\frac{X^{5/6+\epsilon}}{M^{1/6}}.$$
We note that up to the power of $X^\epsilon$, this is expected to be the optimal bound.
For the family $\E_\kappa$, we prove the following result.
\[thunifsmind\] Let $\kappa<7/4$ and $\delta>0$ be positive constants. Then there exists a positive constant $\theta$, depending only on $\delta$ and $\kappa$, such that for every $X>0$, we have $$\#\Bigr\{E\in\E_\kappa:C(E)<X,\;\Delta(E)>X^\delta C(E)\Bigl\}\,\ll_\epsilon
X^{5/6-\theta+\epsilon}.$$
In [@bs2sel], a version of such uniformity estimates were proved. These estimates were used to obtain asymptotics on the number of elliptic curves with bounded height and squarefree discriminant, as well as to compute the average sizes of the $2$-Selmer groups of these elliptic curves. One main input used in proving these estimates was the Ekedahl sieve, as developed by Bhargava in [@manjul-geosieve]. For our applications, this sieve falls short of what is needed since our curves have much larger height than in the previous case. Indeed, the height of $E\in\E_\sf$ with $C(E)=X$ can be as large as $X^2$, in which case the Ekedahl sieve gives rise to an error term of $O(X^{4/3})$ which is much too large. Subsequent improvements to the Ekeshal sieve by Taniguchi–Thorne [@TaniguchiThorneDistLev], in which the sieve is combined with equidistribution methods, are also insufficient for our purposes.
We now describe the proofs of our main theorems. We study the ratios $\Delta(E)/C(E)$ of elliptic curves $E:y^2=f(x)$ in our families by considering the associated family of cubic rings $R_f:=\Z[x]/f(x)$ and cubic algebras $K_f:=\Q[x]/f(x)$ over $\Q$. Let ${{\mathcal O}}_f$ denote the ring of integers of $K_f$. Then $R_f$ is a suborder of $K_f$. Define the invariants $$\begin{array}{rcl}
Q(E)&:=&[{{\mathcal O}}_f:R_f]\\[.05in]
D(E)&:=&\Disc(K_f)
\end{array}$$ which satisfy the relation $$\Delta(E)=\Disc(R_f)=Q(E)^2D(E).$$ For primes $p$, we let $C_p(E)$, $\Delta_p(E)$, $Q_p(E)$, and $D_p(E)$ denote the $p$-parts of $C(E)$, $\Delta(E)$, $Q(E)$, and $D(E)$, respectively. The local invariants $C_p(E)$, $\Delta_p(E)$, $Q_p(E)$, and $D_p(E)$ depend only on the Kodaira symbol of $E$. The starting point of our proof is a determination of these local invariants along with a computation of the density of elliptic curves over $\Q_p$ with fixed Kodaira symbol.
\[propcasessp\] Fix a prime $p\geq 5$ and a Kodaira symbol $T$. Let $E:y^2=f(x)$ be an elliptic curve over $\Z_p$ such that the Kodaira symbol of $E$ is $T$. Then the local invariants of $E$ are as given in Table \[tabloc\]. Furthermore, there exists an element $t\in\Z_p$ such that coefficients of $f(x+t)=x^3+ax^2+bx+c$ are as given in the second column of Table \[tabloc\]. Finally, the density of all elliptic curves with Kodaira symbol $T$ is as given in the last column.
[|c | c| c|c|c|c|c|]{} Kodaira Symbol & Congruence Condition & $C_p(E)$ & $\Delta_p(E)$ & $Q_p(E)$ & $D_p(E)$ & Density\
of $E$ &&&&&&\
$\I_0$ & $p\nmid\Delta(f)$ & $1$ & $1$ & $1$ & $1$ & $(p-1)/p$\
$\I_{n}$ & $p\nmid a, \quad p^{\lceil n/2\rceil}\mid b, \quad p^{n}\mid\mid c$ & $p$ & $p^{n}$ & $p^{\lfloor n/2\rfloor}$ & $p^{n\!\!\!\pmod 2}$ & $(p-1)^2/p^{n+2}$\
$\I\I$ & $p\mid a, \quad p\mid b,\quad p\parallel c$ & $p^2$ & $p^2$ & $1$ & $p^2$ & $(p-1)/p^3$\
$\I\I\I$ & $p\mid a, \quad p\parallel b,\quad p^2\mid c$ & $p^2$ & $p^3$ & $p$ & $p$ & $(p-1)/p^4$\
$\I\rV$ & $p\mid a, \quad p^2\mid b,\quad p^2\parallel c$ & $p^2$ & $p^4$ & $p$ & $p^2$ & $(p-1)/p^5$\
$\I_0^*$ & $p\mid a,\;p^2\mid b,\;p^3\mid c,\;p^7\nmid\Delta(f)$ & $p^2$ & $p^6$ & $p^3$ & $1$ & $(p-1)/p^6$\
$\I_{n}^*$ & $p\parallel a, \; p^{\lceil n/2\rceil+2}\mid b, \; p^{n+3}\parallel c$ & $p^2$ & $p^{n+6}$ & $p^{\lfloor n/2\rfloor+3}$ & $p^{n\!\!\!\pmod 2}$ & $(p-1)^2/p^{n+7}$\
$\I\rV^*$ & $p^2\mid a, \quad p^3\mid b,\quad p^4\parallel c$ & $p^2$ & $p^8$ & $p^3$ & $p^2$ & $(p-1)/p^8$\
$\I\I\I^*$ & $p^2\mid a, \quad p^3\parallel b,\quad p^5\mid c$ & $p^2$ & $p^9$ & $p^4$ & $p$ & $(p-1)/p^9$\
$\I\I^*$ & $p^2\mid a, \quad p^4\mid b,\quad p^5\parallel c$ & $p^2$ & $p^{10}$ & $p^4$ & $p^2$ & $(p-1)/p^{10}$\
These density computations are straightforward, and indeed many of them are implicit in the work of Watkins [@watkins-heuristics § 3.2]. However we include a proof since our use of a $\G_a$-action on the space of monic cubic polynomials simplifies the computations.
We use three different techniques to prove the estimates of Theorems \[thunifsqi\] and \[thunifsmind\]. First, we fix a prime $p\geq 5$ and a Kodaira symbol $T$. The set of elliptic curves that have Kodaira symbol $T$ at $p$ is cut out by certain congruence conditions $S$ modulo $q$, some power of $p$. Working modulo $q$, we compute the Fourier transform of the characteristic function of $S$. An application of Poisson summation then yields baseline estimates for the number of elliptic curves with bounded height having Kodaira symbol $T$ at $p$.
Our next two techniques average over primes $p$ in a crucial way. Suppose that $E:y^2=f(x)$ is an elliptic curve in $\E_\sf$ such that the ratio $\Delta(E)/C(E)$ is large. Then we prove that either the discriminant of the algebra $K_f$ is small, or that the shape of the ring of integers ${{\mathcal O}}_f$ of $K_f$ is very skewed. The work of Bhargava and Harron [@bhargavaharron] proves that the shapes of rings of integers are equidistributed in the family of cubic fields. Furthermore, the forthcoming thesis of Chiche-Lapierre [@Lapierrethesis] determines asymptotics for the number of cubic fields such that the shapes of their ring of integers are constrained to lie within $0$-density sets. Using ideas from these works, we prove bounds on the number of possible cubic algebras $K_f$ corresponding to elliptic curves in $\E_\sf$ with bounded conductor, along with bounds on the average sizes of the $2$-torsion subgroups $\Cl_2(K_f)$ of the class groups of $K_f$. In combination with the work of Brumer–Kramer [@brumerkramer], relating the size of $\Sel_2(E)$ to $\#\Cl_2(K_f)$, we deduce Theorem \[thunifsqi\].
The above method exploits the following crucial fact. If $E:y^2=f(x)$ is an elliptic curve such that the ratio $\Delta(E)/C(E)$ is large, then primes $p$ such that the Kodaira symbol of $E$ at $p$ is $\I_0$, $\I_1$, $\I_2$, $\I\I$, or $\I\I\I$ impose archimedean constraints on the algebras $K_f$. However, primes $p$ with Kodaira symbol $\I\rV$ or $\I_n$ with $n\geq 3$ impose only $p$-adic conditions on $R_f\hookrightarrow K_f$. Namely, the prime $p$ divides the gcd of $Q(E)$ and $D(E)$. To exploit this, we proceed as follows. The set of integer monic traceless cubic polynomials $f$ with $p\mid Q(E_f)$ embeds into the space of binary quartic forms with a rational linear factor. This embedding $\sigma$ is defined in . The group $\PGL_2$ acts on the space of binary quartic forms, and the ring of invariants for this action is freely generated by two polynomials $I$ and $J$. Restricted to the space of reducible binary quartic forms gives an additional invariant $Q$. Explicitly, if $g(x,y)$ is a binary quartic form with coefficients in $\Q$, and $g(\alpha,\beta)=0$, then define $$Q(g(x,y),[\alpha:\beta])=\frac{g(x,y)}{\beta x-\alpha y}(\alpha,\beta).$$ This new invariant $Q$ is an exact analogue of the $Q$-invariants used in [@BSWsf1],[@BSWsf2] to compute the density of polynomials with squarefree polynomials. As there, for every fixed root $[\alpha:\beta]\in\P^1(\Z)$, the discriminant polynomial on the space of integer binary quartic forms $g$ with $g(\alpha,\beta)$ is reducible, and in fact divisible by $Q^2$. We also define $$D(g(x,y),[\alpha:\beta]):=\Delta(g)/(Q(g(x,y),[\alpha:\beta]))^2.$$ Our embedding $\sigma$ satisfies $Q(E)=Q(\sigma(E))$ and $D(E)=D(\sigma(E))$. Then the required estimates on elliptic curves $E\in\E_\kappa$ with large $\Delta(E)/C(E)$, translate to estimates on the number of $\PGL_2(\Z)$-orbits on integral reducible binary quartic forms with bounded height and large $Q$- and $D$-invariants. We prove the required estimates by fibering over roots, and then combining geometry of numbers methods with the Ekedahl sieve.
This paper is organized as follows. In §2 and §3, we work locally, one prime at a time. Theorem \[propcasessp\] is proved in §2, while the Fourier coefficients corresponding to a fixed Kodaira symbol are computed in §3. The computation of the Fourier coefficients are then used to obtain estimates (see Theorem \[thm:equimain\]) on curves with fixed Kodaira symbols at finitely many primes. We prove bounds on the number of cubic fields $K$, weighted by $|\Cl_2(K)|$, in §4, and obtain estimates on the number of reducible integer binary quartic forms with large $Q$- and $D$-invariants in §5. The results of §3, §4, and §5, are combined in §6 to prove the uniformity estimates Theorems \[thunifsqi\] and \[thunifsmind\]. Finally, in §7, we prove the main results Theorems \[thmmain\] and \[thmsel\].
Acknowledgments {#acknowledgments .unnumbered}
---------------
It is a pleasure to thank Manjul Bhargava, Benedict Gross, Hector Pasten, Peter Sarnak, and Jacob Tsimerman for many helpful conversations. The second named author is supported an NSERC Discovery Grant and a Sloan Fellowship. The third named author is supported by an NSERC Discovery Grant.
Reduction types of elliptic curves
==================================
Throughout this section, we fix a prime $p\geq 5$. Let $U$ denote the space of monic cubic polynomials. Then for any ring $R$, we have $$U(R)=\{x^3+ax^2+bx+c:a,b,c\in R\}.$$ We denote the space of traceless elements of $U$ (i.e., $a=0$ in the above equation) by $U_0$. The group $\G_a$ acts on $U$ via $(t\cdot
f)(x)=f(x+t)$. Given any element $f\in U(\Z_p)$, there exists a unique element $\gamma\in\Z_p$ such that $f_0(x)=(\gamma\cdot f)(x)$ belongs to $U_0(\Z_p)$. Thus we may identify the quotient space $\Z_p\backslash
U(\Z_p)$ with $U_0(\Z_p)$. We denote the Euclidean measures on $U(\Z_p)$ and $U_0(\Z_p)$ by $dg=da\,db\,dc$ and $df=db\,dc$, respectively, where $da$, $db$ and $dc$ are Haar measures on $\Z_p$ normalized so that $\Z_p$ has volume $1$. Then the change of measure formula for the bijection $$\label{eqzpaction}
\begin{array}{rcl}
\Z_p\times U_0(\Z_p)&\to& U(\Z_p) \\[.05in]
(t,f(x))&\mapsto& g(x)=(t\cdot f)(x)=f(x+t)
\end{array}$$ is $dt\,df=dg$, where $dt$ is again the Haar measure on $\Z_p$ normalized so that $\Z_p$ has volume $1$.
Given an element $f(x)\in U(\Z_p)$ such that the discriminant $\Delta(f)$ is nonzero, we consider the elliptic curve $E_f$ over $\Q_p$ with affine equation $y^2=f(x)$. An element $f(x)\in U(\Z_p)$ with nonzero discriminant is said to be [*minimal*]{} if $\Delta(f)=\Delta(E_f)$. Equivalently, $f(x)$ is minimal if $f_0(x)=x^3+Ax+B$, the unique element in $U_0(\Z_p)$ in the $\Z_p$-orbit of $f$, does not satisfy $p^4\mid A$ and $p^6\mid
B$. Another equivalent condition is that the roots of $f_0(x)$ are not all multiples of $p^2$. We denote the set of minimal elements in $U(\Z_p)$ by $U(\Z_p)^\min$, and denote $U(\Z_p)^\min\cap U_0(\Z_p)$ by $U_0(\Z_p)^\min$. The map $f\mapsto E_f$ is then a natural surjective map from $\Z_p\backslash U(\Z_p)^\min$ (equivalently $U_0(\Z_p)^\min$) to the set of isomorphism classes of elliptic curves over $\Q_p$.
The twisting-by-$p$ map is a natural involution on the set of isomorphism classes of elliptic curves over $\Q_p$. This yields a natural involution $\sigma$ on $\Z_p\backslash U(\Z_p)^\min$. If $f\in
U(\Z_p)^\min$ such that $f_0(x) = x^3+Ax+B$ with $p^2\nmid A$ or $p^3\nmid B$, then we say $f$ is [*small*]{} and in this case, $\sigma(f)_0(x) = \sigma(f_0)(x)=x^3 + p^2Ax + p^3B$. Otherwise, if $f_0(x) =
x^3+Ax+B$ with $p^2\mid A$ and $p^3\mid B$, then we say $f$ is [ *large*]{} and in this case, $\sigma(f)_0(x)=\sigma(f_0)(x) = x^3 +
p^{-2}Ax + p^{-3}B$. We have $\Delta(E_{\sigma(f)}) = p^6\Delta(E_f)$ if $f$ is small and $\Delta(E_{\sigma(f)}) = p^{-6}\Delta(E_f)$ otherwise. Let $U(\Z_p)^\sm$ denote the set of small elements $f\in U(\Z_p)$.
Let $E$ be an elliptic curve over $\Q_p$, and let $\mathscr{X}$ be a minimal proper regular model of $E$ over $\Z_p$. For brevity, we will say that $T$, the Kodaira symbol associated to the special fiber of $\mathscr{X}$, is the Kodaira symbol of $E$. Define the [*index*]{} of $E$ by $\ind(E):=\Delta(E)/C(E)$. Then the index of $E$ is $1$ if and only if the Kodaira symbol of $E$ is $\I_0$ (when $E$ has good reduction), $\I_1$, or $\I\I$. Given $f\in U(\Z_p)^\min$, we define the [*index*]{} of $f$ to be $\ind(f):=\ind(E_f)$. We also define two other invariants associated to elements $f\in U(\Z_p)^\min$. Let $K_f$ denote the cubic etalé algebra $K_f:=\Q_p[x]/f(x)$, let ${{\mathcal O}}_f$ denote the ring of integers of $K_f$, and let $R_f$ denote the cubic ring $\Z[x]/f(x)$. We define $$\begin{array}{rcl}
Q_p(f)&:=&[{{\mathcal O}}_f:R_f];\\[.05in]
D_p(f)&:=&\Disc(K_f).
\end{array}$$ These quantities are clearly invariant under the action of $\Z_p$ on $U(\Z_p)$ and satisfy the equation $$\Delta(f)=\Delta(R_f)=D_p(f)Q_p(f)^2.$$
The next result gives a criterion for $f\in U(\Z_p)$ to be small in terms of the Kodaira symbol of $E_f$.
\[propKSloc\] Let $f(x)\in U(\Z_p)^\min$ be a monic cubic polynomial corresponding to the elliptic curve $E_f$. Then $f$ is small if and only if the Kodaira symbol of $E_f$ is $\I_n$ for $n\geq 1$, $\I\I$, $\I\I\I$, or $\I\rV$.
We first note that if the Kodaira symbol of $E_f$ is $\I\I$, $\I\I\I$, or $\I\rV$, then $f$ is small since the discriminant is less than $p^6$. If $f$ is not small, then $E_f$ has additive reduction. Hence if the Kodaira symbol of $E_f$ is $\I_n$, then $f$ is small.
Conversely, we start with fixing an element $f\in U(\Z_p)^\sm$. Let $\alpha_1,\alpha_2,\alpha_3$ denote the three roots of $f(x)$ over $\overline{\Q}_p$, the Galois closure of $\Q_p$, and let $\nu_p$ denote the $p$-adic valuation on $\overline{\Q}_p$. We now consider the following four cases: $f(x)$ is irreducible over $\Q_p$; $f(x)$ factors as a product of a linear and a quadratic factor over $\Q_p$, and moreover $E_f$ has additive reduction; $f(x)$ factors as a product of a linear and a quadratic polynomial over $\Q_p$, and moreover $E_f$ has multiplicative reduction; $f(x)$ factors into the product of three distinct linear polynomials over $\Q_p$. In what follows, we will repeatedly use [@silvermanEC2 Table 4.1] to determine the Kodaira symbol of $E_f$ from its reduction type and discriminant.
First suppose $f(x)$ is irreducible over $\Q_p$. The absolute Galois group of $\Q_p$ acts transitively on $\alpha_1,\alpha_2,\alpha_3$. Let $\sigma$ be an element sending $\alpha_1$ to $\alpha_2$. If $\sigma(\alpha_3) = \alpha_3$, then $\sigma(\alpha_1 - \alpha_3) =
\alpha_2 - \alpha_3$. If $\sigma(\alpha_3) = \alpha_1$, then $\sigma(\alpha_2 - \alpha_3) = \alpha_3 - \alpha_1$. Hence in either case $$\nu_p(\alpha_1 - \alpha_3) = \nu_p(\alpha_2 - \alpha_3).$$ Similarly, we have $\nu_p(\alpha_1 - \alpha_3) = \nu_p(\alpha_1 -
\alpha_2)$. Let $m\in\frac13\Z$ be their common value. Let $t =
(\alpha_1 + \alpha_2 + \alpha_3)/3 \in\Z_p$. Then replacing $\alpha_i$ by $\alpha_i - t$, we may assume $\nu_p(\alpha_i)\geq m$ for $i =
1,2,3$. On the other hand, $\nu_p(\alpha_1 -
\alpha_2)\geq\max\{\nu_p(\alpha_1),\nu_p(\alpha_2)\}$. Hence $\nu_p(\alpha_i) = m$ for $i = 1,2,3.$. Since $f$ is integal and small, we have $0\leq m<1$.
If $m = 0$, then $E_f$ has good reduction at $p$, and the Kodaira symbol is $\I_0$. If $m=1/3$, then $E_f$ has additive reduction at $p$ and $\nu_p(\Delta(E_f))=2$. This implies that the Kodaira symbol is $\I\I$. Finally, if $m = 2/3$, then $E_f$ has additive reduction and $\nu_p(\Delta(E_f))=4$. It follows that the Kodaira symbol is $\I\rV$.
Next suppose that $f(x)$ factors as a product of a linear and a quadratic factor over $\Q_p$, and that $E_f$ has additive reduction. Let $\alpha_1$ denote the root of the linear factor and let $\alpha_2$ and $\alpha_3$ denote the conjugate roots of the quadratic factor. Then $\alpha_1,\alpha_2,\alpha_3$ are congruent modulo $p^{1/2}$. Let $t = (\alpha_1 + \alpha_2 + \alpha_3)/3$ as above. Replacing $\alpha_i$ by $\alpha_i-t$, we may assume $$\nu_p(\alpha_1)\geq 1,\quad\nu_p(\alpha_2) = \nu_p(\alpha_3) =
\frac12.$$ The latter equality holds because if $p\mid\alpha_2$, then $f$ is not small. Since $\alpha_2$ and $\alpha_3$ are roots of a quadratic polynomial $q(x)$ with $\Z_p$ coefficients, we have $\alpha_2 +
\alpha_3\in p\Z_p$. Hence $$\nu_p(\alpha_2 - \alpha_3) = \frac12.$$ Clearly, $\nu_p(\alpha_1 -
\alpha_2) = \nu_p(\alpha_1 - \alpha_3) = 1/2$. Hence, $E_f$ has additive reduction and $\nu_p(\Delta(E_f) = 3$. This implies that the Kodaira symbol is $\I\I\I$.
The third case follows immediately: since $E_f$ is assumed to have multiplicative reduction, the Kodaira symbol is $\I_n$ for some $n\geq
1$, which is sufficient. Finally, for the fourth case, suppose that $f$ factors into a product of three linear polynomials over $\Q_p$. If the $\alpha_i$ are all congruent modulo $p$, then replacing each $\alpha_i$ by $\alpha_i-\alpha_1$, we see that that $f$ is not small. Hence $E_f$ does not have additive reduction, and the Kodaira symbol is again $\I_n$ for some $n\geq 0$. This conclude the proof of the proposition.
Next, we prove Theorem \[propcasessp\].
[**Proof of Theorem \[propcasessp\]:**]{} We start by assuming that $E=E_f$ corresponds to $f\in U(\Z_p)^\sm$. By Proposition \[propKSloc\], the associated Kodaira symbol is $\I_n$, $\I\I$, $\I\I\I$, or $\I\rV$. We will begin with verifying the second through sixth columns of the Table \[tabloc\], leaving the density computation to Proposition \[propKSden\]. The result is clear if $E_f$ has good reduction, which happens precisely when $\Delta_p(E_f)=1$.
First assume that $E_f$ has additive reduction, in which case $C_p(E)=p^2$. Then the Kodaira symbol of $E_f$ is $\I\I$, $\I\I\I$, or $\I\rV$, according to whether $\Delta_p(E_f)$ is $p^2$, $p^3$, or $p^4$, respectively. Replacing $f(x)$ with a $\Z_p$-translate, if necessary, we may assume that $f(x)\equiv x^3\pmod{p}$. Write $f(x)=x^3+pa_1x^2+pb_1x+pc_1$ with $a_1,b_1,c_1\in\Z_p$. Then $$\Delta(f)\equiv
4p^3b_1^3-27p^2c_1^2+18p^3a_1b_1c_1\pmod{p^4}$$ and $p^2\parallel
\Delta(f)$ if and only if $p\nmid c_1$. In that case, the paragraph following Lemma 13 in [@bstcubic] implies that $R_f$ is the maximal order of $K_f$. This confirms the second through sixth columns in the case when the Kodaira symbol is $\I\I$.
If $p^3\mid\Delta(f)$, then $p\mid c_1$. We write $c_1=pc_2$ for some $c_2\in\Z_p$, and then $\Delta(f)\equiv 4p^3b_1^3\pmod{p^4}$. Hence $p^3\parallel \Delta(f)$ if and only if $p\nmid b_1$. Suppose $p\nmid
b_1$. Then $R_f$ is a suborder of index $p$ in the cubic ring ${{\mathcal O}}$ corresponding to the binary cubic form $px^3+pa_1x^2y+b_1xy^2+c_2y^3$. The ring ${{\mathcal O}}$ is maximal (from [@bstcubic] as before) with $\Delta_p({{\mathcal O}})=p$, confirming the values of $Q_p$ and $D_p$ when the Kodaira symbol is $\I\I\I$. Finally, suppose that $p\mid b_1$ and write $b_1=pb_2$. Then $f(x)=x^3+pa_1x+p^2b_2+p^2c_2$, and since $f$ is small, we have $p\nmid c_2$. In this case, we see that $\Delta_p(f)=p^4$ and that $R_f$ is a suborder of index $p$ of the maximal order ${{\mathcal O}}$ corresponding to the binary cubic form $px^3+pax^2y+pb_2xy^2+c_2y^3$ with $\Delta_p({{\mathcal O}})=p^2$. This confirms the second through sixth columns of Table \[tabloc\] in the case when $E_f$ has additive reduction and $f\in U(\Z_p)^\sm$.
Next, assume that $E_f$ has multiplicative reduction. From the proof of Proposition \[propKSloc\], it follows that $f(x)$ is not irreducible over $\Q_p$. Suppose that $f(x)$ factors into a product of a quadratic $q(x)$ and a linear polynomial $\ell(x)$ over $\Q_p$ and that $f(x)$ has splitting type $(1^21)$. Let $\alpha_1$ denote the root of the linear factor and let $\alpha_2$ and $\alpha_3$ denote the conjugate roots of the quadratic factor. Let $t = (\alpha_2 +
\alpha_3)/2\in\Z_p$. Replacing $\alpha_i$ by $\alpha_i-t$, we may assume $$\nu_p(\alpha_1) = 0,\quad \nu_p(\alpha_2) = \lambda,\quad
\alpha_3 = -\alpha_2$$ for some positive $\lambda\in\frac{1}{2}\Z$. Then $\nu_p(\alpha_2 - \alpha_3) =
\nu_p(2\alpha_2) = \lambda$. Clearly, $\nu_p(\alpha_1 - \alpha_2) =
\nu_p(\alpha_1 - \alpha_3) = 0$. Thus, $\Delta_p = p^{2\lambda}$ and the Kodaira symbol of $E_f$ is $\I_{2\lambda}$. Moreover, the coefficients $a$, $b$, and $c$ of $f$ satisfy $$\begin{aligned}
\nu_p(a) &=& \nu_p(\alpha_1 + \alpha_2 + \alpha_3) = \nu_p(\alpha_1) = 0,\\
\nu_p(b) &=& \nu_p(\alpha_1(\alpha_2 + \alpha_3) + \alpha_2\alpha_3) = 2\lambda,\\
\nu_p(c) &=& \nu_p(\alpha_1\alpha_2\alpha_3) = 2\lambda.\end{aligned}$$ The cubic order $\Z_p[x]/(f(x))$ is a suborder of index $p^{\lfloor\lambda\rfloor}$ of the cubic order associated to the binary cubic form $$p^{\lfloor\lambda\rfloor}x^3 + ax^2y +
(b/p^{\lfloor\lambda\rfloor})xy^2 + (c/p^{2\lfloor\lambda\rfloor})y^3,$$ which is maximal since its discriminant is $1$ when $\lambda$ is an integer and $p$ when $\lambda$ is a half integer. Hence we have $Q_p(E_f)= p^{\lfloor\lambda\rfloor}$ and $D_p = p^{2\lambda\pmod{2}}$ as necessary.
Suppose instead that $f(x)$ factors as a product of three linear polynomials over $\Q_p$. By assumption, the three roots $\alpha_1$, $\alpha_2$, and $\alpha_3$ of $f(x)$ in $\Z_p$ are not all congruent modulo $p$. After renaming, suppose $\alpha_2$ and $\alpha_3$ are congruent modulo $p$ and $\alpha_1$ is not congruent to them. Let $t =
2\alpha_3-\alpha_2\in\Z_p$. Replacing $\alpha_i$ by $\alpha_i-t$, we may assume $\alpha_1$ is a unit and $\alpha_2 = 2\alpha_3$. That is, $$\nu_p(\alpha_1) = 0,\quad \nu_p(\alpha_2) = \nu_p(\alpha_3) =
\lambda,$$ for some positive integer $\lambda\in\Z$. Thus, $\Delta_p(f) = p^{2\lambda}$, which implies that the Kodaira symbol of $E_f$ is $\I_{2\lambda}$. As a consequence, the coefficients $a$, $b$, and $c$ of $f$ satisfy $$\begin{aligned}
\nu_p(a) &=& \nu_p(\alpha_1 + \alpha_2 + \alpha_3) = 0,\\
\nu_p(b) &=& \nu_p(\alpha_1(\alpha_2+\alpha_3) + \alpha_2\alpha_3) = \lambda,\\
\nu_p(c) &=& \nu_p(\alpha_1\alpha_2\alpha_3) = 2\lambda.\end{aligned}$$ The cubic order $\Z_p[x]/(f(x))$ is a suborder of index $p^{\lambda}$ of the cubic order associated to the binary cubic form $p^\lambda x^3
+ ax^2y + (b/p^\lambda)xy^2 + (c/p^{2\lambda})y^3$, which is maximal since its discriminant is $1$. Therefore, $Q_p(E_f) = p^\lambda$ and $D_p(E_f) = 1$ as required.
We now turn to large elliptic curves. Let $E$ be a large elliptic curve over $\Z_p$. Let $E'$ denote the twist of $E$ by $p$. Then the Kodaira symbol of $E'$ is $\I_n$, $\I\I$, $\I\I\I$, or $\I\rV$, depending on whether the Kodaira symbol of $E$ is $\I_n^*$, $\I\rV^*$, $\I\I\I^*$, or $\I\I^*$, respectively. Let $y^2=f(x)$ be a model for $E'$, where the coefficients of $f(x)=x^3+ax^2+bx+c$ satisfy the congruence conditions of Table \[tabloc\]. Then $y^2=g(x)=x^3+pax^2+p^2bx+p^3c$ is a model for $E$. It is then easy to check that the second column of Table \[tabloc\] is correct for all ten rows. Furthermore, $K_g=K_f$ and $R_g$ has index $p^3$ in $R_f$. It follows that the local invariants of $E$ are as in Table \[tabloc\]. Theorem \[propcasessp\] follows the density computations in the following proposition. $\Box$
\[propKSden\] The density of elliptic curves over $\Z_p$ having a fixed Kodaira symbol is as in Table \[tabloc\].
Let $T$ be a fixed Kodaira symbol. Let $U(\Z_p)^{(T)}$ (resp. $U_0(\Z_p)^{(T)}$) denote the set of elements $f \in U(\Z_p)^\min$ (resp. $f \in U_0(\Z_p)^\min$) such that $E_f$ has Kodaira symbol $T$. Then the density of elliptic curves with Kodaira symbol $T$ is $\Vol(U_0(\Z_p)^{(T)})=\Vol(U(\Z_p)^{(T)})$, where the equality holds since $\Z_p\cdot
U_0(\Z_p)^{(T)}=U(\Z_p)^{(T)}$ and the Jacobian change of variables of the map is $1$.
We start with Kodaira symbol $\I_0$. The set $U(\Z_p)^{(\I_0)}$ consists of those $f\in U(\Z_p)$ such that $f(x)\pmod{p}$ has three distinct roots in $\overline{\F}_p$. Denote these roots by $\alpha_1$, $\alpha_2$, and $\alpha_3$. Either the $\alpha_i$ all belong to $\F_p$, or $\alpha_1\in\F_p$ and $\alpha_2,\alpha_3$ are a pair of conjugate elements in $\F_{p^2}\backslash\F_p$, or the $\alpha_i$ are conjugate elements in $\F_{p^3}\backslash\F_p$. Thus, we have $$\begin{array}{rcl}
\displaystyle\Vol(U(\Z_p)^{(\I_0)})&=&
\displaystyle\frac{p(p-1)(p-2)}{6p^3}
+\frac{p(p^2-p)}{2p^3}+\frac{p^3-p}{3p^3}\\[.2in]
&=&\displaystyle 1-\frac{1}{p},
\end{array}$$ as required.
Second, we consider the Kodaira symbol $\I_n$ for $n\geq 1$. Suppose $f(x)\in U(\Z_p)^{\I_n}$. Then $f(x)$ has exactly one double root modulo $p$. We therefore have $f(x) = g(x)(x-\alpha)$, where $g(x)$ has a double root modulo $p$, and $p \nmid
g(\alpha)$. Clearly, we have $\Delta_p(g)=\Delta_p(f)=p^n$, since $\Delta_p(E_f)=p^n$. We write the quadratic factor $g(x)$ in unique form as $g(x)=(x+\beta)^2+\gamma$. The discriminant condition translates to $p^n\parallel\gamma$, and the condition that $p\nmid
g(\alpha)$ translates to $p\nmid (\alpha+\beta)$. Therefore, every element of $U(\Z_p)^{\I_n}$ can be expressed uniquely in the form $$((x+\beta)^2+\gamma)(x-\alpha)=x^3+(2\beta-\alpha)x^2
+(\beta^2-2\alpha\beta+\gamma)x-\alpha\beta^2-\alpha\gamma,$$ such that $p^n\parallel\gamma$ and $p\nmid(\alpha+\beta)$. The Jacobian change of variables for the map $(\alpha,\beta,\gamma)\mapsto(a,b,c)$ is $-2(\alpha+\beta)^2-2\gamma$ which is always a unit. Thus, we have $$\begin{array}{rcl}
\displaystyle\Vol(U(\Z_p)^{\I_n})&=&
\displaystyle\Vol(p^n\Z_p\backslash
p^{n+1}\Z_p)\Vol(\{(\alpha,\beta)\in\Z_p^2:p\nmid(\alpha+\beta)\})\\[.1in]
&=&\displaystyle (p-1)^2/p^{n+2},
\end{array}$$ as required.
Third, we consider the Kodaira symbols $\I\I$, $\I\I\I$, and $\I\rV$. If $f\in U_0(\Z_p)$ is such that the Kodaira symbol of $E_f$ is one of the three above, then $f(x)=x^3+Ax+B$ has a triple root modulo $p$, which implies that $p$ divides $A$ and $B$. By examining the discriminant of $f$ as in the proof of Proposition \[propKSloc\], we see that the Kodaira symbol of $E_f$ is $\I\I$ if and only if $p\mid A$ and $p\parallel B$; $\I\I\I$ if and only if $p\parallel A$ and $p^2\mid B$; and $\I\rV$ if and only if $p^2\mid A$ and $p^2\parallel B$. Hence the volumes of $U_0(\Z_p)^{(T)}$, for $T=\I\I$, $\I\I\I$, and $\I\rV$, are $(p-1)/p^3$, $(p-1)/p^4$, and $(p-1)/p^5$, as required.
Finally, we turn to the large Kodaira symbols, i.e., those corresponding to large elliptic curves. Consider the following map $$\begin{array}{rcl}
\sigma: U(\Z_p)^\sm&\to& U(\Z_p)\\[.1in]
x^3+ax^2+bx+c&\mapsto& x^3+pax^2+p^2bx+p^3c.
\end{array}$$ Clearly, if $S\subset U(\Z_p)$ is any measurable set, then $\Vol(\sigma(S))=p^{-6}\Vol(S)$. Furthermore, we set $\sigma(\I_n) =
\I_n^*$, $\sigma(\I\I) = \I\rV^*$, $\sigma(\I\I\I) = \I\I\I^*$ and $\sigma(\I\rV)=\I\I^*$. Then $\sigma$ sends $f$ of Kodaira symbol $T$ to $\sigma(f)$ of Kodaira symbol $\sigma(T)$. Moreover, we have $\sigma(t\cdot f)=(pt)\cdot\sigma(f)$. Hence we have $$\sigma\bigl(U(\Z_p)^{(T)}\bigr)=\sigma\Bigl(\Z_p\cdot
U_0(\Z_p)^{(T)}\Bigr)
=(p\Z_p)\cdot\sigma\bigl(U_0(\Z_p)^{(T)}\bigr).$$ Fix any $g\in
U_0(\Z_p)^{(\sigma(T))}$. There exists $t\in\Z_p$ such that the coefficients of $t\cdot g$ are as in the second column of Table \[tabloc\]. Hence there exists $f\in U(\Z_p)^{(T)}$ with $\sigma(f)=t\cdot g$. Then $\sigma(f_0)$ is $\Z_p$-equivalent to $g$. Since $\sigma(f_0)$ and $g$ both belong to $U_0(\Z_p)$, we must have $\sigma(f_0)=g$. Hence we have $\sigma\bigl(U_0(\Z_p)^{(T)}\bigr)=U_0(\Z_p)^{(\sigma(T))}$. Therefore, we have
$$\begin{array}{rcl}
\displaystyle\Vol\Bigl(U(\Z_p)^{(\sigma(T))}\Bigr)&=&
\displaystyle\Vol\Bigl(\Z_p\cdot U_0(\Z_p)^{(\sigma(T))}\Bigr)
\\[.2in]&=&
\displaystyle p\cdot \Vol\Bigl(p\Z_p\cdot U_0(\Z_p)^{(\sigma(T))}\Bigr)
\\[.2in]&=&
p\cdot \Vol\Bigl(\sigma\bigl(U(\Z_p)^{(T)}\bigr)\Bigr)
\\[.2in]&=&
p^{-5}\Vol\bigl(U(\Z_p)^{(T)}\bigr).
\end{array}$$
This concludes the proof of Proposition \[propKSden\], and thus of Theorem \[propcasessp\].
Theorem \[propcasessp\] has the following immediate corollary, which will be useful in what follows.
Let $p\geq5$ be a prime. The density of elliptic curves $E$ over $\Q_p$ with good, multiplicative, or additive reduction, such that $\ind(E)=\Delta_p(E)/C_p(E)=p^k$ is as given in Table \[tabden\].
------------------------------------- ----------- --------------------- ----------------------- -----------------------
Index Good Red. Multiplicative Red. Additive Red. Total
$1$ $(p-1)/p$ $(p-1)^2/p^3$ $(p-1)/p^3$ $(p^2-1)/p^2$
\[.05in\] $p$ 0 $(p-1)^2/p^4$ $(p-1)/p^4$ $(p-1)/p^3$
\[.05in\] $p^2$ 0 $(p-1)^2/p^5$ $(p-1)/p^5$ $(p-1)/p^4$
\[.05in\] $p^3$ 0 $(p-1)^2/p^6$ $0$ $(p-1)^2/p^6$
\[.05in\] $p^4$ 0 $(p-1)^2/p^7$ $(p-1)/p^6$ $(2p-1)(p-1)/p^7$
\[.05in\] $p^k$, $k=6,7,8$ 0 $(p-1)^2/p^{k+3}$ $(2p-1)(p-1)/p^{k+3}$ $(3p-2)(p-1)/p^{k+3}$
\[.05in\] $p^k$, $k=5$ or $k\geq 9$ 0 $(p-1)^2/p^{k+3}$ $(p-1)^2/p^{k+3}$ $2(p-1)^2/p^{k+3}$
------------------------------------- ----------- --------------------- ----------------------- -----------------------
: $p$-adic densities of elliptic curves with given index[]{data-label="tabden"}
Fourier coefficients of polynomials with fixed Kodaira symbol {#sec:equi}
=============================================================
Let $p\geq 5$ be a prime, and let $U(\Z_p)^\min$ and $U(\Z_p)^\sm$ be as in §2. Recall that to each $f(x)\in U(\Z_p)^\min$, we associate the Kodaira symbol of the elliptic curve $E_f$. By Proposition \[propKSloc\] and Theorem \[propcasessp\], an element $f(x)\in
U(\Z_p)^\min$ belongs to $U(\Z_p)^\sm$ and satisfies $\Delta(f) \neq
C(f)$ if and only if the Kodaira symbol of $f$ is $\I\I\I$, $\I \rV$, or $\I_n$ for $n \geq 2$. Denote the set of polynomials $f(x)\in
U(\Z)$ such that $f\in U(\Z_p)^\min$ for all primes $p$ by $U(\Z)^\min$. Given $f(x)\in U(\Z)^\min$ and a prime $p$, we say that the [*Kodaira symbol of $f$ at $p$*]{} is $T$, the Kodaira symbol of $f(x)$ considered as an element in $U(\Z_p)^\min$.
Let $\Sigma$ be a set consisting of the following data: a finite set $\{p_1,\ldots,p_k\}$ of primes $p_i\geq 5$ along with a Kodaira symbol $T(p_i)$ which is $\I\I\I$, $\I\rV$ or $\I_{n\geq 2}$ associated to each prime $p_i$ in the set. We say $f\in U(\Z)$ has splitting type $\Sigma$ if $f$ has Kodaira symbol $T(p_i)$ at each prime $p_i$ in $\Sigma$. Let $U(\Z)_\Sigma$ denote the set of elements $f\in U(\Z)$ with splitting type $\Sigma$. Given such a collection $\Sigma$, we define the constant $Q(\Sigma)$ to be $\prod_{p_i}p_i^{a_i}$, where $a_i=1$ if $T(p_i)$ is $\I\I\I$ or $\I\rV$, and $a_i=\lfloor n/2
\rfloor$ if $T(p_i)$ is $\I_n$. Note that if $f\in U(\Z)_\Sigma$, then $Q(\Sigma)\mid Q(f)$. We define $m_{T}(\Sigma)$ to be the product of all primes $p$ such that $T(p) = T$. We also define $m_\odd(\Sigma)$ to be the product of all primes $p$ in $\Sigma$ such that $\sigma(p)=\I_n$ for some *odd* integer $n$. Finally, we define $\nu(\Sigma)$ to be the product over the primes $p$ in $\Sigma$ of the density $\nu(T_p)$, i.e., the $p$-adic volume of the set of elements in $U(\Z_p)^{\min}$ having Kodaira symbol $T(p)$.
Define the height function $H$ on $U(\R)$ to be $$H(x^3+ax^2+bx+c):= \max\{|a|^6,|b|^3,|c|^2\}.$$ The goal of this section is to obtain a bound on the number of elements in $U(\Z)$ that have bounded height and specified Kodaira symbols $\I\I\I$, $\I\rV$ or $\I_{n\geq 2}$ at finitely many primes. We prove the following theorem.
\[thm:equimain\] Let $\Sigma$ be as above and for every Kodaira symbol $T$, denote $Q(\Sigma)$, $m_\odd(\Sigma)$, and $m_T(\Sigma)$ by $Q$, $m_\odd$, and $m_T$, respectively. Then we have $$\#\{f\in U(\Z)_\Sigma:H(f)<Y\}\ll_\epsilon
\frac{Y}{Q^2m_{\I\I\I}m_{\I\rV}^2m_\odd}+\frac{Qm_\odd}{m_{\I\rV}} Y^\epsilon,$$ where the implied constant is independent of $Y$ and $\Sigma$.
This section is organized as follows. First, in §3.1, we recall some preliminary results from Fourier analysis. In particular, the “twisted Poisson summation” formula of Proposition \[propTPS\] will be our main tool in proving Theorem \[thm:equimain\]. Also, in , we determine how the action of $\G_a$ on $U$ changes the Fourier coefficients of functions. Next, in §3.2, we compute the Fourier coefficients of a slighly modified version of the characteristic functions of the set of monic polynomials having Kodaira symbol $T$, for $T=\I\I\I$, $\I\rV$, and $\I_{n\geq
2}$. Finally, in §3.3, we use these computations and the twisted Poisson summation formula to prove Theorem \[thm:equimain\].
Preliminary results from Fourier analysis
-----------------------------------------
We fix a positive integer $N$ with $(N,6)=1$, and consider the space $\widehat{U(\Z/N\Z)}$ dual to $U(\Z/N\Z)$. We write elements $\chi\in
\widehat{U(\Z/N\Z)}$ as triples $\chi=(\check{a},\check{b},\check{c})\in(\Z/N\Z)^3$, and view $\chi$ as the character given by $$\label{eqchiaction}
\chi(x^3+ax^2+bx+c)=
e\Bigl(\frac{\check{a}\cdot a+\check{b}\cdot b+\check{c}\cdot
c}{N}\Bigr),$$ where $e(x) := \exp(2\pi ix)$. Given a function $\phi:U(\Z/N\Z)\to\C$, we have the Fourier dual $\hat{\phi}:\widehat{U(\Z/N\Z)}\to\C$ defined to be $$\hat\phi(\chi):=\sum_{f\in U(\Z/N\Z)}\phi(f)\chi(f),$$ and Fourier inversion yields the equality $$\frac{1}{N^3}\sum_\chi\hat{\phi}(\chi)\overline{\chi(f)}=\phi(f).$$
The additive group $\Z/N\Z$ acts on the space $U(\Z/N\Z)$ via the action $(r\cdot f)(x)=f(x+r)$. Identifying $U(\Z/N\Z)$ with the coefficient space $(\Z/N\Z)^3$, we write the action explicitly: $$r\cdot (a,b,c)=((a+3r),(b+2ra+3r^2),(c+rb+r^2a+r^3)).$$ Given a function $\phi:U(\Z/N\Z)\to\C$ and an element $r\in\Z/N\Z$, we define $r\cdot\phi:U(\Z/N\Z)\to\C$ to be $(r\cdot\phi)(f):=\phi((-r)\cdot f)$. We also define an action of $\Z/N\Z$ on $\widehat{U(\Z/N\Z)}$ by $$\label{eq:action}
r\cdot\chi:=
((\check{a}+2r\check{b}+r^2\check{c}),(\check{b}+r\check{c}),\check{c}),$$ for $\chi=(\ca,\cb,\cc)$. Then we have $$\label{eqFDtrans}
\begin{array}{rcl}
\displaystyle\widehat{r\cdot\phi}(\chi)&=&
\displaystyle\sum_f (r\cdot\phi)(f)\chi(f)
=\displaystyle\sum_f \phi((-r)\cdot f)\chi(f)
=\displaystyle\sum_f \phi(f)\chi(r\cdot f)\\[.2in]
&=&\displaystyle\sum_{f=(a,b,c)}
\phi(f)e\Bigl(\frac{\check{a}a+\check{b}(b+2ra)+\check{c}(c+rb+r^2a)}{N}\Bigr)
e\Bigl(\frac{3\check{a}r+3\check{b}r^2+\check{c}r^3}{N}\Bigr)\\[.2in]
&=&\displaystyle e\Bigl(\frac{3\check{a}r+3\check{b}r^2+\check{c}r^3}{N}\Bigr)
\sum_{f=(a,b,c)} \phi(f)e\Bigl(\frac{(\check{a}+2r\check{b}+r^2\check{c})a+
(\check{b}+r\check{c})b+\check{c}c}{N}\Bigr)\\[.2in]
&=&\displaystyle \Psi_r(\chi)\hat{\phi}(r\cdot\chi),
\end{array}$$ where we set $$\Psi_r(\chi):=e\Bigl(\frac{3\check{a}r+3\check{b}r^2+\check{c}r^3}{N}\Bigr).$$
Note that if we identify elements $\chi=(\check{a},\check{b},\check{c})\in\widehat{U(\Z/N\Z)}$ with binary quadratic forms $$P_\chi(x,y):=\check{a}x^2+2\check{b}xy+\check{c}y^2,$$ then the action of $\Z/N\Z$ on $\widehat{U(\Z/N\Z)}$ in corresponds exactly to the natural action: $$P_{r\cdot
\chi}(x,y)=P_\chi(x,y+rx).$$ We define $\Delta_2(\chi)=\check{b}^2-\check{a}\check{c}$. Then $\Delta_2$ is invariant under the action of $\Z/N\Z$. Throughout the rest of §4, we will thus identify the space $\widehat{U(\Z/N\Z)}$ with the space $V_2(\Z/N\Z)$, where $V_2=\Sym_2(2)$ is the space of binary quadratic forms with middle coefficient a multiple of $2$.
Finally, we recall the following result which follows from the use of Poisson summation combined with the unfolding technique.
\[propTPS\] Let $\psi:U(\R)\to\R$ denote a smooth function with bounded support. Let $\phi:U(\Z/N\Z)\to\R$ be any function. Then, for every positive real number $Y$, we have $$\label{fourier}
\sum_{(a,b,c)\in U(\Z)}\psi\Bigl(
\frac{a}{Y^{1/6}},\frac{b}{Y^{1/3}},\frac{c}{Y^{1/2}}\Bigr)\phi(a,b,c)=
\frac{Y}{N^3}\sum_{\chi=(\check{a},\check{b},\check{c})\in \widehat{U(\Z)}}\hat{\psi}
\Bigl(\frac{Y^{1/6}\check{a}}{N},\frac{Y^{1/3}\check{b}}{N},
\frac{Y^{1/2}\check{c}}{N}\Bigr)
\hat\phi(\check{a},\check{b},\check{c}).$$ The $\hat\psi$ on the right hand side is the usual Fourier transform over $\R$ and so decays faster than any polynomial.
Bounds on Fourier coefficients
------------------------------
Let $p\geq 5$ be a fixed prime. The conditions imposed by the choice of Kodaira symbol $T$ being equal to $\I\I\I$, $\I\rV$ or $\I_{n\geq
2}$ are defined via congruence conditions modulo $N=N_p(T)$, where $N$ is $p^2$, $p^2$ or $p^n$, respectively. Hence, when we refer to an element $f$ having one of the above Kodaira symbols, we will be implicitly assuming that $f$ belongs to $U(\Z/N\Z)$, where $N$ is the appropriate power of $p$. Naturally, in this context, we will also assume that elements $\chi$ belong to $\widehat{U(\Z/N\Z)}$, and represent them as triplets $(\check{a},\check{b},\check{c})\in(\Z/N\Z)^3$.
For a Kodaira symbol $T\in\{\I\I\I,\I\rV,\I_{\geq 2}\}$, we define the set $\CS_0(T)$ to be $$\begin{array}{rcl}
\{x^3+ax^2+bx+c:p\mid a;p\mid b,p^2\mid c\}\subset U(\Z/p^2\Z)&\mbox{if}&T=\I\I\I;\\[.1in]
\{x^3+ax^2+bx+c:p\mid a;p^2\mid b,p^2 \mid c\}\subset U(\Z/p^2\Z)&\mbox{if}&T=\I\rV;\\[.1in]
\{x^3+ax^2+bx+c:p^{n}\mid b,p^{2n}\mid c\}\subset U(\Z/p^{2n}\Z)&\mbox{if}&T=\I_{2n};\\[.1in]
\{x^3+ax^2+bx+c:p^{n+1}\mid b,p^{2n+1}\mid c\}\subset U(\Z/p^{2n+1}\Z)&\mbox{if}&T = \I_{2n+1}.\\
\end{array}$$ From the second column of Table \[tabloc\], it follows that every element having Kodaira symbol $T$ is contained within some $\G_a$ translate of $\CS_0(T)$. Let $\Phi_{0,T}$ denote the characteristic function of $\CS_0(T)$, and define the function $\Phi_{T}$ by $$\Phi_{T} = \sum_{r\in\Z/M\Z}
r\cdot\Phi_{0,T},$$ where $M=M_p(T)$ is $p$ if $T = \I\I\I, \I\rV$, $p^n$ if $T = \I_{2n}$ and $p^{n+1}$ if $T = \I_{2n +1}$. The next lemma, determining the Fourier transforms of the sets $\Phi_{0,T}$, follows quickly from the definitions.
\[lemequi1\] Let $p\geq 5$ be a prime number. Let $T$ be one the three Kodaira symbols, and let $N=N_p(T)$ denote the appropriate power of $p$. For $\chi=(\check{a},\check{b},\check{c})\in \widehat{U(\Z/N\Z)}$, we have $$\begin{array}{rclrcll}
\displaystyle|\widehat{\Phi_{0,\I\I\I}}(\chi)|&=&\left\{
\begin{array}{rcl}
p^2 &\mbox{ if }& p\mid\check{a},\,p\mid\check{b};\\
0 &\mbox{ else; }&
\end{array}
\right.
&\quad&
\displaystyle|\widehat{\Phi_{0,\I\rV}}(\chi)|&=&\left\{
\begin{array}{rcl}
p &\mbox{ if }& p\mid\check{a};\\
0 &\mbox{ else; }&
\end{array}
\right.\\[.3in]
\displaystyle|\widehat{\Phi_{0,\I_{2n}}}(\chi)|&=&\left\{
\begin{array}{rcl}
p^{3n} &\mbox{if}& p^{2n}\mid\check{a},\,p^n\mid\check{b};\\
0 &\mbox{else;}&
\end{array}
\right.
&\quad&
\displaystyle|\widehat{\Phi_{0,\I_{2n+1}}}(\chi)|&=&\left\{
\begin{array}{rcl}
p^{3n+1} &\mbox{if}& p^{2n+1}\mid\check{a},\,p^n\mid\check{b};\\
0 &\mbox{else.}&
\end{array}
\right.
\end{array}$$
As an immediate consequence, \[eqFDtrans\] yield the inequality $$\label{eqtranslatebound}
|\widehat{\Phi_T}(\chi)|\leq p^{k_T}r_T(\chi),$$ where $k_T$ is $2$, $1$, $3n$, or $3n+1$ depending on whether $T$ is $\I\I\I$, $\I\rV$, $\I_{2n}$, or $I_{2n+1}$, respectively, and $r_T(\chi)$ is the number of $r\in \{0,\ldots,M-1\}$ such that $(r\cdot\chi)$ belongs to the support of $\widehat{\Phi_{0,T}}$. To bound $\widehat{\Phi_T}(\chi)$, it then remains to bound $r_T(\chi)$.
\[lemequi2\] We have
1. Let $T=\I\I\I$. Then $r_T(\chi)=0$ unless $p\mid\Delta_2(\chi)$. In that case, $r_T(\chi)=1$ if $p\nmid\chi$ and $r_T(\chi)=p$ otherwise.
2. Let $T=\I\rV$. Then $r_T(\chi)\leq 2$ if $p\nmid\chi$ and $r_T(\chi)=p$ otherwise.
3. Let $T=\I_{2n}$. Then $r_T(\chi)=0$ unless $\chi$ is $\G_a$-equivalent to some element $(0,p^{n+i}\check{b},p^j\check{c})$, for $i$ and $j$ nonnegative integers and $p\nmid\cb\cc$. Then $r_T(\chi)\ll p^{\min(i,\lfloor j/2\rfloor)}$.
4. Let $T=\I_{2n+1}$. Then $r_T(\chi)=0$ unless $\chi$ is $\G_a$-equivalent to some element $(0,p^{n+i}\check{b},p^j\check{c})$, for $i$ and $j$ nonnegative integers and $p\nmid\cb\cc$. Then $r_T(\chi)\ll p^{\min(i,\lceil
j/2\rceil)}$.
We prove the above lemma in the case when $T=\I_{2n}$. Assume that $\chi$ is $\G_a$-equivalent to $(0,p^{n+i}\cb,p^j\cc)$, for $i$ and $j$ nonnegative integers and $p\nmid\cb\cc$. Note that the entry $p^j\cc$ does not change under the $\G_a$-action. Then by definition, we have $$r_T(\chi)=\#\bigl\{r\in\Z/p^n\Z:p^n\mid rp^j,\,
p^{2n}\mid 2p^{n+i}r\cb+r^2p^j\cc\bigr\}.$$ Write $r\in\Z/p^n\Z$ as $r=sp^k+p^n\Z$ with $p\nmid s$. Then the condition on $r$ translates to $$p^n\mid p^{j+k},\quad\quad p^{2n}\mid (2\cb p^{n+i+k}+s\cc p^{j+2k}).$$ We consider two posible cases. First assume that $p^{2n}$ divides both $2\cb p^{n+i+k}$ and $s\cc p^{j+2k}$. Then we have $k\geq\max(n-i,
n-\lfloor j/2\rfloor)$, which implies that there are $p^{\min(i,\lfloor j/2\rfloor)}$ choices for $r$. Otherwise, we have $n+i+k=j+2k=:\ell<2n$, and $p^{2n-\ell}\mid 2\cb+s\cc$. In this case, $s$ is determined modulo $p^{2n-\ell}$, which implies that there are $p^{n-k-(2n-\ell)}=p^{\ell-n-k}$ choices for $r$. Note that $\ell-n-k=i$. Furthermore, we have $j+2k=\ell<2n$, from which it follows that $2(\ell-n-k)=2j+2k-2n<j$. This proves the lemma in the case when $T=\I_{2n}$. The other three cases are similar, and we omit the proof.
Proof of Theorem \[thm:equimain\]
---------------------------------
Let $\Sigma$ be as before. That is, a finite set of primes $p\geq 5$, and a Kodaira symbol $T_p=\I\I\I$, $\I\rV$, or $\I_{n\geq 2}$ for each prime $p$ in this set. For each prime $p$ of $\Sigma$, set $N_p:=N_p(T_p)$ and set $m_{\odd,p}$ to be $p$ if $T_p=\I_{2n+1}$ and $1$ otherwise. Set $Q_p$ to be the $Q$-invariant associated to $T_p$ in Table \[tabloc\]. We define the quantities $N=N(\Sigma)$, $Q=Q(\Sigma)$, and $m_\odd=m_\odd(\Sigma)$ to be the product over all primes $p$ of $\Sigma$ of $N_p$, $Q_p$, and $m_p$, respectively. Note that $N = Q^2m_{\odd}$. Since $Q^2$ divides $\Delta(f)$ for any $f$ with splitting type $\Sigma$, we may assume that $Q$ and also $N$ are bounded above by some fixed power of $Y$.
Then elements with splitting type $\Sigma$ are defined via congruence conditions modulo $N$. Let $\phi:U(\Z/N\Z)\to\R$ denote the characteristic function of elements with splitting type $\Sigma$. Let $\psi:U(\R)\to\R_{\geq 0}$ be a smooth compactly supported function such that $\psi(f)=1$ for $H(f)\leq 1$. We have
$$\label{eqequibase}
\begin{array}{rcl}
\displaystyle\#\{f\in U(\Z)_\Sigma:H(f)<Y\}&\leq&
\displaystyle\sum_{(a,b,c)\in U(\Z)_\Sigma}\psi
\Bigl(\frac{a}{Y^{1/6}},\frac{b}{Y^{1/3}},\frac{c}{Y^{1/2}}\Bigr)
\phi(a,b,c)\\[.2in] &=&
\displaystyle\frac{Y}{N^3}
\sum_{\chi=(\check{a},\check{b},\check{c})\in \widehat{U(\Z)}}\hat{\psi}
\Bigl(\frac{Y^{1/6}\ca}{N},\frac{Y^{1/3}\cb}{N},\frac{Y^{1/2}\cc}{N}\Bigr)
\hat\phi(\ca,\cb,\cc)\\[.2in] &=&
\displaystyle S_0+S_{\ca\cc=0}+S_{\Delta_2=0}+S_{\neq 0},
\end{array}$$
where $S_0$ is the contribution of the term $\chi=0$, $S_{\ca\cc=0}$ is the contribution from the nonzero terms $\chi$ with $\ca\cc=0$, $S_{\Delta_2=0}$ is the contribution from nonzero terms $\chi$ with $\Delta_2(\chi)=0$, and $S_{\neq 0}$ is the contribution from the terms $\chi$ with $\ca\cc\Delta_2(\chi)\neq 0$. We bound each of these quantities in turn.
To begin with, since $\hat{\phi}(0)/N^3=\nu(\Sigma)$ and $\psi$ is compactly supported, we have $$\label{eqequis0}
S_0=\frac{Y}{N^3}\hat{\psi}(0)\hat{\phi}(0)\ll \nu(\Sigma)Y \ll \frac{Y}{Q^2m_{\I\I\I}m_{\I\rV}^2m_\odd},$$ by Table \[tabloc\]. To bound $S_{\ca\cc=0},$ $S_{\Delta_2=0}$ and $S_{\neq 0}$, we have the following immediate consequence of and Lemma \[lemequi2\].
With notations as above, let $\chi = (\ca,\cb,\cc)\in \widehat{U(\Z)}$ with $\hat{\phi}(\chi)\neq 0$. Let $A$ be the largest divisor of $m_{\I\I\I}m_{\I\rV}$ dividing $\ca$, $\cb$ and $\cc$. For each prime $p$ with $T_p = I_{2n}$ or $T_p = I_{2n+1}$ for some $n\geq1$, let $k_p$ be the nonnegative integer with $p^{2n+k_p}\mid\mid \Delta_2(\chi)$. Then $$\label{eq:mj1}
\hat{\phi}(\chi) \ll A \, m_{\I\I\I}^2 \, m_{\I\rV} \prod_{T_p = I_{2n}} p^{3n+k_p/2} \prod_{T_p = I_{2n+1}} p^{3n+1+k_p/2}.$$
Since $\hat\psi$ decays faster than any polynomial, it suffices to consider characters $\chi = (\ca,\cb,\cc)$ such that $$\ca\ll N^{1+\epsilon}/Y^{1/6}, \quad \cb\ll N^{1+\epsilon}/Y^{1/3}, \quad \cc\ll N^{1+\epsilon}/Y^{1/2}.$$ We consider $S_{\ca\cc=0}$ first. Fix a divisor $A$ of $m_{\I\I\I}m_{\I\rV}$ and a nonnegative integer $k_p$ for every prime $p$ with $T_p = I_{\geq2}$. The number of characters $\chi = (0,\cb,\cc)$ such that $A$ is the largest divisor of $m_{\I\I\I}m_{\I\rV}$ dividing $\chi$ and $m_{\I\I\I}m_{\I\rV}\mid \Delta_2(\chi)$ and $p^{2n+k_p}\mid\mid \Delta_2(\chi)$ for every prime $p$ with $T_p = I_{2n}$ or $T_p = T_{2n+1}$ is $$\ll_\epsilon \frac{N^{1+\epsilon}}{m_{\I\I\I}m_{\I\rV}Y^{1/3}} \frac{N^{1+\epsilon}}{AY^{1/2}} \prod_{T_p = I_{2n}\,or\,I_{2n+1}} p^{-n-k_p/2}.$$ The number of choices for $A$ and the $k_p$’s is $\ll Y^\epsilon$. Combining with the bound , we have $$\frac{Y}{N^3}\sum_{\substack{\cb\ll N^{1+\epsilon}/Y^{1/3}\\\cc\ll N^{1+\epsilon}/Y^{1/2}}}
\hat{\phi}(0,\cb,\cc) \ll_\epsilon \frac{Y^{1/6+\epsilon}}{N}m_{\I\I\I}\prod_{T_p = I_{2n}} p^{2n}\prod_{T_p = I_{2n+1}} p^{2n+1} = \frac{Y^{1/6+\epsilon}}{m_{\I\I\I}m_{\I\rV}^2}.$$ To bound the sum of $\hat{\phi}(\ca,\cb,0)$, we need a slight refinement. Fix again a divisor $A$ of $m_{\I\I\I}m_{\I\rV}$ and a nonnegative integer $\ell_p$ for every prime $p$ with $T_p = I_{\geq2}$. Suppose $\chi = (\ca,\cb,0)$ with $p^{n+\ell_p}\mid\mid \cb$ for every prime $p$ with $T_p = I_{2n}$ or $T_p = T_{2n+1}$. In order for $\widehat{\Phi_{T_p}}(\chi)\neq 0$ at these primes $p$, we need also $p^{n+\ell_p}\mid\ca$ by Lemma \[lemequi2\]. If we further require that $A$ is the largest divisor of $m_{\I\I\I}m_{\I\rV}$ dividing $\chi$ and $m_{\I\I\I}m_{\I\rV}\mid \Delta_2(\chi)$, then the number of such $\chi$ is $$\ll_\epsilon \frac{N^{1+\epsilon}}{AY^{1/6}}\frac{N^{1+\epsilon}}{m_{\I\I\I}m_{\I\rV}Y^{1/3}} \prod_{T_p = I_{2n}\,or\,I_{2n+1}} p^{-2n-2\ell_p}$$ and for any such $\chi$, we have $$\hat{\phi}(\chi) \ll A \, m_{\I\I\I}^2 \, m_{\I\rV} \prod_{T_p = I_{2n}} p^{3n+\ell_p} \prod_{T_p = I_{2n+1}} p^{3n+1+\ell_p}.$$ Combining these two bounds gives $$\frac{Y}{N^3}
\sum_{\substack{\ca\ll N^{1+\epsilon}/Y^{1/6}\\\cb\ll N^{1+\epsilon}/Y^{1/3}}}
\hat{\phi}(\ca,\cb,0) \ll_\epsilon \frac{Y^{1/2+\epsilon}}{N}m_{\I\I\I}\prod_{T_p = I_{2n}} p^{n}\prod_{T_p = I_{2n+1}} p^{n+1} = \frac{Y^{1/2+\epsilon}}{Qm_{\I\rV}}.$$ Hence, we have $$\label{eqequisac}
S_{\ca\cc=0} \ll_\epsilon \frac{Y^{1/6+\epsilon}}{m_{\I\I\I}m_{\I\rV}^2} + \frac{Y^{1/2+\epsilon}}{Qm_{\I\rV}}.$$
Next we consider $S_{\Delta_2=0}$. Third, we consider $S_{\Delta_2=0}$. Note that every $\chi\in
\widehat{U(\Z)}$ with $\Delta_2(\chi)=0$ is of the form $(\alpha^2,\alpha\beta,\beta^2)$ with $\alpha,\beta\in\Z$. Fix a divisor $A$ of $m_{\I\I\I}m_{\I\rV}$ and nonnegative integers $\ell_p$ for each prime $p$ with $T_p=I_{\geq2}$. Suppose $A$ is the largest divisor of $m_{\I\I\I}m_{\I\rV}$ dividing $\chi$ and $p^{\ell_p}\mid\mid\beta$ for all $p$ with $T_p = I_{\geq2}$. Then similar to the case of $\hat{\phi}(\ca,\cb,0)$, we also need $p^{\ell_p}\mid\alpha$ in order for $\widehat{\Phi_{T_p}}(\chi)\neq0$, in which case $$\hat{\phi}(\chi) \ll A\, m_{\I\I\I}^2\, m_{\I\rV} \prod_{T_p=I_{2n}} p^{3n+\ell_p} \prod_{T_p = T_{2n+1}}p^{3n+1+\ell_p}.$$ The number of such characters $\chi$ is $$\ll_\epsilon \frac{N^{1/2+\epsilon}}{AY^{1/12}}\frac{N^{1/2+\epsilon}}{AY^{1/4}}\prod_{T_p = I_{2n}\,or\,I_{2n+1}} p^{-2\ell_p}.$$ Combining these two bounds gives $$\label{eqequidelta}
S_{\Delta_2=0}\ll_\epsilon \frac{Y^{2/3+\epsilon}}{N^2}m_{\I\I\I}^2\, m_{\I\rV}\prod_{T_p=I_{2n}} p^{3n} \prod_{T_p = T_{2n+1}}p^{3n+1}=\frac{Y^{2/3+\epsilon}}{Qm_\odd m_{\I\I\I}m_{\I\rV}^2}.$$
Finally, we turn to $S_{\neq 0}$. Once agian, we fix a divisor $A$ of $m_{\I\I\I}m_{\I\rV}$ and a nonnegative integer $k_p$ for each prime $p$ with $T_p = I_{\geq2}$. The number of characters $\chi=(\ca,\cb,\cc)$ such that $A$ is the largest divisor of $m_{\I\I\I}m_{\I\rV}$ dividing $\chi$, $m_{\I\I\I}m_{\I\rV}\mid \Delta_2(\chi)$ and $p^{2n+k_p}\mid\mid\Delta_2(\chi)$ for any prime $p$ with $T_p = I_{2n}$ or $T_p = T_{2n+1}$ is $$\ll_\epsilon Y^\epsilon \frac{N^{1+\epsilon}}{AY^{1/3}} \frac{N^{2+\epsilon}}{m_{\I\I\I}m_{\I\rV}Y^{2/3}}\prod_{T_p = I_{2n}\,or\,I_{2n+1}}p^{-2n-k_p}.$$ Indeed, the above bounds the number of pairs $(\cb,\Delta_2(\chi))$ satisfying the desired divisibility conditions, and given $\cb$ and $\Delta_2(\chi)$, there are $Y^\epsilon$ choices for $\ca$ and $\cc$. Combining with then gives $$\label{eqequineq0}
S_{\neq0} \ll_\epsilon Y^\epsilon m_{\I\I\I}\prod_{T_p=I_{2n}}p^n\prod_{T_p=I_{2n+1}}p^{n+1} = \frac{Qm_\odd}{m_{\I\rV}}Y^\epsilon.$$
Theorem \[thm:equimain\] now follows from , , , , , and the AM-GM inequality.
The family of cubic fields with prescribed shapes {#seccubicfields}
=================================================
A cubic ring is a commutative ring with unit that is free of rank 3 as a $\Z$-module. Given a cubic ring $R$, the [*trace*]{} $\Tr(\alpha)$ of an element $\alpha\in R$ is the trace of the linear map $\times\alpha:R\to R$. The [*discriminant*]{} $\Disc(R)$ of $R$ is then the determinant of the bilinear pairing $$R\times R\to\Z,\quad\quad (\alpha,\beta):=\Tr(\alpha\beta).$$ Given a [*nondegenerate*]{} cubic ring $R$, i.e., a cubic ring $R$ with nonzero discriminant, we then consider the cubic etalé algebras $R\otimes\Q$ over $\Q$ and $R\otimes\R$ over $\R$. There are two possibilities for $R\otimes\R$, namely, $\R^3$ and $\R\oplus\C$. We have $R\otimes\R\cong\R^3$ when $\Disc(R)>0$ (equivalently, when the signature of $R\otimes\Q$ is $(3,0)$) and $R\otimes\R\cong\R\oplus\C$ when $\Disc(R)<0$ (equivalently, when the signature of $R\otimes\Q$ is $(1,2)$).
The ring $R$ embeds as a lattice into $R\otimes\R$ with covolume $\sqrt{\Disc(R)}$. As regarded as this lattice, the element $1\in R$ is part of any Minkowski basis, and so the first successive minima of $R$ is simply $1$. Let $\ell_1(R)\leq \ell_2(R)$ denote the other two successive minima of $R$. We define the [*skewness*]{} of $R$ by $$\sk(R):=\ell_2(R)/\ell_1(R).$$
Given a field $K$, we denote the ring of integers of $K$ by ${{\mathcal O}}_K$, and the class group of $K$ by $\Cl(K)$. For positive real numbers $X$ and $Z$, let $\RR_3^\pm(X,Z)$ denote the set of cubic fields $K$ that satisfy the following two bounds: $X\leq\pm\Disc({{\mathcal O}}_K)<2X$ and $\sk({{\mathcal O}}_K)>Z$. Set $\RR_3(X,Z)$ to be the union $\RR_3^+(X,Z)\cup
\RR_3^-(X,Z)$. In this section, we prove the following result.
\[thm:skew\] Let $X$ and $Z$ be positive real numbers. Then $$\sum_{K\in\RR_3(X,Z)}|\Cl(K)[2]| \ll X/Z,$$ where the implied constants are independent of $X$ and $Z$.
This section is organized as follows. In Section \[sec:4.1\] we recall the parametrization of cubic rings and of $2$-torsion elements in the class groups of cubic rings, in terms of integral orbits for the action of $\GL_2(\Z)$ on $\Sym^3(\Z^2)$ and of $\GL_2(\Z)\times\SL_3(\Z)$ on $\Z^2\otimes\Sym^2(\Z^3)$, respectively. In §\[sec:4.2\] and §\[sec:4.3\], we then prove Theorem \[thm:skew\] using these parametrizations in conjunction with geometry-of-numbers methods.
The parametrization of cubic rings and the $2$-torsion in their class groups {#sec:4.1}
----------------------------------------------------------------------------
In this section, we recall two parametrizations. First, the parametrization of cubic rings, due to Levi [@levicubicparam], Delone–Faddeev [@delonefaddeev], and Gan–Gross–Savin [@gangrosssavin], and second, Bhargava’s parametrization [@hcl2] of elements in the $2$-torsion subgroups of cubic rings. Let $V_3=\Sym^3(2)$ denote the space of binary cubic forms. We consider the [*twisted action*]{} of $\GL_2$ on $V_3$ given by $$(\gamma\cdot f)(x,y):=\frac{1}{\det\gamma}f((x,y)\cdot\gamma),$$ for $\gamma\in\GL_2$ and $f(x,y)\in V_3$. Then we have the following result.
\[thldf\] There is a natural bijection between the set of $\GL_2(\Z)$-orbits on $V_3(\Z)$ and the set of cubic rings.
We collect some well known facts about the above bijection (for proofs and a more detailed discussion, see [@bstcubic §2]). For an integral binary cubic form $f$, we denote the corresponding cubic ring by $R_f$. The bijection is discriminant preserving, i.e., we have $\Delta(f)=\Disc(R_f)$. The ring $R_f$ is an integral domain if and only if $f$ is irreducible over $\Q$. The group of automorphisms of $R_f$ is isomorphic to the stabilizer of $f$ in $\GL_2(\Z)$.
The bijection of Theorem \[thldf\] can be explicitly described as follows: given a cubic ring $R$, consider the map $R/\Z\to\wedge^2(R/\Z)\cong\Z$ given by $r\mapsto r\wedge r^2$. This map is easily seen to be a cubic map and gives the binary cubic form corresponding to $R$. In fact, this map yields the finer bijection $$\label{eqfinbijcub}
V_3(\Z) \longleftrightarrow \{(R,\omega,\theta)\},$$ where $R$ is a cubic ring and $\langle\omega,\theta\rangle$ is a basis for the 2-dimensional $\Z$-module $R/\Z$. The integral binary cubic form corresponding to $(R,\omega,\theta)$ is $f(x,y)$, where $$\label{eqrefbijcub}
(x\omega+y\theta)\wedge(x\omega+y\theta)^2=f(x,y)(\omega\wedge\theta).$$ It is easily seen that the actions of $\GL_2(\Z)$ on $V_3(\Z)$ and on the set of triples $(R,\omega,\theta)$ agree. Here the latter action is given simply by the natural action of $\GL_2(\Z)$ on the basis $\{\omega,\theta\}$ of $R/\Z$.
Let $f$ be an integral binary cubic form, and let $(R,\omega,\theta)$ be the corresponding triple. Fix an element $\alpha=n+a\omega+b\theta$ of $R$, where $n$, $a$, and $b$ are integers and $(a,b)\neq(0,0)$. The ring $\Z[\alpha]$ is a subring of $R$ having finite index denoted $\ind(\alpha)$. It follows from that we have $$\label{eqindalpha}
\ind(\alpha)=f(a,b).$$ Clearly $\ind(\alpha)=\ind(\alpha+n)$ for $n\in\Z$. Finally, we note that the bijections of Theorem \[thldf\] and continue to hold if $\Z$ is replaced by any principal ideal domain [@bswglobal1 Theorem 5].
Next, we describe the parametrization of $2$-torsion ideals in the class groups of cubic rings. Let $W$ denote the space $2\otimes\Sym^2(3)$ of pairs of ternary quadratic forms. For a ring $S$, we write elements $(A,B)\in W(S)$ as a pair of $3\times
3$ symmetric matrices with coefficients in $S$. The group $G_{2,3}=\GL_2\times\SL_3$ acts on $W$ via the action $(\gamma_2,\gamma_3)\cdot
(A,B):=(\gamma_3A\gamma_3^t,\gamma_3B\gamma_3^t)\gamma_2^t$. We have the [*resolvent map*]{} from $W$ to $V_3$ given by $$\begin{array}{rcl}
W&\to& V_3\\
(A,B)&\mapsto&\det(Ax+By).
\end{array}$$ The resolvent map respects the group actions on $W$ and $V_3$: we have $$\label{eqres}
\Res((\gamma_2,\gamma_3)\cdot
(A,B))=(\det\gamma_2)\cdot\Res(A,B).$$
The following result parametrizing $2$-torsion ideals in cubic rings is due to Bhargava [@hcl2 Theorem 4].
\[thbh2tor\] There is a bijection between $\GL_2(\Z)\times\SL_3(\Z)$-orbits on $W(\Z)$ and equivalence classes of triples $(R,I,\delta)$, where $R$ is a cubic ring, $I\subset R$ is an ideal of $R$ having rank-3 as a $\Z$-module, and $\delta$ is an invertible element of $R\otimes\Q$ such that $I^2\subset (\delta)$ and $N(I)^2=N(\delta)$. Here two triples $(R,I,\delta)$ and $(R',I',\delta')$ are equivalent if there exists an isomorphism $\phi:R\to R'$ and an element $\kappa\in
R\otimes\Q$ such that $I'=\phi(\kappa I)$ and $\delta'=\phi(\kappa^2\delta)$. Moreover, the ring $R$ of the triple corresponding to a pair $(A,B)$ is the cubic ring corresponding to $\Res(A,B)$ under the Delone–Faddeev parametrization.
When $R=R_f$ is the maximal order in a cubic field $K$, the above result gives a bijection between the set of $\GL_2(\Z)\times\SL_3(\Z)$-orbits on the set of pairs $(A,B)\in W(\Z)$ with resolvent $f$, and the set of equivalence classes of pairs $(I,\delta)$, where $I$ is an ideal of $R$, $\delta\in K$ and $I^2=(\delta)$. This latter set is termed the [*$2$-Selmer group*]{} $\Sel_2(K)$ of $K$ (see Definition 5.2.4 and Proposition 5.2.8 of [@cohenadvancedtopics]) and fits into the exact sequence $$1\to R^\times/(R^\times)^2\to\Sel_2(K)\to\Cl(K)[2]\to 1,$$ where $R^\times$ denotes the unit group of $K$. Thus bounds on the $2$-Selmer group of $K$ directly imply bounds on the $2$-torsion subgroup of the class group of $K$.
The number of cubic fields with bounded discriminants and skewed rings of integers {#sec:4.2}
----------------------------------------------------------------------------------
The goal of this section is to prove the following result.
\[propfewskf\] Let $X$ and $Z$ be positive real numbers. There exists some constant $C$ such that $\RR^\pm_3(X,Z)$ is empty if $Z>CX^{1/6}$. Otherwise $|\RR^\pm_3(X,Z)|=O(X/Z)$.
For any subset $S$ of $V_3(\R)$, let $S^\pm$ denote the set of elements $f$ such that $\pm\Delta(f)>0$. Then $V_3(\R)^+$ (resp. $V_3(\R)^-$) consists of a single $\GL_2(\R)$-orbit and corresponds to the cubic algebra $\R^3$ (resp. $\R\oplus\C$). We denote this cubic $\R$-algebra by $R^\pm$. Let $\FF_2$ denote Gauss’ fundamental domain for the action of $\GL_2(\Z)$ on $\GL_2(\R)$. We write elements of $\GL_2(\R)$ in Iwasawa coordinates, in which case we have $$\FF_2=\{n\alpha k\lambda:n\in N'(t),\alpha(t)\in A',k\in K,\lambda\in\Lambda\},$$ where, $$\label{nak}
N'(t)= \left\{\left(\begin{array}{cc} 1 & {} \\ {u} & 1 \end{array}\right):
u\in\nu(t) \right\} , \;\;
A' = \left\{\left(\begin{array}{cc} t^{-1} & {} \\ {} & t \end{array}\right):
t\geq \sqrt[4]3/\sqrt2 \right\}, \;\;
\Lambda = \left\{\left(\begin{array}{cc} \lambda & {} \\ {} & \lambda
\end{array}\right):
\lambda>0 \right\},$$ and $K$ is as usual the (compact) real orthogonal group ${\rm
SO}_2(\R)$; here $\nu(t)$ is a union of one or two subintervals of $[-\frac12,\frac12]$ depending only on the value of $t$. Elements $n\alpha(t) k\lambda$ are expressed in their Iwasawa coordinates as $(n,t,\lambda,k)$. Fix compact sets $B^\pm\subset V_3(\R)^\pm$ that are closures of open bounded sets. Then for every point $v\in B^\pm$, the set $\FF_2\cdot v$, viewed as a multiset, is a cover of a fundamental domain for the action of $\GL_2(\Z)$ on $V_3(\R)^\pm$ of absolutely bounded degree. Recall that for a cubic ring $R$, its skewness $\text{sk}(R)$ is defined to be the quotient $\ell_2(R)/\ell_1(R)$ where $1,\ell_1(R),\ell_2(R)$ are the successive minima of $R$, regarded as a lattice inside $R\otimes\R$. We have the following lemma.
\[lemsk\] Let $v\in B^\pm$ be any binary cubic form. Let $\gamma=(n,t,\lambda,k)\in\FF_2$ be such that $f=\gamma\cdot v$ is an integral binary cubic form. Then we have $$\sk(R_f)\asymp t^2$$ where $R_f$ denotes the cubic ring corresponding to $f$.
Every binary cubic form $v$ in $V_3(\R)^\pm$ gives rise to the cubic algebra $R^\pm$, where $R^+\cong \R^3$ and $R^-\cong\C\oplus\R$, along with elements $\alpha_v$ and $\beta_v$ such that $\langle
1,\alpha_v,\beta_v\rangle$ form a basis for $R^\pm$. Furthermore, the lattice spanned by $1$, $\alpha_v$, and $\beta_v$ has covolume $\sqrt{|\Delta(v)|}$. Since $B^\pm$ is compact, it follows that we have $|\alpha_v|\cdot|\beta_v|\ll\sqrt{|\Delta(v)|}$ for $v\in B^\pm$. Additionally, the action of $\GL_2(\R)$ on $V_3(\R)$ agrees with the action of $\GL_2(\R)$ on pairs $(\alpha_v,\beta_v)$ by linear change of variables. That is, we have $(\alpha_{\gamma\cdot
v},\beta_{\gamma\cdot v})=\gamma\cdot(\alpha_v,\beta_v)$.
Let $f=\gamma\cdot v$ be an integral binary cubic form as in the statement of the lemma. Since $\gamma\in\FF_2$, it follows that $|\alpha_f|\asymp \lambda t^{-1}$ and $|\beta_f|\asymp \lambda t$. As a consequence, $|\alpha_f|\cdot|\beta_f|\asymp\sqrt{\Disc(f)}$. Therefore, we have $\ell_2(R_f)/\ell_1(R_f)\asymp |\beta_f|/|\alpha_f|\asymp t^2$ as necessary.
Next, we have the following lemma, due to Davenport [@davenport-volume1], that estimates the number of lattice points within regions of Euclidlean space.
\[davlem\] Let $\mathcal R$ be a bounded, semi-algebraic multiset in $\R^n$ having maximum multiplicity $m$, and that is defined by at most $k$ polynomial inequalities each having degree at most $\ell$. Then the number of integral lattice points $($counted with multiplicity$)$ contained in the region $\mathcal R$ is $$\Vol(\mathcal R)+ O(\max\{\Vol(\bar{\mathcal R}),1\}),$$ where $\Vol(\bar{\mathcal R})$ denotes the greatest $d$-dimensional volume of any projection of $\mathcal R$ onto a coordinate subspace obtained by equating $n-d$ coordinates to zero, where $d$ takes all values from $1$ to $n-1$. The implied constant in the second summand depends only on $n$, $m$, $k$, and $\ell$.
We are now ready to prove Proposition \[propfewskf\].
**Proof of Proposition \[propfewskf\]:** A general version of the first claim of the proposition, applying to number fields of all degrees, is obtained in [@bstttz Theorem 3.1], and further generalizations are proved in [@Lapierrethesis]. For completeness, we include a proof for our case below. Let $K$ be a cubic field whose ring of integers ${{\mathcal O}}_K$ belongs to $\RR^\pm_3(X,Z)$, and let $\langle 1,\alpha,\beta\rangle$ be a Minkowski basis for ${{\mathcal O}}_K$ with $|\alpha|\leq|\beta|$. Consider the ring $\Z[\alpha]$ which is a suborder of ${{\mathcal O}}_K$. We have $$X^{1/2}\asymp\sqrt{\Disc({{\mathcal O}}_K)}\ll\sqrt{\Disc(\Z[\alpha])}\ll|\alpha|^3,$$ and it follows that $|\alpha|\gg X^{1/6}$. Since $|\alpha||\beta|\asymp
X^{1/2}$, we have $Z=|\beta|/|\alpha|\ll X^{1/2}/X^{2/6}=X^{1/6}$ and the first claim of the proposition follows.
We now estimate $|\RR^\pm_3(X,Z)|$ under the assumption that $Z\ll
X^{1/6}$ following the setup of [@bstcubic §5]. Let $v$ be an element of the compact set $B^\pm$. If $(n,t,\gamma,k)\cdot v$ corresponds to an cubic ring $R$ with $X\leq\Disc(R)<2X$ and $\sk(R)>Z$, then it follows that $\lambda\asymp X^{1/4}$ and $t\gg
Z^{1/2}$, respectively, where the latter fact follows from Lemma \[lemsk\]. Hence we have $$\begin{array}{rcl}
\displaystyle |\RR^\pm_3(X,Z)|&\leq&
\displaystyle\int_{\substack{g=(n,t,\lambda,k)\in\FF_2\\\lambda\asymp X^{1/4}\\t\gg Z^{1/2}}}
\#\{g\cdot B^\pm\cap V_3(\Z)^\irr\}dg\\[.2in]&\leq&
\displaystyle
\int_{\substack{g=(n,t,\lambda,k)\in\FF_2\\\lambda\asymp X^{1/4}\\Z^{1/2}\ll t\ll X^{1/12}}}
\#\{g\cdot B^\pm\cap V_3(\Z)\}dg,
\end{array}$$ where the second inequality follows from the observation that if $t
\gg X^{1/12}$, then every element $f(x,y)$ in $g\cdot B^\pm$ has $x^3$-coefficient less than $1$ in absolute value. Therefore no such integral element $f(x,y)$ can be irreducible since its $x^3$-coefficient must be $0$. Applying Proposition \[davlem\] on the set $g\cdot B^\pm$, we obtain $$\begin{array}{rcl}
|\RR^\pm_3(X,Z)|&\ll&
\displaystyle\int_{\lambda\asymp X^{1/4}}\int_{Z^{1/2}\ll t\ll X^{1/12}}
(\lambda^4+\lambda^3t^3)t^{-2}d^\times td^\times\lambda
\\[.2in]&\ll&\displaystyle\frac{X}{Z}+X^{5/6}\ll\frac{X}{Z},
\displaystyle
\end{array}$$ since $Z\ll X^{1/6}$. The proposition follows. $\Box$
We end this subsection with a counting result on the number of primitive algebraic integers in a cubic field of bounded size to be used in Section \[sec:ellipcount\]. We say an element $\alpha$ in a ring $R$ is [*primitive*]{} if $\alpha\neq n\beta$ for any $\beta\in
R$ and any integer $n\geq2$. We use the superscript $\Tr=0$ to denote the subset of elements of trace $0$.
\[lemkalcountel\] Let $K$ be a cubic field with discriminant $D$. For any real number $Y>0$, let $N_K(Y)$ denote the number of primitive elements $\alpha\in{{\mathcal O}}_K^{\Tr=0}$ with $|\alpha|<Y$. Then $$\label{lem321eqt}
N_K(Y)\leq\left\{
\begin{array}{ccl}
0 &\mbox{if}& Y<\ell_1(K);\\[.1in] 1 &\mbox{if}& \ell_1(K)\leq
Y<\ell_2(K);\\[.1in]
\frac{Y^2}{\sqrt{D}}+O\bigl(\frac{Y}{\ell_1(K)}\bigr) &\mbox{if}&
\ell_2(K)\leq Y.
\end{array}\right.$$
Note that if $\ell_2(K)\leq Y$, then $\frac{Y}{\ell_1(K)}\ll\frac{Y^2}{\sqrt{D}}$ and so we simply have $N_K(Y)\ll\frac{Y^2}{\sqrt{D}}$, which is the best possible bound in this case.
The first two lines of are clearly true. In fact, they are equalities. Then final claim follows from Proposition \[davlem\] by replacing $N_K(Y)$ by the overcount where we count all (not merely primitive) traceless elements $\alpha\in{{\mathcal O}}_K$, since ${{\mathcal O}}_K^{\Tr=0}$ considered as a lattice inside $(K\otimes\R)^{\Tr=0}$ has covolume $\sqrt{D}$.
The 2-torsion subgroups in the class groups of cubic fields {#sec:4.3}
-----------------------------------------------------------
Let $K$ be a cubic field, and let $f\in V_3(\Z)$ be the binary cubic form corresponding to ${{\mathcal O}}_K$, the ring of integers of $K$. A consequence of Theorem \[thbh2tor\] is that the set of $2$-torsion elements in the class group of $K$ injects into the set of $\SL_3(\Z)$-orbits on the elements $(A,B)\in W(\Z)$ satisfying $\Res(A,B)=f$.
Choose Iwasawa coordinates $(n,t,\lambda,k_2)$ for $\GL_2(\R)$ as in the previous subsection and $(u,s_1,s_2,k_3)$ for $\SL_3(\R)$ as in [@manjulcountquartic §2.1]. A Haar-measure for $\SL_3(\R)$ in these coordinates is $s_1^{-6}s_2^{-6}dudk_3d^\times s_1d^\times s_2$. Let $\FF_3$ denote a fundamental domain for the action of $\SL_3(\Z)$ on $\SL_3(\R)$, such that $\FF_3$ is contained within a standard Seigel domain in $\SL_3(\R)$. Then $\FF_{2,3}:=\FF_2\times\FF_3$ is a fundamental domain for the action of $G_{2,3}(\Z)$ on $G_{2,3}(\R)$. There are four $G_{2,3}(\R)$-orbits having nonzero discriminant on $W(\R)$, and we denote them by $W(\R)^{(i)}$, $1\leq
i\leq 4$. For each $i$, let $\B_i\subset W(\R)^{(i)}$ be compact sets, which are closures of open sets, such that $\Res(\B_i)\subset B^+\cup
B^-$, where $B^+$ and $B^-$ are as in the previous subsection. For each element $w\in\B_i$, the set $\FF_{2,3}\cdot w$ is a cover of a fundamental domain for the action of $G_{2,3}(\Z)$ on $W(\R)^{(i)}$. Let $\B$ denote the union of the $\B_i$.
Next, let $W(\Z)^\irr$ denote the set of elements $(A,B)\in W(\Z)$ such that the resolvent of $(A,B)$ corresponds to an integral domain, and such that $A$ and $B$ have no common root in $\P^2(\Q)$. Elements in $W(\Z)$ that are not in $W(\Z)^\irr$ are said to be [*reducible*]{}. Given a reducible element $w$ with resolvent $f$, either $R_f$ is not an integral domain or $w$ corresponds to the identity element in the class group of $R_f$. We now have the following lemmas.
\[lemcgskew\] Let $g=(g_2,g_3)$ be an element in $\FF_{2,3}$, where $g_2=(n,t,k_2,\lambda)\in\FF_2$ and $g_3\in\FF_3$. Let $(A,B)$ be an integral element in $g\cdot\B$ such that $\Res(A,B)=f$. Then we have $$\Delta(f)\asymp \lambda^{12};\quad\sk(R_f)\asymp t^2.$$
The lemma follows immediately from in conjunction with Lemma \[lemsk\] and the fact that $\Delta$ is a degree-$4$ homogeneous polynomial in the coefficients of $V_3$.
\[lemcgred\] Let $(A,B)$ be an element in $W(\Z)$. Denote the coefficients of $A$ and $B$ by $a_{ij}$ and $b_{ij}$, respectively. If $\det(A)=0$ or $a_{11}=b_{11}=0$, then $(A,B)$ is reducible.
If $\det(A)=0$, then the cubic resolvent of $(A,B)$ has $x^3$-coefficient $0$, implying that $(A,B)$ is reducible. If $a_{11}=b_{11}=0$ then $A$ and $B$ have a common zero in $\P^2(\Q)$, implying that $(A,B)$ corresponds to the identity element in the class group of $R_f$.
We are now ready to prove the second claim of Theorem \[thm:skew\].
**Proof of Theorem \[thm:skew\]:** We follow the setup and methods of [@manjulcountquartic]. To begin with, averaging over $w\in \B$ as in [@manjulcountquartic (6) and (8)], we obtain $$\begin{array}{rcl}
&&\displaystyle\sum_{K\in\RR^\pm_3(X,Z)}(|\Cl(K)[2]|-1)\\[.1in]&\ll&
\displaystyle\int_{g\in\FF_{2,3}}
\bigl|\bigl\{w\in g\cdot\B\cap W(\Z)^\irr:K_{\Res(w)}\in\RR_3(X,Z)
\bigr\}\bigr|dg\\[.2in]
&\ll&
\displaystyle\int_{s_1,s_2,t\gg 1}
\bigl|\bigl\{w\in ((\lambda,t),(s_1,s_2))\cdot\B\cap W(\Z)^\irr:
K_{\Res(w)}\in \RR_3(X,Z)
\bigr\}\bigr|\frac{d^\times\lambda d^\times t d^\times s_1 d^\times s_2}{t^{2}
s_1^{6}s_2^{6}},
\end{array}$$ where $K_f$ denotes the algebra $\Q\otimes R_f$ for an integral binary cubic form $f$.
The action of an element $((\lambda,t),(s_1,s_2))\in\FF_{2,3}$ on $W(\R)$ multiplies each coordinate $c_{ij}$ of $W$ by a factor which we denote by $w(c_{ij})$. For example, we have $w(a_{11})=\lambda
t^{-1}s_1^{-4}s_2^{-2}$. The volume of $\B$ is some positive constant, and when $\B$ is translated by an element $((\lambda,t),(s_1,s_2))$, the volume is multiplied by a factor of $\lambda^{12}$, the product of $w(c_{ij})$ over all the coordinates $c_{ij}$. Furthermore, the maximum of the volumes of the projections of $((\lambda,t),(s_1,s_2))\cdot \B$ is $$\ll\prod_{c_{ij}\in S}w(c_{ij})=\prod_{c_{ij}\not\in S}\lambda^{12}w(c_{ij}),$$ where $S$ denotes the set of coordinates $c_{ij}$ of $W(\R)$ such that the length of the projection of $((\lambda,t),(s_1,s_2))\cdot \B$ onto the $c_{ij}$-coordinate is at least $\gg 1$.
For the set $((\lambda,t),(s_1,s_2))\cdot \B\cap W(\Z)^\irr$ to be empty, it is necessary that the projection of $\B':=((\lambda,t),(s_1,s_2))\cdot \B$ onto the $b_{11}$-coordinate is $\gg 1$. Otherwise, every integral element of $\B'$ has $a_{11}=b_{11}=0$, and is hence reducible by Lemma \[lemcgred\]. Similary, the projections of $\B'$ onto the $a_{13}$- and $a_{22}$-coordinates are also $\gg 1$ (since otherwise every integral element $(A,B)$ of $\B'$ satisfy $\det(A)=0$). Finally, for $\B'\cap W(\Z)$ to contain an element whose resolvent cubic form corresponds to a field in $\RR_3(X,Z)$, we must have $\lambda\asymp
X^{1/12}$ and $Z^{1/2}\ll t\ll X^{1/2}$ by Lemma \[lemcgskew\].
Therefore, applying Proposition \[davlem\] to the sets $((\lambda,t),(s_1,s_2))\cdot \B$, we obtain $$\begin{array}{rcl}
\displaystyle\sum_{K\in\RR^\pm_3(X,Z)}(|\Cl(K)[2]|-1)&\ll&
\displaystyle\int_{\substack{\lambda,t,s_1,s_2\\\lambda\asymp X^{1/12}\\Z^{1/2}\ll t\ll X^{1/12}\\s_1,s_2\gg 1}}
\bigl(\lambda^{12}(1+w(a_{11})^{-1}+w(a_{11}a_{12})^{-1}\bigr)
\frac{d^\times\lambda d^\times t d^\times s_1 d^\times s_2}{t^{2}s_1^{6}s_2^{6}}
\\[.3in]&\ll&\displaystyle
\int_{\substack{\lambda,t,s_1,s_2\\\lambda\asymp X^{1/12}\\Z^{1/2}\ll t\ll X^{1/12}\\s_1,s_2\gg 1}}
\bigl(\lambda^{12}+s_1^4s_2^2t\lambda^{11}+s_1^4s_2^4t^2\lambda^{10}\bigr)
\frac{d^\times\lambda d^\times t d^\times s_1 d^\times s_2}{t^{2}s_1^{6}s_2^{6}}
\\[.35in]&\ll&\displaystyle
X/Z+X^{11/12}/Z^{1/2}+X^{5/6+\epsilon},
\end{array}$$ which is sufficient since $Z\ll X^{1/6}$. Theorem \[thm:skew\] now follows from this bound and Proposition \[propfewskf\].
Embedding into the space of binary quartic forms {#sec:Qinv}
================================================
Recall that $U_0(\Z)$ denotes the set of monic cubic polynomials with zero $x^2$-coefficient, and $U_0(\Z)^\min$ denotes the set of elements $f(x)\in U_0(\Z)$ such that the elliptic curve $y^2=f(x)$ has minimal discriminant among all its quadratic twists. We define the height function $H:U_0(\Z)\to\R_{\geq 0}$ by $$H(x^3+Ax+B)=\max\{4|A|^3,27B^2\}.$$ For $f(x)\in U_0(\Z)$, we write $K_f = \Q[x]/(f(x))$, $R_f =
\Z[x]/(f(x))$, and let ${{\mathcal O}}_f$ denote the maximal order in $K_f$. The $Q$-invariant $Q(f)$ of $f$ is defined as the index of $R_f$ in ${{\mathcal O}}_f$, and $D(f)$ is defined to be the discriminant of $K_f$. Observe from Table \[tabloc\] that for primes $p$ of type $\I\I\I$, $\I\rV$ and $\I_{2n+1}$, we have $p\mid Q(f)$ and $p\mid D(f)$. Note also $\gcd(Q(f),D(f))$ is squarefree.
In this section, we obtain a bound on the number of elements $f\in
U_0(\Z)^\min$, having bounded height, such that both $Q(f)$ and $\gcd(Q(f),D(f))$ are large.
\[qinvmt\] Let $Q$ and $q$ be positive real numbers with $Q\geq q$. Let $N_{Q,q}(Y)$ denote the number of elements $f(x)\in U_0(\Z)^\min$ such that $H(f)<Y$, $|Q(f)|>Q$, and $\gcd(Q(f),D(f))>q$. Then $$N_{Q,q}(Y)\ll_\epsilon
\frac{Y^{5/6+\epsilon}}{qQ}
+ \frac{Y^{7/12+\epsilon}}{Q^{1/2}},$$ where the implied constant is independent of $Q$, $q$ and $Y$.
This section is organized as follows. First, in §5.1, we collect classical results on the invariant theory of the action of $\PGL_2$ on the space $V_4$ of binary quartic forms, and summarize the reduction theory of binary quartics developed in [@bs2sel]. Next, in §5.2, we restrict to the space $V_4(\Z)^\red$ of binary quartic forms with a linear factor. We develop the invariant theory for the action of $\PGL_2$ on this space, and construct an embedding $U_0(\Z)^\min\to V_4(\Z)^\red$.
In Sections 5.3, 5.4, and 5.5, we estimate the number of $\PGL_2(\Z)$-orbits on elements in $V_4(\Z)^\red$ with bounded height and large $Q$-invariant and whose $Q$- and $D$-invariants have a large common factor. We do this by fibering the space $V_4(\Z)^\red$ by their roots in $\P^1(\Z)$. Given an element $r\in\P^1(\Z)$, the set of elements in $V_4(\Z)$ that vanish on $r$ is a lattice ${\mathcal{L}}_r$. We then count the number of elements in ${\mathcal{L}}_r$, using the Ekedahl sieve to exploit the condition that $\gcd(Q,D)$ is large.
The action of $\PGL_2$ on the space $V_4$ of binary quartic forms
-----------------------------------------------------------------
Let $V_4$ denote the space of binary quartic forms. The group $\PGL_2$ acts on $V_4$ as follows: given $\gamma\in\GL_2$ and $g(x,y)\in V_4$, define $$(\gamma\cdot g)(x,y):=\frac{1}{(\det \gamma)^2}\,g((x,y)\cdot\gamma).$$ It is easy to check that the center of $\GL_2$ acts trivially. Hence this action of $\GL_2$ on $V_4$ descends to an action of $\PGL_2$ on $V_4$.
The ring of invariants for the action of $\PGL_2(\C)$ on $V_4(\C)$ is freely generated by two elements, traditionally denoted by $I$ and $J$. Explicitly, for $g(x,y)=ax^4+bx^3y+cx^2y^2+dxy^3+ey^4$, we have $$\begin{aligned}
I(g) &=& 12ae-3bd+c^2,\\
J(g) &=& 72ace+9bcd-27ad^2-27eb^2-2c^3.\end{aligned}$$
We collect results from [@bs2sel §2.1] on the reduction theory of integral binary quartic forms. For $i=0,1,2$, we let $V_4(\R)^{(i)}$ to be the set of elements in $V(\R)$ with nonzero discriminant, $i$-pairs of complex conjugate roots, and $4-2i$ real roots. Furthermore, we write $V_4(\R)^{(2)}=V_4(\R)^{(2+)}\cup
V_4(\R)^{(2-)}$ as the union of forms that are positive definite and negative definite. The four sets $L^{(i)}$ for $i\in\{0,1,2+,2-\}$ constructed in [@bs2sel Table 1] satisfy the following two properties: first, $L^{(i)}$ are fundamental sets for the action of $\R_{>0}\cdot\PGL_2(\R)$ on $V_4(\R)^{(i)}$ where $\R$ acts via scaling, and second, the sets $L^{(i)}$ are absolutely bounded. It follows that the sets $R^{(i)}:=\R_{>0}\cdot L^{(i)}$ are fundamental sets for the action of $\PGL_2(\R)$ on $V_4(\R)^{(i)}$, and that the coefficients of an element $f(x,y)\in R^{(i)}$ with $H(f)=Y$ are bounded by $O(Y^{1/6})$.
For $A'$, $N'(t)$, and $K$ defined in , set $$\FF_0=\{n\alpha(t)k:n(u)\in N'(t),\alpha(t)\in A',k\in K\}.$$ Then $\FF_0$ is a fundamental domain for the left multiplication action of $\PGL_2(\Z)$ on $\PGL_2(\R)$; and the multisets $\FF_0\cdot R^{(i)}$ are $n_i$-fold fundamental domains for the action of $\PGL_2(\Z)$ on $V(\R)^{(i)}$, where $n_0=n_{2\pm}=4$ and $n_1=2$. Let $S\subset V_4(\Z)^{(i)}=V_4(\Z)\cap V_4(\R)^{(i)}$ be any $\PGL_2(\Z)$-invariant set. Let $N_4(S;X)$ denote the number of $\PGL_2(\Z)$-orbits on $S$ with height bounded by $X$ such that each orbit $\PGL_2(\Z)\cdot f$ is counted with weight $1/\#\Stab_{\PGL_2(\Z)}(f)$. Let $G_0\subset\PGL_2(\R)$ be a nonempty open bounded $K$-invariant set, and let $d\gamma=t^{-2}dnd^\times tdk$, for $\gamma=ntk$ in Iwasawa coordinates, be a Haar-measure on $\PGL_2(\R)$. Then, identically as in [@bs2sel Theorem 2.5], we have the following result.
\[thbqfcavg\] We have $$N_4(S;X)=\frac{1}{n_i\Vol(G_0)}
\int_{\gamma\in\FF_0}\#\{S\cap \gamma G_0\cdot R^{(i)}_X\}\,d\gamma,$$ where $R^{(i)}_X$ denotes the set of elements in $R^{(i)}$ with height bounded by $X$, the volume of $G_0$ is computed with respect to $d\gamma$, and for any set $T\subset V(\R)$, the set of elements in $T$ with height less than $X$ is denoted by $T_X$.
Apart from its use in this section to obtain a bound on reducible binary quartic forms, Theorem \[thbqfcavg\] will also be used in Section 7 to prove Theorem \[thmsel\].
Embedding $U_0(\Z)^\min$ into the space of reducible binary quartics
--------------------------------------------------------------------
Let $f(x)=x^3+Ax+B$ be an element in $U_0(\Z)^\min$ with $Q(f)=n$. From Theorem \[propcasessp\], it follows that there exists an integer $r$, defined uniquely modulo $n$, such that $f(x+r)$ is of the form $$f(x+r)=x^3+ax^2+bnx+cn^2.$$ Assume that we have picked $r$ so that $0\leq r<n$. The ring of integers ${{\mathcal O}}_f$ in $K_f=\Q[x]/f(x)$ corresponds, under the Delone–Faddeev bijection, to the binary cubic form $$h(x,y)=nx^3+ax^2y+bxy^2+cy^3.$$
Elements in $U_0(\Z)^\min$ with $Q$-invariant $n$ thus correspond to integral binary cubic forms that represent $n$. However, this latter condition is difficult to detect, at least using geometry-of-numbers methods. Instead, we embed the space of binary cubic forms into the space $V_4(\Z)^\red$ of binary quartic forms with a linear factor over $\Q$ by multiplying by $y$. In fact, we will replace $V_4(\Z)^\red$ with its (at most $4$ to $1$) cover $\widetilde{V}_4(\Z)$ consisting of pairs $(g(x,y),[\alpha,\beta])$, where $g$ is a reducible binary quartic forms and $[\alpha,\beta]$ is a root of $f$. Explicitly, $$\widetilde{V}_4(\Z):=\{(g(x,y),[\alpha,\beta]):0\neq g(x,y)\in
V_4(\Z)^\red,\;\alpha,\beta\in\Z,\;\gcd(\alpha,\beta)=1,\;g(\alpha,\beta)=0\}$$ This gives us the following map $\tilde{\sigma}:U_0(\Z)^\min\to
\widetilde{V}_4(\Z)$:
$$\label{eqsig}
\begin{array}{rcccl}
\tilde{\sigma}:U_0(\Z)^\min&\to& V_3(\Z)&\to& \widetilde{V}_4(\Z)\\[.1in]
f(x)&\mapsto &h(x,y)&\mapsto&(yh(x,y),[1,0]).
\end{array}$$
The group $\PGL_2(\Z)$ acts on $\widetilde{V}_4(\Z)$ via $$\gamma\cdot(g(x,y), [\alpha:\beta]) =
((\gamma\cdot g)(x,y),[\alpha:\beta]\gamma^{-1});$$ this is an action since $(\gamma\cdot
g)((\alpha,\beta)\gamma^{-1})=g(\alpha,\beta)=0$. Aside from the classical invariants $I$ and $J$, this action has an extra invariant, which we denote by $Q$, defined as follows. Given $(g,[\alpha:\beta])\in \widetilde{V}_4(\Z)$, let $h(x,y) = g(x,y)/(\beta x-\alpha y)$ be the associated binary cubic form and we define $$\label{eqQexp}
Q(g,[\alpha:\beta]) = h(\alpha,\beta),\qquad D(g,[\alpha:\beta])) = \Delta(h).$$ The $Q$-, $D$-invariants and the discriminant are related by $$\Delta(g) = Q(g,[\alpha:\beta])^2D(g,[\alpha:\beta]).$$
We now have the following result.
There is an injective map $$\sigma: U_0(\Z)^\min\rightarrow \widetilde{V}_4(\Z)\rightarrow \PGL_2(\Z)
\backslash \widetilde{V}_4(\Z),$$ such that for every $f\in U_0(\Z)^\min$, we have $$\label{eq:cond}
I(f)=I(\sigma(f));\qquad J(f)=J(\sigma(f);\qquad Q(f) = Q(\sigma(f));
\qquad D(f)=D(\sigma(f)).$$
The first three equalities of can be checked by a direct computation. The injectivity of $\sigma$ then follows from the fact that $I(f)$ and $J(f)$ determine $f$. Finally, the last equality of can be directly obtained: $$D(f)=\Delta(f)/Q(f)^2=\Delta(\sigma(f))/Q(\sigma(f))^2=D(\sigma(f)),$$ where the second equality follows since $(I(f),J(f))=(I(\sigma(f),J(\sigma(f)))$ and so $\Delta(f) = \Delta(\sigma(f))$.
Therefore, to prove Theorem \[qinvmt\], it suffices to count $\PGL_2(\Z)$-orbits $(g,[\alpha:\beta])$ in $\tilde{V}_4(\Z)$, such that both $Q(g,[\alpha:\beta])$ and the radical $\rad(\gcd(Q(g,[\alpha:\beta]),D(g,[\alpha:\beta])))$ are large.
Counting $\PGL_2(\Z)$-orbits on reducible binary quartic forms
--------------------------------------------------------------
We use the setup of [@bs2sel §2], which is recalled in §5.1. Since the sets $L^{(i)}$ are absolutely bounded, the coefficients of any element in $R^{(i)}=\R_{>0}\cdot L^{(i)}$ having height $Y$ are bounded by $O(Y^{1/6})$. Hence the same is true of every element in $G_0\cdot R^{(i)}_Y$, as $G_0$ is a bounded set. The set $V_4(\Z)^\red$ is not a lattice. To apply geometry-of-numbers methods, we fiber it over the set of possible linear factors. We write $$\label{eqfiberroot}
V_4(\Z)^\red=\bigcup_{r=[\alpha:\beta]}{\mathcal{L}}_r,$$ where $\alpha$ and $\beta$ are coprime integers and for $r=[\alpha:\beta]\in\P^1(\Z)$, we define ${\mathcal{L}}_r$ to be the set of all integral binary quartic forms $f$ such that $f(r)=0$. From Theorem \[thbqfcavg\], in conjunction with the injection $\sigma$ of §5.2, we have $$\label{eqnqqyb1}
N_{Q,q}(Y)\ll
\displaystyle\sum_{r\in\P^1(\Z)}\int_{(ntk)\in\FF_0}
\#\bigl\{g\in{\mathcal{L}}_r\cap
(ntk)G_0R^{(i)}_Y:
Q(g)>Q,\,\rad(\gcd(Q(g),D(g)))>q\bigr\}\,t^{-2}dnd^\times tdk
$$
As $\gamma$ varies over $\FF_0$, the set $\gamma G_0R^{(i)}_Y$ becomes skewed. More precisely, if $\gamma=ntk$ in Iwasawa coordinates, then the five coefficients $a$, $b$, $c$, $d$, and $e$, of any element of $\gamma G_0R^{(i)}(Y)$ satisfy $$\label{eqrbqfcoeffbounds}
a\ll \frac{Y^{1/6}}{t^4};\quad
b\ll \frac{Y^{1/6}}{t^2};\quad
c\ll Y^{1/6};\quad
d\ll t^2Y^{1/6};\quad
e\ll t^4Y^{1/6}.$$ Hence when $t\gg Y^{1/24}$, the $x^4$-coefficient of any integral binary quartic form in $\gamma G_0R^{(i)}(Y)$ is $0$, forcing a root at the point $[1,0]\in\P^1(\Z)$. Moreover we expect it to be rare that such a binary quartic form has another integral root. In what follow, we first consider the lattice ${\mathcal{L}}_{[1,0]}$ in §5.4, and consider the rest of the lattices in §5.5.
The contribution from the root $r = [1:0]$
------------------------------------------
Let $g(x,y)=bx^3y+cx^2y^2+dxy^3+ey^4\in{\mathcal{L}}_{[1:0]}$ be an integral binary quartic form. We write $Q(g)$ for $Q(g,[1:0])$ and $D(g)$ for $D(g,[1:0])$. Then we have $Q(g)=b$ and $D(g)=\Delta(bx^3+cx^2y+dxy^3+ey^3)$, the discriminant of the binary cubic form $g(x,y)/y$. Hence, if a fixed $t\geq 1$ contributes to the estimate $N_{Q,q}(Y)$ in , then we must have $$\label{eq10tbound}
t\ll\frac{Y^{1/12}}{Q^{1/2}}.$$ We now fiber over the $O(Y^{1/6}/t^2)$ choices for $b$. For each such choice, we have $O(Y^\epsilon)$ possible squarefree divisors $m$ of $b$. Fix such a divisor $m>q$ such that $\rad(\gcd(Q(g),D(g)))=m$. Then $m\mid D(g)$ which implies that $$3c^2d^2-4c^3e\equiv 0\pmod{m}.$$ Thus, the residue class of $e$ modulo $m$ is determined by $c$ and $d$, unless $m\mid c$.
From , we see that the number of elements in ${\mathcal{L}}_{[1:0]}\cap(ntk)G_0R^{(i)}_Y$ with $b$ and $m$ fixed as above is bounded by $$O\Bigl(\frac{t^6Y^{1/2}}{m}+t^6Y^{1/3}\Bigr)=
O\Bigl(\frac{t^6Y^{1/2}}{q}+t^6Y^{1/3}\Bigr),$$ where the second term deals with the case $q\gg Y^{1/6}$. It therefore follows that the contribution to $N_{Q,q}(Y)$ in from the root $r=[1:0]$ is bounded by $$\label{eqfinbound10root}
\int_{t=1}^{Y^{1/12}/Q^{1/2}}\frac{Y^{1/6+\epsilon}}{t^2}
\Bigl(\frac{t^6Y^{1/2}}{q}+t^6Y^{1/3}\Bigr)t^{-2}d^\times t
\ll_\epsilon \frac{Y^{5/6+\epsilon}}{qQ}+\frac{Y^{2/3+\epsilon}}{Q},$$ which is sufficiently small. The contribution from the root $r=[0:1]$ can be identically bounded.
The contribution from a general root $r=[\alpha:\beta]$ with $\alpha\beta\neq 0$
--------------------------------------------------------------------------------
Write $r = [\alpha:\beta]$ where $\alpha,\beta$ are coprime integers and $\alpha\beta\neq 0$. Throughout this section, we denote the torus element in $\FF_0$ with entries $t^{-1}$ and $t$ by $a_t$. We have the bijection $$\label{eqskbij}
\begin{array}{rcl}
\theta_t:\{{\mathcal{L}}_r\cap a_tG_0\cdot R^{(i)}_Y\}&\longleftrightarrow&
\{a_t^{-1}{\mathcal{L}}_r \cap G_0 \cdot R^{(i)}_Y\}\\[.1in]
ax^4+bx^3y+cx^2y^2+dxy^3+ey^4&
\longmapsto& t^4ax^4+t^2bx^3y+cx^2y^2+t^{-2}dxy^3+t^{-4}ey^4,
\end{array}$$ which preserves the invariants $I$ and $J$. Define $\tilde{V_4}(\R)$ to be the set of pairs $(g(x,y),r)$, where $g(x,y)\in V_4(\R)$ and $r\in\R^2$ such that $g(r)=0$. We extend the definitions of the $Q$- and $D$-invariants to the space $\tilde{V_4}(\R)$ via . Set $r_t:=r\cdot
a_t=[t^{-1}\alpha,t\beta]$. Then we have $$Q(g,r)=Q(\theta_t\cdot g,r_t),\qquad D(g,r)=D(\theta_t\cdot g,r_t).$$
Identifying the space of binary quartics with $\R^5$ via the coefficients $(a,b,c,d,e)$, we write $$a_t^{-1}{\mathcal{L}}_r = \text{diag}(t^4,t^2,1,t^{-2},t^{-4}) \cdot
\big((\alpha^4,\alpha^3\beta,\alpha^2\beta^2,\alpha\beta^3,\beta^4)^\perp\big),$$ where $(\alpha^4,\alpha^3\beta,\alpha^2\beta^2,\alpha\beta^3,\beta^4)^\perp$ is the sublattice of $\Z^5$ perpendicular to $(\alpha^4,\alpha^3\beta,\alpha^2\beta^2,\alpha\beta^3,\beta^4)$ with respect to the usual inner product on $\R^5$. Since $\alpha$ and $\beta$ are coprime, the following vectors form an integral basis for $a_t^{-1}{\mathcal{L}}_r$: $$w_1 = (t^4\beta,-t^2\alpha,0,0,0),\quad
w_2 = (0,t^2\beta,-\alpha,0,0),\quad
w_3 = (0,0,\beta,-t^{-2}\alpha,0),\quad
w_4 = (0,0,0,t^{-2}\beta,-t^{-4}\alpha).$$ Define the vector $v_t$ to be $v_t:=(t\beta,-t^{-1}\alpha)\in\R^2$. Then it is easy to see that the lengths of $w_i$ are given by: $$\label{eqbasislength}
|w_1| =
t^3|v_t|,\quad |w_2| = t|v_t|,\quad |w_3| = t^{-1}|v_t|,\quad |w_4| =
t^{-3}|v_t|,$$ The next lemma proves that this basis is [*almost-Minkowski*]{}. That is, the quotients $\langle
w_i,w_j\rangle/(|w_i||w_j|)$, for $i\neq j$, are bounded from above by a constant $c<1$ independent of $t$ and $r$.
For $i\neq j$, we have $$\langle w_i,w_{j}\rangle \leq \frac12 |w_i||w_j|.$$
The inner product $\langle w_i,w_j\rangle$ for $i<j$ is $0$ unless $j=i+1$. In those three cases, we have $$\frac{\langle w_i,w_{j}\rangle}{|w_i||w_j|}=\frac{|\alpha\beta|}{t^{-2}\alpha^2+t^2\beta^2}\leq \frac12,$$ by the AM-GM inequality.
We will represent elements in $a_t^{-1}{\mathcal{L}}_r$ by four-tuples $(a_1,a_2,a_3,a_4)\in\Z^4$, where such a tuple corresponds to the element $a_1w_1 + a_2w_2 + a_3w_3 + a_4w_4$. Then we have the following lemma.
\[lemtupinv\] Let $g(x,y)$ be an element in ${\mathcal{L}}_r$, and let $a_t^{-1} g(x,y)$ correspond to the four-tuple $(a_1,a_2,a_3,a_4)$. Then we have $$\begin{aligned}
g(x,y) &=& (\beta x - \alpha y)(a_1 x^3 + a_2 x^2 y + a_3 x y^2 + a_4 y ^3),\\
Q(g,r) &=& a_1\alpha^3 + a_2\alpha^2\beta + a_3\alpha\beta^2 + a_4\beta^3,\\
D(g,r) &=& \Delta_3(a_1,a_2,a_3,a_4),\end{aligned}$$ where $\Delta_3(a_1,a_2,a_3,a_4)$ denotes the discriminant of the binary cubic form with coefficients $a_i$.
The above lemma follows from a direct computation. Next, we determine when an element $(a_1,a_2,a_3,a_4)$ has small length.
\[lemtupbound\] Suppose $g\in a_t^{-1}{\mathcal{L}}_r$, corresponding to $(a_1,a_2,a_3,a_4)$, belongs to $G_0\cdot R^{(i)}_Y$ for some $i$. Then $$\label{eq:bounda}
a_1 \ll \frac{Y^{1/6}}{t^{3}|v_t|};
\quad a_2 \ll \frac{Y^{1/6}}{t|v_t|};
\quad a_3 \ll \frac{Y^{1/6}}{t^{-1}|v_t|};
\quad a_4 \ll \frac{Y^{1/6}}{t^{-3}|v_t|}.$$
Let $|\cdot|$ denote the length of a binary quartic form, where $V_4(\R)$ has been identified with $\R^5$ in the natural way. Then for $g$ to belong in $G_0\cdot R^{(i)}_Y$, it must satisfy $|g|\ll
Y^{1/6}$. For any real numbers $a_1,a_2,a_3,a_4$, we compute $$\begin{aligned}
|a_1w_1+a_2w_2+a_3w_3+a_4w_4|^2 &\geq& a_1^2|w_1|^2 + a_2^2|w_2|^2 + a_3^2|w_3|^2 + a_4^2|w_4|^2 \\[.15in]
&&-|a_1||a_2||w_1||w_2|-|a_2||a_3||w_2||w_3| - |a_3||a_4||w_3||w_4|\\[.15in]
&\geq& \frac{3 - \sqrt{5}}{4}( a_1^2|w_1|^2 + a_2^2|w_2|^2 + a_3^2|w_3|^2 + a_4^2|w_4|^2).\end{aligned}$$ (Of course, the exact constant is not important.) Therefore in order for $|a_1w_1+a_2w_2+a_3w_3+a_4w_4|\ll Y^{1/6}$, Equation must be satisfied.
We now have the following proposition bounding the number of elements in ${\mathcal{L}}_r\cap a_tG_0\cdot R^{(i)}_Y$ whose $Q$- and $D$-invariants share a large common factor.
\[aster\] For $t\gg 1$, we have $$\label{eqaster}
\#\{g(x,y)\in{\mathcal{L}}_r\cap a_tG_0\cdot R^{(i)}_Y:\rad(\gcd(Q(g,r),D(g,r)))>q\}=
\left\{
\begin{array}{lll}
0&\mbox{if}\;\;\;|v_t|\gg Y^{1/6}\\[.1in]
\displaystyle O\Bigl(\frac{Y^{2/3+\epsilon}}{q|v_t|^4}+\frac{Y^{1/2+\epsilon}}{t|v_t|^3}\Bigr)&\mbox{otherwise}
\end{array}
\right.$$ where the implied constant is independent of $r$, $t$, and $Y$.
Using the bijection in conjunction with Lemmas \[lemtupinv\] and \[lemtupbound\], we see that it is enough to prove that the number of four-tuples of integers $(a_1,a_2,a_3,a_4)$, satisfying and $$\rad(\gcd(a_1\alpha^3 + a_2\alpha^2\beta + a_3\alpha\beta^2 + a_4\beta^3,\Delta_3(a_1,a_2,a_3,a_4)))>q,$$ is bounded by the right hand side of . Suppose first $|v_t|\gg Y^{1/6}$. Then any binary quartic form $g(x,y)$ represented by the four-tuple $(a_1,a_2,a_3,a_4)$ satisfying must have $a_1=a_2=0$. From Lemma \[lemtupinv\], it follows that $D(g,r)=0$ and hence $\Delta(g)=0$. Since $G_0\cdot R^{(i)}_Y$ contains no point with $\Delta=0$, it follows that the intersection is empty, proving the first part of the proposition.
The second part of the proposition is proved by using the Ekedhal sieve as developed in [@manjul-geosieve]. We carry the sieve out in detail so as to demonstrate that the implied constant in is indeed independent of $r$ and $t$. Define $$T_{\alpha,\beta}(a_1,a_2,a_3) := \Delta_3(a_1\beta^3, a_2\beta^3,
a_3\beta^3, -(a_1\alpha^3 + a_2\alpha^2\beta + a_3\alpha\beta^2)).$$ It is clear that if $m\mid Q(g,r)$ and $m\mid D(g,r)$ for any integer $m$, then $m\mid
T_{\alpha,\beta}(a_1,a_2,a_3)$.
First, we bound the number of triples $(a_1,a_2,a_3)$ satisfying such that $T_{\alpha,\beta}(a_1,a_2,a_3)=0$. For a fixed pair $(a_1,a_2)\neq (0,0)$, by explicitly writing out $T_{\alpha,\beta}(a_1,a_2,a_3)$, we see that there are at most three possible values of $a_3$ with $T_{\alpha,\beta}(a_1,a_2,a_3)=0$. This gives a bound of $O(Y^{1/3}/(t^4|v_t|^2))$ on the number of triples $(a_1,a_2,a_3)$ with $T_{\alpha,\beta}(a_1,a_2,a_3)=0$. Multiplying with the number of all possibilities for $a_4$, we obtain the bound $$\label{eqgeoTeq0}
O\Bigl(\frac{Y^{1/2}}{t|v_t|^3}\Bigr)$$ on the number of four-tuples of integers $(a_1,a_2,a_3,a_4)$, satisfying and $T_{\alpha,\beta}(a_1,a_2,a_3)=0$.
Next, we fiber over triples $(a_1,a_2,a_3)$ with $T_{\alpha,\beta}(a_1,a_2,a_3)\neq 0$ and satisfying . In this case, we have $(a_1,a_2)\neq (0,0)$. Hence by , we may assume $\alpha,\beta,t\ll
Y^{1/6}$. Hence the value of $T_{\alpha,\beta}(a_1,a_2,a_3)$ is bounded by a polynomial in $Y$ of fixed degree. It follows that the number of squarefree divisors of $T_{\alpha,\beta}(a_1,a_2,a_3)$ is bounded by $O_\epsilon(Y^\epsilon)$. Fix one such divisor $m>q$. We now fiber over a positive squarefree integer $\delta \ll Y^{1/6}/(t^3|v_t|)$ such that $\text{rad}(\gcd(a_1,a_2)) = \delta$. The number of such possible $(a_1,a_2)$ is $$\ll \frac{1}{\delta^2}\frac{Y^{1/6}}{t^3|v_t|}\frac{Y^{1/6}}{t|v_t|}.$$ Fix any such pair. Let $a_3$ be any integer satisfying such that $T_{\alpha,\beta}(a_1,a_2,a_3)\neq 0$. Let $m_1 = \gcd(m,\delta)$ and let $m_2 = m/m_1 > q/\delta$. Then the polynomial $\Delta_3(a_1,a_2,a_3,a_4)$ is identically $0$ modulo $m_1$ and quadratic in $a_4$ modulo any prime factor of $m_2$. Hence the number of these quadruples with the extra condition that $m\mid\Delta_3(a_1,a_2,a_3,a_4)$ is $$\ll\frac{1}{\delta^2}\frac{Y^{1/6}}{t^3|v_t|}\frac{Y^{1/6}}{t|v_t|}\frac{Y^{1/6}}{t^{-1}|v_t|}\left(\frac{1}{q/\delta}\frac{Y^{1/6}}{t^{-3}|v_t|}+1\right) \ll_\epsilon \frac{Y^{2/3}}{\delta q|v_t|^4} + \frac{Y^{1/2}}{\delta^2t^3|v_t|^3}.$$ Summing over $\delta$ and all possible divisors $m$ gives the bound $$\label{eqgeoTneq0}
O\Bigl(\frac{Y^{2/3+\epsilon}}{q|v_t|^4} + \frac{Y^{1/2+\epsilon}}{t^3|v_t|^3}\Bigr).$$
The proposition now follows from and .
We now impose the condition on the $Q$-invariant. From Lemma \[lemtupinv\] and , we obtain $$Q< |Q(g,r)| = |a_1\alpha^3 + a_2\alpha^2\beta + a_3\alpha\beta^2 + a_4\beta^3| \ll Y^{1/6}|v_t|^2.$$ In conjunction with and the estimates of Proposition \[aster\], this yields $$\begin{array}{rcl}
N_{Q,q}(Y)&\ll_\epsilon&
\displaystyle\sum_{k\ll\log Y}\int_{t\gg 1}\sum_{\substack{r=[\alpha:\beta]\\ 2^k<|v_t|\leq 2^{k+1}}}
\Bigl(\frac{Y^{2/3+\epsilon}}{q|v_t|^4}+\frac{Y^{1/2+\epsilon}}{t|v_t|^3}\Bigr)t^{-2}d^\times t
\\[.3in]&\ll_\epsilon&
\displaystyle\sum_{k\ll\log Y}\int_{t\gg 1}
\Bigl(\frac{Y^{5/6+\epsilon}}{qQ}+\frac{Y^{7/12+\epsilon}}{tQ^{1/2}}\Bigr)t^{-2}d^\times t
\\[.3in]&\ll_\epsilon&
\displaystyle\frac{Y^{5/6+\epsilon}}{qQ}+\frac{Y^{7/12+\epsilon}}{Q^{1/2}}.
\end{array}$$ This concludes the proof of Theorem \[qinvmt\].
Uniformity estimates
====================
In this section, we prove Theorems \[thunifsqi\] and \[thunifsmind\], the main uniformity estimates. First, in §6.1, we use the results of §3 to prove Theorem \[thunifsqi\]. Next, in §6.2, we combine the results of §4 and §5 in order to obtain Theorem \[thunifsmind\].
The family of elliptic curves with squarefree index {#sec:ellipcount}
---------------------------------------------------
Recall the family $\E$ defined in the introduction. The assumption that elliptic curves $E\in\E$ satisfy $j(E)\leq\log(\Delta(E))$ implies the height bound $H(E)\ll\Delta(E)^{1+\epsilon}$. Given $E\in\E$, let $E:y^2=f(x)=x^3+Ax+B$ be the minimal Weierstrass model for $E$. Given an etalé algebra $K$ over $\Q$ with ring of integers ${{\mathcal O}}_K$, let ${{\mathcal O}}_K^{\Tr=0}$ denote the set of traceless integral elements in $K$. Consider the map $$\E\to \{(K,\alpha):K\mbox{ cubic algebra over }\Q,\;
\alpha\in{{\mathcal O}}_K^{\Tr=0}\}$$ sending $E:y^2=f(x)$ to the pair $(\Q[x]/f(x),x)$. This map is injective since if $E$ corresponds to the pair $(K,\alpha)$, then $y^2=N_{K/\Q}(x-\alpha)$ recovers $E$. In order to parametrize elements in $\E_\sf$, we will instead use the following modified map: $$\label{eqEEsfparam}
\begin{array}{rcl}
\displaystyle \sigma: \E&\to&
\displaystyle\{(K,\alpha):K\mbox{ cubic \'{e}tale algebra over }\Q,\;
\alpha\in{{\mathcal O}}_K^{\Tr=0}\}
\\[.1in]
\displaystyle E:y^2=f(x)&\mapsto&
\displaystyle (\Q[x]/f(x),{\rm Prim}(x)),
\end{array}$$ where for $0\neq x\in{{\mathcal O}}_K$, the element ${\rm Prim}(x)$ is the unique primitive integer in ${{\mathcal O}}_K$ which is a positive rational multiple of $x$. Note that the map $\sigma$ restricted to $\E_\sf$ is injective due to the squarefree condition on $\Delta(E)/C(E)$ at primes at least $5$ and that $E$ has good reduction at $2$ and $3$. We start with the following lemma.
Let $E$ be an elliptic curve and let $\sigma(E)=(K,\alpha)$. Then $|\Sel_2(E)|\ll_\epsilon
|\Cl(K)[2]|\cdot|\Delta(E)|^\epsilon$. Furthermore, if $E\in\E_\sf$, then $|\alpha|\ll H(E)^{1/6}$.
The first claim is a direct consequence of [@brumerkramer Proposition 7.1]. The second claim is immediate since the minimal Weierstrass model of $E$ is given by $y^2=(x-\beta_1)(x-\beta_2)(x-\beta_3)$, where the $\beta_i$ are the conjugates of an absolutely bounded rational multiple of $\alpha$.
We now prove the following result.
\[prop:dydy\] For positive real numbers $X$ and $Q\leq X$, we have $$\label{eqeezsfprop}
\bigl|\bigl\{(E,\eta):E\in\E_\sf,\;\eta\in\Sel_2(E),
\;X<C(E)\leq 2X,\;QX<\Delta(E)\leq 2QX\bigl\}\bigl|\ll_\epsilon X^{5/6+\epsilon}/Q^{1/6}.$$ where the implied constant is independent of $X$ and $Q$.
Let $E\in\E_\sf$ be an elliptic curve satisfying the conductor and discriminant bounds of , and let $\sigma(E)=(K,\alpha)$. It is easy to verify from Table \[tabloc\] that $\Delta(K) = C(E)^2/\Delta(E)$. Therefore, it follows that $X/(2Q)<\Delta(K)\leq 4X/Q$, and that $|\alpha|\ll
H(E)^{1/6}\ll_\epsilon(QX)^{1/6+\epsilon}$.
Since the map $\sigma$ is injective, it follows that the left hand side of is $$\label{eqprlemsfcf}
\ll_\epsilon X^\epsilon\sum_{\substack{[K:\Q]=3\\\frac{X}{2Q}<\Delta(K)\leq \frac{4X}{Q}}}
N'_K((QX)^{1/6+\epsilon})\,|\Cl(K)[2]|,$$ where $N'_K(Y)$ denotes the number of primitive elements $\alpha$ in ${{\mathcal O}}_K^{\Tr =0}$ such that $|\alpha|<Y$ and the pair $(K,\alpha)$ is in the image of $\sigma$. We now split the above sum over cubic algebras $K$ into three parts, corresponding to the sizes $\ell_1(K)$ and $\ell_2(K)$ of the successive minima of ${{\mathcal O}}_K^{\Tr = 0}$.
First, if $(QX)^{1/6+\epsilon}\ll\ell_1(K)$, then the contribution to is $0$. Second, assume that $\ell_2(K)\ll
(QX)^{1/6+\epsilon}$. Then Lemma \[lemkalcountel\] yields the bound $$N_K'((QX)^{1/6+\epsilon})\ll_\epsilon (QX)^{1/3+\epsilon}/\sqrt{X/Q}\ll \frac{Q^{5/6}}{X^{1/6-\epsilon}}.$$ Using Bhargava’s result [@manjulcountquartic Theorem 5] to bound the sum of $|\Cl(K)[2]|$ over cubic fields $K$ with the prescribed discriminant range, and using the well known genus-theory bounds $\Cl(K)[2]\ll|\Delta(K)|^\epsilon$, for each reducible cubic $K$, we obtain: $$\sum_{\substack{[K:\Q]=3\\\frac{X}{2Q}<\Delta(K)\leq \frac{4X}{Q}\\\ell_2(K)\ll (QX)^{1/6+\epsilon}}}
N'_K((QX)^{1/6+\epsilon})\,|\Cl(K)[2]|\ll_\epsilon X^\epsilon\cdot \frac{Q^{5/6}}{X^{1/6-\epsilon}}\cdot \frac{X}{Q}=\frac{X^{5/6+2\epsilon}}{Q^{1/6}}.$$
Finally, we bound the contribution of cubic étale algebras $K$ such that $\ell_1(K)\ll (QX)^{1/6+\epsilon}\ll\ell_2(K)$. In this case, we have $N_K'((QX)^{1/6+\epsilon})\leq 1$ and $$\sk(K)=\ell_2(K)/\ell_1(K)\gg \sqrt{\Delta(K)}/\ell_1(K)^2
\gg X^{1/6}/Q^{5/6}.$$ Suppose first $K = \Q\oplus L$ is reducible. Then ${{\mathcal O}}_K^{\Tr=0}$ has an integral basis given by $\{(-2,1),(0,\sqrt{d})\}$ where $\Delta(K)
= d$ or $4d$. When $d$ is small, say bounded by $100$, we get an $O(1)$ contribution to . When $d$ is large, the above basis is a Minkowski basis and $(-2,1)$ is the smallest, and hence unique, primitive traceless element. However, this point does not correspond to an elliptic curve since the corresponding cubic polynomial is $(x-2)(x+1)^2$ which has a double root. Hence, we get no contribution in this case. It remains to consider the case where $K$ is a cubic field. Applying Theorem \[thm:skew\], we obtain a bound of $$O_\epsilon\bigl(X^\epsilon(X/Q)/(X^{1/6}/Q^{5/6})\bigr)=
O_\epsilon(X^{5/6+\epsilon}/Q^{1/6}),$$ on the contribution to over cubic fields $K$ with $\ell_1(K)\ll (QX)^{1/6+\epsilon}\ll\ell_2(K)$, as desired.
**Proof of Theorem \[thunifsqi\]:** Note if the conductor $C(E)$ is bounded by $X$ and the index $\Delta(E)/C(E)$ is squarefree, then the index is also bounded by $X$. Divide the conductor range $[1,X]$ into $\log X$ dyadic ranges, and for each such range divide the index range $[M,X]$ into $\log X$ dyadic ranges, and then apply Proposition \[prop:dydy\] on each pair of dyadic ranges. Theorem \[thunifsqi\] follows. $\Box$
The family of elliptic curves with bounded index
------------------------------------------------
As in §3, let $\Sigma$ be a finite set of pairs $(p,T_p)$, where $p$ is a prime number and $T_p=\I\I\I$, $\I\rV$, or $\I_{\geq 2}$ is a Kodaira symbol. Recall the invariants $Q(\Sigma)$, $m_\odd(\Sigma)$ and $m_T(\Sigma)$ for Kodaira symbols $T$. We further define $m_\even(\Sigma)$ to be the product of $p$ over pairs $(p,\I_{2k})$ in $\Sigma$. We define $\E(\Sigma)$ to be the set of elliptic curves $E\in \E$ such that the Kodaira symbol at $p$ of $E$ is $T_p$ for every pair $(p,T_p)\in \Sigma$. Given a set of five positive real numbers $$S=\{m_{\I\I\I},m_{\I\rV},m_{\even},m_{\odd},Q\},$$ we let $\E(S)$ denote the set of elliptic curves $E$ such that the product $P$ of primes at which $E$ has Kodaira symbol $\I\I\I$ (resp. $\I\rV$, $I_{2(k\geq 1)}$, $I_{2(k\geq 1)+1}$) satisfies $m_{\I\I\I}\leq P<2 m_{\I\I\I}$ (resp. $m_{\I\rV}\leq P<2
m_{\I\rV}$, $m_{\even}\leq P<2 m_{\even}$, $m_{\odd}\leq P<2
m_{\odd}$), and $Q\leq Q(E)<2Q$. The following result is a consequence of Theorems \[thm:equimain\] and \[qinvmt\].
\[prop45cb\] Let $S=\{m_{\I\I\I}, m_{\I\rV}, m_{\even}, m_{\odd}, Q\}$ be as above and let $Y$ be a positive real number. Then $$\label{eq45combbound}
\begin{array}{rcl}
&&\displaystyle\#\{E\in \E(S):|\Delta(E)|<Y\}\\[.1in]
&\ll_\epsilon&\displaystyle
Y^\epsilon\,\min\Bigl(
\frac{Y^{5/6}m_{\even}}{Q^2m_{\I\rV}}+\frac{Qm_{\I\I\I}m_{\even}m_{\odd}^2}{Y^{1/6}},
\frac{Y^{5/6}}{Qm_{\I\I\I}m_{\I\rV}m_{\odd}}+\frac{Y^{7/12}}{Q^{1/2}}
\Bigr).
\end{array}$$
First note that if $E\in\E$, then $H(E)\ll
\Delta(E)^{1+\epsilon}$ from the $j$-invariant bound. It is enough to prove that the left hand side of is bounded (up to a factor of $Y^\epsilon$) by both terms in the minimum. For the second term, this is a direct consequence of Theorem \[qinvmt\] and Table \[tabloc\].
For the first term, note that the set of monic cubic polynomials corresponding to curve in $\E(S)$ is clearly the union of $O_\epsilon(Y^\epsilon m_{\I\I\I}m_{\I\rV}m_{\even}m_{\odd})$ sets $U_0(\Z)_\Sigma$, where each such $\Sigma$ satisfies $m_{\I\I\I}(\Sigma)\sim m_{\I\I\I}$, $m_{\I\rV}(\Sigma)\sim
m_{\I\rV}$, $m_{\even}(\Sigma)\sim m_{\even}$, $m_{\odd}(\Sigma)\sim
m_{\odd}$, and $Q(\Sigma)\sim Q$. In §3, we obtained bounds on the number of elements in $U(\Z)_\Sigma$ with height bounded by $Y$. Since the set $U(\Z)_\Sigma$ is invariant under the linear $\Z$-action, we have $$|\{f\in U_0(\Z)_\Sigma:H(f)<Y\}|\ll Y^{-1/6}|\{f\in U(\Z)_\Sigma:H(f)<Y\}|.$$ Combining this with Theorem \[thm:equimain\], and multiplying with the number of different $\Sigma$’s required to cover the set $\E(S)$, we obtain the result.
**Proof of Theorem \[thunifsmind\]:** Given positive real numbers $X$ and $Y$, let $\E(S;X,Y)$ denote the set of $E\in\E(S)$ that satisfy $X\leq C(E)<2X$, and $Y\leq\Delta(E)<2Y$. Fix constants $0<\kappa<7/4$ and $0<\delta$. We first obtain bounds on the sizes of the sets $\E(S;X,Y)$. Let $E$ be an elliptic curve in $\E(S;X,Y)$, and let $P$ be the contribution to the conductor of $E$ that is prime to $m_{\I\I\I}m_{\I\rV}m_{\even}m_{\odd}$. Then we have by Table \[tabloc\] $$\begin{array}{rcccl}
X&\asymp&C(E)&\asymp& m_{\I\I\I}^2m_{\I\rV}^2m_{\even}m_{\odd}P;\\[.1in]
Y&\asymp&\Delta(E)&\asymp& m_{\I\I\I}m_{\I\rV}^2m_{\odd}Q^2P.
\end{array}$$ Therefore, in order for $\E(S;X,Y)$ to be nonempty, we must have $$\label{eqQXY}
\frac{Y}{Q^2}\asymp\frac{X}{m_{\I\I\I}m_{\even}}.$$ First note that we have $$\label{eq6mtb}
\frac{Y^{5/6}m_{\even}}{Q^2m_{\I\rV}}\ll \frac{X}{Y^{1/6}}.$$ Moreover, $$\label{eq6atb}
\begin{array}{rl}
&\displaystyle \min\Bigl(\frac{Qm_{\I\I\I}m_{\even}m_{\odd}^2}{Y^{1/6}}, \frac{Y^{5/6}}{Qm_{\I\I\I}m_{\I\rV}m_{\odd}}\Bigr)
\leq \Bigl(\frac{Y^{15/6-1/6}m_{\even}}{Q^2m_{\I\I\I}^2m_{\I\rV}^3m_{\odd}}\Bigr)^{1/4}\ll X^{1/4}Y^{1/3};\\[.2in]
&\displaystyle \min\Bigl(\frac{Qm_{\I\I\I}m_{\even}m_{\odd}^2}{Y^{1/6}}, \frac{Y^{7/12}}{Q^{1/2}}\Bigr)
\leq(Ym_{\I\I\I}m_{\even}m_{\odd}^2)^{1/3}\ll\frac{Y^{2/3}}{X^{1/3}}.
\end{array}$$
Assume that $Y$ satisfies the bound $X^{1+\delta}\ll Y\ll X^\kappa$ for $\delta>0$ and $\kappa<7/4$. Proposition \[prop45cb\], , and imply that we have $$\label{eq6ftb}
|\E(S;X,Y)|\ll_\epsilon X^{5/6-\theta+\epsilon},$$ for some positive constant $\theta$ depending only on $\delta$ and $\kappa$. It is clear that the set $$\{E\in\E_\kappa:C(E)<X,|\Delta(E)|>C(E)X^\delta\}$$ is the union of $O_\epsilon(X^\epsilon)$ sets $\E(S;X_1,Y_1)$, with $X_1\leq X$ and $X_1^{1+\delta}\ll Y_1\ll X_1^\kappa$. Theorem \[thunifsmind\] now follows from . $\Box$
Additional uniformity estimates
-------------------------------
We will also need (albeit much weaker) estimates on the number of elliptic curves with bounded height and additive reduction, as well as on the number of $\PGL_2(\Z)$-orbits on integral binary quartic forms whose discriminants are divisible by a large square.
We begin with the following result which follows immediately from the proof of [@bs2sel Proposition 3.16].
\[propelemufec\] The number of pairs $(A,B)\in\Z^2$ such that $H(A,B)<X$ and such that $p^2\mid\Delta(A,B)$ is $O(X^{5/6}/p^{3/2})$, where the implied constant is independent of $X$ and $p$.
Next, we have the following estimate which is proved in [@bswglobal2]
\[propelemufbqf\] The number of $\PGL_2(\Z)$-orbits $f$ on $V_4(\Z)$ such that $H(f)<X$ and $n^2\mid\Delta(f)$ for some $n>M$ is bounded by $$O\Bigl(\frac{X^{5/6}}{M^{1-\epsilon}}+X^{19/24+\epsilon}\Bigl),$$ where the error terms are independent of $X$ and $M$.
This is proved in [@bswglobal2], so we merely give a sketch of the proof. The idea is to embed the space of integral binary quartic forms into the space $W_4(\Z)$ of pairs of integral quaternary quadratic forms: $$\begin{array}{rcl}
\pi: V_4(\Z) &\to &W_4(\Z)\\[.1in]
ax^4+bx^3y+cx^2y^2+dxy^3+ey^4&\mapsto&\left(
\left[ \begin{array}{cccc} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 0\end{array} \right],
\left[ \begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & -24a & 6b & 2c\\ 0 & 6b & -4c & -3d\\ 0 & 2c & -3d & -6e\end{array} \right]
\right).
\end{array}$$ Under this map, the cubic resolvents of $f$ and $\pi(f)$ are the same, and hence $f$ and $\pi(f)$ have the same height and discriminant. We also note that $\pi$ has an algebraic interpretation: the $\PGL_2(\Z)$-orbit of a nondegenerate element in $V_4(\Z)$ with cubic resolvent $g(x)$ corresponds to an element in $H^1(\Q,E_g[2])$, while the $\GL_2(\Z)\times\GL_4(\Z)$-orbit of an element in $W_4(\Z)$ corresponds with cubic resolvent $g(x)$ corresponds to an element in $H^1(\Q,E_g[4])$. Then the map $\pi$ simply corresponds to natural map $$H^1(\Q,E_g[2])\rightarrow H^1(\Q,E_g[4]).$$
As proven in [@bs4sel], every element in $W_4(\Z)$ having integral coefficients and discriminant divisible by $n^2$, for some squarefree integer $n$, is $\GL_2(\Q)\times\GL_4(\Q)$-equivalent to some element in $W_4(\Z)$ whose discriminant is divisible by $n^2$ for mod $n$ reasons (in the terminology of [@manjul-geosieve]). Then an application of the Ekedhal sieve in conjunction with geometry-of-numbers methods counting $\GL_2(\Z)\times\GL_4(\Z)$-orbits on $W_4(\Z)$, yields the result.
Asymptotics for families of elliptic curves
===========================================
Let $p$ be a fixed prime. An elliptic curve $E$ over $\Q$ has either good reduction, multiplicative reduction, or additive reduction at $p$. For every prime $p\geq 5$, let $\Sigma_p$ be a nonempty subset of possible reduction types. We say that $\Sigma=(\Sigma_p)_p$ is a [ *collection of reduction types*]{} and that such a collection is [ *large*]{} if for all large enough primes $p$, the set $\Sigma_p$ contains at least the good and multiplicative reduction types.
For a large collection $\Sigma$, let $\E_\sf(\Sigma)$ (resp. $\E_\kappa(\Sigma)$) denote the set of elliptic curves $E\in\E_\sf$ (resp. $E\in\E_\kappa$) such that for all primes $p\geq
5$, the reduction type of $E$ at $p$ belongs to $\Sigma_p$. In this section, we prove the following theorem, from which Theorems \[thmmain\] and \[thmsel\] immediately follow.
\[thmmainlarge\] Let $\Sigma$ be a large collection of elliptic curves. Let $\kappa<7/4$ be a positive constant. Then we have $$\label{eqEFSig}
\begin{array}{rcl}
\displaystyle\#\{E\in \E_\sf(\Sigma)^\pm:\; C(E)<X\}&\sim&
\displaystyle\frac{\alpha^\pm}{60\sqrt{3}}
\frac{\Gamma(1/2)\Gamma(1/6)}{\Gamma(2/3)}
\prod_p\bigl(c_g(p)e_g(p)+c_m(p)e_m(p)+c_a(p)e_a(p)\bigr)\cdot X^{5/6},\\[.2in]
\displaystyle\#\{E\in \E_\kappa(\Sigma)^\pm:\; C(E)<X\}&\sim&
\displaystyle \frac{\alpha^\pm}{60\sqrt{3}}
\frac{\Gamma(1/2)\Gamma(1/6)}{\Gamma(2/3)}
\prod_p\bigl(c_g(p)f_g(p)+c_m(p)f_m(p)+c_a(p)f_a(p)\bigr)\cdot X^{5/6},
\end{array}$$ where $\alpha^+=1$, $\alpha^-=\sqrt{3}$, $c_g(p)$ $($resp. $c_m(p)$, $c_a(p))$ is $1$ or $0$ depending on whether $\Sigma_p$ contains the good $($resp. multiplicative, additive$)$ reduction type, and $e_*(p)$ and $f_*(p)$ are given by $$e_g(p):= \displaystyle 1-\frac{1}{p};\qquad e_m(p):=\frac{1}{p}\Bigl(1+\frac{1}{p^{1/6}}\Bigr)\Bigl(1-\frac{1}{p}\Bigr)^2;\qquad e_a(p):=\displaystyle\frac{1}{p^2}\Bigl(1+\frac{1}{p^{1/6}}\Bigr)\Bigl(1-\frac{1}{p}\Bigr);$$ $$f_g(p):= \displaystyle 1-\frac{1}{p};\qquad
\displaystyle f_m(p):=\frac{1}{p}\Bigl(1-\frac{1}{p^{1/6}}\Bigr)^{-1}\Bigl(1-\frac{1}{p}\Bigr)^2;$$ $$\displaystyle f_a(p):=
\displaystyle \frac{1}{p^{5/3}}\Bigl(1-\frac{1}{p}\Bigr)\Bigl(1+\frac{1}{p^{1/6}}+\frac{1}{p^{7/6}}\Bigr) +
\displaystyle\frac{1}{p^2}\Bigl(1-\frac{1}{p}\Bigr)\Bigl(1-\frac{1}{p^{1/6}}\Bigr)^{-1}
\Bigl(3-\frac{2}{p^{1/2}}\Bigr).$$ Furthermore, when elliptic curves in $\E_\sf(\Sigma)$ are ordered by conductor, the average size of their $2$-Selmer groups is $3$.
The family $\E$ ordered by discriminant
---------------------------------------
We write elliptic curves $E\in\E$ in their minimal short Weierstrass model. In this case, it is easy to check that we have $$\E=\bigl\{E_{AB}:j(E_{AB})<\log(\Delta(E_{AB})),\;\;
16\mid A,\;B\equiv 16\pmod{64},\;\;
3\nmid A\bigr\}$$ Moreover, for every $E_{AB}\in\E$, we have $\Delta(E_{AB})=\Delta(A,B)/2^8$. To count elements in $\E$ with bounded discriminant, we need to incorporate the bound $j(E_{AB})<\log\Delta(E_{AB})$, which is not a semialgebraic condition in $A$ and $B$. However it is clearly definable in an o-minimal structure. Hence we use the following result of Barroero–Widmer [@ominimal Theorem 1.3].
\[thomin\] Let $m$ and $n$ be positive integers, let $\Lambda\subset\R^n$ be a lattice and denote the successive minima of $\Lambda$ by $\lambda_i$. Let $Z\subset\R^{m+n}$ be a definable family, and suppose the fibers $Z_T$ are bounded. Then there exists a constant $c_Z\in \R$, depending only on the family $Z$, such that $$\Bigl|\#\bigl(Z_T\cap\Lambda\bigr)-\frac{\Vol(Z_T)}{\det(\Lambda)}\Bigr|
\leq c_Z\sum_{j=0}^{n-1}\frac{V_j(Z_T)}{\lambda_1\cdots\lambda_j},$$ where $V_j(Z_T)$ is the sum of the $j$-dimensional volumes of the orthogonal projections of $Z_T$ on every $j$-dimensional coordinate subspace of $\R^n$.
For a pair $(A,B)\in\R^2$ with $\Delta(A,B)\neq 0$, let $j(A,B)$ denote $j(E_{AB})$. For any set $S\subset\Z^2$ defined by congruence conditions, let $\nu(S)$ denote the volume of the closure of $S$ in $\hat{\Z}^2$. Equivalently, $\nu(S)$ is the product over the primes $p$ of the closure of $S$ in $\Z_p^2$. We have the following immediate consequence of Theorem \[thomin\].
\[eqabcountdisc\] Let $\Lambda\subset\Z^2$ denote a set of pairs $(A,B)$ defined by congruence conditions on $A$ and $B$ modulo some positive integer $n<X^{1/3-\epsilon}$. Then we have $$\#\bigl\{(A,B)\in\Lambda: j(A,B)<\log(2^{-8}\Delta(A,B)),\;0<\pm\Delta(A,B)<X\bigr\}
=\nu(\Lambda)c_\infty^\pm(X)+O_\epsilon(X^{1/2+\epsilon}),$$ where $c_\infty^\pm(X)$ denotes the volume of the set $$C^\pm(X):=\bigl\{(A,B)\in\R^2: j(A,B)<\log(2^{-8}\Delta(A,B)),\;0<\pm\Delta(A,B)<X\bigr\}$$ computed with respect to Eucledean measure normalized so that $\Z^2$ has covolume $1$.
Since the set $\E$ arises by imposing congruence conditions modulo infinitely many primes, we use a simple sieve to determine asymptotics for the number of elliptic curves in $\E$ with bounded discriminant.
We have $$\#\bigl\{E\in\E:0<\pm\Delta(E)<X\}\sim \frac{\alpha^\pm}{60\sqrt{3}}\cdot\frac{\Gamma(1/2)\Gamma(1/6)}{\Gamma(2/3)}
\cdot\prod_{p\geq 5}\Bigl(1-\frac{1}{p^{10}}\Bigr)X^{5/6},$$ where $\alpha^+=1$ and $\alpha^-=\sqrt{3}$.
First, we describe the set of elliptic curves $E_{AB}:y^2=x^3+Ax+B$ that have good reduction at $2$ and $3$ in Tables \[tabgr2\] and \[tabgr3\], respectively. In both tables, the first column describes the congruence conditions on $A$, the second describes congruence conditions at $B$, the third gives the $2$-part (resp. the $3$-part) of the discriminant $\Delta(A,B)=4A^3+27B^2$, and the fourth column gives the density of these congruence conditions inside the space $(A,B)\in \Z_p^2$ for $p=2$ and $3$. Below, $\delta$ is either $0$ or $1$.
$A$ $B$ $\Delta_2$ Density
----------------------------------------- ------------------------------------------ ------------ -----------
$\equiv 0\pmod{2^4}$ $\equiv 2^4\pmod{2^6}$ $2^8$ $2^{-10}$
$\equiv (5+\delta\cdot 2^6)\pmod{2^7}$ $\equiv (22+\delta\cdot 2^6)\pmod{2^7}$ $2^8$ $2^{-13}$
$\equiv (13+\delta\cdot 2^6)\pmod{2^7}$ $\equiv (14+\delta\cdot 2^6)\pmod{2^7}$ $2^8$ $2^{-13}$
$\equiv (21+\delta\cdot 2^6)\pmod{2^7}$ $\equiv (38+\delta\cdot 2^6)\pmod{2^7}$ $2^8$ $2^{-13}$
$\equiv (29+\delta\cdot 2^6)\pmod{2^7}$ $\equiv (94+\delta\cdot 2^6)\pmod{2^7}$ $2^8$ $2^{-13}$
$\equiv (37+\delta\cdot 2^6)\pmod{2^7}$ $\equiv (54+\delta\cdot 2^6)\pmod{2^7}$ $2^8$ $2^{-13}$
$\equiv (45+\delta\cdot 2^6)\pmod{2^7}$ $\equiv (46+\delta\cdot 2^6)\pmod{2^7}$ $2^8$ $2^{-13}$
$\equiv (53+\delta\cdot 2^6)\pmod{2^7}$ $\equiv (70+\delta\cdot 2^6)\pmod{2^7}$ $2^8$ $2^{-13}$
$\equiv (61+\delta\cdot 2^6)\pmod{2^7}$ $\equiv (126+\delta\cdot 2^6)\pmod{2^7}$ $2^8$ $2^{-13}$
: Elliptic curves $E_{AB}$ with good reduction at $2$[]{data-label="tabgr2"}
$A$ $B$ $\Delta_3$ Density
-------------------------------- --------------------------------------------- ------------ ------------------
$3\nmid A$ - 1 $2\cdot 3^{-1}$
$3^4\parallel A$ $3^6\mid B$ $3^{12}$ $2\cdot 3^{-11}$
$\equiv 2\cdot 3^3\pmod{3^6}$ $\equiv (\pm 20,\pm 34)\cdot 3^3\pmod{3^7}$ $3^{12}$ $4\cdot 3^{-13}$
$\equiv 5\cdot 3^3\pmod{3^6}$ $\equiv (\pm 11,\pm 16)\cdot 3^3\pmod{3^7}$ $3^{12}$ $4\cdot 3^{-13}$
$\equiv 8\cdot 3^3\pmod{3^6}$ $\equiv (\pm 2,\pm 29)\cdot 3^3\pmod{3^7}$ $3^{12}$ $4\cdot 3^{-13}$
$\equiv 11\cdot 3^3\pmod{3^6}$ $\equiv (\pm 7,\pm 20)\cdot 3^3\pmod{3^7}$ $3^{12}$ $4\cdot 3^{-13}$
$\equiv 14\cdot 3^3\pmod{3^6}$ $\equiv (\pm 16,\pm 38)\cdot 3^3\pmod{3^7}$ $3^{12}$ $4\cdot 3^{-13}$
$\equiv 17\cdot 3^3\pmod{3^6}$ $\equiv (\pm 2,\pm 25)\cdot 3^3\pmod{3^7}$ $3^{12}$ $4\cdot 3^{-13}$
$\equiv 20\cdot 3^3\pmod{3^6}$ $\equiv (\pm 7,\pm 34)\cdot 3^3\pmod{3^7}$ $3^{12}$ $4\cdot 3^{-13}$
$\equiv 23\cdot 3^3\pmod{3^6}$ $\equiv (\pm 11,\pm 38)\cdot 3^3\pmod{3^7}$ $3^{12}$ $4\cdot 3^{-13}$
$\equiv 26\cdot 3^3\pmod{3^6}$ $\equiv (\pm 25,\pm 29)\cdot 3^3\pmod{3^7}$ $3^{12}$ $4\cdot 3^{-13}$
: Elliptic curves $E_{AB}$ with good reduction at $3$[]{data-label="tabgr3"}
We now apply Proposition \[eqabcountdisc\]. Let $1\leq i\leq 9$ and $1\leq j\leq 11$ be integers, and consider the set of integers $(A,B)$ that satisfy line $i$ of Table \[tabgr2\] and line $j$ of Table \[tabgr3\]. Let $\nu_{ij}=\nu_2(i)\cdot \nu_3(j)$ denote the density of this set of integers, and let $\Delta_{ij}=\Delta_2(i)\cdot
\Delta_3(j)$ denote the product of the $2$- and $3$-parts of the discriminant $\Delta(A,B)$. It is necessary to count the number of pairs $(A,B)\in\Z^2$ that satisfy the following properties:
1. The pair $(A,B)$ satisfies the $i$th (resp. $j$th) condition of Table \[tabgr2\] (resp. Table \[tabgr3\]);
2. $0<\pm\Delta(A,B)<\Delta_{ij}X$;
3. $j(A,B)<\log(\Delta(A,B)/\Delta_{ij})$;
4. for all primes $p\geq5$, either $p^4\nmid A$ or $p^6\nmid B$.
Counting the pairs $(A,B)$ which satisfy the first four properties is immediate from Proposition \[eqabcountdisc\], and the fifth condition can be imposed by applying a simple inclusion exclusion sieve. We thus obtain $$\#\bigl\{E\in\E:0<\pm\Delta(E)<X\}\sim \sum_{i,j}\nu_{ij}\cdot\prod_{p\geq 5}(1-p^{-10})\cdot c_\infty^\pm(\Delta_{ij}\cdot X).$$ The values $c_\infty^\pm(X)$ scale as follows: we have $$c_\infty^\pm(X)=X^{5/6}c_\infty^\pm(1)+o(X).$$ Furthermore, the values $c_\infty^\pm(1)$ are computed in [@watkins-heuristics §2] to be $$c_\infty^+(1)=\frac{2}{4^{1/3}\cdot 27^{1/2}}\cdot\frac{1}{5}\cdot B(1/2,1/6);\quad\quad
c _\infty^-(1)=\frac{2}{4^{1/3}\cdot 27^{1/2}}\cdot \frac{3}{5}\cdot B(1/2,1/3)=\sqrt{3}\,c_\infty^+(1).$$ Above, $B(x,y)$ denotes the beta function given by $$B(x,y)=\int_{0}^1 t^{x-1}(1-t)^{y-1}dt=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}.$$ We therefore obtain $$\begin{array}{rcl}
\displaystyle\#\bigl\{E\in\E:0<\pm\Delta(E)<X\}&\sim&\displaystyle
\sum_{i,j}\nu_{ij}\Delta_{ij}^{5/6}\cdot\prod_{p\geq 5}(1-p^{-10})\cdot c_\infty^\pm(1)\cdot X^{5/6}
\\&=&\displaystyle
c_\infty^\pm(1)\Bigl(\sum_i\nu_2(i)\Delta_2(i)^{5/6}\Bigr)
\Bigl(\sum_i\nu_3(i)\Delta_3(i)^{5/6}\Bigr)
\prod_{p\geq 5}(1-p^{-10})\cdot X^{5/6}
\\&=&\displaystyle
\frac{2^{2/3}}{4}\frac{2\alpha^\pm}{4^{1/3}\cdot 3^{3/2}\cdot 5}
\frac{\Gamma(1/2)\Gamma(1/6)}{\Gamma(2/3)}
\prod_{p\geq 5}(1-p^{-10})\cdot X^{5/6}
\\&=&\displaystyle
\frac{\alpha^\pm}{60\sqrt{3}}
\frac{\Gamma(1/2)\Gamma(1/6)}{\Gamma(2/3)}
\prod_{p\geq 5}(1-p^{-10})\cdot X^{5/6},
\end{array}$$ as necessary
Ordering elliptic curves by conductor
-------------------------------------
Suppose that $\GG$ is equal to $\E_*(\Sigma)$ for a large collection of reduction types $\Sigma$, where $*$ is either $\sf$ or some positive $\kappa<7/4$. Pick a small positive constant $\delta<1/9$. Then there exists a positive constant $\theta$ such that
$$\label{eqinexec}
\begin{array}{rcl}
\displaystyle\#\{E\in \GG^\pm:C(E)<X\}&=&
\displaystyle\sum_{n\geq 1}
\#\{E\in \GG^\pm:\ind(E)=n;\;\Delta(E)<nX\}\\&=&
\displaystyle\sum_{n,q\geq 1}\mu(q)\,
\#\{E\in \GG^\pm:nq\mid\ind(E);\;\Delta(E)<nX\}\\&=&
\displaystyle\sum_{\substack{n,q\geq 1\\nq<X^\delta}}\mu(q)\,
\#\{E\in \GG^\pm:nq\mid\ind(E);\;\Delta(E)<nX\}+O(X^{5/6-\theta})
\end{array}$$
where we bound the tail using the uniformity estimates in Theorems \[thunifsqi\] and \[thunifsmind\]. We perform another inclusion exclusion sieve to evaluate each summand of the right hand side of the above equation: for each prime $p$, let $\chi_{\Sigma_p,nq}:\Z_p^2\to\R$ denote the characteristic function of the set of all $(A,B)\in\Z_p^2$ that satisfy the reduction type specified by $\Sigma_p$ and satisfy $nq\mid\ind(E_{AB})$. Let $\chi_p$ denote $1-\chi_{\Sigma_p,nq}$, and define $\chi_k:=\prod_{p\mid
k}\chi_p$ for squarefree integers $k$. Then we have $$\prod_{p}\chi_{\Sigma_p,nq}(A,B)=\sum_k\mu(k)\chi_k(A,B)$$ for every $(A,B)\in\Z^2$. Set $\nu_*(nq,\Sigma)$ to be the product over all primes $p$ of the integral of $\chi_{\Sigma_p,nq}$. Therefore, for $nq<X^\delta$, we obtain $$\begin{array}{rcl}
\displaystyle\#\{E\in \GG^\pm:nq\mid\ind(E);\;\Delta(E)<nX\}&=&
\displaystyle\sum_{\substack{(A,B)\in\Z^2\\0<\pm\Delta(E_{AB})<nX}}\sum_{k\geq 1}\mu(k)\chi_k(A,B)\\[.2in]
&=&\displaystyle\sum_{\substack{(A,B)\in\Z^2\\0<\pm\Delta(E_{AB})<nX}}\sum_{k=1}^{X^{4\delta}}\mu(k)\chi_k(A,B)
+O\Bigl(\frac{(nX)^{5/6}}{X^{2\delta}}\Bigr)\\[.4in]&=&
\displaystyle c_\infty^\pm(nX)\nu_*(nq,\Sigma)+O_\epsilon(X^{1/2+2\delta+\epsilon}+X^{5/6-7\delta/6}),
\end{array}$$ where the second equality follows from the uniformity estimate in Proposition \[propelemufec\], and the third follows from Proposition \[eqabcountdisc\] and adding up the volume terms by simply reversing the inclusion exclusion sieve. Note that the constant $\delta$ has been specifically picked to be small enough so that Proposition \[eqabcountdisc\] applies.
For each $n$, let $\lambda_*(n,\Sigma)$ denote the volume of the closure in $\hat{\Z}^2$ of the set of all $(A,B)\in\Z^2$ such that $E_{AB}$ belongs to $\GG=\E_*(\Sigma)$ and $E_{AB}$ has index $n$. Returning to , we obtain $$\begin{array}{rcl}
\displaystyle\#\{E\in \GG^\pm:C(E)<X\}&=&
\displaystyle c^\pm_\infty(1)X^{5/6}
\sum_{\substack{n,q\geq 1\\nq<X^\delta}}\mu(q)n^{5/6}\nu_*(nq,\Sigma)
+o(X^{5/6})\\[.2in]
&=&\displaystyle c_\infty^\pm(1)X^{5/6}\sum_{n\geq 1}n^{5/6}\lambda_*(n,\Sigma),
\end{array}$$ where again, the final equality follows by reversing the inclusion exclusion sieve of .
For each prime $p$ and $k\geq 0$, let $\bar{\nu}_*(p^k,\Sigma)$ denote the $p$-adic density of the set of all $(A,B)\in\Z^2$ such that $E_{AB}\in\E_*(\Sigma)$ and $\ind_p(E_{AB})=p^k$. The constant $\lambda_*(n,\Sigma)$ is a product over all $p$ of local densities: $$\begin{array}{rcl}
\displaystyle \lambda_*(n,\Sigma)&=&
\displaystyle\prod_{p\nmid n}\bar{\nu}_*(p^0,\Sigma)
\prod_{\substack{p^k\parallel n\\k\geq 1}}\bar{\nu}_*(p^k,\Sigma)\\[.2in]
&=&\displaystyle\prod_p\bar{\nu}_*(p^0,\Sigma)
\prod_{\substack{p^k\parallel n\\k\geq 1}}
\frac{\bar{\nu}_*(p^k,\Sigma)}{\bar{\nu}_*(p^0,\Sigma)}.
\end{array}$$ Hence $\lambda_*(n,\Sigma)$ is a multiplicative function in $n$, and we have $$\begin{array}{rcl}
\displaystyle\sum_{n\geq 1}n^{5/6}\lambda_*(n,\Sigma)&=&
\displaystyle\prod_p\bar{\nu}_*(p^0,\Sigma)
\prod_{p}\Bigl(\sum_{k=0}^\infty p^{5k/6}
\frac{\bar{\nu}_*(p^k,\Sigma)}{\bar{\nu}_*(p^0,\Sigma)}\Bigr)\\[.2in]&=&
\displaystyle\prod_p\Bigl(\sum_{k=0}^\infty p^{5k/6}\bar{\nu}_*(p^k,\Sigma)
\Bigr).
\end{array}$$
The values of $\bar{\nu}_*(p^k,\Sigma)$ are easily computed from Table \[tabden\]. We then have , proving the first part of Theorem \[thmmainlarge\].
The average size of the $2$-Selmer groups of elliptic curves in $\E_\sf(\Sigma)$
--------------------------------------------------------------------------------
Let $\Sigma$ be a large collection of reduction types. For a positive integers $n$ and a positive real number $X$, let $\E(\Sigma,n,X)$ denote the set of elliptic curves $E\in\E_\sf(\Sigma)$, such that $n\mid\ind(E)$ and $|\Delta(E)|<X$. Then, as in the previous subsection, we have $$\begin{array}{rcl}
\displaystyle\sum_{\substack{E\in\E(\Sigma)^\pm\\C(E)<X}}(|\Sel_2(E)|-1)&=&
\displaystyle\sum_{n,q\geq 1}\mu(q)\sum_{E\in\E(\Sigma,nq,nX)^\pm}(|\Sel_2(E)|-1)
\\[.2in]&=&
\displaystyle\sum_{\substack{n,q\geq 1\\nq<X^\theta}}\mu(q)\sum_{E\in\E(\Sigma,nq,nX)^\pm}(|\Sel_2(E)|-1)+O_\epsilon(X^{5/6-\theta/6+\epsilon}),
\end{array}$$ for every $\theta>0$, where the second equality is a consequence of Theorem \[thunifsqi\]. Therefore, the final assertion of Theorem \[thmmainlarge\] follows immediately from the following result.
\[propfinalsel\] There exist positive constants $\theta$ and $\theta_1$ such that $$\sum_{E\in\E(\Sigma,qn,nX)^\pm}(|\Sel_2(E)|-1)=2|\E(\Sigma,nq,nX)^\pm|+O(X^{5/6-\theta_1}),$$ for every $nq<X^{\theta}$.
Given the uniformity estimate Proposition \[propelemufbqf\] that we have already proved, the proof of Proposition \[propfinalsel\] very closely follows the proof of [@bs2sel Theorem 3.1]. In what follows, we briefly sketch the proof of Theorem \[thmmainlarge\], indicating the change needed at the places where it differs from [@bs2sel]. The starting point of the proof is the following parametrization of the $2$-Selmer groups of elliptic curves in terms of orbits on integral binary quartic forms. This correspondence is due to Birch and Swinnerton-Dyer, and we state it in the form of [@bs2sel Theorem 3.5].
Let $E:y^2=x^3+Ax+B$ be an elliptic curve over $\Q$, and set $I=I(E):=-3A$ and $J=J(E):=-27B$. Then there is a bijection between $\Sel_2(E)$ and the set of $\PGL_2(\Q)$-equivalence classes of locally soluble integral binary quartic forms with invariants $2^4I$ and $2^6J$.
Moreover, the set of integral binary quartic forms that have a rational linear factor and invariants equal to $2^4I$ and $2^6J$ lie in one $\PGL_2(\Q)$-equivalence class, and this class corresponds to the identity element in $\Sel_2(\Q)$.
The second step in the proof is to obtain asymptotics for the number of $\PGL_2(\Z)$-orbits on the set of integral binary quartic forms whose coefficients satisfy congruence conditions modulo some small number $n$, where these forms have bounded invariants. In [@bs2sel], the invariants were bounded by height. Here instead, we bound their discriminants and corresponding $j$-invariant: for an element $f\in V_4(\R)$ with $\Delta(f)\neq 0$, define $j(f)$ to be $j(E)$ with $E$ given by $$E:y^2=x^3-(I/3)x-J/27.$$ For any $\PGL_2(\Z)$-invariant set $S\subset
V_4(\Z)$, let $N^{(i)}_4(S;X)$ denote the number the number of $\PGL_2(\Z)$-orbits on integral elements $f\in S\subset
V_4^{(i)}(\Z)$, that do not have a linear factor over $\Q$, and satisfy $\Delta(f)<X$ and $j(f)<\log\Delta(f)$.
In §5, we defined the sets $R^{(i)}$ which are fundamental sets for the action of $\PGL_2(\R)$ on $V(\R)^{(i)}$. Then $R^{(i)}$ contains one element $f\in V(\R)^{(i)}$ having invariants $I$ and $J$, for each $(I,J)\in\R^2$ with $4I^3-J^2\in\R_{>0}$ for $i=0,2\pm$ and $4I^3-J^2\in\R_{<0}$ for $i=1$. Furthermore, the coefficients of such an $f$ are bounded by $O(H(f)^{1/6})$. Define the sets $$R^{(i)}(X):=\{f\in R^{(i)}:0<|\Delta(f)|<X;j(f)<\log\Delta(f)\}.$$ Clearly, if $f\in R^{(i)}(X)$ with $\Delta(f)=X$, then $H(f)\ll
X^{1+\epsilon}$ and so the coefficients of $f$ are bounded by $O(X^{1/6+\epsilon})$.
Let $\delta=1/18$ be fixed. Let $L\subset V(\Z)$ be a lattice defined by congruence conditions modulo $n$, where $n<X^\delta$. Denote the set of elements in $L$ that have no linear factor by $L^\irr$ and define $\nu(L)$ to be the volume of the completion of $L$ in $V_4(\hat{\Z})$. Let $G_0\subset\PGL_2(\R)$ be a nonempty bounded open ball, and set $n_1=2$, $n_0=n_{2\pm}=4$. Identically to [@bs2sel §2.3], it follows that $N_4^{(i)}(L,X)$ is given by $$\label{eqbqfcount}
\begin{array}{rcl}
\displaystyle N_4^{(i)}(L,X)&=&
\displaystyle\frac{1}{n_i\Vol(G_0)}\int_{\gamma\in\PGL_2(\Z)\backslash\PGL_2(\R)}
\#\bigl\{\gamma G_0\cdot R^{(i)}(X)\cap L^\irr\}d\gamma
\\[.2in]&=&
\displaystyle\frac{1}{n_i}\int_{\gamma\in\PGL_2(\Z)\backslash\PGL_2(\R)}
\nu(L)\Vol(G_0\cdot R^{(i)}(X))d\gamma+O(X^{7/9}),
\end{array}$$ where the error term is obtained in a similar manner to [@bs2sel (18)–(20)]. There are two differences: first, we use Theorem \[thomin\] (instead of Davenport’s result stated as [@bs2sel Proposition 2.6]) to estimate the number of lattice points in $\gamma G_0\cdot R^{(i)}(X)$. Second, since we are imposing congruence conditions on $L$ modulo $n<X^\delta$ with $\delta=1/18$, we cut off the integral over $\gamma$ when the $t$-coefficient of $\gamma$ in its Iwasawa coordinate is $\gg X^{1/36}$. That way, the coefficients of the ball $\gamma G_0\cdot R^{(i)}(X)$ are always bigger than $n$. The precise values of $\delta=1/18$ and $7/9$, the exponent of the error term, are not important. The third step in the proof is to introduce a bounded weight function $m:V_4(\Z)\to\R$, which is the product $m=\prod_p m_p$ of local weight functions $m_p:V_4(\Z_p)\to\R$, such that for all but negligibly many ($\ll_\epsilon X^{3/4+\epsilon}$) elliptic curves $E_{AB}$, we have $$|\Sel_2(E_{AB})|-1=\sum_{f\in \frac{V_4(\Z)_{A,B}}{\PGL_2(\Z)}}m(f),$$ where $f$ is varying over $\PGL_2(\Z)$-orbits on integral binary quartic forms with no linear factor and invariants $I(f)=-3\cdot 2^4I$ and $J(f)=-27\cdot2^6J$. In our situation, we do not need any changes to this part of the proof.
The fourth and final part of the proof is to perform a sieve so as to count $\PGL_2(\Z)$-orbits on integral binary quartic forms with bounded invariants, so that each form $f$ is weighted by $m(f)$. Performing a standard inclusion-exclusion sieve using together with the uniformity estimate Proposition \[propelemufbqf\] and the volume computations of [@bs2sel §3.3 and §3.6] yields Proposition \[propfinalsel\]. This concludes the proof of Theorem \[thmmainlarge\].
[^1]: See, however, work of Hortsch [@HortschFaltH] obtaining asymptotics for the number of elliptic curves with bounded Faltings height.
|
---
abstract: 'The existence of flat bands is generally thought to be physically possible only for dimensions larger than one. However, by exciting a system with different orthogonal states this condition can be reformulated. In this work, we demonstrate that a one-dimensional binary lattice supports always a trivial flat band, which is formed by isolated single-site vertical dipolar states. These flat band modes correspond to the highest localized modes for any discrete system, without the need of any aditional mechanism like, e.g., disorder or nonlinearity. By fulfilling a specific relation between lattice parameters, an extra flat band can be excited as well, with modes composed by fundamental and dipolar states that occupy only three lattice sites. Additionally, by inspecting the lattice edges, we are able to construct analytical Shockley surface modes, which can be compact or present staggered or unstaggered tails. We believe that our proposed model could be a good candidate for observing transport and localization phenomena on a simple one-dimensional linear photonic lattice.'
address: 'Departamento de Física and Millennium Institute for Research in Optics (MIRO), Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Santiago, Chile'
author:
- 'Gabriel Cáceres-Aravena and Rodrigo A. Vicencio'
title: 'Perfect localization on flat band binary one-dimensional photonic lattices'
---
Introduction
============
The propagation of waves in periodical systems are the natural framework to explore transport and localization phenomena in diverse fields of physics [@rep1; @rep2; @PT]. For example, the first experimental observation of Anderson-like localization in disordered linear systems [@anderson] was made in 2007, in two-dimensional (2D) induced photonic lattices [@moti] and, subsequently, in fabricated one-dimensional (1D) waveguide arrays [@mora]. More recently, an important theoretical and experimental interest on flat band (FB) systems has emerged [@BWB08; @FBluis; @flach1; @flach2], showing interesting localization and transport properties on linear lattices. The current experimental techniques allow direct and indirect excitation of flat band phenomena [@liebprl; @liebseba; @chen1; @chen2; @OLsaw; @OLdiamond; @StubSR; @graphenNJP; @solidstate; @BECStan; @pillarsAmo; @acustic], which is associated to destructive interference on specific lattice geometries. Specificaly, a Lieb photonic lattice was chosen to demonstrate, for the first time in any physical system, the existence of FB localized states [@liebprl; @liebseba]. A FB lattice geometry allows a precise cancellation of amplitudes outside the FB mode area, what effectively cancels the transport of energy across the system. Flat-band systems possess a linear spectrum where at least one band is completely flat or thin compared to the next energy gap. This implies the need of having a system with an unit cell composed of at least two sites and, therefore, at least two bands [@FBluis; @prbpal]. In general, light propagating in FB lattices will experience zero or very low diffraction, when exciting some specific sites at the unit cell [@NJPlieb; @OLsaw].
A very interesting feature of flat band systems is the possibility to construct highly localized eigenmodes by means of a destructive linear combination of extended linear wave functions [@BWB08; @linePRL]. These FB states are spatially compact, occupy only few lattice sites, and rapidly decay to a completely zero tail as soon as a destructive interference condition is fulfilled [@FBluis]. This is a very remarkable property because FB lattices naturally generate localized structures in a linear regime, with a localization length of the order of a single unit cell. Moreover, as these linear localized modes posses all the same frequency, they are completely degenerated and any linear combination of them will also be a stable propagating solution. This can be used to achieve a non-diffractive transmission of optically codified information [@liebprl; @kagsignals; @StubSR; @linePRL; @graphenNJP; @zhensignal1; @zhensignal2].
Almost all studies on photonic lattices have considered single-mode excitation only. This has been reinforced due to the experimental complexity in the excitation of higher-order modes on a given lattice system, which has found a partial solution only by the implementation of a selective p-band population in cold-atom systems [@becdip1; @becdip2] and micropillars arrays [@amo1]. But, optical waveguides can also host higher order modes, depending on the specific experimental parameters used to perform the experiment (waveguide arrays are typically fabricated considering single-mode waveguides at a given wavelength; however, by reducing this parameter higher-order modes can be excited as well). Their excitation could promote richer dynamics and new interesting phenomena, as it has been suggested for cold-atoms loaded in optical potentials [@becdip3; @becdip4; @becdip5; @becdip6]. However, a precise excitation of dipolar states has been possible very recently in optical waveguide lattices by using an image generator setup based on spatial light modulators [@graphenNJP; @dipole1D]. There, a well-defined contrast between the transport of fundamental and dipolar states has been shown clearly. The possibility to experimentally excite and control higher bands excitations, in optical lattice systems, paves the venue in which the study of remarkable properties of correlated systems such as superfluidity, superconductivity, organic ferromagnetic, antiferromagnetic ordering, among others, could become a concrete possibility [@becdip3; @becdip4; @becdip5; @becdip6; @dipoplus1].
As it is well known, diverse interactions have been proposed during several years to achieve stable energy localization on a lattice. For example, disorder [@moti; @mora], impurities and defects [@chen3; @chen4], or even nonlinearity [@PRL98; @PRL03; @OE05]. However, all these mechanisms necessarily destroy the periodicity of the system, what finally has important consequences on the transport of energy across the lattice. In the present work, we propose a new model for the observation of FB properties. We focus on a binary 1D lattice, which to our knowledge corresponds to the simplest physical configuration for studying FB localization. We specifically concentrate on proposing a simple – completely periodic – system which could show the conditions to observe localization and transport of energy on a linear regime. Our model possesses a trivial FB which corresponds to an effectively isolated dipolar mode. This mode is localized at a single waveguide corresponding, therefore, to the most localized FB state ever. By precisely tuning the model parameters, we observe that a second FB can be excited as well, with states occupying only three lattice sites. In addition, we explore edge localization and find analytical Shockley edge modes with different decaying properties. At the end, we find an additional flat band when assuming equal propagation constants on both orthogonal states.
![(a) 1D-binary waveguide array. (b) Coupling interactions for this model (red represents a positive amplitude while blue a negative one). (c) Effective ribbon lattice when considering two modes per waveguide (the shaded area indicates the unit cell).[]{data-label="fig1"}](fig1.pdf){width="8.8cm"}
Model
=====
We study the propagation of light, in weakly coupled waveguide arrays, based on a coupled mode approach which originates from a paraxial wave equation and ends up with a set of Discrete Lineal Schrödinger-like equations [@rep1; @rep2]. This approach assumes an evanescent interaction between the modes of neighboring waveguides, with a coupling coefficient defined via the superposition integral between both mode wave-functions. Obviously, this interaction is negligible when waveguides are far away in distance and becomes physically relevant only when waveguides are close enough. Typical experiments on laser-written photonic lattices [@guiasalex] define a distance of around $20$ microns to correctly describe the system assuming nearest-neighbors interactions only. In this work, we model a binary one-dimensional photonic lattice composed of an alternated configuration of waveguides, as shown in Fig. \[fig1\](a). We assume elliptically oriented waveguides, which are the standard geometry in laser-written systems [@guiasalex], with the propagation coordinate $z$ (dynamical variable) running perpendicular to the transversal waveguide profile. Additionally, we consider that each waveguide supports only two orthogonal modes, the fundamental ($s$) and the dipolar ($p$) ones. In general, a single waveguide possesses always at least one bound state, which corresponds to the symmetric $s$ mode [@snyder; @gloge]. However, depending on the experimental conditions, it is possible to directly excite higher-order states as, e.g., $p$ modes [@dipole1D; @graphenNJP]. For a given waveguide, having a defined geometry and refractive index contrast, the excitation wavelength can be tuned experimentally to excite higher-order states. As different modes have a different spatial configuration, there will be a natural mismatch in their propagation constants. This implies that $\beta_s\neq \beta_p$ [@graphenNJP], where $\beta_i$ is the longitudinal propagation constant of the $i$-mode at any lattice waveguide.
The possible interactions between modes at different neighboring waveguides are depicted in Fig. \[fig1\](b). Considering the symmetry of $s$ and $p$ wave-functions, we construct a general interaction rule for our binary system: the coupling between $s$ modes (defined as $V_s$) is always positive; the coupling between $p$ modes ($V_p$) as well as the coupling between $s$ and vertical $p$ modes ($\bar{V}_{sp}$) are always zero; the coupling between vertical $s$ and horizontal $p$ modes ($V_{sp}$) is defined positive when the $s$ mode is at the left-hand side, if not a minus sign is applied. In general, due to the larger area occupied by $p$ modes, $V_{sp}>V_s$. Having this in mind, we construct an effective ribbon lattice in Fig. \[fig1\](c) and write the effective dynamical equations as follows $$\begin{aligned}
\begin{split}
-i\frac{\partial u_n(z)}{\partial z} &= \beta_s u_n+V_s (v_{n}+v_{n-1}) + V_{sp} (w_n-w_{n-1})\ ,\\
-i\frac{\partial v_n(z)}{\partial z} &= \beta_s v_n+V_s (u_{n+1}+u_{n}) \ ,\\
-i\frac{\partial x_n(z)}{\partial z} &= \beta_p x_n\ ,\\
-i\frac{\partial w_n(z)}{\partial z} &= \beta_p w_n-V_{sp} (u_{n+1}-u_n)\ .\label{eqs}
\end{split}\end{aligned}$$ Here, $u_n$ and $v_n$ ($x_n$ and $w_n$) describe the amplitude of fundamental (dipolar) modes at the $n$th unit cell. The alternated orientation of our 1D binary lattice and the possibility of exciting two modes per waveguide generate a four-state effective system, which is described by these four coupled equations. It is important mentioning that, in order to have an effective dynamical interaction between the $s$ and the $p$ modes, $\Delta \beta\equiv \beta_s-\beta_p$ has to be of the order of $V_{sp}$. If not, this detuning effectively decouples the interaction between these two modes and they simply do not interact [@graphenNJP].
Linear spectrum
===============
![Linear spectrum for the 1D binary lattice. (a) $V_{sp}=1.5$ and (b) $2.5$, for $\Delta\beta=5$. (c) $V_{sp}=1.1$ and (d) $2.0$, for $\Delta\beta=1$. Full black, dashed black, dashed orange, and full orange correspond to $\beta_1$, $\beta_2$, $\beta_3$ and $\beta_4$ bands. (e) FB mode amplitude profile at $\beta_2=0$. In all the figures, we set $V_s=1$.[]{data-label="fig2"}](fig2.pdf){width="8.6cm"}
We solve the linear spectrum of the system by inserting into (\[eqs\]) a standard plane-wave ansatz [@FBluis], of the form $$\{u_n,v_n,x_n,w_n\}(z)=\{A,B,C,D\}\ e^{i k n} e^{i (\beta+\beta_p) z}\ .$$ With this, we assume that the wave propagation occurs along the horizontal direction only, being $k$ equal to the normalized transversal wavector. $\beta$ represents the longitudinal propagation constant of the lattice eigenmodes (also known as supermodes), while $\beta_i$ represents the longitudinal propagation constant of mode $i$ on a single waveguide. Without loss of generality, we include a gauge transformation on $\beta_p$, with the purpose of reducing the model parameters and simplify the overall description. By doing this, we get a set of four coupled equations which can be written as follows $$\begin{aligned}
\label{ep}
\beta \Psi =\hspace{0.cm}\\ \begin{pmatrix}
\Delta \beta & V_s(1+e^{-i k}) & 0 & V_{sp} (1-e^{-i k}) \\
V_s(1+e^{i k}) & \Delta \beta & 0 & 0 \\
0 & 0 & 0 & 0\\
V_{sp} (1-e^{i k}) & 0 & 0 & 0
\end{pmatrix} \Psi\ , \nonumber\end{aligned}$$ with $\Psi\equiv\{A,B,C,D\}$. By solving this eigenvalue problem, we obtain four solutions, with three of them being determined by the following third-order equation $$\left[\beta(\Delta \beta-\beta)^2-4V_s^2 \beta \cos^2\bar{k}+4V_{sp}^2(\Delta \beta-\beta) \sin^2 \bar{k}\right]=0\ ,$$ where $\bar{k}\equiv k/2$. These three bands are analytically non trivial and have no a simple and compact form for arbitrary parameters. Therefore, we show them graphically only in Figs. \[fig2\](a)–(d), in the first Brillouin zone, using full black and orange lines. We also show in these figures (using dashed black lines) a completely constant solution $$\beta_2 =0\ ,$$ which corresponds to the lattice second band, as defined below. This trivial and completely flat band is related to the excitation of isolated vertical $p$ modes only. As these modes possess no coupling at all with nearest-neighbor waveguides, once they are excited they remain localized at the input position as long as the system length. This can be understood easily by directly integrating the third equation in (\[eqs\]), getting $x_n(z)=x_n(0)\exp \{i\beta_p z\}$; being, therefore, a trivial stationary solution. In Fig. \[fig2\](e) we show a sketch of this FB state. This corresponds to the most localized FB state ever, which occupies only one site of the lattice. As this mode can be excited in every vertically oriented waveguide (using arbitrary amplitudes), this trivial band can be used, for example, to transmit optically codified information through this 1D lattice.
We observe that the linear spectrum is quite symmetric; therefore, we analyze the four linear bands considering an increasing order denoted by $\beta_1$, $\beta_2$, $\beta_3$ and $\beta_4$, as indicated in Fig. \[fig2\](a). At $k_x=0$, band edges become $$\{0,0,\Delta\beta-2V_s,\Delta\beta+2V_s\}\ ,$$ for $\Delta\beta\geqslant2V_s$, and $$\{\Delta\beta-2V_s,0,0,\Delta\beta+2V_s\}\ ,$$ for $\Delta\beta<2V_s$. At $k_x=\pi$, band edges are always $$\left\{\frac{\Delta\beta-\sqrt{\Delta\beta^2+16V_{sp}^2}}{2},0,\Delta\beta,\frac{\Delta\beta+\sqrt{\Delta\beta^2+16V_{sp}^2}}{2}\right\},$$ as shown for some specific parameters in Figs. \[fig2\](a)–(d). First of all, we notice that there is no gap between bands $\beta_1$, $\beta_2$ and $\beta_3$ for $\Delta\beta=2V_s$. Then, there is a gap of size $\Delta\beta-2V_s$, between bands $\beta_2$ and $\beta_3$ for $\Delta\beta\geqslant2V_s$, and between bands $\beta_1$ and $\beta_2$ for $\Delta\beta<2V_s$, which does not depend on the interaction between $s$ and $p$ modes. On the contrary, the gap between bands $\beta_3$ and $\beta_4$ changes depending on the curvature of band $\beta_4$, what strongly depends on coupling $V_{sp}$ as shown in Fig. \[fig2\]. This change in the curvature necessarily implies that $\beta_4$ must be flat at some specific value of $V_{sp}$. By demanding that $\beta_4(0)=\beta_4(\pi)$, we obtain the following FB condition $$V_{sp}^{FB}\equiv V_s\sqrt{1+\frac{\Delta\beta}{2V_s}}\ .$$ This mathematical relation is physical and experimentally possible due to the fact that $V_{sp}>V_s$, as expected considering the $s$ and $p$ mode profiles. If we fix coupling $V_s$, $V_{sp}^{FB}$ grows monotonically as a function of detuning $\Delta\beta$. Once this condition is fulfilled, the fourth band becomes completely flat with a value $$\beta_4=\Delta\beta +2V_s,$$ as shown in Figs. \[fig3\](a) and (b) by a straight horizontal full orange line. We look for the eigenmode profile at this new FB condition. We assume a center site $n_0$ and an arbitrary amplitude $A$, obtaining that $$\begin{aligned}
u_{n} = A \delta_{n,n_0},\ v_{n} = \left(\frac{A}{2}\right)\left(\delta_{n,n_0}+\delta_{n,n_{0}-1}\right),\\
x_{n}=0,\ w_{n} = \left(\frac{V_s A}{2V_{sp}^{FB}}\right) \left(\delta_{n,n_0}-\delta_{n,n_{0}-1}\right).\end{aligned}$$ This profile is composed of both, $s$ and $p$, modes simultaneously and a sketch of it, on an effective ribbon lattice, is presented in Fig. \[fig3\](c). We observe that the dipolar mode is smaller in amplitude with a factor $\sim 0.4$, for the parameters used in this figure. As coupling $V_{sp}^{FB}$ is larger than $V_s$, the mode amplitudes are compensated in order to satisfy a FB localization condition, which relies on destructive interference at specific connector sites [@FBluis]. Superposed $s$ and $p$ mode amplitudes give the FB mode intensity profile sketched in Fig. \[fig3\](d), for the 1D binary lattice system. The amplitudes beside the center show a shifted intensity with respect to the center of the waveguide, as expected from the superposition of fundamental and dipolar profiles at those sites. As a consequence, this localized state is very localized in space and perfectly compact.
A study of the transport in this lattice, performed by exciting a single vertical bulk site only (a delta-like input condition) would show a transition between dispersion (transport), localization (insulation), and transport again, while varying parameter $V_{sp}$. Localization would occur close to the FB condition $V_{sp}^{FB}$, while transport would manifest away this value. This behavior is quite similar to the one found for Sawtooth lattices [@OLsaw], where a FB is formed only for a very specific condition between coupling constants. Therefore, our simple 1D binary model could show an insulator transition when coupling interaction $V_{sp}/V_s$ is varied along the experiment. This could be demonstrated by fabricating several lattices having different refractive index profiles or directly shown by varying the temperature of a single binary lattice to achieve a tuning on propagation constants [@bloch].
![Linear spectrum for (a) $\{\Delta\beta,V_{sp}^{FB}\}=\{5,1.87\}$ and (b) $\{\Delta\beta,V_{sp}^{FB}\}=\{1,1.22\}$. Full black, dashed black, dashed orange, and full orange correspond to $\beta_1$, $\beta_2$, $\beta_3$ and $\beta_4$ bands, respectively. (c) Effective amplitude and (d) intensity FB ($\beta_4$) mode profiles for $\Delta\beta=1$. In all the figures, we set $V_s=1$.[]{data-label="fig3"}](fig3.pdf){width="8.6cm"}
Edge states
===========
When solving the eigenvalue problem (\[ep\]), we look for solutions assuming an infinite lattice. Therefore, finite size effects, as for example linear edge modes, will not appear explicitly [@OLsaw]. However, by numerically diagonalizing a finite lattice system, we find that an edge with a vertically oriented waveguide generates an exponentially decaying eigenmode, while a horizontal edge waveguide does not. In order to investigate this edge state, we consider a vertical waveguide at site $n=1$ and assume the following ansatz [@rep1; @rep2; @OLsaw] $$\begin{aligned}
\{u_{n},v_{n},x_{n},w_{n}\}(z) = \{A,B,C,D\} \epsilon^{n-1}e^{i \beta_e z},\end{aligned}$$ for $n\geqslant1$, with $|\epsilon|<1$ (which implies an exponentially decaying state). $A,B$ and $D$ correspond to the amplitudes of this mode to be determined by solving a set of coupled equations. We assume a zero amplitude for mode $x_n$ ($C=0$), due to the no interaction of this mode with the rest of the system. (By taking $x_n\neq 0$ the frequency of this amplitude will be just zero, what not necessarily coincides with the frequency of the edge mode $\beta_e$. Additionally, there is always a perfectly localized edge state $x_n=C\delta_{n,1}$, as a trivial FB solution.) We insert this ansatz into model (\[eqs\]) and write the equations for sites $n=1$ and $n=2$. We obtain two sets of three coupled equations, where the second set is recursively repeated for $n> 2$, what validates the proposed ansatz. These equations are the followings: $$\begin{aligned}
\label{eqeps1}
\beta_{e} A &=&\beta_s A+V_s B+V_{sp} D ,\\
\beta_{e} B &=&\beta_s B+V_s A(1+\epsilon) ,\nonumber\\
\beta_{e} D &=&\beta_p D+V_{sp} A(1-\epsilon), \nonumber\end{aligned}$$ and $$\begin{aligned}
\label{eqeps2}
\beta_{e} A \epsilon &=&\beta_s A\epsilon+V_s B(1+\epsilon)+V_{sp} D(\epsilon-1)\ ,\\
\beta_{e} B \epsilon &=&\beta_s B\epsilon+V_s A\epsilon(1+\epsilon)\ ,\nonumber\\
\beta_{e} D \epsilon &=&\beta_p D\epsilon+V_{sp} A\epsilon(1-\epsilon)\ . \nonumber\end{aligned}$$ By applying some algebra to equations (\[eqeps1\]) and (\[eqeps2\]), we obtain that $$\left(\frac{D}{A}\right)=\left(\frac{V_s}{V_{sp}}\right)\gamma,\ \ \epsilon=2\gamma^2-1,\ \ \beta_{e}=\beta_s+2V_s \gamma,$$ with $$\gamma\equiv \left(\frac{B}{A}\right)=\frac{\sqrt{\Delta\beta^2 V_s^2+16 V_{sp}^2(V_s^2+V_{sp}^2)}-\Delta\beta V_s}{4(V_s^2+V_{sp}^2)}\ .$$ This expression satisfies that $0<\gamma<1$, what implies that $-1<\epsilon<1$; i.e., this edge state is exponentially localized at the surface when this surface has a vertically oriented first waveguide. Additionally, as $V_{sp}>V_s$ then $(D/A)<(B/A)$; therefore, the edge localization is reinforced with a decreasing profile into the bulk of the system. In order to study the effective spatial size of these edge states, we compute an effective participation ratio, defined as $R\equiv [\sum_n(|u_n|^2+|v_n|^2+|x_n|^2+|w_n|^2)]^2/\sum_n(|u_n|^4+|v_n|^4+|x_n|^4+|w_n|^4)$, obtaining $$R=\frac{\left[1+\gamma^2+(V_s/V_{sp})^2\gamma^2\right]^2(1+\epsilon^2)}{\left[1+\gamma^4+(V_s/V_{sp})^4 \gamma^4 \right]\hspace{0.3cm} (1-\epsilon^2)}\ .$$ In order to characterize these states, we plot the decaying factor $\epsilon$, the participation ratio $R$ and the frequency $\beta_{e}$ versus the coupling $V_{sp}$ in Figs. \[fig4\](a)–(c), respectively, for some specific values of $\Delta\beta$ and $V_s$ (the same phenomenology persists for different values).
![(a) Decaying factor $\epsilon$, (b) participation ratio $R$, and (c) propagation constant $\beta_{e}$ versus coupling $V_{sp}$, for the edge mode (full blue line). Bands are plotted in (c) as shaded regions. (d1)–(d3) Effective edge mode amplitude profiles for $V_{sp}:\sqrt{2},\ V_{sp}^{ce}=2.13,\ 3.43$, respectively, labeled by a circle, a diamond and a triangle in (a)–(d). $\Delta\beta=5$ and $V_s=1$.[]{data-label="fig4"}](fig4.pdf){width="8.6cm"}
First of all, we observe that when coupling $V_{sp}\rightarrow 0$, $B$ and $D$ goes to zero as well and the edge state bifurcates at the band center of a standard 1D lattice [@rep1; @rep2], with $\epsilon\rightarrow -1$, $R\rightarrow\infty$, and $\beta_e\rightarrow\beta_s$. This state coincides with the $\pi/2$ linear mode of a standard 1D lattice and have an effective spatial profile of only $s$-mode amplitudes. Once we increase the coupling $V_{sp}$, we observe that the decaying factor $\epsilon$ decreases in magnitude, being for example $-0.5$ for $V_{sp}\approx \sqrt{2}$. In this case, the edge mode has a staggered profile every two sites, as shown in Fig. \[fig4\](d1). After this, we obtain a perfectly localized edge state with an exactly zero tail ($\epsilon=0$), as Fig. \[fig4\](d2) shows. This compact edge mode is obtained for the condition$$V_{sp}^{ce}\equiv V_s\sqrt{1+\frac{\Delta\beta}{\sqrt{2}V_s}}>V_{sp}^{FB}\ .$$ This state is similar to the edge mode found in Sawtooth lattices [@OLsaw], when different amplitudes destructively interfere at the connector sites of the lattice, in this case at the second vertically oriented waveguide. Although this profile corresponds to a perfectly compact edge state, which occupies only two sites of the lattice, with a mixed $s$-$p$ profile, it is not the most localized edge state on this 1D binary system. In fact, for the parameters considered in Fig. \[fig4\], the perfectly compact edge state at $V_{sp}^{ce}=2.13$ has a participation ratio of $R=1.92$, while the minimum participation ratio $R=1.89$ occurs for $V_{sp}=1.98$. After this regime, the decaying factor starts to grow slowly and profiles become completely unstaggered in their phase structure, as the example shown in Fig. \[fig4\](d3) for $\epsilon\approx 0.5$. By a further increment of $V_{sp}$, $\epsilon$ slowly tends to its upper bound $1$, implying a smooth increment of $R$. The propagation constant $\beta_e$ slowly tends to $\Delta\beta+2V_s$, which coincides with the bottom of band $\beta_4$. It is important to notice that the FB condition at $V_{sp}^{FB}=1.87$ \[see dashed vertical line in Fig. \[fig4\](c)\] produces an exchange on band $\beta_4$, in which the fundamental unstaggered mode passes from being at the top of the band for $V_{sp}=0$, to be at the band bottom for $V_{sp}>V_{sp}^{FB}$, as shown in Fig. \[fig4\](c). Finally, a larger $V_{sp}$ coefficient implies that $\delta\rightarrow 1$, with $B/A\rightarrow 1$ and $D/A\rightarrow 0$. Therefore, the lattice effectively transforms into a standard 1D system, with a homogenous spatial profile of $s$-mode amplitudes only, which coincides with a standard unstaggered fundamental mode. Here, although $V_{sp}$ is large compared to $V_s$, there is a consecutive cancellation of dipolar amplitudes $D$, due to the alternated sign of this coupling interaction.
Both limits ($V_{sp}\rightarrow 0,\infty$) gives us an extended mode which coincides with the $\beta_4$-band modes of standard 1D lattices, where no surface states exist without distorting the lattice border [@1DOL; @1DPRL; @1DOLKip]. As we observe in Fig. \[fig4\](c), $\beta_{e}$ is only allowed to exist in the region $\{\Delta\beta,\Delta\beta+2V_s\}$, and edge modes behave quite similar to the one found in a Sawtooth lattice [@OLsaw], including the band twist at the FB critical parameter. As our edge modes do not appear due to any lattice perturbation at the border of the system [@tamm; @1DOL; @1DPRL; @1DOLKip], but due to a crossover (twist) of the fourth linear band, the found edge modes are simply classified as Shockley-surface states [@shock1; @shock2; @shock3].
$\Delta\beta=0$ limit
=====================
![(a) Linear spectrum for $\Delta\beta=0$, $V_s=1$, and $V_{sp}=2.0$. Full black, dashed black, dashed orange, and full orange correspond to $\beta_1$, $\beta_2$, $\beta_3$ and $\beta_4$ bands, respectively. (b) Effective amplitude and (c) intensity FB ($\beta_3$) mode profiles.[]{data-label="fig5"}](fig5.pdf){width="8.6cm"}
Although the case $\Delta\beta=0$ corresponds to a non-physical solution in our photonic system [@snyder; @gloge; @graphenNJP], it becomes interesting to analyze it due to its phenomenology. By applying this condition, the eigenvalue problem (\[ep\]) gives four simple solutions: $$\beta=0,0,\pm2\sqrt{V_s^2 \cos^2\bar{k}+V_{sp}^2 \sin^2\bar{k}}\ .$$ We plot the linear spectrum in Fig. \[fig5\](a) and observe two opposite dispersive bands, showing a particle-hole symmetry [@phs] in which for any value of $k_x$ there are two eigenfrequencies $\pm\beta(k)$. Additionally, we found two flat bands at exactly the same frequency $\beta_2=\beta_3=0$. The first one is the previously found trivial FB $\beta_2=0$, which consists on single-site vertical dipolar states. The second FB is generated by a combination of horizontal $s$ and $p$ modes, whose mode profiles consist on four amplitudes different to zero, having the following structure: $$\begin{aligned}
u_{n} = 0,\ v_{n} = A\left(\delta_{n,n_0}-\delta_{n,n_{0}+1}\right),\\
x_{n}=0,\ w_{n} = -\left(\frac{V_s}{V_{sp}}\right)A\left(\delta_{n,n_0}+\delta_{n,n_{0}+1}\right).\end{aligned}$$ This amplitude profile is sketched in Fig. \[fig5\](b), with the corresponding intensity profile shown in Fig. \[fig5\](c). We observe how a perfect cancellation of amplitudes, at connector sites, allows the formation of a highly localized pattern, which has only two sites different to zero. In terms of localization area, this state is comparable with the one found for diamond lattices [@OLdiamond], which is the most localized FB state observed experimentally up to now.
Conclusions
===========
In conclusion, we have proposed a new model for the study of localization and transport of light in photonic lattices. In particular, our model consists on a rather simple 1D lattice having alternated orientation of elliptical waveguides. We found that, by assuming two – $s$ and $p$ – modes per site, a quasi-1D effective ribbon system emerges, which describes the light dynamics on this lattice. We found that there is always a FB on this system, which corresponds to vertically isolated dipolar states. These FB modes occupy a single site only being, therefore, the most localized FB states of any lattice configuration. By fulfilling a specific relation between lattice parameters, we found that a second non-trivial FB appears, which is composed of an hybridized state with $s$ and $p$ modes excited simultaneously. These FB states occupy only three lattice sites, with a rapidly decaying and perfectly compact profile. By investigating the edges of this lattice, we found that Shockley-like surface states exist on the system for edges having vertically oriented waveguides. We obtained an analytical expression for them and found that they could show different properties depending on the lattice parameters. At the end, we explored the case $\Delta\beta=0$ and found two dispersive and two flat bands for this binary 1D system.
We believe that our simple model could show interesting features for non-diffractive image transmission applications as well as for presenting different transport properties depending on the input condition. For example, by exciting a vertically oriented waveguide with a fundamental state we would simply observe transport, while using a dipolar excitation would produce perfect localization, without the need of applying any extra interaction. This could be useful to excite two completely different states on the system, which could be of interest on optical signal processing.
Acknowledgements {#acknowledgements .unnumbered}
================
Authors acknowledge B. Real for useful comments and discussions in the initial part of this work. This work was supported in part by Programa ICM Millennium Institute for Research in Optics (MIRO) and FONDECYT Grant No.1151444.
[99]{}
F. Lederer, G.I. Stegeman, D.N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. **463**, 1 (2008).
S. Flach, and A.V. Gorbach, Phys. Rep. **467**, 1 (2008).
D.K. Campbell, S. Flach, and Yu.S. Kivshar, Phys. Today [**57**]{} (1), 43 (2004).
P.W. Anderson, Phys. Rev. [**109**]{}, 1492 (1958).
T. Schwartz, G. Bartal, S. Fishman and M. Segev, Nature [**446**]{}, 52 (2007).
Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D.N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. [**100**]{}, 013906 (2008).
D.L. Bergman, C. Wu, and L. Balents, *Phys. Rev. B* [**78**]{}, 125104 (2008).
L. Morales-Inostroza and R.A. Vicencio, Phys. Rev. A [**94**]{}, 043831 (2016).
B. Pal, **98**, 245116 (2018).
D. Leykam, A. Andreanov, and S. Flach, Advances in Physics:X [**3**]{}, 1 (2018).
D. Leykam and S. Flach, APL Photonics [**3**]{}, 070901 (2018).
R.A. Vicencio, C. Cantillano, L. Morales-Inostroza, B. Real, C. Mejía-Cortés, S. Weimann, A. Szameit and M.I. Molina, Phys. Rev. Lett. **114**, 245503 (2015). S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman, P. [Ö]{}hberg, E. Andersson and R.R. Thomson, Phys. Rev. Lett. **114**, 245504 (2015).
Y. Zong, S. Xia, L. Tang, D. Song, Y. Hu, Y. Pei, J. Su, Y. Li, and Z. Chen, Opt. Express [**24**]{}, 8877 (2016).
S. Xia, Y. Hu, D. Song, Y. Zong, L. Tang, and Z. Chen, Opt. Lett. [**41**]{}, 1435 (2016).
S. Weimann, L. Morales-Inostroza, B. Real, C. Cantillano, A. Szameit, and R. A. Vicencio, Opt. Lett. **41**, 2414 (2016).
S. Mukherjee and R.R. Thomson, Opt. Lett. [**40**]{}, 5443 (2015).
B. Real, C. Cantillano, D. López-González, A. Szameit, M. Aono, M. Naruse, S. Kim, K. Wang, and R.A. Vicencio, Sci. Rep. [**7**]{}, 15085 (2017).
C. Cantillano, S. Mukherjee, L. Morales-Inostroza, B. Real, G. Cáceres-Aravena, C. Hermann-Avigliano, R.R. Thomson and R.A. Vicencio, New J. Phys. **20**, 033028 (2018).
Z. Lin, J.-H. Choi, Q. Zhang, W. Qin, S. Yi, P. Wang, L. Li, Y. Wang, H. Zhang, Z. Sun, L. Wei, S. Zhang, T. Guo, Q. Lu, J.-H. Cho, C. Zeng, and Z. Zhang, Phys. Rev. Lett. [**121**]{}, 096401 (2018).
G.-B. Jo, J. Guzman, C.K. Thomas, P. Hosur, A. Vishwanath, and D.M. Stamper-Kurn, Phys. Rev. Lett. [**108**]{}, 045305 (2012).
F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin, A. Lemaître, L. Le Gratiet, I. Sagnes, S. Schmidt, H.E. Türeci, A. Amo, and J. Bloch, Phys. Rev. Lett. [**116**]{}, 066402 (2016).
Y.-X. Xiao, G. Ma, Z.-Q. Zhang, and C.T. Chan, Phys. Rev. Lett. [**118**]{}, 166803 (2017).
D. Guzm‡án-Silva, C. Mejía-Cortés, M.A. Bandres, M.C. Rechtsman, S. Weimann, S. Nolte, M. Segev, A. Szameit, and R. A. Vicencio, New J. Phys. **16**, 063061 (2014).
S. Xia, A. Ramachandran, S. Xia, D. Li, X. Liu, L. Tang, Y. Hu, D. Song, J. Xu, D. Leykam, S. Flach, and Z. Chen, **121**, 263902 (2018).
R.A. Vicencio and C. Mejía-Cortés, J. Opt. **16**, 015706 (2014).
S. Xia, Y. Hu, D. Song, Y. Zong, L. Tang, and Z. Chen, Opt. Lett. **41**, 1435 (2016).
Y. Zong, S. Xia, L. Tang, D. Song, Y. Hu, Y. Pei, J. Su, Y. Li, and Z. Chen, Opt. Express **24**, 8877 (2016).
G. Wirth, M. Ölschläger, and A. Hemmerich, Nat. Phys. **7**, 147 (2011).
S. Klembt, T.H. Harder, O.A. Egorov, K. Winkler, H. Suchomel, J. Beierlein, M. Emmerling, C. Schneider, and S. Höfling, Appl. Phys. Lett. **111**, 231102 (2017).
M. Milićević, T. Ozawa, G. Montambaux, I. Carusotto, E. Galopin, A. Lemaître, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, Phys. Rev. Lett. **118**, 107403 (2017).
C. Wu C, D. Bergman, L. Balents, and S. Das Sarma, Phys. Rev. Lett. **99**, 070401 (2007).
X. Li, E. Zhao, and W.V. Liu, Nat. Commun. **4**, 1523 (2013).
S. Yin, J.E. Baarsma, M.O.J Heikkinen, J.-P. Martikainen, and P. Törmä, Phys. Rev. A **92**, 053616 (2015).
X. Li and W.V. Liu, Rep. Prog. Phys. **79**, 116401 (2016).
C. Cantillano, L. Morales-Inostroza, B. Real, S. Rojas-Rojas, A. Delgado, A. Szameit, and R.A. Vicencio, Sci. Bull. **62**, 339 (2017).
M. Hatanaka, Theor. Chem. Acc. **129**, 151 (2011).
F. Fedele, J. Yang, and Z. Chen, Opt. Lett. [**30**]{}, 1506 (2005).
I. Makasyuk, Z. Chen, and J. Yang, Phys. Rev. Lett. [**96**]{}, 223903 (2006).
H.S. Eisenberg, Y. Silberberg, R. Morandotti, A.R. Boyd, and J.S. Aitchison, Phys. Rev. Lett. [**81**]{}, 3383 (1998).
J.W. Fleischer, T. Carmon, M. Segev, N.K. Efremidis, and D.N. Christodoulides, Phys. Rev. Lett. [**90**]{}, 023902 (2003).
A. Szameit, D. Blömer, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, Opt. Express [**13**]{}, 10552 (2005).
A. Szameit, D. Blömer, J. Burghoff, Appl Phys B [**82**]{}, 507 (2006).
A.W. Snyder and W.R. Young, J. Opt. Soc. Am. [**68**]{}, 297 (1978).
D. Gloge, Appl. Opt. [**10**]{}, 2252 (1971).
T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, , 4752 (1999).
M.I. Molina, R.A. Vicencio, Yu.S. Kivshar, Opt. Lett. [**31**]{}, 1693 (2006).
C.R. Rosberg, D.N. Neshev, W. Krolikowski, A. Mitchell, R.A. Vicencio, M.I. Molina, and Yu.S. Kivshar, Phys. Rev. Lett. [**97**]{}, 083901 (2006).
E. Smirnov, M. Stepić, C.E. Rüter, D. Kip, and V. Shandarov, Opt. Lett. [**31**]{}, 2338 (2006).
I. E. Tamm, Phys. Z. Sowjetunion **1**, 733 (1932).
W. Shockley, Phys. Rev. **56**, 317 (1939).
N. Malkova, I. Hromada, X. Wang, G. Bryant, and Z. Chen, Opt. Lett. **34**, 1633 (2009).
N. Malkova, I. Hromada, X. Wang, G. Bryant, and Z. Chen, Phys. Rev. A **80**, 043806 (2009).
J. He, Y.-X. Zhu, Y.-J. Wu, L.-F. Liu, Y. Liang, and S.-P. Kou, Phys. Rev. B **87**, 075126 (2013).
|
---
abstract: 'Recently, Boutonnet, Chifan, and Ioana proved that McDuff’s examples of continuum many pairwise non-isomorphic separable II$_1$ factors are in fact pairwise non-elementarily equivalent. Their proof proceeded by showing that any ultrapowers of any two distinct McDuff examples are not isomorphic. In a paper by the first two authors of this paper, Ehrenfeucht-Fraïsse games were used to find an upper bound on the quantifier complexity of sentences distinguishing the McDuff examples, leaving it as an open question to find concrete sentences distinguishing the McDuff factors. In this paper, we answer this question by providing such concrete sentences.'
address:
- 'Department of Mathematics, University of California, Irvine, 340 Rowland Hall (Bldg.\# 400), Irvine, CA, 92697-3875.'
- 'Department of Mathematics and Statistics, McMaster University, 1280 Main St., Hamilton ON, Canada L8S 4K1'
- 'Department of Mathematics, University of Pennsylvania, 209 South 33rd Street Philadelphia, PA 19104-6395.'
author:
- 'Isaac Goldbring, Bradd Hart, and Henry Towsner'
title: 'Explicit sentences distinguishing McDuff’s II$_1$ factors'
---
Introduction
============
The first examples of continuum many nonisomorphic separable II$_1$ factors were given by McDuff in [@MD2]. These same examples were shown to be non-elementarily equivalent (in the sense of continuous logic) by Boutonnet, Chifan, and Ioana in [@BCI]. The way they proved that the McDuff factors were not elementarily equivalent was by showing, for any two distinct McDuff examples $\m$ and $\mathcal{N}$ and any two ultrafilters $\u$ and $\mathcal{V}$ on ${\mathbb{N}}$, that the ultrapowers $\m^\u$ and $\mathcal{N}^{\mathcal{V}}$ were not isomorphic; by standard model-theoretic results, it then follows that $\m$ and $\mathcal{N}$ are not elementarily equivalent.
In [@braddisaac], the techniques in [@BCI] were dissected in order to give some information about the sentences distinguishing the McDuff examples. Indeed, if we enumerate the McDuff examples by $\m_{\balpha}$ for $\balpha\in 2^\omega$ and $k\in \omega$ is the least digit such that $\balpha(k)\not=\bbeta(k)$, then it was shown that there must be a sentence $\theta$ with at most $5k+3$ alternations of quantifiers such that $\theta^{\m_{\balpha}}\not=\theta^{\m_{\bbeta}}$. The proof there used Ehrenfeucht-Fraïsse games. The game-theoretic techniques also hinted at a possible strategy of providing concrete sentences distinguishing the McDuff examples *if* concrete sentences distinguishing examples that differed at the first digit could be obtained. In [@braddisaac Section 4.1], such sentences were obtained, but they lacked the uniformity needed to carry out the strategy outlined there.
In this paper, an even finer analysis of the work in [@BCI] is carried out in order to obtain concrete sentences that distinguish McDuff examples that differ at the first digit; this analysis appears in Section 3. In Section 4, the details of the plan outlined in [@braddisaac Section 4.2] are given and the inductive construction of sentences distinguishing all of the McDuff examples is elucidated. We note that the concrete sentences given here that distinguish examples at “level” $k$ also have $5k+3$ alternations of quantifiers, agreeing with the game-theoretic bounds predicted in [@braddisaac].
We list here some conventions used throughout the paper. First, we follow set theoretic notation and view $k\in \omega$ as the set of natural numbers less than $k$: $k=\{0,1\ldots,k-1\}$. In particular, $2^k$ denotes the set of functions $\{0,1,\ldots,k-1\}{\rightarrow}\{0,1\}$. If $\balpha\in 2^k$, then we set $\alpha_i:=\balpha(i)$ for $i=0,1,\ldots,k-1$ and we let $\balpha^\#\in 2^{k-1}$ be such that $\balpha$ is the concatenation of $(\alpha_0)$ and $\balpha^\#$. If $\balpha\in 2^\omega$, then $\balpha|k$ denotes the restriction of $\balpha$ to $\{0,1,\ldots,k-1\}$.
Whenever we write a tuple $\vec x$, it will be understood that the length of the tuple is countable (that is, finite or countably infinite).
We will use uppercase letters to denote variables in formulae while their lowercase counterparts will be elements from algebras. We will use $U$’s and $V$’s (sometimes with subscripts) for variables ranging over the set of unitaries; since unitaries are quantifier-free definable relative to the theory of [$\mathrm{C}^*$]{}-algebras, this convention is harmless. Of course, we will then use $u$’s and $v$’s for unitaries from specific algebras.
Given a group $\Gamma$ and $a\in \Gamma$, we let $u_a\in L(\Gamma)$ be the canonical unitary associated to $a$.
Fix a von Neumann algebra $\m$. For $x,y\in \m$, the commutator of $x$ and $y$ is the element $[x,y]:=xy-yx$. If $A$ is a subalgebra of $\m$, then the relative commutant of $A$ in $\m$ is the set $$A'\cap \m:=\{x\in \m \ | \ [x,a]=0 \text{ for all } a\in A\}.$$In particular, the center of $\m$ is $Z(\m):=\m'\cap \m$. For a tuple $\vec a$ from $\m$, we write $C(\vec a)$ to denote $A'\cap \m$, where $A$ is the subalgebra of $\m$ generated by the coordinates of $\vec a$. Technically, this notation should also mention $\m$, but the ambient algebra will always be clear from context, whence we omit any mention of it in the notation.
Preliminaries
=============
In this section, we gather most of the background material needed in the rest of the paper. First, we recall McDuff’s examples. Let $\Gamma$ be a countable group. For $i\geq 1$, let $\Gamma_i$ denote an isomorphic copy of $\Gamma$ and let $\Lambda_i$ denote an isomorphic copy of $\mathbb{Z}$. Let $\tilde{\Gamma}:=\bigoplus_{i\geq 1}\Gamma_i$. If $S_\infty$ denotes the group of permutations of ${\mathfrak{N}}$ with finite support, then there is a natural action of $S_\infty$ on $\bigoplus_{i\geq 1} \Gamma$ (given by permutation of indices), whence we may consider the semidirect product $\tilde{\Gamma}\rtimes S_\infty$. Given these conventions, we can now define two new groups:
$$T_0(\Gamma):=\langle \tilde{\Gamma}, (\Lambda_i)_{i\geq 1} \ | \ [\Gamma_i,\Lambda_j]=0 \text{ for }i\geq j\rangle$$ and
$$T_1(\Gamma):=\langle \tilde{\Gamma}\rtimes S_\infty, (\Lambda_i)_{i\geq 1} \ | \ [\Gamma_i,\Lambda_j]=0 \text{ for }i\geq j\rangle.$$
Note that if $\Delta$ is a subgroup of $\Gamma$ and $\alpha\in \{0,1\}$, then $T_\alpha(\Delta)$ is a subgroup of $T_\alpha(\Gamma)$. Given a sequence $\balpha\in 2^{\leq \omega}$, we define a group $K_{\balpha}(\Gamma)$ as follows:
1. $K_{\balpha}(\Gamma):=\Gamma$ if $\balpha=\emptyset$;
2. $K_{\balpha}(\Gamma):=(T_{\alpha_0}\circ T_{\alpha_1}\circ \cdots T_{\alpha_{n-1}})(\Gamma)$ if $\balpha\in 2^n$;
3. $K_{\balpha}$ is the inductive limit of $(K_{\balpha|n})_n$ if $\balpha\in 2^\omega$.
We then set $\m_{\balpha}(\Gamma):=L(T_{\balpha}(\Gamma))$. When $\Gamma=\mathbb{F}_2$, we simply write $\m_{\balpha}$ instead of $\m_{\balpha}(\mathbb{F}_2)$; these are the McDuff examples referred to the introduction.
Given $n\geq 1$, we let $\tilde{\Gamma}_{\balpha,n}$ denote the subgroup of $T_{\alpha_0}(K_{\balpha^\#}(\Gamma))$ given by the direct sum of the copies of $K_{\balpha^\#}(\Gamma)$ indexed by those $i\geq n$ and we let $P_{\balpha,n}:=L(\tilde{\Gamma}_{\balpha,n})$. When $\balpha$ has length $1$, we simply refer to $\tilde{\Gamma}_{\emptyset,n}$ as $\tilde{\Gamma}_n$ and $P_{\emptyset,n}$ as $P_n$; if, in addition, $n=1$, then we simply refer to $\tilde{\Gamma}_1$ as $\tilde{\Gamma}$. As introduced in [@braddisaac], we define a *generalized McDuff ultraproduct corresponding to $\balpha$ and $\Gamma$* to be an ultraproduct of the form $\prod_\u \m_{\balpha}(\Gamma)^{\otimes t_s}$, where $(t_s)$ is a sequence of natural numbers and $\u$ is a nonprincipal ultrafilter on ${\mathbb{N}}$. The following definition, implicit in [@BCI] and made explicitly in [@braddisaac], is central to our work in this paper.
We say that a pair of unitaries $u,v $ in a II$_1$ factor $\m$ are [*good unitaries*]{} if, for all $\zeta \in \m$, $$\inf_{\eta\in C(u,v)}\|\zeta -\eta\|_2^2 \leq 100 (\|[\zeta,u]\|_2^2 + \|[\zeta,v]\|_2^2).$$
In the terminology of [@BCI], this says that $C(u,v)$ is a (2,100)-residual subalgebra of $\m$.
We will need the following key facts, whose proofs are outlined in [@braddisaac Facts 2.6].
\[key.fact\] Suppose that $\balpha\in 2^{<\omega}$ is nonempty, $\Gamma$ is a countable group, and $(t_s)$ is a sequence of natural numbers.
1. Suppose that $(m_s)$ and $(n_s)$ are two sequences of natural numbers such that $n_s<m_s$ for all $s$. Further suppose that $\Gamma$ is an ICC group. Then $(\prod_\u P_{\balpha,m_s}^{\otimes t_s})'\cap (\prod_\u P_{\balpha,n_s}^{\otimes t_s})$ is a generalized McDuff ultraproduct corresponding to $\balpha^\#$ and $\Gamma$.
2. For any sequence $(n_s)$, there is a pair of good unitaries $\vec u$ from $\prod_\u \m_{\balpha}(\Gamma)^{\otimes t_s}$ such that $\prod_\u P_{\balpha,n_s}^{\otimes t_s}=C(\vec u)$.
3. Given any separable subalgebra $A$ of $\prod_\u \m_{\balpha}(\Gamma)^{\otimes t_s}$, there is a sequence $(n_s)$ such that $\prod_\u P_{\balpha,n_s}^{\otimes t_s}\subset A'\cap \prod_\u \m_{\balpha}(\Gamma)^{\otimes t_s}$.
Distinguishing examples at level one
====================================
In this section, we will find sentences that distinguish $L(T_0(\Gamma))$ and $L(T_1(\Gamma))$ for nonamenable groups $\Gamma$. For the purposes of the next section, where the main theorem of the paper is proved, we will actually need to prove a bit more.
In the rest of this paper, we set $\chi(X,U_1,U_2):=100(\|[X,U_1]\|^2_2+\|[X,U_2]\|^2_2)$.
\[keylemma\] Let $\Gamma$ be a countable group and $\alpha\in \{0,1\}$. For any $t,n\in {\mathbb{N}}$ with $t\geq 1$, there are $a,b\in \bigoplus_t T_\alpha(\Gamma)$ such that, for any $\zeta\in L(\bigoplus_t T_\alpha(\Gamma))$, we have $$\|\zeta-\mathbb{E}_{L(\bigoplus_t \widetilde{\Gamma_n})}(\zeta)\|_2^2\leq \chi(\zeta,u_a,u_b)^{L(\bigoplus_t T_\alpha(\Gamma))}.$$
This follows from [@BCI Lemmas 2.6-2.10].
We set $\psi_m(V_a,V_b)$ to be the formula $$\sup_{\vec X,\vec Y}((\inf_U \max_{1\leq i,j\leq m}\|[UX_iU^*,Y_j]\|_2){\mathbin{\mathpalette\dotminussym{}}}2\max_{1\leq i\leq m}\sqrt{\chi(X_i,V_a,V_b)})$$ and set $\tau_m:=\inf_{V_a,V_b}\psi_m$.
\[true\] Suppose that $\Gamma$ is a countable group and that $t\geq 1$. Then for any $m\geq 1$, we have $$\tau_m^{L(\bigoplus_t T_1(\Gamma))}=0.$$
Apply Lemma \[keylemma\] with $n=1$, obtaining $a,b\in \bigoplus_t T_1(\Gamma)$. Let $V_a:=u_a$ and $V_b:=u_b$. Fix $m$-tuples $\vec x,\vec y\in L(\bigoplus_t T_1(\Gamma))$ and $\epsilon>0$. For each $i=1,\ldots,m$, we have that $$\|x_i-\mathbb{E}_{L(\bigoplus_t \widetilde{\Gamma})}(x_i)\|_2^2\leq \chi(x_i,u_a,u_b)^{L(\bigoplus_t T_1(\Gamma))}.$$ In particular, there is $k>0$ such that $$\|x_i-\mathbb{E}_{L(\bigoplus_t \bigoplus_{j\leq k}{\Gamma_j})}(x_i)\|_2\leq \sqrt{\chi(x_i,u_a,u_b)^{L(\bigoplus_t T_1(\Gamma))}}+\epsilon.$$ Set $x_i^+:=\mathbb{E}_{L(\bigoplus_t \bigoplus_{j\leq k}{\Gamma_j})}(x_i)$ and $x_i^-:=x_i-x_i^+$. Let $H_p$ be the subgroup of $T_1(\Gamma)$ generated by $\bigoplus_{j\leq p}\Gamma_i \rtimes S_p$ and $\Lambda_1,\ldots,\Lambda_p$. For $p>0$ sufficiently large, setting $y_i^+:=\mathbb{E}_{L(\bigoplus_t H_p)}(y_i)$ and $y_i^-:=y_i-y_i^+$, we have $\|y_i^-\|_2\leq \epsilon$.
Choose $\sigma\in S_\infty$ with $\sigma(j)>p$ for all $j\leq m$. Let $\sigma_1:=(\sigma,\sigma,\ldots,\sigma)\in \bigoplus_t L(T_1(\Gamma))$. Note that $\sigma_1(\bigoplus_t \bigoplus_{j\leq m}\Gamma_j)\sigma_1^{-1}$ commutes with $L(\bigoplus_t H_p)$. Let $u\in U(L(\bigoplus_t T_1(\Gamma)))$ be the unitary corresponding to $\sigma_1$. It follows, for $1\leq i,j\leq m$, that $[ux_i^+u^*,y_j+]=0$, so $$\|ux_iu^*,y_j\|_2\leq \|[ux_i^+u^*,y_j^-]\|_2+\|[ux_i^-u^*,y_j^+]\|_2+\|[ux_i^-u^*,y_j^-]\|_2.$$
Now $$\|[ux_i^+u^*,y_j^-]\|_2\leq \|ux_i^+u^*y_j^-\|_2+\|y_j^-ux_i^+u^*\|_2\leq 2\|y_j^-\|_2\leq 2\epsilon.$$ Here we use that conditional expectation is a contractive map, so $\|x_i^+\|\leq \|x_i\|\leq 1$. Since $\|x_i^-\|\leq 2$, one shows that $\|[ux_i^-u^*,y_j^-]\|_2\leq 4\epsilon$ in a similar fashion. Finally, we have $$\|[ux_i^-u^*,y_j^+]\|_2\leq 2\|x_i^-\|_2\leq 2(\sqrt{\chi(x_i,u_a,u_b)^{L(\bigoplus_t T_1(\Gamma))}}+\epsilon).$$ Letting $\epsilon$ go to $0$, we get the desired result.
The following is probably obvious and/or well-known, but in any event:
There is a function $\upsilon:\r^*{\rightarrow}\r^*$ such that, for every $\epsilon>0$ and an inclusion $N\subseteq M$ of II$_1$ factors, if $x\in N$ is such that $d(x,U(M))<\upsilon(\epsilon)$, then $d(x,U(N))<\epsilon$.
Let $\psi(x):=\max(d(x^*x,1),d(xx^*,1))$. Then $\psi$ is weakly stable, so there is $\eta>0$ such that if $N$ is any II$_1$ factor and $\psi(x)^N<\eta$, then $d(x,U(N))<\epsilon$. Let $\upsilon(\epsilon):=\Delta_\psi(\eta)$, where $\Delta_\psi$ is the modulus of uniform continuity for the formula $\psi$. Now suppose that $N\subseteq M$ are II$_1$ factors and $x\in N$ is such that $d(x,U(M))<\upsilon(\epsilon)$. Then $\psi(x)^N=\psi(x)^M<\eta$, whence $d(x,U(N))<\epsilon$.
The following result, which is Lemma 4.6 in [@BCI], will be very important to us. In what follows, $\pi_n:\Gamma{\rightarrow}\tilde{\Gamma}$ is the canonical embedding with $\pi_n(\Gamma)=\Gamma_n$.
\[bci4.6\] Suppose that $\Gamma$ is a countable non-amenable group and $Q$ is a tracial von Neumann algebra. Then there are $g_1,\ldots,g_m\in \Gamma$ and a constant $C>0$ such that, for any $n\geq 1$, unitaries $v_1,\ldots,v_m\in U(L(\tilde{\Gamma}_{n+1}\otimes Q))$, and $\zeta \in L(T_0(\Gamma))\otimes Q$, we have that $$\|\zeta\|_2\leq C\sum_{k=1}^m \|u_{\pi_n}(g_k)\zeta-\zeta v_k\|_2.$$
Note that in the version of [@BCI] currently available, the lemma only allows for unitaries in $L(\tilde{\Gamma}_{n+1})$ rather than $L(\tilde{\Gamma}_{n+1}\otimes Q)$. However, the proof readily adapts to this more general situation and, indeed, the lemma is used in this more general form in the proof of [@BCI Lemma 4.4].
For a nonamenable group $\Gamma$, let $C(\Gamma)$ and $m(\Gamma)$ be as in Fact \[bci4.6\].
\[false\] Suppose that $\Gamma$ is a nonamenable group. Let $m=m(\Gamma)$, $C=C(\Gamma)$, and $\delta:=\sqrt{\frac{1}{200(30)^2}\upsilon(\frac{1}{2Cm})}$. Then whenever $M$ is an intermediate subalgebra $L(T_0(\Gamma))\subseteq M\subseteq L(T_0(\Gamma))\otimes Q$, it follows that $\tau_m^M\geq \delta.$
Suppose, towards a contradiction, that $v_a,v_b\in U(M)$ are such that $\psi_m(v_a,v_b)^M<\delta$. For each $n$, let $\rho_n:\Gamma{\rightarrow}U(P_n)$ be given by $\rho_n(g):=u_{\pi_n(g)}$. Since $\bigcup_n (P_n'\otimes Q)$ is dense in $L(T_0(\Gamma))\otimes Q$, there is $n$ sufficiently large so that $$\max(\|[\rho_n(g),v_a]\|_2,\|[\rho_n(g),v_b]\|_2)<\delta$$ for all $g\in \Gamma$. Fix such an $n$ and set $\rho:=\rho_n$. It follows that $\chi(\rho(g),v_a,v_b)^M\leq 200\delta^2$ for all $g\in \Gamma$.
By Lemma \[keylemma\], we may find $a',b'\in T_0(\Gamma)$ such that, for all $\zeta\in L(T_0(\Gamma))$, we have $$\|\zeta-\mathbb{E}_{L(\widetilde{\Gamma_{n+1}})}(\zeta)\|_2^2\leq \chi(\zeta,u_{a'},u_{b'})^{L(T_0)(\Gamma)}.$$ For simplicity, write $\mathbb{E}$ instead of $\mathbb{E}_{L(\widetilde{\Gamma_{n+1}})\otimes Q}$. It then follows that, for all $\zeta \in L(T_0(\Gamma))\otimes Q$, we have $$\|\zeta-\mathbb{E}(\zeta)\|_2^2\leq \chi(\zeta,u_{a'},u_{b'})^{L(T_0(\Gamma))\otimes Q}.$$ Let $g_1,\ldots,g_m\in \Gamma$ be as in Fact \[bci4.6\]. Since $\psi_m(v_a,v_b)^M<\delta$, we may find $u\in U(M)$ such that, for all $1\leq k\leq m$, we have $$\max(\|[u\rho(g_k)u^*,u_{a'}]\|_2,\|[u\rho(g_k)u^*,u_{b'}]\|_2)<20\sqrt{2}\delta+\delta\leq 30\delta.$$ Let $v_k:=u\rho(g_k)u^*\in U(L(T_0(\Gamma))\otimes Q)$ and let $v_k':=\mathbb{E}(v_k)$. It follows that $\|v_k-v_k'\|_2^2\leq \chi(v_k,u_{a'},u_{b'})^{L(T_0(\Gamma))\otimes Q}\leq 200(30\delta)^2.$ By the choice of $\delta$, there is $v_k''\in U(L(\widetilde{\Gamma_{n+1}})\otimes Q)$ such that $\|v_k'-v_k''\|_2<\frac{1}{2Cm}$. By Fact \[bci4.6\], we have that $$\|u\|_2\leq C\sum_k \|\rho(g_k)-uv_k''\|_2\leq C\sum_k \|uv_k-uv_k''\|_2<\frac{1}{2},$$ yielding the desired contradiction.
The inductive construction
==========================
In this section, we describe an inductive construction of sentences that allows us to carry out the argument hinted at in [@braddisaac Section 4.2]. By [@braddisaac Section 4.2], we know that centralizers of good unitaries and relative commutants between centralizers of good unitaries are definable sets, whence we can quantify over them. We actually need to know that we can do this in a uniform manner that does not depend on the ambient II$_1$ factor nor the good unitaries at hand. Such uniformity is the content of the next lemma. Note that if $\m$ is a II$_1$ factor, $u_1,u_2\in \m$ are good unitaries and $x\in \m$, then:
- $d(x,C(u_1,u_2))\leq \sqrt{\chi(x,u_1,u_2)^{\m}}$
- if $x\in C(u_1,u_2)$, then $\chi(x,u_1,u_2)^{\m}=0$.
\[quant\]
1. For every formula $\psi(X,\vec Y,\vec U)$, there are formulae $\hat{\psi}_s(\vec Y,\vec U)$ and $\hat{\psi}_i(\vec Y,\vec U)$such that, for any II$_1$ factor $\m$, any pair of good unitaries $\vec u\in \m$, and any tuple $\vec y\in \m$, we have $$\hat{\psi}_s(\vec y,\vec u)^{\m}=\sup\{\psi(x,\vec y,\vec u)^{\m} \ : \ x\in C(\vec u)\}$$ and $$\hat{\psi}_i(\vec y,\vec u)^{\m}=\inf\{\psi(x,\vec y,\vec u)^{\m} \ : \ x\in C(\vec u)\}.$$
2. For every formula $\rho(X,\vec Y, \vec U_1,\vec U_2)$, there are formulae $\overline{\rho}_s(\vec Y,\vec U_1,\vec U_2)$ and $\overline{\rho}_i(\vec Y,\vec U_1,\vec U_2)$ such that, for any II$_1$ factor $\m$ and any two pairs of good unitaries $\vec u_1,\vec u_2\in \m$ with $C(\vec u_2)\subseteq C(\vec u_1)$ and any tuple $\vec y\in \m$, we have $$\overline{\rho}_s(\vec y,\vec u_1,\vec u_2)^{\m}=\sup\{\rho(x,\vec y,\vec u_1,\vec u_2)^{\m} \ : \ x\in C(\vec u_2)'\cap C(\vec u_1))\}$$ and $$\overline{\rho}_i(\vec y,\vec u_1,\vec u_2)^{\m}=\inf\{\rho(x,\vec y,\vec u_1,\vec u_2)^{\m} \ : \ x\in C(\vec u_2)'\cap C(\vec u_1))\}.$$
We only prove the infimum statements. We first prove (1). Let $\alpha$ be a continuous, nondecreasing function such that $\alpha(0)=0$ and $$|\psi(x,\vec y,\vec u)-\psi(x',\vec y, \vec u)|\leq \alpha(d(x,x'))$$ for all $x,x',\vec y,\vec u$. We claim that $$\hat{\psi}_i(\vec Y,U_1,U_2):=\inf_X(\psi(X,\vec Y,U_1,U_2)+\alpha(\sqrt{\chi(X,U_1,U_2)}))$$ works. Fix a II$_1$ factor $\m$, a pair of good unitaries $u_1,u_2\in \m$, and a tuple $\vec y\in \m$. It is clear that $$\hat{\psi}_i(\vec y,u_1,u_2)^{\m}\leq \inf\{\psi(x,\vec y,u_1,u_2)^{\m}\ : \ x\in C(u_1,u_2)\}.$$ To see the other direction, fix $x,x'\in \m$ and note that $$\psi(x,\vec y,u_1,u_2)^{\m}\leq \psi(x',\vec y,u_1,u_2)^{\m}+\alpha(d(x,x')),$$ whence, taking the infimum over $x\in C(u_1,u_2)$, we have $$\inf\{\psi(x,\vec y,u_1,u_2)^{\m}\ : \ x\in C(u_1,u_2)\}\leq \psi(x',\vec y,u_1,u_2)^{\m}+\alpha(\sqrt{\chi(x',u_1,u_2)}^{\m}),$$ whence the desired result follows from taking the infimum over $x'$.
The proof of part (2) proceeds in the same way once we find a formula $\zeta(X,\vec U_1,\vec U_2)$ such that, for any II$_1$ factor $\m$, any two pairs of good unitaries $\vec u_1,\vec u_2\in \m$ such that $C(u_2)\subseteq C(u_1)$, and any $x\in\m$, we have that $d(x,C(u_2)'\cap C(u_1))\leq \zeta(x,\vec u_1,\vec u_2)^{\m}$. Let $$\mathbb{E}:\m{\rightarrow}C(\vec u_2)'\cap C(\vec u_1), \ \mathbb{E}_1:M{\rightarrow}C(\vec u_1), \text{and }\mathbb{E}_2:C(\vec u_1){\rightarrow}C(\vec u_2)'\cap C(\vec u_1)$$ be the usual conditional expecations, so $\mathbb{E}=\mathbb{E}_2\circ \mathbb{E}_1$ and $d(x,C(\vec u_2)'\cap C(\vec u_1))=\|x-\mathbb{E}(x)\|_2$. Note that $$\|x-\mathbb{E}(x)\|_2\leq \|x-\mathbb{E}_1(x)\|_2+\|\mathbb{E}_1(x)-\mathbb{E}_2(\mathbb{E}_1(x))\|_2.$$ Now $\|x-\mathbb{E}_1(x)\|_2\leq \sqrt{\chi(x,u_{11},u_{12})^{\m}}$. As proved in [@braddisaac Section 4.2], $$\|\mathbb{E}_1(x)-\mathbb{E}_2(\mathbb{E}_1(x))\|_2\leq \sqrt{\sup_{y\in C(\vec u_2)}\|[y,\mathbb{E}_1(x)]\|_2}.$$ Now notice that $$\|[y,\mathbb{E}_1(x)]\|_2\leq \|\mathbb{E}_1(x)y-xy\|_2+\|xy-yx\|_2+\|yx-y\mathbb{E}_1(x)\|_2.$$ Let $\psi(X,Y,\vec U_2):=2\chi(X,\vec U_2)+\|XY-YX\|_2$. It follows that $$\sup_{y\in C(\vec u_2)}\|[y,\mathbb{E}_1(x)]\|_2\leq \hat{\psi}_s(x,\vec u_2)^{\m}.$$ Letting $$\zeta(X,\vec U_1,\vec U_2):=\sqrt{\chi(X,\vec U_1)}+\sqrt{\hat{\psi}_s(X,\vec U_2)}$$ yields the desired formula.
Repeatedly applying the Quantification Lemma yields:
For any sentence $\theta$ in prenex normal form, there is a formula $\tilde{\theta}(\vec U_1,\vec U_2)$ such that, for any II$_1$ factor $\m$ and any two pairs of good unitaries $\vec u_1,\vec u_2\in \m$ with $C(\vec u_2)\subseteq C(\vec u_1)$, we have $$\tilde{\theta}(\vec u_1,\vec u_2)^{\m}=\theta^{C(\vec u_2)'\cap C(\vec u_1)}.$$ Moreover, $\tilde{\theta}$ is also in prenex normal form and has the same number of alternations of quantifiers as $\theta$.
We now introduce the formulae
$$\varphi_{\good}(U_1,U_2):=\sup_X\inf_Y \max(\max_{i=1,2}\|[Y,U_i]\|_2,d(X,Y){\mathbin{\mathpalette\dotminussym{}}}\sqrt{\chi(X,U_1,U_2))})$$ and
$$\varphi^n_{\leq}(\vec Y;\vec U):=\sup_{X\in C(\vec U)}\max_{i=1,\ldots,n} \|[X,Y_i]\|_2.$$
In the definition of $\varphi_{\leq}$, we are abusing notation and really mean the formula one obtains from Lemma \[quant\]. In what follows, we will only need to consider $\varphi^3_{\leq}$ and denote this formula simply by $\varphi_{\leq}$.
Note that:
- If $\m$ is an $\aleph_1$-saturated II$_1$ factor, then $\varphi_{\good}(u_1,u_2)^{\m}=0$ if and only if $u_1,u_2$ is a pair of good unitaries.
- If $\m$ is any II$_1$ factor, $\vec u\in \m$ is a pair of good unitaries, and $\vec y\in \m^n$ is arbitrary, then $\varphi_{\leq}(\vec y,\vec u)^{\m}=0$ if and only if $\vec y\leq \vec u$.
Given a sentence $\theta$, we recursively define a sequence of sentences $\theta_n$ as follows: Set $\theta_1:=\theta$. Supposing that $\theta_n$ has been defined, we set $\theta_{n+1}$ to be the sentence $$\inf_{\vec U_1}\max(\varphi_{\good}(\vec U_1),\sup_{A}\inf_{\vec U_2}\max(\varphi_{\good}(\vec U_2),\varphi_{\leq}(A,\vec U_1;\vec U_2),\tilde{\theta_n}(\vec U_1,\vec U_2))).$$
When $\theta=\tau_m$, we write $\theta_{m,n}$ for $(\tau_m)_n$. Here is the main result of this paper:
For each nonamenable group $\Gamma$, there is a sequence $(r_n(\Gamma))$ of positive real numbers such that, for any $n,t\in {\mathbb{N}}$ with $t\geq 1$ and any $\balpha\in 2^n$, we have: $$\begin{array}{lr}
\theta_{m,n}^{L(T_\alpha(\Gamma))^{\otimes t}}=0 \text{ for all }m\geq 1 & \text{ if }\balpha(n-1)=1;\\
\theta_{m(\Gamma),n}^{L(T_\alpha(\Gamma))^{\otimes t}}\geq r_n(\Gamma) &\text{ if }\balpha(n-1)=0.
\end{array}$$
We prove the theorem by induction on $n$. When $n=1$, the theorem holds by Propositions \[true\] and \[false\].
Inductively suppose that the theorem is true for $n$. Fix a non-amenable group $\Gamma$. First suppose that $\balpha\in 2^{n+1}$ is such that $\balpha(n)=1$. Fix also $m,t\geq 1$. Let $\m$ be the ultrapower of $L(T_{\balpha}(\Gamma))^{\otimes t}$; by Łos’ theorem, it suffices to show that $\theta_{m,n+1}^{\m}=0$. Fix a pair of good unitaries $\vec u_1>1$. Given $a\in \m$, we can find a pair of good unitaries $\vec u_2\in \m$ such that $\vec u_2>\{a,\vec u_1\}$. We then have that $C(\vec u_2)'\cap C(\vec u_1)$ is a generalized McDuff ultraproduct corresponding to $\balpha^{\#}$ and $\Gamma$, whence, by the inductive hypothesis, we have that $\tilde{\theta}_{m,n}(\vec u_1,\vec u_2)^{\m}=\theta^{C(\vec u_2)'\cap C(\vec u_1)}=0$. It follows that $\theta_{m,n+1}^{\m}=0$.
Now suppose, towards a contradiction, that there is no constant $r_{n+1}(\Gamma)$. Then for each $l>1$, there is $\balpha_l\in 2^{n+1}$ and $t_l\in {\mathbb{N}}$ with $t_l\geq 1$ such that $\theta_{m(\Gamma),n+1}^{L(T_{\balpha_l}(\Gamma))^{\otimes t_l}}<\frac{1}{l}$. Without loss of generality, each $\balpha_l=\balpha$ for some fixed $\balpha\in 2^{n+1}$. Let $\m:=\prod_\u L(T_{\balpha}(\Gamma))^{\otimes t_l}$, a generalized McDuff ultraproduct corresponding to $\balpha$ and $\Gamma$. We then have that $\theta_{m(\Gamma),n+1}^{\m}=0$. Let $\vec u_1$ be a pair of good unitaries witnessing the infimum. Take any $a>\vec{u_1}$ and then take a pair of good unitaries $\vec u_2>a$ witnessing the infimum for that $a$. We then have that $C(\vec u_2)'\cap C(\vec u_1)$ is a McDuff ultraproduct corresponding to $\balpha^{\#}$ and $\Gamma$, whence $\tilde{\theta}_{m(\Gamma),n}(\vec u_1,\vec u_2)^{\m}=\theta_n^{C(\vec u_2)'\cap C(\vec u_1)}\geq r_n(\Gamma)$, contradicting the fact that $\tilde{\theta}_{m(\Gamma),n}(\vec u_1,\vec u_2)^{\m}=0$.
Note that each $\tau_m$ is equivalent to a formula in prenex normal form that begins with an $\inf$ and has three alternations of quantifiers. By the construction, it is easy to check, by induction on $n$, that each $\theta_{m,n}$ is equivalent to a formula in prenex normal form that begins with an $\inf$ and has $5n+3$ alternations of quantifiers. This agrees with the theoretical bounds given in [@braddisaac].
Suppose that $\Gamma$ is any countable group and $\balpha,\bbeta\in 2^\omega$ are such that $\balpha|n-1=\bbeta|n-1$, $\balpha(n)=1$, and $\bbeta(n)=0$. Write $\bbeta=(\bbeta|n+1) {\widehat{\phantom{\eta}}}\bbeta^*$. Set $m:=m(T_{\bbeta^*}(\Gamma))$ and $r:=r_{n+1}(T_{{\bbeta}^*}(\Gamma))$. Then $\theta_{m,n+1}^{\m_{\balpha}(\Gamma)}=0$ and $\theta_{m,n+1}^{\m_{\bbeta}(\Gamma)}\geq r$.
As pointed out in [@BCI], the results there also show, for any countable group $\Gamma$ and any distinct $\balpha,\bbeta\in 2^{\omega}$, that $\mathrm{C}^*_r(T_{\balpha}(\Gamma))$ and $\mathrm{C}^*_r(T_{\bbeta}(\Gamma))$ are not elementarily equivalent. Our results here do indeed yield concrete sentences distinguishing these algebras. As mentioned in [@BCI], the groups $T_{\balpha}(\Gamma)$ are increasing unions of *Powers groups*, whence, by the proof of [@munster Proposition 7.2.3], the unique trace on $\mathrm{C}^*_r(\Gamma)$ is *definable*, and uniformly so over all $\balpha\in 2^\omega$. Consequently, the $\theta_{m,n}$’s can be construed as formulae in the language of [$\mathrm{C}^*$]{}-algebras with imaginary sorts added and, since the completion of $\mathrm{C}^*_r(T_{\balpha}(\Gamma))$ with respect to the GNS representation induced by the unique trace is $\mathcal{M}_{\balpha}(\Gamma)$, we have that the $\theta_{m,n}$’s distinguish the $\mathrm{C}^*_r(T_{\balpha}(\Gamma))$’s as well.
[@BCI] also show that $\mathrm{C}^*_r(T_{\balpha}(\Gamma))\otimes \mathcal{Z}$ and $\mathrm{C}^*_r(T_{\bbeta}(\Gamma))\otimes \mathcal{Z}$ are also not elementarily equivalent for distinct $\balpha,\bbeta\in 2^\omega$, where $\mathcal{Z}$ is the Jiang-Su algebra. Since the unique trace in a monotracial exact $\mathcal{Z}$-stable algebra is definable (and uniformly so) by [@munster Section 3.5] and the closure of $\mathrm{C}^*_r(T_{\balpha}(\Gamma))\otimes \mathcal{Z}$ in its GNS representation with respect to the unique trace is also $\mathcal{M}(T_{\balpha}(\Gamma))$, we also have concrete sentences distinguishing the $\mathrm{C}^*_r(T_{\balpha}(\Gamma))\otimes \mathcal{Z}$’s *in the case that $\Gamma$ is exact* (e.g. when $\Gamma=\mathbb{F}_2)$. It would be interesting to know if the unique trace on $\mathrm{C}^*_r(T_{\balpha}(\Gamma))\otimes \mathcal{Z}$ is definable in general, that is, for an arbitrary countable group $\Gamma$.
[1]{} R. Boutonnet, I. Chifan, and A. Ioana, *II$_1$ factors with non-isomorphic ultrapowers*, to appear in Duke Math. J. arXiv 1507.06340. I. Farah, B. Hart, M. Lupini, L. Robert, A.P. Tikuisis, A. Vignati, and W. Winter, *Model theory of C$^*$-algebras*, arXiv 1602.08072. I. Goldbring and B. Hart, *On the theories of McDuff’s II$_1$ factors*, International Math Research Notices, to appear. D. McDuff, *Uncountably many II$_1$ factors*, Ann. of Math. **90** (1969) 372-377.
|
---
abstract: 'We study the numerical differentiation formulae for functions given in grids with arbitrary number of nodes. We investigate the case of the infinite number of points in the formulae for the calculation of the first and the second derivatives. The spectra of the corresponding weight coefficients sequences are obtained. We examine the first derivative calculation of a function given in odd-number points and analyze the spectra of the weight coefficients sequences in the cases of both finite and infinite number of nodes. We derive the one-sided approximation for the first derivative and examine its spectral properties.'
author:
- |
Maxim Dvornikov\
Department of Physics, P.O. Box 35, 40014, University of Jyväskylä, Finland[^1] ;\
IZMIRAN, 142190, Troitsk, Moscow region, Russia[^2]
title: Spectral Properties of Numerical Differentiation
---
Mathematics Subject Classification: Primary 65D25; Secondary 65T50
Introduction {#introduction .unnumbered}
============
The problem of numerical differentiation is a long-standing issue. There are plenty of published works devoted to the generation of finite difference formulae in both one and multi dimensional lattices (see, e.g., Ref. [@AbrSti64]). However, many of those methods require preliminary construction of an interpolating polynomial, and hence are very awkward. Moreover, the majority of the previous techniques are valid in the case of a function given in the limited number of nodes.
The finite difference formulae for the calculation of any order derivative in a one dimensional grid with arbitrary spacing were discussed in Refs. [@For88; @For98]. However, only recursion relations for the weight coefficients have been derived. The explicit formulas for the derivatives calculation were recently derived in Ref. [@Li05] on the basis of the generalized Vandermonde determinant.
It should be noted that the low order derivatives (the first and the second ones) as well as equidistant lattices are of the major importance in many problems of applied mathematics and physics. The first and the second numerical derivatives in the equidistant one dimensional grid were studied in Ref. [@Dvo07JCAAM]. The finite difference formulae for the central derivatives of a function given on a lattice with arbitrary number of elements have been derived in that work. It is important that these formulae have been obtained in the explicit form. This method enabled one to examine the spectral properties of weight coefficients sequences as well as to analyze the accuracy of the numerical differentiation.
In the present paper we continue to study the numerical differentiation formulae for functions given in grids with arbitrary number of nodes. On the basis of the results of Ref. [@Dvo07JCAAM] in Sec. \[FSDINP\] we investigate the case of the infinite number of points in the formulae for the calculation of the first and the second derivatives. The spectra of the corresponding weight coefficients sequences are also obtained. Then, in Sec. \[ONP\] we examine the first derivative calculation of a function given in odd-number points. We also analyze the spectra of the weight coefficients sequences in the cases of both finite and infinite number of nodes. In Sec. \[OSD\] we derive the one-sided approximation for the first derivative and examine its spectral properties. It is worth noticing that the derivations of the finite difference formulae in all cases are performed for the arbitrary number of points. Finally, in Sec. \[CONCL\] we resume our results.
Spectral properties of the first and the second derivatives for infinite number of points {#FSDINP}
=========================================================================================
Let us study the function $f(x)$ given in the equidistant points $x_m$, $f(x_m)=f_m$, where $m=0,\dots,\pm n$. It was found in Ref. [@Dvo07JCAAM] that the first and the second derivatives are approximated as $$\label{f'}
f^{\prime}(0)\approx
\frac{1}{2h}
\sum_{m=1}^{n}\alpha_{m}^{(1)}(n)(f_{m}-f_{-m}).$$ and $$\label{f''}
f^{\prime\prime}(0)\approx
\frac{1}{h^{2}}
\sum_{m=1}^{n}\alpha_{m}^{(2)}(n)(f_{m}-2f(0)+f_{-m}),$$ where $h$ is the distance between nodes. The coefficients $\alpha_{m}^{(1)}(n)$ and $\alpha_{m}^{(2)}(n)$ can be calculated explicitly for arbitrary $n$ (see Ref. [@Dvo07JCAAM]).
The spectral properties of the sequences $\alpha_{m}^{(1)}(n)$ and $\alpha_{m}^{(2)}(n)$ in Eqs. and in the case of finite number of interpolation points were carefully examined in Ref. [@Dvo07JCAAM]. We found out that the more points we involved in the sequence $\alpha_{m}^{(1)}(n)$ the more close to linear the corresponding spectrum was. Thus the considered sequence produces more accurate first derivative of a function in the case of great number of points. As for the sequence $\alpha_{m}^{(2)}(n)$, it was also shown in Ref. [@Dvo07JCAAM] that its spectrum approached to parabola if $n>1$. We expect that the corresponding spectra will be exactly linear and parabolic ones if $n\to\infty$.
Let us consider the spectral characteristics of the sequences $\alpha_{m}^{(1)}(n)$ and $\alpha_{m}^{(2)}(n)$ in the case of infinite number of interpolation points. First we remind the result for the $\alpha_{m}^{(1)}(n)$ in the limit $n\to\infty$ (see Ref. [@Dvo07JCAAM]) $$\label{limalpha}
\alpha_{m}^{(1)}=\lim_{n\to\infty}\alpha_{m}^{(1)}(n)=
(-1)^{m+1}\frac{2}{m}.$$ The Fourier transform of a function $f(x)$ can be presented in the form (see, e.g., Ref. [@MunWal00]) $$\label{fourierinf}
c(\omega)=h\sum_{x}e^{-i\omega x}f(x)=
h\sum_{m=-\infty}^{+\infty}
e^{-i\omega mh}f(mh).$$ The inverse Fourier transformation is given by the following expression: $$f(x)=\int_{-\pi/h}^{\pi/h}
\frac{d\omega}{2\pi}
c(\omega)e^{i\omega x},
\quad
x=kh,$$ and has the cutoff at high frequencies, $\vert\omega\vert \leq \pi/h$.
Now we can calculate the spectrum of the sequence $\alpha_{m}^{(1)}$, $$\label{beta1inf}
\beta_1(\omega)= %&
h\sum_{\substack{m=-\infty\\ m\not=0}}^{+\infty}
e^{-i\omega mh}\alpha_{m}^{(1)}
%\notag
%\\
= %&
-4ih\sum_{m=1}^{\infty}
(-1)^{m-1}
\frac{\sin(m\omega h)}{m}=
-2i\omega h^2,$$ where we use Eqs. and . Note that Eq. is valid if $0\leq\omega<\pi/h$. The first derivative of the function can be expressed via the spectra $\beta_1(\omega)$ and $c(\omega)$, $$\label{f'inf}
f'(x)=
\frac{1}{2h}
\int_{-\pi/h}^{\pi/h}
\frac{d\omega}{2\pi h}
\beta^{*}_1(\omega)c(\omega)
e^{i\omega x},
\quad
x=kh.$$ Using the result for the calculation of $\beta_1(\omega)$ presented in Eq. we readily find that $$\label{f'inffin}
f'(x)=
\int_{-\pi/h}^{\pi/h}
\frac{d\omega}{2\pi}
(i\omega)c(\omega)
e^{i\omega x}.$$ Eq. shows that the first derivative calculation with help of the sequence $\alpha_{m}^{(1)}$ gives the exact value of the derivative in the case of infinite number of interpolation points for all frequencies except $\omega_{\mathrm{max}}=\pi/h$. The fact that the first derivative computation does not give correct results at $\omega=\omega_{\mathrm{max}}$ also follows from Fig. \[spbeta12\](a). However, it can be verified directly with help of Eq. for the function $f_m=(-1)^m=\cos(\omega_{\mathrm{max}}mh)$.
![The spectra of differentiating filters, (a) $\alpha_{m}^{(1)}$ and (b) $\alpha_{m}^{(2)}$, in the case of infinite number of points.[]{data-label="spbeta12"}](fig12.eps)
In Ref. [@Dvo07JCAAM] we showed that the use of the coefficients $\alpha_{m}^{(1)}$ give exact value for the first derivative of the function $y(x)=\sin(\omega_\mathrm{max}x/2)$. However, Eq. \[see also Fig. \[spbeta12\](a)\] indicates that this method will give the correct results not only for $\omega=\omega_\mathrm{max}/2$ but also for all frequencies $\omega<\omega_\mathrm{max}$.
The second derivative calculation in the case of infinite number of interpolation points can be analyzed in the similar manner as we have done it for the first derivative. The explicit form of the sequence $\alpha_{m}^{(2)}$ is $$\alpha_{m}^{(2)}=\lim_{n\to\infty}\alpha_{m}^{(2)}(n)=
(-1)^{m+1}\frac{2}{m^2}.$$ For the spectrum of the sequence $\alpha_{m}^{(2)}$ we obtain $$\label{beta2inf}
\beta_2(\omega)=
-\omega^2 h^3+
\frac{\pi^2}{3}h.$$ It should be noted that Eq. is valid for all frequencies $0\leq\omega\leq\pi/h$. The expression for the second derivative takes the form $$\label{f''inf}
f''(x)= %&
\frac{1}{h^2}
\int_{-\pi/h}^{\pi/h}
\frac{d\omega}{2\pi h}
\left[
\beta^{*}_2(\omega)-\beta^{*}_2(0)
\right]
c(\omega)
e^{i\omega x}
%\notag
%\\
= %&
\int_{-\pi/h}^{\pi/h}
\frac{d\omega}{2\pi}
(-\omega^2)c(\omega)
e^{i\omega x},
\quad
x=kh.$$ Eq. demonstrates that the computation of the second derivative with the use of the sequence $\alpha_{m}^{(2)}$ gives the exact results in the case of infinite number of interpolation points for all frequencies even including the maximal one. The spectrum $\beta_2(\omega)$ is depicted in Fig. \[spbeta12\](b).
The first derivative computation of a function given in odd-number points {#ONP}
=========================================================================
In this section we discuss the calculation of the first derivative in the case of a function given in odd-number nodes. Then we discuss the spectral properties of the derived weight coefficients sequences in the case of both finite and infinite number of nodes.
It follows from Fig. \[spbeta12\](a) that the computation of the first derivative gives unsatisfactory results at high frequencies near $\omega_\mathrm{max}$. In order to introduce the numerical differentiation of such rapidly oscillating functions we consider the modified sequence $$\label{alpha1/2}
\alpha_{2m+1}^{(1/2)}(n)=
\frac{1}{(2m+1)\pi_m^{(1/2)}(n)},
\quad
m=0,\dots,n-1,$$ where $$\pi_m^{(1/2)}(n)=\prod_{
\substack{k=0
\\
k{\not=}m}}^{n-1}
\left(
1-\frac{(2m+1)^{2}}{(2k+1)^{2}}
\right),$$ and $\alpha_{2m}^{(1/2)}(n)=0$.
The coefficients in Eq. can be formally derived if we consider the first derivative calculation of a function given in the odd-number points only $$\label{f'1/2}
f'(0)\approx
\frac{1}{2h}
\sum_{m=0}^{n-1}\alpha_{2m+1}^{(1/2)}(n)(f_{2m+1}-f_{-2m-1}).$$ Note that originally the function $f(x)$ was given in $2n+1$ points.
It is worth noticing that the results for the computation of the weights with help of Eq. in some particular cases (namely for $n=3,5,7$ and $9$) coincide with those presented in Ref. [@For88] for the centered approximations at a ’half-way’ point. However, the method for the central derivatives calculation elaborated in our paper enables one to get the expressions for the weight coefficients in the explicit form for any number of nodes.
We consider the spectral properties of the obtained sequence $\alpha_{m}^{(1/2)}(n)$. Using the technique developed in Ref. [@Dvo07JCAAM] one can compute the spectrum of the sequence in question, $$\beta_{1/2}(r)=\sum_{m=0}^{N-1}
\alpha_{m}^{(1/2)}(n)\exp
\left(
-i\frac{2\pi}{N}mr
\right).$$ The spectra of the sequences $\alpha_{m}^{(1/2)}(n)$ are depicted in Fig. \[spbeta1/2\](a) for the various values of $n$ at $N=2000$.
![(a) The spectra of various sequences $\alpha_{m}^{(1/2)}(n)$ at $N=2000$. (b) The example of the use of the sequence $\alpha^{(1/2)}_m(n)$. A rapidly oscillating function corresponds to the dashed lines, its envelope functions are shown by the dotted lines.[]{data-label="spbeta1/2"}](fig34.eps)
The function $y_{1/2}(r)$ has the form $$y_{1/2}(r)=
\begin{cases}
2\pi r/N, & \text{if $0\leq r\leq N/4$}, \\
\pi-2\pi r/N, & \text{if $N/4\leq r\leq N/2$}.
\end{cases}$$ It follows from this figure that for $n=1$ the imaginary part of the spectrum is the function $\sin(2\pi r/N)$. The linearity condition is satisfied only in the vicinity of zero and $N/2$. However at $n=10$ linearity condition remains valid for $r\lesssim
350$ and $r\gtrsim 650$.
We examine the details of the differentiation procedure performed with help of the sequence $\alpha_{m}^{(1/2)}(n)$. If a function is slowly varying, then Eq. gives the approximate value of the first derivative. It also results from Fig. \[spbeta1/2\](a). However, if a function is rapidly oscillating (e.g., at $\omega\lesssim\omega_\mathrm{max}$), we can consider its upper and lower envelope functions. Therefore the use of the sequence $\alpha_{m}^{(1/2)}(n)$ will produce the first derivative of envelope functions since in Eq. we calculate the sum in odd-number points only. Envelope functions can be treated as “smooth” in this case. Fig. \[spbeta1/2\](b) schematically illustrates this process.
Now let us discuss the case of infinite number of the interpolation points. We can treat the sequence $\alpha_{m}^{(1/2)}(n)$ in the similar way as it was done in Ref. [@Dvo07JCAAM]. Indeed, proceeding to the limit $n\to\infty$ in Eq. we find that $$\label{alpha1/2lim}
\alpha_{2m+1}^{(1/2)}=
\lim_{n\to\infty}\alpha_{2m+1}^{(1/2)}(n)=
(-1)^m
\frac{4}{\pi(2m+1)^2}.$$ In Eq. we used the known value of infinite product, $$\prod_{k=0}^{\infty}
\left(
1-\frac{x^{2}}{(2k+1)^{2}}
\right)
=\cos\left(\frac{\pi x}{2}\right).$$ With help of Eq. it is possible to obtain the spectrum of the sequence $\alpha_{m}^{(1/2)}$ $$\label{beta1/2inf}
\beta_{1/2}(\omega)=-2ih\times
\begin{cases}
\omega h, & \text{if $0\leq\omega\leq\pi/2h$}, \\
(\pi-\omega h), & \text{if $\pi/2h\leq\omega\leq\pi/h$}.
\end{cases}$$ The imaginary part of the spectrum $\beta_{1/2}(\omega)$ is presented in Fig. \[spbeta1/2inf\].
![The spectrum of the sequence $\alpha_{m}^{(1/2)}$ in the case of infinite number of points.[]{data-label="spbeta1/2inf"}](fig5.eps)
As it follows from Eq. (see also Fig. \[spbeta1/2inf\]) the sequence $\alpha_{m}^{(1/2)}$ performs the differentiation of a function in question if $\omega\leq\omega_\mathrm{max}/2$, or its envelope functions if $\omega\geq\omega_\mathrm{max}/2$. The of case $\omega=\omega_\mathrm{max}/2$ was also considered in details in Ref. [@Dvo07JCAAM].
One-sided approximation for the first derivative {#OSD}
================================================
In this section we derive the weight coefficients for the one-sided approximation of the first derivative and then we analyze the spectral characteristics of the weight coefficients sequence. It should be noted that the derivation of the weight coefficients is analogous to case the of the central derivatives which was carefully examined in Ref. [@Dvo07JCAAM].
Without restriction of generality we suppose that we approximate the first derivative in the zero point. Let us consider the function $f(x)$ given in the equidistant nodes $x_{m}=mh>0$, where $m=0,\dots,n$, and $h$ is the constant value. We can pass the interpolating polynomial of the $n$th power through these points, $$P_{n}(x)=\sum_{k=0}^{n}c_{k}x^{k}.$$ The values of the function in the nodes $x_{m}=mh$, $f_{m}=f(x_{m})$, should coincide with the values of the interpolating polynomial in these points, $$\label{fm}
f_{m}=\sum_{k=0}^{n}c_{k}h^{k}m^{k}.$$ In order to find the coefficients $c_{k}$, $k=0,\dots,n$, we receive the system of inhomogeneous linear equations with the given free terms $f_{m}$. It will be shown below that this system has the single solution.
We will seek the solution of the system in the following way: $$c_{k}=\frac{1}{h^k}
\sum_{m=0}^{n}f_{m}a_{m}^{(k)}(n),$$ where $a_{m}^{(k)}(n)$ are the undetermined coefficients satisfying the condition, $$\label{a}
\sum_{m=0}^{n}a_{m}^{(l)}(n)m^{k}=\delta_{lk},
\quad l,k=0,\dots n.$$ It is worth to be noted that, if we set $k=0$ and $l{\not=}0$ in Eq. , we obtain the constraint which should be imposed on the coefficients $a_{m}^{(l)}(n)$ $$\label{acond}
\sum_{m=0}^{n}a_{m}^{(l)}(n)=0,
\quad l=1,\dots n.$$ Analogous relation between the weight coefficients was established in Ref. [@For88]. In deriving of Eq. (as well as in all subsequent similar formulae) we suppose that $m^0=1$ if $m=0$.
Let us resolve the system of equations according to the Cramer’s rule $$\label{kram}
a_{m}^{(l)}(n)=
\frac{\Delta_{m}^{(l)}(n)}{\Delta_{0}(n)},$$ where $$\label{delta0}
\Delta_{0}(n)=
\begin{vmatrix}
1 & 1 & 1 & \dots & 1
\\
0 & 1 & 2 & \dots & n
\\
0 & 1 & 2^{2} & \dots & n^{2}
\\
\hdotsfor{5}
\\
0 & 1 & 2^n & \dots & n^n
\\
\end{vmatrix}=
n!\prod_{1\leq i<j\leq n}(j-i){\not=}0,$$ and $$\label{deltam}
\Delta_{m}^{(l)}(n)=
\begin{vmatrix}
1 & 1 & 1 & \dots & 1 & 0 & 1 & \dots & 1
\\
0 & 1 & 2 & \dots & m-1 & 0 & m+1 & \dots & n
\\
\hdotsfor{9}
\\
0 & 1 & 2^{l} & \dots & (m-1)^{l} & 1
& (m+1)^{l} & \dots & n^{l}
\\
\hdotsfor{9}
\\
0 & 1 & 2^{n} & \dots & (m-1)^{n} & 0
& (m+1)^{n} & \dots & n^{n}
\end{vmatrix}.$$ In Eq. we use the formula for the calculation of the Vandermonde determinant. From Eq. it follows that the determinant of the system of equations is not equal to zero, i.e. the system of equations has the single solution.
The most simple expression for $\Delta_{m}^{(l)}(n)$ is obtained in the case of $l=1$ that corresponds to the calculation of the first-order derivative $$\label{deltam1}
\Delta_{m}^{(1)}(n)=(-1)^{m+1}
\left(
\frac{n!}{m}
\right)^{2}
\prod_{
\substack{1\leq i<j\leq n
\\
i,j{\not=}m}}(j-i),
\quad
m=1,\dots,n.$$ From Eq. as well as taking into account Eqs. and we get the expression for the coefficients $a_{m}^{(1)}(n)$ $$\label{am}
a_{m}^{(1)}(n)=(-1)^{m+1}\frac{1}{m}\binom{n}{m},
\quad
m=1,\dots,n,$$ where $$\binom{n}{m}=\frac{n!}{m!(n-m)!},$$ are the binomial coefficients. It is remarkable to note that the coefficient $a_{1}^{(1)}(n)=n$. To simplify numerical calculations (especially the analysis of the spectra of the derived sequences) Eq. should be rewritten in the form $$a_{m}^{(1)}(n)=\frac{1}{m p_m(n)},
\quad
m=1,\dots,n,$$ where $$p_{m}(n)=\prod_{
\substack{k=1
\\
k{\not=}m}}^{n}
\left(
1-\frac{m}{k}
\right).$$
In order to find the coefficient $a_{0}^{(1)}(n)$ we use Eq. rather than compute the determinant . Thus we receive the following expression for this coefficient, $$\label{a0}
a_{0}^{(1)}(n)=-\sum_{m=1}^{n}a_{m}^{(l)}(n)=
-\sum_{m=1}^{n}(-1)^{m+1}\frac{1}{m}\binom{n}{m}=
-\sum_{m=1}^{n}\frac{1}{m}.$$ Eqs. and provide the weight coefficients for the one-sided approximation of the first derivative of the function $f(x)$ given in $n+1$ equidistant nodes, $$f'(0)\approx
\frac{1}{h}
\sum_{m=0}^{n}f_{m}a_m^{(1)}(n).$$ The results for the computation of the weights in some particular cases (namely for $n=1,2,\dots,8$) coincide with those presented in Ref. [@For88]. However, the technique for derivatives calculation developed in the present work allows one to obtain the expressions for the weight coefficients in the explicit form for any $n$.
Without derivation we mention that on the basis of Eqs. - one can find the coefficients $a_{m}^{(n)}(n)$ that correspond to the computation of the $n$th-order derivative $$\label{an}
a_{m}^{(n)}(n)=(-1)^{m+n}
\frac{1}{n!}\binom{n}{m}.
\quad
m=0,\dots,n.$$ It should be noted that Eq. is consistent with Eq. .
Now we consider the spectral properties of the derived sequence $a_{m}^{(1)}(n)$. Using the results of the previous section (see also Ref. [@Dvo07JCAAM]) we readily find the expression for the spectrum of the considered sequence, $$b_{1}(r)=\sum_{m=0}^{N-1}
a_{m}^{(1)}(n)\exp
\left(
-i\frac{2\pi}{N}mr
\right).$$ The imaginary parts, $\Im\mathfrak{m}\big[b^*_1(r)\big]$, of the spectra of the sequences $a_{m}^{(1)}(n)$ for the various values of $n$ at $N=2000$ as well as the linearly growing sequence $I(r)=2\pi r/N$ are presented in Fig. \[sponeside\](a).
![The imaginary parts (a) and the real parts (b) of the spectra of various sequences $a_{m}^{(1)}(n)$ at $N=2000$.[]{data-label="sponeside"}](fig67.eps)
It follows form the this figure that the imaginary parts are close to the linear sequence only in the vicinity of zero ($r\lesssim
200$) even at $n=5$. It points out that the one-sided approximation of the first derivative has worse accuracy in comparison with the central derivatives. This fact was also mentioned in Ref. [@DemMar63eng]. Therefore, the application of the sequence $a_{m}^{(1)}(n)$ for the calculation of the one-sided first derivative will give reliable results only for slowly varying functions.
The spectrum $b_1(r)$ has not only imaginary part, but also nonzero real part since $a_{0}^{(1)}(n){\not=}0$. The real parts, $\Re\mathfrak{e}\big[b^*_1(r)\big]$, of the spectra of the sequences $\alpha_{m}^{(1)}(n)$ for the various values of $n$ at $N=2000$ as well as the constant sequence $R(r)=0$ are shown in Fig. \[sponeside\](b). It can be seen from this figure that the real parts of the spectra are close to zero if $r\lesssim 100$ at $n=1$, and if $r\lesssim
300$ at $n=3$ and $n=5$. The deviation from zero is especially great if $r\gtrsim 700$ at $n=3$, and if $r\gtrsim 500$ at $n=5$. Such a behavior of the real parts of the spectra also reveals the limited level of accuracy of the one-sided first derivative.
Conclusion {#CONCL}
==========
In conclusion we note that in our paper we have studied the numerical differentiation formulae for functions given on grids with arbitrary number of nodes. In Sec. \[FSDINP\] we have investigated the case of the infinite number of points in the formulae for the calculation of the first and the second derivatives. The spectra of the corresponding weight coefficients sequences have been obtained. It has been revealed that the calculation of the first derivative with help of the derived formulae gave reliable results for all spacial frequencies except $\omega_\mathrm{max}$. As for the calculation of the second derivative we have shown that the corresponding formulae were valid for all spacial frequencies including $\omega_\mathrm{max}$. In Sec. \[ONP\] we have examined the first derivative calculation of a function given in odd-number points. We have also analyzed the spectra of the weight coefficients sequences in the cases of both finite and infinite number of nodes. It has been found out that the obtained formulae perform the differentiation of the considered function if $\omega\leq\omega_\mathrm{max}/2$, and its envelope functions if $\omega\geq\omega_\mathrm{max}/2$. In Sec. \[OSD\] we have derived the one-sided approximation for the first derivative and examined its spectral properties. The accuracy of the one-sided first derivative has been discussed. On the basis of the spectral properties of the weight coefficients sequences it has been shown that the accuracy of the one-sided approximation for the first derivative was essentially lower compared to the computation of the central derivatives. This our result is in agreement with previous works (see, e.g. Ref. [@DemMar63eng]). Nevertheless, the obtained one-sided first derivative formulae could be of use in solving differential equations by means of numerical methods. It is also possible to apply the elaborated technique of the numerical differentiation in construction of quantum field theory models of unified interactions.
Acknowledgments {#acknowledgments .unnumbered}
---------------
This research has been supported by the Academy of Finland under the contract No. 108875. The author is indebted to the Russian Science Support Foundation for a grant as well as to Sergey Dvornikov for helpful discussions.
[7]{}
M. Abramowitz and I. A. Stegun, *Handbook of Mathematical Functions*, National Bureau of Standards, Washington D. C., 1964.
B. Fornberg, Generation of Finite Difference Formulas on Arbitrary Spaced Grids, *Math. Comp.*, 51(184), 699-706, (1988).
B. Fornberg, Calculation of Weights in Finite Difference Formulas, *SIAM Rev.*, 40(3), 685-691, (1998).
J. Li, General Explicit Difference Formulas for Numerical Differentiation, *J. Comp. & Appl. Math.*, 183, 29-52, (2005).
M. Dvornikov, Formulae of Numerical Differentiation, *JCAAM*, 5, 77-88, (2007) \[e-print arXiv: math.NA/0306092\].
G. Münster and M. Walzl, Lattice Gauge Theory – A Short Primer, in: *Proceedings of the Summer School on Phenomenology of Gauge Interactions* (D. Graudenz and V. Markushin, eds.), PSI, Villigen, Switzerland, 2000, pp. 127-160 \[e-print arXiv: hep-lat/0012005\].
B. P. Demidovich and I. A. Maron, *Foundations of Computational Mathematics* (2nd ed.), Fiz. Mat. Lit., Moscow, 1963.
[^1]: E-mail: dvmaxim@cc.jyu.fi
[^2]: E-mail: maxdvo@izmiran.ru
|
**Temperature profile in a liquid-vapor interface**
0.1cm
**near the critical point**
0.2cm
Henri Gouin[^1]&Pierre Seppecher[^2] [^3]
0.5cm [**Abstract** ]{}
Thanks to an expansion with respect to densities of energy, mass and entropy, we discuss the concept of *thermocapillary fluid* for inhomogeneous fluids. The non-convex state law valid for homogeneous fluids is modified by adding terms taking into account the gradients of these densities. This seems more realistic than Cahn and Hilliard’s model which uses a density expansion in mass-density gradient only. Indeed, through liquid-vapor interfaces, realistic potentials in molecular theories show that entropy density and temperature do not vary with the mass density as it would do in bulk phases.
In this paper we prove, using a rescaling process near the critical point, that liquid-vapor interfaces behave essentially in the same way as in Cahn and Hilliard’s model.
**Keyword**: Fluid critical-point; temperature profile; phase transition; rescaling process.
**PACS numbers**: 46.15.Cc ; 47.35.Fg ; 47.51.+a ; 64.75.Ef
Introduction
============
Phase separation between liquid and vapor is due to the fact that density of internal energy (i.e. internal energy per unit volume) $\varepsilon_{0} (\rho ,\eta)$ of homogeneous fluids is a non-convex function of mass density $\rho$ and entropy density $\eta$. At a given temperature $T_0$, this non-convexity property is related with the non-monotony of thermodynamical pressure $P(\rho,T_0)$.
The reader may be accustomed to use specific quantities $\alpha={\varepsilon}/{\rho}$, $s={\eta}/{\rho}$ and $v=1/\rho$ instead of volume densities. Indeed the non-convexity property of $\varepsilon_{0}$ is equivalent to the non-convexity of $\alpha$ as a function of $s$ and $v$. In this paper, in accordance with Cahn-Hilliard standard presentation, we privilege volume densities.
In continuum mechanics the simplest model for describing inhomogeneous fluids inside interfacial layers considers an internal-energy density $\varepsilon$ as the sum of two terms: the first one previously defined as $\varepsilon_{0}(\rho ,\eta)$, corresponds to the fluid with an uniform composition equal to its local one, and the second one associated with the non-uniformity of the fluid is approximated by a gradient expansion, $$\varepsilon :=\varepsilon_{0} (\rho ,\eta)+{\frac{1}{2}}\,m\,|\func{grad}\rho\,|^{2}, \label{cahnenergy}$$ where $m$ is a coefficient assumed to be independent of $\rho$, $\eta$ and $\func{grad}\rho $. This form of internal energy density can be deduced from molecular mean-field theory where the molecules are modeled as hard spheres submitted to Lennard-Jones potentials [@Widom; @GouinJCP].
This energy has been introduced by van der Waals [@Waals] and is widely used in the literature [@Korteweg; @Ono; @Cahn; @casal; @casal1]. This model, nowadays known as Cahn-Hilliard fluid model, describes interfaces as diffuse layers. The mass density profile connecting liquid to vapor becomes a smooth function.
The model has been widely used for describing micro-droplets [@Isola2; @Isola3], contact-lines [@casal2; @seppecher; @Seppecher1; @Gouin1], nanofluidics [@Gouin2; @Gouin9; @Garajeu], thin films [@Gouin6], vegetal biology [@gouin7; @gouinbio]. It has been extended to more complex situations e.g. in fluid mixtures, porous materials…, thanks to the so-called second-gradient theory [@Germain; @Isola] which models the behavior of strongly inhomogeneous media [@Gouin-Ruggeri; @Ruggeri; @dell'Isola; @Eremeyev; @Dell'Isola1; @Forest].
It has been noticed that, at equilibrium, expression for the energy density yields an uniform temperature $T_{_0} $ everywhere in inhomogeneous fluids, $$T:=\frac{\partial \varepsilon_{_0} }{\partial \eta}(\rho
,\eta)=T_{_0} . \label{temp0}$$ Let us note that it is not the same for chemical potential $$\mu_{_0}:= \frac{\partial \varepsilon_{_0} }{\partial \rho}(\rho
,\eta) ,$$ which takes the same values in the different bulks but is not uniform inside interfacial layers. From Eq. one can deduce that the entropy density varies with the mass density in the same way as in the bulks and it is a peculiarity of the Cahn-Hilliard model that the configurational $\eta$ and $\varepsilon$ can be written in term of $\rho$, only. The points $(\rho , \eta, \varepsilon)$ representing phase states lie on curve $T=T_0$ and such a model inevitably lead to monotonic variations of all densities [@Widom]. Original assumption of van der Waals which uses long-ranged but weak attractive forces is not exact for more realistic intermolecular potentials [@Ornstein; @Hamak; @Evans]. Aside from the question of accuracy, there are qualitative features like non-monotonic behaviors in transition layers, especially in systems of more than one component, that require two or more independently varying densities - entropy included - (see chapter 3 of [@Rowlinson]). For these reasons, model has been extended in [@Rowlinson; @casal4] by taking into account not only the strong mass density variations through interfacial layers but also the strong variations of entropy associated with latent-heat of phase changes. Rowlinson and Widom in [@Rowlinson] (chapter 3 and chapter 9) noticed that $T=T_{_0}$ is not exact through liquid-vapor interfaces and they introduced an energy arising from the mean-field theory and depending on densities $\rho$ and $\eta $ and also on the gradients of these densities; furthermore, they said that *near the critical point, a gradient expansion typically truncated in second order, is most likely to be successful and perhaps even quantitatively accurate*. This extension has been called *thermocapillary fluid model* in [@casal4] and used in different physical situations when the temperature varies in strongly inhomogeneous parts of complex media [@casal4; @Gouin; @Forest1; @Forest2; @Maitournam].
Near a single-fluid critical point, the mean-field molecular theory yields an approximate but realistic behavior [@Rowlinson; @Domb]. In mean-field theory, the differences of thermodynamical quantities between liquid and vapor phases are expressed in power laws of the difference between temperature and critical temperature. Transformations from liquid to vapor are associated with second-order phase transitions and the mass density difference between the two phases goes to zero as the temperature is converging to the critical one. The same phenomenon holds true for the latent-heat of phase transition and for the difference of entropy densities between liquid and vapor phases.
In this paper we neglect gravity and we use a slightly more general model. We consider state laws which link densities $\varepsilon, \rho , \eta $ and their gradients. We derive the liquid-vapor equilibrium equations of non-homogeneous fluids. As, at equilibrium, a given total mass of the fluid in a fixed domain maximizes its total entropy while its total energy remains constant, the problem can be studied in a variational framework.
We make explicit a polynomial expansion of the homogeneous state law near the critical point. In convenient units, we obtain a generic expression depending only on a unique parameter $\chi$.
We introduce a small parameter $\kappa$ which measures the distance of the considered equilibrium state to the critical point. Using a rescaling process near the critical point we obtain mass and temperature profiles through the liquid-vapor interface. The magnitude orders with respect to $\kappa$ of mass, entropy, temperature are analyzed. The variations of temperature and entropy density inside the interfacial layer appear to be of an order less than the variation of mass density. Consequently, neglecting these variations is well-founded and justifies the utilization of Cahn-Hilliard’s model near the critical point and indeed we prove that the mass density profile converges towards the classical profile obtained by using the Cahn-Hilliard model which does not take account of variations of entropy density. A conclusion highlights these facts.
\[sec2\]Equations of equilibrium
================================
\[subsec2.1\]Preliminaries
--------------------------
When homogeneous simple fluids are considered, a state law $${\cal L}_0(\varepsilon,\eta,\rho)=0$$ links internal energy density $\varepsilon$, entropy density $\eta$ and mass density $\rho$. This local law is generally made explicit under the form $$\varepsilon-\varepsilon_{_0}(\eta,\rho)=0 .$$ In other words, it is assumed without loss of generality that $\partial {\cal L}_0/{\partial \varepsilon}=1$. Then, as usual, one introduces the Kelvin temperature $T:=-\partial {\cal
L}_0/{\partial \eta}$, the chemical potential ${\mu }:=-\partial
{\cal L}_0/{\partial \rho}$ and the thermodynamical pressure $$P:=\rho\,{\mu } -\varepsilon+\eta\,T.$$ These notations can be resumed as $$d\varepsilon- {\mu }\, d\rho-{ T}\, d\eta=0, \quad dP- \eta\,dT
-\rho\, d\mu=0. \label{chempot}$$
However, when the state of the material endows strong spatial variations of the thermodynamical variables - as it is the case near a liquid vapor interface - the locality of the state law has to be questioned. This is what we do in this paper by considering a general law of the type $${\cal L}(\varepsilon,\eta,\rho,\nabla \varepsilon,
\nabla \eta,\nabla \rho )=0 ,\label{relation}$$ where $\nabla$ denotes the spatial gradient. For the sake of simplicity, we study in this paper the particular form () : $$\begin{aligned}
{\cal L}(\varepsilon,\eta,\rho,\nabla \varepsilon,\nabla \eta,\nabla \rho )= &
{\cal L}_0(\varepsilon,\eta,\rho) - \frac{1}{2}
\Big(
C_0\, |\nabla\rho|^2 + E_0\, |\nabla\eta|^2 + H_0\, |\nabla\varepsilon|^2\nonumber \\
+ & 2\, D_0\, \nabla\rho\cdot\nabla\eta + 2\, F_0\, \nabla\rho \cdot \nabla\varepsilon + 2\, G_0 \, \nabla\eta \cdot \nabla \varepsilon
\Big),
\label{specialvolumeenergy}\end{aligned}$$ where $$\left[\begin{matrix}C_0 & D_0 & F_0 \cr D_0 & E_0 & G_0
\cr F_0 & G_0 & H_0 \end{matrix}\right]$$ is a constant positive symmetric matrix. This is the simplest extension of the classical model when one wants to take account of the spatial variations of $\eta$, $\varepsilon$ and $\rho$. Generalization is widely studied [@Waals; @Cahn] in the particular case ${\cal L}(\varepsilon,\eta,\rho,\nabla \rho )=0$; that is when one sets $D_0=E_0=F_0=G_0=H_0=0$ in . This special case coincides with the well-known model of Cahn-Hilliard’s fluids [@Cahn].
In our framework, we still call temperature, chemical potential, thermodynamical pressure the quantities $$T:=-\partial {\cal L}_0/{\partial \eta},\quad {\mu }:=-\partial
{\cal L}_0/{\partial \rho}\quad {\rm and}\quad P:=\rho\,{\mu }
-\varepsilon+\eta\,T.$$ Thus, the state law reads in differential form : $$d\varepsilon- \mu\,d\rho - {T}\, d\eta - \boldsymbol{\Phi}\cdot
d(\nabla \rho) - \boldsymbol{\Psi}\cdot d(\nabla\eta) -
\boldsymbol{\Xi}\cdot d(\nabla\varepsilon) =0 \label{diffenergy}$$ with $$\begin{split}
\boldsymbol{\Phi} &= C_0\, \nabla\rho + D_0 \, \nabla\eta+ F_0 \, \nabla\varepsilon,\quad \\
\boldsymbol{\Psi} &= D_0\, \nabla\rho + E_0\, \nabla\eta + G_0 \, \nabla\varepsilon,\quad \\
\boldsymbol{\Xi} &= F_0\, \nabla\rho + G_0\, \nabla\eta + H_0 \,
\nabla\varepsilon.
\end{split}
\label{phipsi}$$
\[subsec2.2\] The variational method
------------------------------------
The total mass and the total energy of an isolated and fixed domain ${\cal D}$ are $$M= \int_{\cal D}\rho \ dx,\qquad E = \int_{\cal D}\varepsilon \ dx,$$ where $dx$ is the volume element. They remain constant during the evolution of the system towards equilibrium. The equilibrium is reached when the total entropy $$S=\int_{{\cal D}}\eta \,dx=\int_{{\cal D}}\rho\, s \,dx$$ of the system is maximal. With classical notations, at equilibrium we get the variational equation $$\delta S - T_0^{-1} (\delta E - \mu_0\, \delta M)=0$$ where $T_0^{-1}$ and $\mu_0$ are constant Lagrange multipliers ($T_0$ has the physical dimension of a temperature while $\mu_0$ has the physical dimension of a chemical potential). This equation is valid for all variations $(\delta\varepsilon, \delta\eta,
\delta\rho)$ compatible with the state law i.e. $\delta{\cal L}=0$. We can take this constraint into account by introducing a Lagrange multiplier field $\Lambda$ (with no physical dimension) and write that $$T_0 \delta S - \delta E + \mu_0\, \delta M + \int_{{\cal D}}\Lambda\, \delta{\cal L}\, dx =0$$ for all triple field $(\delta\varepsilon, \delta\eta,
\delta\rho)$. This equation reads $$\begin{aligned}
\int_{\cal D} \Big( (T_0-\Lambda\, T)\,\delta \eta&+ (\Lambda- 1)\, \delta\varepsilon +
(\mu_0-\Lambda\,\mu)\,\delta\rho \\
& - \Lambda\,\big( \boldsymbol{\Phi}\cdot (\nabla \delta\rho) +
\boldsymbol{\Psi}\cdot (\nabla\delta\eta) + \boldsymbol{\Xi}\cdot
(\nabla\delta\varepsilon) \big) \Big)\, dx=0\end{aligned}$$ Using the divergence theorem and considering only variations with compact support in ${\cal D}$, we have $$\begin{aligned}
\int_{\cal D} \Big((T_0- \Lambda\, T+\,\func{div}(\Lambda\boldsymbol{\Psi}) )\,\delta \eta&+(\Lambda-1+
\func{div}(\Lambda\boldsymbol{\Xi}) ) \delta\varepsilon \\
& + (\mu_0-\Lambda\,\mu+ \func{div}(\Lambda\boldsymbol{\Phi}))\,\delta\rho \Big)\,
dx=0,\end{aligned}$$ and we deduce the local equations in ${\cal D}$ : $$\begin{split}
& \func{div}(\Lambda\boldsymbol{\Phi})=\Lambda\mu- \mu_0,\\
&\func{div}(\Lambda\boldsymbol{\Psi}) =\Lambda T-T_0,\\
&\func{div}(\Lambda\boldsymbol{\Xi})=1-\Lambda.
\end{split}$$ In the special case of a energy density of form , the system reads $$\left\{
\begin{array}{l}
\displaystyle C_0\, \func{div}(\Lambda\nabla\rho) + D_0\,\func{div}(\Lambda\nabla\eta) +F_0\, \func{div}(\Lambda\nabla\varepsilon) =\Lambda\mu- \mu_{_0}, \\
\displaystyle D_0\,\func{div}(\Lambda\nabla\rho) + E_0\,\func{div}(\Lambda\nabla\eta) +G_0\, \func{div}(\Lambda\nabla\varepsilon)= \Lambda T-T_0 , \\
\displaystyle F_0\,\, \func{div}(\Lambda\nabla\rho) + G_0\,\func{div}(\Lambda\nabla\eta) +H_0\, \func{div}(\Lambda\nabla\varepsilon)= 1- \Lambda , \label{€qphases}
\end{array}\right.$$
\[sec3\]Thermodynamical potentials near a critical point
========================================================
Let $(\varepsilon_c,\eta_c,\rho_c)$ be an admissible homogeneous state indexed by $c$. Then, $${\cal L}_0(\varepsilon_c,\eta_c,\rho_c)=0.$$ Let $P_c$, $T_c$, $\mu_c$ be the associated thermodynamical quantities. At point $(\varepsilon_c,\eta_c,\rho_c)$, we assume that $\partial^2{\cal L}_0/\partial \eta^2\not=0$ and we introduce the quantity $$a_c:= \frac{\partial^2{\cal L}_0/\partial \eta\partial \rho}
{\partial^2{\cal L}_0/\partial \eta^2}(\varepsilon_c,\eta_c,\rho_c).$$ If the studied fields remain close to point $(\varepsilon_c,\eta_c,\rho_c)$, it is natural to make a change of variables in order to work in the vicinity of zero; we set $$\begin{aligned}
&\tilde \rho:=\rho-\rho_c,\ \tilde \eta:=\eta-\eta_c + a_c \tilde \rho, \ \tilde \varepsilon:=\varepsilon- \varepsilon_c- (\mu_c-T_c a_c) \tilde \rho - T_c \tilde \eta,\label{change1}\\
&\tilde{\cal L}_0(\tilde \varepsilon,\tilde \eta,\tilde \rho):={\cal L}_0( \varepsilon_c
+\tilde \varepsilon+T_c \tilde \eta+(\mu_c-T_c a_c)\tilde \rho, \eta_c+ \tilde \eta-a_c \tilde \rho, \rho_c+\tilde\rho).\label{change2}\end{aligned}$$ The change of variables - may seem unnecessarily complicated : its aim is, like in classical nondimensionalization process, to reduce the number of parameters of the problem to the minimal set of parameters which actually affect the qualitative features of the solution. We show below that a unique dimensionless parameter $\tilde \chi$ is enough for describing the shape of the energy function in the vicinity of the critical point.
It is clear that maximizing $\int_{\cal D} \eta\, dx$ under the constraints $\int_{\cal D} \rho\, dx=M$ and $\int_{\cal D}
\varepsilon\, dx=E$ is equivalent to maximizing $\int_{\cal D}
\tilde \eta\, dx$ under the constraints $\int_{\cal D} \tilde
\rho\, dx=M-\rho_c |{\cal D}| $ and $\int_{\cal D} \tilde
\varepsilon\, dx=E-\mu_c M -(\varepsilon_c+\mu_c\rho_c) |{\cal D}|
$. Therefore the variational analysis performed in the previous section remains unchanged if we replace all quantities by their $ \widetilde{\ }\,$- equivalent. Of course this property is only true if we replace the derivative quantities $T$, $\mu$ by the quantities derived from $\tilde{\cal L}$. We set: $$\tilde T:= T-T_c,\quad \tilde \mu:= \mu-\mu_c- a_c \tilde T \label{tempchem}$$ The constants $(C_0,\dots ,H_0)$ have also to be modified but it is not worth writing the expressions of the new constants $(\tilde
C_0,\dots ,\tilde H_0)$ in terms of $(C_0,\dots ,H_0)$, $\varepsilon_c$, $\rho_c$, $T_c$, $\mu_c$ and $a_c$. We have $\tilde {\cal L}_0(0,0,0)=0$, $ {\partial \tilde {\cal
L}_0}(0,0,0)/{\partial \tilde\eta}=0$, $ {\partial \tilde {\cal
L}_0}(0,0,0)/{\partial \tilde\rho}= 0 $ and, owing to the particular choice we made by introducing $a_c$ in the change of variables, we have also $$\frac{\partial^2 \tilde {\cal L}_0}{\partial
\tilde\eta\partial \tilde\rho}(0,0,0)=0 \label{partialder}.$$ Consequently, from and , we can write the Taylor expansion of $\tilde{\cal
L}_0$ in the vicinity of point $(0,0,0)$ under the form $$\begin{aligned}
\tilde{\cal L}_0=\tilde \varepsilon -a_{20}\, \tilde \eta^2- a_{02}\,\tilde \rho^2 -
a_{30}\, \tilde \eta^3 - a_{21}\, \tilde \eta^2\tilde \rho - a_{12}\, \tilde \eta\tilde \rho^2
- a_{03}\, \tilde \rho^3 +o(\tilde\tau^3)\end{aligned}$$ where $\tilde \tau$, which stands for $\max(\tilde \eta,\tilde \rho)$, is a measure of the distance to point $(\eta_c,\rho_c)$ in the space $(\eta,\rho)$. Indeed $
\tilde \tau \leq (1+|a_c|)\max(\eta-\eta_c,\rho-\rho_c)$. Accordingly, we obtain: $$\tilde T= 2 a_{20} \tilde \eta +o(\tilde \tau).$$ Recalling that we have assumed that $a_{20}=\partial^2\tilde {\cal
L}_0/\partial \tilde \eta^2\not=0$, we have $\tilde \tau \sim \max(\tilde
T,\tilde \rho)$ and $\tilde \eta= {\tilde T}/{(2 a_{20})}
+o(\tau)$. Hence $$\begin{aligned}
\tilde \mu&= 2 a_{02}\, \tilde \rho + a_{21}\, \tilde \eta^2 + 2 a_{12}\, \tilde \eta\tilde \rho +
3 a_{03}\,\tilde \rho^2 + o(\tilde \tau^2) \\
&= 2 a_{02}\, \tilde \rho + \frac{ a_{21}}{ 4 a_{20}^2}\, \tilde T^2 +
\frac{ a_{12}}{ a_{20}}\,\, \tilde T\tilde \rho+ 3 a_{03}\,\tilde \rho^2 +o(\tilde \tau^2).\end{aligned}$$
Now, we assume that $(\varepsilon_c,\eta_c,\rho_c)$ corresponds to the critical point of ${\cal L}_0$. Equivalently, $(0,0,0)$ is the critical point of $\tilde {\cal L}_0$.The critical conditions state that, at [*fixed*]{} critical temperature $\tilde T=0$, the first and second derivatives of $\tilde \mu$ with respect to $\tilde \rho$ vanish. In view of the previous equation these conditions state that $a_{02}=a_{03}=0$. Let us now go a bit further in the expansions of $\tilde L$, $\tilde T$ and $\tilde \mu$. In the generic case, when the coefficients $ a_{12}$ and $a_{04}$ like $a_{20}$ do not vanish, we get $$\begin{aligned}
&\tilde{\cal L}_0=\tilde \varepsilon - a_{20} \tilde \eta^2 - a_{12} \tilde \eta\tilde \rho^2 - a_{04} \tilde \rho^4 +o( \tilde \xi^2), \\
&\tilde T= 2 a_{20} \tilde \eta + a_{12} \tilde \rho^2 +o( \tilde \xi),\\
&\tilde \mu= 2 a_{12} \tilde \eta\tilde \rho+ 4 a_{04} \tilde \rho^3+\tilde \rho\ o( \tilde \xi),\end{aligned}$$ where $\tilde\xi$ stands for $\max\, (\tilde \eta, \tilde
\rho^2)$. Furthermore, we can use a mass unit such that $a_{04}=1$ and an entropy unit such that $a_{12}=1$. We denote $\tilde\chi$ the value of $a_{20}$ in such an unit system. We finally get $$\begin{aligned}
&\tilde{\cal L}_0=\tilde \varepsilon - \tilde\chi \tilde \eta^2 -
\tilde \eta\tilde \rho^2 - \tilde \rho^4 +o(\tilde\xi^2) \nonumber\\
&\tilde T= 2 \tilde\chi \tilde \eta + \tilde \rho^2 +o(\tilde\xi)\\
&\tilde \mu= 2 \tilde \eta\tilde \rho+ 4 \tilde \rho^3+\tilde\rho \ o(\tilde\xi)\nonumber\end{aligned}$$ These equations are the generic asymptotic form of the thermodynamic potentials near a critical point in an adapted system of coordinates. Note that $\tilde \chi$ has to satisfy $4\tilde\chi-1 > 0$ in order to ensure the positivity of $\tilde\chi \tilde \eta^2 + \tilde \eta\tilde \rho^2 + \tilde
\rho^4$. Otherwise no homogeneous phase could be stable in the studied zone.
From now on, we study the equilibrium of two phases by assuming that $$\begin{aligned}
&\tilde{\cal L}_0=\tilde \varepsilon - \tilde\chi \tilde \eta^2 - \tilde \eta\tilde \rho^2 - \tilde \rho^4 \label{Lzero}\end{aligned}$$ and consequently $$\begin{aligned}
&\tilde T= 2 \tilde\chi \tilde \eta + \tilde \rho^2, \label{temperature}\\
&\tilde \mu= 2 \tilde \eta\tilde \rho+ 4 \tilde \rho^3.\label{chempote}\end{aligned}$$ Relations and are the associated temperature and chemical potential. Function $$\tilde \varepsilon_0(\tilde \eta, \tilde \rho)=
\tilde\chi \tilde \eta^2 + \tilde \eta\tilde \rho^2 + \tilde
\rho^4 \label{energyinterne}$$ is represented in Fig. \[fig1\] where one can check that the critical point lies on the boundary of the domain where $\tilde\varepsilon $ does not coincide with its lower convex envelope.
![Internal energy density $\tilde \varepsilon_0(\tilde \eta,\tilde \rho)$ of a homogeneous fluid near critical point corresponding to $(\tilde\eta_c,\tilde\rho_c,\tilde\varepsilon_c)=(0,0,0)$ when we chose $\tilde\chi=0.35$. []{data-label="fig1"}](1.eps){width="10cm"}
Integration of equations in planar interfaces
=============================================
We consider a planar interface and assume that all fields depend only on transverse space-variable $z$. We denote $\varphi'$ the derivative of any field $\varphi$ with respect to $z$.
System of equilibrium equations
-------------------------------
[[System of equilibrium equations (\[€qphases\]) completed by the state law reads in term of new $ \widetilde{\ }\,$- equivalent quantities, $$\left\{
\begin{array}{l}
\tilde C_0\, (\tilde\Lambda \tilde \rho')' + \tilde D_0\,(\tilde\Lambda \tilde
\eta')'+\tilde F_0\, (\tilde\Lambda \tilde \varepsilon')' =\tilde\Lambda \tilde \mu- \tilde \mu_{_0}, \\
\tilde D_0\,(\tilde\Lambda\tilde \rho')' + \tilde E_0\,(\tilde\Lambda \tilde
\eta')' +\tilde G_0\, (\tilde\Lambda \tilde \varepsilon')'=\tilde\Lambda
\tilde T- \tilde T_0, \\
\tilde F_0\, (\tilde\Lambda\tilde \rho')' + \tilde G_0\,(\tilde\Lambda\tilde
\eta')' +\tilde H_0\, (\tilde\Lambda \tilde \varepsilon')'= 1- \tilde
\Lambda, \\
\tilde{\cal L}_0(\tilde \varepsilon,\tilde \eta,\tilde \rho) - \tilde Q( \tilde \rho',\tilde \eta', \tilde \varepsilon') = 0, \label{systemA}
\end{array}\right.$$]{}]{} where $\tilde Q( \tilde \rho',\tilde \eta', \tilde \varepsilon'):= \frac{1}{2}
\big(\tilde C_0\, \tilde \rho'^2 + \tilde E_0\,\tilde \eta'^2
+\tilde H_0\, \tilde \varepsilon'^2 +
2 \tilde D_0\,\tilde
\eta'\tilde \rho'
+ 2 \tilde F_0\, \tilde \rho'\tilde \varepsilon'+2 \tilde G_0\, \tilde \varepsilon'\tilde \eta'\big) $.
Multiplying the three first equations respectively by $\tilde \rho'$, $\tilde \eta'$, $\tilde \varepsilon'$, summing and using the fourth equation derived with respect to $z$, leads to $$2 \Big(\tilde\Lambda\, \tilde Q( \tilde \rho',\tilde \eta', \tilde \varepsilon')\Big)'=\tilde \varepsilon'-\tilde \mu_{_0}\tilde \rho'- \tilde T_0 \tilde
\eta' ,$$ which gives the first energy integral $$2 \tilde\Lambda\, \tilde Q( \tilde \rho',\tilde \eta', \tilde \varepsilon')=\tilde \varepsilon-\tilde \mu_{_0}\tilde \rho- \tilde T_0 \tilde
\eta +\tilde P_0,$$ or equivalently, by using , $$(2 \tilde\Lambda-1)\, \tilde \varepsilon=2\tilde\Lambda\, \tilde \varepsilon_0-\tilde \mu_{_0}\tilde \rho- \tilde T_0 \tilde
\eta +\tilde P_0 , \label{IP2}$$ where the constant $\tilde P_0$ has the dimension of a pressure.
In the bulk the fields become constant and the equilibrium equations lead to $$\tilde\Lambda\tilde \mu- \tilde \mu_{_0}=0,\quad
\tilde\Lambda \tilde T-\tilde T_0 =0 ,\quad
1- \tilde\Lambda=0, \quad \tilde \varepsilon-\tilde \mu_{_0}\tilde \rho- \tilde T_0 \tilde
\eta +\tilde P_0=0.$$ Hence $\tilde \Lambda=1$ and $\tilde \mu_{_0}$, $\tilde T_0$, $\tilde P_0$ are respectively the common values of the chemical potential, temperature and pressure in both bulk phases and we recover the usual global equilibrium conditions for planar interfaces.
We denote by superscripts ${}^+$ and ${}^-$ the values of the fields in the two bulk phases. From , , we deduce the equalities of thermodynamical quantities $\tilde\mu_0,
\tilde T_0, \tilde P_0$ in the two bulks phases $$\begin{aligned}
2 \tilde \eta^+\tilde \rho^++ 4 (\tilde \rho^+)^3&= 2 \tilde \eta^-\tilde \rho+
4 (\tilde \rho^-)^3=\tilde\mu_0\label{eqmu}\\
2 \tilde\chi \tilde \eta^+ + (\tilde \rho^+)^2&=2 \tilde\chi \tilde
\eta^- + (\tilde \rho^-)^2=\tilde T_0\label{eqT}\\
\tilde\chi (\tilde \eta^+)^2+2 \tilde \eta^+ (\tilde \rho^+)^2
+ 3 (\tilde \rho^+)^4&=\tilde\chi (\tilde \eta^-)^2+2 \tilde \eta^-
(\tilde \rho^-)^2 + 3 (\tilde \rho^-)^4=\tilde P_0. \label{eqP}\end{aligned}$$ Using (\[eqT\]), equations (\[eqmu\]) and (\[eqP\]) can be written $$\begin{aligned}
\tilde T_0 \tilde \rho^+ + (4\tilde\chi-1)(\tilde \rho^+)^3&=
\tilde T_0 \tilde \rho^- + (4\tilde\chi-1)(\tilde \rho^-)^3=\tilde\chi \tilde
\mu_0 ,\\
2\tilde T_0 (\tilde \rho^+)^2 + 3(4\tilde\chi-1) (\tilde \rho^+)^4&= 2\tilde T_0 (\tilde \rho^-)^2 + 3(4\tilde\chi-1) (\tilde \rho^-)^4=4\tilde\chi\tilde P_0- \tilde T_0^2 ,\end{aligned}$$ which implies $$\begin{aligned}
&\Big(\tilde T_0 + (4\tilde\chi-1)\big((\tilde \rho^+)^2+\tilde \rho^+ \tilde \rho^-+ (\tilde \rho^-)^2\big)
\Big) \big(\tilde \rho^+- \tilde \rho^-\big)= 0,\\
&\Big(2\tilde T_0 + 3(4\tilde\chi-1) \big((\tilde \rho^+)^2+ (\tilde \rho^-)^2\big)\Big)
\big((\tilde \rho^+)^2- (\tilde \rho^-)^2\big)= 0.\end{aligned}$$ Considering an interface between two distinct phases, we have $\tilde
\rho^+\not= \tilde \rho^-$, thus $$\begin{aligned}
&\tilde T_0 + (4\tilde\chi-1)\big((\tilde \rho^+)^2+\tilde \rho^+ \tilde \rho^-+ (\tilde \rho^-)^2\big) = 0,\\
&\Big(2\tilde T_0 + 3(4\tilde\chi-1) \big((\tilde \rho^+)^2+ (\tilde \rho^-)^2\big)\Big) \big(\tilde \rho^++ \tilde \rho^-\big)= 0.\end{aligned}$$ Subtracting to the second equation the product of the first one by $2 \big(\tilde \rho^++ \tilde \rho^-\big)$ the system becomes $$\begin{aligned}
& (4\tilde\chi-1) \big(\tilde \rho^+- \tilde \rho^-\big)^2 \big(\tilde \rho^+ + \tilde \rho^-\big)= 0 \\
&\tilde T_0 + (4\tilde\chi-1)\big((\tilde \rho^+)^2+\tilde \rho^+ \tilde
\rho^-+ (\tilde \rho^-)^2\big) = 0.\end{aligned}$$ As expected this system admits no solution when $\tilde T_0>0$, or equivalently when the temperature in the phases is greater than the critical one. Let us set $\tilde T_0:=-(4\tilde\chi-1)\, \kappa^2$, i.e. $$\kappa:=\sqrt{\frac {-\tilde T_0}{4\tilde\chi-1}}. \label{estimation0}$$ The small quantity $\kappa$ measures the distance from the critical point. Using again $\tilde \rho^+\not= \tilde \rho^-$ we find $$\tilde \rho^+=\kappa\quad {\rm and}\quad \tilde \rho^-=-\kappa, \label{estimation1}$$ from which we directly deduce, $$\tilde \eta^+= \tilde \eta^-=-2\,\kappa^2 ,\quad \tilde
\mu_0=0,\quad \tilde \varepsilon^+= \tilde \varepsilon^-=\tilde P_0=(4\chi-1)\kappa^4 .
\label{estimation2}$$
The rescaling process
---------------------
[In view of Eqs. , , the values of $\tilde \rho$ and $\tilde \eta$ in the phases lead to the natural rescaling $$\check \rho:= \kappa^{-1} \tilde\rho,\quad \check \eta :=
\kappa^{-2} \tilde\eta,\quad \check \varepsilon:= \kappa^{-4} \tilde\varepsilon, \quad\check z:= \kappa z\label{rescal1}$$ and system becomes $$\left\{
\begin{array}{l}
\tilde C_0\,(\tilde \Lambda \check \rho')' + \tilde D_0\,\kappa\, (\tilde \Lambda\,\check \eta')'+\tilde F_0\, \kappa^3\,(\tilde \Lambda\check \varepsilon')' =\tilde \Lambda \big(2 \check \eta\check \rho+ 4 \check \rho^3\big), \\
\tilde D_0\,\kappa\, (\tilde \Lambda\,\check \rho')' + \tilde E_0\,\kappa^2\,(\tilde \Lambda\check
\eta')' +\tilde G_0\, \kappa^4\,(\tilde \Lambda\check \varepsilon')'=
\tilde \Lambda \big( 2 \tilde\chi \check \eta+ \check \rho^2 \big)+ (4\tilde\chi-1), \\
\tilde F_0\, \kappa^3\, (\tilde \Lambda\check \rho')' + \tilde G_0\,\kappa^4\, (\tilde \Lambda\check
\eta')' +\tilde H_0\, \kappa^6\, (\tilde \Lambda\check\varepsilon')'= 1- \tilde
\Lambda , \label{systemC}
\end{array}\right.$$ where the space derivatives are now relative to $\check z$. Hence $\tilde\Lambda=1+O(\kappa^3)$ and at the first order with respect to the small parameter $\kappa$, $$\left\{
\begin{array}{l}
\displaystyle \tilde C_0\, \check \rho''=2 \check \eta\check \rho+ 4 \check \rho^3, \\
\displaystyle 0 = 2 \tilde\chi \check \eta + \check \rho^2
+(4\tilde\chi-1),
\end{array}\right.
\label{doubleetoile}$$ which gives by elimination of $\check \eta$, $$\begin{aligned}
&\frac{\tilde\chi\tilde C_0 }{(4\tilde\chi-1)}\, \check \rho''=\check \rho^3 - \check \rho.\label{eqdif}\end{aligned}$$ Multiplying by $\check \rho'$, integrating and taking into account , we get $$\begin{aligned}
&\frac{\tilde\chi\tilde C_0 }{(4\tilde\chi-1)}\,\frac { \check \rho'^2} 2=\frac 1 4 \left(\check \rho^2 - 1 \right)^2.\label{IPrho}\end{aligned}$$ Hence the mass density profile $\check \rho_{eq}$ at equilibrium across an interface has the classical representation (cf. [@Rowlinson] p. 251)]{} $$\check \rho_{eq} (\check z)=\tanh(\frac {\check z}{\ell})
\label{densityprofile}$$ where $$\ell=\sqrt{\frac{2 \tilde\chi\check C_0
}{(4\tilde\chi-1)}}.\label{ell}$$ Note that this well known profile is an exact solution of but results from several approximations. It is valid only for a planar interface lying far from the boundaries of the domain. Moreover considering the polynomial form for the energy is clearly an approximation as well as neglecting the terms of lower order in , and .
Using and we obtain that the temperature through the interface is constant at the first order: $$\check T_{eq}= 2 \tilde\chi \check \eta_{eq} + (\check
\rho_{eq})^2 =-(4\tilde\chi-1) .$$
![Classical density profile of the normalized density ($\check \rho \in ]-1,+1[$) associated with and ; the $ x$-axis unit is $\ell$.[]{data-label="fig2"}](2.eps){width="8cm"}
However the second equation of system gives a more accurate information about the temperature profile through the interface; indeed, at order $\kappa$, $$\tilde D_0\, \kappa\, \check \rho'' = 2 \tilde\chi \check \eta + \check \rho^2
+(4\tilde\chi-1)+ O(\kappa^2). \label{system1}$$ That is $$\begin{aligned}
\check T_{eq}&=-(4\tilde\chi-1)+ \kappa\, \tilde D_0\, {\check\rho_{eq}''}+O(\kappa^2)\\
&=-(4\tilde\chi-1)+\kappa\, \frac {(4\tilde\chi-1)}{\tilde\chi} \, \frac {\tilde D_0} {\tilde C_0}
\,\left[ {\check\rho_{eq}^3}- {\check\rho_{eq}}\right]+O(\kappa^2).\end{aligned}$$ Consequently, $$\check T_{eq} = (4\tilde\chi-1)\Bigg( -1+
\frac{\kappa}{\tilde\chi} \, \frac {\tilde D_0}{\tilde C_0} \left( {\tanh^3\left(\frac {\check z}{\ell}\right)} -
{\tanh\left(\frac {\check z}{\ell}\right)}\right)\Bigg)+O(\kappa^2).
\label{temprofile}$$
![Variation of normalized temperature $\check T_{eq}+(4\chi-1)$ through the interface near the critical point. The $x$-axis unit is $\ell$ and the $y$-axis unit is $\displaystyle \kappa\,
\frac{(4\tilde\chi-1)}{\tilde\chi} \, \frac {\tilde D_0}{\tilde
C_0}$. \[fig3\]](3.eps){width="9cm"}
Note that in Eq. the variation of the temperature across the interface is no more monotonic (see Fig. \[fig3\]). Moreover, the variation of temperature $\check T_{eq}$ is multiplied by the small parameter $\kappa$ and is negligible with respect to the variation of $\check \rho_{eq}$.
Surface tension
===============
Surface tension $\sigma $ of a plane liquid-vapor interface corresponds to the excess of free energy $ \tilde e:= \tilde \varepsilon- \tilde T \tilde \eta$ inside the interface. Using and , we have $$\tilde e=\frac {4\tilde \chi-1} {4\tilde \chi} \big( \tilde \rho^4-2\tilde\kappa^2 \tilde \rho^2-\kappa^4 (4\tilde \chi-1) \big) +\tilde Q( \tilde \rho',\tilde \eta', \tilde \varepsilon')$$ As, in the bulk, we have $$\tilde e^+= \tilde e^-=-(4 \tilde \chi-1)\kappa^4,$$ surface tension is $$\begin{aligned}
\tilde\sigma &:=\int_{-\infty}^{+\infty} \left( \tilde e+ (4 \tilde \chi-1)\kappa^4\right) dz
=\int_{-\infty}^{+\infty} \left(\frac {4\tilde \chi-1} {4\tilde \chi}( \tilde \rho^2-\tilde\kappa^2)^2 +\tilde Q( \tilde \rho',\tilde \eta', \tilde \varepsilon')\right) dz\nonumber \\
&=\kappa^3 \int_{-\infty}^{+\infty} \left(\frac {4\tilde \chi-1} {4\tilde \chi}( \check \rho^2-1)^2 +\tilde Q(\check \rho',\kappa \check \eta',\kappa^3 \check \varepsilon')\right) d\check z .\end{aligned}$$ At the first order with respect to $\kappa$, we obtain $$\begin{aligned}
\tilde\sigma &=\kappa^3 \int_{-\infty}^{+\infty} \left(\frac {4\tilde \chi-1} {4\tilde \chi}( \check \rho^2-1)^2 +\frac 1 2 \tilde C_0\, \check \rho'^2\right) d\check z +O(\kappa^4) \label{34} \\
&= \kappa^3 \int_{-1}^{+1} \left(\sqrt{
\frac {(4\tilde \chi-1) \tilde C_0} {2\tilde \chi}}\, (1- \check \rho^2)
\right) d\check \rho +O(\kappa^4)\nonumber \\
&= \kappa^3 \, \frac 4 3 \, \sqrt{
\frac {(4\tilde \chi-1) \tilde C_0} {2\tilde \chi}}\, +O(\kappa^4)
\end{aligned}$$ Thus, at the leading order, equilibrium values and surface tension are those given by the Cahn-Hilliard theory : the effect of the gradients of entropy and energy densities are negligible. A more accurate description could be obtained : terms of order $\kappa^4$ would come from *(i)* the perturbation of system by taking into account the coupling term $\tilde D_0$ and *(ii)* the introduction of the same coupling term in .\
Conclusion
==========
We have obtained the mass density and temperature profiles through an interface near the critical point. Our results present some similarities with the ones obtained in [@Ruggeri] for fluid mixtures where two mass densities have the role played here by mass and entropy densities. The differences lie in the fact that we are not here impelled to deal with combinations of densities and also in the fact that the notion of critical point is more complex in the case of a mixture where non-monotonic profiles can be obtained at the leading order.
We have introduced a state law in which all gradients are considered with respect to mass, entropy and *energy* densities. At our knowledge, it is the first time that this though natural assumption is used. In this framework, we confirm the conjecture made by Rowlinson and Widom [@Rowlinson] that, near the critical point, the variations of temperature inside the interfacial layer are negligible. This result is mainly due to the fact that the variations of entropy density are negligible with respect to the variations of mass density.\
**Data accessibility statement**. This work does not have any experimental data.\
**Competing interests statement**. We have no competing interest.\
**Author’s contribution**. H.G. and P.S. conceived the mathematical model, interpreted the results, and wrote together the paper.\
**Acknowledgements**. P.S. thanks the Laboratoire de Mécanique et d’Acoustique (Marseille) for its hospitality.\
**Funding statement.** This work was supported by C.N.R.S.\
**Ethics statement**. This work is original, has not previously been published in any Journal and is not currently under consideration for publication elsewhere.\
[99]{}
B. Widom, What do we know that van der Waals did not know?, Physica A, 263 (1999) 500–515.
H. Gouin, Energy of interaction between solid surfaces and liquids, The Journal of Physical Chemistry B, 102 (1998) 1212–1218 & arXiv:0801.4481.
J.D. van der Waals, The thermodynamic theory of capillarity under the hypothesis of continuous variation of density, translation by J.S. Rowlinson, Journal of Statistical Physics, 20 (1979) 200–244.
J. Korteweg, Sur la forme que prennent les équations du mouvement des fluides si l’on tient compte des forces capillaires, [Archives Néerlandaises]{}, [2, n]{}$^{o}\ $[6]{} [(1901)]{} [1–24]{}.
.
J.W. Cahn, J.E. Hilliard, Free energy of a nonuniform system, III. Nucleation in a two-component incompressible fluid, Journal of Chemical Physics, 31 (1959) 688–699.
P. Casal, La capillarité interne, Cahier du Groupe Fran[ç]{}ais de Rhéologie. CNRS VI (1961) 31-–37.
P. Casal, H. Gouin, Connexion between the energy equation and the motion equation in Korteweg’s theory of capillarity, Comptes Rendus de l’Académie des Sciences de Paris II, 300 (1985) 231–-233.
F. dell’Isola, H. Gouin, P. Seppecher, Radius and surface tension of microscopic bubbles by second gradient theory, Comptes Rendus de l’Académie des Sciences de Paris IIb, 320 (1995) 211–216.
F. dell’Isola, H. Gouin, G. Rotoli, Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations, European Journal of Mechanics, B/Fluids, 15 (1996) 545–568.
P. Casal, La théorie du second gradient et la capillarité, Comptes Rendus de l’Académie des Sciences de Paris A 274, (1972) 1571–-1574.
P. Seppecher, Equilibrium of a Cahn and Hilliard fluid on a wall: influence of the wetting properties of the fluid upon stability of a thin liquid film, European Journal of Mechanics B/fluids, 12 (1993) 61–84.
P. Seppecher, Moving contact line in the Cahn-Hilliard theory, International Journal of Engineering Science, 34 (1995) 977–992.
H. Gouin, W. Kosiński, Boundary conditions for a capillary fluid in contact with a wall, Archives of Mechanics, 50 (1998) 907–916 & arXiv:0802.1995
H. Gouin, Liquid Nanofilms. A mechanical model for the disjoining pressure, International Journal of Engineering Science, 47 (2009) 691–699 & arXiv:0904.1809
H. Gouin, Statics and dynamics of fluids in nanotubes. Note di Matematica, 32 (2012) 105–124 & arXiv:1311.2303.
M. Gărăjeu, H. Gouin, G. Saccomandi, Scaling Navier-Stokes equation in nanotubes, Physics of fluids, 25 (2013) 082003 & arXiv:1311.2484
H. Gouin, S. Gavrilyuk, Dynamics of liquid nanofilms, International Journal of Engineering Science, 46 (2008) 1195–1202 & arXiv:0809.3489
H. Gouin, Solid-liquid interaction at nanoscale and its application in vegetal biology, Colloids and Surfaces A, 383 (2011) 17–22 & arXiv:1106.1275
H. Gouin, The watering of tall trees - Embolization and recovery, Journal of Theoretical Biology, 369 (2015) 42–50 & arXiv:1404.4343
P. Germain, The method of the virtual power in continuum mechanics - Part 2: microstructure, SIAM Journal of Applied Mathematics, 25 (1973) 556–575.
F. Dell’Isola, P. Seppecher, A. Della Corte, The postulations à la d’Alembert and à la Cauchy for higher gradient continuum theories are equivalent: a review of existing results, Proceedings of the Royal Society A, 471 (2015) 20150415.
H. Gouin, T. Ruggeri, Mixtures of fluids involving entropy gradients and acceleration waves in interfacial layers, European Journal of Mechanics B/Fluids, 24 (2005) 596-613 & http://arXiv:0801.2096
H. Gouin, A. Muracchini, T. Ruggeri, Travelling waves near a critical point of a binary fluid mixture, International Journal of Non-Linear Mechanics, 47 (2012) 77–84 & http://arXiv:1110.5137
F. dell’Isola, K. Hutter, Continuum mechanical modelling of the dissipative processes in the sediment-water layer below glaciers, Comptes Rendus de l’Académie des Sciences de Paris IIb, 325 (1997) 449–456.
V.A. Eremeyev, F.D. Fischer, On the phase transitions in deformable solids, Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM), 90 (2010) 535–536.
F. dell’Isola, P. Seppecher, A. Madeo, How contact interactions may depend on the shape of Cauchy cuts in N-th gradient continua: approach “à la D’Alembert”, Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 63 (2012) 1119–1141.
A. Bertram, S. Forest, The thermodynamics of gradient elastoplasticity, Continuum Mechanics and Thermodynamics, 26 (2014) 269–286.
Ornstein L.S., Statistical theory of capillarity, in: KNAW, Proceedings, 11, Amsterdam, (1909) 526-542.
H.C. Hamaker, The London-van der Waals attraction between spherical particles, Physica, 4 (1937) 1058–1072.
R. Evans, The nature of liquid-vapour interface and other topics in the statistical mechanics of non-uniform classical fluids, Advances in Physics, 28 (1979) 143–200.
Clarendon Press, Oxford and Google books, 2012.
P. Casal, H. Gouin, Equation of motion of thermocapillary fluids, Comptes Rendus de l’Académie des Sciences de Paris II, 306 (1988) 99–-104.
, Properties of thermocapillary fluids and symmetrization of motion equations, International Journal of Non-Linear Mechanics, 85 (2016) 152–160.
S. Forest, M. Amestoy, Hypertemperature in thermoelastic solids, Comptes Rendus Mécanique, 336 (2008) 347–353.
S. Forest, E. C. Aifantis, Some links between recent gradient thermo-elasto-plasticity theories and the thermomechanics of generalized continua, International Journal of Solids and Structures, 47 (2010) 3367–3376.
M.H. Maitournam, Entropy and temperature gradients thermomechanics: Dissipation, heat conduction inequality and heat equation, Comptes Rendus Mécanique, 340 (2012) 434–443.
C. Domb, The critical point, Taylor and Francis, London, 1996.
[^1]: Aix-Marseille Univ, Centrale Marseille, CNRS, M2P2 UMR 7340, 13451 Marseille France\
E-mail: henri.gouin@univ-amu.fr, henri.gouin@yahoo.fr
[^2]: IMATH, Institut de Mathématiques, Université de Toulon, BP 132, 83957 La Garde, Cedex, France\
E-mail: seppecher@imath.fr
[^3]: LMA, CNRS, UPR 7051, Aix-Marseille Univ, Centrale Marseille, 13402 Marseille, France\
E-mail: seppecher@lma.cnrs-mrs.fr
|
---
abstract: |
Let $D$ be an integral domain, $S(D)=I(D)$ $(I_{t}(D))$ the set of proper nonzero ideals (proper $t$-ideals) of $D,$ $Max(D)$ $(t$-$Max(D)$ the set of maximal ($t$-) ideals of $D,$ and let $P$ be a predicate on $S(D)$ with nonempty truth set $\Pi _{S(D)}\subseteq S(D)$, where $P$ can be: “—is invertible” or “—is divisorial” etc.$.$ We say $S(D)$ meets $P$ $%
(S(D)\vartriangleleft P)$ if $\forall s\in S(D)\exists \pi \in \Pi _{S(D)}(P)
$ $(s\subseteq $ $\pi )$. Clearly $S(D)\vartriangleleft P\Leftrightarrow
Max(D)$ ($t$-$Max(D))\subseteq $ $\Pi _{S(D)}(P)$. We show that if $S(D)$ $%
\vartriangleleft P,$ we have no control over $\dim D$. We also show that $%
I(D)$ $\vartriangleleft P$ does not imply $I(R)$ $\vartriangleleft P,$ while $I_{t}(D)$ $\vartriangleleft P$ implies $I_{t}(R)$ $\vartriangleleft P,$ for most choices of $P,$ when $R=D[X]$ and have examples to show that generally $%
S(D)\vartriangleleft P$ does not extend to rings of fractions. We study restrictions that may control the dimension of $D$ when $S(D)%
\vartriangleleft P.$ We also say $S(D)\vartriangleleft P$ with a twist $%
(S(D)\vartriangleleft ^{t}P)$ if $\forall s\in S(D)$ $\exists \pi \in $ $\Pi
_{D}(P)(s^{n}$ $\subseteq \pi $ for some $n\in N)$ and study $%
S(D)\vartriangleleft ^{t}P,$ along the same lines as $S(D)\vartriangleleft P$ and provide examples.
address: 'Department of Mathematics, Idaho State University, Pocatello, 83209 ID'
author:
- Muhammad Zafrullah
title: Domains whose ideals meet a universal restriction
---
Introduction\[S1\]
==================
Let $D$ be an integral domain with quotient field $K\neq D$ and let $F(D)$ be the set of nonzero fractional ideals of $D.$ For $I\in F(D),$ the set $%
I^{-1}=\{x\in K|xI\subseteq D\}$ is again a fractional ideal and thus the relation $v$: $I\mapsto I_{v}$ is a function on $F(D).$ This function is called the $v$-operation on $D.$ Similarly the relation $t$: $I\mapsto
I_{t}=\cup \{F_{v}|$ $0\neq F$ is a finitely generated subideal of $I\}$ is a function on $F(D)$ and is called the $t$-operation on $D.$ These and the operation $d$: $I\mapsto I$ are examples of the so called star operations. The reader may consult sections 32 and 34 of [@Gil; @1972] or the first chapter of [@Jess; @2019] for these operations. However, for the purposes of this introduction, we note that $I\in F(D)$ is a $v$-ideal ($t$-ideal) if $I=I_{v}$ (resp. $I=I_{t})$ and if $I$ is finitely generated, $I_{v}=I_{t}.$ The rather peculiar definition of the $t$-operation allows one to use Zorn’s Lemma to prove that each integral domain that is not a field has at least one integral $t$-ideal maximal among integral $t$-ideals, that this maximal $%
t$-ideal is prime and that every proper, integral $t$-ideal is contained in at least one maximal $t$-ideal. The set of all maximal $t$-ideals of a domain $D$ is denoted by $t$-$Max(D).$ It can be shown that $D=\cap _{M\in t%
\text{-}Max(D)}D_{M}.$ While we are at it let’s also denote by $I(D)$ the set of all nonzero proper integral ideals of $D$ and by $I_{t}(D)$ the set of all proper $t$-ideals of $D.$
Now let $S(D)$ represent $I(D)$ ( or $I_{t}(D)$). Let $P$ be a predicate that defines a non-empty truth set $\Pi _{S(D)}(P)$ $\subseteq S(D),$ where $%
P$ can be: “—is invertible” or “—is divisorial”, —is finitely generated" etc.. We say $S(D),$ for a given value or both values meets $P$ $(S(D)\vartriangleleft P)$ if $\forall s\in S(D)\exists \pi \in
\Pi _{S(D)}(P)$ $(s\subseteq $ $\pi )$.
From an abstract point of view we are actually dealing with a non-empty poset $(A,\leq )$ such that every member of $A$ precedes at least one maximal element of $A$. Suppose further that we designate a non-empty subset $\Pi $ of $A$ by some rule. Then every maximal member of $A$ is in $\Pi $ if and only if every member of $A$ precedes some member of $\Pi .$ Thus $%
S(D)\vartriangleleft P\Leftrightarrow Max(D)$ $(t$-$Max(D))\subseteq $ $\Pi
_{S(D)}(P)$. That is easy enough, but the trouble starts when we ask questions like: Suppose for example $I(D)\vartriangleleft P$ and suppose $R$ is an extension of $D$ must $I(R)\vartriangleleft P?$ (Same question for $%
S(D)=I_{t}(D).)$ On the other hand we get the following benefit from carrying out this study: Take a property $P~$say $"$ finitely generated“, that characterizes commutative Noetherian rings. Then $I(D)\vartriangleleft
P $ gives us a, ring each of whose maximal ideal is finitely generated. It turns out that his ring is non-Noetherian unless it is of dimension one. We shall however restrict our attention to integral domains and note that $D$ is a Krull domain if and only every $t$-ideal of $D$ is $t$-invertible. If $%
P $ stands for ”is $t$-invertible" then, as we shall see, $%
I_{t}(D)\vartriangleleft P$ is a domain charaterized by the property that every maximal $t$-ideal of $D$ is $t$-invertible. Now you can set $P$ as every nonzero ideal of $D$ is invertible and check for yourself that $%
I(D)\vartriangleleft P$ delivers a domain whose maximal ideals are all invertible but such a domain is not Dedekind unless it is of dimension one. In fact for each natural number $n$ we can find an $n$ dimensional domain with each maximal ideal invertible. This fascinating uncontrollability of Krull dimension is shared by most of $I(D)\vartriangleleft P$ and $%
I_{t}(D)\vartriangleleft P$ etc..
We show in section \[Section S2\] that if $X$ is an indeterminate over $L$ a field extension of $K,$ and $R=D+XL[X],$ $S(D)$ $\vartriangleleft P$ if and only if $S(R)\vartriangleleft P$ for a $P$ that holds (returns the truth value $T)$ for principal ideals as well. Since $\dim R=\dim D+1,$ [@CMZ; @1986 Corollary 1.4], this shows that if $S(D)$ $\vartriangleleft P$ and $P$ returns the truth value $T$ for each principal ideal, then one can expect no restriction on the Krull dimension of $D$. Next we show, in section 2, that if $R=D[X]$ and $I(D)$ $\vartriangleleft P,$ then $I(R)$ $%
\ntriangleleft P$ in cases that we have considered, yet if $I_{t}(D)$ $%
\vartriangleleft P$, then $I_{t}(R)$ $\vartriangleleft P$ almost always. We give examples to show that generally $S(D)\vartriangleleft P$ does not extend to rings of fractions. We study restrictions, such as requiring the domain to be completely integrally closed or to be Noetherian etc., that control the dimension of $D$ when $S(D)\vartriangleleft P,$ in some cases. In section \[Section S3\] we study $S(D)\vartriangleleft P$ with a twist $%
(S(D)\vartriangleleft ^{t}P)$ if $\forall s\in S(D)$ $\exists \pi \in $ $\Pi
_{D}(P)(s^{n}$ $\subseteq \pi $ for some $n\in N)$ and study $%
S(D)\vartriangleleft ^{t}P$ along the same lines as $S(D)\vartriangleleft P$, providing necessary examples. (Here $N$ denotes the set of natural numbers.)
Effects of a Universal Restriction on $S(D)$ \[Section S2\]
===========================================================
Let us start with an introduction to general star operations so that we can reap full benefits from our toils. A star operation $\ast $ on $D$ is a function on $F(D)$ that satisfies the following properties for every $I,J\in
F(D)$ and $0\neq x\in K$:
\(i) $(x)^{\ast} = (x)$ and $(xI)^{\ast} = xI^{\ast}$,
\(ii) $I \subseteq I^{\ast}$, and $I^{\ast} \subseteq J^{\ast}$ whenever $I
\subseteq J$, and
\(iii) $(I^{\ast })^{\ast }=I^{\ast }$.
Now, an ideal $I\in F(D)$ is a $\ast $-ideal if $%
I^{\ast }=I,$ so a principal ideal is a $\ast $-ideal for every star operation $\ast .$ Moreover $I\in F(D)$ is called a $\ast $-ideal of finite type if $I=J^{\ast }$ for some $J\in f(D)$. It can be shown that (a) for every star operation $\ast $ and $I,J\in F(D),~(IJ)^{\ast }=(IJ^{\ast
})^{\ast }=(I^{\ast }J^{\ast })^{\ast },$ (the $\ast $-multiplication), (b) $%
(I+J)^{\ast }=(I+J^{\ast })^{\ast }=(I^{\ast }+J^{\ast })^{\ast }$ (the $%
\ast $-sum) and (c) $(I^{\ast }\cap J^{\ast })^{\ast }=I^{\ast }\cap J^{\ast
}$($\ast $-intersection).
To each star operation $\ast $ we can associate a star operation $\ast _{s}$ defined by $I^{\ast _{s}}=\bigcup \{\,J^{\ast }\mid J\subseteq I$ and $J\in
f(D)\,\}.$ A star operation $\ast $ is said to be of finite type, or of finite character, if $I^{\ast }=I^{\ast _{s}}$ for all $I\in F(D).$ Indeed for each star operation $\ast ,$ $\ast _{s}$ is of finite character. Thus if $\ast $ is of finite character $I\in F(D)$ is a $\ast $-ideal if and only if for each finitely generated subideal $J$ of $I$ we have $J^{\ast }\subseteq
I.$ Also it is easy to see that $I_{t}=\bigcup \{\,J_{v}\mid J\subseteq I$ and $J\in f(D)\,\}=I_{v_{s}}$ and so the $t$-operation is an example of a star operation of finite character. Star operations of finite character, especially the $t$-operation, will figure prominently in our discussions. A fractional ideal $I$ is called $\ast $-invertible if $(II^{-1})^{\ast }=D.$ It is well known that if $I$ is $\ast $-invertible for a finite character star operation $\ast $ then $I^{\ast }$ and $I^{-1}$ are of finite type and that every $\ast $-invertible $\ast $-ideal is divisorial [@Zaf; @2000]. If $\ast $ is a star operation of finite character then just like the $t$-operation, every nonzero proper integral $\ast $-ideal is contained in a maximal integral $\ast $-ideal that is prime and just like the $t$-ideals $%
D=\cap D_{M}$ where $M$ varies over the maximal $\ast $-ideals of $D.$ We shall be mostly concerned with the two values of $S(D)$ but will use occasionally $I_{\ast }(D)$ the set of proper, integral, $\ast $-ideals when we want to go general and not lose sight of the two values of $S(D).$ (Since $I_{\ast }(D)=I(D)$ $(I_{t}(D))$ for $\ast =d$ (resp., $\ast =t$)). Let’s note that while $I_{\ast }(D)\cup \{D\}$ is a monoid under the usual $\ast $-multiplication of $\ast $-ideals with multiplicative identity $D.$From the poset angle $(I_{\ast }(D)\cup \{D\},+^{\ast },\times ^{\ast }$ $\leq ),$ with $A\leq B$ $\Leftrightarrow A\supseteq B,$ is a p.o. monoid and a lattice where $A+^{\ast }B=(A,B)^{\ast }=\inf (A,B)=A\wedge B$ and $\sup
(A,B)=A\cap B.$ The idea of using a universal restriction via a predicate germinated in [@DZ; @2010] where we studied the set $I_{\ast }^{f}(D)$ of proper $\ast $-ideals of finite type with a preassigned non-empty subset $%
\Gamma $ of $I_{\ast }^{f}(D),$ requiring that every pair of members with $%
A+^{\ast }B\in $ $I_{\ast }^{f}(D),$ $A,B$ be contained in some member of $%
\Gamma .$ (This is equivalent to saying that every proper ideal in $I_{\ast
}^{f}(D)$ is contained in a member of $\Gamma ,$ hence the current approach.) As these studies appeal mostly to partial order, they stand to have applications in other areas, as well.
We start with a simple example to set the scene. Let’s consider, for a star operation $\ast $ of finite character, $I_{\ast }(D)$ and define $\Pi
_{I_{\ast }(D)}(P)$ with $P=$ “—is principal” and and suppose that $%
I_{\ast }(D)\vartriangleleft P.$ Then every maximal $\ast $-ideal of $D$ is principal, as we have already observed. But the story doesn’t end here. The event of $I_{\ast }(D)\vartriangleleft P$ imparts some properties to $D,$ such as: the only atoms (irreducible elements) in $D$ are primes and hence generators of maximal $\ast $-ideals. For this let $d$ be an atom and let $%
d|ab$ for some $a,b\in D.$ If $d\nmid a$ and $d\nmid b$, then, $%
D=((d,a)^{\ast }(d,b)^{\ast })^{\ast }=(d^{2},da,db,ab)^{\ast }\subseteq dD$ a contradiction, because $d|ab$. Thus an irreducible element is a prime in $%
D,$ if $I_{\ast }(D)\vartriangleleft P$ for any star operation $\ast $ of finite character. Now for $\ast =d$ the identity operation $%
I(D)\vartriangleleft P$ gives a domain $D$ in which every proper nonzero ideal is contained in a principal ideal, something stronger than what Cohn [@Coh; @1968] called a pre-Bezout domain. In fact $I(D)\vartriangleleft P$ gives a domain something that is even stronger than what was called a special pre-Bezout, or spre-Bezout domain in [@DZ; @2010]. Similarly if $%
I_{t}(D)\vartriangleleft P$, then $D$ is something stronger than a PSP-domain (every primitive polynomial over $D$ is super-primitive), also discussed in [@DZ; @2010]. Recall that a polynomial $f$ is super primitive if $(A_{f})_{v}=D$, where $A_{f}$ is the content, the ideal generated by the coefficients of $f.$ Now it is easy to see that if such a domain is atomic it is at least a UFD (when $I_{t}(D)\vartriangleleft P)$ and at most a PID (When $I(D)\vartriangleleft P).$ (If the last sentence is not clear wait till the paper unfolds itself.) Now, can we find domains that satisfy these properties and yet are not atomic? Yes indeed!
\[Example TX0\] Let $Z,Q$ denote the ring of integers and its quotient field respectively and let $X$ be an indeterminate over $Q,$ then the ring $%
D=Z+XQ[X]$ is such that $I(D)\vartriangleleft P$, where $P=$ “—is principal”.
Illustration: According to [@CMZ; @1978 Theorem 4.21] the nonzero prime ideals of $D$ are of the form $pZ+XQ[X],XQ[X]$ and maximal height one principal primes of the form $f(X)D$ where $f(X)$ is irreducible in $Q[X]$ and $f(0)=1.$Now $XQ[X]$ is not maximal and the rest of them are. So all the maximal ideals are principal and so $I(D)\vartriangleleft P$ with $P$ given above. That $D$ is not atomic can be concluded from the fact that $X$ cannot be expressed as a product of atoms.
Now according to [@CMZ; @1978], $\dim D=2$ and we said that if $%
I(D)\vartriangleleft P,$ then there maybe no restriction on $\dim D.$ The answer to this question is provided in a more general form below.
Let’s first collect some simple results, observations and notation. We say that $P$ returns $T$ on an ideal of $I(D)$ if the truth value of $P$ for that ideal is $T.$ For the sake of easy reference, let’s start with an observation that we have already made.
** **\[Lemma TX1\]Let $(A,\leq )$ be a non-empty poset such that every element of $A$ precedes some maximal element of $A$ and suppose that we can designate a non-empty subset $\Pi $ of $A$ by some rule. Also let $%
Max(A)$ denote the set of all maximal elements of $A.$ Then every member of $%
A$ precedes some member of $\Pi $ if and only if $Max(A)\subseteq \Pi .$ Thus $I(D)\vartriangleleft P$ if and only if $P$ returns $T$ for each member of $Max(D)$ and $I_{t}(D)\vartriangleleft P$ if and only if $P$ returns $T$ for each member of $t$-$Max(D).$
This, somewhat simple observation may, in some instances, have some interesting consequences.
\[Lemma TX2\](1) If a maximal ideal $M$ of $D$ is a $t$-invertible $t$-ideal, then $M$ is invertible. (2) If $P_{1}=$ “—is $t$-invertible” and $%
P_{2}=$ “— is invertible”, then $I(D)\vartriangleleft P_{1}\Leftrightarrow
I(D)\vartriangleleft P_{2}$ and (3) $I(D)\vartriangleleft P$ $\Rightarrow
I_{t}(D)\vartriangleleft P$ for any predicate $P$ whose truth set consists of $t$-ideals.
\(1) Suppose $M$ is a $t$-invertible $t$-ideal then $(MM^{-1})_{t}=D.$ If $%
MM^{-1}\neq D$ then $MM^{-1}$ must be contained in a maximal ideal $N.$ But since $M\subseteq MM^{-1},N=M.$ So $MM^{-1}\subseteq M.$ But as $M$ is also a $t$-ideal, $D=(MM^{-1})_{t}\subseteq M,$ a contradiction.
\(2) By Lemma \[Lemma TX1\], $I(D)\vartriangleleft P_{i}$ $\Leftrightarrow
P_{i}$ returns $T$ for each maximal ideal $M$ and for each $i=1,2.$ So $%
I(D)\vartriangleleft P_{1}\Rightarrow $ every maximal ideal is a $t$-invertible $t$-ideal and by (1) every maximal ideal is invertible. So $%
I(D)\vartriangleleft P_{1}\Rightarrow I(D)\vartriangleleft P_{2}.$ The converse is obvious because every invertible ideal is a $t$-invertible $t$-ideal.
\(3) Suppose that $I(D)\vartriangleleft P$ then, in particular, for every maximal $t$-ideal $M,$ $P$ returns $T$.
\[Proposition UX0\](1) Let, on $I(D),$ $P=$ “— is a principal ideal (resp., $t$-invertible $t$-ideal, $t$-ideal of finite type, $t$-ideal, finitely generated ideal,divisorial ideal). Then $I(D)\vartriangleleft P$ if and only if every maximal ideal of $D$ is a principal ideal (resp., invertible ideal, $t$-ideal of finite type, $t$-ideal, finitely generated ideal, divisorial ideal) of $D.$ (2) Let, on $I(D),$ $P=$ ”— is a principal ideal (resp., invertible ideal, $t$-invertible $t$-ideal, $t$-ideal of finite type, finitely generated ideal, divisorial ideal). Then $%
I_{t}(D)\vartriangleleft P\Leftrightarrow $ every maximal $t$-ideal is a principal ideal (resp., invertible ideal, $t$-invertible $t$-ideal, $t$-ideal of finite type, finitely generated ideal, divisorial ideal).
In the presence of Lemma \[Lemma TX1\] and Lemma \[Lemma TX2\], it appears totally unnecessary to repeat the arguments required for the proofs of (1) and (2).
Note that in case of (1) every maximal ideal being a $t$-ideal of finite type ensures that every maximal $t$-ideal of $D$ is actually a maximal ideal. Indeed if we suppose that $\wp $ is a maximal $t$-ideal that is not maximal, then $\wp $ is contained in a maximal ideal, say $M,$ but $M$ is already a $t$-ideal.
We have restricted our attention to the star operations that are easily defined for usual extensions. One of the usual extensions is the $D+XL[X]$ construction, where $L$ is an extension of $K$ and $X$ an indeterminate over $L.$ It is a special case of the $D+M$ construction of [@BR; @1976]. To be able to fully appreciate how it works, one needs to learn a little about the construction $D+XL[X].$ Let $D,L,X$ be be as above$.$ Then $R=D+XL[X]$ $%
=\{f\in L[X]|f(0)\in D\}$ is an integral domain. Indeed $R$ has two kinds of nonzero prime ideals $P$ , ones that intersect $D$ trivially and ones that don’t. If $P\cap D\neq (0)$ then $P=P\cap D+XL[X]$ [@CMZ; @1986 Lemma 1.1] and obviously $P$ is maximal if and only if $P\cap D$ is. It can be shown, as was indicated prior to the proof of Corollary 16 in [@ACZ; @2015], that if $P=P\cap D+XL[X]$, then $P$ is a maximal $t$-ideal of $R$ if and only if $%
P\cap D$ is a maximal $t$-ideal of $D$ and indeed as $P_{v}=(P\cap
D)_{v}+XL[X],$ $P$ is divisorial if and only if $(P\cap D)$ is. Moreover, prime ideals of $R$ that are not comparable with $XL[X]$ are of the form $%
(1+Xg(X))R$ where $1+Xg(X)$ is an irreducible element of $L[X],$ [@CMZ; @1986 Lemmas 1.2, 1.5]. Also as $XL[X]$ is of height one $XL[X]$ is a $t$-ideal and $\dim R=\dim D+1,$ by [@CMZ; @1986 Corollary 1.4]. Let us say that a predicate $P$ respects principals if $P$ returns $T$ on principals as well.
\[Theorem UX0A\]Let $P$ be a predicate that respects principals on $%
S(D), $ $L$ an extension field of $K,$ $X$ an indeterminate over $L$ and let $R=D+XL[X]$. Then (i) given that $P$ returns $T$ on a maximal ideal $M$ of $%
D $ if and only if $P$ returns $T$ on $M+XL[X],$ $I(D)\vartriangleleft P$ $%
\Leftrightarrow I(R)\vartriangleleft P$ (ii) given that $P$ returns $T$ on a maximal $t$-ideal $M$ of $D$ if and only if $P$ returns $T$ on $M+XL[X],$ $%
I_{t}(D)\vartriangleleft P\Leftrightarrow I_{t}(R)\vartriangleleft P.$
\(i) Suppose $I(D)\vartriangleleft P,$ then $P$ returns $T$ for every maximal ideal $M$ of $D$ and hence for every maximal ideal of $R$ of the form $%
M+XL[X].$ Since $P$ respects principal ideals we conclude that $P$ returns $%
T $ for every maximal ideal of $R.$ That is $I(R)\vartriangleleft P.$ Conversely suppose that $I(R)\vartriangleleft P.$ Then $P$ returns $T$ for all maximal ideals $\mathcal{M}$ of $R,$ in particular for the ones that intersect $D$ non-trivially. But those are precisely of the form $\mathcal{M}%
=\mathbf{m}+XL[X]$ where $\mathbf{m}=\mathcal{M}\cap D$ is maximal and as $P$ returns $T$ for $\mathbf{m}+XL[X]$ if and only if $P$ returns $T$ for $%
\mathbf{m},$ and as the $\mathbf{m}s$ are precisely the maximal ideals of $D$ we conclude that $I(D)\vartriangleleft P.$ The proof of (ii) follows the same lines as those adopted in the proof of (i). However, just for completeness we include it. Suppose $I_{t}(D)\vartriangleleft P$ then $P$ returns $T$ for every maximal $t$-ideal $M$ of $D$ and hence for every maximal $t$-ideal of $R$ of the form $M+XL[X].$ Since $P$ respects principal ideals we conclude that $P$ returns $T$ for every maximal $t$-ideal of $R.$ That is $I_{t}(R)\vartriangleleft P.$ Conversely suppose that $%
I_{t}(R)\vartriangleleft P.$ Then $P$ returns $T$ for all maximal $t$-ideals $\mathcal{M}$ of $R,$ in particular for the ones that intersect $D$ non-trivially. But those are precisely of the form $\mathcal{M}=\mathbf{m}%
+XL[X]$ where $\mathbf{m}=\mathcal{M}\cap D$ is a maximal $t$-ideal and as $%
P $ returns $T$ for $\mathbf{m}+XL[X]$ if and only if $P$ returns $T$ for $%
\mathbf{m},$ and as the $\mathbf{m}s$ are precisely the maximal $t$-ideals of $D$ we conclude that $I_{t}(D)\vartriangleleft P.$
The above “theorem” is not much of a theorem, really. But it tells us what to check for, before making a statement such as $I(D)\vartriangleleft P$ $%
\Leftrightarrow I(R)\vartriangleleft P.$
\[Corollary UX0B\](i)with $D,L,X,R$ as in Theorem \[Theorem UX0A\] and with $P=$ “— is a principal ideal (resp., $t$-invertible $t$-ideal, $t$-ideal of finite type, $t$-ideal, finitely generated ideal,divisorial ideal) $I(D)\vartriangleleft P\Leftrightarrow I(R)\vartriangleleft P$ and (ii) with $D,L,X,R$ as in Theorem \[Theorem UX0A\] and with $P=$ ”— is a principal ideal (resp., invertible ideal, $t$-invertible $t$-ideal, $t$-ideal of finite type, finitely generated ideal,divisorial ideal) $I_{t}(D)%
\vartriangleleft P\Leftrightarrow I_{t}(R)\vartriangleleft P$
(i)Note that in each case $P$ returns $T$ for a principal ideal in $I(D)$. Moreover for $A$ an ideal of $D,$ because $A_{v}+XL[X]=(A+XL[X])_{v}$ and $%
A_{t}+XL[X]=(A+XL[X])_{t}$ and because $A+XL[X]=A(D+XL[X]),$ $A$ being finitely generated, invertible (or being a $v$-ideal of finite type) results in $A+XL[X]$ being of that kind and vice versa, we conclude that the requirements of Theorem \[Theorem UX0A\] are met. (Indeed as a maximal ideal being a $t$-invertible $t$-ideal is invertible, we haven’t let anything unverified.) For (ii) note that all the checking is as in (i), even the $t$-invertible $t$-ideal case falls under $t$-ideals of finite type and $%
t$-ideals of finite type are all $v$-ideals. So nothing more needs be done. $%
+$
\[Remaek UX0C\]Note that if $D$ is not a field, as we have assumed from the start, then, whatever be $D,$ $D+XL[X]$ is not Noetherian. This is because $D+XL[X]$ affords a strictly ascending chain of ideals such as $%
(X)\subseteq (X/d)\subseteq (X/d^{2})\subseteq $ $...\subseteq (X/d^{n})$ for any nonzero non unit $d$ of $D.$ Now as the maximal ideals of a Noetherian domain $D$ are finitely generated so are the maximal ideals of $%
D+XL[X]$, by Corollary \[Corollary UX0B\]. This gives us an example (a) of a non-Noetherian domain whose maximal ideals are all finitely generated. That is not all, we can construct chains of domains, of any length, starting with a domain whose maximal ideals are all finitely generated. To make things simple let $L=K.$ Let $R_{0}$ be a domain with the property that every maximal ideal of $R_{0}$ is finitely generated and let $%
R_{1}=R_{0}+X_{0}qf(R_{0})[X_{0}],$ where $X_{0}$ is an indeterminate over $%
qf(R_{0}),$ $R_{2}=R_{1}+X_{1}qf(R_{1})[X_{1}],$ where $X_{1}$ is an indeterminate over $qf(R_{1})$ and obviously every maximal ideal of $R_{2}$ is finitely generated because $R_{1}$ has this property. If proceeding in this manner, we reach $R_{n}=R_{n-1}+X_{n-1}qf(R_{n-1})[X_{n-1}]$, where $%
X_{n-1}$ is an indeterminate over $qf(R_{n-1})$ we can construct the next. As a result of this recursive procedure we have a chain of domains: $%
R_{0}\subseteq R_{1}\subseteq ...$ $\subseteq R_{n}\subseteq
R_{n+1}\subseteq ...,$ where each of $R_{i}$ gets the property of having all maximal ideals finitely generated from the previous, for $i>0$. Next recall that (b) $D$ is a Mori domain if $D$ has ACC on integral divisorial ideals. Obviously Noetherian domains and less obviously Krull domains are Mori. It can be shown that $D$ is a Mori domain if and only if for every nonzero integral ideal $A$ of $D$ there is a finitely generated ideal $F\subseteq A$ such that $A_{v}=F_{v}$ [@Nishi; @1963 Lemma 1]. This translates to: every $t$-ideal is a $t$-ideal of finite type [@AA; @1988 Corollary 1.2]. Thus if $D$ is Mori, then every maximal $t$-ideal of $D$ is of finite type. To show that the property of having every maximal $t$-ideal of finite type does not characterize Mori domains one can construct $R=D+XK[X]$ indicating, via Corollary \[Corollary UX0B\], that every maximal $t$-ideal of $R$ is of finite type but $R$ is not Mori because $R$ affords an ascending chain like: $(X)\subseteq (X/d)\subseteq (X/d^{2})\subseteq $ $...\subseteq
(X/d^{n})$ for any nonzero non unit of $D.$ We can actually construct, as in (a) above, chains of domains satisfying this property.
There are other uses Corollary \[Corollary UX0B\] can be put to, but we shall let the reader discover those, if need arises. We now concentrate on the next extension $R=D[X]$ where $X$ is the usual indeterminate over $D.$
\[Proposition UXA\] (1) Let $I(D)\vartriangleleft P$ where $P=$ “— is a proper nonzero principal ideal (resp., $t$-invertible $t$-ideal, $t$-ideal, $t$-ideal of finite type, divisorial ideal), let $X$ be an indeterminate over $D$ and let $R=D[X].$ Then it never is the case that $%
I(R)\vartriangleleft P$ for $P=$ ”— is a proper nonzero principal ideal (resp., $t$-invertible $t$-ideal, $t$-ideal, $t$-ideal of finite type, divisorial ideal ) and (2) Let $I_{t}(D)\vartriangleleft P$ where $P=$ “— is a $t$-invertible $t$-ideal (resp., $t$-ideal, $t$-ideal of finite type, divisorial ideal), let $X$ be an indeterminate over $D$ and let $R=D[X].$ Then $I_{t}(R)\vartriangleleft P$ where $P=$ ”— is a $t$-invertible $t$-ideal (resp., $t$-ideal, $t$-ideal of finite type, divisorial ideal) and conversely.
\(1) Let $I(D)\vartriangleleft P$ where $P=$ “— is a proper nonzero principal ideal (resp., $t$-invertible $t$-ideal, $t$-ideal, $t$-ideal of finite type, divisorial ideal). Then every maximal ideal $\wp $ of $D$ is a $%
t$-ideal. Now consider the prime ideal $\wp \lbrack X]$ in $R[X]$ and note that $\wp \lbrack X]$ can never be a maximal ideal because $R[X]/\wp \lbrack
X]\cong (R/\wp )[X]$ is a polynomial ring over a field and so must have an infinite number of maximal ideals. This forces $\wp \lbrack X]$ to be properly contained in an infinite number of maximal ideals $M_{\alpha }$ of $%
R[X].$ Let $M$ be one of them$.$ Then $M=(f,\wp \lbrack X]).$ Now, if it were the case that $I(R)\vartriangleleft P$ for $P=$ ”— is a proper $t$-ideal", then every maximal ideal of $R$ would be a $t$-ideal. This would make $M$ a $t$-ideal with $M\cap D=\wp \neq (0).$ But then, according to Proposition 1.1 of [@HZ; @1989], $M=(M\cap D)[X]=\wp \lbrack X],$ a contradiction to the fact that $\wp \lbrack X]\subsetneq M$. For (2) note that if $I_{t}(D)\vartriangleleft P$ where $P$ is as specified, then every maximal $t$-ideal $\wp $ of $D$ is a $t$-invertible $t$-ideal (resp., $t$-ideal, $t$-ideal of finite type, divisorial ideal). Now let $M$ be a maximal $t$-ideal of $R.$ If $M\cap D=(0)$, then $M$ is a $t$-invertible $t$-ideal and hence a $t$-ideal (and divisorial, being a finite type $t$-ideal), by Theorem 1.4 of [@HZ; @1989]. Next if $M$ is such that $M\cap
D\neq (0),$ then $M=(M\cap D)[X]$ where $M\cap D$ is a maximal $t$-ideal of $%
D$ and hence a $t$-ideal, and obviously is divisorial if and only if $M$ is divisorial [@HH; @1980 Proposition 4.3]. Conversely suppose that $%
I_{t}(R)\vartriangleleft P$ for $P$ as specified. Then every maximal $t$-ideal $M$ of $R$ is a $t$-invertible $t$-ideal (resp., $t$-ideal, $t$-ideal of finite type, divisorial ideal). Now let $\wp $ be a maximal $t$-ideal of $%
D.$ Then $\wp \lbrack X]$ is a maximal $t$-ideal of $R$ by Proposition 1.1 of [@HZ; @1989] and hence divisorial. But this leads to $\wp \lbrack
X]=(\wp \lbrack X])_{v}=\wp _{v}[X]$ ($(\wp \lbrack X])_{t}=\wp _{t}[X])$ and hence to $\wp =\wp _{v}.$ (We have chosen to focus of divisorial ideals ($t$-ideals), as all the other cases are divisorial (or $t$-ideals) and a maximal $t$-ideal of $R$ that intersects $D$ trivially is divisorial of finite type and hence a $t$-ideal.) Moreover if a maximal $t$-ideal $M$ of $%
R $ intersects $D$ non-trivially then $M=(M\cap D)[X]$ as above and of course $M$ is a $t$-ideal ( $t$-ideal of finite type, divisorial) if and only if $M\cap D$ is) [@HH; @1980 Proposition 4.3].
I cannot find a way to prove or disprove the following: Let $R=D[X],$ and let $P=$ “— is a finitely generated ideal” then $I(D)\vartriangleleft
P\nRightarrow I(R)\vartriangleleft P.$
Now we are ready to show that if $R=D_{\mathcal{S}},$ for a multiplicative set $\mathcal{S}$ of $D$ where $S(D)\vartriangleleft P$ for $P=$ "— is a proper nonzero principal ideal (resp., $t$-invertible $t$-ideal, $t$-ideal, $%
t$-ideal of finite type, divisorial ideal), then it may not generally be the case that $S(R)\vartriangleleft P$. Let’s first recall from Lemma \[Lemma TX2\] that if a maximal ideal is a $t$-invertible $t$-ideal then it is actually invertible. Before we start constructing examples, let’s take a look at the tool that we use in the following example. Let $K$ be a subfield of a field $L$, let $X$ be an indeterminate over $L$ and let $T=K+XL[X]$. The ring $T$ is an example of an atomic domain that is not a UFD (see [@Coh; @1989 page 353]) and an example of a $D+M$ construction. That $T$ is one dimensional follows from [@CMZ; @1986 Corollary 1.4], that every maximal ideal of $T$ different from $XL[X]$ is principal of height one follows from Lemmas 1.2 and 1.5 of [@CMZ; @1986] and that $XL[X]$ is divisorial can be easily checked, because $XL[X]=(X,lX)_{v}$ where $l\in L\backslash K.$
\[Example UXD\]Let $L$ be a field extension of $K$ with $[L:K]=\infty ,$ let $X$ be an indeterminate over $L$ and consider $R=D+XL[X].$ Set $%
S=D\backslash (0).$ If every maximal ideal of $D$ is principal (invertible, finitely generated) then so is every maximal ideal of $R.$ But that is not the case for every maximal ideal of $R_{S}.$ For $R_{S}=K+XL[X]$ has a maximal ideal that is a $t$-ideal but neither principal nor finitely generated, because $[L:K]=\infty .$ (It is easy to see that every invertible ideal is principal in $T$, [@BZ; @1988 Example 1.10].)
The following example has been taken, almost verbatim, from [@HZ; @2015 Example 3.3]. To decipher this example, recall that $D$ is a PVMD (Prufer $v$-multiplication domain) if every nonzero finitely generated ideal of $D$ is $%
t$-invertible. A good source for this concept is [@MZ; @1981].
\[Example UXE\]. There does exist at least one example of a domain $D$ such that each maximal ideal of $D$ is a $t$-ideal but for some maximal $M$ we have $MD_{M}$ not a $t$-ideal. One such example is that of an essential domain that is not a PVMD. (Recall that an integral domain $D$ is essential if $D$ has a set $F$ of primes such that $D_{p}$ is a valuation domain for each $P\in F$ and $D=\cap _{P\in F}D_{P}.).$ Now the example in question was constructed by Heinzer and Ohm in [@HO; @1973] and further analyzed in [@MZ; @1981] and [@GHL; @2004]. As it stands, the example has all except one maximal ideals height one primes and hence $t$-ideals and the other maximal ideal $M$ is a height $2$ prime $t$-ideal. Indeed this is the maximal ideal $M$ such that $D_{M}$ is a $2$-dimensional regular local ring and so with a maximal ideal that is not a $t$-ideal. Showing that while $%
I(D)\vartriangleleft P$ for $P=$ "— is a $t$-ideal of $D$, $%
I(D_{M})\ntriangleleft P$.
For the next example recall from [@Zaf; @1987] that an integral domain $D$ is a pre-Schreier domain if for all $a,b_{1},b_{2}\in D\backslash \{0\},$ $%
a|b_{1}b_{2}$ implies that $a=a_{1}a_{2},$ with $a_{i}\in D$ such that $%
a_{i}|b_{i}.$ Also call a $D$-module $M$ locally cyclic if for any elements $%
x_{1},x_{2},...,x_{n}\in M$ there is a $d\in M$ such that $x_{i}=r_{i}d.$
\[Example UXF\]For $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ the field of real numbers, let $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
+M$ , be a non-discrete rank one valuation domain, as constructed in say Example 4.5 of [@Zaf; @1987]. As decided in the above-mentioned example, $%
\,T=%
%TCIMACRO{\U{211a} }%
%BeginExpansion
\mathbb{Q}
%EndExpansion
+M$ (where $%
%TCIMACRO{\U{211a} }%
%BeginExpansion
\mathbb{Q}
%EndExpansion
$ is the field of rational numbers) is a pre-Screier domain with $M$ divisorial and by [@Zaf; @1987 Theorem 4.4 ] locally cyclic. But then $M$ cannot be a $v$-ideal of finite type. For if $M=(x_{1},x_{2},...,x_{n})_{v},$ then there would be a $d\in M$ such that $M=(x_{1},x_{2},...,x_{n})_{v}%
\subseteq (d)\subseteq M,$ contradicting the construction in Example 4.5 of [@Zaf; @1987]. Now let $p$ be a prime element in $%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
$, the ring of integers, and consider the local ring $R=%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
_{(p)}+M.$ Indeed the maximal ideal of $R$ is principal and hence can pass as a $t$-ideal of finite type, a $t$-invertible $t$-ideal. But if $S$ is the multiplicative set of $R$ generated by $p,$ neither of these properties are shared by the maximal ($t$-) ideal $M$ of $R_{S}=%
%TCIMACRO{\U{211a} }%
%BeginExpansion
\mathbb{Q}
%EndExpansion
+M.$
Now the fact that $I(D)\vartriangleleft P$ can go through the $D+XL[X]$ construction with the various descriptions of $P$ can be used to construct, for example a domain of any (finite) dimension with $t$-maximal ideals principal. If that reminds an attentive reader of comments (3) and (4) of Remarks 8 of [@MZ; @1990], then so be it. The point however is that the events of $I(D)\vartriangleleft P$ and $I_{t}(D)\vartriangleleft P,$ with suitable descriptions of $P$, do not have the usual Ascending Chain Conditions on ideals (principal or $t$-)ideals. One may wonder if there are any simple restriction that will get the beast under control. Yet to prepare to see that, here is another simple set of results that can come in handy when we are dealing with completely integrally closed integral domains. Of course before we bring in those results some introduction is in order. Recall that an integral domain $D$ with quotient field $K$ is completely integrally closed if whenever $rx^{n}$ $\in D$ for $x\in K$, $0\neq $ $r\in
D $, and every integer $n\geq 1$, we have $x\in D$. It can be shown that an intersection of completely integrally closed domains is completely integrally closed. The go to reference for Krull domains is Fossum’s book [@Fos; @1973] where you can find that $D$ is a Krull domain if $D$ is a locally finite intersection of localizations at height one primes such that $%
D_{P}$ is a discrete valuation domain at each height one prime. Thus a Krull domain is completely integrally closed. Glaz and Vasconcelos [@GV; @1977] called an integral domain $D$ an H-domain if for an ideal $A$ with $%
A^{-1}=D, $ (or equivalently $A_{v}=D)$ then $A$ contains a finitely generated subideal $F$ such that $A^{-1}=F^{-1}$. They showed that a completely integrally closed H-domain is a Krull domain. In [@HZ; @1988 Proposition 2.4] it was shown that $D$ is an H-domain if and only if every maximal $t$-ideal of $D$ is divisorial. We have in the following a basic result and some of its derivatives.
\[Proposition UX1\](a) Let $D$ be a completely integrally closed domain. Then (1) $D$ is a Krull domain if and only if $I_{t}(D)\vartriangleleft P$ for $P=\,$“— is a proper divisorial ideal, (2) $D$ is a locally factorial Krull domain if and only if $I_{t}(D)\vartriangleleft P$ for $P=\,$”— is a proper invertible integral ideal of $D,$ (3) $D$ is a Krull domain if and only if $I_{t}(D)\vartriangleleft P$ for $P=\,$“— is a proper $t$-invertible $t$-ideal of $D,$ (b) (4) Let $D$ be such that $D_{M}$ is a Krull domain for each maximal ideal $M$ of $D.$ Then $D$ is a Krull domain if and only if $I_{t}(D)\vartriangleleft P$ for $P=\,$”— is a proper divisorial ideal of $D$ [@EIT; @2019] (5) Let $D$ be an intersection of rank one valuation domains. Then $D$ is a Krull domain if and only if $%
I_{t}(D)\vartriangleleft P$ for $P=\,$“— a proper divisorial ideal of $D,$ (6) Let $D$ be an almost Dedekind domain. Then $D$ is a Dedekind domain if and only if $I(D)\vartriangleleft P$ for $P=\,$”— is a proper divisorial ideals of $D.$
The idea of proof, in each case, is that every maximal $t$-ideal (maximal ideal) being contained in a proper divisorial ideal must be equal to it and combining this with the fact that $D$ is completely integrally closed we get the Krull domain conclusion. For the locally factorial domain conclusion in (2) we note that every maximal $t$-ideal of $D$ is invertible and so divisorial. This gives the Krull conclusion and a Krull domain is locally factorial if and only if every height one prime of $D$ is invertible [@And; @1978 Theorem 1]. For the Dedekind domain conclusion in (6), we note that every maximal ideal is of height one and divisorial, being invertible, so every maximal ideal is a $t$-ideal and so the domain is Krull and one dimensional. The converse in each case is obvious, in that if $D$ is a Krull domain then $D$ is completely integrally closed and every maximal $t$-ideal of $D$ is, a $t$-invertible $t$-ideal and hence, divisorial. (If $D$ is locally factorial, as in (2), every maximal $t$-ideal of $D$ is invertible and hence divisorial.) And if $D$ is Dedekind, then $D$ is completely integrally closed and every maximal ideal is invertible and hence divisorial.
It is well known that $D$ is a Krull domain if and only if every $t$-ideal of $D$ is a $t$-product of prime $t$-ideals of $D$ [@Nishi; @1973]. As we have seen, the prime $t$-ideals in a Krull domain happen to be all $t$-invertible $t$-ideals, and hence maximal $t$-ideals and divisorial [@HZ; @1989 Proposition 1.3]. Also, according to [@Zaf; @1986 Theorem 1.10], $D$ is a locally factorial Krull domain if, and only if, every $t$-ideal of $%
D$ is invertible. Finally, $D$ being completely integrally closed may not control the dimension of $D$ when every maximal ideal is a $t$-ideal. Because the ring of entire functions is an infinite dimensional Bezout domain and completely integrally closed [@Gil; @1972 page 146]. (Also, in a Bezout domain every maximal ideal is a $t$-ideal.)
Another condition that helps control the dimension is requiring some kind of an ascending chain condition. Call $D$ a $t$-ACC domain if $D$ satisfies ACC on its $t$-invertible $t$-ideals.
\[Lemma UX2\]Let $D$ be a $t$-ACC domain and let $I$ be a proper $t$-invertible $t$-ideal of $D.$ Then $\cap (I^{n})_{t}=(0).$ Consequently, in a domain satisfying $t$-ACC, if $A$ is a proper divisorial ideal of $D$ and $%
I$ a $t$-invertible $t$-ideal then $(AI)_{v}=A$ implies $I=D$.
Because a $t$-invertible $t$-ideal is a $v$-ideal of finite type with $%
I^{-1} $ of finite type there is no harm in using $v$ for $t.$ Now let $\cap
(I^{n})_{v}\neq 0$ and let $x$ be a nonzero element in $\cap (I^{n})_{v}.$ Then there is a chain of $t$-invertible $t$-ideals $xI^{-1}\subseteq
(xI^{-2})_{v}\subseteq ...\subseteq x(I^{-r})_{v}$ $...$ which must stop after a finite number of steps, because of the $t$-ACC restriction. Say $%
x(I^{-n})_{v}=x(I^{-n-1})_{v}.$ Cancelling $x$ from both sides we get $%
(I^{-n})_{v}=(I^{-n-1})_{v}.$ Multiplying both sides by $I^{n+1}$ and applying the $v$-operation we get $I=D,$ a contradiction that arises from assuming that there is a nonzero element in $\cap (I^{n})_{v}.$ For the consequently part note that $(AI)_{v}=A$ implies that $A\subseteq
(I^{n})_{v} $ for all positive integers $n$.
\[Proposition UX3\]Let $D$ be a $t$-ACC domain. Then (1) $D$ is a PID if and only if $I(D)\vartriangleleft P$ for $P=\,$“— is a proper nonzero principal ideal” and (2) $D$ is a Dedekind domain if and only if $%
I(D)\vartriangleleft P$ for $P=\,$“— is a proper invertible ideal” and (3) $D$ is a Krull domain if and only if $I_{t}(D)\vartriangleleft P$ for $P=\,$“— is a proper $t$-invertible $t$-ideal”$.$
We shall prove (3) and explain why it should work for the other two cases. For (3) note that $I_{t}(D)\vartriangleleft P$ for $\Leftrightarrow \forall
A\in I_{t}(D)$ $(A\neq D$ $\Rightarrow \exists \pi \in \Pi (A\subseteq \pi
)) $ where $\Pi $ is the set determined by $P=\,$“— is a proper $t$-invertible $t$- (resp., nonzero principal, invertible) ideal”$.$ Then, by the condition, every maximal $t$-ideal (maximal ideal) of $D$ is $t$-invertible (resp., principal, invertible). By Lemma \[Lemma UX2\] we have for each maximal $t$-ideal $M$ (maximal ideal $M)$ $\cap (M^{n})_{v}=(0)$ (resp., $\cap M^{n}=(0),$ since powers of principal (invertible) ideals are $%
v$-ideals). Thus each maximal $t$-ideal (maximal ideal) is of height one. Thus $D$ is of $t$-dimension one (resp., of dimension one). Now, in each case, $MD_{M}$ is of height one and principal, forcing $D_{M}$ to be a rank one valuation domain for each maximal $t$-ideal (maximal ideal) $M$. This makes $D$ completely integrally closed, for $D=\cap D_{M}$ where $M$ ranges over maximal $t$-ideals (maximal ideals). Now apply Proposition [Proposition UX1]{}, using the fact that each maximal $t$-ideal (maximal ideal) is divisorial, being a $t$-invertible $t$-ideal (principal (invertible) ideal). The converse is obvious in each case.
\[Proposition UX4\]Let $D$ be a $t$-ACC domain. Then (1) $D$ is a UFD if and only if $I_{t}(D)\vartriangleleft P$ for $P=\,$“— is a proper nonzero principal ideal” of $D$ and (2) $D$ is a locally factorial Krull domain if and only if $I_{t}(D)\vartriangleleft P$ for $P=\,$“— is a proper invertible ideal”.
We shall prove (1) and explain why it should work for the other case. For (1) note that $I_{t}(D)\vartriangleleft P$ for $P=\,$“— proper nonzero principal (invertible) ideal” $\Leftrightarrow \forall A\in I_{t}(D)$ $%
(A\neq D$ $\Rightarrow \exists \pi \in \Pi (A\subseteq \pi ))$ where $\Pi $ is the set determined by $P$ returning $T.$ Then, by the condition, every maximal $t$-ideal of $D$ is principal (invertible). By Lemma \[Lemma UX2\] we have for each maximal $t$-ideal $M$, $\cap M^{n}=(0)$ $,$ since powers of principal (invertible) ideals are $v$-ideals. Thus each maximal $t$-ideal is of height one. Thus $D$ is of $t$-dimension one. Now, in each case, $MD_{M}$ is of height one and principal, forcing $D_{M}$ to be a rank one valuation domain for each maximal $t$-ideal. This makes $D=\cap D_{M},$ where $M$ ranges over maximal $t$-ideals, a completely integrally closed domain. Now apply Proposition \[Proposition UX1\], using the fact that each maximal $t$-ideal is divisorial, being principal or invertible. This gets us the Krull conclusion. Now recall that in a Krull domain $D,$ $%
A_{t}=(P_{1}^{n_{1}}...P_{r}^{n_{r}})_{t}.$ Then, in case of (2), $D$ is locally factorial by [@Zaf; @1986 Theorem 1.10] and, in case of (1), $D$ is factorial because every principal ideal is a product of prime powers. The converse, in each case, is obvious in that a UFD (locally factorial Krull domain) is Krull every maximal $t$-ideal of whose is principal (resp., invertible).
As already mentioned, an integral domain $D$ that satisfies ACC on integral divisorial ideals is called a Mori domain. Obviously a Noetherian domain is a Mori domain. It is easy to check that for every nonzero integral ideal $A$ of a Mori domain $D$ there are elements $a_{1},..,a_{r}\in A$ such that $%
A_{v}=(a_{1},..,a_{r})_{v}.$ So the inverse of a nonzero ideal of a Mori domain is a $v$-ideal of finite type. Hence a $v$-invertible ideal in a Mori domain is $t$-invertible. It is well known that a domain $D$ is a Krull domain if, and only if, every nonzero ideal of $D$ is $t$-invertible (see e.g. [@MZ; @1991 Theorem 2.5]) and thus a Krull domain is Mori too. Noting that a Mori domain is a $t$-ACC domain and that Noetherian is Mori too, we have the following direct corollaries.
\[Corollary UX5\]Let $D$ be a Mori domain. Then (1) $D$ is a PID if and only if $I(D)\vartriangleleft P$ for $P=\,$“— is a proper nonzero principal ideal”, (2) $D$ is a Dedekind domain if and only if $%
I(D)\vartriangleleft P$ for $P=\,$“— is a proper invertible ideal”, (3) $D$ is a Krull domain if and only if $I_{t}(D)\vartriangleleft P$ for $P=\,$“— is a proper $t$-invertible $t$-ideal”$,$ (4) $D$ is a UFD if and only if $%
I_{t}(D)\vartriangleleft P$ for $P=\,$“— is a proper nonzero principal ideal” and (5) $D$ is a locally factorial Krull domain if and only if $%
I_{t}(D)\vartriangleleft P$ for $P=\,$“— is proper invertible ideal”.
\[Corollary VX\]Let $D$ be a Noetherian domain. Then (1) $D$ is a PID if and only if $I(D)\vartriangleleft P$ for $P=\,$“— is a proper nonzero principal ideal” and (2) $D$ is a Dedekind domain if and only if $%
I(D)\vartriangleleft P$ for $P=\,$“— is a proper invertible ideal”.
\[Corollary WX\]Let $D$ be a Mori domain. Then (1) $D$ is a UFD if and only if $I_{t}(D)\vartriangleleft P$ for $P=\,$“— is a proper nonzero principal ideal”, (2) $D$ is a locally factorial Krull domain if and only if $I_{t}(D)\vartriangleleft P$ for $P=\,$“— is a proper invertible integral ideal”$,$ (3) $D$ is a Krull domain if and only if $I_{t}(D)\vartriangleleft
P$ for $P=\,$“— is a proper $t$-invertible $t$-ideal”$.$
Finally, consider the following scheme of results.
\[Proposition XX\]Suppose that $D$ satisfies ACCP (ACC on principal ideals). Then (1) $D$ is a PID if and only if $I(D)\vartriangleleft P$ for $%
P=\,$“— is a proper nonzero principal ideal” and (2) $D$ is a UFD if and only if $I_{t}(D)\vartriangleleft P$ for $P=\,$“— is a proper nonzero principal ideal”$.$
$I_{t}(D)\vartriangleleft P$ for $P=\,$“— is a proper nonzero principal ideal”$\Leftrightarrow \forall A\in I_{\ast }(D)$ $(A\neq D$ $\Rightarrow
\exists \pi \in \Pi (A\subseteq \pi ))$ where $\Pi $ is the set of proper nonzero principal ideals of $D$ fixed by $P$ and $\ast =d$ or $t$. Then, by the condition, for any maximal (maximal $t$-ideal) $M$ of $D$ we have $%
M\subseteq \pi $ for some $\pi \in \Pi $ and so $M=$ $\pi D.$ Claim that, because of the ACCP, $M$ is of height one. (For if not, then there is $%
Q\subseteq \cap \pi ^{n}D$. So for every nonzero $x\in Q,$ $x$ is divisible by every power of $\pi ,$ giving rise to an infinite ascending chain $%
xD\subsetneq \frac{x}{\pi }D\subsetneq \frac{x}{\pi ^{2}}\subsetneq
...\subsetneq \frac{x}{\pi ^{n}}D\subsetneq ...$ which is impossible in the presence of ACCP on $D.)$ Now $MD_{M}$ is principal and of height one, making $D_{M}$ a rank one discrete valuation domain and making $D=\cap D_{M}$ completely integrally closed with every maximal ($t$-) ideal principal. This makes $D$ a Krull domain with every height one prime a principal ideal and so a UFD. Finally, a UFD with every height one prime maximal is a PID. The converse, in each case is straightforward.
The above Proposition may revive an old question touched on in [@MZ; @1991]: If $D$ has ACCP and $M$ a maximal $t$-ideal, must $M$ be of height one? We couldn’t answer it then and we had to resort to using the strong ACCP": $D$ has ACCP and $D_{M}$ has ACCP for every maximal $t$-ideal $%
M.$ Now I have taken the route of using the $t$-ACC and this gives rise to: If $D$ has $t$-ACC, must $D_{M}$ have ACCP for each maximal $t$-ideal $M$?
A Universal Restriction with Conditions [Section S3]{}
======================================================
Call a directed p.o. group $G$ an almost l.o. group if for each finite subset $X=\{x_{1},...,x_{r}\}\subseteq G^{+}$ there is a positive integer $%
n=n(X)$ such that $inf(x_{1}^{n},...x_{r}^{n})\in G^{+}.$ Almost l.o. groups were introduced in [@DLMZ; @2001] and further studied in [@Yang; @2008]. One can talk about a commutative p.o. monoid $M$ with least element $1$ and a pre-assigned set $\Pi $ such that for all $x_{1},...,x_{r}\in M$ with $%
\mathcal{L}(x_{1},...,x_{r})\neq 1,$ there being a $\pi \in \Pi $ such that $%
x_{1}^{n},...,x_{r}^{n}\geq \pi .$ As ring theory provides a plethora of examples of this concept, we turn to ring theory.
Let $D$ be a domain with a finite type star operation $\ast $ defined on it, let $I_{\ast }(D)$ be the set of proper $\ast $-ideals of $D$ and let $\Pi
_{I_{\ast }(D)}$ be a non-empty subset of $I_{\ast }(D)$ defined by a predicate $P$ such that for each $A\in $ $I_{\ast }(D)$ there is $n=n(A)$ with $A^{n}\subseteq \pi $ for some $\pi \in \Pi _{I_{\ast }(D)}.$ Let us say $I_{\ast }(D)$ meets $P$ with a twist when this happens and denote it by $I_{\ast }(D)\vartriangleleft ^{t}P.$ We start with a motivating example of this notion.
\[Example A\]Let $R$ be a Dedekind domain with torsion class group, let $%
K$ be the quotient field of $R$ and let $X$ be an indeterminate over $K.$ Then the ring $D=R+XK[X]$ is such that $I(D)\vartriangleleft ^{t}P$ where $%
P= $ “— is a principal ideal”$.$
Illustration: Recall, as we have already done, from [@CMZ; @1978 Theorem 4.21] that maximal ideals of $D$ are of the form $M+XK[X],$ where $M$ is a maximal ideal of $R,$ or of the form $(1+Xf(X))D$ where $1+Xf(X)$ is irreducible in $K[X].$ Now since for each maximal ideal $M$ of $R$ we have $%
M^{n}\subseteq dR$ for some positive integer $n$ and some nonzero $d\in R$ we have $(M+XK[X])^{n}=$ $M^{n}+XK\{X]\subseteq dR+XK[X].$ Next since for each maximal ideal $\mathcal{M}$ of $D$, either $\mathcal{M}$ is principal, and hence is contained in a principal ideal in $\Pi _{I(D)}$ or $\mathcal{M}$ is such that $\mathcal{M}^{n}$ is contained in a principal ideal for some positive integer $n,$ the same must hold for every ideal $I$ of $D$.
The above example leads to the following statement.
\[Proposition A1\](1)$I(D)\vartriangleleft ^{t}P$ where $P=$ “—- is a proper nonzero finitely generated ideal” if and only if for every maximal ideal $M$ of $D$ we have $M^{n}\subseteq \pi \in \Pi _{I(D)}(P)$ and (2) let $L$ be an extension field of $K=qf(D)$ and let $X$ be an indeteminate over $%
L.$ Then $I(D)\vartriangleleft ^{t}P$ where $P=$ “—- is a proper nonzero finitely generated ideal” if and only if $I(R)\vartriangleleft ^{t}P$ where $%
R=D+XL[X].$
\(1) Suppose that for every ideal $A$ of $D$ we have some $n=n(A)$ and a $\pi
_{A}\in \Pi _{I(D)}$ such that $A^{n}\subseteq \pi _{A}$ then the same holds if $A$ is a maximal ideal of $D.$ Conversely suppose that for every maximal ideal $M$ of $D$ we have some $n=n(M)$ and some $\pi _{M}\in \Pi _{I(D)}$ such that $M^{n(M)}\subseteq \pi _{M}$ and let $A$ be a proper nonzero ideal of $D.$ Then $A\subseteq M$ for some maximal ideal $M$ of $D$ and $%
A^{n(M)}\subseteq M^{n(M)}\subseteq \pi _{M}.$ For (2) let $%
I(D)\vartriangleleft ^{t}P$, where $P$ is as given, then, by (1), for every maximal ideal $\wp $ of $D$ there is $n=n(\wp )$ such that $\wp
^{n}\subseteq \pi _{\wp }$ for some $\pi _{\wp }\in \Pi _{I(D)}.$ Since every maximal ideal of $D+XL[X]$ is either principal, and hence finitely generated, or of the form $\wp +XL[X]$ where $\wp $ is a maximal ideal of $D$ [@CMZ; @1986 Lemmas 1.2, 1.5], for every ideal $A$ of $R$ there is $n=1$ or $n(\wp )$ such that $A^{n}\subseteq \pi _{A}\in \Pi _{I(R)},$ so $%
I(R)\vartriangleleft ^{t}P.$ Conversely suppose that $I(R)\vartriangleleft
^{t}P.$ Then, in particular, for maximal ideals $\mathcal{M}$ of the form $%
\wp +XL[X]$ there are positive integers $n(\mathcal{M})$ such that $(\wp
+XL[X])^{n(\mathcal{M})}$ $=\wp ^{n(\mathcal{M})}+XL[X]$ $\subseteq \pi _{%
\mathcal{M}}\in \Pi _{I(R)}.$ But then $\pi _{\mathcal{M}}\cap D\neq (0)$ forcing $\pi _{\mathcal{M}}=\pi +XL[X]=\pi (D+XL[X])$ [@CMZ; @1986 Lemma1.1], where $\pi $ is finitely generated because $\pi _{\mathcal{M}}$ is. This gives $\wp ^{n(\mathcal{M})}+XL[X]\subseteq \pi +XL[X]$ and modding out $XL[X]$ we get $\wp ^{n(\mathcal{M})}\subseteq \pi \in \Pi _{I(D)}=\{\pi
\neq (0)|\pi +XL[X]\in \Pi _{I(R)}\}.$
\[Proposition A2\](1) $I(D)\vartriangleleft ^{t}P$ where $P=$ “— is a proper $t$-ideal of finite type” if and only if for every maximal ideal $M$ of $D,$ there is $n=n(M)$ such that $M^{n}\subseteq \pi \in \Pi _{I(D)},$ (2) $I_{t}(D)\vartriangleleft ^{t}P$ where $P=$ “— is a proper $t$-ideal of finite type” if and only if for every maximal $t$-ideal $M$ of $D$ we have $M^{n}\subseteq \pi \in \Pi _{I(D)}$, (3) let $L$ be an extension field of $K$ and let $X$ be an indeteminate over $L.$ Then $I(D)\vartriangleleft
^{t}P$ where $P=$ “— is a proper $t$-ideal of finite type” if and only if $%
I(R)\vartriangleleft ^{t}P$ where $R=D+XL[X]$ and (4) let $L$ be an extension field of $K$ and let $X$ be an indeteminate over $L.$ Then $%
I_{t}(D)\vartriangleleft ^{t}P$ where $P=$ “— is a proper $t$-ideal of finite type” if and only if $I_{t}(R)\vartriangleleft ^{t}P$ where $%
R=D+XL[X].$
\(1) The proof works as the proof of (1) of Proposition \[Proposition A1\]. (2) The proof works in the same manner as that of (1) of Proposition [Proposition A1]{}, except that here the maximal $t$-ideals are in the focus. (3) Let $I(D)\vartriangleleft ^{t}P$ where $P=$ “— is a proper $t$-ideal of finite type”. To show that $I(R)\vartriangleleft ^{t}P$ all we need show is that for every maximal ideal $\mathcal{M}$ of $R$, there is a positive integer $n=n(\mathcal{M)}$ such that $\mathcal{M}^{n}\subseteq \pi _{%
\mathcal{M}}\in \Pi _{I(R)}.$ Now, as we have shown in the proof of (2) of Proposition \[Proposition A1\], a maximal ideal $\mathcal{M}$ of $R$ is either principal and hence contained in some member of $\Pi _{I(R)}$ or of the form $\mathcal{M}=M+XL[X],$ where $M$ is a maximal ideal of $D.$ But then, for $n=n(M)$ we have $M^{n}\subseteq \pi D,$ where $\pi $ is a $t$-ideal of finite type in $\Pi _{I(D)}$, forcing $\mathcal{M}%
^{n}=M^{n}+XL[X]\subseteq \pi R.$ Because $\pi $ is a $t$-ideal of finite type of $D,$ so is $\pi R=\pi +XL[X],$ see e.g. proof of Lemma 3.5 of [Zaf 2019]{}. Conversely, suppose that $I(R)\vartriangleleft ^{t}P$ where $P=$ “— is a proper $t$-ideal of finite type”$.$ Here, in particular, for a maximal ideal $\mathcal{M}$ of the form $\mathcal{M}=M+XL[X]$ we have a positive integer $n=n(\mathcal{M})$ such that $\mathcal{M}^{n(\mathcal{M)}%
}=M^{n(\mathcal{M)}}+XL[X]$ $\subseteq \pi _{\mathcal{M}}$ where $\pi _{%
\mathcal{M}}$ is a $t$-ideal of finite type of $R.$ Obviously as $M^{n(%
\mathcal{M)}}=(M^{n(\mathcal{M)}}+XL[X])\cap D\subseteq \pi _{\mathcal{M}%
}\cap D,$ and as $M^{n(\mathcal{M)}}\cap D\neq (0),$ we conclude that $\pi _{%
\mathcal{M}}\cap D\neq (0).$ Thus $\pi _{\mathcal{M}}=\pi _{\mathcal{M}}\cap
D+XL[X]$ by [@CMZ; @1986 Lemma 1.1]. And as observed in the proof of Lemma 3.5 of [@Zaf; @2019] $\pi _{\mathcal{M}}=\pi _{\mathcal{M}}\cap
D+XL[X]$ is a $t$-ideal of $R$ if and only if $\pi _{\mathcal{M}}\cap D$ is a $t$-ideal of $D.$ That $\pi _{\mathcal{M}}$ is of finite type if and only if $\pi _{\mathcal{M}}\cap D$ is, follows from the fact that $\pi _{\mathcal{%
M}}=(a_{1},...,a_{n})_{v}+XL[X].$ Finally, for (4), let $I_{t}(D)%
\vartriangleleft ^{t}P$ where $P=$ “— is a proper $t$-ideal of finite type” and as maximal $t$-ideals of $R$ that intersect $D$ trivially are prime ideals of $R$ that intersect $D$ trivially, are not comparable with $%
XL[X],$ and hence are principal we need to concentrate on maximal $t$-ideals $\mathcal{M}$ of $R$ that intersect $D$ non-trivially. But those are precisely $\mathcal{M=(M\cap }$ $D)+XL[X]$ and as $\mathcal{M=M}_{t}\mathcal{%
=(M\cap }$ $D)_{t}~+XL[X]$ we have $\mathcal{(M\cap }$ $D)_{t}~=\mathcal{%
(M\cap }$ $D).~$Thus $\mathcal{M}=M+XL[X]$ where $M$ is a maximal $t$-ideal of $D.$ But, by the hypothesis, there is a positive integer $n=n(M)$ such that $M^{n(M)}\subseteq \pi _{M}$ for some $\pi _{M}\in \Pi _{D}.$ This forces $\mathcal{M}^{n(M)}=M^{n(M)}+XL[X]\subseteq \pi _{M}+XL[X]$ which is a $t$-ideal of finite type and hence in $\Pi _{I(R)}.$ For the converse we take the same line as in the proof of (3) and note that for each maximal $t$-ideal $M$ of $D,$ $\mathcal{M=}M+XL[X]$ is a maximal $t$-ideal of $R$ and as $\mathcal{M}^{n}=(M+XL[X])^{n}\subseteq \pi _{\mathcal{M}}$ for some $\pi
_{\mathcal{M}}\in \Pi _{R},$ $M^{n}\subseteq \pi _{\mathcal{M}}\cap D\neq
(0).$ Now, as in (3), $\pi _{\mathcal{M}}\cap D$ can be shown to be a $t$-ideal of finite type and hence in $\Pi _{I(D)}.$
Apart from the examples constructed in the above proposition there are examples of domains $I_{\ast }(D)\vartriangleleft ^{t}P$ for $P=$ “— is a $%
\ast $-ideal of finite type”. Some of these examples are simple and straightforward and some are not so simple. Presented in the following is a sampling of them. If $D$ is Noetherian and $P=$ “— is a finitely generated ideal, then $I(D)\vartriangleleft ^{t}P.$ Recall, again, that $D$ is a Mori domain if it satisfies ACC on its integral divisorial ideals. Obviously Noetherian domains are Mori and less obviously Krull domains are Mori. Recall also that $D$ is Mori if and only if for every nonzero integral ideal $A$ of $D$ there is a finitely generated ideal $F\subseteq A$ such that $%
A_{v}=F_{v},$ if and only if every $t$-ideal of $D$ is a $t$-ideal of finite type [@Zaf; @1989]. Thus if $D$ is a Mori domain then $I_{t}(D)%
\vartriangleleft ^{t}P$ where $P=$ ”— is a $t$-ideal of finite type". Note that since for a finitely generated nonzero ideal $A$ of any domain $%
A_{t}=A_{v}$, every $t$-ideal of a Mori domain is divisorial. In what follows we shall also need the fact that if $I$ is a $\ast $-ideal for some star operation $\ast $, then $\sqrt{I}$ is a $\ast _{s}$-ideal (see Theorem 1 of [@Zaf; @2005]). Thus if $I$ is divisorial, or a $t$-ideal then $\sqrt{%
I}$ is a $t$-ideal.
\[Proposition A3\]Let $D$ be a Mori domain. Then $I(D)\vartriangleleft
^{t}P$ with $P=$ “— is a $t$-ideal” if and only if every maximal ideal of $%
D$ is divisorial.
If every maximal ideal $M$ of $D$ is a $t$-ideal then, being a $t$-ideal of finite type, $M$ is a $t$-ideal of finite type and hence in $\Pi _{I(D)},$ returning $T$ for $P.$ Whence $I(D)\vartriangleleft ^{t}P$. Conversely suppose that $D$ is Mori and $I(D)\vartriangleleft ^{t}P$ where $P$ is a given and let $M$ be a maximal ideal of $D.$ Then by the condition $%
M^{n}\subseteq A$ where $A$ is a $t$-ideal. This gives $M=\sqrt{M^{n}}%
\subseteq \sqrt{A}.$ Since $M$ is maximal, we have $M=\sqrt{A}$ which is a $%
t $-ideal. Since $M$ is arbitrary we have the result.
The event of $I(D)\vartriangleleft ^{t}P$ for $P=$ “—- is a $t$-ideal of finite type” does not put any constraint on the height of maximal ideals of a Mori domain. Indeed there do exists examples of Noetherian domains with maximal $t$-ideals of height greater than one, see e.g. [FZ 2019]{}.
\[Corollary A4\]Let $D$ be a Noetherian integral domain. Then $%
I(D)\vartriangleleft ^{t}P$ with $P=$ “—- is a $t$-ideal of finite type” if and only if every maximal ideal of $D$ is divisorial.
Indeed as in a polynomial ring over $D\neq K$, every maximal ideal being a radical of a $t$-ideal of any kind is not possible because that would make every maximal ideal of the polynomial ring a $t$-ideal as we have seen in section 2. On the other hand, we have the following statement.
\[Proposition A4\]Let $R=D[X]$ (a) if $P=$ “—- is a $t$-ideal (resp., $%
t$-invertible $t$-ideal, divisorial ideal)” Then $I_{t}(D)\vartriangleleft
^{t}P\Rightarrow I_{t}(R)\vartriangleleft ^{t}P$ and if $D$ is integrally closed, $I_{t}(R)\vartriangleleft ^{t}P\Rightarrow I_{t}(D)\vartriangleleft
^{t}P$ and (b) suppose that $D$ is integrally closed and $P=$ “—- is a principal ideal”, then $I_{t}(D)\vartriangleleft ^{t}P\Leftrightarrow
I_{t}(R)\vartriangleleft ^{t}P.$
(a). Let $M$ be a maximal $t$-ideal of $D[X]$ and suppose that $M\cap D\neq
(0).$ Then $M=\wp \lbrack X]$ where $\wp =M\cap D$ is a maximal $t$-ideal of $D$ [@HZ; @1989]. Since $I_{t}(D)\vartriangleleft ^{t}P$ we conclude that for some $n=n(\wp ),$ $\wp ^{n}$ is contained in a $t$-invertible $t$-ideal (resp. $t$-ideal, divisorial ideal) $A$. But then, $M^{n}=\wp
^{n}[X]\subseteq A[X].$ Next let $M$ be a maximal $t$-ideal of $D[X]$ such that $M\cap D=(0).$ Then $M$ is a $t$-invertible $t$-ideal and hence divisorial by Theorem 1.4 of [@HZ; @1989] and $M^{n}\subseteq M$ for all $%
n.$ Next suppose that $I_{t}(R)\vartriangleleft ^{t}P$ for the specified $P.$ Then, in particular, for every maximal $t$-ideal $\wp $ of $D$ we have the maximal $t$-ideal $M=\wp \lbrack X]$ and, by the condition, there is $n=n(M)$ such that $M^{n}$ is contained in a $t$-ideal (resp., $t$-invertible $t$-ideal, divisorial ideal) $A$ of $D[X].$ Since $M^{n}\cap D\neq (0),$ $A\cap
D\neq (0)$ and since $D$ is integrally closed $A=(A\cap D)[X]$ and $A\cap D$ is a $t$-ideal (resp., $t$-invertible $t$-ideal, divisorial ideal), if $A$ is [@AKZ; @1995 Corollary 3.1].
(b). Suppose that $D$ is integrally closed, $P$ is as given and that $%
I_{t}(D)\vartriangleleft ^{t}P.$ Then for each maximal $t$-ideal $M$ of $D$ we have that $M^{n}$ is contained in a proper principal ideal, say $\pi $. Now let $\mathcal{M}$ be a maximal $t$-ideal such that $\mathcal{M}\cap
D\neq (0).$ Then $\mathcal{M}=M[X]$ where $M$ is a maximal $t$-ideal of $D$ and so $\mathcal{M}^{n}=M^{n}[X]\subseteq \pi \lbrack X].$ If, on the other hand, $\mathcal{M}\cap D=(0),$ by Lemma 4.5 of [@HZ; @1988], we have $%
\mathcal{M}=fJ[X]$ where $f$ is a non-constant polynomial and $J$ is a fractional ideal of $D$. Setting up $J=\frac{A}{d},$ where $A$ is an integral ideal and using the fact that $I_{t}(D)\vartriangleleft ^{t}P,$ we conclude that $J^{m}=\frac{A^{m}}{d^{m}}\subseteq \frac{c}{d^{m}}D$ for some positive integer $m.$ But then $\mathcal{M}^{m}=f^{m}J^{m}[X]\subseteq f^{m}%
\frac{c}{d^{m}}R.$ Thus we have $I_{t}(D)\vartriangleleft ^{t}P\Rightarrow
I_{t}(R)\vartriangleleft ^{t}P.$ For the converse suppose that $%
I_{t}(R)\vartriangleleft ^{t}P$ and let $M$ be a maximal $t$-ideal of $D.$ Then $M[X]$ is a maximal $t$-ideal of $R$ and because $I_{t}(R)%
\vartriangleleft ^{t}P$, there is a positive integer $n$ such that $%
M^{n}[X]=(M[X])^{n}\subseteq fR.$ Since $M^{n}[X]\cap D\neq (0),$ $fR\cap
D\neq (0)$ and $f$ is a constant. Whence $M^{n}[X]\subseteq cD[X],$ forcing $%
M^{n}\subseteq cD.$ As $M$ is arbitrary, we conclude that for each maximal $%
t $-ideal $M$ of $D$ there is a positive integer $n$ such that $M^{n}$ is contained in a principal ideal of $D.$ But that means $I_{t}(R)%
\vartriangleleft ^{t}P\Rightarrow I_{t}(D)\vartriangleleft ^{t}P.$
Indeed as the behavior of $D+XL[X]$ is the same under $S(D)\vartriangleleft
^{t}P$ as it was under $S(D)\vartriangleleft P$, one can construct examples to show that if $R$ is a ring of fractions of $D,$ $S(D)\vartriangleleft
^{t}P$ may not imply $S(R)\vartriangleleft ^{t}P$ in general. This leaves us to check what happens if we restrict a domain to be completely integrally closed and satisfy $S(D)\vartriangleleft ^{t}P$ for a suitable $P.$ To appreciate the following proposition we need to have an idea of the divisor class group of a Krull domain being torsion. For this too the reference to go to is [@Fos; @1973]. For our purposes the divisor class group being torsion means that for each proper divisorial ideal $I$ there is some positive integer $n$ such that $(I^{n})_{v}$ is principal. The other concept to know is the local class group $G(D)=Cl(D)/Pic(D)$ of a Krull domain $D,$ introduced and studied by Bouvier in [@Bou; @1983]. Now $G(D)$ being torsion is equivalent to $(I^{n})_{v}$ being invertible, for some integer $%
n, $ for each proper divisorial ideal $I.$
\[Proposition A5\](a) Let $D$ be a completely integrally closed domain. Then (1) $D$ is a Krull domain if and only if $I_{t}(D)\vartriangleleft
^{t}P $ for $P=\,$“— is a proper divisorial ideal”, (2) $D$ is a Krull domain if and only if $I_{t}(D)\vartriangleleft ^{t}P$ for $P=\,$“— is a proper $t$-invertible $t$-ideal”, (3) $D$ is a Krull domain, with torsion divisor class group, if and only if $I_{t}(D)\vartriangleleft ^{t}P$ for $%
P=\,$“— is a proper principal ideal”$,$ (b) Let $D$ be an intersection of rank one valuation domains. Then (4) $D$ is a Krull domain, if and only if $%
I_{t}(D)\vartriangleleft ^{t}P$ for $P=\,$“— a proper $v$-ideal of finite type” and (5) $D$ is a Krull domain, with torsion local class group, if and only if $I_{t}(D)\vartriangleleft ^{t}P$ for $P=\,$“— a proper invertible ideal”, (c) Let $D$ be completely integrally closed. Then (6) $D$ is a Dedekind domain if and only if $I(D)\vartriangleleft ^{t}P$ for $P=\,$“— is a proper divisorial ideal” (resp. invertible ideal) and (7) $D$ is a Dedekind domain with torsion class group if and only if $I(D)%
\vartriangleleft ^{t}P$ for $P=\,$“— is a proper principal ideal”.
(1). Let $D$ be a completely integrally closed domain and let $%
I_{t}(D)\vartriangleleft ^{t}P$ for $P=\,$“— is a proper divisorial ideal”. Now let $M$ be a maximal $t$-ideal of $D.$ We claim that $M$ is divisorial, for if not then $M_{v}=D.$ But, by the condition, $M^{n}$ is contained in a proper divisorial ideal $\pi .$ Thus $(M^{n})_{v}\subseteq
\pi $ because $\pi $ is a divisorial ideal. On the other hand $%
(M^{n})_{v}=((M_{v})^{n})_{v}=D,$ contradicting the assumption that $\pi $ is a proper divisorial ideal. Whence $M_{v}\neq D,$ forcing $M=M_{v}.$ Now as $M$ is arbitrary, we conclude that $D$ is an H domain [@HZ; @1988]. Finally, according to [@GV; @1977], $D$ is Krull. Conversely if $M$ is a maximal $t$-ideal of a Krull domain then $M$ is divisorial and so is $%
(M^{n})_{v}$ which returns $T$ for $P$ for any $n$. (2). Because a proper $t$-invertible $t$-ideal is divisorial too and because every prime $t$-ideal of a Krull domain is $t$-invertible and so must be every maximal $t$-ideal $M,$ with $(M^{n})_{v}$ a $t$-invertible $t$-ideal, we conclude that the proof of (1) applies. (3). For sufficiency, note that proper principal ideal is divisorial. So $D$ is at least a Krull domain, by part (1). Now let $M$ be a maximal $t$-ideal of $D.$ Then, by the condition, $M^{n}$ is contained in a proper nonzero principal ideal $\pi $ and clearly $M^{n}\subseteq \pi
\subseteq M.$ Thus $M$ is the radical of a principal ideal and Theorem 3.2 of [@And; @1982] applies to give the conclusion that the divisor class group of $D$ is torsion. Conversely if $D$ is a Krull domain whose divisor class group is torsion, then via Theorem 3.2 of [@And; @1982] (or via \[Proposition 6.8\][Fos 1973]{}) one finds that for each maximal $t$-ideal $M$ we have $(M^{n})_{v}=\pi $ a principal ideal verifying that $M^{n}$ is contained in a proper principal ideal for each maximal $t$-ideal $M$ of $D.$ Note in part (b) that $D$ being completely integrally closed is provided by the given. Then (4) can be proved just like (1) and that leaves (5). Now in (5) we prove just like (3) that $D$ is a Krull domain and then use the condition to show that $M$ is the radical of an invertible ideal. This would give, via Theorem 3.3 of [@And; @1982] the conclusion that $G(D)$ is torsion. For necessity in this case we appeal to Theorem 3.3 of [@And; @1982] to conclude that $I_{t}(D)\vartriangleleft ^{t}P.$ For (6) and (7) note that every maximal $t$-ideal is maximal, and divisorial, because every maximal ideal is divisorial. So, in each cas$D$ is a one dimensional Krull domain and hence a Dedekind domain. Now in case of (7) we can conclude, as in the proof of (3), that every maximal ideal is the radical of a principal ideal. The converse in each case is obvious, if not dealt with.
For a star operation $\ast $ of finite type, defined on $D,$ call $D$ of finite $\ast $-character if every nonzero non unit of $D$ belongs to at most a finite number of maximal $\ast $-ideals of $D.$ We shall be mostly concerned with $\ast =t$ or $d$ though some of the considerations here may apply to the general approach. In any case we may define $\ast $-dimension as the supremum of the lengths of chains of $\ast $-ideals that are prime. Call $D$ a weakly Krull domain (WKD) if $D=\cap _{P\in X^{1}(D)}D_{P}$ and the intersection is locally finite. It turns out that $D$ is of finite $t$-character and of $t$-dimension one [@AMZ; @1992]. We shall also need to use the $nth$ symbolic power $Q^{(n)}$of a prime $Q$ defined by $%
Q^{(n)}=Q^{n}D_{Q}\cap D=\{x\in Q|sx\in Q^{n}\}$. We shall need also to recall that a nonzero finitely generated ideal $I$ is said to be rigid ($t$-rigid) if $I$ is contained in a unique maximal ($t$-) ideal. A maximal ($t$-) ideal is said to be ($t$-) potent if it contains a ($t$-) rigid ideal. Finally a domain $D$ is said to be ($t$-) potent if each of its maximal ($t$-) ideals is $(t$-) potent.
\[Proposition B\](1)Let $I(D)\vartriangleleft ^{t}P$ where $P=$ “— is a proper nonzero principal ideal” (resp. invertible ideal, $t$-invertible $t$-ideal) $.$ If $D$ has $t$-ACC, then $D$ is a $t$-potent domain whose maximal ideals $M$ are divisorial such that $\cap (M^{n})_{v}=(0)$ and (2) Let $I_{t}(D)\vartriangleleft ^{t}P$ where $P=$ “— is a proper nonzero principal ideal” (resp. invertible ideal, $t$-invertible $t$-ideal) $.$ If $%
D $ has $t$-ACC, then $D$ is a $t$-potent domain whose maximal $t$-ideals $M$ are divisorial such that $\cap (M^{n})_{v}=(0)$
For (1) let $I(D)\vartriangleleft ^{t}P$ where $P=$ “— is a proper nonzero principal ideal” (resp. invertible ideal, $t$-invertible $t$-ideal) and suppose that $D$ has $t$-ACC. As we concluded in the proof of Proposition \[Proposition A5\], every maximal ideal $M$ is divisorial. Next, for every maximal ideal $M$ we have $M^{n}\subseteq \pi \in \Pi _{I(D)}(P).$ This shows also that $M$ is $t$-potent. Next $(M^{n})_{v}\subseteq \pi ,$ because $\pi $ is divisorial. So $\cap (M^{nr})_{v}\subseteq \cap (\pi ^{n})_{v}.$ Since $\pi $ is a $t$-invetible $t$-ideal and since $D$ is $t$-ACC, Lemma \[Lemma UX2\] applies to give $\cap (\pi ^{n})_{v}=(0).$ Whence $\cap
(M^{n})_{v}=(0).$For (2) note that $I_{t}(D)\vartriangleleft ^{t}P$ implies that $M^{n}\subseteq \pi \in \Pi _{I_{t}(D)}(P)$ for each maximal $t$-ideal $%
M$. Since $\pi $ is divisorial, $M$ must be. The rest of proof follows the same lines as taken in the proof of (1).
The above result does not give much. But with some give and take it can.
\[Proposition B1\] (a)Let $I(D)\vartriangleleft ^{t}P$ where $P=$ “— is a proper nonzero principal ideal” and suppose that $D$ has $t$-ACC. Then the following are equivalent: (1) $D$ is one dimensional, (2) for every maximal ideal $M,$ $M^{n}$ being contained in a principal ideal $dD$ implies $Q^{(n)}\subseteq dD$ for every nonzero prime $Q$ contained in $M$, (3) $D$ is a one dimensional WKD and (4) Every power of every nonzero prime ideal $Q$ of $D$ is a primary ideal and (b) Let $I_{t}(D)\vartriangleleft ^{t}P$ where $P=$ “— is a proper nonzero principal ideal” and suppose that $D$ has $t$-ACC. Then the following are equivalent: (1) $D$ has $t$-dimension one, (2) for every maximal $t$-ideal $M,$ $M^{n}$ being contained in a principal ideal $dD$ implies $Q^{(n)}\subseteq dD$ for every nonzero prime $Q$ contained in $M$, (3) $D$ is a WKD.
\(a) That (1) $\Rightarrow $ (2) is clear. For (2) $\Rightarrow $ (3), we show that there $D$ is one dimensional. Assume by way of contradiction that there is a nonzero non-maximal prime $Q$ contained in a maximal ideal $M.$ Let $M^{n}\subseteq dD$ for a non unit $d\in D$ and let $0\neq x\in Q^{(n)}.$ Then $x\in dD.$ Since $d\notin Q,$ $(x/d)d\in Q^{(n)}$ forces $x/d\in
Q^{(n)}.$ Repeating the argument over and over again we get $\frac{x}{d}%
D\subseteq \frac{x}{d^{2}}D\subseteq \frac{x}{d^{3}}D\subseteq ...\subseteq
\frac{x}{d^{n}}D\subseteq \frac{x}{d^{n+1}}D\subseteq ...$ which is impossible in the presence of $t$-ACC. Thus $D$ is one dimensional and hence of $t$-dimension one. Now a $t$-potent domain of $t$-dimension one is a WKD by [@HZ; @2019 Theorem 5.3]. That (3) $\Rightarrow $ (4), is direct because $D$ is one dimensional. For (4) $\Rightarrow $ (1), suppose that there is a nonzero non-maximal prime ideal $Q$ and proceed as in the proof of (2) $\Rightarrow $ (3) to get the desired contradiction. For the proof of (b) note that (1) $\Rightarrow $ (2) is obvious and (2) $\Rightarrow $ (3) goes exactly along the lines taken in the proof of (2) $\Rightarrow $ (3) of (a), while (3) $\Rightarrow $ (1) is obvious too.
Lest a reader considers Proposition \[Proposition B1\] an empty result we hasten to give examples to allay such feelings. For the following set of examples we need to know that an extension of domains $A\subseteq B$ is called a root extension if for each $b\in B$ there is a positive integer $%
n=n(b)$ such that $b^{n}\in A.$ Let’s call $A\subseteq B$ a fixed root extension if there is a fixed positive integer $n$ such that $b^{n}\in A,$ for all $b\in B.$ Also an integral domain $D$ is called an Almost Principal Ideal (API-)domain if for each subset $\{a_{\alpha }\}$ of $D\backslash
\{0\} $ there is a positive integer $n$ such that $(\{a_{\alpha }^{n}\})$ is principal. According to [@AZ; @1991 Theorem 4.11] if $A\subseteq B$ is a fixed root extension and $B$ is a subring of the integral closure of $A$, then $A$ is an API domain if and only if $B$ is.
\[Example B2\]Of course (1) every Dedekind domain $D$ with torsion class group is an example of a one dimensional WKD such that $I(D)\vartriangleleft
^{t}P$ where $P=$ “— is a proper nonzero principal ideal” . (2) In section 4 of [@AZ; @1991] there are studied several examples of Noetherian API domains that are not integrally closed. The simplest of these being $%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
\lbrack 2i]=%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
+2i%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
.$ Since for each $a+bi\in $ $%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
\lbrack i]$ we have $(a+bi)^{2}=a^{2}-b^{2}+2abi\in
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
\lbrack 2i],$ this gives the conclusion that $%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
\lbrack 2i]$ is Noetherian and that $%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
\lbrack 2i]$ $\subseteq
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
\lbrack 2i]$ is a fixed root extension. Because $%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
\lbrack i]$ is a PID, Corollary 4.13 of [@AZ; @1991] applies to give the conclusion that $%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
\lbrack 2i]$ is an API domain. That $%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
\lbrack 2i]$ is one dimensional, follows from Theorem 2.1 of [@AZ; @1991]. Now let $M$ be a maximal ideal of $%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
\lbrack 2i].$ Then $M$ is finitely generated, say $M=(x_{1},x_{2},...,x_{r})$ then $(x_{1}^{n},...,x_{r}^{n})$ is principal and, using Lemma 2.3 of [XAZ ARXIV]{}, we conclude that $M^{nr}\subseteq (x_{1}^{n},...,x_{r}^{n}).$ (3) Finally, let $K$ be a field of characteristic $p>0$ and let $L$ be a purely inseparable field extension of $K$ such that $L^{p}\subseteq K$ and consider $T=K+XL[X].$ According to the information gathered prior to Example \[Example UXD\], the only non-principal maximal ideal of $T$ is $%
XL[X]=(X,lX)_{v}$ where $l\in L\backslash K.$ Obviously $%
(X^{p},(lX)^{p})_{v}=X^{p}$ and an application of Lemma 2.3 of [@XAZ; @ARXIV] or direct computation gives $(XL[X])^{2p}\subseteq ((XL[X])^{2p})_{v}$ $=((X,lX)^{2p})_{v}\subseteq X^{p}.$ The above can serve also as examples for part (b), but all fastfaktorielle rings of [@Stor; @1967] dubbed as almost factorial domains in [@Fos; @1973] can serve as examples as almost factorial domains are nothing but Krull domains with torsion divisor class groups. For non-Krull examples for (b) recall that, according to [@Zaf; @1985], an integral domain $D$ is called an AGCD domain if for each pair $%
a,b\in D\backslash \{0\}$ there is a positive integer $n=n(a,b)$ such that $%
a^{n}D\cap b^{n}D$ is principal (equivalently for every nonzero finitely generated ideal $(a_{1},...,a_{r})$ there is $n=n$ $(a_{1},...,a_{r})$ such that $(a_{1}^{n},...,a_{r}^{n})_{v}$ is principal). Any Noetherian AGCD domain would serve as an example for (b). Reason: take a maximal $t$-ideal $%
M,$ it’s finitely generated. Say $M=$ $(a_{1},...,a_{r}),$ for some $n$ we must have $(a_{1}^{n},...,a_{r}^{n})_{v}=dD,$ principal. But then $%
M^{nr}\subseteq (a_{1}^{n},...,a_{r}^{n})\subseteq
(a_{1}^{n},...,a_{r}^{n})_{v}=dD,$ by Lemma 2.3 of [@XAZ; @ARXIV].
[99]{} D.D. Anderson, $\pi $-domains, overrings and divisorial ideals, Glasgow Math J. (1978), 199-203.
\_\_\_\_\_\_\_\_\_\_\_, Globalization of some local properties in Krull domains, Proc. Amer. Math. Soc. 85 (1982) 141–145.
D.D. Anderson and D.F. Anderson, Some remarks on star operations and the class group, J. Pure Appl. Algebra 51 (1988), 27-33.
D.D. Anderson, G.W. Chang, M. Zafrullah, Nagata-like theorems for integral domains of finite character and finite t-character. J. Algebra Appl. 14(8) (2015)
D.D. Anderson, T. Dumitrescu and M. Zafrullah, Quasi-Schreier domains II, Comm. Algebra 35 (2007), 2096-2104.
D.D. Anderson, D. Kwak and M. Zafrullah, Agreeable domains, Comm. Algebra 23 (1995), 4861-4883.
D.D. Anderson, J. Mott and M. Zafrullah, Finite character representations for integral domains, Bollettino U.M.I (7) 6-B (1992) 613-630.
D.D. Anderson and M. Zafrullah, Almost Bezout domains, J. Algebra 142 (1991), 285-309.
V. Barucci, On a class of Mori domains. Comm. Algebra (1983), 1989-2001 .
A. Bouvier, The local class group of a Krull domain, Canad. Math. Bull. 26 Z (1983) , 13-19.
A. Bouvier and M. Zafrullah, On some class groups of an integral domain, Bull. Soc. Math. Grece 29(1988) 45-59.
J. Brewer, E. Rutter, $D+M$ constructions with general overrings. Michigan Math. J. 23 (1976), no. 1, 33—42.
P.M. Cohn, Bezout rings and their subrings, Proc. Cambridge Philos. Soc. 64 (1968) 251-264.
\_\_\_\_\_\_\_\_, Algebra Volume 2, (2nd edition), John Wiley & Sons, 1989.
D.L. Costa, J.L. Mott and M. Zafrullah,The construction $%
D+XD_{S}[X]$, J. Algebra 53(1978) 423-439.
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_, Overrings and dimensions of general $D+M$ constructions, J. Natur. Sci. and Math. 26(2) (1986), 7-14.
T, Dumitrescu, Y. Lequain, J., Mott and M. Zafrullah, Almost GCD domains of finite $t$-character, J. Algebra 245 (2001), 161–181.
T. Dumitrescu and M. Zafrullah, Characterizing domains of finite $\ast $-character, J. Pure Appl. Algebra, 214 (2010), 2087-2091.
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_, $t$-Schreier domains, Comm. Algebra, (2011), (39), 808-818.
S. El-Baghdadi, L. Izelgue, A. Tamoussit, Almost Krull domains and their rings of integer-valued polynomials, J. Pure Appl. Algebra (2019) in press.
J. Elliott, Rings, modules and closure operations, Springer Monographs in Mathematics. Springer, Cham.
M. Fontana and M. Zafrullah, $t$-local domains and valuation domains, in Advances in commutative ring theory, Ayman Badawi and James Barker Coykendall, Editors, Springer 2019.
R. Fossum, The divisor class group of a Krull domain, Ergebnisse der Mathematik und ihrer grenzgebiete B. 74, Springer-Verlag, Berlin, Heidelberg, New York, 1973.
S. Gabelli, E. Houston, and T. Lucas, The t\#-property for integral domains, J. Pure Appl. Algebra 194 (2004), 281-298.
R. Gilmer, Multiplicative Ideal Theory, Marcel-Dekker 1972.
S. Glaz and W. Vascocelos, Flat ideals II, Manuscripta Math. 22(1977) 325-341.
J.R. Hedstrom and E.G. Houston, Some remarks on star-operations, J. Pure Appl. Algebra 18(1980) 37-44.
W. Heinzer and J. Ohm, An essential ring which is not a v-multiplication ring, Canad. J. Math. 25 (1973), 856-861.
E. Houston and M. Zafrullah, Integral domains in which every $t$-ideal is divisorial, Michigan Math. J. 35(1988) 291-300.
\_\_\_\_\_\_\_\_\_\_, On $t$-invertibility II, Comm. Algebra 17(8)(1989) 1955-1969.
\_\_\_\_\_\_\_\_\_\_, Integral domains in which any two $v$-coprime elements are comaximal, J. Algebra 423 (2015), 93-115.
\_\_\_\_\_\_\_\_\_\_, $\ast $-super potent domains, J. Commut. Algebra, Forthcoming (2019).
I. Kaplansky, Commutative Rings, Allyn and Bacon, Boston (1970).
J. Mott and M. Zafrullah, On Prufer v-multiplication domains, Manuscripta Math. 35 (1981), 1-26.
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_, Unruly Hilbert domains, Canad. Math. Bull. 33 (1) (1990), 106-109.
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_, On Krull domains, Arch. Math. 56(1991),559-568.
T. Nishimura, Unique factorization of ideals in th e sense of quasi-equality, J. of Math. of Kyoto Univ., Vol. 3, No. 1 (1963), 115-125.
\_\_\_\_\_\_\_\_\_\_, On $t$-ideals of an integral domain, J. Math. Kyoto Univ. (JMKYAZ) 13-1 (1973) 59-65.
U. Storch, Fastfaktorielle Ringe, Schrift. Math. Ins. Univ. M"unster 36 (1967).
S.Q. Xing, D.D. Anderson and M. Zafrullah, Two generalizations of Krull domains, http://arxiv.org/abs/1912.02306
Y.C. Yang, Some remarks on almost lattice-ordered groups, Archiv der Mathematik, 91 (2008), no. 5, 392-398.
M. Zafrullah, A general theory of almost factoriality,Manuscripta Math. 15(1985) 29-62.
\_\_\_\_\_\_\_\_, On generalized Dedekind domains, Mathematika 33(1986), 285-295.
\_\_\_\_\_\_\_\_, On a property of pre-Schreier domains, Comm. Algebra 15 (1987), 1895-1920.
\_\_\_\_\_\_\_\_, Ascending chain conditions and star operations, Comm. Algebra 17(6)(1989) 1523-1533.
\_\_\_\_\_\_\_\_, Putting t-invertibility to use, Non-Noetherian Commutative Ring Theory, in: Math. Appl., vol. 520, Kluwer Acad. Publ., Dordrecht, 2000, pp. 429-457.
\_\_\_\_\_\_\_\_, Help Desk, https://lohar.com/mithelpdesk/hd0504.pdf
\_\_\_\_\_\_\_\_, On $\ast $-homogeneous ideals, https://arxiv.org/pdf/1907.04384.pdf
|
---
author:
- 'V. Buat'
- 'S. Heinis'
- 'M. Boquien'
- 'D. Burgarella'
- 'V. Charmandaris'
- 'S. Boissier'
- 'A. Boselli'
- 'D. Le Borgne'
- 'G. Morrison'
date: 'Received ; accepted '
title: ' Ultraviolet to infrared emission of $z>1$ galaxies: Can we derive reliable star formation rates and stellar masses?'
---
[Our knowledge of the cosmic mass assembly relies on measurements of star formation rates (SFRs) and stellar masses ($M_{\rm star}$), of galaxies as a function of redshift. These parameters must be estimated in a consistent way with a good knowledge of systematics before studying their correlation and the variation of the specific star formation rate. Constraining these fundamental properties of galaxies across the Universe is of utmost importance if we want to understand galaxy formation and evolution.]{} [We seek to derive star formation rates and stellar masses in distant galaxies and to quantify the main uncertainties affecting their measurement. We explore the impact of the assumptions made in their derivation with standard calibrations or through a fitting process, as well as the impact of the available data, focusing on the role of infrared (IR) emission originating from dust.]{} [We build a sample of galaxies with $z>1$, all observed from the ultraviolet to the infrared in their rest frame. The data are fitted with the code CIGALE, which is also used to build and analyse a catalogue of mock galaxies. Models with different star formation histories are introduced: an exponentially decreasing or increasing star formation rate and a more complex one coupling a decreasing star formation rate with a younger burst of constant star formation. We define different set of data, with or without a good sampling of the ultraviolet range, near-infrared, and thermal infrared data. Variations of the metallicity are also investigated. The impact of these different cases on the determination of stellar mass and star formation rate are analysed.]{} [Exponentially decreasing models with a redshift formation of the stellar population $z_{\rm f} \simeq 8$ cannot fit the data correctly. All the other models fit the data correctly at the price of unrealistically young ages when the age of the single stellar population is taken to be a free parameter, especially for the exponentially decreasing models. The best fits are obtained with two stellar populations. As long as one measurement of the dust emission continuum is available, SFR are robustly estimated whatever the chosen model is, including standard recipes. The stellar mass measurement is more subject to uncertainty, depending on the chosen model and the presence of near-infrared data, with an impact on the SFR-$M_{\rm star}$ scatter plot. Conversely, when thermal infrared data from dust emission are missing, the uncertainty on SFR measurements largely exceeds that of stellar mass. Among all physical properties investigated here, the stellar ages are found to be the most difficult to constrain and this uncertainty acts as a second parameter in SFR measurements and as the most important parameter for stellar mass measurements. ]{}
Introduction
============
Star formation rates (SFR) and stellar masses ($M_{\rm star}$) are the main parameters estimated from large samples of galaxies and they can be used to constrain their star formation history and the evolution of their baryonic content. A large number of works found a tight relation between SFR and $M_{\rm star}$ both at low and high redshift (e.g., [@brinchmann04; @daddi07; @elbaz07; @rodighiero11]), which is often called main sequence (MS) of galaxies ([@noeske07]). The slope and the scatter of this relation as well as its evolution with redshift put constraints on the star formation history of the galaxies as a function of their mass. The galaxies located on this MS may experience a rather smooth star formation evolution during several Gyr ([@heinis13b]) and the starburst mode seems to play a minor role in the production of stars ([@rodighiero11]).\
The degree to which we can interpret these observations depends on our ability to estimate SFR and $M_{\rm star}$. Two major methods (not independent) are commonly used to measure SFR and $M_{\rm star}$. The first approach consists of using empirical recipes. The SFR is deduced by applying conversion factors between an observed emission coming mostly from young stars and the SFR (e.g.,[@kennicutt98]). The impact of dust attenuation has long been identified as a major issue. To overcome it, some calibrations combine different wavelengths and account for all the star formation directly observed in ultraviolet (UV)-optical or reprocessed in thermal infrared (IR) (e.g.,[@hao11; @KennEvans12; @calzetti12]). These relations rely on strong assumptions on star formation history ([@boissier12; @calzetti12]), which are valid for local, normal galaxies, but may well break down for more extreme objects and at high redshift ([@kobayashi12; @schaerer13]). The $M_{\rm star}$ estimations are also based on tabulated relations between mass to light ($M/L$) ratios and colors ([@bell03; @zibetti09]). The accuracy of their determination when rest-frame near-infrared (NIR) data are either included or not included remains an open issue: optical colors-M/L relations are less sensitive to the uncertain thermally pulsing asymptotic giant branch evolutionary phase, but the uncertainty due to dust reddening is strongly minimized in NIR ([@conroy13 and references therein]. The determination of SFR and $M_{\rm star}$ is also strongly dependent on the choice of the stellar libraries and initial mass function ([@bell03; @muzzin09; @marchesini09]).\
Another widespread method to derive these physical parameters is to exploit the full panchromatic information available for a given sample by fitting the spectral energy distributions (SEDs). The first step is to model the stellar emission using an evolutionary population synthesis method, assuming a star formation history with some recipes for dust attenuation. We then compare these theoretical SEDs to data. This method is particularly convenient when a large range of redshift is studied and when the wavelength coverage is wide. Without strong constraints on the amount of dust attenuation, an age-extinction degeneracy cannot be avoided ([@pforr12; @conroy13]). In parallel to the development of models for the stellar emission, IR ($> \sim 5 \mu$m) SEDs resulting from dust emission have been built and models have been developed that predict the full UV to IR SEDs of galaxies in a self-consistent manner for galaxies where most of the energy is produced by stars ([@dacunha08; @noll09]). The coupling of emissions both from stars and dust is physically due to the absorption of UV-optical photons by dust. The exact process of dust and star interaction is highly complex as it depends on many parameters, such as the dust grain composition and distributions, geometry, and age of the stellar populations ([@popescu11]). At the scale of entire galaxies, it is impossible to model this process and the net effect of dust attenuation on the emission of a galaxy is usually described by an attenuation curve, the most popular one being that of [@calzetti00]. The $M_{\rm star}$ derivations are not sensitive to dust attenuation, but strongly depend on the assumed stellar population synthesis model and star formation history (e.g., [@papovich01; @salim07; @pforr12; @maraston10; @lee09]). For a given set of assumptions about the stellar population synthesis model, including metallicity and star formation history, and adopting an initial mass function, the stellar masses can be estimated robustly. However, systematic differences appear when different assumptions are made and these input parameters are very poorly constrained ([@muzzin09]). [@marchesini09] showed that these systematic uncertainties contribute at the same level as random errors in the derivation of stellar mass functions. The uncertainty about star formation history itself induces large variations in stellar mass derivation ([@bell03; @lee09; @pforr12]): the high luminosity of the young stellar population may hide an old stellar population. The effect can be very strong in bursty systems and leads to an underestimation of stellar ages [@maraston10]. As a consequence, stellar masses are likely to be more reliably measured in quiescent systems than in bursty galaxies ([@wuyts09; @pforr12; @conroy13]).\
In this paper, we aim to measure the SFR and $M_{\rm star}$ in a consistent way using different assumptions about star formation history and different wavelength coverages. We focus on galaxies with a redshift larger than 1, which are intensively forming stars. Our analysis is based on a unique stellar population synthesis model ([@maraston05]). We refer to [@conroy10] for a comparison of the most popular population synthesis models in the framework of SED modelling. Numerous papers have been published to explore the reliability of SED fitting techniques, most of which are based on artificial catalogues of galaxies drawn from semi-analytical models or hydrodynamical codes ([@wuyts09; @lee09; @pforr12; @pacifici12; @mitchell13]). These studies are all based on UV-optical and NIR data, but do not include IR emission. Our approach is somewhat different and complementary to these previous studies. First, we consider the whole electromagnetic spectrum from the UV to the IR and we estimate dust attenuation by balancing the energy absorbed by the dust in the UV/optical to the energy re-emitted in the thermal infrared. The strong constraints put on dust attenuation are expected to reduce the uncertainty of recent star formation history retrieved by the SED fitting process. We combine true data and mock catalogues built with our fitting code. The priors of our artificial sources created with the fitting code are well defined and completely known, and their influence can be analysed at the price of over-simplistic modelling. Models based on semi-analytical codes or hydrodynamical simulations are certainly more realistic, but also depend on the assumptions made to produce them and it is more difficult to quantify the influence of these priors on the results of the SED fitting analysis.\
We work between z=1 and z= 3 in order to sample the UV continuum and the dust emission well. All galaxies are detected in the IR and, in particular, at 24 $\mu$m. The choice of the redshift range is essentially motivated by the quality of the photometric data and the performance of the SED fitting code as described in Sect. 2 and 3.1. The dataset is described in detail in Sect. 2, with a particular attention paid to the sampling of the rest-frame UV continuum emission using a large number of intermediate band filters. These filters have been proven to be very effective in measuring photometric redshifts ([@ilbert09; @benitez09; @cardamone10]) as well as in characterising the dust attenuation ([@buat11; @buat12]) but, to our knowledge, their influence in retrieving physical parameters has not been studied yet. The fitting process, based on our modelling code, CIGALE ([@noll09]), is presented in Sect. 3 along with the adopted star formation models and the determination of the main physical parameters (SFR, $M_{\rm star}$, and stellar ages). The influence of specific datasets (UV sampling, NIR, and IR) in the determination of these quantities are explored in Sect. 4. In Sect. 5 we generate mock catalogues and use them to explore possible systematic effects in the estimations of the physical parameters. Finally, our conclusions are presented in Sect. 6. We assume that $\Omega_{\rm m} = 0.3$, $\Omega_{\Lambda} = 0.7$, and $H_0 = 70\,{\rm km\,s^{-1}\,Mpc^{-1}}$. The luminosities are defined in solar units with ($L_\odot = 3.83~ 10^{33} ~ \rm erg s^{-1}$ and the adopted solar luminosity in the K band used to define $M/L_{\rm K}$ ratios is 5.13 $\rm 10^{32} erg s^{-1} Hz^{-1}$ (corresponding to $M_{\rm K} = 5.19 ~{\rm ABmag})$.\
Observations and sample used
============================
The Great Observatories Origins Deep Survey Southern (GOODS-S) field is among the best observed fields for the purpose of cosmological studies. Its wavelength coverage is exceptional, combining photometric observations from the UV to the IR [^1] and spectral surveys to measure as many redshifts as possible [@kurk13 and reference therein]. We select galaxies in this field with accurate measurements of the UV, visible, NIR, and IR rest frame emissions.\
In the framework of the MUSYC project, [@cardamone10] compiled a uniform catalogue of multi-wavelength photometry for sources in GOODS-S, incorporating the GOODS [*Spitzer*]{} IRAC and MIPS data ([@dickinson03]). In addition to broadband optical data, they used deep intermediate-band (IB) imaging from the Subaru telescope to provide photometry with fine wavelength sampling and to estimate more accurate photometric redshifts. In a previous work, we used these data to trace the detailed shape of the UV rest-frame continuum and to perform an accurate measurement of the dust attenuation curve ([@buat11; @buat12]). As mentioned by [@cardamone10] , the fluxes of extended sources may be underestimated in the MUSYC catalogue. The reason is that total fluxes are deduced from aperture fluxes using the SExtractor’s AUTO fluxes and a correction based on point source (stellar) growth curves. For extended sources this correction underestimates the total flux by a factor that depends on both the size and magnitude of the sources . Our selection of sources with $z > 1$ ensures that we avoid this potential problem.\
We restrict the field to the $10' \times 10'$ observed by the PACS instrument ([@poglitsch10]) on board the $Herschel$ Space Observatory ([@pilbratt10]) at 100 and 160$\mu$m, as part of the GOODS-$Herschel$ key programme ([@elbaz11]). The GOODS-$Herschel$ catalogue is obtained from source extraction on the PACS images performed at the prior position of ${\it Spitzer}$ 24 $\mu$m sources, as described in [@elbaz11] and in the documentation provided with the data release [^2].
We start with the sources detected at 24 $\mu$m with a S/N ratio larger than 3 and we adopt the Spitzer/IRAC detections used to extract the 24 $\mu$m sources ([@elbaz11]). These sources are cross-correlated with the MUSYC catalogue with a tolerance radius between the IRAC and optical coordinates equal to 1 arcsec. We further restrict the sample to sources that are not detected in the MUSYC X-ray catalogue of [@cardamone08]. Our field corresponds to the very deep $Chandra$ Deep Field-South survey reaching a limiting flux of $\rm 1.7 10^{-16} erg s^{-1} cm^{-2}$ in the 0.5-2 keV band (CDFS-S A03). We also select sources such that they have a single optical/UV counterpart within 3 arcsec. Particular care is given to the redshift of the sources: we select only galaxies of the MUSYC catalogue with a reliable spectroscopic redshift. All the spectra were taken with VLT spectrometers FORS and VIMOS. The quality of the redshift corresponds to more than 60$\%$ confidence level for the VIMOS surveys and labelled to be ’high quality, secure, and likely’ determinations for the FORS data (more detail can be found in the readme of the MUSYC catalogue and reference therein). As in Buat et al. (2012) (hereafter Paper I), we consider only sources with at least two measurements at $\lambda<1800\AA$ in the rest frame of galaxies (broad or IB bands ) with a S/N ratio larger than 5. This condition ensures that we have a good definition of the UV continuum in the wavelength range mostly used to measure star formation rates ($1500-1600 \AA$) and for which calibrations are available in the literature. We obtain 312 galaxies that satisfy all these conditions. To combine the PACS observations with this sample we need to apply de-blending techniques to measure fluxes because of the large size of the PACS beam ([@elbaz11]). We follow the prescription given in the documentation provided with the data release and only consider PACS measurements (with S/N > 3) for sources without bright neighbours defined as being brighter than half the flux density of the source and closer than 1.1 of the FWHM of the PSF. We add a further condition that no 24 $\mu$m bright neighbour must be found close to the 100 and 160 $\mu$m detections (i.e. no source brighter than half the flux density of the source at 24 $\mu$m and closer than 1.1 of the full width half maximum (FWHM) of the point spread function (PSF) at 100 or 160 $\mu$m). In the end, we obtain 100 $\mu$m fluxes for 92 out of the 312 sources and 160 $\mu$m fluxes for 54. Finally, for z $>$ 2, we only keep the sources detected with PACS since the 24 $\mu$m data correspond to rest frame wavelength lower than 8 $\mu$m and are not considered reliable to measure total IR emission from dust. With only two sources at $z > 3$, we reduce the sample to the 236 sources with $1<z<3$. We detect 32 objects with PACS at 100 and 160 $\mu$m, 40 only at 100 $\mu$m, and eight only at 160 $\mu$m. The redshift distribution is shown in Fig.\[zdist\]. We compile 28 photometric bands from U to 24 $\mu$m: the optical and NIR broadbands ($U, U38, B, V, R, I, z, J, H, K$), 13 IB bands from 427 to 738 nm, the four IRAC bands, and the MIPS1 24 $\mu$m band. We apply the revised IRAC selection criteria of [@donley12] to check that the 230 galaxies detected in the four IRAC bands all fall out of the AGN selection region defined in the $f_{5.8}/f_{3.6}$ and $f_{8}/f_{4.5}$ colour plot. The rest-frame luminosity at 1530$\AA$ (corresponding to the FUV $GALEX$ filter) of each galaxy is obtained by modelling a powerlaw, $f_\lambda ({\rm erg\,cm^{-2}\,s^{-1}\,\AA^{-1}}) \propto \lambda^{\beta}$ between 1300 and 2500 $\AA$, in the rest frame of the sources. In the following, we define $L_{\rm FUV}$ as $\nu L_{\nu}$ at 1530$\AA$.
![Redshift distribution of the sources. The light blue histogram is for sources detected in at least one PACS filter.[]{data-label="zdist"}](figure1.pdf){width="\columnwidth"}
Parameter Symbol Range
----------------------------------------------------------- --------------- -----------------------------------------------------------------------------------------
Amount of dust attenuation $A_V$ [**0.25, 0.5, 0.75**]{}, 0.9,[**1.05**]{}, 1.2, 1.35, [**1.5**]{}, 1.65, [**1.8**]{}mag
Two stellar populations , age free
age of the stellar population $t_f$ [**1, 2, 3, 4, 5**]{}Gyr
$e$-folding rate of the old stellar population $\tau$ [**1, 3**]{}Gyr
age of the young stellar population $t_{\rm ySP}$ [**0.01, 0.03, 0.1, 0.3**]{}Gyr
stellar mass fraction due to the young stellar population $f_{\rm ySP}$ [**0.01, 0.02, 0.05, 0.1, 0.2**]{}, 0.5
One stellar population with an exponential SFR, age free
age of the stellar population $t_f$ 0.05,0.1,0.25, 0.5,1,2,3, 4, 5Gyr
$e$-folding rate of the stellar population $\tau_1$ 0.5,1, 3, 5, 10Gyr
Models with a fixed age for the stellar population
$1<z<1.2$ $t_f$ 5 Gyr
$1.2\le z\le 1.5$ $t_f$ 4 Gyr
$1.5\le z \le 2$ $t_f$ 3 Gyr
$2\le z \le 3$ $t_f$ 2 Gyr
SED fitting
===========
The SED fitting is performed with the CIGALE code (Code Investigating GALaxy Emission)[^3] developed by [@noll09]. The code CIGALE combines a UV-optical stellar SED with a dust component emitting in the IR and fully conserves the energy balance between the dust absorbed stellar emission and its re-emission in the IR. In the present work, we do not perform as detailed a study of dust attenuation as in Paper I. Instead, we keep the parameters of the attenuation curve (amplitude of the $2175 \AA$ bump and slope of the UV attenuation curve) free since dust attenuation curves are expected to vary among galaxies ([@witt00; @inoue06]). We have checked that the estimated output parameters are fully consistent with the results obtained in Paper I. We adopt the stellar populations synthesis models of [@maraston05], a Kroupa ([@kroupa01]) initial mass function (IMF), and a solar metallicity as our baseline. Different star formation histories are considered as described below.\
Dust luminosities (L$_{\rm IR}$ between 8 and 1000$\mu$m) are computed by fitting [@dale02] templates and are linked to the attenuated stellar population models: the stellar luminosity absorbed by the dust is re-emitted in the IR. The validity of [@dale02] templates for measuring total IR luminosities of sources detected by $Herschel$ is confirmed by the studies of [@elbaz10; @elbaz11]. A single parameter $\alpha$ describes these templates, defined as the exponent of the distribution of dust mass over heating intensity. When the source is detected in at least one PACS band, the input values of $\alpha$ are 1, 1.5, 2, and 2.5. As in Paper I, when PACS data are not available $\alpha$ is assumed to be 2, which corresponds to the average value found for galaxies detected with PACS. We have checked that the predicted fluxes in the PACS bands are consistent with a non-detection of these galaxies with PACS at a 3$\sigma$ level quoted by [@elbaz11]. The input parameter measuring dust attenuation is the attenuation in the V band (for the young stellar population if two stellar populations are involved), the input values used in this work are listed in Table 1. Dust attenuation in the FUV band is also defined as an output parameter of the code.\
The parameters are all estimated from their probability distribution function (PDF) with the expectation value and its standard deviation, it is described as the ’sum’ method in [@noll09] and [@giovannoli11]. In addition to the input parameters, the output parameters considered in this work will be $M_{\rm star}$, SFR, instantaneous and averaged over 100 Myr, ages of the stellar populations, and dust attenuation in the FUV ($A_{\rm FUV}$).
Star formation histories
------------------------
We use different star formation histories (SFHs) since we want to test their influence on parameter estimations. All the input parameters related to the SFH are listed in Table 1. Three scenarios are implemented in CIGALE and we consider all of them. Models are built with an age of creation of the first stars that can be either a free or fixed parameter. To fix the age of the stellar population we follow [@maraston10] (see also [@pforr12]) and adopt a redshift formation $z_f \simeq 8$. Practically the whole redshift range is split to four intervals with different age models corresponding to a redshift formation of 8 for the central redshift of each interval (Table 1). The different SFH considered in this work are:
1. A single stellar population with exponentially decreasing SFR and an e-folding rate $\tau_1$, called hereafter decl.-$\tau$ model. The age $t_f$ of the stellar population is left a free parameter since adopting a fixed age would be unrealistic for active star forming galaxies and would not produce reliable results. This model is still commonly used in the literature ([@ilbert13; @muzzin13]) although it has long been identified to be unrealistic ([@boselli01]), leading to very young ages for the stellar populations ([@maraston10]).
2. A single stellar population with exponentially increasing SFR, called hereafter rising-$\tau$ model. The parameters are identical to those for the decl.-$\tau$ model, but with the age of the stellar population left either free or fixed. Several recent studies of distant galaxies prefer similar scenarios with rising SFR ([@maraston10; @papovich11; @pforr12; @reddy12]).
3. Two stellar populations: a recent stellar population with a constant SFR on top of an older stellar population created with an exponentially declining SFR. The parameter $t_f$ is the age of the older stellar population, it is chosen to be either free or fixed. The age $t_{\rm ySP}$ of the young component is always a free parameter. The two stellar components are linked by their mass fraction $f_{\rm ySP}$. Such models are introduced since they are expected to better reproduce real systems, which experience several phases of star formation ([@papovich01; @borch06; @gawiser07; @lee09]).
The complete grid of values is used to build the PDFs. In what follows we sometimes refer to Decl.-$\tau$ and rising-$\tau$ models as $\tau$-models. For the two-populations model, a reduction factor of the visual attenuation, $f_{\rm att}$, is applied to the old stellar population to account for the distributions of stars of different ages ([@charlot00; @panuzzo07]). From our previous analyses, we adopt $f_{\rm att}=0.5$. The results are not sensitive to the exact value of this parameter.\
More complex SFHs are also proposed in the literature. Delayed SFR ($ t \exp(-t/\tau$)) are not fully conclusive overall ([@lee10; @lee11; @schaerer13]). More realistic histories derived from modelling of galaxy evolution combining power law and exponential variations [@buat08; @behroozi12] are promising. They are not yet implemented in CIGALE, but will be in its future version (Burgarella et al. in preparation). The treatment of emission lines is not optimal in the version of the code used in here. As a result we do not consider sources with $z<1$ and those for which IB filters enter the rest-frame optical range where emission lines can have a large impact ([@kriek11; @schaerer13]).
![Distributions of $\chi^2$ values obtained for the best fits and plotted in a logarithmic scale. Black: two-populations model, green: decl.-$\tau$ model, blue: rising-$\tau$ model. The solid lines are obtained for models with the age of the stellar population left free, the dashed lines correspond to a fixed age for the stellar population. []{data-label="Khi2dist"}](figure2.pdf){width="\columnwidth"}
Results of the spectral energy distribution fitting process
-----------------------------------------------------------
An initial, global, comparison of the three models of SFH can be performed by comparing the reduced best-fit $\chi^2$ distribution given by the code as shown in Fig. 2. Note that the derived physical parameters are not directly retrieved from the best model, but from the analysis of their probability distribution function calculated with all the input models. The number of degrees of freedom to calculate the reduced $\chi^2$ varies from 4 for the fixed-age $\tau$-models to 7 for the free-age two-populations model. The best-fit $\chi^2$ distributions appear similar for all models. Best-fit $\chi^2$ values lower than 3 are found for more than $\sim 90 \%$ of the sample with median value of $\chi^2$ equal to 1 for both fixed-age and free-age two-populations models and 1.3 for the one-population models (age-free decl.$\tau$ and age free and fixed rising.$\tau$ models). We conclude that the models considered reproduce our data satisfactorily with a slightly better fit obtained with the two-populations models.
### Stellar ages
![Stellar ages estimated with CIGALE. Upper panel: ages of the (oldest) stellar population for free-age models as a function of redshift. Black triangles: two-populations model, green circles: decl.-$\tau$ model, magenta crosses: rising-$\tau$ model. The red steps correspond to the age adopted for the models with fixed ages. Lower panel: mass weighted ages for all the models: same symbols as in the upper panel are used for the free-age models, blue squares and red diamonds are added for the fixed-age rising-$\tau$ model and two-populations models, respectively. The curves correspond to the age at redshift z for a redshift formation $z_f =8$ (thick line), $z_f=5$ (thin line), and $z_f =2.5$ (dotted line). []{data-label="z-age"}](figure3a.pdf "fig:"){width="\columnwidth"} ![Stellar ages estimated with CIGALE. Upper panel: ages of the (oldest) stellar population for free-age models as a function of redshift. Black triangles: two-populations model, green circles: decl.-$\tau$ model, magenta crosses: rising-$\tau$ model. The red steps correspond to the age adopted for the models with fixed ages. Lower panel: mass weighted ages for all the models: same symbols as in the upper panel are used for the free-age models, blue squares and red diamonds are added for the fixed-age rising-$\tau$ model and two-populations models, respectively. The curves correspond to the age at redshift z for a redshift formation $z_f =8$ (thick line), $z_f=5$ (thin line), and $z_f =2.5$ (dotted line). []{data-label="z-age"}](figure3b.pdf "fig:"){width="\columnwidth"}
In the upper panel of Fig. \[z-age\] we compare the ages $t_f$ when star formation begins, obtained for each model considered. The estimated ages are reported with the ones adopted for the fixed-age models. Very young and quite unrealistic ages for the beginning of the star formation are found with the free-age decl.-$\tau$ model. With the free-age rising-$\tau$ model the estimated ages are higher, yet remain quite low. It is an illustration of the well-known outshining of the young populations in galaxies forming stars actively as nicely illustrated by [@maraston10] (their Fig. 12) . Ages obtained with the free-age two-populations model are more realistic: combining a several billions-years-old population with a much more recent one ($< 300$ Myr) gives more flexibility to account for the main light production by young stars with an underlying older population dominating the mass. The ages of the old population found with the free-age two-populations model correspond to a redshift formation from 2.5 (for galaxies at $z=1$) to five (for galaxies at $z=2$). If we assume that we are observing the same population of galaxies from $z=2-3$ to $z=1$, the age of the stellar populations found at different redshifts must be consistent: it is not the case for our free-age models. However, we may observe different galaxies at $z=1$ and $z=2$ since galaxies active in star formation at $z=2$ may become quiescent at $z=1$. Combining the average SFH [@heinis13b] derive from the SFR-$M_{\rm star}$ relation at different redshifts and assuming that galaxies exit the main sequence when they reach a mass characteristic of quiescent systems at the corresponding redshift, [@heinis13b] estimate the time galaxies can stay on the main sequence. Galaxies with $M_{\rm star} > 10^{10} M_{\odot}$ (which correspond to the bulk of our sample as shown in Section 3.2.2) located on the main sequence at $z\ge 2$ reach their quenching mass before $z=1$ and may well have evolved out of the main sequence . [@heinis13b] also find that time spent on the main sequence decreases when $M_{\rm star}$ increases.\
The mass weighted age is also an output parameter of CIGALE. This age is representative of the average age of stellar populations. The values are reported in the lower panel of Fig. \[z-age\] as a function of redshift. The general trends are similar to those found in the upper panel with global shifts to lower ages. The free-age rising.$\tau$ and decl.-$\tau$ models lead to similar weighted ages whereas $t_f$ values were found to be larger for the free-age rising.$\tau$ models. For these models, the bulk of the stellar mass is built well after the beginning of the star formation. The fixed-age rising.$\tau$ model leads to slightly larger, but still very short, mass weighted ages.\
The free-age decl.-$\tau$ models lead to very unrealistic ages, but we keep them, because of their popularity. Concerning the rising-$\tau$ models, the free-age option also leads to very young ages. Rising-$\tau$ models are commonly used by adopting a fixed formation redshift, which allows a high redshift formation with an average age for the bulk of the stellar population that remains young ([@maraston10; @papovich11; @pforr12]). Therefore, we decided to keep only the fixed-age rising-$\tau$ model. For the two-populations models, the free-age option leads to plausible ages as discussed above, therefore we will keep both free-age and fixed-age models.
For the purpose of comparison between all these models, we adopt the free-age two-populations model as our baseline in the following.
### Stellar mass and star formation rate determinations
#### Variation of the star formation history:
SFR and $M_{\rm star}$ estimates depend a priori on the assumed SFH. In this section, we consider the instantaneous SFR.\
In Fig. \[Mstars\], we compare $M_{\rm star}$ values for all the models considered in this work, with our baseline model taken as the reference (x-axis). A good agreement is found with the rising-$\tau$ model with $ \Delta(\log(M_{\rm star-ref.mod} )-\log(M_{\rm star-rising-\tau }))~= 0.03\pm 0.08 $ dex. Substantial differences are found between the reference model and the same model with $t_f$ fixed: fixing the age of the oldest stellar population increases the stellar mass by $0.12\pm 0.04$ dex. The lowest values of $M_{\rm star}$ are found for the free-age decl.-$\tau$ models with $\Delta(\log(M_{\rm star-ref.mod} )-\log(M_{\rm star-decl.-\tau }))~~= 0.17 \pm 0.09$ dex. The difference between the extreme cases (free-age decl.-$\tau$ and fixed-age two-populations models) reaches 0.3 dex.\
SFR determinations are much more consistent with each other. The agreement is almost perfect between both two-populations models. The star formation is dominated by the young stars formed in the more recent burst, which is fitted in the same way whether or not the age of the older population is free or fixed. A very slight shift towards higher SFR is found for the two-populations models as compared to the two other ones: $\Delta(\log(SFR_{\rm 2 pop})-\log(SFR_{\rm decl.-\tau})~=~0.04$ dex and $\Delta(\log(SFR_{\rm 2 pop})-\log(SFR_{\rm rising-\tau})~=~0.06$ dex.\
We confirm that the choice of the SFH changes the $M_{\rm star}$ determinations significantly ([@pforr12]). Because they are outshined by young stars, older stellar populations are hidden and $\tau$- models are likely to give only lower limits to the total stellar mass ([@papovich01; @pforr12]). Models considering an old and young population are known to yield higher masses ([@papovich01; @borch06; @lee09; @pozzetti07]). The measurement of SFR are found to be robust against changes in the SFH. $\tau$ models yield results consistent with two-populations models with a constant star formation for the recent burst. Actually, [@reddy12] have shown that for ages larger than $\sim 100$ Myr rising-$\tau$ models lead to a constant production of the UV light. In the present study stellar populations are older than 2 Gyr for the rising-$\tau$ model, so we expect a robust determination of the SFR with the fit of the intrinsic UV continuum.
The effective SFH we obtain for the decl.-$\tau$ model is close to a constant SFR. The $t_{\rm f}/\tau$ ratio is always lower than unity and the age of the stellar population is larger than 100 Myr for $98\%$ of our galaxies, ensuring a stationarity in the production of the UV light. Therefore we expected an agreement between our SFR estimations. The presence of IR data provides a robust measurement of the dust attenuation, allowing us to fit the true intrinsic UV continuum of our galaxies. The large impact of dust attenuation corrections will be discussed in Sect. 4 and 5.
#### Variation of the IMF:
Throughout this paper we adopt a Kroupa IMF. CIGALE allows us to use a Salpeter IMF ([@salp55]). Several authors have studied the influence of the choice of the IMF on the SFR and $M_{\rm star}$ determinations ([@bell03; @brinchmann04; @pforr12]). We confirm that changing the IMF from Kroupa to Salpeter increases SFR and stellar masses by a constant factor found equal to $0.17\pm0.01$ dex and $0.21\pm0.01$ dex respectively, independent of the adopted SFH.
#### Variation of the metallicity:
Throughout the paper we adopt a solar metallicity ($\rm Z_\odot$), but we can run CIGALE with Maraston models with $\rm Z_\odot/2$ and $\rm 2 Z_\odot$. The comparison of the SFR and $M_{\rm star}$ determinations for these different metallicities are shown in Fig. \[metal\]. The values obtained with non-solar metallicities correlate very well with those obtained with $\rm Z_\odot$. A very small systematic shift is found for $M_{\rm star}$ determinations, the stellar masses measured with a non-solar metallicity being lower than the reference metallicity by $\sim 0.05$ dex. This is likely due to the fact that stars in both sub-solar and super-solar models have slightly smaller turnoff masses (Maraston, private communication). The effect is larger for SFR measurements in the case of super-solar metallicity. The SFR is increased by 0.20 dex for $\rm 2 Z_\odot$ models (and decreased by only 0.03 dex for $\rm Z_\odot/2$). This shift is attributed to a decrease of the UV photons for a given SFR in super-solar environment.
![ Comparison of $M_{\rm star}$ (upper panel) and SFR (lower panel) determinations from the different models. The x-axis is from the baseline model (free-age 2-populations), the y-axis corresponds to free-age decl.-$\tau$ (green circles), fixed-age rising-$\tau$ (blue dots), and two-populations (red diamonds) models. Typical uncertainties on parameter estimations are indicated by a black cross. $\Delta(SFR) = \Delta(\log(SFR_{\rm 2 pop})-\log(SFR_{\rm \tau-model})$ and $\Delta(M_{\rm star} = \Delta(\log(M_{\rm star-2 pop} )-\log(M_{\rm star-rising-\tau }) ).$[]{data-label="Mstars"}](figure4a.pdf "fig:"){width="\columnwidth"} ![ Comparison of $M_{\rm star}$ (upper panel) and SFR (lower panel) determinations from the different models. The x-axis is from the baseline model (free-age 2-populations), the y-axis corresponds to free-age decl.-$\tau$ (green circles), fixed-age rising-$\tau$ (blue dots), and two-populations (red diamonds) models. Typical uncertainties on parameter estimations are indicated by a black cross. $\Delta(SFR) = \Delta(\log(SFR_{\rm 2 pop})-\log(SFR_{\rm \tau-model})$ and $\Delta(M_{\rm star} = \Delta(\log(M_{\rm star-2 pop} )-\log(M_{\rm star-rising-\tau }) ).$[]{data-label="Mstars"}](figure4b.pdf "fig:"){width="\columnwidth"}
![ Comparison of $M_{\rm star}$ (upper panel) and SFR (lower panel) determinations for different metallicities. The quantities plotted on the x-axis are obtained with the baseline model assuming a solar metallicity, the y-axis corresponds to determinations with a half solar (dots) and twice solar (crosses) metallicity.[]{data-label="metal"}](figure5a.pdf "fig:"){width="8cm" height="6cm"} ![ Comparison of $M_{\rm star}$ (upper panel) and SFR (lower panel) determinations for different metallicities. The quantities plotted on the x-axis are obtained with the baseline model assuming a solar metallicity, the y-axis corresponds to determinations with a half solar (dots) and twice solar (crosses) metallicity.[]{data-label="metal"}](figure5b.pdf "fig:"){width="8cm" height="6cm"}
![Comparison between $SFR_{\rm IR,FUV}$ (x-axis) and $SFR_{\rm SED}$ (y-axis): the two upper panels are for $\tau$-models , the two lower panels for the two-populations model, the difference between $SFR_{\rm SED}$ and $SFR_{\rm IR,FUV }$ is plotted against the age of the (youngest) stellar population. Symbols are the same as in Fig. \[z-age\].[]{data-label="comp-SFR"}](figure6a.pdf "fig:"){width="\columnwidth"} ![Comparison between $SFR_{\rm IR,FUV}$ (x-axis) and $SFR_{\rm SED}$ (y-axis): the two upper panels are for $\tau$-models , the two lower panels for the two-populations model, the difference between $SFR_{\rm SED}$ and $SFR_{\rm IR,FUV }$ is plotted against the age of the (youngest) stellar population. Symbols are the same as in Fig. \[z-age\].[]{data-label="comp-SFR"}](figure6b.pdf "fig:"){width="\columnwidth"}
![Difference between the SFR averaged over 100 Myr and the instantaneous one given by CIGALE plotted as a function of the age of the youngest stellar population ($t_f$ for the $\tau$-models and $t_{\rm ySP}$ for the two-populations model). Triangles are for the two-populations model, small circles for the rising-$\tau$ model and large circles for the $\tau$ model. The absolute value of the decreasing rate $\tau$ is color coded for $\tau$-models. []{data-label="sfr100"}](figure7.pdf){width="\columnwidth"}
### Star formation rate calibrations
We now compare the SFR found with our fitting method ($SFR_{\rm SED}$) to the one deduced directly from recipes converting FUV and IR luminosities into SFRs. The standard calibrations assume a constant SFR over $\sim 10^8$ years, which is sufficient to reach a stationary state for the UV production. Under this assumption, the intrinsic FUV luminosity is directly proportional to the current SFR. The total IR ($\rm \sim 5 - 1000 \mu m$) luminosity is related to the SFR by assuming that the bolometric emission of young stars is absorbed by dust and re-emitted in IR ([@kennicutt98]). Here we consider the total SFR as the sum of SFR obtained with the IR and FUV (not corrected for dust attenuation) luminosities. We do not account for heating of dust by older stars since all stellar populations are quite young. [@buat08] obtained a calibration for a Kroupa IMF and a constant SFR over $10^8$ years. The observed FUV luminosity is calculated at 1530 $\AA$ and the total IR luminosity is integrated over the whole IR SED. It is an output of the CIGALE code, which is found independent on the assumed SFH. The SFR is calculated with the formula: $$SFR_{\rm IR,FUV} = SFR_{\rm IR}+SFR_{\rm FUV} = L_{\rm IR}/10^{9.97}+L_{\rm FUV}/10^{9.69}$$ where the SFR is expressed in $\rm M_\odot yr^{-1}$ and the luminosities in $\rm L_\odot$\
The comparison between $SFR_{\rm IR, FUV} $ and $SFR_{\rm SED}$ is shown in Fig.\[comp-SFR\]. Both estimates are very consistent for $\tau$-models since most of the sample stellar populations are older than $10^8$ years with a SFH close to a constant one. There is a small shift between $ SFR_{\rm IR,FUV}$ and $SFR_{\rm SED}$ for the two-populations model: the young population is younger than $10^8$ years ($t_{\rm ySP} = 76 \pm 38$ Myr)) and the UV production does not reach a fully steady production rate, leading to a slight underestimate of the instantaneous SFR with $ SFR_{\rm IR,FUV}$. The difference is tightly correlated to $t_{\rm ySP}$, the age of the young stellar population, but remains very modest on average ($0.05\pm 0.03$ dex) (Fig.\[comp-SFR\], lower panel). The same effect is observed for decl.-$\tau$ models with very young stellar populations, leading to $SFR_{\rm SED}$ larger than $ SFR_{\rm IR,FUV}$ (Fig. \[comp-SFR\], upper panel). The few galaxies for which $SFR_{\rm SED}<SFR_{\rm IR,FUV}$ correspond to objects with a very low mass fraction locked in the young stellar component ($\simeq 2\%$): the SFH is dominated by the decreasing old component, which is not constant over the last $10^8$ years but, instead, has slightly decreased.\
CIGALE also provides the SFR averaged over the last 100 Myr $<SFR_{100}>$. The difference between $<SFR_{100}>$ and $SFR_{\rm SED}$ are plotted in Fig.\[sfr100\] against the age of the (youngest) stellar population. For $\tau$-models, the difference between the averaged and instantaneous SFR depends on the age of the stellar population and on the absolute value of e-folding rate $\tau$, but the mean difference remains very small (-0.01 dex for the decl.-$\tau$ model, -0.03 dex for the rising-$\tau$ one). The situation is very different for the two-populations model: the difference between the averaged and instantaneous SFR strongly depends on the age of the young stellar population, which dominates the current star formation. The mean difference reaches -0.20 dex, but much larger differences can be found for individual objects. Very similar trends are found when $ SFR_{\rm IR,FUV} $ is used instead of the instantaneous SFR (not shown here). This clearly demonstrates that the SFR measured with UV and IR data, either with SED fitting or with empirical calibrations, are not equivalent to a SFR averaged over 100 Myr.
Star formation rate-stellar mass relation
-----------------------------------------
{width="15cm"}
We have shown that the values of SFR and $M_{\rm star}$ depend on the assumptions made to derive them. We now explore the consequences of these variations on the SFR-$M_{\rm star}$ scatter plot. The SFR-$M_{\rm star}$ relation is expected to evolve with redshift ([@noeske07; @wuyts11; @karim11]). We split the sample into four redshift bins $z<1.2$, $1.2<z<1.7$, $1.7<z<2$, and $2<z<3$. The SFR and $M_{\rm star}$ deduced from the SED fitting are plotted in Fig. \[MS-allSFH\]. Very similar plots are found using $SFR_{\rm IR, FUV} $ as expected from the very good correlation found between both SFR estimates. Different relations from the literature and covering our redshift range are overplotted: the relations of [@elbaz07] and [@daddi07] at $z=1$ and 2 respectively, and the relations of [@heinis13b] at $z=1.5$ and 3.
Slight variations are seen between both two-populations and decl.-$\tau$ models, caused by the difference in $M_{\rm star}$ measurements. The dispersion is large in all the scatter plots (except for the rising-$\tau$ model as discussed below) and we do not try to provide any relation. Our galaxy sample is built to have the best wavelength coverage, but is not complete in any sense and is not suited to derive a representative SFR- $M_{\rm star}$ relation. Our selection of galaxies observed both in FUV and IR (rest-frame) is likely to bias towards objects with a high star formation activity. Free-age and fixed-aged two-populations models lead to a similar dispersion with a slight shift towards larger $M_{\rm star}$ and lower specific star formation rate (SSFR) for the fixed-age model. Lower $M_{\rm star}$ and similar SFR are obtained with the decl.-$\tau$-model as compared to the results obtained with the other models and the discrepancy is larger when $M_{\rm star}$ decreases (Section 3 and Fig.\[Mstars\]). It induces a flatter SFR-$M_{\rm star}$ relation as compared to the other models. The residual RMS error obtained with a linear fit for each redshift bin is $\sim 0.3$ for both two-populations and decl.$\tau$ models. The rising-$\tau$ model leads to a steeper, well-defined relation SFR-$M_{\rm star}$, which does not much evolve with redshift and the residual RMS error to linear fits is reduced to $\sim 0.2$. This reduction is expected since rising SFHs have been introduced to explain the small scatter found in the SFR-$M_{\rm star}$ relation and the non-evolution of the SSFR at very high redshift ([@maraston10; @papovich11]). This variation of the RMS error illustrates that the tightness of the derived SFR-$M_{\rm star}$ relation also depends on the way these two quantities are derived.
Fitting spectral energy distributions with a reduced dataset
============================================================
{width="17.5cm" height="12.5cm"}
We now check the importance of the wavelength coverage to estimate our physical parameters. We identify three sets of data whose influence has to be investigated: the IB filters, sampling the UV continuum and sensitive to the recent SFH; the NIR rest frame data (K and all IRAC bands), often presented as the best tracers of $M_{\rm star}$; and the IR emission from dust sampled by IRAC4, MIPS, and PACS data, which is expected to provide a strong constraint to dust attenuation. We perform fits (i) omitting data obtained with all IB filters, (ii) omitting NIR data and (iii) omitting IR data. When one dataset is omitted, all the other data are kept. It may appear unrealistic from an observational point of view since galaxies detected in the IR are all detected with IRAC. Nevertheless, our aim is to test the influence of a specific wavelength domain on the determination of the physical parameters and to disentangle the different potential biases. The tests are performed for all SFH models, but are presented here for the two-populations model only. Similar trends are found for the other models.
In Fig. \[comp-bands\], the estimated values of the age of the stellar populations ($t_{\rm f}$ and $t_{\rm ySP}$), the mass fraction in the youngest population ($f_{\rm ySP}$), the dust attenuation in FUV ($A_{\rm FUV}$), SFR and $M_{\rm star}$, obtained for cases (i), (ii), and (iii), are compared to those obtained with the full dataset (from Section 3). With CIGALE, the internal accuracy of the different parameter estimations is measured by the standard deviation of the probability distribution function of each parameter and for each object. The mean values of this standard deviation, averaged over the full sample of 236 galaxies are listed in Table 2 for each dataset.\
All parameter estimates and the associated averaged dispersions are very similar with and without IB data, which play a minor role in these estimations. Only the values of $t_{\rm f}$ are slightly larger without IB data, with no consequence in the determination of SFR and $M_{\rm star}$. We recall that the IB filters cover only the UV rest-frame range, which is essentially featureless. The situation is likely to be very different for IB sampling of the visible range with strong emission lines, which can strongly modify the overall spectrum ([@schaerer13]). A good sampling of the UV continuum is useful to constrain the dust attenuation curve (Paper I) but does not add any useful information on the recent SFH provided that broadband photometry is available to measure the UV emission.\
The decision of whether or not to include the NIR rest-frame data modifies most of the parameters, except $A_{\rm FUV}$ and SFR, which are exclusively related to the young stellar population. We find moderate shifts for the ages of the stellar populations. When NIR data are excluded, $t_{\rm f}$ is lower by up to 0.5 Gyr when $t_{\rm f} > 2.5$ Gyr and the young stellar populations are found to be 10% younger. The impact on the determination of $M_{\rm star}$ is a small systematic shift, $M_{\rm star}$ being lower by $15\%$ on average when NIR data are excluded, with a dispersion reaching 0.16 dex between estimations with and without NIR data. The intrinsic uncertainty in the determination of $M_{\rm star}$ also increases from 0.21 dex for the full dataset to 0.33 dex without NIR data.\
Without IR data, results substantially change for all parameters linked to recent star formation. As expected, the parameters related to the old component ($t_{\rm f}$, $M_{\rm star}$) are much less affected. A substantial dispersion is observed in Fig. \[comp-bands\] between the values of $t_{\rm ySP}$, $f_{\rm ySP}$, SFR and $A_{\rm FUV}$, estimated with and without IR data (corresponding to 0.14 dex, 0.16 dex, 0.18 dex, and 0.44 mag, respectively). Looking at Table 2 it is clear that omitting IR data affects the accuracy of $A_{\rm FUV}$ and SFR. The distribution of all these parameters is flatter without IR data and SFR are larger by $20\%$ on average when IR data are excluded. This is illustrated in Fig.\[deltasfr\] with a dependence of the SFR estimates when IR data are not present on the SFR measured with the whole dataset. Under the assumption that SFR obtained with the whole dataset are reliable, in the absence of IR data, low SFR are overestimated and large SFR are underestimated, by a factor that can reach 2.5. The difference in SFR estimations strongly depends on the estimation of dust attenuation as seen in the lower panel of Fig.\[deltasfr\]. If we assume that secure measurements are obtained when including IR data, large attenuations are underestimated implying an underestimation of the SFR, the inverse trend being observed for low attenuations. This is consistent with the conclusion of [@burgarella05 and erratum] that without IR data, the SED fitting process biases towards average values of dust attenuation.
---------------------------------------------- ----------- ------- -------- -------
[**parameter** ]{} full data no IB no NIR no IR
(i) (ii) (iii)
$t_f$(Gyr) 1.09 1.07 1.10 1.10
$\log(t_{\rm ySP})$ (Gyr) 0.45 0.44 0.47 0.49
$\log(f_{\rm ySP})$ 0.49 0.48 0.52 0.53
$A_{\rm FUV}$ (mag) 0.52 0.47 0.54 0.86
$\log(SFR) ({\rm M_\odot yr^{-1}})$ 0.18 0.16 0.19 0.33
$\log(M_{\rm star}) ({\rm M_\odot yr^{-1}})$ 0.21 0.19 0.33 0.24
---------------------------------------------- ----------- ------- -------- -------
: Intrinsic uncertainty of the parameter estimation using the different datasets, measured as the dispersion of the probability function of each parameter. []{data-label="tab:dispersion-parameters"}
![ Difference in SFR estimates when IR data are omitted. Upper panel: The SFR reported on the x-axis is calculated with the whole dataset. The $\Delta(\log(SFR)$ on the y-axis is defined as the $\log(SFR)$(with the reduced dataset)-$\log(SFR)$ (with the whole dataset). Lower panel: $\Delta(\log(SFR)$ as a function of $A_{\rm FUV}$ measured with the full dataset. The difference between $A_{\rm FUV}$ is color coded. []{data-label="deltasfr"}](figure10a.pdf "fig:"){width="\columnwidth"} ![ Difference in SFR estimates when IR data are omitted. Upper panel: The SFR reported on the x-axis is calculated with the whole dataset. The $\Delta(\log(SFR)$ on the y-axis is defined as the $\log(SFR)$(with the reduced dataset)-$\log(SFR)$ (with the whole dataset). Lower panel: $\Delta(\log(SFR)$ as a function of $A_{\rm FUV}$ measured with the full dataset. The difference between $A_{\rm FUV}$ is color coded. []{data-label="deltasfr"}](figure10b.pdf "fig:"){width="\columnwidth"}
Mock galaxies
=============
We checked the internal consistency and limitations of the method we used to retrieve physical quantities of galaxies. To this aim we built a sample of artificial SEDs defined in the same filters as those used for real sources. Then we attempted to retrieve the known properties of these pseudo-galaxies and to check the reliability of parameter estimation as well as the presence of potential systematic biases.
Definition of the mock catalogue with $1<z<2$ and fitting process
-----------------------------------------------------------------
Given the few sources with $z>2$, we restrict the mock catalogue to $1<z<2$. The actual redshift is not important since the models are only dependent on the physical ages of the stellar populations. The redshift range is introduced to reproduce data corresponding to the real sample and to account for the redshifting of the rest-frame emission of galaxies.\
We only consider the free-age two-populations model. If we were to use all the parameters considered in Table 1 to simulate the artificial SEDs the total number of sources is very large and the computational time prohibitively long. Therefore, we restrict the sampling of the input parameters to the values appearing with bold characters in Table 1. We choose a single distribution of dust emission ($\alpha=2$), the dust attenuation law is fixed with parameters chosen from our previous study (Paper I). The sampling of the amount of dust attenuation is also reduced (but with the same total amplitude of variation), and we limit the mass fraction in the young stellar population to 20$\%$ at most. The SEDs are generated for all of the 30 bands (including the PACS bands) considered in this work, and a random noise, typically of the order of 10$\%$, is added to the simulated fluxes. A similar, relatively small, error is applied to all the data since our aim is not to measure the effect of varying observational conditions, but to understand the limitations intrinsic to our modelling. All galaxies are created with a total galaxy mass of $10^{10} M_\odot$: we cannot check the dynamical range of SFR and $M_{\rm star}$ but only systematic changes for these parameters. A total of 4970 artificial galaxies are created.\
We fit the mock data, using CIGALE with the whole set of input parameters, as for real galaxies and not using the reduced set used to create the catalogue (cf. Table 1) We again perform again four distinct fits: using the whole dataset, excluding IB, NIR, or IR data. Note that when considering the influence of NIR rest-frame data we are not testing the role of the stellar population models since we are creating and fitting our artificial data with the same code.\
![Upper panels: Histogram of the difference between the estimated values of $M_{\rm star}$ and SFR and the exact values from the mock catalogue. Black line: whole dataset, blue line: without IB data, green filled histogram: without NIR data, red filled histogram: without IR data, Lower panel: SSFR estimated with SED fitting are compared to the true values of the SSFR. Black point: whole dataset, green dots: without NIR data, red dots: without IR data. Results without IB are not shown since they are indistinguishable from those without IR data. []{data-label="Mstarhist"}](figure11a.pdf "fig:"){width="\columnwidth"} ![Upper panels: Histogram of the difference between the estimated values of $M_{\rm star}$ and SFR and the exact values from the mock catalogue. Black line: whole dataset, blue line: without IB data, green filled histogram: without NIR data, red filled histogram: without IR data, Lower panel: SSFR estimated with SED fitting are compared to the true values of the SSFR. Black point: whole dataset, green dots: without NIR data, red dots: without IR data. Results without IB are not shown since they are indistinguishable from those without IR data. []{data-label="Mstarhist"}](figure11b.pdf "fig:"){width="\columnwidth"}
Stellar mass and star formation rate determinations.
-----------------------------------------------------
In Fig. \[Mstarhist\] the difference between all the estimated and true values of SFR and $M_{\rm star}$ are represented for the three combinations of data[^4]. No effect on $M_{\rm star}$ and SFR is found due to the lack of introduction of IB data, which confirms the findings with the real sample, the case without IB data will not be discussed. The impact of IR data is clearly seen, especially for the SFR determinations and the lack of introduction of NIR data modifies the $M_{\rm star}$ distribution but in a much smaller amount than the IR data for the SFR.\
When all the data are considered, there is only a small systematic difference between the estimated and true values of $M_{\rm star}$: $<\Delta(\log(M_{\rm star})> =-0.07$ dex with a $1\sigma$ dispersion equal to 0.14 dex. Without NIR data the systematic shift and the dispersion are larger: $<\Delta(\log(M_{\rm star})> =-0.13$ dex and $ \sigma=0.20$ dex. We confirm the difference of 15$\%$ found in Section 4.1, with and without NIR data for real galaxies.\
The situation is worse for the SFR, when IR data are omitted: if the mean systematic difference, $<\Delta(\log(SFR)> $, remains consistent with 0, the dispersion varies from 0.05 dex with IR data to 0.30 dex without IR data. This poor determination of the SFR without IR data is clearly visible when SSFR are compared (lower panels of Fig. \[Mstarhist\] ). We confirm that the range of values of SSFR found without IR data is reduced as compared to that of the true values, which are well estimated when IR data are considered. Intrinsically, small SSFR are overestimated without NIR data, this is due to underestimation of stellar masses for these objects, since SFR are well estimated in this case.
![Upper panel: distribution of the relative difference between the estimated and true age of the old stellar population. The sample is split as a function of the true age: $t_f= 1$ Gyr, $t_f=2$ Gyr, $t_f=3$ Gyr, and $t_f=4,5$ Gyr. Lower panel: distribution of the ratio of the estimated and true age of the young stellar population in logarithmic units. The sample is split as a function of the true age: $t_{\rm ySP} = 0.01$ Gyr, $t_{\rm ySP}= 0.03$ Gyr , $t_{\rm ySP}=0.1$ Gyr, and $t_{\rm ySP}=0.3$ Gyr. []{data-label="age-mock"}](figure12.pdf){width="\columnwidth"}
![Difference between estimated and true values of SFR and $M_{\rm star}$ as a function of the difference between estimated and true ages. The colors are the same as in Fig.\[age-mock\][]{data-label="mock-2"}](figure13.pdf){width="\columnwidth"}
![Difference between estimated and true values of SFR as a function of the difference between estimated and true ages, dot are for fits with IR data, crosses correspond to fits without IR data. The difference between estimated and true values of $A_{\rm FUV}$ is shown with a color scale.[]{data-label="age-dust-sfr-mock"}](figure14a.pdf "fig:"){width="\columnwidth"} ![Difference between estimated and true values of SFR as a function of the difference between estimated and true ages, dot are for fits with IR data, crosses correspond to fits without IR data. The difference between estimated and true values of $A_{\rm FUV}$ is shown with a color scale.[]{data-label="age-dust-sfr-mock"}](figure14b.pdf "fig:"){width="\columnwidth"}
Stellar ages
------------
We explored the different estimations of stellar ages, with and without IR and NIR data. We found very similar trends whether IR data are omitted or not. Therefore, in this sub-section, we begin by studying the fit with the whole dataset, and then we focus on the difference found between the fits with and without NIR data.\
We have seen that the determination of the age of the stellar population is crucial to describing the SFH and to measuring $M_{\rm star}$, and that the impact on SFR is less important, as discussed in Section 3 and in the next sub-section. Our artificial sources are created with stellar populations spanning a large range of ages, between 1 and 5 Gyr for the old component and 0.01 to 0.3 Gyr for the young component. The relative difference between the estimated and true values of $t_{\rm f}$ is shown in Fig.\[age-mock\]. Ages lower than $\simeq 2$ Gyr are overestimated (by 127$\%$ for $t_{\rm f}=1$ Gyr and by 20$\%$ for $t_{\rm f}=2$ Gyr) whereas larger ages are underestimated (by 16$\%$ for $t_{\rm f}=3$ Gyr and by 30$\%$ for $t_{\rm f} =$ 4 and 5 Gyr). The strongest discrepancy is found for the very small ages of the oldest stellar population. [@lee09] also found that the ages of model Lyman Break Galaxies , at $z>3$ and lower than 1 Gyr, were strongly overestimated with SED fitting methods. We confirm that the ages of the old stellar populations are underestimated in typical galaxies with an old ($> 3$ Gyr) underlying stellar population. The same effect is found for the age of the young stellar population.\
These systematic shifts in age determination have a direct impact on $M_{\rm star}$ measurements as shown in Fig.\[mock-2\]. Underestimations of $t_{\rm f}$ and $M_{\rm star}$ are correlated, which explains the large dispersion found in the $M_{\rm star}$ distribution. The effect of $t_{\rm ySP}$ is less important, as expected, but the presence of a very young stellar population increases the uncertainty in the mass determination (Fig.\[mock-2\], lower panel).\
Dust attenuation
----------------
We have seen in Section 4 that dust attenuation has a major impact on SFR determinations. We re-investigate its role with our mock catalogue. The estimates of $A_{\rm FUV}$, with and without IR data, are compared in Fig.\[age-dust-sfr-mock\]. Whereas dust attenuation is very well measured with IR data, when these IR data are missing $A_{\rm FUV}$ is overestimated in systems with low attenuation and underestimated in very dusty systems, confirming the flattening of the distribution of $A_{\rm FUV}$ found in Section 4 in the absence of observed IR data. It can be seen in Fig.\[age-dust-sfr-mock\] that dust attenuation plays a crucial role for SFR determinations whereas the age of the stellar population dominating the current SFR ($t_{\rm ySP}$) only acts as a secondary parameter. When this age is overestimated the recent star formation is diluted over too large atimescale and the SFR is underestimated, as also found by [@lee09], but the effect remains very modest for our sample.\
Conclusions
===========
We measured SFR, $M_{\rm star}$, and stellar ages for a sample of 236 galaxies at $1<z<3$ in the GOODS-S field observed with an excellent wavelength coverage, including UV and IR rest-frame data. We used 28 photometric bands from U to 24 $\mu$m, and 80 sources are detected with PACS at 100 or 160 $\mu$m. We also considered intermediate band (IB) filters which sample the UV rest-frame. We performed SED fitting with the code CIGALE, which implements an energy budget between dust and stellar emission. We explored different SFH: exponentially increasing and decreasing $\tau$-models and a model combining an old decreasing SFR and a younger population of constant SFR. In a first step, the age of the main stellar population was left either fixed or free. The Decl.-$\tau$ models with a fixed redshift formation ($z_{\rm f}\simeq 8$) did not fit the data well. All the other models yield much better results at the price of young ages for free-age models, which were unrealistic with $\tau$-models. The best fits are obtained with two-populations models. Ages were higher for the free-age two-populations model. In a second step, the models selected for the study were the free-age decl.$\tau$ model (because of its popularity), the fixed-age rising-$\tau$ model, and both fixed and free-age two-populations models. We investigated the impact of the coverage of different wavelength ranges. The analysis was also based on an artificial catalogue of 4970 sources built with the input parameters of the CIGALE code. Our main results can be summarized as follows:
1. Instantaneous SFR are found independent of SFH assumptions with systematic differences lower than 10$\%$. The instantaneous SFR measured by fitting the whole SED with $\tau$-models are found fully consistent with $SFR_{\rm IR,FUV}$ measured with empirical recipes, and $15\%$ higher on average than $SFR_{\rm IR,FUV}$ for the two-populations models, with a dependence on the age of the younger stellar population. The SFR averaged over 100 Myr and instantaneous SFR agree well for $\tau$-models but not for the two-populations models.
2. The stellar masses depend on the assumed SFH and are systematically lower by a factor of 1.5-2 for the decl.-$\tau$ model as compared to all the other models considered. It is caused by an obvious underestimate of the age of the stellar population for the free-age decl.$\tau$ model. Fixed-age rising-$\tau$ and free-age two-populations models are fully consistent with $M_{\rm star}$ values lower on average by a factor of 1.3 than those obtained with the fixed-age two-populations model.
3. The IB filters sample the UV rest-frame of our galaxies, which is almost featureless. As a consequence these data are found to play a minor role, if any, in the determination of SFR, $M_{\rm star}$ as well as of stellar ages.
4. Whether or not we include NIR and IR data modify parameter estimations substantially . With IR data, SFR are measured with a dispersion of 50$\%$. Without IR data, the intrinsic dispersion reaches a factor of two and the range of estimated values is reduced when compared to the true values. The difference between estimations with and without IR is tightly correlated to the uncertainty of dust attenuation measurements. Excluding NIR data lowers $M_{\rm star}$ estimates by 15$\%$ with an increase of the intrinsic dispersion from 60$\%$ with NIR data to a factor $\sim 2$ without them.
5. Stellar age estimates are analysed with our mock catalogue. Systematic shifts are found: the shortest ages are overestimated and the largest ones underestimated with a direct impact on $M_{\rm star}$ derivations, explaining the moderate systematic shift and dispersion found between the estimated and true values of $M_{\rm star}$. The impact of stellar age uncertainties on SFR measurements is much lower, SFR being far more sensitive to dust attenuation.
6. These results have some impact on the SFR-$M_{\rm star}$relation. The assumption of different SFH modifies the SFR-$M_{\rm star}$ scatter plot. When two stellar populations are introduced, whether or not we fix the age of the oldest population has only a modest impact. The decl.-$\tau$ model leads to larger SSFR, especially for low-mass galaxies. The assumption of a rising SFH with a fixed age implies a well-defined SFR-$M_{\rm star}$ which does not evolve much with z. A smaller range of SFR is found without IR data as well as a flatter variation of the SSFR than when IR data are introduced.
This work is supported by the French National Agency for Research (ANR-09-BLAN-0224) and CNES. The GOODS-$Herschel$ data were accessed through the HeDaM database (http://hedam.lam.fr) operated by CeSAM and hosted by the Laboratoire d’Astrophysique de Marseille. The authors thank D. Elbaz, E. Daddi, and M. Bethermin for very fruitful discussions, and C. Maraston and J. Pforr for their help with stellar models. The anonymous referee’s suggestions have greatly helped the authors to improve the paper.
Bell E. F., McIntosh D. H., Katz N., Weinberg M. D., 2003, ApJS, 149, 289 Benitez, N., Moles, M., Aguerri, J.A.L. et al 2009, ApJ 692, L5 Berhoozi, P., Wechsler, R., Conroy, C., 2012, arXiv:1207.6105 Boissier, S. 2013, Planets, Stars and Stellar Systems. Volume 6: Extragalactic Astronomy and Cosmology, 141 Borch A., Meisenheimer, K., Bell, E., et al., 2006, A&A, 453, 869
Boselli, A., Gavazzi, G., Donas, J., & Scodeggio, M. 2001, , 121, 753 Brinchmann J., Charlot S., White S. D. M., Tremonti C., Kauffmann G., Heckman T., Brinkmann J., 2004, MNRAS, 351, 1151 Bruzual, G., & Charlot, S. 2003, MNRAS, 344, 1000 Buat, V., Boissier, S., Burgarella, D. et al. 2008, A&A, 483, 107 Buat, V., Giovannoli, E., Heinis, S. et al. 2011, A&A, 533, 93 Buat, V., Noll, S.., Heinis, S., et al. 2012, A&A, 545, 141 (paper I) Burgarella, D., Buat V., & Iglesias-Páramo, J. 2005, MNRAS, 360, 1413 Calzetti, D., Armus, L., Bohlin, R.C., et al. 2000, ApJ, 533, 682 Calzetti, D. 2012, Proceedings of the XXIII Canary Islands Winter School of Astrophysics: ‘Secular Evolution of Galaxies’, edited by J. Falcon-Barroso and J.H. Knapen Cardamone, C.N., Urry, P.G., Damen, M., et al. 2008, ApJ, 680, 130 Cardamone, C.N., van Dokkum, P.G., Urry, C.M., et al. 2010, ApJSS, 189, 270 Charlot, S., & Fall, S.M. 2000, ApJ, 539, 718 Conroy, C.,Gunn, J.E., 2010, APJ, 712, 833 Conroy, C.,2013, arXiv:1301.7095 da Cunha, E., Charlot, S., & Elbaz, D. 2008, MNRAS, 388, 1595 Daddi E., Dickinson, M., Morrison, G., et al. 2007, ApJ, 670, 156 Dale, D.A., & Helou, G. 2002, ApJ, 576, 159 Dickinson, M., Giavalisco, M., & the GOODS team, 2003, in “The Mass of Galaxies at Low and High Redshift”, eds. R. Bender & A. Renzini, Springer, p. 324. Donley, J. L., Koekemoer, A. M., Brusa, M., et al. 2012, , 748, 142 Elbaz, D., Daddi, E., Le Borgne, D., et al. 2007, A&A, 468, 33 Elbaz, D., Hwang, H.S., Magnelli, B., et al. 2010, A&A, 518, L29 Elbaz, D., Dickinson, M., Hwang, H.S., et al. 2011, A&A, 533, 119 Lucia, G., Monaco, P., Somerville, R. S., & Santini, P. 2009, , 397, 1776 Gawiser, E., Francke, H., Lai, K., et al. 2007, , 671, 278 Giovannoli, E., Buat, V., Noll, S., Burgarella, D., & Magnelli, B. 2011, , 525, A150 Finkelstein, F.S., Papovich, C., Salmon, B., et al. 2011, ApJ, 756, 164 Hao, C.-N., Kenniccutt, R.C., Jonhson, B.D., et al. 2011, ApJ, 741, 124 Heinis, S., Buat, V., Bethermin, M., et al. 2013, arXiv:1310.3227
Ilbert, O., Capak, P., Salvato, M., et al. 2009, ApJ, 690, 1236 Ilbert, O., Salvato, M., Le Floc’h, E., et al. 2010, ApJ, 709, 644 Ilbert, O., McCracken, H. J., Le Fevre, O., et al. 2013, arXiv:1301.3157 Inoue, A., Buat, V., Burgarella, D., et al. 2006, MNRAS, 370, 380 Karim, A., Schinnerer, E., Mart[í]{}nez-Sansigre, A., et al. 2011, , 730, 61 Kennicutt, R.C. 1998, ARA&A 36, 189 Kennicutt, R.C. & Evans, N.J. 2012, ARA&A 50, 531 Kobayashi, M. A. R., Inoue, Y., & Inoue, A. K. 2013, , 763, 3 Kriek, M., van Dokkum, P. G., Whitaker, K. E., et al. 2011, , 743, 168 Kroupa, P. 2001, MNRAS, 322, 231
Kurk, J., Cimatti, A., Daddi, E., et al. 2013, , 549, A63 Lee, S.K., Idzi, R., Ferguson, H.C. et al. 2009, ApJSS, 184, 100 Lee, S.K., Ferguson, H.C., Somerville, R., Wiklind, T., Giavalisco, M. 2010, ApJ, 725, 1644 Lee, K.S., Dey, A. Reddy, N. et al. 2011, ApJ 733, 99 Maraston, C. 2005, , 362, 799 Maraston, C., Pforr, J., Renzini, A., et al. 2010, , 407, 830 Marchesini, D., van Dokkum, P. G., F[ö]{}rster Schreiber, N. M., et al. 2009, , 701, 1765 Mitchell, P. D., Lacey, C. G., Baugh, C. M., & Cole, S. 2013, , 1998 Muzzin, A., Marchesini, D., van Dokkum, P.G., et al. 2009, ApJ, 701, 1839 Muzzin, A., Marchesini, D., Stefanon, M., et al. 2013, arXiv:1303.4409 Noeske, K. G., Weiner, B. J., Faber, S. M., et al. 2007, , 660, L43 Noll, S., Burgarella, D., Giovannoli, E., et al. 2009, A&A, 507, 1793 Pacifici, C., Charlot, S., Blaizot, J., Brinchmann, J. 2012, MNRAS, 421, 2002 Panuzzo, P., Granato, G.L., Buat, V., et al. 2007, MNRAS, 375, 640 Papovich, C., Dickinson, M., & Ferguson, H. C. 2001, , 559, 620 Papovich, C., Finkelstein, s.L., Ferguson, H.C., Lotz, J.M. Giavalisco M. 2011, MNRAS, 412, 1123 Pforr, J., Maraston, C., & Tonini, C. 2012, , 422, 3285 Pilbratt, G., Riedinger, J.R., Passvogel, T., et al. 2010, A&A 518, L1 Poglitsch, A., Waelkens, C., Geis, N., et al. 2010, A&A, 518, L2 Popescu, C. C., Tuffs, R. J., Dopita, M. A., et al. 2011, , 527, A109 Pozzetti, L., Bolzonella, M., Lamareille, F., et al. 2007, , 474, 443 Reddy, N., Pettini, M., Steidel,C. et al. 2012, ApJ, 754, 25 Rodighiero, G., Daddi, E., Baronchelli, I., et al. 2011, , 739, L40 Salim, S., Rich, R. M., Charlot, S., et al. 2007, , 173, 267 Salpeter, E. E. 1955, , 121, 161 Schaerer, D., de Barros, S., Sklias, P. 2013, A&A 549, 4 Taylor, E. N., Franx, M., van Dokkum, P. G., et al. 2009, , 694, 1171 Witt, A.N., & Gordon, K.D. 2000, ApJ, 528, 799 Wuyts, S., Franx,M., Cox, T. et al. 2009, ApJ, 696, 348 Wuyts, S., Forster-Schreiber, N., Lutz, D., et al. 2011, ApJ, 738, 106 Zibetti, S., Charlot, S., & Rix, H.-W. 2009, , 400, 1181
[^1]: http://www.stsci.edu/science/goods/
[^2]: http://hedam.lam.fr/GOODS-Herschel
[^3]: http://cigale.lam.fr
[^4]: In Section 5, all the difference between the estimated and true values of a parameter are defined as ’estimated value-true value’.
|
---
abstract: 'Graphene has raised high expectations as a low-loss plasmonic material in which the plasmon properties can be controlled via electrostatic doping. Here, we analyze realistic configurations, which produce inhomogeneous doping, in contrast to what has been so far assumed in the study of plasmons in nanostructured graphene. Specifically, we investigate backgated ribbons, co-planar ribbon pairs placed at opposite potentials, and individual ribbons subject to a uniform electric field. Plasmons in backgated ribbons and ribbon pairs are similar to those of uniformly doped ribbons, provided the Fermi energy is appropriately scaled to compensate for finite-size effects such as the divergence of the carrier density at the edges. In contrast, the plasmons of a ribbon exposed to a uniform field exhibit distinct dispersion and spatial profiles that considerably differ from uniformly doped ribbons. Our results provide a road map to understand graphene plasmons under realistic electrostatic doping conditions.'
author:
- Sukosin Thongrattanasiri
- Iván Silveiro
- 'F. Javier García de Abajo'
title: Plasmons in electrostatically doped graphene
---
Plasmons[@R1988] –the collective oscillations of conduction electrons in metals– are capable of confining electromagnetic energy down to deep sub-wavelength regions. They can also enhance the intensity of an incident light wave by several orders of magnitude. These phenomena are the main reason why the field of plasmonics is finding a wide range of applications that include single-molecule sensing,[@paper125] nonlinear optics,[@DN07] and optical trapping of nanometer-sized objects.[@JRQ11]
Recently, confined plasmons have been observed and spatially mapped in doped graphene.[@graphene1st] The level of doping in this material can be adjusted by exposing it to the electric fields produced by neighboring gates. Electrostatic doping has actually been used to demonstrate plasmon-frequency tunability[@graphene1st] and induced optical modulations in the THz[@JGH11] and infrared[@FAB11] response of graphene.
The two-dimensional (2D) band structure of pristine graphene consists of two cones filled with valence electrons and two empty inverted cones joining the former at the so-called Dirac points, which mark the Fermi level. Extra electrons or holes added to this structure form a 2D electron or hole gas that can sustain surface plasmons.[@S1986; @WSS06] Compared to noble-metal plasmons, graphene modes are believed to be long-lived excitations.[@JBS09] But most importantly, their frequency can be controlled via the above-mentioned electrostatic doping.[@JGH11; @FAB11; @graphene1st] For example, in homogeneous suspended graphene, a perpendicular DC electric field $\mathcal{E}$ applied to one side of the carbon sheet is completely screened by an induced surface charge density $-en=\mathcal{E}/4\pi$, and this in turn situates the Fermi level at an energy $E_F=\hbar v_F \sqrt{\pi|n|}$ relative to the Dirac points.[@CGP09] Here, $v_F\approx10^6\,$m/s is the Fermi velocity of graphene.
![Electrostatic doping and plasmon modes in backgated graphene ribbons. [**(a)**]{} Average Fermi energy $\langle|E_F|\rangle$ as a function of width-to-distance ratio $W/d$, normalized to the value $E_F^\infty$ obtained in the $W\gg d$ limit. The upper inset shows the $E_F$ distribution across the ribbon, normalized to $\langle|E_F|\rangle$. The lower inset shows a sketch of the geometry. [**(b)**]{} Frequency $\omega$ of the dipolar and quadrupolar plasmons, normalized to $\omega_0=(e/\hbar)\sqrt{\langle|E_F|\rangle/W}$, as obtained from the Drude model. The insets show the surface charge-density oscillating at frequency $\omega$ and corresponding to these plasmons (vertical axis) as a function of position across the ribbon (horizontal axis). The dashed curves indicate the $W\gg d$ limit.[]{data-label="Fig1"}](Fig1){width="85mm"}
Plasmons in doped graphene nanostructures have been generally studied by assuming a uniform doping electron density $n$.[@VE11; @paper176] But in practice $n$ is inhomogeneous and depends on the actual geometrical configuration. For example, in a ribbon of width $W$ placed at a distance $d$ from a planar biased backgate, $n$ shows a dramatic pileup near the edges, as it is well known in microstrip technology,[@W1964; @W1977; @SE08; @VZ10] leading to divergent local $E_F$ levels, as shown in the inset of Fig. \[Fig1\](a) for various values of $W/d$. More precisely, $n$ diverges as $\propto1/\sqrt{x}$ with the distance to the edge $x$ (and hence, $E_F\propto1/x^{1/4}$). The average of the Fermi energy over the ribbon area \[$\langle|E_F|\rangle$, see Fig. \[Fig1\](a)\] is very different from the $W/d\gg1$ limit ($E_F^\infty=\hbar v_F\sqrt{|V|/4ed}$) and diverges as[@analimit] $\sqrt{d/W}/\log(2d/W)$ in the narrow ribbon limit ($W\ll d$) for constant bias potential $V$. Therefore, the question arises, how different are the plasmon energies and field distributions in actual doped graphene nanostructures compared to those obtained for uniformly doped graphene?
Here, we analyze plasmons in doped graphene ribbons under different geometrical configurations. Specifically, we study backgated single ribbons, co-planar parallel ribbon pairs of opposite polarity, and single ribbons immersed in a uniform external electric field. For simplicity, we describe the frequency-dependent conductivity of doped graphene in the Drude model as $\sigma(\omega)=(ie^2|E_F|/\pi\hbar^2)/(\omega+i\gamma)$, where $\gamma\ll\omega$ is a small relaxation rate. The doping electron density $n$ is obtained from electrostatic boundary-element calculations, while the plasmon frequencies are computed using a discrete surface-dipole approximation (DSDA), as explained in Appendix \[appcomputation\].
[*Backgated ribbons.-*]{} A first conclusion extracted from Fig. \[Fig1\](a) is that the level of doping, quantified in the Fermi energy, is not well described by the simple capacitor analysis of the $W\gg d$ geometry. Normalizing to the average value $\langle|E_F|\rangle$, we find Fermi-energy profiles that vary between a well of sharp corners for large $W/d$ and a smoother, converged shape for small $W/d$ \[upper inset of Fig. 1(a)\]. The former limit corresponds to the ribbon in close proximity to the backgate, in which $E_F$ is nearly uniform. In contrast, at large separations ($d\gg W$) we find a profile determined by the interaction with a distant image, which converges to a well-defined shape up to an overall factor $\langle |E_F|\rangle$ evolving as shown in the main plot of Fig. 1(a).
It is convenient to normalize the ribbon plasmon frequencies to $\omega_0=(e/\hbar)\sqrt{\langle|E_F|\rangle/W}$, so that $\omega/\omega_0$ is a dimensionless number, independent of the specific ribbon width $W$ and gate voltage (i.e., $\langle|E_F|\rangle$), as proved in Appendix \[appscaling\]. For example, with $W=100\,$nm and $\langle|E_F|\rangle=0.5\,$eV, we find $\hbar\omega_0=0.085\,$eV and a dipole plasmon energy $\hbar\omega\sim0.17\,$eV (wavelength $\sim7.3\,\mu$m). With this normalization, $\omega/\omega_0$ shows just a mild dependence on $W/d$ for the dipolar and quadrupolar modes \[Fig. \[Fig1\](b)\]. The corresponding induced densities (insets) are only slightly affected by the change in doping profile relative to uniform doping (i.e., the average level of doping is a dominant parameter, and the effect of edge divergences is only marginal). In conclusion, the plasmon frequencies and induced densities can be approximately described by assuming a uniform Fermi energy in backgated ribbons, thus supporting the validity of previous analyses for this configuration,[@paper176; @NGG11; @paper181] although the Fermi energy has to be appropriately scaled as shown in Fig. \[Fig1\](a) to compensate for the effect of finite $W/d$ ratios.
![Plasmons in pairs of co-planar parallel graphene ribbons of opposite polarity. [**(a)**]{} Fermi energy distribution across pairs of ribbons for different ratios of the ribbon width $W$ to the gap distance $d$. The ribbons are placed at potentials $-V$ and $V$, respectively (see inset), and the Fermi energy $E_F$ is normalized to the value $E_F^\infty=\hbar v_F\sqrt{|V|/4ed}$. [**(b)**]{} Frequency $\omega$ of the dipolar and quadrupolar modes as a function of $W/d$, normalized to $\omega_0=(e/\hbar)\sqrt{\langle|E_F|\rangle/W}$. Solid (dashed) curves correspond to inhomogeneous (uniform) doping. The insets show the plasmon charge-density associated with both modes (vertical axis) as a function of position across the ribbon on the right (horizontal axis, with the position of the gap indicated by an arrow) for $W/d=0.2,1,3,$ and 10 (curves evolving in the direction of the arrows).[]{data-label="Fig2"}](Fig2){width="85mm"}
[*Two co-planar parallel graphene ribbons.-*]{} Two neighboring ribbons can act both as plasmonic structures and as gates. We explore this possibility in Fig. \[Fig2\], where the ribbons are taken to be oppositely polarized. This produces doping profiles as shown in Fig. \[Fig2\](a), which evolve from a shape similar to the one obtained for the single ribbon of Fig. \[Fig1\] in the small-ribbon limit at large ribbon-pair separations, towards a converged profile near the gap in the $W/d\rightarrow\infty$ limit. Again, plasmons in this structure are very similar to those of neighboring uniformly doped ribbons \[see Fig. \[Fig2\](b)\], provided one compares for the same value of the average Fermi energy $\langle|E_F|\rangle$. The separation dependence of $\langle|E_F|\rangle$ is shown in Appendix \[calculationof\]. Incidentally, plasmons in pairs of uniform ribbons have been thoroughly described and the evolution of the plasmon frequency with distance explained in a recent publication,[@paper181] including the redshift with decreasing $d$.
![Plasmons in individual ribbons subject to a uniform external electric field $\mathcal{E}$. [**(a)**]{} Fermi energy distribution normalized to $\langle|E_F|\rangle=0.6\,\hbar v_F\sqrt{|\mathcal{E}|/e}$. The inset shows a sketch of the geometry. [**(b)**]{} Surface charge associated with the dipolar plasmon mode (solid curve). The plasmon frequency is $\omega\approx1.45\,\omega_0$, where $\omega_0=(e/\hbar)\sqrt{\langle|E_F|\rangle/W}$. The dashed curve shows the charge density profile for a uniform doping density (i.e., $E_F=\langle|E_F|\rangle$). [**(c)**]{} Plasmon dispersion diagram representing the dependence of the density of optical states on frequency $\omega$ and wave vector parallel to the ribbon $k_\parallel$. [**(d)**]{} Same as (c) for uniform doping.[]{data-label="Fig3"}](Fig3){width="85mm"}
[*Doping through a uniform electric field.-*]{} A disadvantage of the above doping schemes is the fabrication process involved in adding contacts that allow electrically charging the graphene, which can be a source of defects in the carbon layer. This could be avoided by doping through an external electric field $\mathcal{E}$ produced by either distant gates or low-frequency radiation. The graphene would then remain globally neutral. This possibility is analyzed in Fig. \[Fig3\], where we consider a ribbon subject to a uniform electric field directed across its width. The doping profile \[Fig. \[Fig3\](a)\] is again exhibiting a divergence of $E_F$ at the edges, and it vanishes at the center of the ribbon, where the doping density changes sign (see Appendix \[ribbonin\]). Following the methods described in Appendix \[ribbonin\], we find the relation $\langle|E_F|\rangle=0.6\,\hbar v_F\sqrt{|\mathcal{E}|/e}$. The resulting dipolar plasmon \[Fig. \[Fig3\](b), solid curve\] displays a large concentration of induced charges near the center of the ribbon, in contrast to the dipole plasmon obtained for a homogeneous doping density (dashed curve). This inhomogeneous dipole-charge concentration is induced by the vanishing of the doping charge density, which can be understood as a thinning of the effective graphene-layer thickness, similar to what happens near the junction of two barely touching metallic structures (e.g., two spheres[@paper075]).
We have so far discussed plasmons that are invariant along the length of the ribbon (i.e., as those excited by illuminating with light incident perpendicularly to the graphene). But now, we show in Fig. \[Fig3\](c),(d) the full plasmon dispersion relation as a function of frequency $\omega$ and wave vector $k_\parallel$ parallel to the ribbon, both for inhomogeneous doping produced by an external uniform field \[Fig. \[Fig3\](c)\] and for a ribbon with uniform doping density \[Fig. \[Fig3\](d)\]. The dispersion relations are rather different in both situations, with the inhomogeneous ribbon showing a denser set of modes, as well as more localization in the lowest-energy plasmons for large $k_\parallel$, as we show in Appendix \[appcomputation\] by means of near-field plots for the lowest-energy modes of both types of ribbons.
Finally, let us mention that the inelastic plasmon decay rate is given by $\gamma$ within the Drude model in uniformly doped structures.[@paper181] However, $\gamma$ depends on position for inhomogeneous doping. Using the DC mobility $\mu$, one can estimate $\gamma=ev_F^2/\mu|E_F(x)|\approx2\times10^{12}\,$s$^{-1}$ for $E_F=0.5\,$eV and a typical measured mobility $\mu=10,000\,$cm$^2/$Vs.[@NGM04; @ZTS05] Noticing that the local contribution to inelastic losses is proportional to ${\rm Re}\{\sigma\}\approx(e^2/\pi\hbar^2)|E_F|\gamma/\omega^2$ (i.e., independent of $x$), we conclude that the inhomogeneity of $\gamma$ is however translated into a uniform spatial distribution of losses.
[*Conclusions.-*]{} We have shown that the plasmons of doped graphene ribbons are highly sensitive to the inhomogeneities of the doping charge density produced by realistic electrostatic landscapes. The doping profile can be engineered by adjusting the configuration of the gates relative to the graphene. We find an interesting scenario when a uniform external electric field is used to dope the graphene, leading to plasmons with very different characteristics (e.g., induced charges piling up near the center of the ribbon) compared to those of uniformly doped graphene (in which plasmons pile up at the edges). This configuration can be used to avert losses associated with nonlocal effects at the edges, which are expected to be significant.[@paper183] The present study can be straightforwardly extended to other configurations, such as finite graphene nanoislands exposed to either backgates or side gates. Electrostatic charge accumulation at sharp edges can offer an additional handle to manipulate plasmon modes. In addition to the possibilities explored in this paper, one can use biased tips to produce localized disk-like doping areas at designated positions targeted by simply moving the tips above a graphene flake. In conclusion, the design of electrostatic landscapes becomes a useful tool to engineer graphene plasmons.
We would like to thank Enrique Bronchalo for helpful discussions. This work has been supported by the Spanish MICINN (MAT2010-14885 and Consolider NanoLight.es) and the European Commission (FP7-ICT-2009-4-248909-LIMA and FP7-ICT-2009-4-248855-N4E).
{width="170mm"}
 for different values of the ratio of the ribbon width $W$ to the gap distance $d$. The electron density $n$ is normalized to the value $n^\infty=-V/4\pi ed$ corresponding to the $W/d\gg1$ limit.[]{data-label="FigSM4"}](FigSM4){width="85mm"}
![Electrostatic doping electron density across pairs of ribbons for different ratios of the ribbon width $W$ to the gap distance $d$. The ribbons are placed at potentials $-V$ and $V$, respectively (see inset), and the electron density $n$ is normalized to the value $n^\infty=-V/4\pi ed$.[]{data-label="FigSM3"}](FigSM3){width="85mm"}
. The Fermi energy is normalized to $E_F^\infty=\hbar v_F\sqrt{|V|/4ed}$.[]{data-label="FigSM5"}](FigSM5){width="85mm"}
Electrostatic calculation of doping surface charge distributions {#calculationof}
================================================================
We consider three different geometries for electrically doping graphene, as illustrated in Fig. \[FigSM1\]. Geometries in Figs. \[FigSM1\](a),(c) refer to doping of biased graphene ribbons, which accumulate a net electric charge. In contrast, the ribbon of Fig. \[FigSM1\](b), exposed to an external uniform field, has zero total charge. The charge distribution in a backgated ribbon \[Fig. \[FigSM1\](a)\] has been reported in the past,[@W1964; @W1977; @SE08; @VZ10] particularly in the context of microstrip technology,[@W1964; @W1977] but we provide here a general procedure to calculate it based upon boundary elements, which we extend to other geometries under consideration.
Backgated graphene ribbon {#backgatedribbon}
-------------------------
A ribbon placed at a potential $V$ relative to a backgate \[Fig. \[FigSM1\](a)\] displays a charge density $-en(x)$ that can be calculated using the method of images (notice that $n$ is the doping electron density). The charge depends on the coordinate across the ribbon $x$, which varies in the range $0<x<W$, where $W$ is the width. Using the method of images, this problem is equivalent to two parallel ribbons vertically separated by a distance $2d$ and placed at potentials $V$ (upper ribbon) and $-V$ (lower ribbon), so that the backgate plane ($z=0$) is at zero potential. The lower ribbon is thus represented by a charge density $en(x)$. The potential at $x$ in the upper ribbon is then given by $$V=\int_0^W dx'\int_{-\infty}^\infty dy \,[-en(x')]\, \left[\frac{1}{\sqrt{(x-x')^2+y^2}}-\frac{1}{\sqrt{(x-x')^2+y^2+4d^2}}\right].\label{self1}$$
Analytically performing the integral along the $y$ coordinate (perpendicular to the plane of Fig. \[FigSM1\]) and using the notation $\eta=W/d$, $\theta=x/d$, and $$u=-V/ed,$$ Eq. (\[self1\]) reduces to $$u=\int_0^\eta d\theta' \,n(\theta'd)\, F(\theta,\theta'), \label{self}$$ where $$F(\theta,\theta')=\ln\left[1+\frac{4}{(\theta-\theta')^2}\right].\nonumber$$ We solve this integral equation by discretizing $\theta$ through a set of $N$ equally spaced points $\theta_j=(j+1/2)\eta/N$, with $j=0,\dots,N-1$. Equation (\[self\]) is then approximated as $$u\approx\sum_{j'} \,n(\theta_{j'}d)\, M_{jj'}, \label{uu}$$ where $$M_{jj'}=\int_{\theta_{j'}-\eta/2N}^{\theta_{j'}+\eta/2N} d\theta' \, F(\theta_j,\theta') \label{MM}$$ is an integral over the interval surrounding point $\theta_{j'}$. We have assumed that $n(x)$ is a smooth function, and although we find later that it diverges as $\propto1/\sqrt{x}$ near the edges (see main paper), this divergence is integrable and contributes negligibly to the total integral for $N\gg1$. From here, the charge distribution is found by inverting the matrix $M$, so that $$n(\theta_jd)=u\sum_{j'} \left[M^{-1}\right]_{jj'}. \label{solution}$$ In practice, this method converges for $N\sim100$. Similar convergence is obtained for the geometries considered in Secs. \[twoco\] and \[ribbonin\]. Each curve in Fig. 1(a) of the main paper is actually consisting of two curves obtained with $N=100$ and $N=500$, and one cannot tell the difference between them on the scale of the plot. Finally, notice that the uniform electron density in the $W/d\gg1$ limit is given by $n^\infty=u/4\pi$. Also, we obtain the average of the Fermi energy $\langle |E_F|\rangle$ normalized to the $W/d\gg1$ limit $E_F^\infty=\hbar v_F\sqrt{|u|/4}$ as $$\frac{\langle |E_F|\rangle}{E_F^\infty}\approx\frac{1}{N}\sum_j\sqrt{4\pi |n(\theta_jd)/u|}, \nonumber$$ where $v_F\approx10^6\,$m/s is the Fermi velocity. (Incidentally, we consider the absolute value of $E_F$ because the graphene response is nearly insensitive to the sign of $E_F$.)
Figure \[FigSM4\] shows examples of doping charge densities from which we have extracted the Fermi energies shown in the inset of Fig. 1(a) of the main paper.
Two co-planar parallel ribbons set at opposite potentials {#twoco}
---------------------------------------------------------
We can repeat the same analysis as in Sec. \[backgatedribbon\] for two ribbons arranged as shown in Fig. \[FigSM1\](c) and set at potentials $-V$ (left ribbon) and $V$ (right ribbon). The separation between ribbons is $d$, and $x=0$ is chosen at the left edge of the right ribbon. Equations (\[self\])-(\[solution\]) remain valid when we consider the charge distribution in the right ribbon, but now the kernel of Eq. (\[self\]) becomes $$F(\theta,\theta')=2\ln\left|\frac{\theta+\theta'+1}{\theta-\theta'}\right|.\nonumber$$ The charge density in the left ribbon is found from the symmetry $n(x)=-n(-x-d)$.
Figure \[FigSM3\] shows examples of doping charge densities from which we have extracted the Fermi energies shown in Fig. 2(a) of the main paper. The average Fermi energy $\langle |E_F|\rangle$ is shown in Fig. \[FigSM5\] as a function of $d/W$.
![Doping charge density in a graphene ribbon subject to an external uniform electric field $\mathcal{E}$.[]{data-label="FigSM2"}](FigSM2){width="85mm"}
Graphene ribbon in a uniform electric field {#ribbonin}
-------------------------------------------
For a ribbon subject to an external uniform electric field, only the component parallel to the ribbon $\mathcal{E}$ can produce charge redistributions and local doping \[see Fig. \[FigSM1\](b)\]. Following the same procedure as in Sec. \[backgatedribbon\], and taking the ribbon to be placed at zero potential, we can write $$0=-\mathcal{E}x+\int_{-W/2}^{W/2} dx'\int_{-\infty}^\infty dy \,[-en(x')]\, \frac{1}{\sqrt{(x-x')^2+y^2}},\nonumber$$ where the first term is the scalar potential produced by the external field. Now, using the normalization $\theta=x/W$, the above equation reduces to $$\mathcal{E}\theta=\int_{-1/2}^{1/2} d\theta' \,[-en(\theta'W)]\, F(\theta,\theta'), \nonumber$$ where $$F(\theta,\theta')=\ln\left[\frac{1}{(\theta-\theta')^2}\right].\nonumber$$ We use the discretization $\theta_j=-1/2+(j+1/2)/N$ to write an expression similar to Eq. (\[uu\]), where $M_{jj'}$ is still given by Eq. (\[MM\]) with $\eta=1$. Finally, the electron density is obtained from $$n(\theta_jd)=(-\mathcal{E}/e)\sum_{j'} \left[M^{-1}\right]_{jj'}\theta_{j'}. \nonumber$$ The doping charge density obtained from this equation is represented in Fig. \[FigSM2\]. From here, we find a local Fermi energy $E_F(x)=\hbar v_F\sqrt{\pi|n(x)|}$ as shown in Fig. 3(a) of the main paper. The average Fermi energy is found to be $\langle |E_F|\rangle=0.6\,\hbar v_F\sqrt{|\mathcal{E}|/e}$.
Scaling of plasmon frequencies in the Drude model {#appscaling}
=================================================
The electric scalar potential $\phi$ associated with the plasmon modes in a doped graphene planar nanostructure satisfies the self-consistent electrostatic equation $$\phi({{\bf R}})=\frac{i\chi}{\omega}\int \frac{d^2{{\bf R}}'}{|{{\bf R}}-{{\bf R}}'|}\,\nabla\cdot\left[\sigma({{\bf R}}',\omega)\,\nabla\phi({{\bf R}}')\right],\label{self2}$$ where an $\exp(-i\omega t)$ time dependence is undertood and all quantities are evaluated at the graphene plane (i.e., coordinates ${{\bf R}}$ and ${{\bf R}}'$ are in that plane). This equation represents the field induced by the self-consistent surface-charge density, which is calculated from the continuity equation as $(-i/\omega)\nabla\cdot{{\bf j}}$ in terms of the current ${{\bf j}}$, and this is in turn proportional to the in-plane electric field via $-\sigma\nabla\phi$, where $\sigma$ is the conductivity. Incidentally, $\sigma=0$ outside the graphene, so that $\sigma$ presents a jump at the graphene edge, the gradient of which generates an edge charge density. Equation (\[self2\]) is valid for small patterning lengths compared to the light wavelength. The constant $\chi$ takes the values $\chi=1$ for suspended graphene in vacuum, $\chi=1/\epsilon$ for graphene embedded in a uniform dielectric of permittivity $\epsilon$, or $\chi=2/(\epsilon+1)$ for graphene supported on a substrate.[@paper181] Now, we consider the Drude model for the conductivity, $$\sigma({{\bf R}},\omega)=\frac{e^2}{\pi\hbar^2}\frac{i|E_F({{\bf R}})|}{(\omega+i\gamma)},\label{Drude}$$ where $0<\gamma\ll\omega$ and the ${{\bf R}}$ dependence comes from the inhomogeneous electron density $n({{\bf R}})$ (see Sec. \[calculationof\]), which locally situates the Dirac points at an energy $E_F({{\bf R}})=\hbar v_F\sqrt{\pi|n({{\bf R}})|}$ relative to the Fermi level.[@CGP09]
From the analysis of Sec. \[calculationof\], we can write the Fermi energy distribution as $E_F({{\bf R}})=\langle |E_F|\rangle\,f({{\bf R}})$, where $f$ is a dimensionless envelope function. Likewise, distances can be scaled with a characteristic length $W$ as ${{\bf U}}={{\bf R}}/W$. Then, Eq. (\[self2\]) becomes $$\phi({{\bf U}})=\zeta\int \frac{d^2{{\bf U}}'}{|{{\bf U}}-{{\bf U}}'|}\,\nabla_{{{\bf U}}'}\cdot\left[f({{\bf U}}')\,\nabla_{{{\bf U}}'}\phi({{\bf U}}')\right],\nonumber$$ where $$\zeta=\frac{-e^2\chi}{\pi\hbar^2W}\frac{\langle |E_F|\rangle}{\omega(\omega+i\gamma)}\nonumber$$ is a dimensionless eigenvalue. The modes of the system satisfy this equation for specific choices of $\zeta$, and therefore, we conclude that the plasmon energies $\omega$ can be naturally normalized to the frequency $$\omega_0=(e/\hbar)\sqrt{\chi \langle |E_F|\rangle/W}. \label{w0}$$ Obviously, $\omega/\omega_0$ does not depend on the specific choice of $\langle |E_F|\rangle$ (the doping level), $W$ (the size of the system), or $\chi$ (the dielectric environment), and consequently, we present results normalized in this way in the main paper, which are universal for the kind of geometries under consideration, provided we stay within the limits of validity of the Drude model (i.e., $\omega<E_F$, and $E_F$ smaller than the optical phonon energy $\sim0.2\,$eV[@JBS09]). Additionally, because the electrostatic eigensystem is Hermitian,[@OI1989] the eigenvalues $\zeta$ are real, and therefore, the plasmon frequencies have imaginary part ${\rm Im}\{\omega\}=-\gamma/2$, so that the plasmon lifetime is $1/\gamma$, independent of geometrical and physical parameters within the Drude approximation.
Computation of plasmon frequencies and near fields {#appcomputation}
==================================================
We solve the electrostatic problem of Eq. (\[self2\]) by describing the graphene as a periodic array of surface dipoles with a small period compared to the characteristic lengths of the structure. This is the discrete surface-dipole approximation[@DSDA] (DSDA), in which the polarizability of each element is taken such that a layer formed by a uniform lattice of dipoles has the same conductivity as a uniform layer of graphene. The sum over dipole elements along $y$ is performed before a self-consistent solution is sought, and therefore, the numerical problem reduces to solving a set of $2N$ linear equations with $2N$ variables (the dipole components along both $x$ and $y$ directions), where $N$ is the number of dipoles across $x$. In practice, convergence is achieved with a few hundred dipoles for the dimensions considered in this work. Here, we have modified this method by allowing each element to depend through $E_F$ on the spatial position along $x$.
{width="110mm"}
As an example of calculation, we show in Fig. \[FigSM6\](a),(b) plasmon dispersion diagrams for both an inhomogeneously doped ribbon under the same conditions as in Fig. \[FigSM2\] and a homogeneously doped ribbon. These plots are the same as Fig. 3(c),(d) of the main paper. We also show the near fields calculated for the three lowest-energy modes in each case, with $k_\parallel W=10$. These are plasmons of monopole (c,f), dipole (d,g), and quadrupole (e,h) character. In the inhomogeneous ribbon, the modes are associated with large field enhancement near the center. In contrast, the uniform ribbon hosts modes with large field enhancement near the edges.
[26]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, **, vol. of ** (, , ).
, , , , , , , , ****, ().
, ****, ().
, , , ****, ().
, , , , , , , , , , , ****, ().
, , , , , , , , , , , ****, ().
, ****, ().
, , , , ****, ().
, , , ****, ().
, , , , , ****, ().
, ****, ().
, , , ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, , , , ****, ().
, , , , , ****, ().
, , , , ****, ().
, , , , , , , , ****, ().
, , , , ****, ().
, , , ****, ().
, ****, ().
|
---
abstract: 'We propose to measure the photo-production cross section of [$J/\psi$]{}near threshold, in search of the recently observed LHCb hidden-charm resonances $P_c$(4380) and $P_c$(4450) consistent with ‘pentaquarks’. The observation of these resonances in photo-production will provide strong evidence of the true resonance nature of the LHCb states, distinguishing them from kinematic enhancements. A bremsstrahlung photon beam produced with an $11\,\text{GeV}$ electron beam at CEBAF covers the energy range of [$J/\psi$]{}production from the threshold photo-production energy of $8.2\,\text{GeV}$, to an energy beyond the presumed [$P_c(4450)$]{}resonance. The experiment will be carried out in Hall C at Jefferson Lab, using a $50\,\mu\text{A}$ electron beam incident on a 9% copper radiator. The resulting photon beam passes through a $15\,\text{cm}$ liquid hydrogen target, producing [$J/\psi$]{}mesons through a diffractive process in the $t$-channel, or through a resonant process in the $s$- and $u$-channel. The decay $e^+e^-$ pair of the [$J/\psi$]{}will be detected in coincidence using the two high-momentum spectrometers of Hall C. The spectrometer settings have been optimized to distinguish the resonant $s$- and $u$-channel production from the diffractive $t$-channel [$J/\psi$]{}production. The $s$- and $u$-channel production of the charmed 5-quark resonance dominates the $t$-distribution at large $t$. The momentum and angular resolution of the spectrometers is sufficient to observe a clear resonance enhancement in the total cross section and $t$-distribution. We request a total of 11 days of beam time with 9 days to carry the main experiment and 2 days to acquire the needed $t$-channel elastic $J/\psi$ production data for a calibration measurement. This calibration measurement in itself will greatly enhance our knowledge of $t$-channel elastic [$J/\psi$]{}production near threshold.'
address:
- 'Alikhanyan National Science Laboratory, Yerevan, Armenia'
- 'Argonne National Laboratory, Chicago, IL'
- 'Brookhaven National Laboratory, Upton, NY'
- 'California State University, Los Angeles, CA'
- 'Duke University, Durham, NC'
- 'Florida International University, Miami, FL'
- 'Mississippi State University, Starkville, MS'
- 'Old Dominion University, Norfolk, VA'
- 'Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, China'
- 'Institut de Physique Nucléaire d’Orsay, Orsay, France'
- 'University of Regina, SK, Canada'
- 'Seoul National University, Seoul, Korea'
- 'Thomas Jefferson National Accelerator Facility, Newport News, VA'
- 'Temple University, Philadelphia, PA'
author:
- 'Z.-E. Meziani'
- 'S. Joosten'
- 'M. Paolone'
- 'E. Chudakov'
- 'M. Jones'
- 'K. Adhikari'
- 'K. Aniol'
- 'W. Armstrong'
- 'J. Arrington'
- 'A. Asaturyan'
- 'H. Atac'
- 'S. Bae'
- 'H. Bhatt'
- 'D. Bhetuwal'
- 'J.-P. Chen'
- 'X. Chen'
- 'H. Choi'
- 'S. Choi'
- 'M. Diefenthaler'
- 'J. Dunne'
- 'R. Dupré'
- 'B. Duran'
- 'D. Dutta'
- 'L. El-Fassi'
- 'Q. Fu'
- 'H. Gao'
- 'H. Go'
- 'C. Gu'
- 'J. Ha'
- 'K. Hafidi'
- 'O. Hansen'
- 'M. Hattawy'
- 'D. Higinbotham'
- 'G. M. Huber'
- 'P. Markowitz'
- 'D. Meekins'
- 'H. Mkrtchyan'
- 'R. Nouicer'
- 'W. Li'
- 'X. Li'
- 'T. Liu'
- 'C. Peng'
- 'L. Pentchev'
- 'E. Pooser'
- 'M. Rehfuss'
- 'N. Sparveris'
- 'V. Tadevosyan'
- 'R. Wang'
- 'F. R. Wesselmann'
- 'S. Wood'
- 'W. Xiong'
- 'X. Yan'
- 'L. Ye'
- 'Z. Ye'
- 'A. Zafar'
- 'Y. Zhang'
- 'F. Zhao'
- 'Z. Zhao'
- 'S. Zhamkochyan'
bibliography:
- 'Pc\_main.bib'
date: 'August 31, 2016'
title: 'A Search for the LHCb Charmed ‘Pentaquark’ using Photo-Production of $J/\psi$ at Threshold in Hall C at Jefferson Lab'
---
Introduction and motivation {#sec:intro}
===========================
Photo-production of $J/\psi$ on a nucleon very close to threshold is an important subject in the field of non-perturbative QCD in its own right [@Brodsky:2015zxu] and is already planned to be investigated at Jefferson Lab as the 12 GeV upgrade of CEBAF is completed [@CLAS12-tcs:proposal; @SoLIDjpsi:proposal]. Oddly enough the potential of discovery of hidden charm baryon resonances via photo-production was discussed in 2014 [@Huang:2013mua] inspired in part by the SoLID-$J/\psi$ approved proposal at Jefferson Lab [@SoLIDjpsi:proposal]. However, CERN’s recent experimental discovery [@cernpr:2016] has spurred a new excitement and a sense of urgency to carry out measurements of photo-production at threshold in a timely manner.
Less than a year ago, on July 14, 2015, a press release from the CERN press office announced the observation of exotic pentaquark particles [@cernpr:2016] just a day after the manuscript describing the discovery was posted on the arXiv.org [@Sheldon:2016] website by the collaboration. A month later, on August 12, 2015 the announcement was followed by the publication of the manuscript describing the discovery in Physical Review Letters [@Aaij:2015tga]. This announcement was received with both excitement and a healthy dose of skepticism due to the early saga of ‘pentaquarks,’ in the beginning of the new millennium, which proved inconclusive. Unlike these earlier announced pentaquarks, which consisted of four light quarks and one strange quark, the resonant state observed by the LHCb includes two heavy quarks, namely charm and anti-charm quarks and thus must be different in nature.
Subsequent to the announcement a series of theoretical papers [@Liu:2016fe; @Karliner:2015ina; @Chen:2015loa; @Eides:2015dtr; @Wang:2015jsa; @Karliner:2015voa; @Kubarovsky:2015aaa; @Guo:2015umn] appeared in the literature with possible interpretations of the observed resonance. A range of explanations was invoked, from a possible true pentaquark resonant state to a kinematic enhancement like those observed in other experiments close to kinematic thresholds [@Bai:2003sw], such as a bound state of charmonium$
(2S)$ and the proton [@Eides:2015dtr], or a molecule composed of $\Sigma_c$ and $\bar D^*~$[@Lu:2016nnt; @Huang:2016tcr]. But without further experimental measurements it is not clear whether the formed exotic resonance can be unambiguously identified as a resonance. Some authors suggested that effects of final state interactions are responsible for the LHCb observed rate enhancements [@Guo:2015umn]. While the interest of the theory community has produced more than 200 citations up to date, LHCb is the only experiment that has observed these states. The hadronic physics community is eager to see these possible resonant states confirmed in more than one experiment and proceed with a detailed investigation of the quantum numbers of such states.
In summary, to resolve the true nature of the $P_c^+(4380)$ and $P_c^+(4450)$ states it is proposed to study these pentaquark candidates in direct photo-production of $J/\psi$ on the proton and provide not only further evidence of their existence but also investigate their spin and parity, as noted in several papers, such as [@Wang:2015jsa; @Karliner:2015voa; @Kubarovsky:2015aaa; @Blin:2016dlf]. This proposal is more specifically about a direct search of the higher mass narrow width $P_c^+(4450)$ and follows Wang et al. [@Wang:2015jsa] using the different spins and parity described in the paper but with the less optimistic assumptions about the coupling to the resonant states during our complete simulations, namely a 5% coupling.
We believe that the results from this search at Jefferson Lab will have a high impact on the broader physics community.
Present data status
-------------------
The photo-production of $J/\psi$ has been measured in many experiments at high invariant mass of the photon-proton system ($W_{\gamma-p}$ at HERA [@Chekanov:2002xi; @Alexa:2013xxa], and more recently at LHCb [@Aaij:2013jxj] (see right Fig. \[fig:xsection\]). The total elastic $J/\psi$ production at high photon-nucleon invariant mass $W_{\gamma p}$ is well described by the $t$-channel exchange of a colorless object between the photon and the proton [@frankfurt:2002], in this case two-gluon exchange. The differential cross section in the proton momentum transfer variable $t$ is usually described by $d\sigma/dt \propto e^{bt}$ with a value of $b$ that depends on $W_{\gamma p}$. As $W_{\gamma p}$ decreases towards the threshold region of $J/\psi$ production, the mechanism is described by a Pomeron exchange or two-gluon exchange [@Brodsky:2000zc] or perhaps a more complicated multi-gluon exchange carrying the non-perturbative information of the gluonic fields in the nucleon. The new LHCb resonance happens to be in this threshold region of invariant mass, a region that has been poorly explored in modern times. It is worth pointing out that the few measurements of this region occurred in the 1970s at Cornell and SLAC and in the 80s at Fermilab ( see left Fig. \[fig:xsection\]). In those experiments, issues of unambiguously defining the elastic process of $J/\psi$ production were hampered in some cases by the use of nuclear targets, detector resolution and the detection of one lepton only in the case of the $J/\psi$ pair decay.
In Hall C at Jefferson Lab, a photo-production experiment (E03-008) was performed in the [*subthreshold*]{} regime, but unfortunately no signal was observed after one week of beam scattering off a $^{12}$C target [@Bosted:2008mn]. The experiment used a bremsstrahlung beam produced in a copper radiator by the 6 GeV incident electron beam at CEBAF. The pair of spectrometers (HMS and SOS) of Hall C were used to detect the pair of leptons resulting from the decay of the $J/\psi$. This experiment allowed an upper limit to be set on the cross section which was found to be consistent with the quasi-free production. More recently a proposal [@Chudakov:2007] for the 12 GeV upgrade of Hall C was considered by the PAC and conditionally approved. The authors proposed again the use of bremsstrahlung photon beam created in a radiator to look at the photo-production at threshold in a series of nuclei. The physics goal was to measure the photo-production cross section in order to investigate the A-dependence of the propagation of the $J/\psi$ in the nuclear medium as well as extract the $J/\psi-N$ interaction. In the latter proposal, the $J/\psi$ decay pair was to be detected by the HMS and SHMS similar to what is proposed here, however the optimization of the spectrometer settings was related to enhancing the rate of pairs detected from $J/\psi$ decays in primarily diffractive $J/\psi$ production off nuclei, no resonant production was considered.
In summary, the near threshold region of elastic $J/\psi$ production has yet to be fully explored in the context of understanding the non-perturbative gluonic $J/\psi$-nucleon interaction. At Jefferson Lab there are approved proposals to measure this region using the CLAS12 detector in Hall B [@CLAS12-tcs:proposal] and the SoLID detector in Hall A [@SoLIDjpsi:proposal]. In this proposal our focus is to confirm the observation of LHCb through a resonant production of the $P_c(4450)$ in the $s$- and $u$-channel.
![Compilation of world data for the electro- and photo-production of elastic [$J/\psi$]{}. Shown in the left panel are Cornell data from [@Gittelman:1975ix], SLAC data from [@Camerini:1975cy; @Anderson:1976sd], CERN NA14 data from [@Barate:1986fq], FNAL data from [@Binkley:1981kv; @Frabetti:1993ux], H1 data from [@Adloff:2000vm; @Alexa:2013xxa], ZEUS data from [@Chekanov:2002xi] and LHCb data from [@Aaij:2013jxj]. Legend in the figure with $\gamma^*$ refer to electro-production data and thus an effective photon energy defined by $E_{\gamma}^{eff} = (W_{\gamma p}^2-M_p^2 )/2M_p$ was used. The right panel zooms on the region of interest near the [$J/\psi$]{}production threshold region. The red curve on the right figure is the result of a 2-gluon fit. []{data-label="fig:xsection"}](figures/jpsi-xsec-full.pdf "fig:"){width="66.00000%"} ![Compilation of world data for the electro- and photo-production of elastic [$J/\psi$]{}. Shown in the left panel are Cornell data from [@Gittelman:1975ix], SLAC data from [@Camerini:1975cy; @Anderson:1976sd], CERN NA14 data from [@Barate:1986fq], FNAL data from [@Binkley:1981kv; @Frabetti:1993ux], H1 data from [@Adloff:2000vm; @Alexa:2013xxa], ZEUS data from [@Chekanov:2002xi] and LHCb data from [@Aaij:2013jxj]. Legend in the figure with $\gamma^*$ refer to electro-production data and thus an effective photon energy defined by $E_{\gamma}^{eff} = (W_{\gamma p}^2-M_p^2 )/2M_p$ was used. The right panel zooms on the region of interest near the [$J/\psi$]{}production threshold region. The red curve on the right figure is the result of a 2-gluon fit. []{data-label="fig:xsection"}](figures/jpsi-xsec-zoom.pdf "fig:"){width="33.00000%"}
The proposed measurement in Hall C at Jefferson Lab
===================================================
We propose to measure the elastic $J/\psi$ photo-production cross section as a function of $t$ and photon energy $E_{\gamma}$ in the near threshold region in Hall C. A bremsstrahlung photon beam will be created using a 9% copper radiator in front of a liquid hydrogen target, similar to the E-05-101 experiment [@Fanelli:2015eoa]. The optimal placement of the radiator will be chosen to account for the closer proximity of the flow diverters to the beam.
Both high momentum spectrometers of Hall C along with their associated detectors will be used to detect the di-lepton pair decay, namely e$^+$e$^-$. The photon beam mixed with the primary electron beam will strike a 15 cm liquid hydrogen target. The electron-positron decay pair will be detected in coincidence between the high momentum spectrometer (HMS) set for electron detection and the super-high momentum spectrometer (SHMS) set for positron detection. Both spectrometer arms will be used in their standard configuration.
The proposed measurement is designed to search for the highest mass narrow exotic resonant state discovered at LHCb, namely the $P_c(4450)$. The spectrometer settings (shown in Tab. \[table:kin\]) are optimized to be most sensitive to the possible resonant production of $P_c$(4450) in the $s$- and $u$-channel. The two spectrometers will detect the $J/\psi$ decay into e$^+$e$^-$ from either the diffractive channel or resonant $P_c$ channel production. However, we will take advantage of the different $t$-dependence of the two processes to optimize the spectrometers’ angle and momentum settings to enhance the $P_c(4450)$ signal relative to that of the $t$-channel production.
The experiment in Hall C
------------------------
![The experimental layout of the HMS (for $e^-$ detection ) and SHMS (for e$^+$ detection) and associated detectors combined with a liquid 15 cm hydrogen target and a 9% copper radiator. We will detect the $J/\psi$ decay e$^+$e$^-$ pair in coincidence between the two spectrometers. From the scattering angle and momentum determination of the lepton pairs, we are able to reconstruct the invariant mass of the $J/\psi$ as well as its three-momentum. From this information the four-momentum transfer to the proton $t$ is calculated and the real photon energy $E_{\gamma}$ is determined.[]{data-label="exp:layout"}](figures/layout_HallC.eps){width="\textwidth"}
The layout of the proposed experiment is shown in Fig. \[exp:layout\] where the HMS is set at an angle of 34$^{\circ}$ to detect the electrons of the e$^+$e$^-$ decay pairs while the SHMS is set at angle of 13$^{\circ}$ to detect the corresponding positrons. This configuration has been optimized to reduce the accidental coincidences between the two spectrometers as well as minimize the absolute background in each spectrometer. Part of the momentum acceptance of the spectrometer will allow for the detection of the Bethe-Heitler process in a kinematic region forbidden to the diffractive or resonant production of $J/\psi$. The HMS is chosen to detect electrons, because the inclusive inelastic electron scattering cross section drops rapidly with increasing scattering angle. On the other hand the SHMS would have a very large rate of inclusive electron scattering if it were to be used to detect electrons at the small angle setting of 13$^{\circ}$. It is thus run in positive polarity to detect positrons, but it will also accept positive pions and protons.
Each spectrometer has a similar set of standard detectors to identify electrons/positrons and reject charged pions and protons. In each case the momentum of the particles is provided through tracking by a set of drift chambers, and electron identification is ensured by a light gas Čerenkov counter and an electromagnetic calorimeter. The trigger in each spectrometer is defined by a coincidence between a set of 2 hodoscope scintillator planes along the path of the particles. These configurations offer an electron or positron detection efficiency greater than 98% and a pion rejection factor of about a thousand. The timing resolution online will be defined by the time coincidence between the hodoscopes in each spectrometer first and then a time coincidence between both spectrometers. We expect it to be on the range of few nanoseconds, however improvements will be possible when the tracking information is corrected for offline. A gate of about 50 ns will be used between the two spectrometers.
--------- --------- -------------- --------- -------------- ----------- -------------
$p$ $\theta$ $p$ $\theta$ t-channel $P_c(4450)$
Setting GeV/$c$ GeV/$c$ % %
\#1 3.25 34.5$^\circ$ 4.5 13.0$^\circ$ 0.0004 0.003
\#2 4.75 20.0$^\circ$ 4.25 20.0$^\circ$ 0.01 0.003
--------- --------- -------------- --------- -------------- ----------- -------------
: Kinematic setting of the HMS and SHMS spectrometers to measure in coincidence the decay-pair of the [$J/\psi$]{}. The main spectrometer setting (1) is optimized to measure the $P_c(4450)$ with minimal $t$-channel background production, while the additional setting (2) is chosen to allow for a precise determination of the $t$-channel background left of the $P_c(4450)$ resonance.
\[table:kin\]
We point out that a lower electron beam energy of 10.7 GeV, rather than 11.0 GeV, with the same spectrometer settings, will result in a more suppressed $t$-channel production, while it should not affect the $s$ and $u$ channel $P_c(4450)$ production.
Kinematics
----------
The kinematics were optimized using a full simulation of the experiment and focused on enhancing the resonant production of $J/\psi$ through the $P_c(4450)$ relative to the diffractive production. This is done by taking advantage of the $t$-dependence of the diffractive production since the latter is suppressed at large values of $t$ while the resonant production in the $s$-channel of $P_c$ is rather flat across the same $t$ range. The spectrometer settings are chosen to take advantage of this difference in $t$-dependence. In Tab. \[table:kin\] we list the spectrometers’ momentum and angle settings converged upon and the resulting acceptance for a coincident detection of the di-lepton pair. Also shown is the additional spectrometer setting needed for a precise determination of the $t$-channel background left of the [$P_c(4450)$]{}peak.
Shown in Fig. \[fig:kin-t\] left and right are the distributions of angle versus momentum of the decay pair of leptons, the full correlated phase space of the $t$-channel production of the pair is shown on the left figure while the similar phase space of the resonant $P_c$ production is shown on the right.
![Angular distribution of the [$J/\psi$]{}production for the $t$-channel, normalized to the same area for each curve in arbitrary units, in comparison with the $J/\psi$ production through the exotic $P_c$ resonant state with various possible spin/parity assumptions. The angle $\theta$ is the relative angle between the J/psi and the photon in the center of mass. Note that, for the $t$-channel this is directly related to the $t$-dependence of the cross section.[]{data-label="fig:kin-theta"}](figures/jpsi-angular-dependence-linear.pdf){width="60.00000%"}
It is important to take into account the different possibilities of spin of the exotic charmed resonance. In all cases we found that it is best to keep kinematics that correspond to $\cos{\theta}$ between $-0.4$ and $0.2$ as shown in Fig. \[fig:kin-theta\], or, in other words, close to the 90$^{\circ}$ range in the center of mass frame. This maximizes the $P_c$ production rate, relative to the $t$-channel production rate.
![The acceptance for setting \#1 (left) and \#2 (right) from Tab. \[table:kin\] as a function of the photon energy $E_\gamma$. The $t$-channel is shown in red, and the [$P_c(4450)$]{}is shown in blue (the angular decay distribution was taken to be consistent with the 5/2+ assumption shown in Fig. \[fig:kin-theta\]). The acceptance for setting \#1 is perfectly optimized to measure the [$P_c(4450)$]{}.[]{data-label="fig:acc-e"}](figures/acceptance-setting1.pdf "fig:"){width="47.00000%"} ![The acceptance for setting \#1 (left) and \#2 (right) from Tab. \[table:kin\] as a function of the photon energy $E_\gamma$. The $t$-channel is shown in red, and the [$P_c(4450)$]{}is shown in blue (the angular decay distribution was taken to be consistent with the 5/2+ assumption shown in Fig. \[fig:kin-theta\]). The acceptance for setting \#1 is perfectly optimized to measure the [$P_c(4450)$]{}.[]{data-label="fig:acc-e"}](figures/acceptance-setting2.pdf "fig:"){width="47.00000%"}
Physics and accidental backgrounds
==================================
The typical process of elastic $J/\psi$ production is usually described by a Feynman diagram represented in Fig. \[fig:t-chan-diag\] and is well understood at high energies using perturbation theory [@Donnachie:2002en]. This process at threshold is usually described by a two-gluon exchange [@Brodsky:2000zc; @frankfurt:2002] although at threshold the gluons could be an effective representation of an interaction that conserves color but is much more complicated. A full experimental physics program to explore the threshold region of $J/\psi$ production completely is planned by gathering large amount of data through electro-production and photo-production using the CLAS12 [@CLAS12-tcs:proposal] and SoLID [@SoLIDjpsi:proposal] detectors in the coming years.
Production of $e^+e^-$ through elastic $J/\psi$ production and decay
--------------------------------------------------------------------
![t-channel 2-gluon exchange elastic $J/\psi$ photo-production mechanism.[]{data-label="fig:t-chan-diag"}](figures/Jpsi-t-channel-diag.eps){width="50.00000%"}
In the proposed experiment we are not concerned with the elastic $t$-channel production of $J/\psi$, which we consider to be a physics background, but are rather interested in confirming the possible resonance production of the $J/\psi$ through the decay of the newly discovered states at LHCb, namely $P_c(4450)$ and $P_c(4380)$. This production is typically described by an $s$- and $u$-channel production of these resonances according to the diagrams of Fig. \[fig:s-u-chan-diag\]. More specifically it is a search for $P_c(4450)$ that we are focused on in this proposal. $P_c(4380)$ is broader with a lower cross section, and thus requires a more challenging setup to be determined cleanly.
![$s-$ (a) and $u-$ (b) channel resonant production of $J/\psi$ through $P_c$.[]{data-label="fig:s-u-chan-diag"}](figures/Jpsi-s-u-channel-diag.eps){width="75.00000%"}
Therefore, in this experiment the challenge is to separate the two different processes, one that we consider to be a physics background ($t$-channel production of $J/\psi$) and one that is our important signal ($s$- and $u$-channel resonant production through $P_c$). We propose to use spectrometer settings that will dramatically reduce the acceptance of the $t$-channel production of the [$J/\psi$]{}relative to the $s$- and $u$-channel resonant production of [$P_c(4450)$]{}. These setting were optimized using a multidimensional scan of the acceptance for both spectrometers in the full phase space.
Bethe-Heitler pair production
-----------------------------
To evaluate the Bethe-Heitler background represented by the processes described in Fig. \[fig:bh-t\] we used the calculations of Pauk and Vanderhaeghen [@Pauk:2015oaa; @Vander:2016] but with $M_{l^+l^-}$ evaluated within the acceptance of our spectrometer settings centered around the mass of $P_c (4450)$. We use the dipole electromagnetic form factor for the range of momentum transfers covering the proposed experiment. We find that this background is over 10 times smaller as shown in Fig. \[fig:bh-rate\], nevertheless it can be calculated and controlled for.
![Bethe-Heitler (BH) mechanism producing a background process to the $t$-channel and $P_c$ resonant production.[]{data-label="fig:bh-t"}](figures/BH-diagrams.eps){width="70.00000%"}
![B-H rate relative to the elastic [$J/\psi$]{}production in the $t$-channel. []{data-label="fig:bh-rate"}](figures/background-bh.pdf){width="50.00000%"}
Single $e^\pm$ background
-------------------------
The electron rate in the HMS was estimated using CTEQ5 [@Lai:2000im], and cross checked using the F1F209 program [@Bosted:2012qc]. The positron rate in the SHMS was estimated using the EPC program [@Lightbody:1988ke] combined with a positron background program written for the E94-010 experiment at JLab. The rate can be found in Tab. \[table:singles\].
Single $\pi^\pm$ background
---------------------------
The charged pion singles rate was estimated using the Wiser program [@wiser:1977]. Its results can be found in Tab. \[table:singles\].
Accidental coincidence rate
---------------------------
Using the results from Tab. \[table:singles\], the accidental coincidence rate for a $50\,\text{ns}$ trigger window between the HMS and SHMS, was found to be of the order of $10^{-5}\,\text{Hz}$. This is two full orders of magnitude lower than the expected signal rate and therefore negligible. For this calculation we assumed a pion rejection larger than 10$^3$ from the combined Cerenkov and calorimeter system.
--------- --------------------- --------------------- --------------------- -------------------
Setting $e^-$ (kHz) $\pi^-$ (kHz) $e^+$ (kHz) $\pi^+$ (kHz)
\#1 $6.9\times 10^{-3}$ $7.5\times 10^{-2}$ $6.5\times 10^{-4}$ $1.95\times 10^2$
\#2 $9.7\times 10^{-1}$ $2.2\times 10^{0}$ $7.5\times 10^{-4}$ $10.5\times 10^0$
--------- --------------------- --------------------- --------------------- -------------------
: Singles rates
\[table:singles\]
Background from $\gamma p\rightarrow J/\psi p \pi $
---------------------------------------------------
The inelastic channel of [$J/\psi$]{}production, where an additional final state pion is produced but not detected, might contaminate the kinematic region where the [$P_c(4450)$]{}is produced. The cross section of these inelastic channels was found to be less than 30% of the elastic $t$-channel cross section at high energies, and is expected to be even smaller near the [$J/\psi$]{}threshold [@Binkley:1981kv; @Chudakov:2007]. In this region, the dominant contribution is the resonant channel $\gamma
p\rightarrow {\ensuremath{J/\psi}\xspace}\Delta(1320)$, with a threshold at approximately $E_\gamma>9\,\text{GeV}$.
In the acceptance of our proposed setting \#1, and for a photon endpoint energy of $10.66\,\text{GeV}$, the photon energy spectrum for this inelastic $t$-channel process occupies the same reconstructed energy range as the elastic $t$-channel events. These reconstructed energies corresponds to a true photon energies of $1\,\text{GeV}$ higher, where the cross is approximately four times larger. This fourfold rise in the cross section is almost exactly compensated by the acceptance, which is four times lower for this process, due to a corresponding shift of the reconstructed $t$ of $1\,\text{GeV}^2$. Ultimately, this background is expected to increase the elastic $t$-channel background by at most 30%.
Background from lepto-production
--------------------------------
To estimate the background due to lepto-production, we simulated the $e p
\rightarrow e \gamma^* p \rightarrow e {\ensuremath{J/\psi}\xspace}p$ process for a $50\,\mu\text{A}$ electron beam at $11\,\text{GeV}$. We found only quasi-real photons, up to a virtuality of $Q^2 ~ 0.01\,\text{GeV}^2$ to have a significant impact. Photons with a higher virtuality are highly suppressed because of the following reasons:
- the virtual photon flux drops with $Q^2$
- Higher $Q^2$ means lower $W^2$ for fixed values of $\nu$, and the $t$-channel cross section drops for lower $W^2$. Furthermore, close to threshold, the available phase space shrinks rapidly for lower $W^2$.
- The highly-tuned spectrometer acceptance drops with $Q^2$.
Because only the quasi-real photons play a role, the contribution due to lepto-production will lead to a (small) enhancement of the count rates. We will verify the impact of this contribution by conducting a dedicated measurement without the radiator.
Simulation of the experiment
============================
We use a custom Monte-Carlo generator to obtain a realistic estimate of the [$J/\psi$]{}photo-production rates. This generator uses the bremsstrahlung spectrum for a 10% radiator, appropriate models for the $t$-channel and $P_c$ resonant channel, and the HMS and SHMS spectrometer acceptance with realistic smearing effects. The leptonic ${\ensuremath{J/\psi}\xspace}\rightarrow e^+e^-$ decay is simulated using a $(1+cos^2\theta_{e})$ angular distribution in the [$J/\psi$]{}helicity frame. More details about the simulation can be found below.
Model for the $t$-channel cross section
---------------------------------------
In order to calculate the cross section for the $t$-channel production, we fit the cross section ansatz for two gluon exchange from Brodksy et al. [@Brodsky:2000zc] (equation (3)) to the available world data. The result of this fit (in $\text{nb}/\text{GeV}^2)$ is given by, $$\begin{aligned}
\frac{d\sigma}{dt} =
A v \frac{(1-x)^2}{M_{{\ensuremath{J/\psi}\xspace}}^2}(s-M_p^2)^2
\exp{bt},
\label{eq:tchannel-xsec}\end{aligned}$$ where $x$ is given by a near-threshold definition of the fractional momentum carried by the valence quark, $v$ is a kinematic factor, $b$ the impact parameter and $A$ an overall normalization constant that was determined by a fit to the world data, $$\begin{aligned}
x&=\frac{2M_{{\ensuremath{J/\psi}\xspace}}M_p +M_{{\ensuremath{J/\psi}\xspace}}^2}{s-M_p^2}\\
v&=\frac{1}{16\pi(s-M_p^2)^2}\\
b&=1.13\,\text{GeV}^{-2},\\
A&=6.499\times10^3\,\text{nb}.\end{aligned}$$ Additionally, $M_p$ and $M_{{\ensuremath{J/\psi}\xspace}}$ are respectively the proton mass and [$J/\psi$]{}mass in GeV. The curve from Eq. \[eq:tchannel-xsec\] is shown as a red line in Figures \[fig:xsection\] and \[fig:xsec-comp\].
Model for the $P_c\rightarrow{\ensuremath{J/\psi}\xspace}p$ cross section
-------------------------------------------------------------------------
Several equivallent approaches to calculate the $\gamma p\rightarrow P_c\rightarrow{\ensuremath{J/\psi}\xspace}p$ cross section can be found in the literature [@Wang:2015jsa; @Karliner:2015voa; @Kubarovsky:2015aaa; @Blin:2016dlf]. We based our model of the cross section on the work by Wang et al. [@Wang:2015jsa]. Note that this cross section depends quadratically on the coupling to the ${\ensuremath{J/\psi}\xspace}p$ channel. We considered the (5/2+) and (5/2-) spin/parity assumptions for the narrow [$P_c(4450)$]{}state, with the corresponding (3/2-) and (3/2+) assumption for the [$P_c(4380)$]{}state. The angular distribution for the [$J/\psi$]{}production for each of the spin-parity assumptions can be found in Fig. \[fig:kin-theta\]. The contributions of the (5/2+,3/2-) channels to the [$J/\psi$]{}photo-production cross section as a function of photon energy $E_\gamma$ are shown in Fig. \[fig:xsec-comp\].
We optimized the spectrometer settings for a (5/2+) [$P_c(4450)$]{}case with 5% coupling to the ${\ensuremath{J/\psi}\xspace}p$ channel, as it agrees well with the existing photo-production data. This setting, also has a good sensitivity to a (5/2-) [$P_c(4450)$]{}as its production cross section is a full order of magnitude larger. To perform this optimization, a total of 3.4 million possible spectrometer settings were considered. We selected a setting that maximizes the acceptance for [$J/\psi$]{}produced with a $\cos\theta$ between $-0.4$ and $0.2$ in the center-of-mass frame, as shown in Fig.\[fig:kin-theta\]. This corresponds to a setting that selects the high-$t$ region, where there is a maximum sensitivity to the (5/2+) [$P_c(4450)$]{}resonant production, while simultaneously the sensitivity to the $t$-channel [$J/\psi$]{}production is highly suppressed. This setting is listed on the first line of Tab. \[table:kin\].
![$J/\psi$ production cross section as a function of the photon energy. The $P_c$ resonant production is shown for the (5/2+, 3/2-) case assuming 3% coupling, compared with the available measurements in this region [@Gittelman:1975ix; @Anderson:1976sd].[]{data-label="fig:xsec-comp"}](figures/jpsi-xsec-comparison.pdf){width="50.00000%"}
Bremsstrahlung spectrum
-----------------------
The generator uses equation (24) from Tsai [@Tsai:1966js] to evaluate for the bremsstrahlung spectrum. For a 9% radiator combined with 1% from the target (a total of 10% radiator), the photon beam has an integrated intensity of 2.3% of the primary electron beam.
Detector acceptance and resolution
----------------------------------
The spectrometer acceptance and realistic smearing are simulated using the parameters listed in Tab. \[table:spec\]. An $e^+e^-$ invariant mass spectrum that was generated using the optimized setting listed on the first line of Tab. \[table:kin\] can be found in Figure \[fig:invmass\]. The reconstructed [$J/\psi$]{}mass resolution is $5\,\text{MeV}$.
[cccccccccc]{} & $P$ & $\Delta P/P$ & $\sigma P/P$ & $\theta^\text{in}$ & $\Delta\theta^\text{in}$ & $\Delta\theta^\text{out}$ & $\Delta\Omega$ & $\sigma\theta^\text{in}$ & $\sigma\theta^\text{out}$\
& GeV/$c$ & %& %& & mrad & mrad& msr & mrad & mrad\
\
HMS & 0.4-7.4 & -10 +10 & 0.1 & 10.5$^\circ$-90$^\circ$ & $\pm 24$ & $\pm 70$ & 8 & 0.8 & 1.0\
SHMS & 2.5-11. & -15 +25 & 0.1 & 5.5$^\circ$-25$^\circ$ & $\pm 20$ & $\pm 50$ & 4 & 1.0 & 1.0\
\[table:spec\]
![ Invariant mass of the detected lepton pair with realistic smearing. The invariant mass resolution is $5\,\text{MeV}$. []{data-label="fig:invmass"}](figures/jpsi-invariant-mass.pdf){width="50.00000%"}
Projected results {#sec:res}
=================
In this section we describe the results of our simulation and the expected results for 9 days of beam on target. We discuss the projected yields in case of 5% coupling for the ${\ensuremath{P_c(4450)}\xspace}\rightarrow J/\psi p$ channel. Additionally, we will quantify the statistical precision with which we can identify the $P_c$ resonance for different values of the coupling. Finally, we will show the estimated impact of this experiment on the available world data for the [$J/\psi$]{}photo-production cross section.
Projected results in case of 5% coupling
----------------------------------------
As shown in Fig. \[fig:yieldone\] and Fig. \[fig:yieldtwo\] the results, clearly reveal the resonant structure of the pentaquark assuming a 5% coupling.
![Expected results for the reconstructed $t$ and $E_\gamma$ spectrum for 9 days of beam on target, assuming the less probable (5/2-, 3/2+) $P_c$ from [@Wang:2015jsa] with 5% coupling. Due to the larger cross section for the 5/2-, the separation in both spectra is even better than for the 5/2+ assumption shown in Fig. \[fig:yieldone\].[]{data-label="fig:yieldtwo"}](figures/yield-trc-52p32m.pdf "fig:"){width="45.00000%"} ![Expected results for the reconstructed $t$ and $E_\gamma$ spectrum for 9 days of beam on target, assuming the less probable (5/2-, 3/2+) $P_c$ from [@Wang:2015jsa] with 5% coupling. Due to the larger cross section for the 5/2-, the separation in both spectra is even better than for the 5/2+ assumption shown in Fig. \[fig:yieldone\].[]{data-label="fig:yieldtwo"}](figures/yield-Egamma-52p32m.pdf "fig:"){width="45.00000%"}
![Expected results for the reconstructed $t$ and $E_\gamma$ spectrum for 9 days of beam on target, assuming the less probable (5/2-, 3/2+) $P_c$ from [@Wang:2015jsa] with 5% coupling. Due to the larger cross section for the 5/2-, the separation in both spectra is even better than for the 5/2+ assumption shown in Fig. \[fig:yieldone\].[]{data-label="fig:yieldtwo"}](figures/yield-trc-52m32p.pdf "fig:"){width="45.00000%"} ![Expected results for the reconstructed $t$ and $E_\gamma$ spectrum for 9 days of beam on target, assuming the less probable (5/2-, 3/2+) $P_c$ from [@Wang:2015jsa] with 5% coupling. Due to the larger cross section for the 5/2-, the separation in both spectra is even better than for the 5/2+ assumption shown in Fig. \[fig:yieldone\].[]{data-label="fig:yieldtwo"}](figures/yield-Egamma-52m32p.pdf "fig:"){width="45.00000%"}
The projected results from the calibration measurement of the $t$-channel [$J/\psi$]{}background to the $P_c$ resonant channel, for 2 days of beam on target, can be found in Fig. \[fig:yieldbg\]. Note that in addition to providing the necessary leverage for the $t$-channel background subtraction, this calibration measurement will greatly impact our knowledge of the $t$-channel [$J/\psi$]{}photo-production near threshold, where currently no world data exist.
![Expected results for the reconstructed $E_\gamma$ spectrum for the calibration measurement with 2 days of beam on target. The left panel shows the (5/2+, 3/2-) case , and the right panel shows the (5/2-, 3/2+) case, both with 5% coupling. []{data-label="fig:yieldbg"}](figures/tchannel-yield-Egamma-52p32m.pdf "fig:"){width="45.00000%"} ![Expected results for the reconstructed $E_\gamma$ spectrum for the calibration measurement with 2 days of beam on target. The left panel shows the (5/2+, 3/2-) case , and the right panel shows the (5/2-, 3/2+) case, both with 5% coupling. []{data-label="fig:yieldbg"}](figures/tchannel-yield-Egamma-52m32p.pdf "fig:"){width="45.00000%"}
Sensitivity to the $P_c$ resonant production
--------------------------------------------
To obtain an estimate of the sensitivity to the $P_c$ resonant process as a function of the coupling to the ${\ensuremath{J/\psi}\xspace}p$ channel, we calculated the log-likelihood difference $\Delta\log\mathcal{L}$ between the hypothesis that the simulated spectra can be described by just a $t$-channel process, and the hypothesis that the $P_c$ resonances are present on top of the $t$-channel production. We assumed 9 days of beam at $50\,\mu A$ for setting \#1. We then used Wilk’s theorem [@Wilks:1938dza] to relate the value of $2\Delta\log\mathcal{L}$ to a value of $\chi^2$ with 5 degrees of freedom (one for the coupling, and 4 for the mass and width of each of the $P_c$). Note that a binned likelihood approach was used, which yields a conservative estimate compared to the results of a full unbinned extended maximum likelihood procedure.
The results of this sensitivity study can be found in Fig. \[fig:sensitivity\]. We found that, for values of the coupling of 1.3% and higher, we have a sensitivity of more than the 5 standard deviations for discovery. Fig. \[fig:sensitivity\] also shows the projected results in case of a 1.3% coupling. For a coupling of 5%, our sensitivity far exceeds 20 standard deviations.
In the proposal, we assumed a realistic coupling of 5% from Wang [@Wang:2015jsa], which they found to be compatible with the currently existing ${\ensuremath{J/\psi}\xspace}$ photo-production data. A more recent statistical analysis by Blin [@Blin:2016dlf] found an upper limit of the coupling values to be between $8-17\%$ at the 95% confidence level for the [$P_c(4450)$]{}(5/2+). Furthermore, Karliner [@Karliner:2015ina] argues that the coupling cannot be too small, as the ${\ensuremath{P_c(4450)}\xspace}\rightarrow {\ensuremath{J/\psi}\xspace}p$ signal is 4.1% of the ${\ensuremath{J/\psi}\xspace}p$ final state in $\Lambda_b\rightarrow K^-{\ensuremath{J/\psi}\xspace}p$. If the coupling were too small, the value of $\Lambda_b\rightarrow K^- P_c$ with the $P_c$ decaying to final states other than ${\ensuremath{J/\psi}\xspace}p$, would become unreasonably large in comparison with the measured branching fraction of $\Lambda_b \rightarrow K^-{\ensuremath{J/\psi}\xspace}p$. This means that, due to the sensitivity of the proposed experiment down to very low values of the coupling, we will have the ability to provide a very strong exclusion of the charmed-pentaquark assumption in case it is not found.
![ The left figure shows the sensitivity to the $P_c$ as a function of the coupling to the ${\ensuremath{J/\psi}\xspace}p$ channel, obtained from a log-likelihood analysis. The dashed line shows the $5\sigma$ level of sensitivity necessary for discovery. This level is reached starting from a coupling of 1.3%. The right panel shows the expected results for the reconstructed $E_\gamma$ spectrum for this 1.3% coupling for the [$P_c(4450)$]{}(5/2+).[]{data-label="fig:sensitivity"}](figures/sensitivity.pdf "fig:"){width="45.00000%"} ![ The left figure shows the sensitivity to the $P_c$ as a function of the coupling to the ${\ensuremath{J/\psi}\xspace}p$ channel, obtained from a log-likelihood analysis. The dashed line shows the $5\sigma$ level of sensitivity necessary for discovery. This level is reached starting from a coupling of 1.3%. The right panel shows the expected results for the reconstructed $E_\gamma$ spectrum for this 1.3% coupling for the [$P_c(4450)$]{}(5/2+).[]{data-label="fig:sensitivity"}](figures/yield-min-Egamma-52p32m.pdf "fig:"){width="45.00000%"}
Projected impact on the world data for [$J/\psi$]{}production
-------------------------------------------------------------
![ Projected impact of this experiment assuming, the (5/2+, 3/2-) case with 5% coupling, for 9 days of data taking in setting \#1 (solid circles) and 2 additional days of data taking in setting \#2 (open circles). The existing data points from Cornell data from [@Gittelman:1975ix] and SLAC (unpublished) [@Anderson:1976sd] are also shown. []{data-label="fig:impact"}](figures/impact.pdf){width="80.00000%"}
The projected impact of the proposed experiment, assuming the (5/2+, 3/2-) case with 5% coupling is shown in Fig. \[fig:impact\]. These results will dramatically enhance our knowledge of [$J/\psi$]{}photo-production near threshold. The absolute cross section measurements from this experiment will provide valuable input for future experimental endeavors at CLAS12 and SoLID [@CLAS12-tcs:proposal; @SoLIDjpsi:proposal].
Run plan and beam request
=========================
We propose to carry the measurement of elastic photo-production of [$J/\psi$]{}in the threshold region with the aim to confirm the LHCb $P_c(4450)$ discovery. The experiment uses the standard equipment of the upgraded Hall C apparatus at Jefferson Lab. We request 11 days (264 hours) of beam time. The first 40 hours will focus on measuring the shape of the $t$ distribution with high statistics, using setting \#2 to maximize the combined acceptance for this process. We will take an additional 8 hours of data in this setting without the radiator, in order to assess the contribution from lepto-production. Finally, we will conduct our main measurement in setting \#1 for the remaining 216 hours. See Tab. \[table:kin\] for the definitions of the spectrometer settings \#1 and \#2. Accidental coincidences between the two spectrometers will be measured at the same setting and the same time in the momentum acceptance of the spectrometers outside the true physics events.
We request 11 days to perform this high-impact measurement in search of the LHCb charmed exotic resonances consistent with “pentaquarks”.
|
---
abstract: 'We study the rational Kontsevich integral of torus knots. We construct explicitely a series of diagrams made of circles joined together in a tree-like fashion and colored by some special rational functions. We show that this series codes exactly the unwheeled rational Kontsevich integral of torus knots, and that it behaves very simply under branched coverings. Our proof is combinatorial. It uses the results of Wheels and Wheeling and various spaces of diagrams.'
address: |
Institut de Mathématiques de Jussieu, Équipe “Topologie et Géométries Algébriques”\
Case 7012, Université Paris VII, 75251 Paris CEDEX 05, France
title: |
A computation of the Kontsevich integral\
of torus knots
---
Introduction and notation
=========================
In 1998, in [@bgrt], a conjecture was formulated about a precise expression for the Kontsevich integral of the unknot (later it was proved, see for instance [@blt]). Until now, we do not know any complete formula for this powerful invariant for non-trivial knots. Let ${\mathcal{B}}$ be the usual space of uni-trivalent diagrams, which is the target space of the Kontsevich integral. We define a localization of ${\mathcal{B}}$ called ${\mathcal{B}}_s$. Unfortunately, the localization map is not injective in high loop degrees. Then, we give a formula for the family of torus knots which takes its values in ${\mathcal{B}}_s$ and which we state in a “rational” form.
The starting point of our proof is a well-known formula: it has been used by Christine Lescop (see [@les]) and Dror Bar-Natan in unpublished work. From this expression, we construct a sequence of series of diagrams which are all obtained by “gluing wheels” and which converges to the unwheeled Kontsevich integral of torus knots. (See Section 1.1 for the precise meaning of “unwheeled” here.) From this we present the result in a compact way by using “gluing graphs”, which we will define along the way.
Then, in the third part, we compute a rational form of the preceding expression and show that only tree-like gluing diagrams appear as is suggested by figure \[loop\]. More precisely, we show the following theorem:
Let $D$ be the operator on ${\mathbb{Q}}(t)$ defined by $Dg (t)=tg'(t)$ and let $h(t)=\frac{t+1}{t-1}$ (the operator $D$ acts as $\frac{{\text{d}}}{{\text{d}}x}$ on $g(\exp(x))$).
There is a series of diagrams $Y^{rat}_{p,q}$ obtained by inserting circles in vertices of tree graphs, such that circles corresponding to vertices of valence $k$ are colored by $D^{k-1}h(t^p), D^{k-1} h(t^q)$ or $D^{k-1} h(t^{pq})$.
Applying the “hair map” (in other words, substituting $t$ with the exponential of a small leg attached to the circle), we obtain a series in ${\mathcal{B}}_s$ which is equal to the logarithm of the unwheeled Kontsevich integral of the torus knot of parameters $p,q$ plus a fixed series $\log \langle \Omega,\Omega \rangle$ (see \[intro\] for precise definitions and theorem \[theoreme\] for an explicit expression of the first terms).
As a consequence of this computation, we show that the operator $\operatorname{Lift}_r$ which corresponds to cyclic branched coverings of $S^3$ along the knot simply acts on $Y_{p,q}^{rat}$ by multiplying a diagram $D$ by $r^{-\chi(D)}$ where $\chi$ is the Euler characteristic.
This article extends the result of our previous article [@moi] where we computed the unwheeled Kontsevich integral of torus knots up to degree 3. Here, we give a formula for all degrees using quite different methods.
Lev Rozansky also computed formulas for the loop expansion of torus knots in the weight system associated to $\rm{sl}_2$, see [@roz2]. The computation of the 2-loop part of torus knots has been done independantly by Tomotada Ohtsuki in [@oht] who computed more generally a formula for 2-loop part of knots cabled by torus knots.
We would like to thank Stavros Garoufalidis, Marcos Marino, Tomotada Ohtsuki and Pierre Vogel for useful remarks. We also thank the referee and Gregor Masbaum for their remarks and their careful reading.
Normalizations of the Kontsevich integral {#intro}
-----------------------------------------
Let $K$ be a knot in $S^3$ and suppose that $K$ has a banded structure with self-linking 0. We will denote by $Z(K)$ the Kontsevich integral of $K$ in the algebra ${\mathcal{A}}$ of trivalent diagrams lying on a circle.
Let ${\mathcal{B}}$ be the algebra of uni-trivalent diagrams. It is well known that the Poincaré-Birkhoff-Witt map $\chi:{\mathcal{B}}\to{\mathcal{A}}$ is an isomorphism but not an algebra isomorphism. We will denote by $\sigma$ its inverse.
If $U$ is the trivial knot, we define $\Omega=\sigma Z(U)$. The series $\Omega$ is the exponential of a series of connected graphs whose first terms are $\frac{1}{48}\twowheel-\frac{1}{5760}\fourwheel+\cdots$.
The map $\Upsilon=\chi \circ \partial_{\Omega}:{\mathcal{B}}\to {\mathcal{A}}$ defined for instance in [@th] is known to be an algebra isomorphism. The quantity $Z^{\fourwheel}(K)=\Upsilon^{-1}Z(K)$ will be called unwheeled Kontsevich integral. It behaves better than $\sigma Z(K)$ under connected sum and cyclic branched coverings.
For each knot $K$, the quantities $Z(K)$, $\sigma Z(K)$ and $Z^{\fourwheel}(K)$ are group-like, which means that they are exponentials of a series of connected diagrams. We will denote by respectively $z(K)$, $\sigma z(K)$ and $z^{\fourwheel}(K)$ the logarithms of these quantities.
Loop degree and rationality
---------------------------
If $D$ is a connected diagram of ${\mathcal{B}}$, its first Betti number defines a degree called loop degree. The loop degree 1 part of $\sigma Z(K)$ or $Z^{\fourwheel}(K)$ is well-known: it only depends on the Alexander polynomial of $K$. For the higher degrees, very little is known. There are formulas for the 2-loop part of small knots in Rozansky’s table (see [@roz]), and we can find in [@double] a formula for the 2-loop part of untwisted Whitehead doubles. In the sequel, we give a formula for the full Kontsevich integral of torus knots.
In order to make precise computations, we will need the following formalism which is yet another version of “diagrams with beads” used in [@roz; @kr; @rat] for instance, and generalized by P. Vogel in [@vog].
Let ${\mathcal{C}}$ be the category whose objects are free abelian groups of finite rank and morphisms are linear isomorphisms. We call ${\mathcal{C}}$-module a functor from ${\mathcal{C}}$ to the category of ${\mathbb{Q}}$-vector spaces. In the sequel, we will associate to any ${\mathcal{C}}$-module $F$ a ${\mathbb{Q}}$-vector space ${\mathcal{D}}(F)$ which will be called the space of diagrams decorated by $F$. This construction will be a functor from the category of ${\mathcal{C}}$-modules to the category of ${\mathbb{Q}}$-vector spaces.
\[espdediag\] Let $F$ be a ${\mathcal{C}}$-module and $\Gamma$ be a finite trivalent graph with local orientations at vertices (we allow $\Gamma$ to have connected components which are circles).
We define ${\mathcal{D}}(F)$ as the quotient of $\bigoplus_{\Gamma} F(H^1(\Gamma,{\mathbb{Z}}))$ by the following relations:
- If $\Gamma$ is isomorphic to $\Gamma'$ via a map $\phi$, then we identify $x\in F(\Gamma')$ and $F(\phi^*)(x)\in F(\Gamma)$ for any $x$ in $F(\Gamma')$.
- If $\Gamma$ and $\Gamma'$ just differ by the orientation of a vertex, then we identify $x$ in $F(H^1(\Gamma,{\mathbb{Z}}))$ with $-x$ in $F(H^1(\Gamma',{\mathbb{Z}}))$.
- If $\Gamma$ is a graph with one four-valent vertex, we note $\Gamma_I, \Gamma_H, \Gamma_X$ the three standard resolutions of this vertex. We have canonical identifications between $H^1(\Gamma,{\mathbb{Z}}), H^1(\Gamma_I,{\mathbb{Z}}),H^1(\Gamma_H,{\mathbb{Z}})$ and $H^1(\Gamma_X,{\mathbb{Z}})$. For every $x\in F(H^1(\Gamma,{\mathbb{Z}}))$, we add the relation $x_I=x_H-x_X$.
Let us give the main example: we define $F(H)={\mathbb{Q}}[[H]]=\prod_{n\ge 0}S^n(H\otimes {\mathbb{Q}})$. The space ${\mathcal{D}}(F)$ is obtained by coloring graphs with 1-cohomology classes which can be materialized by small legs attached to the edges. It is not hard to see that ${\mathcal{D}}(F)\simeq{\mathcal{B}}$. This isomorphism is a convenient way to express series of diagrams.
Let $f(x)$ be the power series defined by $\frac{1}{2}\log{\frac{\sinh{x/2}}{x/2}}$. The famous wheel formula (see [@th]) states that $\sigma z(U)=f(x)$ (we suppose that the variable $x$ is a generator of $H^1(\bigcirc,{\mathbb{Z}})$). Further, it was shown in [@kr] that the loop degree 1 part of $\sigma z(K)$ is $f(x)+Wh_K(x)$ where $Wh_K(x)=-\frac{1}{2}\log \Delta(e^x)$ and $\Delta$ is the Alexander polynomial of $K$. As we are interested in the higher loop degree part, we need to recall the rationality theorem which was proved in [@rat]. For that, we define a new space of diagrams with the help of a new functor.
Let ${\mathbb{Q}}[\exp(H)]$ be the polynomial algebra on elements on the form $\exp(\lambda), \lambda\in H$ with the relation $\exp(\lambda+\mu)=\exp(\lambda)\exp(\mu)$ for all $\lambda$ and $\mu$ in $H$.
Then, $F(H)={\mathbb{Q}}[\exp(H)]_{loc}$ is the localization of the preceding space by expressions such as $P_1(\exp(h_1))\cdots P_k(\exp(h_k))$ with $h_1,\ldots,h_k \in H$, $P_1,\ldots,P_k\in {\mathbb{Q}}[h]$ and $P_1(1)\cdots P_k(1)\ne 0$.
It defines a new space of diagrams ${\mathcal{B}}^{\rm rat}={\mathcal{D}}(F)$ with a map $\operatorname{Hair}:{\mathcal{B}}^{\rm rat}\to{\mathcal{B}}$ induced by the Taylor expansion: ${\mathbb{Q}}[\exp(H)]_{loc}\to{\mathbb{Q}}[[H]]$ which is a map of ${\mathcal{C}}$-modules.
The rationality theorem tells us that there is an element ${Z^{\rm rat}}(K)\in {\mathcal{B}}^{\rm rat}$ such that $$\sigma Z(K)=\exp(f(x)+Wh_K(x)) \operatorname{Hair}{Z^{\rm rat}}(K).$$ Respectively, there is an element ${Z^{\fourwheel {\rm rat}}}(K) \in {\mathcal{B}}^{\rm rat}$ such that $$Z^{\fourwheel}(K)=
\frac{1}{\langle \Omega,\Omega\rangle}
\exp(f(x)+Wh_K(x)) \operatorname{Hair}{Z^{\fourwheel {\rm rat}}}(K).$$ A construction, developed in [@rat] gives ${Z^{\rm rat}}(K)$ as an invariant of $K$; this is not automatic because the $\operatorname{Hair}$ map is not injective, although it is so in small degrees (see [@ber]). We refer to [@lift] for a construction of ${Z^{\fourwheel {\rm rat}}}(K)$.
Diagrammatic expressions of the integral
========================================
First diagrammatic expression
-----------------------------
Let $p$ and $q$ be two coprime integers such that $p>0$. We note $K_{p,q}$ the torus banded knot with parameters $p$ and $q$ and self-linking 0, and $L_{p,q}$ the torus banded knot with banding parallel to the torus on which it lies. This knot has self-linking $pq$, and its Kontsevich integral is a bit easier to compute.
The method of computation is inspired from [@les]: we first compute the Kontsevich integral of the following braid.
Let $p$ points be lying on the vertices of a regular $p$-gon. We note $\gamma$ the braid obtained by rotating the whole picture by an angle $2\pi\frac{q}{p}$ as is shown in figure \[polygone\].
Let us associate to any one dimensional manifold $\Gamma$ the space ${\mathcal{A}}(\Gamma)$ of trivalent diagrams lying on $\Gamma$. This defines a contravariant functor with respect to continuous maps relative to boundaries. Let $\phi_p^*$ be the map induced by the projection onto the first factor $\phi_p:[0,1]\times\{1,\ldots,p\}\to [0,1]$ and $\isolatedchord$ be the only degree 1 diagram in ${\mathcal{A}}([0,1])$. Then a direct computation of monodromy of the K-Z connection shows that $\gamma$ has Kontsevich integral equal to $\phi_p^*( \exp_{\#}(\frac{q}{2p}\isolatedchord))$.
The banded knot $L_{p,q}$ is obtained by closing the previous braid: this translates diagrammatically to the following: let $\psi_p$ be the map from $S^1$ to itself defined by $\psi_p(z)=z^p$. Then, $Z(L_{p,q})=\psi_p^*(\nu\# \exp_{\#}(\frac{q}{2p}\isolatedchord))$, where $\nu=Z(U)$.
By lemma 4.10 of [@th], the map $\psi_p^*$ viewed in ${\mathcal{B}}$ has the following form: if $D\in {\mathcal{B}}$ has $k$ legs (i.e. univalent vertices) then $\sigma \psi_p^* \chi D = p^k D$. We will note more simply $D_p$ the result of this operation which looks like a change of variable.
Then, to compute $Z(K_{p,q})$ from $Z(L_{p,q})$, we only need to change the framing, that is $Z(K_{p,q})=\exp_{\#}(-\frac{pq}{2}\isolatedchord)\#Z(L_{p,q})$. We will transform this product into the usual one by applying the unwheeling map $\Upsilon^{-1}$. As a result, we will have a formula for $Z^{\fourwheel}(K_{p,q})$.
We now sum up the steps of the computation:
1. Computation of $\sigma(\nu\# \exp_{\#}(\frac{q}{2p}\isolatedchord))$
2. Change of variables $x\mapsto px$
3. Unwheeling
We recall that $\Upsilon=\chi\circ\partial_{\Omega}$ is an algebra isomorphism and that $\Upsilon^{-1}\nu = \frac{\Omega}{\langle \Omega,\Omega\rangle}$ and $\Upsilon^{-1}\isolatedchord= \frown-\ThetaGraph/24$. Here $\langle A,B\rangle$ is the sum over all ways of gluing all legs of $A$ to all legs of $B$.
Then, to achieve the first step, we have to compute $$\label{expr}
\sigma(\nu\# \exp_{\#}(\frac{q}{2p}\isolatedchord))=
\frac{\partial_{\Omega} (\Omega\exp(\frac{q}{2p}{\!\!\frown}))}{\langle\Omega,\Omega\rangle\exp(\frac{q}{48p}\ThetaGraph)}.$$ We explain below what will be a diagram colored by some parameters. In this setting, we will recall how usual operations on diagrams are obtained. Finally we will describe a new operation.
\[definition\]$\phantom{99}$
- Let $P$ be a set of parameters. We denote ${\mathcal{B}}(P)$ the space of couples $(D,f)$ where $D$ is a uni-trivalent graph and $f$ is a map from legs of $D$ to $P$. We will say that legs of $D$ are labeled or colored by elements of $P$. If $D\in{\mathcal{B}}$ and $x\in P$, we will write $D_x$ the diagram $D$ whose legs are colored by $x$.
- If $D\in {\mathcal{B}}(x)$, we define the diagram $D_{x+y}\in{\mathcal{B}}(x,y)$ by replacing legs of $D$ by the same legs colored by $x$ or $y$ in all possible ways.
- If $D,E\in{\mathcal{B}}(P)$ and $x\in P$ we define ${\langle}D,E {\rangle}_x$ as the sum over all gluings of all $x$-legs of $D$ on all $x$-legs of $E$. We also define $\partial_D E$ as the sum over all gluings of all $x$-legs of $D$ on some $x$-legs of $E$. This operator satisfies $\partial_{D_x} E_x = {\langle}D_y,E_{x+y}{\rangle}_y$ and for $F\in{\mathcal{B}}(P)$, ${\langle}D_x,E_x F_x{\rangle}_x={\langle}\partial_{E_x}D_x,F_x{\rangle}_x$.
- Let $A_x$ and $B_x$ be two series of diagrams in ${\mathcal{B}}(P)$ where $P$ contains at least three colors: $\{x,y,z\}$. We define $A\cdot B=\langle A_{y+x},B_{x+z}\rangle_x$. The $y$-legs of $A\cdot B$ will be called left legs and $z$-legs of $A\cdot B$ will be called right legs for obvious reasons. For rational numbers $r$ and $r'$, the diagram ${\vphantom{A\cdot B}_{r} A\cdot B_{r'}}$ will be the result of multiplying all left legs by $r$ and all right legs by $r'$.
Moreover, if $a_x$ and $b_x$ are two series of connected diagrams, then the series $\exp(a), \exp(b)$ and $\exp(a)\cdot\exp(b)$ are group-like, hence, we define ${\vphantom{a\times b}_{y} a\times b_{z}}=\log({\vphantom{\exp(a)\cdot\exp(b)}_{y} \exp(a)\cdot\exp(b)_{z}})$.
Using the notation of definition \[definition\], we compute $$\partial_{\Omega}(\Omega\exp(\frac{q}{2p}{\!\!\frown}))=\langle \Omega_x,\Omega_{x+y}\exp(\frac{q}{2p}{\vphantom{{\!\!\frown}}_{x+y} {\!\!\frown}_{x+y}})\rangle_x.$$ We use the fact that ${\vphantom{{\!\!\frown}}_{x+y} {\!\!\frown}_{x+y}}={\vphantom{{\!\!\frown}}_{x} {\!\!\frown}_{x}}+2{\vphantom{{\!\!\frown}}_{x} {\!\!\frown}_{y}}+{\vphantom{{\!\!\frown}}_{y} {\!\!\frown}_{y}}$ and properties recalled in the preceding definition to obtain that the series $\partial_{\Omega}(\Omega\exp(\frac{q}{2p}{\!\!\frown}))$ is equal to $\langle \partial_{\exp(\frac{q}{2p}{\vphantom{{\!\!\frown}}_{x} {\!\!\frown}_{x}})}\Omega_x,\Omega_{x+y}\exp(\frac{q}{p}{\vphantom{{\!\!\frown}}_{x} {\!\!\frown}_{y}})\rangle_x \exp(\frac{q}{2p}{\vphantom{{\!\!\frown}}_{y} {\!\!\frown}_{y}})$.
Now we use the fundamental formula $\partial_{D_x}\Omega_x = {\langle}D_x,\Omega_x{\rangle}_x \Omega_x$ which is true if $D$ contains only $x$-legs. The preceding expression reduces to $$\langle \exp(\frac{q}{2p}{\!\!\frown}),\Omega\rangle \left({\vphantom{\Omega\cdot\Omega}_{\frac{q}{p}} \Omega\cdot\Omega_{}}\right)\exp(\frac{q}{2p}{\!\!\frown}).$$ Thanks to the identity $\langle \exp(\frac{q}{2p}{\!\!\frown}),\Omega\rangle=\exp(\langle \frac{q}{2p}{\!\!\frown},\Omega\rangle)=\exp(\frac{q}{48p}\ThetaGraph)$, we may cancel this factor in the expression above.
Multiplying by $p$ to the power the number of legs, we obtain a formula for $Z(L_{p,q})$: $$Z(L_{p,q})= \frac{{\vphantom{\Omega\cdot\Omega}_{q} \Omega\cdot\Omega_{p}}\exp(\frac{qp}{2}{\!\!\frown})}{\langle\Omega,\Omega\rangle}.$$ We obtain a formula for $Z^{\fourwheel}(K_{p,q})$ by unwheeling $Z(L_{p,q})$ and multiplying by $\Upsilon^{-1} \exp_{\#}(-\frac{pq}{2}\isolatedchord)=\exp(-\frac{pq}{2}{\!\!\frown}+\frac{pq}{48}\ThetaGraph)$: $$\label{fff}
Z^{\fourwheel}(K_{p,q})=
\partial_{\Omega}^{-1}(
{\vphantom{\Omega\cdot\Omega}_{q} \Omega\cdot\Omega_{p}}\exp(\frac{qp}{2}{\!\!\frown})
)
\frac{\exp(-\frac{pq}{2}{\!\!\frown}+\frac{pq}{48}\ThetaGraph)}{\langle\Omega,\Omega\rangle}.$$
A sequence converging to the integral of torus knots
----------------------------------------------------
In this part, we will start from formula to deduce new formulas by an induction process.
We define ${\mathcal{B}}^c= {\mathcal{B}}(\{\text{active,inert}\})$. There is a forgetful map ${\mathcal{B}}^c\to{\mathcal{B}}$.
Let us define three operators analogous to those of definition \[definition\] but for diagrams in ${\mathcal{B}}^c$ in the following way:
\[definition2\]$\phantom{99}$
- For $A\in {\mathcal{B}}^c$ and $r\in{\mathbb{Q}}$, we set $A_r$ to be the diagram $A$ whose active legs are multiplied by $r$, viewed as an element of ${\mathcal{B}}$.
- If $A\in{\mathcal{B}}^c$ and $B\in{\mathcal{B}}$, the diagram $\partial_A B \in{\mathcal{B}}$ is the sum over gluings of all active legs of $A$ on some legs of $B$. This is an element of ${\mathcal{B}}$.
- For $A\in{\mathcal{B}}$ and $B\in {\mathcal{B}}^c$ two diagrams, we set $A\cdot B\in{\mathcal{B}}^c$ to be the sum over all gluings of some active legs of $B$ with some legs of $A$. The active legs of $A\cdot B$ are the remaining active legs of $B$. We also define $a\times b=\log(\exp(a)\times\exp(b))$. See figure \[active\] for both definitions.
\[lm\] For all $A\in {\mathcal{B}}^c$ and $B,C$ in ${\mathcal{B}}$ we have the identity $\partial_A(BC)=\partial_{C\cdot A}B$.
In the formula $\partial_A(BC)$, the active legs of $A$ are glued to legs of $B$ and $C$. We can split these legs in two parts and glue the first part to $C$ and the second part to $B$. Before gluing the second part to $B$, we obtain exactly the diagram $C\cdot A$. The remaining active legs of $A$ should be glued to $B$: we write the result of this operation $\partial_{C\cdot A}B$. This proof is summed up in figure \[lmfig\].
We now define a sequence of series $\omega^n$ in ${\mathcal{B}}^c$. Let us put $\omega^{-1}={\vphantom{\omega\times\omega}_{\frac{1}{p}} \omega\times\omega_{\frac{1}{q}}}$ and $\omega^0=\omega$. We declare that all their legs are active.
We suppose all legs of $\omega^{-1}$ and $\omega^0$ are active, and for all $n\ge 0$, we set $\omega^{n+1}=(\omega^{n-1}_{pq}-\omega^{n}_{pq})\times\omega^n -\omega^{n-1}_{pq}+\omega^{n}_{pq}$. In this formula, the added term $\omega^{n-1}_{pq}-\omega^{n}_{pq}$ is considered inert.
Let us prove the following proposition:
For all $n\ge 0$, we have
1. The series $\exp(\frac{pq}{2}{\!\!\frown})$ acts by derivation on active legs of $\exp(\omega^n)$ by multiplication by $\exp(\frac{pq}{48}\ThetaGraph)$. In formulas, we mean that $\partial^{\text{active}}_{\exp(\frac{pq}{2}{\!\!\frown})}\exp(\omega^n)=\exp(\frac{pq}{48}\ThetaGraph)\exp(\omega^n)$. Here we have written $\partial^{\text{active}}$ in order to mean that only active legs are glued in this process, contrary to the definition \[definition2\].
2. $\phantom{99}$ $$Z^{\fourwheel}(K_{p,q})=\left(\partial^{-1}_{\exp(\omega^n)}\exp(\frac{pq}{2}{\!\!\frown}+\omega^{n-1}_{pq})\right)\frac{\exp(-\frac{pq}{2}{\!\!\frown}\!+\!\omega^{-1}_{pq}\!-\!\omega^{n-1}_{pq}\!+\!\frac{pq}{48}\ThetaGraph)}{\langle\Omega,\Omega\rangle}$$
3. $\phantom{99}$ $$\begin{gathered}
Z^{\fourwheel}(K_{p,q})=\partial^{-1}_{\exp(\omega^n)}\left[\partial_{\exp(\omega^n)}\exp(\frac{pq}{2}{\!\!\frown})\exp(\omega^{n-1}_{pq}-\omega^n_{pq})\right]\cdot\\
\frac{\exp(-\frac{pq}{2}{\!\!\frown}+\omega^{-1}_{pq}-\omega^{n-1}_{pq})}{\langle\Omega,\Omega\rangle}.\end{gathered}$$
4. Moreover the sequence $\omega^{-1}_{pq}-\omega^n_{pq}$ converges to $\log\left(Z^{\fourwheel}(K_{p,q})\langle\Omega,\Omega\rangle\right)$ with respect to loop degree.
(i)Let us prove the first formula by induction.
The assertion for $n=0$ is just $\partial_{\exp(\frac{pq}{2}{\!\!\frown})}\Omega=\exp(\frac{pq}{48}\ThetaGraph) \Omega$. This is a consequence of the fundamental relation $\partial_D\Omega = {\langle}D,\Omega{\rangle}\Omega$.
Suppose that for some $n\ge 0$ we have the identity $$\partial^{\text{active}}_{\exp(\frac{pq}{2}{\!\!\frown})}\exp(\omega^n)=\exp(\frac{pq}{48}\ThetaGraph)\exp(\omega^n).$$ Then by definition, we have the identity $$\exp(\omega^{n+1})=\frac{\exp(\omega^{n-1}_{pq}-\omega^n_{pq})\cdot \exp(\omega^n)}{\exp(\omega^{n-1}_{pq}-\omega^n_{pq})}.$$ But all active legs of the series $\exp(\omega^{n+1})$ come from the series $\exp(\omega^n)$. This proves the identity $$\partial^{\text{active}}_{\exp(\frac{pq}{2}{\!\!\frown})}\exp(\omega^{n+1})=\frac{\exp(\omega^{n-1}_{pq}-\omega^n_{pq})\cdot \partial^{\text{active}}_{\exp(\frac{pq}{2}{\!\!\frown})}\exp(\omega^n)}{\exp(\omega^{n-1}_{pq}-\omega^n_{pq})}$$ because all “derivation-like” operations commute (see for instance lemma \[lm\]).
Using the induction assumption, we finally prove the following formula: $$\partial^{\text{active}}_{\exp(\frac{pq}{2}{\!\!\frown})}\exp(\omega^{n+1})=\exp(\frac{pq}{48}\ThetaGraph) \exp(\omega^{n+1}).$$
(ii)The second formula for $n=0$ is nothing but a version of formula . Suppose the second formula is true for some $n\ge 0$. To prove the second formula for $n+1$, we will prove that the expression at rank $n+1$ is equal to the expression at rank $n$. This is equivalent to the following identity: $$\begin{gathered}
\partial^{-1}_{\exp(\omega^{n+1})}
\exp(\frac{pq}{2}{\!\!\frown}+\omega^{n}_{pq})\exp(-\omega^n_{pq})
=\\
\partial^{-1}_{\exp(\omega^n)}
(\exp(\frac{pq}{2}{\!\!\frown}+\omega^{n-1}_{pq}))\exp(-\omega^{n-1}_{pq}).\end{gathered}$$ By applying the operator $\partial_{\exp(\omega^{n+1})}$, this identity is equivalent to the following equation: $$\exp(\frac{pq}{2}{\!\!\frown}+\omega^{n}_{pq})=\partial_{\exp(\omega^{n+1})}[
\partial^{-1}_{\exp(\omega^n)}
(\exp(\frac{pq}{2}{\!\!\frown}+\omega^{n-1}_{pq}))\exp(\omega^{n}_{pq}-\omega^{n-1}_{pq})
]$$ Using lemma \[lm\], we have the following equalities: $$\begin{aligned}
\partial_{\exp(\omega^{n+1})} A
&=& \partial_{\frac{\exp(\omega^{n-1}_{pq}-\omega^n_{pq})\cdot \exp(\omega^n)}{\exp(\omega^{n-1}_{pq}-\omega^n_{pq})}}A \\
&=& \left(\partial_{\exp(\omega^{n-1}_{pq}-\omega^n_{pq})\cdot \exp(\omega^n)} A \right) \exp(\omega^n_{pq}-\omega^{n-1}_{pq})\\
&=& \partial_{\exp(\omega^n)}\left(A\exp(\omega^{n-1}_{pq}-\omega^n_{pq})\right) \exp(\omega^n_{pq}-\omega^{n-1}_{pq})\end{aligned}$$ Replacing $A$ by its value, we obtain the desired identity.
(iii)We obtain the third formula from the second one by using the following formula: $$\begin{aligned}
\partial_{\exp(\omega^n)}\exp(\frac{pq}{2}{\!\!\frown})&=&\langle \exp(\omega^n_a),\exp(\frac{pq}{2}{\vphantom{{\!\!\frown}}_{a+b} {\!\!\frown}_{a+b}})\rangle_a\\
&=& \left(\partial^{\text{active}}_{\exp(\frac{pq}{2}{\!\!\frown})} \exp(\omega^n)\right)_{pq}\exp(\frac{pq}{2}{\!\!\frown}) \\
&=& \exp(\omega^n_{pq})\exp(\frac{pq}{48}\ThetaGraph+\frac{pq}{2}{\!\!\frown})\end{aligned}$$ The last step uses the first part of the proposition.
(iv)Let us prove the last property: we will use mainly the third formula of the proposition.
We want to show that all diagrams in the series $$\label{rappel}
\partial^{-1}_{\exp(\omega^n)}\left[\partial_{\exp(\omega^n)}\exp(\frac{pq}{2}{\!\!\frown})\exp(\omega^{n-1}_{pq}-\omega^n_{pq})\right]\exp(-\frac{pq}{2}{\!\!\frown}).$$ have loop degree greater or equal to $n$. This will prove that the expression $\omega^{-1}_{pq}-\omega^{n-1}_{pq}$ and the expression $\log(Z^{\fourwheel}(K_{p,q})\langle\Omega,\Omega\rangle)$ coincide up to loop degree $n$, and then that the series $\omega^{-1}_{pq}-\omega^{n-1}_{pq}$ converges to $\log(Z^{\fourwheel}(K_{p,q})\langle\Omega,\Omega\rangle)$ with respect to loop degree when $n$ goes to infinity.
Let us prove by induction that for all $n$, the series $\omega^{n-1}_{pq}-\omega^n_{pq}$ contains diagrams with loop degree greater or equal to $n$. This is obvious for $n=0$. Suppose it is true for some $n$, then by definition of $\omega^{n+1}$, we have $$\omega^{n+1}_{pq} -\omega^n_{pq}= \left((\omega^{n-1}_{pq}-\omega^n_{pq})\times \omega^n\right)_{pq}-\omega^{n-1}_{pq}.$$ This proves that the series $\omega^{n+1}_{pq} -\omega^n_{pq}$ is obtained by non trivial gluings of $\omega^{n-1}_{pq}-\omega^n_{pq}$ and $\omega^n$. As by assumption the series $\omega^{n-1}_{pq}-\omega^n_{pq}$ has loop degree greater or equal to $n$, the series $\omega^{n+1}_{pq} -\omega^n_{pq}$ must have loop degree greater than $n$.
Now, the operator $\partial_{\exp(\omega^n)}$ always increases the loop degree, and so does its inverse. This proves that diagrams of $\exp(\omega^{n-1}_{pq}-\omega^n_{pq})$ cannot create diagrams of loop degree less than $n$ in the expression . Hence, up to degree $n$, we can ignore the series $\exp(\omega^{n-1}_{pq}-\omega^n_{pq})$, and the preceding expression reduces to $$\partial^{-1}_{\exp(\omega^n)}\left(\partial_{\exp(\omega^n)}\exp(\frac{pq}{2}{\!\!\frown})\right)\exp(-\frac{pq}{2}{\!\!\frown})=1.$$ This proves the last part of the proposition.
An expression with gluing graphs
--------------------------------
We remark that all diagrams appearing in $\omega^n$ are made by “gluing wheels”, i.e. they are constructed from wheel series as $\omega,\omega_p,\omega_q$ by applying some gluing operators. We make this statement precise by introducing gluing graphs and we give a presentation of $\log(Z^{\fourwheel}(K_{p,q})\langle\Omega,\Omega\rangle)$ with gluing graphs. These graphs will be useful for finding a rational expression of the preceding expression. As a non-trivial application, we will show that only tree-like graphs appear in this expression.
Let $P$ be a set of parameters. We denote by ${\mathcal{S}}(P)$ the ${\mathbb{Q}}$-vector space generated by unoriented finite graphs $(V,E,h)$ with a map $V\to P$ where $V$ is the set of vertices and $E$ the set of edges and a map $h:P\to{\mathbb{Q}}[[x]]$. Elements of this space will be called gluing graphs colored by $P$. The number of edges is a degree on ${\mathcal{S}}(P)$. We complete ${\mathcal{S}}(P)$ with respect to this degree. This space has an obvious Hopf algebra structure whose primitive elements are formed of connected graphs.
We define a substitution map $s:{\mathcal{S}}(P)\to {\mathcal{B}}$ in the following way: if $(X,h)$ is a diagram in ${\mathcal{S}}(P)$, we define $s(X,h)$ by gluing for all $a\in P$ and vertices $v$ decorated by $a$ the edges adjacent to $v$ to wheels generated by $h(a)$ in all possible ways. Sometimes, we call $a$-colored the remaining legs generated by $h(a)$. It is clear that $s$ is a map of Hopf algebras.
Let us give some examples: if $P=\{a\}$ $X=\bullet_a$ and $h(a)=f(x)$, then $s(X,h)=\omega$. In the same way, $s(\exp(\bullet_a),h)=\Omega$ and $$s({}_a\bullet\!\!-\!\!\bullet_b\!\!\!\!\!-\!\!\!-\!\!\!\bullet_c,h)=16 -\!\!\bigcirc\!\!-\!\!\bigcirc\!\!-\!\!\bigcirc\!\!\equiv$$ where $h(a)=x^2, h(b)=x^2$ and $h(c)=x^4$.
Let us define two operations on ${\mathcal{S}}(P)$:
\[operations\]$\phantom{99}$
- Given two connected gluing graphs $X$ and $Y$ with $A$ some set of parameters of $X$ and $B$ some set of parameters of $Y$, we define $X{\vphantom{\times}_{A} \times_{B}}Y=\log(\exp(X)\cdot\exp(Y))$ where $\exp(X)\cdot\exp(Y)$ is obtained by adding in all ways a finite number of edges from $A$-colored vertices of $\exp(X)$ to $B$-colored vertices of $\exp(Y)$.
- Let $(X,h)$ be a diagram in ${\mathcal{S}}(P{\small{\amalg}}\{a\})$, $h(x)\in {\mathbb{Q}}[[x]]$, and $r\in{\mathbb{Q}}\setminus\{0\}$. We note $a^{r}$ the operator which divides $X$ by $r^N$ where $N$ is the sum of valences of $a$-colored vertices of $X$.
Let us look at an example: take two formal parameters $a$ and $b$, then $$\exp(\bullet_a)\cdot\exp(\bullet_b)=\exp(\bullet_a+\bullet_b+{}_a\bullet\!\!-\!\!\bullet_b+{}_a\bullet\!\!\!=\!\!\!\bullet_b
+\frac{1}{2}{}_a\bullet\!\!-\!\!\bullet_b\!\!\!\!\!-\!\!\!-\!\!\!\bullet_a
+\frac{1}{2}{}_b\bullet\!\!-\!\!\bullet_a\!\!\!\!\!-\!\!\!-\!\!\!\bullet_b
+{}_a\bullet\!\!\!\equiv\!\!\!\bullet_b+ \cdots).$$ The operations of definition \[operations\] are a version for gluing graphs of usual operations on diagrams as is shown in the next proposition.
\[compat\] Let $X$ and $Y$ be two gluing graphs in ${\mathcal{S}}(P)$. Let $A$ and $B$ be two disjoint subsets of $P$. Then, we have $s(X{\vphantom{\times}_{A} \times_{B}}Y)=s(X){\vphantom{\times}_{A} \times_{B}}s(Y)$ (we omit the decorations of the vertices are they remain the same).
Let $X$ be a graph in ${\mathcal{S}}(P{\small{\amalg}}\{a\})$. We have $s(X,h)_{ra}=s(a^r X, h(rx))$ (in this formula, we only change the decoration of the $a$-vertex).
The proof of this proposition comes from a direct combinatorial description of the gluing operations that are involved, hence we omit it.
The main result of this section is the following:
\[Xpq\] There is an explicit series of gluing graphs $X_{p,q}$ whose substitution is $\log(Z^{\fourwheel}(K_{p,q})\langle \Omega,\Omega \rangle)$.
We will show by induction that for all $n\ge 0$, $\omega^n=s(X^n)$ for some $X^n\in{\mathcal{S}}(P)$ where $P=\{*\}\cup\{a,b,c\}$. The first parameter is active and corresponds to $f(x)$, the three last parameters are inert and correspond to $f(px),f(qx)$ and $f(pqx)$.
Firstly, the diagram $\omega^{-1}_{pq}$ has only inert legs and is obtained from the substitution of the gluing graph: $X^{-1}_{pq}=a^p b^q \bullet_a \times \bullet_b \in {\mathcal{S}}(a,b)$ where $a$ and $b$ are respectively associated to $f(px)$ and $f(qx)$. We deduce directly this construction from the formula $\omega^{-1}_{pq}={\vphantom{\omega\times\omega}_{p} \omega\times\omega_{q}}$ and proposition \[compat\].
We start the recursion at $n=0$ by setting $X^0=\bullet_*$. We have $s(X^0)=\omega^0$.
Take $n\ge 0$ and suppose we have constructed $X^k$ for all $k\le n$. Then, we set $X_{pq}^{k}= *^{pq} X^k |_{*\to c}$ for all $0\le k \le n$ in such a way that we have $\omega^k_{pq}=s(X_{pq}^k)$ for all $-1\le k \le n$ where $c$ is associated to $f(pqx)$.
Then we set $X^{n+1}=(X^{n-1}_{pq}-X^n_{pq}){\vphantom{\times}_{a,b,c} \times_{*}}X^n-(X^{n-1}_{pq}-X^{n}_{pq})$. This definition satisfies $\omega^{n+1}=s(X^{n+1})\in {\mathcal{B}}^c$. It gives a recursive way for finding all diagrams $X^n$ and then we set $X_{p,q}=\lim_n (X^{-1}_{pq}-X^n_{pq})$.
Let us give all terms with less than two edges as an example. We use some notations to give a more compact form. Edges may be oriented or not, and vertices are expressed as a sum of integers in a box. To obtain the result, we color a vertex $\fbox{m+n}$ by $f(mx)+f(nx)$. Then, we obtain a sum of diagrams colored by integers. We divide each diagram of vertices $x_1,\ldots,x_n$ by the product of colors of each $x_i$ to the power the number of adjacent edges which are not coming to this vertex.
For instance the diagram $\fbox{pq}\!\!{\rightarrow}\!\!\fbox{p}\!\!{\leftarrow}\!\!\fbox{q}$ is a graphical expression for $\frac{1}{pq^2}{}_c\bullet\!\!-\!\!\bullet_a\!\!\!\!-\!\!-\!\!\!\bullet_b$ and the dots mean terms with more than two edges. $$\begin{aligned}
\omega^{-1}_{pq}&=\fbox{p}+\fbox{q}+\fbox{p}\!\!-\!\!\fbox{q}+\fbox{p}\!\!=\!\!\fbox{q}+\frac{1}{2}\fbox{p}\!\!-\!\!\fbox{q}\!\!-\!\!\fbox{p}
+\frac{1}{2}\fbox{q}\!\!-\!\!\fbox{p}\!\!-\!\!\fbox{q}+\cdots\\
\omega^0&=\fbox{1}\\
\omega^1&=
\fbox{1}+\fbox{p+q-pq}\!\!{\leftarrow}\!\!\fbox{1}+
\fbox{p+q-pq}\!\!\leftleftarrows\!\!\fbox{1}+\frac{1}{2}\fbox{1}\!\!{\rightarrow}\!\!\fbox{p+q-pq}\!\!{\leftarrow}\!\!\fbox{1}\\
&+\frac{1}{2}\fbox{p+q-pq}\!\!{\leftarrow}\!\!\fbox{1}\!\!{\rightarrow}\!\!\fbox{p+q-pq}
+\fbox{1}\!\!{\rightarrow}\!\!\fbox{p}\!\!-\!\!\fbox{q}+\fbox{1}\!\!{\rightarrow}\!\!\fbox{q}\!\!-\!\!\fbox{p}+\cdots\\
\omega^2&=\omega^1 - \fbox{p+q-pq}\!\!{\leftarrow}\!\!\fbox{pq}\!\!{\leftarrow}\!\!\fbox{1} - \fbox{1}\!\!{\rightarrow}\!\!\fbox{p+q-pq}\!\!{\leftarrow}\!\!\fbox{pq}+\cdots\\
X_{p,q}&=\lim_{n}(\omega^{-1}_{pq}-\omega^n_{pq})=\fbox{p}+\fbox{q}-\fbox{pq}+\fbox{p}\!\!-\!\!\fbox{q} -\fbox{p+q-pq}\!\!{\leftarrow}\!\!\fbox{pq}+
\fbox{p}\!\!=\!\!\fbox{q}\\
&-\fbox{p+q-pq}\!\!\leftleftarrows\!\!\fbox{pq}
+\frac{1}{2}\fbox{p}\!\!-\!\!\fbox{q}\!\!-\!\!\fbox{p}
+\frac{1}{2}\fbox{q}\!\!-\!\!\fbox{p}\!\!-\!\!\fbox{q}\\
&+\frac{1}{2}\fbox{pq}\!\!{\rightarrow}\!\!\fbox{p+q-pq}\!\!{\leftarrow}\!\!\fbox{pq}
-\frac{1}{2}\fbox{p+q-pq}\!\!{\leftarrow}\!\!\fbox{pq}\!\!{\rightarrow}\!\!\fbox{p+q-pq}\\
&-\fbox{pq}\!\!{\rightarrow}\!\!\fbox{p}\!\!-\!\!\fbox{q}-\fbox{pq}\!\!{\rightarrow}\!\!\fbox{q}\!\!-\!\!\fbox{p}
+\fbox{p+q-pq}\!\!{\leftarrow}\!\!\fbox{pq}\!\!{\leftarrow}\!\!\fbox{pq} +\cdots\end{aligned}$$
Rationality
===========
The expression we have obtained until now is not rational. When trying to find a rational expression, we will show that expressions coming from non-tree gluing graphs vanish. For example, the graph $\fbox{p}\!\!=\!\!\fbox{q}$ above will not appear in the final expression of the unwheeled Kontsevich integral of torus knots.
Diagrams with singular colorings {#diag}
--------------------------------
We need to work with diagrams which are colored by some singular algebraic expressions. For that we define two spaces of diagrams ${\mathcal{B}}_s$ and ${\mathcal{B}}_s^{rat}$ which fit in the following commutative diagram: $$\xymatrix{
{\mathcal{B}}^{\rm rat}\ar[d]^{\operatorname{Hair}}\ar[r] & {\mathcal{B}}^{\rm rat}_s \ar[d]^{\operatorname{Hair}} \\
{\mathcal{B}}\ar[r] & {\mathcal{B}}_s
}$$ Here we define ${\mathcal{B}}_s={\mathcal{D}}(F)$ where $F(H)=(H\setminus\{0\})^{-1} {\mathbb{Q}}[[H]]$ , the localization of ${\mathbb{Q}}[[H]]$ by elements of $H\setminus\{0\}$. The space ${\mathcal{B}}^{rat}_s$ is constructed from the functor $F(H)=S^{-1}{\mathbb{Q}}[\exp(H)]$, where $S$ is made of non zero expressions of the form $P_1(\exp(h_1))\cdots P_k(\exp(h_k))$ for $P_1,\ldots,P_k\in{\mathbb{Q}}[h]$ and $h_1,\ldots,h_k\in H$.
The $\operatorname{Hair}$ map is again defined by the map induced by the Taylor expansion.
These constructions motivate the definition \[espdediag\]. In the article [@moi], we used a different construction using coloring of edges by elements of ${\mathbb{Q}}[[h]][h^{-1}]$. This construction was not convenient because the map from non-singular diagrams to singular ones fails to be injective even for small degrees as is shown in the following identity: $$\xymatrix{
-\!\!\bigcirc\!\!-\!\!\bigcirc\!\!- =
-\!\!\bigcirc\!\!\raisebox{2pt}{$\overset{\frac{1}{h} h}{\perp\!\!\perp}$}\!\!\bigcirc\!\!- = 0.
}$$ To avoid this, we have changed the construction of singular diagrams in some way which avoids inverting null-homologous legs.
A formula for the substitution map
----------------------------------
Let $X$ be a connected gluing graph, with vertices $x_1,\ldots,x_N$ and power series $f_1,\ldots,f_N$ corresponding to these vertices. The aim of this part is to give an explicit formula for $s(X)\in{\mathcal{B}}_s$. We describe it in the following proposition:
\[subst\] Given a gluing graph $X$ with power series $f_1,\ldots,f_N$, its substitution is a finite combination of diagrams $\Gamma$ obtained by gluing at each vertex the incoming edges to a circle, and decorating by expressions of the form $$\prod\limits_{i=1}^{N}\left(\frac{f_i'(y_i)}{{y_i}^{p_i}}\right)^{(k_i-p_i)}/D_i$$ where the $y_i$s are non zero cohomology classes, $k_i$ is the valency of the $i$-th vertex, $p_i$ is an integer satisfying $0\le p_i\le k_i$ and $D_i$ is a product of $p_i$ linear terms. Moreover, we have the inequality $\sum_i p_i>0$ unless $X$ is a tree.
To compute $s(X)$, we may glue at each vertex $x$ edges incoming to this vertex to wheels generated by $f(x)$. Hence, we can choose a cycling ordering of edges around each vertex, compute the result of the gluings which repect this order, and sum over such orderings.
Fix an ordering $e^x_1,\ldots,e^x_{k_x}$ of edges around each vertex $x$. Let $\Gamma$ be the trivalent graph obtained by replacing $x$ by a circle attached to the edges in the prescribed order, and $H=H^1(\Gamma,{\mathbb{Z}})$. Fix the order $n_x$ of the wheel of $f(x)$ we will glue at $x$. Write $x_1,\ldots,x_{k_x}$ the elements of $H$ coming from the edges $e_1$ to $e_2$, ..., $e_{k_x}$ to $e_1$ (see figure \[subs\]).
First, we glue the edge $e_1$ to any leg, so we have a factor $n_{x}$. Then all gluings are listed in the following formula: $$\label{ratexp}
\sum\limits_{\text{orderings},x}n_x\sum\limits_{i_1+\cdots+i_{k_x}=n_x-k_x}x_1^{i_1}\cdots x_{k_x}^{i_{k_x}}.$$ We explain an algorithmic reduction for this formula. Suppose there are two indices $l$ and $m$ such that $x_l\ne x_m \in H^1(\Gamma,{\mathbb{Z}})$, write $i_{lm}=i_l+i_m$. Then, $\sum\limits_{i_1+\cdots+i_{k_x}=n_x-k_x}x_1^{i_1}\cdots x_{k_x}^{i_{k_x}}$ may be replaced by the sum $$\sum\limits_{i_1+\cdots+i_{k_x}+i_{lm}=n_x-k_x}x_1^{i_1}\cdots\widehat{x_l^{i_l}}\cdots \widehat{x_m^{i_m}} \cdots x_{k_x}^{i_{k_x}} \frac{x_l^{i_{lm}}-x_m^{i_{lm}}}{x_l-x_m} \in (H\setminus\{0\})^{-1}{\mathbb{Q}}[[H]]$$ This is a difference of sums of the same form but where each summand is now a product of $k_x-1$ monomials, and with a degree one denominator. Observe that the sum of the exponents of the monomials in each summand is unchanged and equal to $n_x-k_x$ as before. We can perform such a reduction again for each one of the sums thus obtained a different number of times. This algorithm stops if all cohomology classes involved in the resulting expression are the same.
Suppose that an expression has been obtained form by reducing it $p_x$ times. Then this expression is a sum of identical terms each of which is equal to a cohomology class $y$ to the power $n_x-k_x$ and divided by a product of elements of $H$. Counting the number of times this term occurs, the resulting expression is finally equal to $$\frac{\binom{n_x-p_x-1}{k_x-p_x-1}y^{n_x-k_x}}{D_x}$$ where $D_x$ is a product of $p_x$ linear terms.
Let us prove the last assertion of the proposition. If all the $p_i$s are 0, it means that there is some gluing of $X$ such that for all vertices $x$ of $X$, the resulting cohomology classes $x_1,\ldots,x_{k_x}$ are identical. It means that all edges in $X$ correspond to a null cohomology class, and hence $X$ is a tree.
Application to the integral of torus knots
------------------------------------------
We recall that in [@lift], the invariant ${Z^{\fourwheel {\rm rat}}}$ is defined such that $\operatorname{Hair}{Z^{\fourwheel {\rm rat}}}(K)=\langle\Omega,\Omega\rangle Z^{\fourwheel}(K)$ for any knot $K$.
Our aim would be to compute ${Z^{\fourwheel {\rm rat}}}(K_{p,q})$ but we have information only on $\operatorname{Hair}Z^{rat\fourwheel}(K_{p,q})$. Hence we will compute it only up to the kernel of the $\operatorname{Hair}$ map. We have shown that $\langle\Omega,\Omega\rangle Z^{\fourwheel}(K)=s(X_{p,q})$, hence we are interested in finding a rational expression for $s(X_{p,q})$. Using the preceding section and rationality results of [@rat], we will prove the following theorem:
\[theoreme\] Let $X_{p,q}$ be the gluing graph of proposition \[Xpq\]. It is a gluing graph decorated by the parameters $p,q$ and $pq$ associated respectively to the series $f(px),f(qx)$ and $f(pqx)$. We construct from $X_{p,q}$ a rational singular diagram $Y^{\rm rat}_{p,q}\in{\mathcal{B}}^{\rm rat}_s$ in the following way.
We consider a tree component $T$ of $X_{p,q}$ and at each vertex $x$, we glue a circle. As the diagram is a tree, the ordering has no importance, and the cohomology class of the circle is well defined: we note it $h_x$. If the vertex $x$ has valence $k_x$ and is colored by $n_x\in\{p,q,pq\}$, we decorate the “bubble-tree” diagram with the series $$\prod_x \frac{1}{4}\left(\frac{e^{n_x h_x}+1}{e^{n_x h_x}-1}\right)^{(k_x-1)}.$$ Therefore, we have constructed from $T$ an element $T^{\rm rat}\in{\mathcal{B}}^{\rm rat}_s$. We set $Y^{\rm rat}_{p,q}=\sum\limits_{T\text{ in }X_{p,q}} T^{\rm rat}$.
Then we have $\operatorname{Hair}z^{\rm rat\fourwheel}(K_{p,q})=\operatorname{Hair}Y^{\rm rat}_{p,q}$.
In figure \[develop\], we give the beginning of the expansion of the unwheeled Kontsevich integral of torus knots obtained thanks to the gluing graph $Y^{\rm rat}_{p,q}$. The parameter $x$ corresponds to the cohomology class of the circle on which the series is located.
Our strategy is to apply the results of the preceding section to $X_{p,q}$: we recall that $X_{p,q}$ is a gluing graph whose power series attached to vertices are either $f(px), f(qx)$ or $f(pqx)$, where $f(x)$ is the series $\frac{1}{2}\log\frac{\sinh(x/2)}{x/2}$. We compute that the derivative of $f$ is $\frac{1}{4}\coth\frac{x}{2}-\frac{1}{2x}$.
Thanks to proposition \[subst\], we can conclude that $s(X_{p,q})$ is obtained by decorating graphs by expressions such as $g(\exp(h))/D$ where $g\in{\mathbb{Q}}(t)$, $h\in H$ and $D$ is a homogeneous polynomial in $H$.
Hence it is natural to define the following space:
Let $H$ be a free abelian group of finite rank. We note $G(H)$ the image of $S^{-1}{\mathbb{Q}}[\exp(H)]\otimes (H\setminus\{0\})^{-1}{\mathbb{Q}}[H]$ in $(H\setminus\{0\})^{-1}{\mathbb{Q}}[[H]]$. This is again a ${\mathcal{C}}$-module and hence, we can define a space ${\mathcal{B}}'={\mathcal{D}}(G)$.
The map $S^{-1}{\mathbb{Q}}[\exp(H)]\otimes (H\setminus\{0\})^{-1}{\mathbb{Q}}[H]\to G(H)$ is an isomorphism of ${\mathcal{C}}$-modules. We define the degree of $\frac{P}{Q}\in (H\setminus\{0\})^{-1}{\mathbb{Q}}[H]$ by $\deg(\frac{P}{Q})=\deg(P)-\deg(Q)$. This degree extends to $G(H)$ and to ${\mathcal{B}}'$.
This statement is purely algebraic, it comes from the fact that polynomials in elements of $H$ and exponentials of elements of $H$ are algebraically independant as power series in $H$.
But we know from [@rat] that there is a diagram $z^{\rm rat \fourwheel}(K_{p,q})$ in ${\mathcal{B}}^{\rm rat}$ such that $\operatorname{Hair}z^{\rm rat \fourwheel}(K_{p,q}) = s(X_{p,q})$. We interpret this result saying that the series $\operatorname{Hair}Z^{rat\fourwheel}(K_{p,q})$ lies in the degree 0 part of ${\mathcal{B}}'$.
But in the process of substitution of a connected gluing graph, we add denominators unless all edges of the graph are null-homologous (see proposition \[subst\]). In the same way, all terms containing fractions in $f'(px), f'(qx)$ and $f'(pqx)$ will have a negative degree.
Keeping diagrams in $X_{p,q}$ with degree 0 and substituting them, we obtain exactly the series $\operatorname{Hair}Y^{\rm rat}_{p,q}$ described in the theorem. As this series made of exactly all gluing graphs producing degree 0 diagrams in ${\mathcal{B}}'$ by substitution, we conclude that $\operatorname{Hair}Y^{\rm rat}_{p,q}=s(X_{p,q})=\operatorname{Hair}z^{\rm rat\fourwheel}(K_{p,q})$. This states the theorem.
There are two natural questions for which we have no answer at the moment:
- How can we apply a weight system to singular diagrams in order to obtain formulas for the Jones functions of torus knots from $Y^{\rm rat}_{p,q}$?
- The series $ z^{\rm rat\fourwheel}(K_{p,q})$ and $Y^{\rm rat}_{p,q}$ agree up to the kernel of the $\operatorname{Hair}$ map. Are they equal?
Branched coverings
==================
A great interest for rational expression of Kontsevich integral comes from its relation with branched coverings. More precisely, if $K_{p,q}$ is the torus knot of parameters $p$ and $q$, and $r$ is an integer, let us note $\Sigma^r(K_{p,q})$ be the pair formed of the cyclic branched covering of $S^3$ of order $r$ over $K_{p,q}$ and the ramification link.
If $r$ is coprime with $p$ and $q$, the ramification locus is a knot, and the underlying 3-manifold is a rational homology sphere, the Brieskorn manifold $\Sigma(p,q,r)$.
In [@lift], a map $\operatorname{Lift}_r$ is described which intertwines rational invariant of the cyclic branched coverings and rational invariant of the initial knot in the following way: $$Z^{\rm rat\fourwheel}(\Sigma^r(K))= \exp(\frac{\sigma_r(K)}{16}\ThetaGraph)\operatorname{Lift}_r Z^{\rm rat\fourwheel}(K).$$ We now study this map in the case of torus knots and prove the following proposition:
Call $\Pi_r$ the operator on ${\mathcal{B}}^{\rm rat}_s$ which multiplies any diagram $D$ by $r^{-\chi(D)}$ where $\chi$ is the Euler characteristic. then we have $$\operatorname{Lift}_r Y^{\rm rat}_{p,q}=\Pi_r Y^{\rm rat}_{p,q}$$
The $\operatorname{Lift}_r$ map was defined only for diagrams decorated by fractions without poles at $r$-roots of unity. As we extended the decorations to all fractions, the definition of $\operatorname{Lift}_r$ makes sense for any diagram. In the definition of the $\operatorname{Lift}_r$ map, we need to express all denominators as polynomials of $t^r$. Then, we look at the numerators as a coloring by monomials, which is the same as a linear combination of 1-cohomology classes of the underlying graph. We keep only the classes divisible by $r$ and divide them, then we put back denominators replacing $t^r$ by $t$. Finally we multiply the result by $r$.
This construction is very easy in our case because we only need to know how the map $\operatorname{Lift}_r$ acts on derivatives of the fraction $\frac{t^n+1}{t^n-1}$ where $n=p,q$ or $pq$. Let us write $h(t)=\frac{t^n+1}{t^n-1} \in {\mathbb{Q}}(t)$. The operator $\operatorname{Lift}_r$ and the derivation operator $Dg(t) =tg'(t)$ act on the space ${\mathbb{Q}}(t)$.
We can develop $h$ in formal series: $h(t)=-1-2\sum\limits_{k\ge 1}t^{nk}$. This expression shows that we have $$\operatorname{Lift}_r h(t)=-1 -2\sum\limits_{k\ge 1, r|nk}t^{nk/r}=h(t).$$ because $n$ is coprime with $r$.
Then, for $i>0$, we also have $D^i h(t)=-2\sum\limits_{k\ge 1}(nk)^it^{nk}$. We check in the same way the following formula: $$\operatorname{Lift}_r D^i h(t) = -2\sum\limits_{k\ge 1, r|nk}(nk)^it^{nk/r}=r^i D^i h(t).$$ This shows finally that $\operatorname{Lift}_r$ acts on a diagram of $Y^{\rm rat}_{p,q}$ by multiplying it by $r$ to the power $1+\sum_i(v_i-1)$ where $v_i$ is the valence of the $i$-th vertex. This expression is the number of vertices of the diagram minus 1, hence the number of loops minus one.
This ends the proof of the proposition.
Of course, we have not proved that $\operatorname{Lift}_r z^{\rm rat\fourwheel}(K_{p,q})=\Pi_r z^{\rm rat\fourwheel}(K_{p,q})$. Although the first diagram we know in the kernel of the $\operatorname{Hair}$ map has loop degree 17 (see [@ber]), we do not know if the $\operatorname{Hair}$ map is injective in degree greater than 3.
**D Bar-Natan**, **S Garoufalidis**, **L Rozansky**, **DP Thurston**, *[Wheels, wheeling, and the Kontsevich integral of the unknot]{}*, Israel J. Math. 119 (2000) 217–237
**D Bar-Natan**, **TQT Le**, **DP Thurston**, *[Two applications of elementary knot theory to Lie algebras and Vassiliev invariants]{}*, 7[2003]{}11[31]{}
**S Garoufalidis**, [*Whitehead doubling persists*](http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-40.abs.html), Algebr. Geom. Topol. 4 (2004) 935–942
**S Garoufalidis**, **A Kricker**, *Finite type invariants of cyclic branched covers*, Topology 43 (2004) 1247–1283
**S Garoufalidis**, **A Kricker**, *A rational noncommutative invariant of boundary links*, 8[2004]{}4[115]{}[204]{}
**A Kricker**, *[The lines of the Kontsevich integral and Rozansky’s rationality conjecture]{}*,
**C Lescop**, *[Introduction to the Kontsevich integral of framed tangles]{}*, Technical report, Grenoble Summer School (1999)
**J Marché**, *[On Kontsevich integral of torus knots]{}*, Topology Appl. 143 (2004) 15–26
**T Ohtsuki**, *A cabling formula for the 2-loop polynomial of knots*, Publ. Res. Inst. Math. Sci. 40 (2004) 949–971
**B Patureau-Mirand**, *[Non-injectivity of the Hair map]{}*,
**L Rozansky**, *[Higher order terms in the Melvin-Morton expansion of the colored Jones polynomial]{}*, Comm. Math. Phys. 183 (1997) 291–306
**L Rozansky**, *[A rationality conjecture about Kontsevich integral of knots and its implications to the structure of the colored Jones polynomial]{}*, from: “Invariants of Three-Manifolds (Calgary, AB, 1999)”, Topology Appl. 127 (2003) 47–76
**D Thurston**, *[Wheeling: a diagrammatic analogue of the Duflo isomorphism]{}*, PhD thesis, UC Berkeley (2000),
**P Vogel**, *Vassiliev theory*, Technical report, MaPhySto (2000)
|
---
abstract: 'Two-stage sampling designs are commonly used for household and health surveys. To produce reliable estimators with assorted confidence intervals, some basic statistical properties like consistency and asymptotic normality of the Horvitz-Thompson estimator are desirable, along with the consistency of assorted variance estimators. These properties have been mainly studied for single-stage sampling designs. In this work, we prove the consistency of the Horvitz-Thompson estimator and of associated variance estimators for a general class of two-stage sampling designs, under mild assumptions. We also study two-stage sampling with a large entropy sampling design at the first stage, and prove that the Horvitz-Thompson estimator is asymptotically normally distributed through a coupling argument. When the first-stage sampling fraction is negligible, simplified variance estimators which do not require estimating the variance within the Primary Sampling Units are proposed, and shown to be consistent. An application to a panel for urban policy, which is the initial motivation for this work, is also presented.'
author:
- |
Guillaume Chauvet$^{(1)}$ and Audrey-Anne Vallée$^{(2)}$\
$^{(1)}$ Ensai (Irmar), Campus de Ker Lann, Bruz - France\
$^{(2)}$ Institut de Statistique, Université de Neuchâtel, Switzerland
nocite: '[@rao:har:coc:62]'
title: 'Inference for two-stage sampling designs with application to a panel for urban policy'
---
[**Keywords:**]{} Asymptotic normality, coupling method, rejective sampling, simplified variance estimator.
Introduction {#sec1}
============
In household and health surveys, the population is often sparse over a large territory and there is regularly no sampling frame. Two-stage sampling designs are convenient in such situations. The population are grouped into large blocks (e.g., municipalities or counties), called Primary Sampling Units (PSUs), which are sampled at the first stage. Only a frame of these PSUs is needed at this stage, which is easier to create. At the second stage, a list of population units is obtained inside the selected PSUs, and a sample of these population units is selected. Despite its convenience, multistage sampling has the drawback to lead to estimators with inflated variance, compared to sampling designs where the population units are directly selected. A detailed treatment of multistage sampling may be found in [@coc:77], [@sar:swe:wre:92] and [@ful:11].\
To produce reliable estimators with assorted confidence intervals, some statistical properties are needed for a sampling design: (a) the Horvitz-Thompson estimator should be consistent for the true total; also, (b) this estimator should be asymptotically normally distributed, and (c) consistent variance estimators should be available, to be able to produce normality-based confidence intervals. General conditions for the consistency of the Horvitz-Thompson estimator are given in [@isa:ful:82] and [@rob:82], see also [@pra:sen:09]. The asymptotic normality is usually studied design by design, see for example [@haj:64] for rejective sampling, [@ros:72] for successive sampling or [@ohl:86] for the Rao-Hartley-Cochran (1972) procedure; see also [@bic:fre:84] for stratified simple random sampling and [@che:rao:07] for two-phase sampling designs. These properties are also studied in [@bre:ops:00] for the class of local polynomial regression estimators and in [@bre:ops:san:16], but under assumptions that are not generally applicable for multistage sampling designs. More recently, [@boi:lop:rui:17] and [@ber:cha:cle:17] established functional central limit theorems for Horvitz-Thompson empirical processes. In summary, these properties have been mainly studied in the literature for one-stage sampling designs.\
In two-stage sampling, the asymptotic properties of estimators are more difficult to study, due to the dependence introduced in the selection of the sampling units. [@kre:rao:81] studied the case when the primary units are selected with replacement, and [@ohl:89] derived a general central limit theorem for such designs. Recently, [@cha:15] considered coupling methods to prove the asymptotic normality of the Horvitz-Thompson estimator and the validity of a bootstrap procedure for stratified simple random sampling at the first stage. However, there is a lack of general conditions ensuring that properties (a)-(c) hold for general two-stage sampling designs, and this is the purpose of the present paper. A notable exception is [@bre:ops:08], who obtain the consistency of the Horvitz-Thompson estimator under very weak conditions. This is discussed in Section \[sec4\].\
In this paper, the properties of estimators and variance estimators are studied for a general class of two-stage sampling designs. The framework is introduced in Section \[sec2\] and the variance of the Horvitz-Thompson estimator is decomposed in a sum of three components. In Section \[sec3\], the assumptions used to establish the asymptotic properties are defined. In Section \[sec4\], the Horvitz-Thompson estimator is shown to be consistent under our conditions, and the order of magnitude of the three components of the variance is determined. The consistency of two unbiased variance estimators is established in Section \[sec41\]. A simplified variance estimator which does not require estimating the variance within the PSUs can be produced. We prove in Section \[sec42\] that this variance estimator is consistent when the total variance within the PSUs is negligible. In Section \[sec5\], the specific case of large-entropy sampling designs at the first-stage is considered. When rejective sampling is used at the first-stage, the consistency of a Hájek-type variance estimator is established under reduced assumptions, along with the asymptotic normality of the Horvitz-Thompson estimator. We define a coupling procedure to extend these results to a more general class of large-entropy sampling designs at the first-stage. In Section \[sec6\], the properties of the Hájek-type variance estimators are evaluated in a simulation study. An application to a panel for urban policy, which is the initial motivation for this work, is presented in Section \[appli:ppv\].
Notation {#sec2}
========
We are interested in a finite population $U$ of size $N$, in which a sample is selected by means of a two-stage sampling design. The units in $U$, called Secondary Sampling Units (SSUs) are partitioned into a population $U_I$ of $N_I$ Primary Sampling Units (PSUs). A sample $S_I$ of $n_I$ PSUs is selected in $U_I$. We are interested in estimating the population total $$\begin{aligned}
\label{eq1:sec2}
Y & = & \sum_{i=1}^{N_I} \sum_{k =1}^{N_i} y_{ik}
=\sum_{i=1}^{N_I} Y_i,
\end{aligned}$$ for some variable of interest $y$, where $Y_i=\sum_{k =1}^{N_i} y_{ik}$ is the sub-total of the variable $y$ on the PSU $i$ and $N_i$ is the number of SSUs inside the PSU $i$.\
We assume that the population $U$ belongs to a nested sequence $\{U_t\}$ of finite populations with increasing sizes $N_t$, and that the population vector of values $y_{Ut}=(y_{1t},\ldots,y_{Nt})^{\top}$ belongs to a sequence $\{y_{Ut}\}$ of $N_t$-vectors. The index $t$ is suppressed in what follows but all limiting processes are taken as $t \to \infty$. We assume that $N_I \to \infty$ and $n_I \to \infty$ as $t \to \infty$. We consider a single stratum of PSUs, but our results may be easily generalized to the case of a finite number of strata, see the application to the panel for urban policy in Section \[appli:ppv\]. An alternative asymptotic set-up is possible, under which the number of strata tends to infinity while the sample size per stratum remains bounded, see [@kre:rao:81] and [@bre:ops:san:16].\
We note $I_{Ii}$ for the sample membership indicator of the PSU $i$ into $S_I$, $\pi_{Ii}=E(I_{Ii})$ for the inclusion probability of the PSU $i$, and $\pi_{Iij} = E(I_{Ii} I_{Ij})$ for the probability that the PSUs $i$ and $j$ are selected jointly in $S_I$. Inside any PSU $i \in S_I$, a sample $S_i$ of $n_i$ SSUs is selected at the second stage. We note $$\begin{aligned}
\label{eq2:sec2}
N_0 = \frac{1}{N_I} \sum_{i=1}^{N_I} N_i & \textrm{ and } & n_0 = \frac{1}{N_I} \sum_{i=1}^{N_I} n_i
\end{aligned}$$ for the average size of the PSUs and for the average sample size selected inside the PSUs. We do not need particular assumptions on the limit behaviour of $n_0$ and $N_0$, and $n_0$ may be either bounded or unbounded. Our set-up covers in particular the case when the SSUs are comprehensively surveyed inside a selected PSU, which amounts to single-stage sampling on the population of PSUs.\
For any SSU $k$ in the PSU $i$, we note $I_{k}$ for the sample membership indicator of $k$ in $S_i$. Also, we note $\pi_{k|i}=E(I_{k}|i \in S_I)$ for the conditional inclusion probability of $k$, and $\pi_{kl|i} = E(I_{k} I_{l}|i \in S_I)$ for the conditional joint probability that two SSUs $k,l \in i$ are selected together in $S_i$. We assume invariance of the second-stage designs, as defined by [@sar:swe:wre:92]: the second stage of sampling is independent of $S_I$. Also, we assume that the second-stage designs are independent from one PSU to another, conditionally on $S_I$.\
The Horvitz-Thompson (HT) estimator of $Y$ is $$\begin{aligned}
\label{eq3:sec2}
\hat{Y}_{\pi} = \sum_{i \in S_I} \frac{\hat{Y}_i}{\pi_{Ii}} & \textrm{ with } & \hat{Y}_i = \sum_{k \in S_i} \frac{y_{ik}}{\pi_{k|i}}.
\end{aligned}$$ The variance of $\hat{Y}_{\pi}$ may be written as $$\begin{aligned}
\label{eq4:sec2}
V(\hat{Y}_{\pi}) & = & \sum_{i=1}^{N_I} \sum_{j=1}^{N_I} \Delta_{Iij} \frac{Y_i}{\pi_{Ii}} \frac{Y_j}{\pi_{Ij}} + \sum_{i=1}^{N_I} \left(\frac{1-\pi_{Ii}}{\pi_{Ii}}\right) V_i + \sum_{i=1}^{N_I} V_i \nonumber \\
& = & V_1(\hat{Y}_{\pi})+V_2(\hat{Y}_{\pi})+V_3(\hat{Y}_{\pi})
\end{aligned}$$ with $\Delta_{Iij}=\pi_{Iij}-\pi_{Ii}\pi_{Ij}$, and $$\begin{aligned}
\label{eq5:sec2}
V_i \equiv V(\hat{Y}_i) & = & \sum_{k=1}^{N_i}\sum_{l=1}^{N_i} \Delta_{kl|i} \frac{y_{ik}}{\pi_{k|i}} \frac{y_{il}}{\pi_{l|i}},
\end{aligned}$$ with $\Delta_{kl|i}=\pi_{kl|i}-\pi_{k|i}\pi_{l|i}$. The term $V_1(\hat{Y}_{\pi})$ is the variance due to the first stage. The sum of the two last terms in (\[eq4:sec2\]) may be simplified as $$\begin{aligned}
\label{eq6:sec2}
V_2(\hat{Y}_{\pi})+V_3(\hat{Y}_{\pi}) & = & \sum_{i=1}^{N_I} \frac{V_i}{\pi_{Ii}}.
\end{aligned}$$ This is the variance due to the second stage of sampling.\
When estimating the variance, the terms $V_1(\hat{Y}_{\pi})+V_2(\hat{Y}_{\pi})$ and $V_3(\hat{Y}_{\pi})$ are handled separately. Variance estimators for these two terms are considered in Section \[sec4\], and proved to be consistent under assumptions which are stated and discussed in Section \[sec3\]. In case of large entropy sampling designs at the first-stage, consistent variance estimators can be produced under reduced assumptions, and without using second-order inclusion probabilities. This is studied in Section \[sec5\] .
Assumptions {#sec3}
===========
To study the asymptotic properties of the estimators and variance estimators that we consider below, a number of assumptions are needed. We present in Section \[sec21\] the assumptions on the first-stage sampling design, and in Section \[sec22\] the assumptions on the second-stage sampling designs. The assumptions related to the variable of interest are presented in Section \[sec23\].
Assumptions on the first-stage sampling design {#sec21}
----------------------------------------------
- Some constant $f_{I0}<1$ exists s.t. $$\begin{aligned}
\label{eq1:sec3}
N_I^{-1} n_I & \leq & f_{I0}.
\end{aligned}$$ Some constants $c_{I1},C_{I1}>0$ exist s.t. for any PSU $i$ $$\begin{aligned}
\label{eq2:sec3}
c_{I1} & \leq N_I n_I^{-1} \pi_{Ii} \leq & C_{I1}.
\end{aligned}$$
- Some constants $C_{I2},C_{I3}>0$ exist s.t. for any PSUs $i \ne j \ne {i'}$ $$\begin{aligned}
& \pi_{Iij} \leq & C_{I2} N_I^{-2} n_I^2, \label{eq3:sec3} \\
& \pi_{Iiji'} \leq & C_{I3} N_I^{-3} n_I^3, \label{eq4:sec3}
\end{aligned}$$ with $\pi_{Iiji'}$ the probability that the PSUs $i,j,{i'}$ are selected together in $S_I$. Some constants $C_{I4},C_{I5}$ exist s.t. $$\begin{aligned}
\Delta_{I1} \equiv \max_{i \neq {j}=1,\ldots,N_I} \left| \pi_{Iij}-\pi_{Ii} \pi_{Ij} \right| \leq C_{I4} N_I^{-2} n_I, \label{eq5:sec3} \\
\Delta_{I2} \equiv \max_{i \neq {j} \neq {i'} \neq {j'} =1,\ldots,N_I} \left| \pi_{Iiji'j'}-\pi_{Ii} \pi_{Ij}\pi_{Ii'} \pi_{Ij'} \right| \leq C_{I5} N_I^{-4} n_I^3, \nonumber \end{aligned}$$ with $\pi_{Iiji'j'}$ the probability that the PSUs $i,j,{i'},{j'}$ are selected together in $S_I$.
- Some constant $c_{I2}>0$ exists s.t. for any $i \ne j =1,\ldots,N_I$ $$\begin{aligned}
\label{eq6b:sec3}
c_{I2} N_I^{-2} n_I^2 & \leq \pi_{Iij}.
\end{aligned}$$
The Assumption (FS1) is related to the order of magnitude of the first-stage sample size $n_I$, and to the first-order inclusion probabilities. Equation (\[eq1:sec3\]) ensures that the first-stage sample is not degenerate, in the sense that the PSUs are not comprehensively surveyed. This assumption is compatible with the case $n_I/N_I \to 0$ (negligible first-stage sampling fraction). A similar condition is considered in [@bre:ops:00 assumption A5], and in [@boi:lop:rui:17 assumption HT3]. Equation (\[eq2:sec3\]) states that the first-order inclusion probabilities do not depart much from that obtained under simple random sampling. The same condition is considered in [@boi:lop:rui:17 assumption C1]. Overall, (FS1) is under the control of the survey sampler.\
The Assumption (FS2) is related to the inclusion probabilities of order $2$ to $4$. If (FS1) holds, equations (\[eq3:sec3\]) and (\[eq4:sec3\]) will automatically hold for negatively associated sampling designs [e.g. @bra:jon:12] which includes simple random sampling, rejective sampling [@haj:64], Sampford sampling [@sam:67] and pivotal sampling [@dev:til:98; @cha:12], for example. In equation (\[eq5:sec3\]), the quantities $\Delta_{I1}$ and $\Delta_{I2}$ are two measures of dependency in the selection of units. These quantities will be equal to $0$ when the units are selected independently, which is known as Poisson sampling [see for example @ful:11 p. 13]. Equation (\[eq5:sec3\]) is respected for simple random sampling. If (FS1) holds, it is also respected under rejective sampling [see @boi:lop:rui:12 Theorem 1], and it can be proved that it holds for the Rao-Sampford sampling design (see Hajek, 1981, Chapter 8). Similar conditions are considered in [@bre:ops:00 assumption A7], and in [@boi:lop:rui:17 conditions C2-C4].\
The Assumption (FS3) provides a uniform lower bound for the second-order inclusion probabilities. A similar condition is considered in [@bre:ops:00 assumption A6]. This assumption holds for simple random sampling, but is more difficult to prove for unequal probability sampling designs. On the other hand, it is needed to prove the consistency of the Horvitz-Thompson variance estimator and of the Yates-Grundy variance estimator, see our Section \[sec41\] and Theorem 3 in [@bre:ops:00]. We consider in Section \[sec5\] the specific case of large entropy sampling designs at the first-stage, for which alternative consistent variance estimators are possible, and for which the assumption (FS3) can be suppressed.
Assumptions on the second-stage sampling design {#sec22}
-----------------------------------------------
- Some constants $\lambda_1,\Lambda_1>0$ and $\phi_1,\Phi_1>0$ exist s.t. for any PSU $i$ $$\begin{aligned}
\lambda_1 n_0 & \leq n_i \leq & \Lambda_1 n_0, \label{eq8:sec3} \\
\phi_1 N_0 & \leq N_i \leq & \Phi_1 N_0. \label{eq9:sec3}
\end{aligned}$$
- Some constants $c_{1},C_{1}>0$ exist s.t. for any PSU $i$ and for any $k$ inside: $$\begin{aligned}
\label{eq10:sec3}
c_{1} & \leq N_0 n_0^{-1} \pi_{k|i} \leq & C_{1}.
\end{aligned}$$
- Some constants $C_2,C_3>0$ exists s.t. for any PSU $i$ and any $k \ne l \ne k'$ inside: $$\begin{aligned}
& \pi_{kl|i} \leq & C_{2} N_0^{-2} n_0^2, \label{eq11:sec3} \\
& \pi_{klk'|i} \leq & C_{3} N_0^{-3} n_0^3, \label{eq12:sec3}
\end{aligned}$$ with $\pi_{klk'|i}$ the conditional probability that the SSUs $k,l,k'$ are selected together in $S_i$. Also, some constants $C_{4},C_{5}$ exist s.t. $$\begin{aligned}
\Delta_1 &\equiv& \max_{i=1,\ldots,N_I} \max_{k \neq l=1,\ldots,N_i} \left| \pi_{kl|i}-\pi_{k|i} \pi_{l|i} \right| \leq C_4 N_0^{-2} n_0, \label{eq13:sec3}\\
\Delta_2 &\equiv& \max_{i=1,\ldots,N_I} \max_{k \neq l \neq k' \neq l'=1,\ldots,N_i} \left| \pi_{klk'l'|i}-\pi_{k|i} \pi_{l|i} \pi_{k'|i} \pi_{l'|i}\right| \leq C_5 N_0^{-4} n_0^3, \nonumber \end{aligned}$$ with $\pi_{klk'l'|i}$ the conditional probability that the SSUs $k,l,k',l'$ are selected together in $S_i$.
- Some constant $c_{2}>0$ exists s.t. for any PSU $i$ and for any $k \ne l$ inside: $$\begin{aligned}
\label{eq14b:sec3}
c_{2} N_0^{-2} n_0^{2} & \leq \pi_{kl|i}. &
\end{aligned}$$
It is assumed in (SS0) that the sizes $N_i$ of the PSUs are comparable, and that the numbers $n_i$ of SSUs selected inside the PSUs are also comparable. In practice, to reduce the variance associated to the first stage of sampling, the PSUs are usually grouped into strata in such a way that the PSUs inside one stratum are of similar sizes. Also, the number of selected SSUs is commonly the same for any PSU, so that all the interviewers have a comparable workload. Equations (\[eq8:sec3\]) and (\[eq9:sec3\]) seem therefore reasonable in practice. The assumptions (SS1)-(SS3) are similar to the assumptions (FS1)-(FS3) made for the first-stage sampling design.\
As previously mentioned, one-stage sampling designs are a particular case of our set-up. They are obtained when $N_i=1$ for any PSU $i$ and when $n_i=1$ for any unit $i \in S_I$. In such case, assumptions (SS0)-(SS1) automatically hold while assumptions (SS2)-(SS3) vanish.
Assumptions on the variable of interest {#sec23}
---------------------------------------
- There exists some constants $M_1$ and $m_1>0$ such that $$\begin{aligned}
& \displaystyle N^{-1} \sum_{i=1}^{N_I} \sum_{k =1}^{N_i} y_{ik}^4 & \leq M_1, \label{eq16:sec3} \\
m_1 \leq & \displaystyle N^{-1} \sum_{i=1}^{N_I} \sum_{k =1}^{N_i} y_{ik}. & \label{eq17:sec3}
\end{aligned}$$
- There exists some constant $m_2>0$ such that $$\begin{aligned}
\label{eq18:sec3}
m_2 & \leq & N^{-2} n_I \left\{V_1(\hat{Y}_{\pi})\right\}.
\end{aligned}$$
It is assumed in (VAR1) that the variable of interest has a bounded moment of order four, and a mean bounded away from $0$. It is assumed in (VAR2) that the first-stage sampling variance is non-vanishing. These assumptions are fairly weak, although we may find situations under which they are not respected. The condition (\[eq16:sec3\]) is not fulfilled for heavily skewed populations, where a non-negligible part of the individuals exhibit particularly large values for the variable of interest. This may be the case in wealth surveys, for example. Equations (\[eq17:sec3\]) and (\[eq18:sec3\]) are not fulfilled when we are interested in domain estimation, and when the domain size $N_d$ is negligible as compared to the population size.
Consistency of estimators {#sec4}
=========================
We begin with determining the orders of magnitude of the components of the variance decomposition in (\[eq4:sec2\]). The proof of Proposition \[prop1\] follows from some moment inequalities, which are given in Section 1 of the Supplementary Material.
\[prop1\] Suppose that assumptions (FS1)-(FS2),(SS0)-(SS2) and (VAR1) hold. Then $$\begin{aligned}
V_1(\hat{Y}_{\pi}) & = & O\left(N^2 n_I^{-1}\right), \nonumber \\
V_2(\hat{Y}_{\pi}) & = & O\left(N^2 n_I^{-1} n_0^{-1}\right), \label{eq1:sec4} \\
V_3(\hat{Y}_{\pi}) & = & O\left(N^2 N_I^{-1} n_0^{-1}\right). \nonumber
\end{aligned}$$
When $n_0 \to \infty$, the first variance component is the leading term and the two last ones are negligible. When $n_0$ is bounded, the first and second component have the same order of magnitude. The third component is negligible if $N_I^{-1} n_I \to 0$, and has the same order of magnitude otherwise. In practice, the third term is expected to be small as compared to the two first ones.\
The consistency of the HT estimator is established in Proposition \[prop2\]. The proof follows from Proposition \[prop1\], and is therefore omitted.
\[prop2\] Suppose that assumptions (FS1)-(FS2),(SS0)-(SS2) and (VAR1) hold. Then the HT estimator is design-unbiased. Also, we have $$\begin{aligned}
E\left[ N^{-1} \left\{\hat{Y}_{\pi}-Y\right\}\right]^2 = O(n_I^{-1})
& \textrm{ and } & \frac{\hat{Y}_{\pi}}{Y} \longrightarrow_{Pr} 1, \label{eq2:sec4}
\end{aligned}$$ where $\rightarrow_{Pr}$ stands for the convergence in probability.
Proposition \[prop2\] implies that the HT estimator is $\sqrt{n}$-consistent for the true total. Also, it is important to note that the consistency of the HT-estimator requires that the sampled number of PSUs $n_I$ tends to infinity, while the consistency is not related to the behaviour of $n_0$. For example, suppose that a sample of same size $n_i=n_0$ is selected inside any PSU, so that the total number of SSUs selected is $n=n_I n_0$. Then, even if $n \to \infty$, the HT-estimator may be inconsistent if $n_I$ is bounded. In practice, it is therefore important that a large number of PSUs is selected at the first stage.\
The consistency of the HT-estimator is proved in [@bre:ops:08] under the alternative assumptions:
- For any population $U$, $\min_{k \in U} \pi_k \geq \pi^* >0$ where $N \pi^* \to \infty$, and there exists $\kappa \geq 0$ such that $$\begin{aligned}
N^{0.5+\kappa} (\pi^*)^2 \to \infty & \textrm{ and } & \max_{k \in U} \sum_{l \in U;l \ne k} (\Delta_{kl})^2 = O(N^{-2\kappa}),
\end{aligned}$$ where $\Delta_{kl}=\pi_{kl}-\pi_k \pi_l$.
- The variable of interest satisfies $$\begin{aligned}
\limsup \frac{1}{N} \sum_{i=1}^{N_I} \sum_{k=1}^{N_i} y_{ik}^2 \leq \infty.
\end{aligned}$$
Under (D4) and (D5), we have $E\left[ N^{-1} \left\{\hat{Y}_{\pi}-Y\right\}\right]^2 = o(1)$ [@bre:ops:08 Lemma A.1]. Clearly, our condition in (VAR1) on the fourth moment implies (D5). Also, it can be shown that if $n_0$ is bounded, our assumptions (FS1)-(FS2) and (SS0)-(SS2) imply (D4) with $\kappa=0$. Our stronger conditions are needed in particular to get the consistency of variance estimators, see Sections \[sec41\] and \[sec42\].
Unbiased variance estimators {#sec41}
----------------------------
We first consider the so-called Horvitz-Thompson variance estimator $$\begin{aligned}
\hat{V}_{HT}(\hat{Y}_{\pi}) & = & \sum_{i,j \in S_I} \frac{\Delta_{Iij}}{\pi_{Iij}} \frac{\hat{Y}_i}{\pi_{Ii}} \frac{\hat{Y}_j}{\pi_{Ij}} + \sum_{i \in S_I} \frac{\hat{V}_{HT,i}}{\pi_{Ii}} \nonumber \\
& = & \hat{V}_{HT,A}(\hat{Y}_{\pi})+\hat{V}_{HT,B}(\hat{Y}_{\pi}), \label{eq4:sec4}
\end{aligned}$$ where $$\begin{aligned}
\hat{V}_{HT,i} & = & \sum_{k,l \in S_i} \frac{\Delta_{kl|i}}{\pi_{kl|i}} \frac{y_{ik}}{\pi_{k|i}} \frac{y_{il}}{\pi_{l|i}}. \label{eq5:sec4}
\end{aligned}$$
\[prop3\] If assumptions (FS1)-(FS3),(SS0)-(SS2) and (VAR1) hold, we have: $$\begin{aligned}
E\left[ N^{-2} n_I \left\{\hat{V}_{HT,A}(\hat{Y}_{\pi})-V_1(\hat{Y}_{\pi})-V_2(\hat{Y}_{\pi})\right\}\right]^2 & = & O(n_I^{-1}). \label{eq8:sec4}
\end{aligned}$$ If assumptions (FS1)-(FS2),(SS0)-(SS3) and (VAR1) hold, we have: $$\begin{aligned}
E\left[ N^{-2} N_I n_0 \left\{\hat{V}_{HT,B}(\hat{Y}_{\pi})-V_3(\hat{Y}_{\pi})\right\}\right]^2 & = & O(n_I^{-1}). \label{eq9:sec4}
\end{aligned}$$ If assumptions (FS1)-(FS3),(SS0)-(SS3),(VAR1)-(VAR2) hold, we have: $$\begin{aligned}
\qquad E\left[ N^{-2} n_I \left\{\hat{V}_{HT}(\hat{Y}_{\pi})-V(\hat{Y}_{\pi})\right\}\right]^2 = O(n_I^{-1}) \textrm{ and }
\frac{\hat{V}_{HT}(\hat{Y}_{\pi})}{V(\hat{Y}_{\pi})} \rightarrow_{Pr} 1. \label{eq10:sec4}
\end{aligned}$$
The proof of Proposition \[prop3\] is tedious but standard, and is therefore omitted. It implies that $\hat{V}_{HT}(\hat{Y}_{\pi})$ is a term by term unbiased and $\sqrt{n}$-consistent variance estimator, in the sense that $\hat{V}_{HT,A}(\hat{Y}_{\pi})$ is unbiased and $\sqrt{n}$-consistent for $V_1(\hat{Y}_{\pi})+V_2(\hat{Y}_{\pi})$, and $\hat{V}_{HT,B}(\hat{Y}_{\pi})$ is unbiased and $\sqrt{n}$-consistent for $V_3(\hat{Y}_{\pi})$. In their Theorem 3, [@bre:ops:00] state a similar result in case of one-stage sampling designs, for a more general class of estimators that they call local polynomial estimators. In the literature, the consistency of the HT-variance estimator is often stated as an assumption; e.g., [@kim:par:lee:17] for two-stage sampling designs.\
If the sampling designs used at both stages are of fixed size, we may alternatively use the Yates-Grundy variance estimator $$\begin{aligned}
\hat{V}_{YG}(\hat{Y}_{\pi}) & = & -\frac{1}{2} \sum_{i \neq j \in S_I} \frac{\Delta_{Iij}}{\pi_{Iij}} \left(\frac{\hat{Y}_i}{\pi_{Ii}} - \frac{\hat{Y}_j}{\pi_{Ij}}\right)^2 + \sum_{i \in S_I} \frac{\hat{V}_{YG,i}}{\pi_{Ii}} \nonumber \\
& = & \hat{V}_{YG,A}(\hat{Y}_{\pi})+\hat{V}_{YG,B}(\hat{Y}_{\pi}), \label{eq6:sec4}
\end{aligned}$$ with $$\begin{aligned}
\hat{V}_{YG,i} & = & -\frac{1}{2} \sum_{k \neq l \in S_i} \frac{\Delta_{kl|i}}{\pi_{kl|i}} \left(\frac{y_{ik}}{\pi_{k|i}} - \frac{y_{il}}{\pi_{l|i}} \right)^2. \label{eq7:sec4}
\end{aligned}$$ We prove in Proposition \[prop4\] that $\hat{V}_{YG}(\hat{Y}_{\pi})$ is also a term by term unbiased and $\sqrt{n}$-consistent variance estimator. The proof is similar to that of Proposition \[prop3\].
\[prop4\] If assumptions (FS1)-(FS3),(SS0)-(SS2) and (VAR1) hold, we have: $$\begin{aligned}
E\left[ N^{-2} n_I \left\{\hat{V}_{YG,A}(\hat{Y}_{\pi})-V_1(\hat{Y}_{\pi})-V_2(\hat{Y}_{\pi})\right\}\right]^2 & = & O(n_I^{-1}). \label{eq11:sec4}
\end{aligned}$$ If assumptions (FS1)-(FS2),(SS0)-(SS3) and (VAR1) hold, we have: $$\begin{aligned}
E\left[ N^{-2} N_I n_0 \left\{\hat{V}_{YG,B}(\hat{Y}_{\pi})-V_3(\hat{Y}_{\pi})\right\}\right]^2 & = & O(n_I^{-1}). \label{eq12:sec4}
\end{aligned}$$ If assumptions (FS1)-(FS3),(SS0)-(SS3), (VAR1)-(VAR2) hold, we have: $$\begin{aligned}
~\qquad E\left[ N^{-2} n_I \left\{\hat{V}_{YG}(\hat{Y}_{\pi})-V(\hat{Y}_{\pi})\right\}\right]^2 = O(n_I^{-1}) \textrm{ and }
\frac{\hat{V}_{YG}(\hat{Y}_{\pi})}{V(\hat{Y}_{\pi})} \rightarrow_{Pr} 1. \label{eq13:sec4}
\end{aligned}$$
Simplified one-term variance estimators {#sec42}
---------------------------------------
Both the variance estimators $\hat{V}_{HT}(\hat{Y}_{\pi})$ and $\hat{V}_{YG}(\hat{Y}_{\pi})$ may be cumbersome in practice, since they require an unbiased and consistent variance estimator $\hat{V}_{HT,i}$ or $\hat{V}_{YG,i}$ inside any of the selected PSUs. Consider the example of self-weighted two-stage sampling designs, which are common in practice. They consist in selecting a sample of PSUs, with probabilities $\pi_{Ii}$ proportional to the size of the PSUs, and a sample of $n_0$ SSUs inside any of the selected PSUs. This leads to equal sampling weights for all the SSUs in the population, hence the name. In case of self-weighted two-stage sampling designs, systematic sampling is frequently used at the second stage. In such case, the assumption (SS3) is usually not respected.\
A simplified variance estimator can be obtained by using $\hat{V}_{HT,A}(\hat{Y}_{\pi})$ only, or for a fixed-size sampling design $\hat{V}_{YG,A}(\hat{Y}_{\pi})$ only, see for instance @sar:swe:wre:92. Proposition \[prop5\] states that these simplified estimators are consistent when $$\begin{aligned}
\frac{V_3(\hat{Y}_{\pi})}{V_1(\hat{Y}_{\pi})+V_2(\hat{Y}_{\pi})} & \to & 0, \label{eq14:sec4}
\end{aligned}$$ i.e. when the third component of the variance in the decomposition (\[eq4:sec2\]) is negligible. Note that in Proposition \[prop5\] we do not need the assumption (SS3) which guarantees a lower bound for the second-order inclusion probabilities at the second stage.
\[prop5\] Suppose that assumptions (FS1)-(FS3), (SS0)-(SS2), (VAR1)-(VAR2) hold. Suppose that equation (\[eq14:sec4\]) holds. Then $$\begin{aligned}
E\left[ N^{-2} n_I \left\{\hat{V}_{HT,A}(\hat{Y}_{\pi})-V(\hat{Y}_{\pi})\right\}\right]^2 & = & o(1), \label{eq15:sec4} \\
\frac{\hat{V}_{HT,A}(\hat{Y}_{\pi})}{V(\hat{Y}_{\pi})} & \longrightarrow_{Pr} & 1. \label{eq15b:sec4}
\end{aligned}$$ If in addition the first-stage sampling design is of fixed-size, we have $$\begin{aligned}
E\left[ N^{-2} n_I \left\{\hat{V}_{YG,A}(\hat{Y}_{\pi})-V(\hat{Y}_{\pi})\right\}\right]^2 & = & o(1), \label{eq16:sec4} \\
\frac{\hat{V}_{YG,A}(\hat{Y}_{\pi})}{V(\hat{Y}_{\pi})} & \longrightarrow_{Pr} & 1. \label{eq17:sec4}
\end{aligned}$$
The proof is immediate from Propositions \[prop3\] and \[prop4\], and by using equation (\[eq14:sec4\]). The simplified variance estimators $\hat{V}_{HT,A}(\hat{Y}_{\pi})$ and $\hat{V}_{YG,A}(\hat{Y}_{\pi})$ are simpler to compute, since they do not involve variance estimators $\hat{V}_i$ inside PSUs, but only unbiased estimators $\hat{Y}_i$ for the sub-totals over the PSUs.\
Under the assumptions (FS1)-(FS3), (SS0)-(SS2) and (VAR1)-(VAR2), a sufficient condition for equation (\[eq14:sec4\]) to hold is that $N_I^{-1} n_I \to 0$ (negligible first-stage sampling rate). In practice, we expect the term $V_3(\hat{Y}_{\pi})$ to have a small contribution in the overall variance even if the first-stage sampling rate is not negligible. This is illustrated in Section \[sec6\] through a simulation study, and in Section \[appli:ppv\] in the application to the panel for urban policy. The two simplified variance estimators $\hat{V}_{HT,A}(\hat{Y}_{\pi})$ and $\hat{V}_{YG,A}(\hat{Y}_{\pi})$ may therefore be reasonable choices for variance estimation in practice.
Case of large entropy sampling designs {#sec5}
======================================
In this Section, we focus on the situation when large entropy sampling designs are used at the first stage. We consider a Hájek-type variance estimator, and prove its consistency with limited assumptions, namely by dropping the conditions (FS2) and (FS3). Building on the work of [@ohl:89], we also prove that the HT-estimator is asymptotically normally distributed. The rejective sampling design [@haj:64] is first considered in Section \[sec51\]. The results are extended in Section \[sec52\] to a class of large entropy sampling designs by using a coupling algorithm. The properties of a simplified variance estimator are studied in Section \[sec53\].
Rejective sampling {#sec51}
------------------
The rejective (or conditional Poisson) sampling design was introduced by [@haj:64]. Rejective sampling in $U_I$ consists in repeatedly selecting samples by means of Poisson sampling, until the sample has the required size $n_I$. The inclusion probabilities of the Poisson sampling design are chosen so that the required inclusion probabilities $\pi_{Ii},~i \in U_I$ are respected; see for example [@dup:75]. The rejective sampling design has been extensively studied in the literature, see [@til:06] for a review. Under a rejective sampling design at the first-stage, the assumption (FS2) is implied by the assumption (FS1), see our discussion in Section \[sec21\].\
We note $p_r(\cdot)$ the rejective sampling design with inclusion probabilities $\pi_{Ii}$ in the population $U_I$. Also, we note $S_{rI}$ a first-stage sample selected by means of $p_r$, and $$\begin{aligned}
\hat{Y}_{r\pi} & = & \sum_{i \in S_{rI}} \frac{\hat{Y}_i}{\pi_{Ii}}
\end{aligned}$$ the associated HT-estimator. Making use of a uniform approximation of the second-order inclusion probabilities, [@haj:64] proposed a very simple variance estimator for which these second-order inclusion probabilities are not needed. In our two-stage sampling context, this leads to replacing in (\[eq4:sec4\]) the term $\hat{V}_{HT,A}(\hat{Y}_{r\pi})$ with $$\begin{aligned}
\qquad \hat{V}_{HAJ,A}(\hat{Y}_{r\pi}) & = &
\left\{\begin{array}{ll}
\displaystyle \sum_{i \in S_{rI}} (1-\pi_{Ii}) \left(\frac{\hat{Y}_i}{\pi_{Ii}} - \hat{\hat{R}}_{r\pi} \right)^2 & \textrm{if } \hat{d}_{rI} \geq \frac{c_{I0}}{2} n_I, \\
0 & \textrm{otherwise},
\end{array} \right.
\label{eq1:sec5}
\end{aligned}$$ with $$\begin{aligned}
\hat{\hat{R}}_{r\pi} = \hat{d}_{rI}^{-1} \sum_{i \in S_{rI}} (1-\pi_{Ii}) \frac{\hat{Y}_i}{\pi_{Ii}} & \textrm{ and } & \hat{d}_{rI} = \sum_{i \in S_{rI}} (1-\pi_{Ii}), \label{eq2:sec5}
\end{aligned}$$ and where $c_{I0}$ is defined in Lemma \[lem6b\] (see the Supplementary Material). This leads to the global variance estimator $$\begin{aligned}
\hat{V}_{HAJ}(\hat{Y}_{r\pi}) & = & \hat{V}_{HAJ,A}(\hat{Y}_{r\pi})+\hat{V}_{HT,B}(\hat{Y}_{r\pi}), \label{eq3:sec5}
\end{aligned}$$ where $\hat{V}_{HT,B}(\hat{Y}_{r\pi})$ is defined in equation (\[eq4:sec4\]). If the second-stage sampling designs are all of fixed-size, we could alternatively replace $\hat{V}_{HT,B}(\hat{Y}_{r\pi})$ with $\hat{V}_{YG,B}(\hat{Y}_{r\pi})$ given in equation (\[eq6:sec4\]).\
Note that the variance estimator $\hat{V}_{HAJ}(\hat{Y}_{r\pi})$ is truncated to avoid extreme values for $\hat{\hat{R}}_{r\pi}$. This is needed to establish its consistency, which is done in Proposition \[prop6\]. An advantage of this variance estimator is that the first-stage second-order inclusion probabilities are not required. In particular, the condition (FS3) is not needed to prove the consistency. We also prove in Proposition \[prop6\] that the HT-estimator is asymptotically normally distributed, by using Theorem 2.1 in [@ohl:89].
\[prop6\] Suppose that a rejective sampling design is used at the first stage. Suppose that assumptions (FS1), (SS0)-(SS2) and (VAR1) hold. Then $$\begin{aligned}
\label{eq4:sec5}
E\left[ N^{-2} n_I \left\{\hat{V}_{HAJ,A}(\hat{Y}_{r\pi})-V_1(\hat{Y}_{r\pi})-V_2(\hat{Y}_{r\pi})\right\}\right]^2 & = & o(1).
\end{aligned}$$ If in addition the assumption (VAR2) holds, then $$\begin{aligned}
\label{eq4b:sec5}
\frac{\hat{Y}_{r\pi}-Y}{\sqrt{V(\hat{Y}_{r\pi})}} & \longrightarrow_{\mathcal{L}} & \mathcal{N}(0,1),
\end{aligned}$$ where $\rightarrow_{\mathcal{L}}$ stands for the convergence in distribution. If in addition the assumption (SS3) holds, then $$\begin{aligned}
\qquad E\left[ N^{-2} n_I \left\{\hat{V}_{HAJ}(\hat{Y}_{r\pi})-V(\hat{Y}_{r\pi})\right\}\right]^2 = o(1)
\textrm{ and } \frac{\hat{V}_{HAJ}(\hat{Y}_{r\pi})}{V(\hat{Y}_{r\pi})} \rightarrow_{Pr} 1. \label{eq5:sec5}
\end{aligned}$$
The proof is given in Section 2 of the Supplementary Material. The asymptotic normality of the HT-estimator has been proved by [@haj:64] for a single stage rejective sampling design, but the consistency of the Hájek-type variance estimator has not been rigorously established previously. Proposition \[prop6\] has therefore its own interest, even for one-stage sampling designs. It follows that under rejective sampling at the first-stage, an approximate two-sided $100(1-2\alpha) \% $ confidence interval for $Y$ is obtained as $$\begin{aligned}
\label{eq6:sec5}
\left[\hat{Y}_{r\pi} \pm u_{1-\alpha} \{\hat{V}_{HAJ}(\hat{Y}_{r\pi})\}^{0.5}\right],
\end{aligned}$$ with $u_{1-\alpha}$ the quantile of order $1-\alpha$ of the standard normal distribution.
Other sampling designs {#sec52}
----------------------
We consider a more general class of sampling designs at the first-stage, which are close to the rejective sampling design with respect to the Chi-square distance. Other distance functions have been considered in the literature, such as the Hellinger distance [@con:14] or the total variation distance [@ber:cha:cle:17]. We note $p(\cdot)$ for a fixed-size sampling design with inclusion probabilities $\pi_{Ii}$ in the population $U_I$. It is said to be close to the rejective sampling design $p_r(\cdot)$ with respect to the Chi-square distance if $$\begin{aligned}
\label{eq1:sec52}
\qquad d_2(p,p_r) \rightarrow 0 & \textrm{where} & d_2(p,p_r)=\sum_{s_I \subset U_I;~p_r(s_I)>0} \frac{\left\{p(s_I)-p_r(s_I)\right\}^2}{p_r(s_I)}.
\end{aligned}$$ Equation (\[eq1:sec52\]) holds for the Rao-Sampford [@sam:67] sampling design, for example. We note $S_{pI}$ a first-stage sample selected by means of $p(\cdot)$, and the associated HT-estimator is $$\begin{aligned}
\hat{Y}_{p\pi} & = & \sum_{i \in S_{pI}} \frac{\hat{Y}_i}{\pi_{Ii}}.
\end{aligned}$$
We introduce in Algorithm \[algo:1\] a coupling procedure to obtain the estimators $\hat{Y}_{p\pi}$ and $\hat{Y}_{r\pi}$ jointly, which is the main tool in extending the results in Proposition \[prop6\] to $\hat{Y}_{p\pi}$. We note $$\begin{aligned}
\qquad \alpha=1-d_{TV}(p,p_r) & \textrm{where} & d_{TV}(p,p_r) = \frac{1}{2} \sum_{s_I \in U_I} |p(s_I)-p_r(s_I)|
\end{aligned}$$ is the total variation distance between $p(\cdot)$ and $p_r(\cdot)$. By using Lemma 11 in Section 3 of the Supplementary Material, it can be proved that the coupling procedure in Algorithm \[algo:1\] leads to estimators $\hat{Y}_{r\pi}$ and $\hat{Y}_{p\pi}$ associated to the required two-stage sampling designs; see also [@van:16], Theorem 2.9.
1. Draw $u$ from a uniform distribution $U[0,1]$.
2. If $u \leq \alpha$, then:
1. Select a sample $s_I$ with probabilities $\displaystyle \frac{p(s_I) \wedge p_r(s_I)}{\alpha}$, and take $S_{rI}=S_{pI}=s_I$.
2. For any $i \in S_{rI}=S_{pI}$, select the same second-stage sample $S_{i}$ for both $\hat{Y}_{r\pi}$ and $\hat{Y}_{p\pi}$.
3. If $u > \alpha$, then:
1. Select the sample $S_{pI}$ with probabilities $\displaystyle \frac{p(s_{I}) - p_r(s_{I})}{1-\alpha}$ in the set $\{s_I \in U_I;~p(s_I)>p_r(s_I)\}$. For any $i \in S_{pI}$, select a second-stage sample $S_{i}$ for $\hat{Y}_{p\pi}$.
2. Independently of $S_{pI}$ and of the associated second-stage samples $S_{i}$’s, select the sample $S_{rI}$ with probabilities $\displaystyle \frac{p_r(s_{I}) - p(s_{I})}{1-\alpha}$ in the set $\{s_I;~p(s_I) \leq p_r(s_I)\}$. For any $i \in S_{rI}$, select a second-stage sample $S_{i}$ for $\hat{Y}_{r\pi}$.
\[prop7a\] Suppose that the samples $S_{rI}$ and $S_{pI}$ are selected by means of the coupling procedure in Algorithm \[algo:1\]. Then: $$\begin{aligned}
E \left(\hat{Y}_{p\pi}-\hat{Y}_{r\pi}\right)^2 & \leq & \sum_{s_I \in U_I} |p(s_I)-p_r(s_I)| \left\{\left(\sum_{i \in s_I} \frac{Y_i}{\pi_{Ii}}-Y \right)^2 + \sum_{i \in s_I} \frac{V_i}{\pi_{Ii}^2} \right\}.
\end{aligned}$$
\[prop7\] Suppose that the samples $S_{rI}$ and $S_{pI}$ are selected by means of the coupling procedure in Algorithm \[algo:1\]. Suppose that assumptions (FS1), (SS0)-(SS2) and (VAR1) hold. Suppose that $d_2(p,p_r) \to 0$. Then $$\begin{aligned}
\label{eq1:prop7}
E \left(\hat{Y}_{p\pi}-\hat{Y}_{r\pi}\right)^2 & = & o\left(N^2 n_I^{-1}\right).
\end{aligned}$$ If in addition the assumption (VAR2) holds, then $$\begin{aligned}
\label{eq2:prop7}
\frac{V\left(\hat{Y}_{p\pi}\right)}{V\left(\hat{Y}_{r\pi}\right)} & \rightarrow & 1.
\end{aligned}$$
The proofs of Propositions \[prop7a\] and \[prop7\] are given in Sections 3.2 and 3.3 of the Supplementary Material. These propositions state that if the sampling designs $p(\cdot)$ and $p_r(\cdot)$ are close with respect to the Chi-square distance, then $E \left(\hat{Y}_{p\pi}-\hat{Y}_{r\pi}\right)^2$ is smaller than the rate of convergence of $\hat{Y}_{r\pi}$. Consequently, the results in Proposition \[prop6\] can be extended to the sampling design $p(\cdot)$, as stated in Proposition \[prop8\]. Similar coupling arguments are used by [@cha:15] to obtain asymptotic results for multistage sampling designs with stratified simple random without replacement sampling at the first stage.
\[prop8\] Suppose that assumptions (FS1), (SS0)-(SS2), (VAR1)-(VAR2) hold, and that $d_2(p,p_r) \to 0$. Then $$\begin{aligned}
\label{eq1:prop8}
\frac{\hat{Y}_{p\pi}-Y}{\sqrt{V(\hat{Y}_{p\pi})}} & \longrightarrow_{\mathcal{L}} & \mathcal{N}(0,1).
\end{aligned}$$ If in addition the assumption (SS3) holds, we have $$\begin{aligned}
\label{eq2:prop8}
\qquad E\left[ N^{-2} n_I \left|\hat{V}_{HAJ}(\hat{Y}_{p\pi})-V(\hat{Y}_{p\pi}) \right| \right] = o(1) & \textrm{and} &
\frac{\hat{V}_{HAJ}(\hat{Y}_{p\pi})}{V(\hat{Y}_{p\pi})} \rightarrow_{Pr} 1.
\end{aligned}$$
The proof is given in Section 3.4 of the Supplementary Material. From Proposition \[prop8\], the two-sided $100(1-2\alpha) \% $ confidence interval given in (\[eq6:sec5\]) is also asymptotically valid for $\hat{Y}_{p\pi}$.\
We now turn back to the choice of the distance function. Let $X(s_I)$ denote some function of a sample $s_I$. Equation (\[eq1:prop7\]) in Proposition \[prop7\] is based on the inequality $$\begin{aligned}
\label{eqa:sec52}
\sum_{s_I \subset U_I} |p(s_I)-p_r(s_I)| X(s_I) & \leq & \sqrt{d_2(p,p_r)} \times \sqrt{\sum_{s_I \subset U_I} p_r(s_I) X(s_I)^2} \nonumber \\
& \leq & \sqrt{d_2(p,p_r)} \times \sqrt{E\{X(S_{rI})^2\}}.
\end{aligned}$$ From equation (\[eqa:sec52\]) and Proposition \[prop7a\], $X(S_{rI})$ and $X(S_{pI})$ are asymptotically equivalent if (a) $d_2(p,p_r) \to 0$, and if (b) we can control the second moment of $X(S_{rI})$. This last point may be obtained through standard algebra for rejective sampling, see Lemma 8 for example.\
If we rather resort to the Kullback-Leibler divergence $$\begin{aligned}
d_{KL}(p,p_r) & = & \sum_{s_I \subset U_I;~p_r(s_I)>0} p(s_I) \log\left\{\frac{p(s_I)}{p_r(s_I)}\right\},
\end{aligned}$$ we can obtain the similar inequality $$\begin{aligned}
\sum_{s_I \subset U_I} |p(s_I)-p_r(s_I)| X(s_I) \leq
\sqrt{d_{KL}(p,p_r)} \times \sqrt{\frac{4}{3} E\{X(S_{rI})^2\} + \frac{2}{3} E\{X(S_{pI})^2\}}.
\end{aligned}$$ Consequently, we may alternatively demonstrate that $X(S_{rI})$ and $X(S_{pI})$ are asymptotically equivalent if (a’) $d_{KL}(p,p_r) \to 0$, if (b) we can control the second moment of $X(S_{rI})$, and if (c) we can control the second moment of $X(S_{pI})$. This last point is difficult to prove for a general sampling design.
A simplified variance estimator {#sec53}
-------------------------------
The variance estimator $\hat{V}_{HAJ}(\hat{Y}_{r\pi})$ proposed in (\[eq3:sec5\]) has been proved to be consistent for large entropy sampling designs, with limited assumptions on the first-stage sampling design. However, unbiased and consistent variance estimators are required inside the PSUs, which can be cumbersome for a data user. It is stated in Proposition \[prop9\] that the simplified one-term variance estimator $\hat{V}_{HAJ,A}(\hat{Y}_{r\pi})$ is consistent, provided that the third component of the variance in the decomposition (\[eq4:sec2\]) is negligible. The proof readily follows from Propositions \[prop6\] and \[prop8\], and is therefore omitted. Note that the assumption (SS3) providing a lower bound for the second order inclusion probabilities at the second stage is not needed any more.
\[prop9\] Suppose that assumptions (FS1), (SS0)-(SS2), (VAR1)-(VAR2) hold. Suppose that equation (\[eq14:sec4\]) holds. If a rejective sampling design $p_r$ is used at the first-stage, we have $$\begin{aligned}
E\left[ N^{-2} n_I \left\{\hat{V}_{HAJ,A}(\hat{Y}_{r\pi})-V(\hat{Y}_{r\pi})\right\}\right]^2 & = & o(1), \label{eq1:prop9} \\
\frac{\hat{V}_{HAJ,A}(\hat{Y}_{r\pi})}{V(\hat{Y}_{r\pi})} & \longrightarrow_{Pr} & 1. \label{eq2:prop9}
\end{aligned}$$ If the first-stage sampling design $p$ is such that $d_2(p,p_r) \to 0$, then $$\begin{aligned}
E\left[ N^{-2} n_I \left|\hat{V}_{HAJ,A}(\hat{Y}_{p\pi})-V(\hat{Y}_{p\pi})\right|\right] = o(1) & \textrm{and} &
\frac{\hat{V}_{HAJ,A}(\hat{Y}_{p\pi})}{V(\hat{Y}_{p\pi})} \rightarrow_{Pr} 1. \label{eq3:prop9}
\end{aligned}$$
Simulation study {#sec6}
================
A simulation study was conducted to evaluate the asymptotic properties of the Hájek-type variance estimators $\hat{V}_{HAJ}(\hat{Y}_{\pi})$ and $\hat{V}_{HAJ,A}(\hat{Y}_{\pi})$. Three populations $U_1,U_2,U_3$ of $N_I=2,000$ PSUs were generated. The number of SSUs per PSU were randomly generated, with mean $N_0=40$ and with a coefficient of variation equal to 0, 0.03 and 0.06 for population 1, 2, and 3 respectively. The PSUs are therefore of equal size in the first population.\
In each population, a value $\nu_i$ was generated for any PSU $i$ from a standard normal distribution. Three variables were generated, for any SSU $k$ inside PSU $i$, in each population according to the model $$\begin{aligned}
y_{ikh} = \lambda + \sigma \nu_i + [ \rho_h^{-1} (1-\rho_h)]^{0.5} \sigma \varepsilon_k,
\end{aligned}$$ where $\lambda = 20$, $\sigma = 2$, where $\varepsilon_k$ was generated from a standard normal distribution, and $\rho_h$ was such that the intra-cluster correlation coefficient ($ICC$) was approximately 0.1, 0.2 and 0.3 for $h=1,2$ and 3 respectively.\
From each population, we repeated $R=1,000$ times the following two-stage sampling design. A first-stage sample $S_I$ of $n_I=20, 40, 100$ or 200 PSUs was selected by means of a rejective sampling design, with inclusion probabilities $\pi_{Ii}$ proportional to the size $N_i$. A second-stage sample $S_i$ of $n_i=n_0=5$ or 10 was selected inside any $i \in S_I$ by simple random sampling without replacement. In each sample, we computed the HT-estimator $\hat{Y}_\pi$ and the Hájek-type variance estimators $\hat{V}_{HAJ,A}(\hat{Y}_\pi)$ and $\hat{V}_{HAJ}(\hat{Y}_\pi)$.\
As a measure of bias of a variance estimator $\hat{V}$, we computed the Monte Carlo percent relative bias $$\begin{aligned}
\mathrm{RB}_{MC}(\hat{V}) = \frac{ \displaystyle \frac{1}{R} \sum_{r=1}^R \hat{V}^{(r)} - V(\hat{Y}_\pi) }{ V(\hat{Y}_\pi) } \times 100,
\end{aligned}$$ with $\hat{V}^{(r)}$ the value of the estimator in the $r$th sample, and $\mathrm{V}(\hat{Y}_\pi)$ the exact variance. The Monte Carlo percent relative stability, $$\begin{aligned}
\mathrm{RS}_{MC}(\hat{V}) =
\frac{ \displaystyle \left\{ \frac{1}{R} \sum_{r=1}^R \left[ \hat{V}^{(r)} - \mathrm{V}(\hat{Y}_\pi) \right]^2 \right\}^{1/2} }{\mathrm{V}(\hat{Y}_\pi)} \times 100,
\end{aligned}$$ was calculated as a measure of variability of $\hat{V}$. We also calculated the error rates of the normality-based confidence interval given in (\[eq6:sec5\]), with nominal one-tailed error rate of 2.5 % in each tail.\
The results are presented in Table \[tab:pop3\] for the population 3. We observed no qualitative difference with populations 1 and 2, and the results are therefore omitted for conciseness. As expected, the variance estimator $\hat{V}_{HAJ}(\hat{Y}_\pi)$ is almost unbiased in any case, with $\mathrm{RB}_{MC}$ lower than 2% in absolute value. The stability $\mathrm{RS}_{MC}$ decreases with $n_I$ but not with $n_i$, as expected. The bias of the simplified variance estimator $\hat{V}_{HAJ,A}(\hat{Y}_\pi)$ is comparable with a small first-stage sampling fraction, but increases with $n_I/N_I$. Even with the largest sampling fraction, the bias of $\hat{V}_{HAJ,A}(\hat{Y}_\pi)$ is limited and no greater than $7 \% $. This supports the fact that the term of variance $V_3(\hat{Y}_{\pi})$ in the decomposition (\[eq1:sec4\]) has a small contribution to the global variance. Both variance estimators perform similarly in terms of stability, with $\mathrm{RS}_{MC}$ being slightly larger for $\hat{V}_{HAJ,A}(\hat{Y}_\pi)$ with the largest sampling fraction. The coverage probabilities are well respected in any case, lying between 93% and 95%.
------- ------- ------- ---------------------------------------- -------------------------------------- ---------------------------------------- -------------------------------------- ---------------------------------------- --------------------------------------
$ICC$ $n_I$ $n_i$ $\widehat{V}_{HAJ,A}(\widehat{Y}_\pi)$ $\widehat{V}_{HAJ}(\widehat{Y}_\pi)$ $\widehat{V}_{HAJ,A}(\widehat{Y}_\pi)$ $\widehat{V}_{HAJ}(\widehat{Y}_\pi)$ $\widehat{V}_{HAJ,A}(\widehat{Y}_\pi)$ $\widehat{V}_{HAJ}(\widehat{Y}_\pi)$
0.1 20 5 0.08 0.70 33.58 33.59 0.94 0.94
10 -0.98 -0.57 31.30 31.30 0.93 0.93
40 5 -1.00 0.24 21.59 21.56 0.94 0.94
10 -2.66 -1.84 21.85 21.77 0.93 0.93
100 5 -3.23 -0.08 14.02 13.64 0.94 0.94
10 -2.36 -0.27 14.34 14.15 0.95 0.95
200 5 -6.59 -0.19 11.17 9.03 0.94 0.94
10 -4.15 0.17 10.42 9.57 0.94 0.95
0.2 20 5 -0.37 0.05 33.13 33.13 0.93 0.93
10 -0.80 -0.57 32.03 32.02 0.93 0.93
40 5 -0.82 0.01 22.20 22.18 0.94 0.94
10 -2.17 -1.71 21.99 21.94 0.93 0.93
100 5 -2.25 -0.13 14.07 13.89 0.95 0.95
10 -1.75 -0.56 14.34 14.25 0.94 0.95
200 5 -4.54 -0.17 10.20 9.14 0.94 0.94
10 -2.22 0.28 9.96 9.72 0.94 0.94
0.3 20 5 -0.72 -0.43 32.89 32.88 0.94 0.94
10 -0.69 -0.54 32.39 32.39 0.93 0.93
40 5 -0.77 -0.19 22.58 22.56 0.94 0.94
10 -1.85 -1.55 22.02 21.99 0.93 0.93
100 5 -1.63 -0.14 14.09 14.00 0.95 0.95
10 -1.44 -0.67 14.29 14.24 0.95 0.95
200 5 -3.26 -0.16 9.80 9.25 0.95 0.95
10 -1.29 0.32 9.83 9.75 0.95 0.95
------- ------- ------- ---------------------------------------- -------------------------------------- ---------------------------------------- -------------------------------------- ---------------------------------------- --------------------------------------
: Percent relative biases, percent relative stabilities and coverage probabilities of $\widehat{V}_{HAJ,A}(\widehat{Y}_\pi)$ and $\widehat{V}_{HAJ}(\widehat{Y}_\pi)$ in population 3[]{data-label="tab:pop3"}
Illustration on the panel for urban policy {#appli:ppv}
==========================================
We consider an application to the Panel for Urban Policy (PUP), which is the original motivation for this work. This is a panel survey in four waves, performed by the French General Secretariat of the Inter-ministerial Committee for Cities (SGCIV) and conducted between 2011 and 2014. The scope of the survey is the collection of various information about security, employment, precariousness, schooling and health, for people living in the Sensitive Urban Zones (ZUS). The initial panel $S_I$ is selected through two-stage sampling, with districts as PSUs and households as SSUs. The individuals in the selected households are comprehensively surveyed.\
At the first stage, the population $U_I$ of districts is partitioned into $H=4$ strata defined according to the progress of the urban renewal program. A stratified sample $S_I$ of $n_I=40$ districts is selected, with probabilities proportional to the number of main dwellings. The first-stage inclusion probabilities range from $0.04$ to $0.67$, for a first-stage sampling rate of approximately $0.09$. Inside any selected district $i$, a sample $S_i$ of $n_i$ households is selected with equal probabilities. The sample of households is prone to unit non-response, but this issue is not considered here for the sake of simplicity. In this illustration, the sample of responding households is viewed as the true sample. In summary, the data set is a sample of $1,065$ households obtained by stratified two-stage sampling.\
We are interested in four variables related to security, town planning and residential mobility. The variable $y_1$ gives the perceived reputation of the district (good, fair, poor, no opinion). The variable $y_2$ indicates if a member of the household has witnessed trafficking (never, rarely, sometimes, no opinion). The variable $y_3$ indicates if some significant roadworks have been done in the neighborhood in the twelve last months (yes, no, no opinion). The variable $y_4$ indicates if the households intends to leave the district during the next twelve next months (certainly/probably, certainly not, probably not, no opinion). For any possible characteristic $c$ of some variable $y$, we are interested in the proportion $$\begin{aligned}
\label{eq1:appli}
p_{c} = \frac{\sum_{h=1}^H \sum_{i=1}^{N_{Ih}} Y_{i}}{\sum_{h=1}^H \sum_{i=1}^{N_{Ih}} N_{i}} & \textrm{ with } & Y_{i}=\sum_{k=1}^{N_i} 1(y_{ik}=c),
\end{aligned}$$ and where $N_{Ih}$ is the number of PSUs in the stratum $h$. The proportion $p_c$ is estimated by its substitution estimator $$\begin{aligned}
\label{eq2:appli}
\hat{p}_{c} = \frac{\sum_{h=1}^H \sum_{i \in S_{Ih}} \frac{\hat{Y}_i}{\pi_{Ii}}}{\hat{N}_{\pi}} & \textrm{ with } & \hat{N}_{\pi} \equiv \sum_{h=1}^H \sum_{i \in S_{Ih}} \sum_{k \in S_i} \frac{1}{\pi_{Ii} \pi_{k|i}},
\end{aligned}$$ and where $S_{Ih}$ is the sample of PSUs in the stratum $h$.
For each proportion, we consider the two variance estimators presented in Section \[sec5\]. We first compute the linearized variable of $p_c$, which is $$\begin{aligned}
\label{eq3:appli}
e_{ik} & = & \frac{1}{\hat{N}_{\pi}} \{1(y_{ik}=c)-\hat{p}_{c}\}.
\end{aligned}$$ We then compute the variance estimator in (\[eq3:sec5\]) by replacing the variable $y_{ik}$ with $e_{ik}$, and without truncating the first term of variance for simplicity. With stratified sampling at the first stage, and since the second-stage samples are selected with equal probabilities, this leads to the variance estimator $$\begin{aligned}
\label{eq4:appli}
\hat{V}_{HAJ}(\hat{p}_{c}) & = & \hat{V}_{HAJ,A}(\hat{p}_{c}) + \hat{V}_{HT,B}(\hat{p}_{c}), \\
\textrm{with } \hat{V}_{HAJ,A}(\hat{p}_{c}) & = & \sum_{h=1}^4 \sum_{i \in S_{Ih}} (1-\pi_{Ii}) \left(\frac{\hat{E}_i}{\pi_{Ii}} - \hat{\hat{R}}_{eh\pi} \right)^2, \nonumber \\
\textrm{with } \hat{V}_{HT,B}(\hat{p}_{c}) & = & \sum_{h=1}^4 \sum_{i \in S_{Ih}} \frac{N_i^2}{\pi_{Ii}} \left(\frac{1}{n_i}-\frac{1}{N_i}\right) s_{ei}^2, \nonumber
\end{aligned}$$ and where $$\begin{aligned}
\label{eq5:appli}
\hat{\hat{R}}_{eh\pi} = \frac{\sum_{i \in S_{Ih}} (1-\pi_{Ii}) \frac{\hat{E}_i}{\pi_{Ii}}}{\sum_{i \in S_{Ih}} (1-\pi_{Ii})} & \textrm{ with } & \hat{E}_i = \sum_{k \in S_i} \frac{e_{ik}}{\pi_{k|i}}, \\
s_{ei}^2 = \frac{1}{n_i-1} \sum_{k \in S_i} (e_{ik}-\bar{e}_i)^2 & \textrm{ with } & \bar{e}_i = \frac{1}{n_i} \sum_{k \in S_i} e_{ik}. \nonumber
\end{aligned}$$ The second, simplified variance estimator is $\hat{V}_{HAJ,A}(\hat{p}_{c})$, obtained from equation (\[eq4:appli\]) by dropping the second component.\
The two variance estimators are then plugged into a normality-based confidence interval, with a nominal one-tailed error rate of 2.5 % . The results are presented in Table \[result:illust\], and show almost identical performance of both variance estimators.
--------------------------- -------------------- ----------------- ----------------- -----------------
Good Fair Poor No opinion
Estimator $\hat{p}_c$ $0.218$ $0.227$ $0.527$ $0.028$
CI with $\hat{V}_{HAJ}$ \[0.182,0.253\] \[0.205,0.250\] \[0.485,0.569\] \[0.018,0.038\]
CI with $\hat{V}_{HAJ,A}$ \[0.183,0.252\] \[0.206,0.248\] \[0.486,0.568\] \[0.019,0.038\]
Never Rarely Sometimes No opinion
Estimator $\hat{p}_c$ $0.582$ $0.053$ $0.163$ $0.049$
CI with $\hat{V}_{HAJ}$ \[0.537,0.628\] \[0.037,0.068\] \[0.135,0.192\] \[0.036,0.063\]
CI with $\hat{V}_{HAJ,A}$ \[0.538,0.627\] \[0.038,0.068\] \[0.136,0.191\] \[0.037,0.062\]
Yes No No opinion
Estimator $\hat{p}_c$ $0.463$ $0.503$ $0.034$
CI with $\hat{V}_{HAJ}$ \[0.398,0.528\] \[0.434,0.572\] \[0.022,0.045\]
CI with $\hat{V}_{HAJ,A}$ \[0.399,0.527\] \[0.435,0.572\] \[0.023,0.044\]
Certainly/Probably Probably not Certainly not No opinion
Estimator $\hat{p}_c$ $0.275$ $0.129$ $0.562$ $0.034$
CI with $\hat{V}_{HAJ}$ \[0.255,0.295\] \[0.098,0.159\] \[0.531,0.594\] \[0.025,0.043\]
CI with $\hat{V}_{HAJ,A}$ \[0.257,0.292\] \[0.099,0.158\] \[0.532,0.593\] \[0.036,0.042\]
--------------------------- -------------------- ----------------- ----------------- -----------------
Discussion {#sec7}
==========
In this article, we proposed an asymptotic set-up for the study of two-stage sampling designs. We gave general conditions under which the Horvitz-Thompson estimator is consistent, and under which usual variance estimators are consistent. In case of large entropy sampling designs at the first stage, we also proved that the Horvitz-Thompson estimator is asymptotically normally distributed and that a truncated Hájek-like variance estimator is consistent. When the first-stage sampling fraction is negligible, simplified variance estimators are also shown to be consistent, under limited assumptions.\
Multistage sampling designs are often used at baseline for longitudinal household surveys. If we wish to perform longitudinal estimations, individuals from the initial sample are followed over time. If we also wish to perform cross-sectional estimations at several times, additional samples are selected at further waves and mixed with the individuals originally selected. Even in the simplest case when estimations are produced at baseline with a single sample, variance estimation is challenging due to the different sources of randomness which need to be accounted for: this includes not only the sampling design, but also unit non-response, item non-response and the corresponding statistical treatments. Variance estimation in such more realistic context is an important matter for further investigation.
[9]{}
Bertail, P., Chautru, E., and Cl[é]{}men[ç]{}on, S. (2017). Empirical processes in survey sampling with (conditional) poisson designs. , 44(1):97–111.
Bickel, P. J. and Freedman, D. A. (1984). Asymptotic normality and the bootstrap in stratified sampling. , pages 470–482.
Boistard, H., Lopuha[ä]{}, H. P., Ruiz-Gazen, A., et al. (2012). Approximation of rejective sampling inclusion probabilities and application to high order correlations. , 6:1967–1983.
Boistard, H., Lopuha[ä]{}, H. P., Ruiz-Gazen, A., et al. (2017). Functional central limit theorems for single-stage sampling designs. , 45(4):1728–1758.
Br[ä]{}nd[é]{}n, P. and Jonasson, J. (2012). Negative dependence in sampling. , 39(4):830–838.
Breidt, F. J. and Opsomer, J. D. (2000). Local polynomial regresssion estimators in survey sampling. , 28(4):1026–1053.
Breidt, F. J. and Opsomer, J. D. (2008). Endogenous post-stratification in surveys: classifying with a sample-fitted model. , 36(1):403–427.
Breidt, F. J., Opsomer, J. D., and Sanchez-Borrego, I. (2016). Nonparametric variance estimation under fine stratification: An alternative to collapsed strata. , 111(514):822–833.
Chauvet, G. (2012). On a characterization of ordered pivotal sampling. , 18(4):1320–1340.
Chauvet, G. (2015). Coupling methods for multistage sampling. , 43(6):2484–2506.
Chen, J. and Rao, J. N. K. Asymptotic normality under two-phase sampling designs. , 17(3):1047–1064.
Cochran, W. G. (1977). . John Wiley & Sons, New York-London-Sydney, third edition. Wiley Series in Probability and Mathematical Statistics.
Conti, P. L. (2014). On the estimation of the distribution function of a finite population under high entropy sampling designs, with applications. , 76(2):234–259.
Deville, J.-C. and Till[é]{}, Y. (1998). Unequal probability sampling without replacement through a splitting method. , 85(1):89–101.
Dupakova, J. (1975). A note on rejective sampling. .
Fuller, W. A. (2011). , volume 560. John Wiley & Sons.
H[á]{}jek, J. (1964). Asymptotic theory of rejective sampling with varying probabilities from a finite population. , 35:1491–1523.
Isaki, C. T. and Fuller, W. A. (1982). Survey design under the regression superpopulation model. , 77(377):89–96.
Kim, J. K., Park, S., and Lee, Y. (2017). Statistical inference using generalized linear mixed models under informative cluster sampling. , 45(4):479–497.
Krewski, D. and Rao, J. N. K. (1981). Inference from stratified samples: properties of the linearization, jackknife and balanced repeated replication methods. , 9(5):1010–1019.
Ohlsson, E. (1986). Asymptotic normality of the [R]{}ao-[H]{}artley-[C]{}ochran estimator: an application of the martingale [CLT]{}. , 13(1):17–28.
Ohlsson, E. (1989). Asymptotic normality for two-stage sampling from a finite population. , 81(3):341–352.
Pr[á]{}[š]{}kov[á]{}, Z. and Sen, P. K. (2009). Asymptotics in finite population sampling. In [*Handbook of Statistics*]{}, volume 29, pages 489–522. Elsevier.
Qualit[é]{}, L. (2008). A comparison of conditional poisson sampling versus unequal probability sampling with replacement. , 138(5):1428–1432.
Rao, J. N. K., Hartley, H. O., and Cochran, W. G. (1962). On a simple procedure of unequal probability sampling without replacement. , 24:482–491.
Robinson, P. (1982). On the convergence of the horvitz-thompson estimator. , 24(2):234–238.
Ros[é]{}n, B. (1972). Asymptotic theory for successive sampling with varying probabilities without replacement. [I]{}, [II]{}. , 43:373–397; ibid. 43 (1972), 748–776.
Sampford, M. (1967). On sampling without replacement with unequal probabilities of selection. , 54(3-4):499–513.
S[ä]{}rndal, C.-E., Swensson, B., and Wretman, J. (1992). . Springer Series in Statistics.
Till[é]{}, Y. (2006). . Springer, New York.
van Der Hofstad, R. (2016). . Cambridge series in statistical and probabilistic mathematics.
|
---
abstract: 'We argue that the intractable part of the measurement problem—the ‘big’ measurement problem—is a pseudo-problem that depends for its legitimacy on the acceptance of two dogmas. The first dogma is John Bell’s assertion that measurement should never be introduced as a primitive process in a fundamental mechanical theory like classical or quantum mechanics, but should always be open to a complete analysis, in principle, of how the individual outcomes come about dynamically. The second dogma is the view that the quantum state has an ontological significance analogous to the significance of the classical state as the ‘truthmaker’ for propositions about the occurrence and non-occurrence of events, i.e., that the quantum state is a representation of physical reality. We show how both dogmas can be rejected in a realist information-theoretic interpretation of quantum mechanics as an alternative to the Everett interpretation. The Everettian, too, regards the ‘big’ measurement problem as a pseudo-problem, because the Everettian rejects the assumption that measurements have definite outcomes, in the sense that one particular outcome, as opposed to other possible outcomes, actually occurs in a quantum measurement process. By contrast with the Everettians, we accept that measurements have definite outcomes. By contrast with the Bohmians and the GRW ‘collapse’ theorists who add structure to the theory and propose dynamical solutions to the ‘big’ measurement problem, we take the problem to arise from the failure to see the significance of Hilbert space as a new *kinematic* framework for the physics of an indeterministic universe, in the sense that Hilbert space imposes kinematic (i.e., pre-dynamic) objective probabilistic constraints on correlations between events.'
author:
- |
**Jeffrey Bub**\
[Department of Philosophy, University of Maryland, College Park, MD 20742[^1] ]{}\
**Itamar Pitowsky**\
[Department of Philosophy, The Hebrew University, Jerusalem, Israel[^2]]{}
bibliography:
- 'Jeff.bib'
title: '**Two Dogmas About Quantum Mechanics**'
---
Oxford Everett
==============
The salient difference between classical and quantum mechanics is the noncommutativity of the operators representing the physical magnitudes (‘observables’) of a quantum mechanical system—or, equivalently, the transition from a classical event space, represented by the Boolean algebra of (Borel) subsets of a phase space, to a non-Boolean quantum event space represented by the projective geometry of closed subspaces of a Hilbert space, which form an infinite collection of intertwined Boolean algebras, each Boolean algebra corresponding to a resolution of the identity: a partition of the Hilbert space representing a family of mutually exclusive and collectively exhaustive events.
Probabilities in quantum mechanics are, as von Neumann put it [@Neumann1954 p. 245], ‘uniquely given from the start’ as a nonclassical relation between events represented by the angles between the 1-dimensional subspaces representing atomic (elementary) events in the projective geometry of subspaces of Hilbert space. If $e$ and $f$ are atomic events, the ‘transition probability’ (Born probability) between the events is: $$\mbox{prob}(e,f) = |\langle e|f\rangle|^{2} = |\langle f|e\rangle|^{2} = \cos^{2}\theta_{ef}$$ The transition probability can be expressed as: $$\mbox{prob}_{e}(f) = {\mbox{$\mathrm{Tr}$}(P_{e}P_{f})}$$ where $P_{e}$ and $P_{f}$ are the projection operators onto the 1-dimensional subspaces representing the events $e$ and $f$, respectively. Uniqueness is shown by Gleason’s theorem [@Gleason]:[^3] in a Hilbert space ${\mbox{$\mathcal{H}$}}$ of dimension greater than 2, if $\sum_{i}\mbox{prob}(f_{i}) = 1$ for the atomic events $f_{i}$ in each Boolean algebra generated by a partition of the Hilbert space into orthogonal 1-dimensional subspaces, then the probabilities of events $f$ represented by subspaces of ${\mbox{$\mathcal{H}$}}$ are uniquely represented as: $$\mbox{prob}_{\rho}(f) = {\mbox{$\mathrm{Tr}$}(\rho P_{f})}$$ where $P_{f}$ is the projection operator onto the subspace representing the event $f$ and $\rho$ is a density operator representing a pure state ($\rho = P_{e}$, for some atomic event $e$) or a mixed state ($\rho = \sum_{i}w_{i}P_{e_{i}}$).
It is assumed that the assignment of probabilities satisfies a condition that Barnum et al [@BCFFS2000] call ‘the noncontextuality of probability,’ that the probability assigned to an event $f$ depends only on $f$ and is independent of the Boolean algebra to which the event belongs. Note that if ‘$f$ in context 1’ and ‘$f$ in context 2’ represented two distinct events, we could not represent the structure of quantum events as the projective geometry of subspaces of a Hilbert space: we would have to enlarge the structure.
The question is: what do these ‘transition probabilities’ or ‘transition weights’ mean? The probabilities are probabilities of—*what*? Evidently, $|\langle e|f\rangle|^{2}$ does not represent the probability of a spontaneous transition from an event $e$ to the event $f$. The textbook answer is that $|\langle e|f\rangle|^{2}$ represents the probability, for a system in the state ${| e \rangle}$ in which the event $e$ has probability 1, of finding the event $f$ in a measurement of an observable of the system, where the set of possible outcomes of the measurement generates a Boolean algebra, representing a partition of the Hilbert space containing the event $f$ (but note, not the event $e$).
The textbook answer by itself, without adding anything more to the story of how these events are supposed to come about in a measurement process, is adequate only if we are content with an instrumentalist interpretation of the theory. Why? The structure of the quantum event space determines the kinematic part of quantum theory. This includes the association of Hermitian operators with observables, the Born probabilities, the von Neumann-Lüders conditionalization rule, and the unitarity constraint on the dynamics, which is related to the event structure via a theorem of Wigner [@Wigner1959],[@Uhlhorn1963]. The transition from the state ${| e \rangle}$, in which the event $e$ has probability 1, to the state ${| f \rangle}$, in which the event $f$ has probability 1, with probability $|\langle e|f\rangle|^{2}$ in a measurement process is a non-unitary stochastic transition that is not described by the unitary dynamics. Since the probability of the event $e$ was 1 before the measurement and is now, in the state ${| f \rangle}$ after the occurrence of the measurement outcome $f$, less than 1, there is a loss of information on measurement or—as Bohr put it—an ‘irreducible and uncontrollable’ measurement disturbance. Without a dynamical explanation of this measurement disturbance, or an analysis of what is involved in a quantum measurement process that addresses the issue (including, possibly, rejecting the ‘eigenvalue-eigenstate rule’—the association of the outcome event $f$ with the state ${| f \rangle}$—as in Bohm’s theory or modal interpretations), the theory qualifies as an algorithm for predicting the probabilities of measurement outcomes, but cannot be regarded as providing a realist account, in principle, of how events come about in a measurement process.
This is the measurement problem. Proposed solutions to the problem, such as Bohm’s ‘hidden variable’ theory [@GoldsteinSEP] or the GRW ‘dynamical collapse’ theory [@GhirardiSEP], add structure to the theory: particle trajectories in the case of Bohm’s theory or a non-unitary stochastic dynamics for the ‘primitive ontology’ of the GRW theory: mass density in the GRWm version, or ‘flashes’ in the GRWf version (see, e.g., /cite[Allori2007]{}). The Everett interpretation purports to solve the problem without adding any new structural elements to quantum mechanics.
The central claims of the Everett interpretation in the ‘Oxford’ version developed by Deutsch [@Deutsch1999], Saunders [@Saunders1995; @Saunders1998; @Saunders2004], Wallace [@Wallace2003a; @Wallace2003b; @Wallace2003c; @Wallace2005], Greaves and others [@Greaves2004; @Greaves2007a; @Greaves2007b] can be outlined as follows:
Ontology
: At the most fundamental level, what there *is* is described by the quantum state of the universe—so whatever is true or false is determined by the quantum state as the ‘truthmaker’ for propositions about the occurrence and non-occurrence of events.
Branching
: A family of effectively non-interfering or decoherent histories of coarse-grained events associated with relatively stable systems at the macrolevel emerges through the dynamical process of decoherence, as a consequence of the Hamiltonian that characterizes the dynamical evolution of the universal quantum state. With respect to the coarse-grained basis selected by decoherence, the quantum state decomposes into a linear superposition that can be interpreted as describing an emergent branching structure of non-interfering quasi-classical histories or ‘worlds,’ identified with the familiar classical macroworlds of our experience, weighted by the Born probabilities. The alternative outcomes of a quantum measurement process are associated with different branches in the decomposition of the quantum state with respect to the decoherence basis. There is no fact of the matter as to the number of branches: the history space is a quasi-classical probability space that is inherently vaguely defined (appropriately so, given the vague specification of macro-configurations). The coarse-graining of the event space can be refined or coarsened to a certain extent without compromising effective decoherence, and the decoherence basis can be unitarily transformed (e.g., rotated) over a certain range of transformations without compromising decoherence.
Uncertainty/Caring
: There is a sense in which a rational agent on a branch, faced with subsequent branching, can be uncertain about the future (i.e., uncertain about ‘which branch the agent will subsequently occupy’). Such an agent can have rational credences (degrees of belief that satisfy the axioms of probability theory) about the outcomes of quantum measurements, even though all outcomes occur on different branches. Alternatively, even without uncertainty, an agent faced with multiple futures will care about what happens on a branch, and so will have a ‘caring measure’ for decision-making that quantifies the extent of caring for different branches and satisfies the axioms of probability theory.
Probability
: To achieve a realist interpretation of quantum mechanics that solves the measurement problem, it suffices to postulate that an agent’s credence function or caring measure conforms to the objective quantum mechanical weights of the different branches. In fact, it is possible to prove that this must be so, given standard rationality constraints on an agent’s preferences, and a measurement neutrality assumption: that a rational agent is indifferent between two quantum wagers that agree on the quantum state, the observable measured, and the payoff function on the outcomes, i.e., the agent is indifferent between alternative measurement procedures; alternatively, the result follows from a related equivalence assumption: that a rational agent assigns equal credences to events that are assigned equal quantum weights. These additional assumptions can be justified as rationality constraints, but only on the Everett interpretation, in which all possible measurement outcomes occur, relative to different branches.
The Everettian aims to show that standard quantum mechanics can be understood as a complete theory in a realist sense—that the measurement problem does not reduce the theory to an instrument for the probabilistic prediction of measurement outcomes. The basic problem for the Everettian is to ‘save the appearances,’ given the radical difference between our experience of a stable macroworld and the ontological assumption. The dynamics of decoherence yields an emergent weighted branching structure of quasi-classical histories at the macrolevel. So what has to be explained is how uncertainty or caring makes sense when all alternatives occur relative to different branches, and how the quantum weights—which are a feature of the quantum state, i.e., the ontology—are associated with the credence function or caring measure of rational agents. The measurement problem is the problem of explaining the apparently ‘irreducible and uncontrollable disturbance’ in a quantum measurement process, the ‘collapse’ of the wave function described by von Neumann’s projection postulate. The Everettian’s solution is to show how appearances can be saved by denying that there is any such disturbance, on the basis that no definite outcome is selected in a measurement—all outcomes are selected relative to different branches, according to the quantum theory. The appearance of disturbance on a single branch is a reflection of how the quantum weights are distributed in the emergent process of branching, and if we either assume or prove that our credence function or caring measure should conform to these weights, then we have an explanation for the appearance of disturbance in a realist interpretation of quantum mechanics as a complete dynamical theory.
Of course, everything hinges on whether the different components of the intepretation can be established satisfactorily, and there is now an extensive literature challenging and defending these claims, especially **Uncertainty/Caring** and **Probability**. Here we simply list these components[^4] and note that the claim is that the Everett interpretation solves the measurement problem on the basis of (i) the weighted branching structure of quasi-classical histories that emerges through the dynamical process of decoherence, (ii) an argument that rational agents can be uncertain or care differently about different futures in a branching universe, and (iii) the proposal that the credence function or caring measure of rational agents should conform to the weights of the branches. For the Everettian, the icing on the cake is that the interpretation yields a derivation of Lewis’s Principal Principle: the identification of an objective feature of the world—the quantum weights—with the credence function or caring measure of rational agents, and hence the interpretation of the quantum weights as objective chances. But the cake itself, so to speak, is independent of this additional feature. (See Wallace [@Wallace2005].)
In a previous publication [@Pitowsky07], one of us characterized debates about the foundations of quantum mechanics in terms of two assumptions or dogmas, and distinguished two measurement problems: a ‘big’ measurement problem and a ‘small’ measurement problem. The first dogma is Bell’s assertion (defended in [@Bellmeas]) that measurement should never be introduced as a primitive process in a fundamental mechanical theory like classical or quantum mechanics, but should always be open to a complete analysis, in principle, of how the individual outcomes come about dynamically. The second dogma is the view that the quantum state has an ontological significance analogous to the ontological significance of the classical state as the ‘truthmaker’ for propositions about the occurrence and non-occurrence of events, i.e., that the quantum state is a representation of physical reality. The ‘big’ measurement problem is the problem of explaining how measurements can have definite outcomes, given the unitary dynamics of the theory: it is the problem of explaining *how individual measurement outcomes come about dynamically.* The ‘small’ measurement problem is the problem of accounting for our familiar experience of a classical or Boolean macroworld, given the non-Boolean character of the underlying quantum event space: it is the problem of explaining the *dynamical emergence of an effectively classical probability space of macroscopic measurement outcomes* in a quantum measurement process.
The ‘big’ measurement problem depends for its legitimacy on the acceptance of the two dogmas. We argue below that both dogmas should be rejected, and that the ‘big’ measurement problem is a pseudo-problem. In a sense, the Everettian, too, regards the ‘big’ measurement problem as a pseudo-problem, because the Everettian rejects the assumption that measurements have definite outcomes, in the sense that one particular outcome, as opposed to other possible outcomes, actually occurs in a quantum measurement process. By contrast with the Everettians, we accept that measurements have definite outcomes. By contrast with the Bohmians and the GRW ‘collapse’ theorists who add structure to the theory and propose dynamical solutions to the ‘big’ measurement problem, we take the problem to arise from the failure to see the significance of Hilbert space as a new *kinematic* framework for the physics of an indeterministic universe, in the sense that Hilbert space imposes kinematic (i.e., pre-dynamic) objective probabilistic constraints on correlations between events. By ‘predynamic’ here, we refer to generic features of quantum systems, independent of the details of the dynamics (see Jannsen [@Jannsen2007a] for a similar kinematic-dynamic distinction in the context of special relativity). The ‘small’ measurement problem is resolved by considering the dynamics of the measurement process and the role of decoherence in the emergence of an effectively classical probability space of macro-events to which the Born probabilities refer (alternatively, by considering certain combinatorial features of the probabilistic structure: see Pitowsky [@Pitowsky07 §4.3]).
In the following section, we list the essential features of the proposed information-theoretic interpretation, somewhat more extensively than our brief sketch of the Everett interpretation. Further discussion follows in a subsequent Commentary.
An Information-Theoretic Interpretation of Quantum Mechanics
============================================================
The elements of the information-theoretic interpretation we propose[^5] can be set out as follows:
‘No Cloning’
: The empirical discovery underlying the transition from classical to quantum mechanics is the discovery that chance set-ups behave differently than we thought they did. More precisely: there are information sources that cannot be broadcast—there is no universal cloning machine capable of copying the outputs of an arbitrary information source.
Kinematics
: Hilbert space as a projective geometry (i.e., the subspace structure of Hilbert space) represents a non-Boolean event space, in which there are built-in, structural probabilistic constraints on correlations between events (associated with the angles between events)—just as in special relativity the geometry of Minkowski space-time represents spatio-temporal constraints on events. Certain principles characterizing physical processes motivate the choice of Hilbert space as the representation space for the correlational structure of events, just as Einstein’s principle of special relativity and the light postulate motivate the choice of Minkowski space-time as the representation space for the spatio-temporal structure of events. In the case of quantum mechanics, these principles are information-theoretic and include a ‘no signaling’ principle and a ‘no cloning’ principle. The structure of Hilbert space imposes kinematic (i.e., pre-dynamic) objective probabilistic constraints on events to which a quantum dynamics of matter and fields is required to conform, through its symmetries, just as the structure of Minkowski space-time imposes kinematic constraints on events to which a relativistic dynamics is required to conform. In this sense *Hilbert space provides the kinematic framework for the physics of an indeterministic universe*, just as Minkowski space-time provides the kinematic framework for the physics of a non-Newtonian, relativistic universe. There is no deeper explanation for the quantum phenomena of interference and entanglement than that provided by the structure of Hilbert space, just as there is no deeper explanation for the relativistic phenomena of Lorentz contraction and time dilation than that provided by the structure of Minkowski space-time.
Dynamics
: The unitary quantum dynamics evolves the whole structure of events with probabilistic correlations in Hilbert space (in the Heisenberg picture), not the evolution from one configuration of the universe to another, i.e., not the evolution from one actual co-occurrence of events to a subsequent actual co-occurrence of events. This means that there can be a real change in the correlations between events at the microlevel without a change in the occurrence of events at the macrolevel (as in the evolution of a quantum system through the unitary gates of a quantum computer, prior to the final measurement).
Probability
: By Gleason’s theorem, there is a unique assignment of credences conforming to the structural probabilistic constraints (the objective chances) of Hilbert space (see Pitowsky [@PitowskyBetting]). These credences are encoded in the quantum state. So the quantum state is a credence function.
Information Loss
: The salient principle marking the transition from classical to nonclassical theories of information is the ‘no cloning’ principle: there is no universal cloning machine capable of copying the outputs of an arbitrary information source.[^6] This principle entails a loss of information in a measurement process—an ‘irreducible and uncontrollable disturbance’—*irrespective of how the measurement process is implemented dynamically*. The loss of information is to be understood, ultimately, as a kinematic effect of the nonclassical quantum event space, just as Lorentz contraction is, ultimately, a kinematic effect in special relativity.
Completeness
: Conditionalizing on a measurement outcome leads to a nonclassical updating of the credence function represented by the quantum state via the von Neumann-Lüders rule, which expresses the information loss on measurement. This updating is consistent with a dynamical account of the correlations between micro and macro-events in a quantum measurement process. The Hamiltonians characterizing the interactions between microsystems and macrosystems, and the interactions between macrosystems and their environment, are such that certain relatively stable structures of events associated with the familiar macrosystems of our experience emerge at the macrolevel, forming an effectively classical probability space. This amounts to a consistency proof that, say, a Stern-Gerlach spin-measuring device or a bubble chamber behaves dynamically according to the kinematic constraints represented by the projective geometry of Hilbert space, as these constraints manifest themselves at the macrolevel. Such a consistency proof demonstrates the completeness of quantum mechanics. Given the ‘no cloning’ principle underlying the kinematics of Hilbert space, there is no further story to be told about how individual measurement outcomes come about dynamically (assuming we don’t add structure to the theory, such as Bohmian trajectories or dynamical ‘collapses’). Similarly, the dynamical explanation of relativistic phenomena like Lorentz contraction in terms of forces, insofar as the forces are required to be Lorentz covariant, amounts to a consistency proof. There is no further story to be told about Lorentz contraction, once it is shown how to provide a dynamical account consistent with the kinematic constraints of Minkowski geometry (assuming we don’t add structure to the theory, such as the ether).
Realism
: The possibility of a dynamical analysis of measurement processes consistent with the Hilbert space kinematic constraints justifies the information-theoretic interpretation of quantum mechanics as realist and not merely a predictive instrument for updating probabilities on measurement outcomes.
Commentary
==========
On the information-theoretic interpretation, the quantum state is a credence function, a bookkeeping device for keeping track of probabilities—the universe’s objective chances—not the quantum analogue of the dynamically evolving classical state understood as the ‘truthmaker’ for propositions about the occurrence and non-occurrence of events.
Conditionalization on the occurrence of an event $a$, in the sense of a minimal revision—consistent with the subspace structure of Hilbert space—of the probabilistic information encoded in a quantum state given by a density operator $\rho$, is given by the von Neumann-Lüders rule:[^7] $$\rho \rightarrow \rho_{a} \equiv \frac{P_{a}\rho P_{a}}{\mbox{$\mathrm{Tr}$}
(P_{a} \rho P_{a})} \label{eqn:luders}$$ where $P_{a}$ is the projection operator onto the subspace representing the event $a$. That is, $\rho_{a}$ is the conditionalized density operator, conditional on the event $a$, and the normalizing factor $
\mbox{$\mathrm{Tr}$}(P_{a} \rho P_{a}) = \mbox{$\mathrm{Tr}$}(\rho P_{a})$ is the probability assigned to the event $a$ by the state $\rho$.
If we consider a pair of correlated systems, A and B, then conditionalization on an A-event, for the probabilistic information encoded in the density operator $\rho_{B}$ representing the probabilities of events at the remote system B, will always be an updating, in the sense of a refinement.
For example, suppose the system A is associated with a 3-dimensional Hilbert space $\mbox{$\mathcal{H}$}_{A}$ and the system B is associated with a 2-dimensional Hilbert space $\mbox{$\mathcal{H}$}_{B}$. Suppose the composite system AB is in an entangled state: $$\begin{aligned}
| \psi^{AB} \rangle & = & \frac{1}{\sqrt{3}}(| a_{1} \rangle| b_{1} \rangle
+ | a_{2} \rangle| c \rangle + | a_{3} \rangle| d \rangle) \notag \\
& = & \frac{1}{\sqrt{3}}(| a^{\prime}_{1} \rangle| b_{2} \rangle + |
a^{\prime}_{2} \rangle| e \rangle + | a^{\prime}_{3} \rangle| f \rangle)\end{aligned}$$ where $| a_{1} \rangle,| a_{2} \rangle,| a_{3} \rangle$ and $|
a^{\prime}_{1} \rangle,| a^{\prime}_{2} \rangle,| a^{\prime}_{3} \rangle$ are two orthonormal bases in $\mbox{$\mathcal{H}$}_{A}$ and $| b_{1}
\rangle, | b_{2} \rangle$ is an orthonormal basis in $\mbox{$\mathcal{H}$}
_{B}$. The triple $| b_{1} \rangle,| c \rangle,| d \rangle$ and the triple $
| b_{2} \rangle,| e \rangle,| f \rangle$ are nonorthogonal triples of vectors in $\mbox{$\mathcal{H}$}_{B}$.[^8]The state of B (obtained by tracing over $\mbox{$\mathcal{H}$}_{A}$) is the completely mixed state $
\rho_{B} = \frac{1}{2}I_{B}$: $$\frac{1}{3}| b_{1} \rangle\langle b_{1} | + \frac{1}{3}| c \rangle\langle c
| + \frac{1}{3}| d \rangle\langle d | = \frac{1}{3}| b_{2} \rangle\langle
b_{2} | + \frac{1}{3}| e \rangle\langle e | + \frac{1}{3}| f \rangle\langle
f | = \frac{I_{B}}{2}$$
Conditionalizing on one of the eigenvalues $a_{1},a_{2},a_{3}$ or $
a_{1}^{\prime },a_{2}^{\prime },a_{3}^{\prime }$ of an A-observable $A$ or $
A^{\prime }$ via (\[eqn:luders\]), i.e., on the occurrence of an event corresponding to $A$ taking the value $a_{i}$ or $A^{\prime }$ taking the value $a_{i}^{\prime }$ for some $i$, changes the density operator $\rho _{B}
$ of the remote system B to one of the states $|b_{1}\rangle ,|c\rangle
,|d\rangle $ or to one of the states $|b_{2}\rangle ,|e\rangle ,|f\rangle $. Since the mixed state $\rho _{B}=\frac{1}{2}I_{B}$ can be decomposed as an equal weight mixture of $|b_{1}\rangle ,|c\rangle ,|d\rangle $ and as an equal weight mixture of $|b_{2}\rangle ,|e\rangle ,|f\rangle $, the change in the state of B is an updating, in the sense of a refinement of the information about B encoded in the state $|\psi ^{AB}\rangle $, taking into account the new information $a_{i}$ or $a_{i}^{\prime }$. In fact, the mixed state $\rho _{B}=\frac{1}{2}I_{B}$ corresponds to an infinite variety of mixtures of pure states in $\mbox{$\mathcal{H}$}_{B}$ (not necessarily equal weight mixtures, of course). The effect at the remote system B of conditionalization on any event at A will always be an updating, in the sense of a refinement, with respect to one the these mixtures.[^9] This is the content of the Hughston-Jozsa-Wootters theorem [@HJW]. It is what Schrödinger called ‘remote steering’ and is the basis of quantum teleportation, quantum dense coding, and other peculiarities of quantum information, including the impossibility of unconditionally secure bit commitment (see Bub [@BubQIC] for a discussion).
The effect of conditionalization at a remote system (the system that is not directly involved in the conditionalizing event) is then consistent with a ‘no signaling’ principle: $$\begin{aligned}
\sum_{b}p(ab|AB) \equiv p(a|AB) & = & p(a|A) \\
\sum_{a}p(ab|AB) \equiv p(b|AB) & = & p(b|B) \ \end{aligned}$$ where $a$ represents a value of $A$ and $b$ represents a value of $B$. If conditionalization on the value of an A-observable changed the probabilities at a remote system B in a way that could *not* be represented as an updating in the sense of a refinement of the prior information about B expressed in terms of correlations between A-observables and B-observables (as encoded in the entangled state $| \psi^{AB} \rangle$), then conditionalization would allow instantaneous signaling between A and B. The occurrence of a particular sort of event at A—corresponding to a determinate value for the observable $A$ as opposed to a determinate value for some other observable $A^{\prime}$—would produce a detectable change in the B-probabilities, and so Alice at A could signal instantaneously to Bob at B merely by performing an $A$ -measurement and gaining a specific sort of information about A (the value of $A$ or the value of $A^{\prime}$).
The ‘no signaling’ principle is a special case of what Barnum et al [@BCFFS2000] call ‘the noncontextuality of probability,’ which can be expressed as a condition on the probabilities assigned to the eigenvalues of any two commuting observables $[X,Y] = 0$: $$\begin{aligned}
\sum_{y}p(xy|XY) \equiv p(x|XY) & = & p(x|X) \\
\sum_{x}p(xy|XY) \equiv p(y|XY) & = & p(y|Y) \ \end{aligned}$$ This formulation of the noncontextuality of probability follows from the representation of an observable in terms of its spectral measure.[^10] We obtain the ‘no signaling’ condition if we take $X = A\otimes I$ and $Y =
I \otimes B$. Note that ‘no signaling’ is not specifically a relativistic constraint on superluminal signaling. It is simply a condition imposed on the marginal probabilities of events for separated systems, requiring that the marginal probability of a B-event is independent of the particular set of mutually exclusive and collectively exhaustive events selected at A, and conversely, and this might well be considered partly constitutive of what one means by separated systems.
To preserve the ‘no signaling’ principle, quantum probabilities must also satisfy a ‘no cloning’ principle: there can be no universal cloning machine, i.e., it is impossible to construct a cloning machine that will clone the output of an arbitrary information source. More precisely, there can be no universal broadcasting machine—no device that takes a probability distribution over an event space to a new probability distribution over a product space of events, where the marginal probability distributions over each factor space is the same as the original distribution. We will continue to use the term ‘cloning’ rather than ‘broadcasting’ because it is more intuitive and more familiar, but note that we have in mind copying the *outputs* of an information source, not the information source itself (defined by the probability distribution).
Suppose a universal cloning machine were possible. Then such a device could copy any state in the orthogonal triple $| b_{1} \rangle, | c \rangle, | d
\rangle$ as well as any state in the orthogonal triple $| b_{2} \rangle, | e
\rangle, | f \rangle$. It would then be possible for Alice at A to signal to Bob at B. If Alice obtains the information given by an eigenvalue $a_{i}$ of $A$ or $a^{\prime}_{i}$ of $A^{\prime}$, and Bob inputs the system B into the cloning device $n$ times, he will obtain one of the states $| b_{1}
\rangle^{\otimes n}, | c \rangle^{\otimes n}, | d \rangle^{\otimes n}$ or one of the states $| b_{2} \rangle^{\otimes n}, | e \rangle^{\otimes n}, | f
\rangle^{\otimes n}$, depending on the nature of Alice’s information. Since these states tend to mutual orthogonality in $\otimes^{n}\mbox{$\mathcal{H_{B}}$}$ as $n \rightarrow \infty$, they are distinguishable in the limit. So, even for finite $n$, Bob would in principle be able to obtain some information instantaneously about a remote event.
More fundamentally, the existence of a universal cloning machine is inconsistent with the interpretation of Hilbert space as providing the kinematic framework for an indeterministic physics, in which probabilities (objective chances) are ‘uniquely given from the start’ by the geometry of Hilbert space. For such a device would be able to distinguish the equivalent mixtures of nonorthogonal states represented by the same density operator $\rho_{B} = \frac{1}{2}I_{B}$. If a quantum state prepared as an equal weight mixture of the states $| b_{1}
\rangle, | c \rangle, | d \rangle$ could be distinguished from a state prepared as an equal weight mixture of the states $| b_{2} \rangle, | e
\rangle, | f \rangle$, the representation of quantum states by density operators would be incomplete.
Now consider the effect of conditionalization on the state of A. The state of AB can be expressed as the biorthogonal (Schmidt) decomposition: $$| \psi^{AB} \rangle = \frac{1}{\sqrt{2}} (| g \rangle| b_{1} \rangle + | h
\rangle| b_{2} \rangle)$$ where $$\begin{aligned}
| g \rangle & = & \frac{2| a_{1} \rangle - | a_{2} \rangle -| a_{3} \rangle}{
\sqrt{6}} \\
| h \rangle & = & \frac{| a_{2} \rangle - | a_{3} \rangle}{\sqrt{2}}\end{aligned}$$ The density operator $\rho_{A}$, obtained by tracing $| \psi^{AB} \rangle$ over B, is: $$\rho_{A} = \frac{1}{2}| g \rangle\langle g | + \frac{1}{2}| h \rangle\langle
h |$$ which has support on a 2-dimensional subspace in the 3-dimensional Hilbert space $\mbox{$\mathcal{H}$}_{A}$: the plane spanned by $| g \rangle$ and $|
h \rangle$ (in fact, $\rho_{A} = \frac{1}{2}P_{A}$, where $P_{A}$ is the projection operator onto the plane). Conditionalizing on a value of $A$ or $
A^{\prime}$ yields a state that has a component outside this plane. So the state change on conditionalization cannot be interpreted as an updating of information in the sense of a refinement, i.e., as the selection of a particular alternative among a set of mutually exclusive and collectively exhaustive alternatives represented by the state $\rho_{A}$.
This is the notorious ‘irreducible and uncontrollable disturbance’ arising in the registration of new information about the occurrence of an event that underlies the measurement problem: the loss of some of the information encoded in the original state (in the above example, the probability of the A-event represented by the projection operator onto the 2-dimensional subspace $P_{A}$ is no longer 1, after the registration of the new information about the observable $A$ or $A^{\prime}$). If the registration of new information is the outcome of a measurement then, since the state change on measurement will have to be stochastic and non-unitary, it cannot be described by the deterministic dynamics of the theory, which must be unitary (for closed systems) for consistency with the Hilbert space representation of probabilities. A solution to the problem is generally understood to require amending the theory in such a way that the loss of information can be accounted for dynamically, and the quantum probabilities can be reconstructed dynamically as measurement probabilities. Then the quantum probabilities are not ‘uniquely given from the start’ as kinematic features of an appropriately represented event structure, i.e., they do not arise kinematically but are derived dynamically, as artifacts of the measurement process or of decoherence. Even on the Everett interpretation, where Hilbert space is interpreted as the representation space for a new sort of ontological entity, represented by the quantum state, and no definite outcome out of a range of alternative outcomes is selected in a quantum measurement process (so no explanation is required for such an event), probabilities arise as a feature of the branching structure that emerges in the dynamical process of decoherence.
From the perspective of the information-theoretic interpretation, the ‘disturbance’ involved in conditionalization is a kinematic phenomenon associated with the non-Boolean quantum event space. If there were no information loss in the conditionalization of quantum probabilities, then cloning would be possible, and equivalent mixtures associated with the same density operator would be distinguishable, in which case Hilbert space would not be an appropriate representation space for quantum events and their probabilistic correlations.[^11] In the Appendix, we show that this follows directly from the ‘no cloning’ principle for a large class of theories. We prove that in this class of theories the ‘no cloning’ principle demarcates the boundary between classical theories and theories in which measurement involves an ‘irreducible and uncontrollable disturbance’. It seems plausible, therefore, that this principle should play a central role in a derivation of the Hilbert space structure from information theory.
It is instructive here to recall Einstein’s distinction between ‘principle’ theories, like the special theory of relativity, formulated in terms of the relativity principle and the light postulate (empirical regularities raised to the level of postulates), and ‘constructive’ theories, like Lorentz’s theory, formulated in terms of a rich ontology of objects like particles, fields, and the ether. Einstein compared thermodynamics as a principle theory (‘no perpetual motion machines of the first and and second kind’) to the kinetic theory of gases as a constructive theory (where the mechanical and thermal behavior of a gas is reduced to the motion of molecules, modeled as little billiard balls). He proposed special relativity as a kinematic replacement for Lorentz’s dynamical interpretation of what we now refer to as Lorentz covariance, which he saw as unsatisfactory, not as a rival theory of matter and radiation. One might say that what eventually replaced Lorentz’s theory was relativistic quantum theory. From this perspective, Minkowski space-time is the constructive theory corresponding to Einstein’s principle theory formulation of special relativity: it is a component of the kinematic part of the constructive theory of the constitution of matter provided by relativistic quantum theory. (See Janssen for an account along these lines.)
In an article entitled ‘How to Teach Special Relativity’ [@BellRelativity], John Bell considers the following puzzle: Three identical spaceships, $A, B$, and $C$, are at rest relative to one other, drifting freely far from other matter without rotation, with $A$ equidistant from $B$ and $C$. The spaceships $B$ and $C$ are connected by a fragile thread, which is just long enough to span the distance between them. On reception of a signal from $A$, the spaceships $B$ and $C$ start their engines and accelerate gently. Since $B$ and $C $ are assumed to be identical, with identical acceleration programs, they will have the same velocity and so remain separated by the same distance relative to $A$. When $B$ and $C$ reach a certain velocity, the thread breaks. The question is: why does the thread break? Note that the thread would not break under similar assumptions in a Newtonian universe.
The relativistic kinematic explanation goes along the following lines:
Let $F1$ be the inertial frame in which the spaceships $A, B, C$ are *initially* at rest (and $A$ remains at rest). In $F1$, the distance between $
B$ and $C$, as the spaceships begin to move and continue moving, remains the same as the initial resting distance. But the moving thread undergoes a Lorentz contraction in the direction of its motion in $F1$. The explanation, in $F1$, of why the thread breaks is just this: the thread breaks because it is contracting, and this contraction is resisted by the thread being tied to $B$ and $C$, which maintain a distance apart greater than the contraction requires. The thread will break when $B$ and $C$ reach a sufficiently high velocity in $F1$ and the prevention of the Lorentz contraction produces sufficient stress to break the thread.
Let $F2$ be the inertial frame in which $B$ and $C$ are *finally* at rest again, after their engines have been shut off. From the perspective of $
F2$, there is a different explanation for the thread breaking. In $F2$, the two spaceships $B$ and $C$ are decelerating, and eventually come to rest. However, they are not decelerating at the same rate (they would be if $B$ and $C$ were connected by a rigid rod). It is this difference in deceleration that is responsible for the stress in the thread, which eventually causes the thread to break.
To clarify further, one might consider two additional spaceships, $E$ and $F$, identical to $B$ and $C$, with identical acceleration programs, initially at rest in $F1$ (before $B$ and $C$ start their engines), with $E$ adjacent to $B$, and $F$ adjacent to $C$. Suppose $E$ and $F$ are connected rigidly, so that $EF$ behaves like a rigid rod with the two spaceships as endpoints, initially at rest in $F1$. Suppose also that $EF$ starts accelerating at the same time as $B$ and $C$ in $F1$, and that the rod connecting $E$ and $F$ is strong enough to remain rigid under the acceleration. Bell’s characterization of the setup requires that, in $F1$, the distance between $
B $ and $C$, as the spaceships begin to move and continue moving, remains the same as the initial resting distance. So, in $F1$, this distance will become greater than the distance between $E$ and $F$, once the spaceships start moving, since $EF$ will suffer a Lorentz contraction in the direction of its motion. In the explanation in frame $F1$, the thread breaks because it is contracting by as much as $EF$ contracts. In the explanation in frame $
F2$, $B$ and $C$ are not decelerating at the same rate—rather, the endpoints of $EF$ are decelerating at the same rate—and this difference in deceleration, relative to the deceleration of $EF$, is responsible for the stress in the thread, which eventually causes it to break.
The explanations are frame-dependent, insofar as they involve elements that are frame-dependent notions in special relativity. However, the increasing stress in the thread that causes it to break, and the fact that the thread breaks when the stress exceeds the tensile strength of the thread, are frame-independent features common to all explanations. What Bell pointed out was that one ought to be able to provide an explanation for the thread breaking in terms of an explicit calculation of the forces involved, and the tensile strength of the thread. He suggests that such a dynamical explanation is a deeper or at least more informative explanation than the kinematic explanation. Harvey Brown’s book *Physical Relativity* [@BrownBook] develops this theme.
In Bell’s spaceship example, the dynamical explanation for the thread breaking in terms of forces, insofar as the forces are Lorentz covariant, shows the possibility of a dynamics consistent with the kinematics of special relativity. The only factor relevant to the thread breaking is the Lorentz contraction, a feature of the geometry of Minkowski space-time which is quite independent of the material constitution of the thread and the nature of the specific interactions involved. Given Einstein’s two principles, there is no deeper explanation for the thread breaking than the kinematic explanation provided by the structure of Minkowski space-time. [^12] The demonstration that a dynamical explanation yields the same result as the kinematic explanation sketched above amounts to a consistency proof that a relativistic dynamics—a dynamics that conforms to the structure of Minkowski space-time—is possible.
If we take special relativity as a template for the analysis of quantum conditionalization and the associated measurement problem,[^13] the information-theoretic view of quantum probabilities as ‘uniquely given from the start’ by the structure of Hilbert space as a kinematic framework for an indeterministic physics is the proposal to interpret Hilbert space as a constructive theory of information-theoretic structure or probabilistic structure, part of the kinematics of a full constructive theory of the constitution of matter, where the corresponding principle theory includes information-theoretic constraints such as ‘no signaling’ and ‘no cloning.’[^14] Lorentz contraction is a physically real phenomenon explained relativistically as a kinematic effect of motion in a non-Newtonian space-time structure. Analogously, the change arising in quantum conditionalization that involves a real loss of information is explained quantum mechanically as a kinematic effect of *any* process of gaining information of the relevant sort in the non-Boolean probability structure of Hilbert space (irrespective of the dynamical processes involved in the measurement process). Given ‘no cloning’ as a fundamental principle, there can be no deeper explanation for the information loss on conditionalization than that provided by the structure of Hilbert space as a probability theory or information theory. The definite occurrence of a particular event is constrained by the kinematic probabilistic correlations encoded in the structure of Hilbert space, and only by these correlations—it is otherwise ‘free.’
The Born weights are probabilities in a purely formal sense unless they are related to experience by some explicitly formulated principle. The cash value of the ‘transition probability’ $|\langle e|f\rangle|^{2}$ is that $|\langle e|f\rangle|^{2}$ represents the probability, in the state $| e \rangle$, of finding the outcome corresponding to the state $| f \rangle$ in a measurement of an observable of which $| f \rangle$ is an eigenstate. But if quantum mechanics is more than an instrument for predicting the probabilities of measurement outcomes, it must be possible, in principle, to locate structures that represent macroscopic measuring instruments and recording devices in Hilbert space, where the dynamical behavior of such structures is consistent with the kinematic information-theoretic (probabilistic) principles encoded in the structure of Hilbert space.
In special relativity one has a consistency proof that a dynamical account of relativistic phenomena in terms of forces, like the breaking of the thread in Bell’s spaceship example, is consistent with the kinematic account in terms of the structure of Minkowski space-time. An analogous consistency proof for quantum mechanics would be a dynamical explanation for the effective emergence of classicality, i.e., Booleanity, at the macrolevel, because it is with respect to the Boolean algebra of the macroworld that the Born weights of quantum mechanics have empirical cash value.
In classical mechanics, taking a Laplacian view, one can consider the phase space of the entire universe, in principle. The classical state, represented by a point in phase space that evolves dynamically, defines a 2-valued homomorphism on the Boolean algebra of (Borel) subsets of phase space, distinguishing events that occur at a particular time from events that don’t occur. In this sense, the classical state is the ‘truthmaker’ for propositions about the occurrence or non-occurrenc of events, for all possible events.
Similarly, in quantum mechanics one can consider the Hilbert space of the entire universe, in principle. This is a space of possible events, with a certain kinematic structure of probabilistic correlations between events, represented by the subspace structure or projective geometry of the space (different from the classical correlational structure represented by the subset structure of phase space). On the usual view, the quantum analogue of the classical state is a pure state represented by a ray or 1-dimensional subspace in Hilbert space. There is, of course, no 2-valued homomorphism on the non-Boolean algebra of subspaces of Hilbert space, but a pure state can be taken as distinguishing events that occur at a particular time (events represented by subspaces containing the state, and assigned probability 1 by the state) from events that don’t occur (events represented by subspaces orthogonal to the state, and assigned probability 0 by the state). This leaves all remaining events represented by subspaces that neither contain the state nor are orthogonal to the state (i.e., events assigned a probability $p$ by the state, where $0 < p < 1$) in limbo: neither occurring nor not occurring. The measurement problem then arises as the problem of accounting for the fact that an event that neither occurs not does not occur when the system is in a given quantum state can somehow occur when the system undergoes a measurement interaction with a macroscopic measurement device—giving measurement a very special status in the theory. Once the pure state is taken as the analogue of the classical state in this sense, the only way out of this problem, without adding structure to the theory, is the Everettian manoeuvre.
On the information-theoretic interpretation, the quantum state is a derived entity, a credence function that assigns probabilities to events in alternative Boolean algebras associated with the outcomes of alternative measurement outcomes. The measurement outcomes are macro-events in a particular Boolean algebra, and the macro-events that actually occur, corresponding to a particular measurement outcome, define a 2-valued homomorphism on this Boolean algebra. What has to be shown is how this occurrence of events in a particular Boolean algebra is consistent with the quantum dynamics.
It is a contingent feature of the dynamics of our particular quantum universe that events represented by subspaces of Hilbert space have a tensor product structure that reflects the division of the universe into microsystems (e.g., atomic nuclei), macrosystems (e.g., macroscopic measurement devices constructed from pieces of metal and other hardware), and the environment (e.g., air molecules, electromagnetic radiation). The Hamiltonians characterizing the interactions between microsystems and macrosystems, and the interactions between macrosystems and their environment, are such that a certain relative structural stability emerges at the macrolevel as the tensor-product structure of events in Hilbert space evolves under the unitary dynamics. Symbolically, an event represented by a 1-dimensional projection operator like $P_{{| \psi \rangle}} = {| \psi \rangle}{\langle \psi |}$, where $${| \psi \rangle} = {| s \rangle}{| M \rangle}{| \varepsilon \rangle}$$ and $s, M, \varepsilon$ represent respectively microsystem, macrosystem, and environment, evolves under the dynamics to $P_{{| \psi(t) \rangle}}$, where $${| \psi(t) \rangle} = \sum_{k}c_{k}{| s_{k} \rangle}{| M_{k} \rangle}{| \varepsilon_{k}(t) \rangle}, \label{eq:correlation}$$ and $${| \varepsilon_{k}(t) \rangle} = \sum_{\nu}\gamma_{\nu}e^{-ig_{k\nu}t}{| e_{\nu} \rangle}$$ if the interaction Hamiltonian $H_{M\varepsilon} $ between a macrosystem and the environment takes the form $$H_{M\varepsilon} = \sum_{k\gamma}g_{k\nu}{| M_{k} \rangle}{\langle M_{k} |}\otimes{| e_{\nu} \rangle}{\langle e_{\nu} |}$$ with the ${| M_{k} \rangle}$ and the ${| e_{k} \rangle}$ orthogonal. That is, the ‘pointer’ observable $\sum_{k}m_{k}{| M_{k} \rangle}{\langle M_{k} |}$ commutes with $H_{M\varepsilon}$ and so is a constant of the motion induced by the Hamiltonian $H_{M\varepsilon}$.
Here $P_{{| M_{k} \rangle}}$ can be taken as representing, in principle, a configuration of the entire macroworld, and $P_{{| s_{k} \rangle}}$ a configuration of all the micro-events correlated with macro-events. The dynamics preserves the correlation represented by the superposition $\sum_{k}c_{k}{| s_{k} \rangle}{| M_{k} \rangle}{| \varepsilon_{k}(t) \rangle}$ between micro-events, macro-events, and the environment for the macro-events $P_{{| M_{k} \rangle}}$, even for nonorthogonal ${| s_{k} \rangle}$ and ${| \varepsilon_{k} \rangle}$, but not for macro-events $P_{{| M'_{l} \rangle}}$ where the ${| M'_{l} \rangle}$ are linear superpositions of the ${| M_{k} \rangle}$. Since the tri-decomposition $\sum_{k}c_{k}{| s_{k} \rangle}{| M_{k} \rangle}{| \varepsilon_{k}(t) \rangle}$ is unique (unlike the biorthogonal Schmidt decomposition; see Elby and Bub [@ElbyBub]), a correlation of the form ${| s \rangle}{| M \rangle}{| \varepsilon \rangle}$ evolves to a linear superposition in which the macro-events $P_{{| M'_{l} \rangle}}$ become correlated with entangled system-environment events represented by subspaces (rays) spanned by linear superpositions of the form $\sum_{k}c_{k}d_{lk}{| s_{k} \rangle}{| \varepsilon_{k}(t) \rangle}$. (See Zurek [@Zurek2005 p. 052105-14].)
It is characteristic of the dynamics that correlations represented by (\[eq:correlation\]) evolve to similar correlations (similar in the sense of preserving the micro-macro-environment division), and the macro-events represented by $P_{{| M_{k} \rangle}}$, at a sufficient level of coarse-graining, can be associated with structures at the macrolevel—the familiar macro-objects of our experience—that remain relatively stable under the dynamical evolution. So a Boolean algebra ${\mbox{$\mathcal{B_{M}}$}}$ of macro-events $P_{{| M_{k} \rangle}}$ correlated with micro-events $P_{{| s_{k} \rangle}}$ in (\[eq:correlation\]) is emergent in the dynamics. Note that the emergent Boolean algebra is not the same Boolean algebra from moment to moment, because the correlation between micro-events and macro-events changes under the dynamical evolution induced by the micro-macro interaction (e.g., corresponding to different measurement interactions). What remains relatively stable under the dynamical evolution are the *macrosystems* associated with macro-events in correlations of the form (\[eq:correlation\]), even under a certain vagueness in the coarse-graining associated with these macro-events: macrosystems like grains of sand, tables and chairs, macroscopic measurement devices, cats and people, galaxies, etc.
It is further characteristic of the dynamics that the environmental events represented by $P_{{| \varepsilon_{k}(t) \rangle}}$ very rapidly approach orthogonality, i.e., the ‘decoherence factor’ $$\zeta_{kk'} = \langle\varepsilon_{k}|\varepsilon_{k'}\rangle = \sum_{\nu}|\gamma_{\nu}|^{2}e^{i(g_{k'\nu}-g_{k\nu})t}$$ becomes negligibly small almost instantaneously. When the environmental events $P_{{| \varepsilon_{k}(t) \rangle}}$ correlated with the macro-events $P_{{| M_{k} \rangle}}$ are effectively orthogonal, the reduced density operator is effectively diagonal in the ‘pointer’ basis ${| M_{k} \rangle}$ and there is effectively no interference between elements of the emergent Boolean algebra ${\mbox{$\mathcal{B_{M}}$}}$. That is, the conditional probabilities of events associated with a subsequent emergent Boolean algebra (a subsequent measurement) are additive on ${\mbox{$\mathcal{B_{M}}$}}$. (See Zurek [@Zurek2005 p. 052105-14], [@Zurek2003a].)
The Born probabilities are probabilities of events in the emergent Boolean algebra, i.e., the Born probabilities are probabilities of ‘pointer’ positions, the coarse-grained basis selected by the dynamics. Applying quantum mechanics kinematically, say in assigning probabilities to the possible outcomes of a measurement of some observable of a microsystem, we consider the Hilbert space of the relevant degrees of freedom of the microsystem and treat the measuring instrument as simply selecting a Boolean subalgebra in the non-Boolean event space of the microsystem to which the Born probabilities apply. In principle, we can include the measuring instrument in a dynamical analysis of the measurement process, but such a dynamical analysis—even though complete in terms of the quantum dynamics—does not provide a dynamical explanation of how individual outcomes come about. In such a dynamical analysis, the Born probabilities are probabilities of the occurrence of events in an emergent Boolean algebra. The information loss on conditionalization relative to classical conditionalization is a kinematic feature of the the structure of quantum events, not accounted for by the unitary quantum dynamics, which conforms to the kinematic structure. This is analogous to the situation in special relativity, where Lorentz contraction is a kinematic effect of relative motion that is *consistent* with a dynamical account in terms of Lorentz covariant forces, but is not explained in Einstein’s theory—by contrast with Lorentz’s theory—as a dynamical effect in a Newtonian space-time structure, in which this sort of contraction does not arise as a purely kinematic effect. That is, the dynamical explanation of Lorentz contraction in special relativity involves forces that are Lorentz covariant—in effect, the dynamics is assumed to have symmetries that respect Lorentz contraction as a kinematic effect of relative motion. In quantum mechanics, the possibility of a dynamical analysis of the measurement process conforming to the kinematic structure of Hilbert space provides a consistency proof that the familiar objects of our macroworld behave dynamically in accordance with the kinematic probabilistic constraints on correlations between events.
A physical theory of an indeterministic universe is primarily a theory of probability (or information). Probabilities are defined over an event structure, which in the quantum case is a family of Boolean algebras forming a particular sort of non-Boolean algebra. On the information-theoretic interpretation, no assumption is made about the fundamental ‘stuff’ of the universe. So, one might ask, what do tigers supervene on?[^15] In the case of Bohm’s theory or the GRW theory, the answer is relatively straightforward: tigers supervene on particle configurations in the case of Bohm’s theory, and on mass density or ‘flashes’ in the case of the GRW theory, depending on whether one adopts the GRWm version or the GRWf version. In the Everett interpretation, tigers supervene on features of the quantum state, which describes an ontological entity. In the case of the information-theoretic interpretation, the ‘supervenience base’ is provided by the dynamical analysis: tigers supervene on events defining a 2-valued homomorphism in the emergent Boolean algebra.
It might be supposed that this involves a contradiction. What is contradictory is to suppose that a correlational event represented by $P_{{| \psi(t) \rangle}}$ actually occurs, where ${| \psi(t) \rangle}$ is a linear superposition $\sum_{k}c_{k}{| s_{k} \rangle}{| M_{k} \rangle}{| \varepsilon_{k}(t) \rangle}$, as well as an event represented by $P_{{| s_{k} \rangle}{| M_{k} \rangle}{| \varepsilon_{k}(t) \rangle}}$ for some specific $k$. We do not suppose this. On the information-theoretic interpretation we propose, there is a kinematic structure of possible correlations (but no particular atomic correlational event is selected as the ‘state’ in a sense analogous to the pure classical state), and a particular dynamics that preserves certain sorts of correlations, i.e., correlational events of the sort represented by $P_{{| \psi(t) \rangle}}$ with ${| \psi(t) \rangle} = \sum_{k}c_{k}{| s_{k} \rangle}{| M_{k} \rangle}{| \varepsilon_{k}(t) \rangle}$ evolve to correlational events of the same form. What can be identified as emergent in this dynamics is an effectively classical probability space: a Boolean algebra with atomic correlational events of the sort represented by orthogonal 1-dimensional subspaces $P_{{| s_{k} \rangle}{| M_{k} \rangle}}$, where the probabilities are generated by the reduced density operator obtained by tracing over the environment, when the correlated environmental events are effectively orthogonal.
The dynamics does not describe the (deterministic or stochastic) evolution of the 2-valued homomorphism on which tigers supervene to a new 2-valued homomorphism (as in the evolution of a classical state). Rather, the dynamics leads to the relative stability of certain event structures at the macrolevel associated with the familiar macrosystems of our experience, and to an emergent effectively classical probability space whose atomic events are correlations between events associated with these macrosystems and micro-events.
It is part of the information-theoretic interpretation that events defining a 2-valued homomorphism on the Boolean algebra of this classical probability space actually occur with the emergence of the Boolean algebra at the macrolevel. This selection of actually occurring events is only in conflict with the quantum pure state if the quantum pure state is assumed to have an ontological significance analogous to the ontological significance of the classical pure state as the ‘truthmaker’ for propositions about the occurrence and non-occurrence of events, and if the quantum pure state evolves unitarily—in particular, if it is assumed that the quantum pure state partitions all events into events that actually occur, events that do not occur, and events that neither occur nor do not occur, as on the usual interpretation. We argued that this assumption is one of the dogmas about quantum mechanics that should be rejected. Rather, we take the quantum state, pure or mixed, to represent a credence function: the credence function of a rational agent (an information-gathering entity ‘in’ the emergent Boolean algebra) who is updating probabilities on the basis of events that occur in the emergent Boolean algebra.
Concluding Remarks
==================
We have argued that the ‘big’ measurement problem is like the problem for Newtonian physics raised by relativistic effects such as length contraction and time dilation, and that the solution to both problems involves the recognition of a fundamental change in the underlying *kinematics* of our physics, represented by the transition from a Newtonian space-time to Minkowski space-time in the case of special relativity, and from the set-theoretic structure of classical phase space to the subspace structure of Hilbert space in the case of quantum mechanics. So the two assumptions, about the ontological significance of the quantum state and about the dynamical account of how measurement outcomes come about, should be rejected as unwarranted dogmas about quantum mechanics.
The solutions to the ‘big’ measurement problem provided by Bohm’s theory and the GRW theory are dynamical and involve adding structure to quantum mechanics. There is a sense in which adding structure to the theory to solve the measurement problem dynamically—insofar as the problem arises from a failure to recognize the significance of Hilbert space as the kinematic framework for the physics of an indeterministic universe—is like Lorentz’s attempt to explain relativistic length contraction dynamically, taking the Newtonian space-time structure as the underlying kinematics and invoking the ether as an additional structure for the propagation of electromagnetic effects. In this sense, Bohm’s theory and the GRW theory are ‘Lorentzian’ interpretations of quantum mechanics.
The Everettian rejects the legitimacy of the problem by simply denying that measurements have definite outcomes, i.e., by denying that the pure states in a superposition describe alternative event complexes, only one of which actually occurs. This requires showing that a *particular* decomposition of the quantum state corresponding to our experience has a preferred significance, and that weights can be assigned to the individual terms in the preferred superposition that have the significance of probabilities, even though no one definite event complex is selected as actually occurring in contrast to the other event complexes in the superposition. The Everettian’s solution to *this* problem is dynamical. So the Everettian, too, sees the underlying problem as dynamical.
We reject the legitimacy of the ‘big’ measurement problem on the basis of an information-theoretic interpretation of quantum mechanics, in terms of which the problem arises from the failure to see the significance of Hilbert space as the kinematic framework for an indeterministic physics. The dynamical analysis we provide is a solution to a consistency problem: the ‘small’ measurement problem. The analysis shows that a quantum dynamics, consistent with the kinematics of Hilbert space, suffices to underwrite the emergence of a classical probability space for the familiar macro-events of our experience, with the Born probabilities for macro-events associated with measurement outcomes derived from the quantum state as a credence function. The explanation for such nonclassical effects as the loss of information on conditionalization is not provided by the dynamics, but by the kinematics, and given ‘no cloning’ as a fundamental principle, there can be no deeper explanation. In particular, there is no dynamical explanation for the definite occurrence of a particular measurement outcome, as opposed to other possible measurement outcomes in a quantum measurement process—the occurrence is constrained by the kinematic probabilistic correlations encoded in the projective geometry of Hilbert space, and only by these correlations.
Acknowledgements
================
Jeffrey Bub acknowledges support from the National Science Foundation under Grant No. 0522398. Itamar Pitowsky’s research is supported by the Israel Science Foundation, Grant 744/07.
Appendix: The Information Loss Theorem
======================================
We show that *it follows from the ‘no cloning’ principle* that information cannot be extracted from a nonclassical source without changing the source irreversibly. (We prove this theorem for quantum information sources, but note that the proof does not depend on specific features of the Hilbert space formalism.)
We assume:
- The ‘no cloning’ principle: there is no universal cloning machine.
- Every (quantum) state $\rho$ is specified by the probabilities of the measurement outcomes of a finite, informationally complete (or ‘fiducial’) set of observables.
Assumption (2) holds for a large class of theories, including quantum and classical theories. Note that an informationally complete set is not unique. For example, in the case of a qubit, the probabilities for spin ‘up’ and spin ‘down’ in three orthogonal directions suffice to define a direction on the Bloch sphere and hence to determine the state, so the spin observables $
\sigma_{x}, \sigma_{y}, \sigma_{z}$ form an informationally complete set. (For a classical system or a classical information source, an informationally complete set is given by of a single observable, with $n$ possible outcomes, for some $n$.)
Let $\mbox{$\mathcal{F}$} = \{A,B,C,\ldots\}$ be an informationally complete set of observables represented by a finite set of Hermitian operators on an $
n$-dimensional Hilbert space $\mbox{$\mathcal{H}$}_{n}$. A quantum state $
\rho$ assigns a probability distribution to every outcome of any measurement of an obervable in $\mbox{$\mathcal{F}$}$. Measuring $A$ yields one of the outcomes $a_{1}, a_{2}, \ldots$ with a probability distribution $
P_{\rho}(a_{1}|A), P_{\rho}(a_{2}|A), \ldots$. Similarly, measuring $B$ yields one of the outcomes $b_{1}, b_{2}, \ldots$ with a probability distribution $P_{\rho}(b_{1}|A), P_{\rho}(b_{2}|A), \ldots$, and so on. If $
\mbox{$\mathcal{F}$}$ is informationally complete, the finite set of probabilities completely characterizes $\rho$ as the state on $
\mbox{$\mathcal{H}$}$.
Assuming that all measurement outcomes are independent and ignoring any algebraic relations among elements of $\mbox{$\mathcal{F}$}$, a classical probability measure on a classical (Kolmogorov) probability space can be constructed from these probabilities: $$P_{\rho}(a,b,\ldots|A,B,\ldots) = P_{\rho}(a|A)P_{\rho}(b|B)\ldots$$ (cf. the ‘trivial’ hidden variable construction of Kochen and Specker in [@KochenSpecker]). Note that the probability space is finite since $
\mbox{$\mathcal{F}$}$ is finite and $\mbox{$\mathrm{dim}$}{
\mbox{$\mathcal{H}$}} < \infty$. (The number of atoms in the probability space is at most $\mbox{$\mathrm{dim}$}{\mbox{$\mathcal{H}$}}^{|
\mbox{$\mathcal{F}$}|}$.) The quantum state $\rho$ can be reconstructed from $P_{\rho}$ (given as a classical information source, or rationally approximated in the memory of a classical computer).
We now prove:
Assumptions (1) and (2) entail that extracting information from a quantum information source given by a quantum state $\rho$, sufficient to generate the probabilities of an informationally complete set of observables, is either impossible or necessarily changes the state $\rho$ irreversibly, i.e., there must be information loss in the extraction of such information.
Step 1: begin with a quantum source in the state $\rho$ and measure $A,B,\ldots$ sufficiently many times to generate the classical probability measure $P_{\rho}$, to as good an approximation as required, without destroying $\rho$. Step 2: from $P_{\rho}$ construct a copy of $\rho$. $$\rho \stackrel{\mbox{\scriptsize measure}}{\longrightarrow} P_{\rho} \stackrel{\mbox{\scriptsize prepare}}{\longrightarrow} \rho$$
This procedure defines a universal cloning machine, which we assume to be impossible. Since Step 2 is possible by assumption (2), the ‘no cloning’ assumption (1) entails that Step 1 is blocked.
We are left with two options: either there is no way to generate $P_{\rho}$ from $\rho$ (which is the case in quantum mechanics if we have only one copy of $\rho$, or too few copies of $\rho$), or else, if we can generate $P_{\rho}$ from $\rho$, assumption (1) entails that the original ‘blueprint’ $\rho$ must have been changed irreversibly by the process of extracting the information to generate $P_{\rho}$ (if not, the change in $\rho$ could be reversed dynamically and cloning would be possible): $$\xcancel{\rho} \stackrel{\mbox{\scriptsize measure}}{\longrightarrow} P_{\rho} \stackrel{\mbox{\scriptsize prepare}}{\longrightarrow} \rho \label{eqn:dist}$$
Since we can prepare multiple copies of the state $\rho$ from $P_{\rho}$, one might think that even if the original state is destroyed in generating $
P_{\rho}$, we still end up with multiple copies of $\rho$: $$\begin{aligned}
& \xcancel{\rho} \overset{\mbox{\scriptsize measure}}{\longrightarrow} &
P_{\rho} \overset{\mbox{\scriptsize prepare}}{\longrightarrow} \rho \notag
\\
& & \hspace{.2in} \overset{\mbox{\scriptsize prepare}}{\searrow} \rho \notag
\\
& & \hspace{.35in} \vdots\end{aligned}$$ But note that to generate $P_{\rho}$, we need to begin with multiple copies of $\rho$, i.e., we need to begin with a state $\rho \otimes \rho \cdots$, so what we really have is: $$\xcancel{\rho} \otimes \xcancel{\rho} \cdots \overset{
\mbox{\scriptsize
measure}}{\longrightarrow} P_{\rho} \overset{\mbox{\scriptsize prepare}}{
\longrightarrow} \rho \otimes \rho \cdots$$ which simply re-states (\[eqn:dist\]).
No complete dynamical (i.e., unitary) account of the state transition in a measurement process is possible in quantum mechanics, in general.
Any measurement can be part of an informationally complete set, so any measurement must lead to an irreversible (hence non-unitary) change in the quantum state of the measured system.
We conclude—essentially from the ‘no cloning ’ principle—that there can be no measurement device that functions dynamically in such a way as to identify with certainty the output of an arbitrary quantum information source without altering the source irreversibly or ‘uncontrollably,’ to use Bohr’s term—no device can distinguish a given output from every other possible output by undergoing a dynamical (unitary) transformation that results in a state that represents a distinguishable record of the output, without an irreversible transformation of the source.
[^1]: *E-mail address:* jbub@umd.edu
[^2]: *E-mail address:* itamarp@vms.huji.ac.il
[^3]: For von Neumann, uniqueness is a consequence of invariance under the unitary symmetries of the projective lattice representing events.
[^4]: For a critique of **Probability** by one of us, see Hemmo and Pitowsky [@HemmoPitowsky2007].
[^5]: For related views, see Demopoulos [@Demopoulos2008], Pitowsky [@PitowskyBetting; @Pitowsky07].
[^6]: More precisely, there is no universal broadcasting machine. See below.
[^7]: See [@BubProjPost] for a discussion.
[^8]: The vectors in each triple are separated by an angle $2\pi/3$. For a precise specification of these vectors, see Bub [@Bub2007].
[^9]: Fuchs makes a similar point in [@FuchsInfo4].
[^10]: Barnum et al formulate noncontextuality as the requirement that the probability assigned to an event $e$ depends only on $e$ and is independent of the other events in each mutually exclusive and collectively exhaustive set of events $\{e_{i}\}$ containing $e$, i.e., that the probability of an event is independent of the Boolean subalgebra to which the event belongs.
[^11]: For the Everettian, there is the appearance of measurement disturbance on each branch, or rather, on ‘most’ branches, because there will always be some branches on which it appears that there is no measurement disturbance—and on these branches it will appear that cloning is possible.
[^12]: Harvey Brown’s book [@BrownBook] presents an extended argument for the contrary view.
[^13]: See Brown and Timpson [@BrownTimpson2007] for a contrary view.
[^14]: While the ‘no cloning’ principle demarcates classical from non-classical theories, we require some further principle or principles to recover Hilbert space and exclude ‘superquantum’ theories for which the correlation of entangled states violates the Tsirelson bound for quantum states, while conforming to the ‘no signaling’ constraint. See Barnum *et al* [@BBLW2006; @BBLW2007].
[^15]: We thank Allen Stairs for raising the realism question in this form.
|
---
abstract: 'This paper presents a study on data dissemination in unstructured Peer-to-Peer (P2P) network overlays. The absence of a structure in unstructured overlays eases the network management, at the cost of non-optimal mechanisms to spread messages in the network. Thus, dissemination schemes must be employed that allow covering a large portion of the network with a high probability (e.g. gossip based approaches). We identify principal metrics, provide a theoretical model and perform the assessment evaluation using a high performance simulator that is based on a parallel and distributed architecture. A main point of this study is that our simulation model considers implementation technical details, such as the use of caching and Time To Live (TTL) in message dissemination, that are usually neglected in simulations, due to the additional overhead they cause. Outcomes confirm that these technical details have an important influence on the performance of dissemination schemes and that the studied schemes are quite effective to spread information in P2P overlay networks, whatever their topology. Moreover, the practical usage of such dissemination mechanisms requires a fine tuning of many parameters, the choice between different network topologies and the assessment of behaviors such as free riding. All this can be done only using efficient simulation tools to support both the network design phase and, in some cases, at runtime.'
address: 'Department of Computer Science and Engineering. University of Bologna, Italy.'
author:
- 'Gabriele D’Angelo'
- Stefano Ferretti
bibliography:
- 'paper.bib'
title: Highly intensive data dissemination in complex networks
---
Data dissemination ,Simulation ,Complex Networks ,Performance Evaluation
Acronyms {#acronyms .unnumbered}
========
=LUNES Large Unstructured NEtwork Simulator\
FP Fixed Probability (dissemination protocol)\
PB Probabilistic Broadcast (dissemination protocol)\
DDF1 Degree Dependent Function 1 (dissemination protocol)\
DDF2 Degree Dependent Function 2 (dissemination protocol)\
|
---
abstract: 'Protein fibril accumulation at interfaces is an important step in many physiological processes and neurodegenerative diseases as well as in designing materials. Here we show, using $\beta$-lactoglobulin fibrils as a model, that semiflexible fibrils exposed to a surface do not possess the Gaussian distribution of curvatures characteristic for wormlike chains, but instead exhibit a spontaneous curvature, which can even lead to ring-like conformations. The long-lived presence of such rings is confirmed by atomic force microscopy, cryogenic scanning electron microscopy and passive probe particle tracking at air- and oil-water interfaces. We reason that this spontaneous curvature is governed by structural characteristics on the molecular level and is to be expected when a chiral and polar fibril is placed in an inhomogeneous environment such as an interface. By testing $\beta$-lactoglobulin fibrils with varying average thicknesses, we conclude that fibril thickness plays a determining role in the propensity to form rings.'
author:
- Sophia Jordens
- 'Emily E. Riley'
- Ivan Usov
- Lucio Isa
- 'Peter D. Olmsted'
- Raffaele Mezzenga
title: Adsorption at Liquid Interfaces Induces Amyloid Fibril Bending and Ring Formation
---
[{width="50.00000%"}]{}
This document is the unedited Author’s version of a Submitted Work that was subsequently accepted for publication in ACS ©American Chemical Society after peer review. To access the final edited and published work see <http://pubs.acs.org/doi/abs/10.1021/nn504249x>.
Introduction
============
Polymers exposed to an unfavorable environment can collapse or change shape in order to minimize surface energy [@deGennes; @Doi; @Pereira]. Examples of unfavorable environments include a poor solvent or a hydrophilic-hydrophobic interface like the one between water and either air or oil. Examples of conformations driven by such energy minimization are rings, loops, coils, spools, tori/toroids, hairpins or tennis rackets [@Cohenmorph]. In filaments comprising aggregated proteins or peptides, ring formation falls into two main classes: fully annealed rings occasionally observed as intermediate states during protein fibrillation, like in apolipoprotein C-II [@Hatters] and A$\beta_{1-42}$ [@Mustata]; or ring formation in actively driven systems, where the energy required for filament bending is provided by GTP or ATP [@Paez; @Tang; @Kabir; @Sumino]. Insulin has been shown to form open-ring shaped fibrils when pressure was applied during fibrillation [@Jansen], which was explained by an anisotropic distribution of void volumes in fibrils and therefore aggregation into bent fibrils.
We study amyloid fibrils, which are linear supramolecular assemblies of proteins/peptides that, despite a large diversity in possible peptide sequences, show remarkable structural homogeneity. Peptides form $\beta$-sheets that stack, often with chiral registry, to form a filament whose main axis is perpendicular to the $\beta$-strands [@Dobson; @Eichner]. Fully formed fibrils can consist of one or, more commonly, multiple filaments, assembled into twisted ribbons with a twist pitch determined by the number of filaments in the fibrils [@Adamcik]. Their high aspect ratio (diameter usually less than $10$ nm, total contour length up to several $\mu$m) leads to liquid crystalline phases in both three (3D) [@Jung] and two dimensions (2D) [@IsaSM; @Jordens]. Amyloid fibrils were initially studied due to their involvement in many different degenerative diseases such as diabetes II or Parkinson’s disease [@DobsonNature]. However, protein fibrils have recently experienced a surge of interest in potential applications in materials [@Mankar], and functional roles have been identified in biological processes such as hormone storage [@Riek], emphasizing the importance of understanding their structure and properties in 2D.
Here, we present experimental evidence for the development of *curved* fibrils at interfaces. Semiflexible $\beta$-lactoglobulin fibrils are found to undergo a shape change and passively form open rings upon adsorption to an interface (liquid-liquid or liquid-air). We show that this cannot be described by a simple bending modulus; this bending can instead be understood in terms of a *spontaneous curvature* induced on symmetry grounds by the chiral and polar nature of the fibril, when interacting with the heterogeneous environment provided by an interface. A comparison of different fibril batches of the same protein shows that the probability of forming rings depends on the average fibril thickness, with batches of thicker fibrils not forming loops. These results imply that flexible non-symmetric bodies embedded in heterogeneous media such as the physiological environment can be expected to deform, bend, and twist, depending on the specific surface interaction with the environment. For example, concentration gradients of ions or pH could enhance shape changes necessary for locomotion in flexible nanoswimmers [@Sengupta; @Keaveny], or be used to promote or control self-assembly through shape changes. One could even envision high surface to volume materials such as bicontinuous phases with large length scales being used to process large amounts of flexible shape changers.
Results and discussion
======================
Morphology
----------
[{width="90.00000%"}]{}
When imaging the air-water interfacial fibril layer by AFM using a modified Langmuir-Schaefer horizontal transfer technique (see Materials and Methods) to resolve 2D liquid crystallinity, we found that, in addition to nematic and isotropic fibril domains [@Jordens], some $\beta$-lactoglobulin fibrils were present in circular conformations. These rings appear at the lowest interfacial density investigated, where fibril alignment is still negligible [@Jordens], and persist in the presence of nematic fibril domains up to high densities \[see Supplementary Note 1, Supplementary Fig. S1 and S2\]. Ring diameters range from $0.5-2$ $\mu\text{m}$ (Fig. \[fig:AFM\] and \[fig:PTSEM\]), and are consistent whether observed *via AFM* at the air-water interface, cryogenic Scanning Electron Microscopy (cryo-SEM) or passive probe particle tracking at the oil-water interface, confirming that fibrils have a similar tendency to bend at air-liquid and liquid-liquid interfaces. A small selection of the vast variety of ring morphologies is presented in Fig. \[fig:AFM\]. Highly complex structures involving several fibrils are quite common (Fig. \[fig:AFM\]a, b, S1 and S2), whereas relatively few distinct rings or tennis rackets comprise a single fibril and can rather be thought to be intermediate assembly states $en route$ to final ring structures (Fig. \[fig:AFM\]c and d) [@Schnurr]. Short fibrils, which could be the result of fracture due to the bending strain, exposure to air or inhomogeneous strong surface tension, also assemble into rings (Fig. S3). Alternatively, short fibrils frequently accumulate within an outer ring and align either along the circumference of this ring or parallel to each other in the center, with minimal contact with the ring itself (Fig. \[fig:AFM\]b and e).
[{width="100.00000%"}]{}
The long-lived presence, and hence inferred stability, of these self-organized conformations was confirmed by passive probe particle tracking experiments performed at the oil-water interface, where fluorescently-labelled spherical tracer particles (diameter $\approx{774}$ nm) were observed to move in near-perfect circles or sickle-shaped trajectories over the course of three to four minutes. A simple pathway for ring formation could be the presence of nano- or microbubbles at the liquid surface, which give fibrils the opportunity to bend around their circumference [@Martel]. This would, however, also lead to a distortion of the peptide layer (see Materials and Methods) at the interface; once the sample has dried, the bubble would have disappeared but still be visible in AFM images as a height discontinuity through the ‘bubble’. The absence of such observations in AFM (Fig. S4), or of bubbles (cavities) in the cryo-SEM images (Fig. \[fig:PTSEM\]), indicates that there is an inherent predisposition of the fibrils to bend, which then leads to circle formation upon interaction with a liquid surface.
Fibril Free Energy
------------------
Understanding these data requires a study of how surface effects influence the shape of fibrils (or indeed filaments). We consider an inextensible fibril of length $L$, represented as a twisted ribbon with chiral wavelength $\lambda$ and pitch angle $\theta_p=\cot^{-1}(2\pi R/\lambda)$, where $R$ is the inscribing radius of the twisted ribbon (see Supplementary Note 2). We parametrize the shape by ${\mathbf{\hat{{t}}}}(s)$, the direction parallel to the central axis of the ribbon, or equivalently the tangent vector of the fibril. The ribbon twists around its axis ${\mathbf{\hat{{t}}}}(s)$ by the angle $\phi(s)$. We will parametrize the bending in terms of the angular rate of deflection $\dot{\boldsymbol{\Theta}}={\mathbf{\hat{{t}}}}\times\dot{\hat{\textbf{t}}}$, where $\mathbf{\kappa}(s)=d{\mathbf{\hat{{t}}}}/ds\equiv\dot{\hat{\textbf{t}}}$ is the local curvature. Hence, $\dot{\boldsymbol{\Theta}}=\kappa {\mathbf{\hat{{n}}}}$, where ${\mathbf{\hat{{n}}}}$ is the axis about which the tangent vector is deflected during a bend. For a fibril confined to bend on a surface, we take ${\mathbf{\hat{{n}}}}$ to be outward surface normal vector (pointing *into* the liquid), so that $\kappa$ can be either positive or negative. The free energy is given by [@MarkSigg94b] $$\begin{aligned}
G_{\textrm{fib}}&=\int_{0}^{L}ds
\left\{\frac{B}{2}{\dot{\Theta}}^{2} +
\frac{C}{2}\left(\dot{\boldsymbol{\phi}} - {\mathbf{q}}\right)^{2}
+ {\mathbf{D}}\cdot\dot{\hat{\textbf{t}}}\times(\dot{\boldsymbol{\phi}}
- \mathbf{q})\right\} \label{FullBulk}\\
&=\int_{0}^{L}ds \left\{\frac{B}{2}\,\kappa^2 +\ldots\right\} . \end{aligned}$$ The first term penalizes bending, and $B$ is the bending modulus. The second term penalizes twist relative to the native helical twist, which is parametrized by the chiral wavenumber $q=2\pi/\lambda$. Here, $C$ is the twist modulus. The vector ${\mathbf{D}}$ represents the twist-bend couplings allowed by a polar fibril with a non-symmetric local cross section [@MarkSigg94b]. In this work we will focus on the bend degrees of freedom, since in filaments with free ends, such as those considered here, the twist degrees of freedom will relax to accomodate any imposed bend.
A polar twisted fibril has an anisotropy that distinguishes ‘head’ from ‘tail’ directions along the fibril axis; in F-actin this ‘polarization’ arises from the orientations required of G-actin monomers to effect self-assembly [@Howard2001]; in an $\alpha$-helix the N-C polymerisation breaks the polar symmetry and in cross-$\beta$ amyloid fibrils such as those studied here the polarity is due to the molecular packing of $\beta$-sheets [@rogers2006investigating; @fitzpatrick2013atomic; @cohen2013proliferation]. The polarity is reflected in variations in molecular structure along the exposed surface of the twisted ribbon. When this structure is placed in a heterogenous environment, as occurs near a solid surface or when immersed within a meniscus between two fluids (or fluid and gas), the inhomogeneity of the environment generally leads to unbalanced torques on the body (see Supplementary Note 2, Fig. S5 and S6 for details), even when local forces have balanced to place the fibril at the interface. A non-symmetric body, such as a chiral and polar fibril, can thus experience an effective spontaneous curvature [@isambert1995bending].
To demonstrate this effect, we consider a fibril adsorbed *onto* a planar surface with which it interacts, rather than immersed *within* a meniscus. The effects are qualitatively the same, but the details are easier to understand in the adsorbed case. The surface and the adsorbed ribbon interact *via* numerous molecular interactions [@israelachvili]. Although in principle *all* atoms in the fibril interact with every point on the surface due to Coulomb interactions, screening limits the interaction to only the adsorbing surface. Long-range dispersion interactions are also irrelevant for fibrils that are induced to bend or twist within the plane, since the change in this energy will be negligible. Hence, we consider the following surface free energy $$\begin{aligned}
G_{\textrm{surf}}&=\frac{2L}{\lambda} \int_{S} \left[\bar{\gamma} + \delta\gamma(\mathbf{r})\right]d^2r, \end{aligned}$$ where $\lambda$ is the twist pitch or wavelength, the average surface energy $\bar{\gamma}$ controls adsorption, and $S$ is the contact area of a the ribbon, which occurs every half wavelength. The asymmetry $\delta\gamma(\mathbf{r})$ reflects the polar nature of the interaction and can vary from repulsive to attractive along the repeat patch. A polar moment (with dimensions of energy) of the interaction can be defined by $${\mathbf{P}}=\frac{2}{\lambda}\int_{S} \,{\mathbf{r}}\,\delta\gamma({\mathbf{r}})\,d^2r,$$ where $S$ is the area of the patch where the fibril contacts the surface. The polar moment ${\mathbf{P}}$ is determined by the nature of the interaction with the surface, and is thus not an intrinsic property of the fibril alone. Fig. \[fig:surfacecartoon\] shows an example in which the surface patch is a parallelogram with length $\ell$ and width $\omega$. For a simple surface potential $\delta\gamma({\mathbf{r}})=\varepsilon (x\cos\Phi+y\sin\Phi)$, where the coordinate $x$ is parallel to the fibril axis coordinate $s$, the polar moment (see Supplementary Note 2) has magnitude $P=\alpha(\theta_p,\Phi)\omega^3\ell\varepsilon/\lambda$. Here, $\alpha(\theta_p,\Phi)$ is a geometric prefactor whose sign depends on the polarization and chirality, and parametrizes the degree to which the symmetric parellelogram is deformed into a non-symmetric shape to favor one sign of surface ‘charge’.
[![Twisted ribbon against a surface. ([**a**]{}) After horizontal transfer of the interfacial fibril layer, the AFM tip probes the fibrils from the side that was originally pointing towards the water phase. ([**b, c**]{}) The contact area as seen through the interface from the air-side is a parallelogram ([**b**]{}), which deforms asymmetrically when the fibril is bent ([**c**]{}). This leads to a greater contact area by one ’charge’ (indicated by color) of the polar interaction, which implies a preference for one sign of bend and thus a spontaneous curvature. The example shown is that of a bend that decreases the contact energy. The symmetry breaking of the polar region upon bending has been amplified for visualisation purposes.[]{data-label="fig:surfacecartoon"}](./fig3.jpg "fig:"){width="50.00000%"}]{}
When the twisted ribbon is bent the ribbon-surface contact area changes shape, so that either the repulsive or attractive part of the polar interaction has more contact with the surface, depending on the sign of the bend (Fig. \[fig:surfacecartoon\]c). This leads to a spontaneous curvature. The contribution of bending to the overall interaction energy can then be written as a chiral coupling between the bending rate $\dot{\boldsymbol{\Theta}}$ and the polar moment ${\mathbf{P}}$: $$\begin{aligned}
G_{\textrm{surf}} &= \int_0^{L}\!\!ds\left\{ -{A}\,\dot{\boldsymbol{\Theta}}\cdot{\mathbf{\hat{{t}}}}\times
{\mathbf{P}} + \ldots\right\} \label{eq:surf} \\
&= \int_0^{L}\!\!ds\left\{-{A}\,\kappa\,{\mathbf{\hat{{n}}}} \cdot {\mathbf{\hat{{t}}}}\times{\mathbf{P}}
+ \ldots\right\} . \label{eq:surf2} \end{aligned}$$ The vector product is the simplest term which has no mirror symmetry, and is thus appropriate for a chiral filament. Moreover, under $s\rightarrow -s$ both $\kappa$ and ${\mathbf{t}}$ change sign, whereas ${\mathbf{P}}$ does not, so that the free energy is also reparametrization-invariant. The dimensionless geometric factor ${A}$ and the moment ${\mathbf{P}}$ depend on the details of the surface free energy $\delta\gamma({\mathbf{r}})$ interaction potential $U$, the contact area shape, and its deformation under bending. The polar moment ${\mathbf{P}}$ depends on the surface normal vector through its vector nature and the details of the surface-fibril interaction. The ellipses indicate other terms induced by the surface, such as contributions to the bend-twist or curvature moduli, or a spontaneous twist. We choose the convention that the surface normal vector ${\mathbf{\hat{{n}}}}$ points *away* from the surface and thus into the fibril. An example free energy $G_{\textrm{fib,P}}$ is calculated in the Supplementary Information for a simple model contact potential.
The curvature in Eq. \[eq:surf2\] carries a sign: for $\kappa>0$ the fibril bends in a right-handed sense around the surface normal vector ${\mathbf{\hat{{n}}}}$, while for $\kappa<0$ the fibril bends in a left-handed sense. The process of transferring the surface layer for AFM observation orients the surface normal towards the AFM observer, so that observation is from the liquid side towards the air side (Fig. \[fig:surfacecartoon\]). Consider a polarization such that ${\mathbf{P}}\cdot{\mathbf{\hat{{y}}}}=\sin\Phi$, where $\Phi=+\pi/4$, and choose ${\mathbf{\hat{{t}}}}\parallel{\mathbf{\hat{{x}}}}$ (as observed in the AFM image; see Fig. \[fig:surfacecartoon\]), where ${\mathbf{\hat{{x}}}}\times{\mathbf{\hat{{y}}}}={\mathbf{\hat{{n}}}}$. This implies ${\mathbf{\hat{{n}}}}\cdot{{\mathbf{\hat{{t}}}}}\times{\mathbf{\hat{{P}}}}>0$. Consider a bend as shown in Fig. \[fig:surfacecartoon\], in which $\dot{\boldsymbol{\Theta}}\cdot{\mathbf{\hat{{n}}}}=\kappa$, where $\kappa<0$. In Supplementary Note 2 we find ${A}>0$, so that this bend ($\kappa<0$) increases the energy, and thus $\kappa>0$ is favored. Similarly, for the opposite sign of ${\mathbf{\hat{{t}}}}\times{\mathbf{P}}$ a negative curvature $\kappa<0$ is favored.
The competition between the surface energy (Eq. \[eq:surf2\]) and the ordinary fibril bending energy (Eq. \[FullBulk\]) leads, by minimization, to a spontaneous curvature $\kappa_0$ given by (see SI) $$\kappa_0= \frac{{A}}{B}\, {\mathbf{\hat{{n}}}}\cdot{\mathbf{\hat{{t}}}}\times{\mathbf{P}}.
\label{eq:c0}$$ This is equal to $\varepsilon \omega^3 \ell\,\alpha(\theta_p,\Phi)/B\,\lambda\sin^2\theta_p$ for the simple surface potential $\delta\gamma({\mathbf{r}})=\varepsilon (x\cos\Phi+y\sin\Phi)$. Isambert and Maggs [@isambert1995bending] articulated how a surface can induce spontaneous curvature in a polar and chiral filament. They proposed a phenomenological free energy with an explicit spontaneous curvature that depends on the twist angle, and a surface interaction that breaks polar symmetry. Hence, they have actually introduced a spontaneous curvature ‘by hand’. Conversely, we present a model in which a polar surface interaction is itself chiral by virtue of the local chirality of the filament, and this gives rise to an effective spontaneous curvature as a result of total energy minimization. Therefore, the functional form of the resulting spontaneous curvature differs from that proposed in Ref. \[30\].
Enhanced curvature is expected for amyloid fibrils with fewer filaments (as confirmed in Fig. \[fig:thickness\]), which will have smaller bending moduli $B$, or for fibrils with larger polar moments $P $ and thus stronger surface interactions. In addition, the specific details of the surface deformation encapsulated in the function $\alpha(\theta_p,\Phi)$ play an important role: fibrils for which the deformation leads to a more symmetric contact area will have a stronger geometric factor and thus a greater expected spontaneous curvature.
Non-Gaussian Curvature Distributions
------------------------------------
Consider a segment of arc length $ds$ of a wormlike chain (WLC). The probability ${\cal P}(\kappa)$ of finding this segment curved with curvature $\kappa=1/R_\kappa$, where $R_\kappa$ is the radius of curvature, is governed by the bending modulus and should be Gaussianly distributed, ${\cal P}(\kappa)\sim \exp\left\{-ds\,\ell_p \kappa^2/2\right\}$, where $\ell_p=B/{k_{\scriptscriptstyle\rm B}T}$ is the persistence length. Deviations from the WLC model can be quite common, as with toroidal DNA [@Noy; @Seaton], in which the nucleic acids have a smaller persistence length at short length scales [@Noy]. The presence of rings in our system suggests a characteristic intrinsic curvature or length scale, in addition to the usual $\ell_p$. For quantitative analysis, we have extracted the $xy$ coordinates of fibrils from images acquired at low interfacial fibril densities after short adsorption times, where interactions and contact between fibrils are still minimal, and calculated ${\cal
P}(\kappa)$ (Fig. \[fig:curvature\]; see Materials and Methods). Any rings present on the image were excluded from the analysis, since their closed topology would introduce an additional constraint. To benchmark this approach, we first generate conformations based on the discrete WLC model with the $\ell_p$ obtained from the 2D mean squared end-to-end distance of fibrils at the air-water interface [@Rivetti]. These conformations are used to create artificial images of WLC polymers with the same resolution as the AFM images and then subjected to the same tracking algorithm used for analyzing the real fibril image. Fig. \[fig:curvature\] shows the normalized probability distribution of curvatures ${\cal P}(\kappa)$/${\cal
P}_{max}(\kappa)$ for both the original WLCs and the corresponding tracked conformations (see Methods). In the tracked conformations the distribution shifts towards lower curvatures: this change is due to finite image resolution (Fig. \[fig:curvature\]a). Importantly, however, both distributions are Gaussian. In contrast, and as expected from the theoretical considerations put forth above, the normalized ${\cal
P}(\kappa)$ for real fibrils adsorbed at the air-water interface can indeed not be fitted with a single Gaussian distribution function but has a pronounced fat tail instead.
[![ Fibrils exhibit a spontaneous curvature when adsorbed to a surface. Upper panel: zoomed in images of $\beta$-lactoglobulin fibrils ([**a**]{}) at the air-water interface after $t=10$ minutes of adsorption from a $c_{\text{init}}=0.001\%$ w/w fibril suspension and ([**b**]{}) deposited onto mica for $2$ minutes from the bulk with $c_{\text{init}}=0.1\%$ w/w. Lower panel: probability distributions of normalized absolute local curvatures $\kappa$ extracted from the full ([**a**]{}) $30\times30$ $\mu\text{m}$ (see Appendix for full image) and ([**b**]{}) $5\times5$ $\mu\text{m}$ images (green diamonds) with a $ds$ of $24$ and $9.8$ nm, respectively. The curvature distribution of simulated WLCs generated using all relevant parameters from the corresponding AFM image (see Methods) is shown as purple crosses and is successfully fitted with a Gaussian probability distribution function (purple line). Tracking these WLCs results in a change in the probability densities (blue crosses) but the values are still Gaussianly distributed (blue line). Plotting the normalized probabilities in logarithmic scale as a function of $\kappa^2$ clearly shows fat tails and thus the presence of spontaneous curvature only in real fibrils (Insets in the lower panel).[]{data-label="fig:curvature"}](./fig4.jpg "fig:"){width="65.00000%"}]{}
It has been argued that differences in $\kappa_0$ are to be expected depending on the strength of adsorption to the surface [@Joanicot] and on whether the polymer is in 3D or 2D [@Rappaport]. To test this, we compare the curvature distributions from fibrils adsorbed to the air-water interface and transferred horizontally to mica (Fig. \[fig:curvature\]a) to fibrils deposited onto mica from a drop of the bulk solution (Fig. \[fig:curvature\]b). The modified Langmuir-Schaefer AFM sample preparation is a 2D to 2D transfer from a liquid onto a solid surface, which is much faster (milliseconds) than the slower (seconds) 3D to 2D equilibration obtained by depositing onto a solid substrate from bulk [@Rivetti]. The bending probability of fibrils adsorbed from the bulk to mica, where no rings are observed, was also found to deviate from a typical Gaussian distribution (Fig. \[fig:curvature\]b). Fibrils hence bend as a result of their exposure to the inhomogeneous environment of solid-liquid, liquid-liquid, and gas-liquid interfaces, independently of how they initially adsorbed at these phase boundaries.
Average Fibril Thickness Determines Propensity to Bend
------------------------------------------------------
[{width="65.00000%"}]{}
As noted above, we predict a larger fibrillar diameter to imply a larger bending modulus, and hence a smaller likelihood of bending spontaneously (according to Eq.\[eq:c0\]). This was confirmed by studying fibrils from different batches of preparation, as well as from different suppliers. Fig. \[fig:thickness\] shows ratios of double- to triple-stranded fibrils for $\beta$-lactoglobulin fibrils produced from native protein obtained from three different suppliers. Non-identical distributions can be expected due to different fibril processing conditions (sample volumes, shearing and stirring histories) between batches, and/or genetic variants between suppliers [@Qin]. This then affects the individual filament thickness, and number of filaments per fibril, due to subtle differences in proteolysis. Thicker filaments, with larger bending moduli, should have much smaller spontaneous curvatures, and not be visibly curved if thick enough. Fig. \[fig:thickness\] shows the distribution of number of strands per fibril, which is proportional to thickness, as determined from the AFM images. The batch with the highest number of rings (Fig. \[fig:AFM\]) contains the largest amount of double-stranded fibrils (Fig. \[fig:thickness\]a). By contrast, for batches of fibrils formed with the same protocol but from protein obtained from a different supplier, primarily three-stranded fibrils were found, which did not assemble into rings (Fig. \[fig:thickness\]d). Both a second batch of fibrils from the first source as well as a batch from a third supplier containing a more even mix of double- and triple-stranded fibrils yielded curved conformations (Fig. \[fig:thickness\]b and c). By separating the data used to calculate the normalized distribution of ${\cal
P}(\kappa)$ presented in Fig. \[fig:curvature\]a into double- and triple-stranded fibrils (Fig. \[fig:thickness\]e), we confirm that the normalized ${\cal
P}(\kappa)$ distribution of thick fibrils has a less pronounced fat tail and these fibrils thus bend less than their thinner counterparts. A similar trend is observed for the different batches in Fig. \[fig:thickness\]a-d, where a higher fraction of thicker fibrils in the sample results in less curved structures at the air-water interface and less spontaneous curvature (Fig. \[fig:thickness\]f).
Conclusions
===========
We provide evidence from three different and independent experimental techniques for the presence of complex self-assembled amyloid fibril structures at air-water and oil-water interfaces. It has previously been reported that fibril ends are particularly reactive, as shown in the disruption of liposomes occurring preferentially at fibril ends [@Milanesi]. Their enhanced fibrillation properties as compared to the rest of the fibrils [@Tyedmers; @Carulla; @Knowles; @Xue] in addition to possible capillary interactions [@Botto] may play a role in the observed tendency of fibrils to form almost-closed rings. The genesis of these rings and loops is explained by a spontaneous curvature arising from the interaction of polar, chiral and semiflexible fibrils with an interface. Because a spontaneous curvature but no ring formation was determined in fibrils at the solid mica-liquid interface, it can be concluded that a certain degree of mobility at the interface supports the assembly of fibrils into such geometries. This is in agreement with the fact that amyloid fibrils adsorbed onto a mica surface from bulk can asymptotically reach the expected $3/4$ exponent for a self-avoiding random walk in 2D [@Lara; @Usov]. The ability of fibrils to form rings correlates with the average fibril height distribution, with loops only observed in systems where single- and double-stranded fibrils dominate. A shift in fibril height towards more triple-stranded populations reduces the number of high curvature counts and thus the amount of ring structures present. It is noteworthy, however, that a spontaneous curvature is expected also for thicker fibrils but at lower $\kappa$ because of their higher bending modulus, meaning that only thick fibrils which are long enough ($L \ge 2\pi/\kappa $) will be able to form full rings. These findings have consequences for the understanding of how fibrils deposit $in$ $vivo$, the morphology of plaques, biomechanical interactions of chiral filaments with surrounding tissues, and ultimately their effect on cells and organisms. A larger natural dynamic analogue in the form of the circular motion of polarly flagellated bacteria near solid surfaces has been described in the literature [@Lauga] and together, these results could be seen as a new approach for the controlled design, fabrication or improvement of nanoswimmers and -robots.
Experimental
============
Fibril Formation
----------------
Amyloid $\beta$-lactoglobulin fibrils were prepared according to the protocol of Jung [*et al.*]{} [@Jung]. The native, freeze-dried protein was obtained from three different sources: Davisco, Sigma, and TU Munich [@Toro-Sierra]. A $2$% w/w solution of purified and dialyzed $\beta$-lactoglobulin was stirred during 5 hours at 90 $^{\circ}$C and $p\text{H}$ $2$. The resultant fibrils were then dialyzed against $p\text{H}$ $2$ MilliQ water for 5 to 7 days to remove unconverted proteinaceous material. There is, however, evidence that even after complete removal of non-fibrillar material, the system will go back to an equilibrium point where both fibrils and ”free” peptides are present. This has been proposed for the case of A$\beta_{1-40}$ and SH3 domain fibrils [@ONuallain; @Carulla] and recently for $\beta$-lactoglobulin [@Jordens; @RuehsJRheol]. Another pathway for the accumulation of peptides may be the disaggregation of fibrils upon adsorption to the air-water interface.
Atomic Force Microscopy
-----------------------
Sample preparation and atomic force microscopy (AFM) were performed as described previously [@Jordens]. All samples contained no added salt. For the modified Langmuir-Schaefer technique, a $2$ $\mu$L aliquot of a fibril solution of desired concentration $c_{\text{init}}$ was carefully pipetted into a small glass vial and left to stand for time $t$. For a given $c_{\text{init}}$ the interfacial fibril density increases with $t$ as more fibrils adsorb to the interface. A freshly cleaved mica sheet glued to a metal support was lowered towards the liquid surface horizontally and retracted again immediately after a brief contact. The mica was then dipped into ethanol ($\ge99.8$% v/v) to remove any unadsorbed bulk material before drying the sample under a weak clean air flow. Alternatively, images of fibrils in the bulk were collected by pipetting $20$ $\mu$L of the sample onto a freshly cleaved mica. After two minutes, the mica was gently rinsed with MilliQ water and dried with pressurized air. Sample scanning in air was performed on a Nanoscope VIII Multimode Scanning Probe Microscope (Veeco Instruments) in tapping mode.
Passive Probe Particle Tracking
-------------------------------
A volume of $15$ $\mu$L of a $c_{\text{init}}=0.001\%$ w/w fibril sample seeded with $0.075\%$ w/v fluorescein isothiocyanate labelled, positively charged silica tracer particles of diameter $\approx{774}$ nm, was pipetted into an epoxy resin well on a thoroughly cleaned and plasma-treated glass coverslide. Medium chain triglycerides were poured on top so as to create a flat oil-water interface. The motion of tracers trapped at this interface was then recorded on an inverted microscope (Leica DM16000B) equipped with a $63\times1.4$ NA oil HCX PlanApo DIC objective for up to $700$ frames at a rate of $0.374$ s. Images were analysed with standard as well as custom-written software in IDL (ITT Visual Information Solutions) [@Besseling; @IsaSM; @Jordens]
Cryogenic Scanning Electron Microscopy
--------------------------------------
Samples for freeze-fracture cryogenic Scanning Electron Microscopy (FreSCa cryo-SEM [@IsaNatCommun]) were prepared by creating a flat medium chain triglycerides (MCT)-fibril solution interface in clean, small copper holders. The fibril solution contained the same concentration of fluorescent tracer particle as in passive probe particle tracking experiments and were added here for easier location of the interface during imaging. The samples were then frozen at a cooling rate of $30000$ Ks$^{-1}$ in a liquid propane jet freezer (Bal-Tec/Leica JFD 030) and fractured under high vacuum at $-140$ $^{\circ}$C (Bal-Tec/Leica BAF060). After partial freeze-drying at $-110$ $^{\circ}$C for 3 minutes to remove ice crystals and condensed water from the sample surfaces, they were coated with a $2$-nm thin layer of tungsten at $-120$ $^{\circ}$C. All samples were transferred to the precooled cryo-SEM (Zeiss Gemini 1530) under high vacuum ($\lesssim{5\times10^{-7}}$ mbar) with an air-lock shuttle. Imaging was performed at $-120$ $^{\circ}$C with a secondary electron detector.
Local Curvature Determination
-----------------------------
A home-built fibril tracking routine based on open active contours [@Rivetti] was used to extract the fibrils’ $xy$ coordinates from AFM images with a tracking step length $\Delta s\approx1$ pixel between two subsequent points along a tracked fibril. Any fibrils involved in ring formation as well as those deposited from the subphase (for example the bright ones running from top left to bottom right of the image in Fig. S7) were discarded from the analysis. The absolute local curvature $\kappa=|1/R_\kappa|$ with $R_\kappa$ being the radius of curvature between two vectors **v**$_\textbf{1}$ and **v**$_\textbf{2}$ of equal length along the fibril contour with a distance $ds$ between them, was calculated for all fibril segment pairs in the image of interest. The curvature is given by $1/R_\kappa=($**v**$_\textbf{2}(s+ds)-$**v**$_\textbf{1}(f))/|$**v**$|ds$, where we chose $ds=2\Delta s$. For a fibril penalized by only a bending energy, the probability of a curved segment is given by $${\cal P}(\kappa)={\cal N}e^{-\tfrac12 \ell_p\kappa^2 ds},$$ where ${\cal N}$ is a normalization factor, and $\ell_p=B/k_{\scriptscriptstyle B} T$ is the persisence length [@Doi]. The distribution depends on the segment length $ds$ chosen for the calculation of bending. Of course, the intrinsic persistence length is a material property and cannot depend on this discretization. Hence, the distribution of the quantity $\tilde{\kappa}=\kappa\sqrt{ds}$ is independent of the image resolution, and was used to parametrize the distribution of curvatures.
Images of WLCs were generated using the following parameters obtained from real AFM fibril images:
1. the mean and variance of the length distribution,
2. the average fibril radius,
3. the number of fibrils per image,
4. $\ell_p$ determined from the fit of the average 2D mean squared end-to-end distance $$\langle R^{2}_{2D}
\rangle=4L_c\ell_p\left[1-2\frac{\ell_p}{L_c}\left(1-e^{-L_c/2\ell_p}\right)\right],$$ where $L_c$ is the internal contour length,
5. fibril tracking step $\Delta s$,
6. and discretization $ds$.
The WLC coordinates from which the artifical images were created, were used as such for the calculation of ${\cal P}(\kappa)$. Additionally, the generated chains were tracked with the same algorithm used for real AFM images to illustrate the change in ${\cal P}(\kappa)$ due to resolution limits in the imaging and the apparently lower but purely Gaussian curvature distribution in tracked WLCs compared to untracked WLCs.
To calculate the curvature distribution for either double- or triple-stranded fibrils, the tracked fibril data set was separated into two based on a cut-off height obtained from the average fibril height histogram.
Support by the Electron Microscopy of ETH Zurich (EMEZ) is acknowledged and the authors thank A. Schofield for the silica tracers. L. Böni is thanked for his help with figure design. The authors acknowledge financial support for S.J. from ETH Zurich (ETHIIRA TH 32-1), I.U. from SNF (2-77002-11), P.D.O. from an SNSF visiting fellowship (IZK072\_141955), and L.I. from SNSF grants PP00P2\_144646/1 and PZ00P2\_142532/1.
[51]{}
deGennes, P.-G. [*Scaling Concepts in Polymer Physics*]{} Cornell University Press: Ithaca, 1979.
Doi, M.; Edwards, S. F. [*The Theory of Polymer Dynamics*]{} Clarendon Press: Oxford, 1988.
Pereira, G. G. Charged, Semi-Flexible Polymers under Incompatible Solvent Conditions. [*Curr. Appl. Phys.*]{} [**2008**]{}, [*8*]{}, 347–350.
Cohen, A. E.; Mahadevan, L. Kinks, Rings, and Rackets in Filamentous Structures. [*Proc. Natl. Acad. Sci. USA*]{} [**2003**]{}, [*100*]{}, 12141–12146.
Hatters, D. M.; MacPhee, C. E.; Lawrence, L. J.; Sawyer, W. H.; Howlett, G. J. Human Apolipoprotein C-II Forms Twisted Amyloid Ribbons and Closed Loops. [*Biochemistry*]{} [**2000**]{}, [*39*]{}, 8276–8283.
Mustata, G.-M.; Shekhawat, G. S.; Lambert, M. P.; Viola, K. L.; Velasco, P. T.; Klein, W. L.; Dravid, V. P. Insights into the Mechanism of Alzheimer’s $\beta$-Amyloid Aggregation as a Function of Concentration by Using Atomic Force Microscopy. [*Appl. Phys. Lett.*]{} [**2012**]{}, [*100*]{}, 133704–133704-4.
Paez, A.; Tarazona, P.; Mateos-Gil, P.; Vélez, M. Self-Organization of Curved Living Polymers: FtsZ Protein Filaments. [*Soft Matter*]{} [**2009**]{}, [*5*]{}, 2625–2637.
Tang, J. X.; Käs, J. A.; Shah, J. V.; Janmey, P. A. Counterion-Induced Actin Ring Formation. [*Eur. Biophys. J.*]{} [**2001**]{}, [*30*]{}, 477–484.
Kabir, A. M. R.; Wada, S.; Inoue, D; Tamura, Y.; Kajihara, T.; Mayama, H.; Sada, K.; Kakugo, A.; Gong, J. P. Formation of Ring-Shaped Assembly of Microtubules with a Narrow Size Distribution at an Air$-$Buffer Interface. [*Soft Matter*]{} [**2012**]{}, [*8*]{}, 10863–10867.
Sumino, Y.; Nagai, K. H.; Shitaka, Y.; Tanaka, D.; Yoshikawa, K.; Chaté, H.; Oiwa, K. Large-Scale Vortex Lattice Emerging from Collectively Moving Microtubules. [*Nature*]{} [**2012**]{}, [*483*]{}, 448–452.
Jansen, R.; Grudzielanek, S.; Dzwolak, W.; Winter, R. High Pressure Promotes Circularly Shaped Insulin Amyloid. [*J. Mol. Biol.*]{} [**2004**]{}, [*338*]{}, 203–206.
Dobson, C. The Structural Basis of Protein Folding and Its Links with Human Disease. [*Phil. Trans. R. Soc. Lond. B*]{} [**2001**]{}, [*356*]{}, 133–145.
Eichner, T.; Radford, S. E. A Diversity of Assembly Mechanisms of a Generic Amyloid Fold. [*Mol. Cell*]{} [**2011**]{}, [*43*]{}, 8–18.
Adamcik, J.; Jung, J.-M.; Flakowski, J.; De Los Rios, P.; Dietler, G.; Mezzenga, R. Understanding Amyloid Aggregation by Statistical Analysis of Atomic Force Microscopy Images. [*Nature Nanotech.*]{} [**2010**]{}, [*5*]{}, 423–428.
Jung, J.-M.; Mezzenga, R. Liquid Crystalline Phase Behavior of Protein Fibers in Water: Experiments [*versus*]{} Theory. [*Langmuir*]{} [**2010**]{}, [*26*]{}, 504–514.
Isa, L.; Jung, J.-M.; Mezzenga, R. Unravelling Adsorption and Alignment of Amyloid Fibrils at Interfaces by Probe Particle Tracking. [*Soft Matter*]{} [**2011**]{}, [*7*]{}, 8127–8134.
Jordens, S.; Isa, L.; Usov, I.; Mezzenga, R. Non-Equilibrium Nature of Two-Dimensional Isotropic and Nematic Coexistence in Amyloid Fibrils at Liquid Interfaces. [*Nat. Commun.*]{} [**2013**]{}, [*4*]{}, 1917–1917-8.
Dobson, C. M. Protein Folding and Misfolding. [*Nature*]{} [**2003**]{}, [*426*]{}, 884–890.
Mankar, S.; Anoop, A.; Sen, S.; Maji, S. K. Nanomaterials: Amyloids Reflect Their Brighter Side. [*Nano Rev.*]{} [**2011**]{}, [*2*]{}, 6032.
Maji, S. K.; Perrin, M. H.; Sawaya, M. R.; Jessberger, S.; Vadodaria, K.; Rissman, R. A.; Singru, P. S.; Nilsson; K. P. R.; Simon, R.; Schubert, D. [*et al.*]{} Functional Amyloids as Natural Storage of Peptide Hormones in Pituitary Secretory Granules. [*Science*]{} [**2009**]{}, [*325*]{}, 328–332.
Sengupta, S.; Ibele, M. E.; Sen, A. Fantastic Voyage: Designing Self-Powered Nanorobots. [*Angew. Chem. Int. Ed.*]{} [**2012**]{}, [*51*]{}, 8434–8445.
Keaveny, E. E., Walker, S. W.; Shelley, M. J. Optimization of Chiral Structures for Microscale Proplusion. [*Nano Lett.*]{} [**2013**]{}, [*13*]{}, 531–537.
Schnurr, B.; Gittes, F.; MacKintosh, F. C. Metastable Intermediates in the Condensation of Semiflexible Polymers. [*Phys. Rev. E*]{} [**2002**]{}, [*65*]{}, 061904-1–061904-13.
Martel, R.; Shea, H. R.; Avouris, P. Rings of Single-Walled Carbon Nanotubes. [*Nature*]{} [**1999**]{}, [*398*]{}, 299.
Marko, J. F.; Siggia, E. D. Bending and Twisting Elasticity of [D]{}[N]{}[A]{}. [*Macromolecules*]{} [**1994**]{}, [*27*]{}, 981–988.
Howard, J. [*Mechanics of Motor Proteins and the Cytoskeleton*]{} Sinauer Associates: Sunderland, 2001.
Rogers, S. S.; Venema, P.; van der Ploeg, J. P. M.; van der Linden, E.; Sagis, L. M. C.; Donald, A. M. Investigating the Permanent Electric Dipole Moment of $\beta$-Lactoglobulin Fibrils, Using Transient Electric Birefringence. [*Biopolymers*]{} [**2006**]{}, [*82*]{}, 241–252.
Fitzpatrick, A. W. P.; Debelouchina, G. T.; Bayro, M. J.; Clare, D. K.; Caporini, M. A.; Bajaj, V. S.; Jaroniec, C. P.; Wang, L.; Ladizhansky, V.; Müller, S. A. [*et al.*]{} Atomic Structure and Hierarchical Assembly of a Cross-$\beta$ Amyloid Fibril. [*Proc. Natl. Acad. Sci. USA*]{} [**2013**]{}, [*110*]{}, 5468–5473.
Cohen, S. I. A.; Linse, S.; Luheshi, L. M.; Hellstrand, E.; White, D. A.; Rajah, L.; Otzen, D. E.; Vendruscolo, M.; Dobson, C. M.; Knowles, T.P. Proliferation of Amyloid-$\beta$42 Aggregates Occurs through a Secondary Nucleation Mechanism. [*Proc. Natl. Acad. Sci. USA*]{} [**2013**]{}, [*110*]{}, 9758–9763.
Isambert, H.; Maggs, A. C. Bending of Actin Filaments. [*Europhys. Lett*]{} [**1995**]{}, [*31*]{}, 263–267.
Israelachvili, J.N. [*Intermolecular and Surface Forces*]{} Academic Press: London, 1992.
Noy, A.; Golestanian, R. Length Scale Dependence of DNA Mechanical Properties. [*Phys. Rev. Lett.*]{} [**2012**]{}, [*109*]{}, 228101–228101-5.
Seaton, D. T.; Schnabel, S.; Landau, D. P.; Bachmann, M. From Flexible to Stiff: Systematic Analysis of Structural Phases for Single Semiflexible Polymers. [*Phys. Rev. Lett.*]{} [**2013**]{}, [*110*]{}, 028103-1–028103-5.
Rivetti, C.; Guthold, M.; Bustamante, C. Scanning Force Microscopy of [D]{}[N]{}[A]{} Deposited onto Mica: Equilibration [*versus*]{} Kinetic Trapping Studied by Statistical Polymer Chain Analysis. [*J. Mol. Biol.*]{} [**1996**]{}, [*264*]{}, 919–932.
Joanicot, M.; Revet, B. DNA Conformational Studies from Electron Microscopy. I. Excluded Volume Effect and Structure Dimensionality. [*Biopolymers*]{} [**1987**]{}, [*26*]{}, 315–326.
Rappaport, S. M.; Medalion, S.; Rabin, Y. Curvature Distribution of Worm-Like Chains in Two and Three Dimensions. Preprint at $<$arXiv:0801.3183$>$ [**2008**]{}.
Qin, B. Y.; Bewley, M. C.; Creamer, L. K.; Baker, E. N.; Jameson, G. B. Functional Implications of Structural Differences between Variants A and B of Bovine $\beta$-Lactoglobulin. [*Protein Sci.*]{} [**1999**]{}, [*8*]{}, 75–83.
Milanesi, L.; Sheynis, T.; Xue, W. F.; Orlova, E. V.; Hellewell, A. L.; Jelinek, R:; Hewitt, E. W.; Radford, S. E.; Saibil, H. R. Direct Three-Dimensional Visualization of Membrane Disruption by Amyloid Fibrils. [*Proc. Natl. Acad. Sci. USA*]{} [**2012**]{}, [*109*]{}, 20455–20460.
Tyedmers, J.; Treusch, S.; Dong, J.; McCaffery, J. M.; Bevis, B.; Lindquist, S. Prion Induction Involves an Ancient System for the Sequestration of Aggregated Proteins and Heritable Changes in Prion Fragmentation. [*Proc. Natl. Acad. Sci. USA*]{} [**2010**]{}, [*107*]{}, 8633–8638.
Carulla, N.; Caddy, G. L.; Hall, D. R.; Zurdo, J.; Gairí, M.; Feliz, M.; Giralt, E.; Robinson, C. V.; Dobson, C. M. Molecular Recycling within Amyloid Fibrils. [*Nature*]{} [**2005**]{}, [*436*]{}, 554–558.
Knowles, T. P. J.; Waudby, C.A.; Devlin, G. L.; Cohen, S. I. A.; Aguzzi, A.; Vendruscolo, M.; Terentjev, E. M.; Welland, M. E.; Dobson, C. M. An Analytical Solution to the Kinetics of Breakable Filament Assembly. [*Science*]{} [**2009**]{}, [*326*]{}, 1533–1537.
Xue, W.-F.; Homans, S. W.; Radford, S. E. Systematic Analysis of Nucleation-Dependent Polymerization Reveals New Insights into the Mechanism of Amyloid Self-Assembly. [*Proc. Natl. Acad. Sci. USA*]{} [**2008**]{}, [*105*]{}, 8926–8931.
Botto, L.; Yao, L.; Leheny, R. L.; Stebe, K. J. Capillary Bond between Rod-Like Particles and the Micromechanics of Particle-Laden Interfaces. [*Soft Matter*]{} [**2012**]{}, [*8*]{}, 4971–4979.
Lara, C.; Usov, I.; Adamcik, J.; Mezzenga, R. Sub-Persistence-Length Complex Scaling Behavior in Lysozyme Amyloid Fibrils. [*Phys. Rev. Lett.*]{} [**2011**]{}, [*107*]{}, 238101-1–238101-5.
Usov, I.; Adamcik, J.; Mezzenga, R. Polymorphism Complexity and Handedness Inversion in Serum Albumin Amyloid Fibrils. [*ACS Nano*]{} [**2013**]{}, [*7*]{}, 10465–10474.
Lauga, E.; DiLuzio, W. R.; Whitesides, G. M.; Stone, H. A. Swimming in Circles: Motion of Bacteria near Solid Boundaries. [*Biophys. J.*]{} [**2006**]{}, [*90*]{}, 400–412.
Toro-Sierra, J.; Tolkach, A.; Kulozik, U. Fractionation of $\alpha$-Lactalbumin and $\beta$-Lactoglobulin from Whey Protein Isolate Using Selective Thermal Aggregation, an Optimized Membrane Separation Procedure and Resolubilization Techniques at Pilot Plant Scale. [*Food Bioprocess Tech.*]{} [**2013**]{}, [*6*]{}, 1032–1043.
O’Nuallain, B.; Shivaprasad, S.; Kheterpal, I.; Wetzel, R. Thermodynamics of A$\beta$ (1-40) Amyloid Fibril Elongation. [*Biochemistry*]{} [**2005**]{}, [*44*]{}, 12709–12718.
Rühs, P. A.; Affolter, C.; Windhab, E. J.; Fischer, P. Shear and Dilatational Linear and Nonlinear Subphase Controlled Interfacial Rheology of $\beta$-Lactoglobulin Fibrils and Their Derivatives. [*J. Rheol.*]{} [**2013**]{}, [*57*]{}, 1003–1022.
Besseling, R.; Isa, L.; Weeks, E. R.; Poon, W. C. K. Quantitative Imaging of Colloidal Flows. [*Adv. Colloid Interface Sci.*]{} [**2009**]{}, [*146*]{}, 1–17.
Isa, L.; Lucas, F.; Wepf, R.; Reimhult, E. Measuring Single-Nanoparticle Wetting Properties by Freeze-Fracture Shadow-Casting Cryo-Scanning Electron Microscopy. [*Nat. Commun.*]{} [**2011**]{}, [*2*]{}, 438–438-9.
Appendix – Supplementary Information
====================================
Persistence of Rings in the Presence of Nematic Domains
-------------------------------------------------------
Rings can be observed even at high interfacial fibril densities, where nematic domains cover most of the observed area as shown in Fig. \[Sfig1\] and \[Sfig2\]. At this point, the rings are usually composed of many fibrils or are completely filled by short fibrils. It is worth noting, however, that some regions on the same sample can be void of rings. There is a population of fibrils in all four batches investigated that is not consistent with the height and pitch distributions observed in [@Adamcik]. These fibrils are very tightly wound with a half-pitch length around $40$ nm and a maximum height between $4$ and $7$ nm and can also be seen to partake in ring formation.
Spontaneous Bending of a Polar Twisted Ribbon at an Interface
-------------------------------------------------------------
### Surface Interaction
Most particles, including proteins, adsorb to a hydrophobic-hydrophilic interface in order to reduce the nascent hydrophobic surface tension [@Pickering1907]. In addition to this, a protein will interact specifically with the two media according to the nature of the amino acids. Such interactions are both short-range (charge, hydrophobic effect, steric shapes) and long-range (dispersion interactions) [@israelachvili]. Long range interactions depend weakly on the nature of the surface, as they typically include the bulk of the two interface materials and the entire protein. However, the short range surface interactions depend critically on the details of the surface of the protein. The inhomogeneous surface of a protein results in a local moment or torque applied by the fluid at each point on the surface. For a helical protein immersed in a homogeneous fluid, this local torque will sum to zero across the entire surface of the protein. However, for a protein in an inhomogeneous environment, such as one confined to an interface, will experience a non-zero total torque $\Gamma$. This can induce a spontaneous curvature or twist depending on both the direction of $\Gamma$ and the strength of the intrinsic bend and twist moduli.
The net torque on the protein due to its environment can be separated into contributions from short range and long range forces: $$\begin{aligned}
{\Gamma} &= \int_V \!d^3r\, {\mathbf{r}}\times {\mathbf{f}}_{LR}({\mathbf{r}}) +
\Delta\int_S d^2r \,{\mathbf{r}}\times {\mathbf{f}}_{SR}({\mathbf{r}},z)\,,\\[5.0truept]
\noalign{\noindent where $S$ and $V$ are respectively the surface and volume of the protein. The force densities are given by}\nonumber\\[-4.0truept]
{\mathbf{f}}({\mathbf{r}}) &= - \int_{\textrm{env}}d^3r' \,
\frac{\partial {\cal U}\left({\mathbf{r}}-{\mathbf{r}}'\right)}{\partial ({\mathbf{r}}-{\mathbf{r}}')},
$$ where the energy density ${\cal U}({\mathbf{r}}-{\mathbf{r}}')$ of interaction (energy per volume squared) between material in the environment at ${\mathbf{r}}'$ and in the protein at ${\mathbf{r}}$ can be separated into long range (*e.g.* dispersion or Coulomb) and short-range (*e.g.* hydrophobic or steric) interactions. Here, $\Delta$ is the interaction depth within the protein (of order an amino acid in size), and the forces are obtained by integrating over points ${\mathbf{r}}'$ in the environment external to the protein. Although the net torque will generally depend on the entire shape and volume of the protein (because of long range dispersion and Coulomb interactions), we will illustrate the example where the effects of long range forces are negligible compared to those of the short range interactions. For example, an unbalanced torque that leads to a bend in the plane of the interface will not perturb the long range energy of interaction appreciably, since there will be neligible response perpendicular to the interface.
In the case of short range interactions, we can approximate the integral over the environment as $\int_{\textrm{env}}d^3r'\simeq a\int dz'$, where the coordinate $z'$ is along the surface normal and $a$ is the lateral area of the short interaction. By integrating the short range potential and using the reference ${\cal U}(z=\infty)=0$, we can write the torque exerted on the surface as $$\begin{aligned}
{\Gamma} &= a \Delta\int_S d^2r \,({\mathbf{r}}\times{\mathbf{\hat{{n}}}})\,{\cal U}({\mathbf{r}}),\\
&\equiv\int_S d^2r \,({\mathbf{r}}\times{\mathbf{\hat{{n}}}})\,\left[\bar{\gamma} + \delta\gamma\left({\mathbf{r}}\right)\right],\end{aligned}$$ The quantity $(a\Delta) {\cal U}({\mathbf{r}})\equiv\bar{\gamma} + \delta\gamma({\mathbf{r}})$ is the surface energy density of interaction introduced in Eq. \[3\] of the main text.
**Fluid-fluid interface** – At fluid-fluid interfaces an adsorbed fibril will be surrounded by both fluids, according to the (inhomogeneous) degree of wettability of the fibril on the two fluids. This inhomogeneous environment leads to a net uncompensated moment when averaged over the inhomogenous solvent environment around the fibril. Although this applies to the problem at hand, we will take the a pragmatic approach and illustrate the method for the simpler example of a fluid-solid interface with short-range interactions.
**Fluid-solid interface** – Consider a fibril adsorbed to a fluid-solid interface. Material within a short range $\Delta$, set by Coulomb screening, shapes of asperities, or hydrophobic effects, will interact with the solid substrate on a strip. For short range interactions a surface interaction that is symmetric from head to tail (a non-polar interaction) will lead to zero applied total torque, as the local torque will sum to zero, as in a homogeneous environment. However, a non-symmetric interaction will lead to uncompensated torques, or bending moments, all along the length of the adsorbed fibril.
### Twisted Ribbon of Fixed Radius
To make progress, we approximate the fibril of length $L$ as a twisted ribbon with wavelength $\lambda$, which makes contact every half wavelength with a solid surface on the exposed edges at the ribbon radius $R$ (Figures \[fig1\], \[fig2\]). The wavelength is related to the helical angle $\theta_p$ by $$\begin{aligned}
\cos{\theta_{p}}&=\frac{qR}{\sqrt{1+(qR)^2}}&
\sin{\theta_{p}}&=\frac{1}{\sqrt{1+(qR)^{2}}},
\label{pitch}\end{aligned}$$ where $q=2\pi/\lambda$. The centerline of the undeformed fibril defines a tangent vector ${\mathbf{\hat{{t}}}}_0$, which upon bending becomes ${\mathbf{\hat{{t}}}}(s)$, with local curvature $\kappa=|d{\mathbf{\hat{{t}}}}/ds|\equiv|\dot{\hat{\textbf{t}}}|$. Equivalently, we can parametrize the curvature in terms of the vector angular rotation of the tangent vector, defined by $\dot{\boldsymbol{\Theta}}={\mathbf{\hat{{t}}}}\times\dot{\hat{\textbf{t}}}$.
Rather than work in terms of torques exerted across the body, we will calculate the surface energy of the adsorbed fibril as a function of the fibril shape. Minimizing this energy with respect to in-plane bending will lead to an induced spontaneous curvature, which is equivalent to finding an uncompensated torque for a straight fibril.
For a small interaction range, $\Delta\ll R$, the interaction between the surface and the twisted ribbon can be approximated by the surface energy of series of strips of thickness $ \omega=2\sqrt{2R\Delta-\Delta^2}\simeq\sqrt{8R\Delta}$ (Fig. \[fig1\]). The ribbon-surface energy is given by $$\begin{aligned}
G_{\textrm{surf}} &= \sum_{j=1}^{2L/\lambda}\int_{S_{j}} \left[\bar{\gamma} + \delta\gamma({\mathbf{r}})\right]d^2r\\
&\equiv \sum_{j=1}^{2L/\lambda}G_{\textrm{polar},j},\end{aligned}$$ where $S_{j}$ is the surface area of the $j$th interaction strip, the average surface energy $\bar{\gamma}$ represents the absorption properties of the ribbon, and $\delta\gamma$ captures the polar nature of the interaction. There are $2L/\lambda$ distinct interaction strips. We assume that the strip has an anisotropic interaction potential that is polar along the direction ${\mathbf{\hat{{u}}}}$ within the strip, and assume the simple form
$$\begin{aligned}
\delta\gamma({\mathbf{r}}) &= \varepsilon\,{\mathbf{r}}\cdot{\mathbf{\hat{{u}}}}\\
&= \varepsilon\left(x\cos\Phi + y\sin\Phi\right),\end{aligned}$$
where ${\mathbf{\hat{{u}}}}$ is at an angle $\Phi$ with respect to the tangent vector ${\mathbf{\hat{{t}}}}$.In the limit of $R\gg\Delta$ the strips can be approximated as flat, taking ${\mathbf{r}}$ as a two-dimensional vector in the plane of the surface. For short range interactions these flat strips constitute the primary interaction between the surface and the twisted ribbon.
Amyloid fibrils are composed of protofilaments, which in turn comprise layers of aligned beta sheets that are twisted about their central axis. A given fibril contains a number of protofilaments that form a ribbon, which we approximate as shown in Figure \[fig2\](A). The ribbon diameter $D$ is given by the number of protofilaments in the fibrils, while the ribbon thickness $d$ is determined by the diameter of an individual protofilament. The ribbon length $L$ is determined by the total number of aligned beta strands.
For an undeformed fibril the interaction strip is a parallelogram tilted at an angle $\theta_p$ determined by the pitch of the ribbon, and with lengths determined by the thickness $d$ of the ribbon (the perpendicular distance between the edges) and the strip thickness $\omega$, as shown in Figure \[fig2\](B). Two sides of length $\ell=d/\sin\theta_p$ are parallel to the tangent vector ${\mathbf{\hat{{t}}}}_0$, while the other two sides have length $ \omega/\cos\theta_p$.
When the ribbon is bent the ribbon thickness $d$ is fixed due to the fixed radius, but it curves to follow the deformed tangent vector ${\mathbf{\hat{{t}}}}$. Given that we are in the small bend regime, we approximate these sides as straight, but tilted additionally by $\bar{\phi}=\tfrac12(\phi_R+\phi_L)$ according to the average tilt of the interaction strip (Figure \[fig2\](C)). Here $\phi_L$ and $\phi_R$ represent the additional tilts on the left and right hand sides of the interaction strip.
When the strip is bent downwards the top of the interaction strip is under tension whereas the bottom of the strip is under compression. Although the center of the strip is not under tension or compression, bend-stretch coupling terms may cause the ribbon to stretch or compress, leading to a new strip length $\ell'=d/\sin(\theta_p-\bar{\phi})$. This change in length contributes to the bend-stretch coupling, which is not of interest here.
Initially, the polarity vector ${\mathbf{\hat{{u}}}}_0$ is at an angle $\Phi$ with respect to the tangent vector ${\mathbf{\hat{{t}}}}_0$. When the twisted ribbon is bent, then to first order the all vectors in the interaction strip rotate with the average rotation $\bar{\phi}$ of a particular segment; this includes both the polarity vector and the local tangent vector. However, the stretching and compression on either side of the bend cause the polarity vector to deflect non-affinely aross the strip; e.g the tilt of the polarity vector should vary smoothly between $\phi_L$ and $\phi_R$, when moving from left to right across the strip. For simplicity we will take the polarity vector to be tilted by $\bar{\phi}$ everywhere on the interaction strip. With this notation, the polar surface potential becomes $$\left.\delta\gamma({\mathbf{r}})\right|_{\textrm{bent}}=\varepsilon\left[x\cos(\Phi-\bar{\phi})+ y\sin(\Phi-\bar{\phi})\right].$$
### Polar Free Energy
The polar energy across a single interaction strip, or equivalently the energy per helical repeat, is then given by $$\begin{aligned}
G_{\textrm{polar}}&=\varepsilon \int^{\tfrac12 \omega}_{-\tfrac12 \omega}\,dy\,\int^{f_R(y)}_{f_L(y)} \,dx
\left[x\cos\left(\Phi-\bar{\phi}\right)+ y\sin\left(\Phi-\bar{\phi}\right)\right],
\\
\noalign{\noindent where}
f_L(y)&=y\cot[\theta_{p}+\tfrac12(\phi_R-\phi_L)]-\tfrac12\ell \\
f_R(y)&=y\cot[\theta_{p}-\tfrac12(\phi_R-\phi_L)]+\tfrac12\ell \,,\end{aligned}$$ and $\ell$ is the length of center of the interaction strip parallel to ${\mathbf{\hat{{t}}}}_0$. This evaluates to $$\begin{aligned}
G_{\textrm{polar}}=&\frac{\varepsilon\omega^3}{12}\left[\cot\left(\theta_{p}-\tfrac12\Theta\right)-\cot\left(\theta_{p}+\tfrac12\Theta\right)\right]
\left\{{\sin}\left(\Phi-\bar{\phi}\right) \right.\nonumber\\
& \left.\tfrac12{{\cos}\left(\Phi-\bar{\phi}\right)}\left[\cot\left(\theta_{p}-\tfrac12\Theta\right)+\cot\left(\theta_{p}+\tfrac12\Theta\right)\right]\right\},
\label{eq:polar}\end{aligned}$$ where ${\Theta=\phi_R-\phi_L}$ is the angular deflection associated with the bend. The energy of deformation vanishes for zero bend $\Theta=0$. A positive bend $\Theta>0$ corresponds to a right hand bend, when travelling parallel to the chosen direction fo the tangent vector.
Our goal is to study the lowest order effects of the surface, which induce a spontaneous curvature signified by the term linear in bend $\Theta$ that arises from the small $\Theta$ approximation to $G_{\textrm{polar}}$. The average tilt $\bar{\phi}$ can be related, geometrically, to a combination of twist and stretch, which leads to surface-induced bend-twist and bend-stretch couplings. Thus, we will expand Eq. \[eq:polar\] to first order in $\Theta$, and set $\bar{\phi}=0$ because we are not interested in higher order bend-twist or bend-stretch couplings (the effects of these would only be visible upon observing changes in total fibril length, or in local chirality). To lowest order in the deflection we find $$\begin{aligned}
G_{\textrm{polar}}&=\frac{\varepsilon \omega^3}{12\sin^2\theta_p}
(\cos\Phi\,{\cot\theta_p}+\sin\Phi)\,\Theta+\ldots\,\\
&\simeq\frac{\varepsilon \omega^3\ell}{12\sin^2\theta_p}
(\cos\Phi\,{\cot\theta_p}+\sin\Phi)\,\frac{d\Theta}{ds}.\end{aligned}$$ In performing this expansion we have assumed that the polar direction ${\mathbf{\hat{{u}}}}$ (or $\Phi$) rotates affinely with the tangent; deviations from this will lead to higher order couplings $\Theta\,\delta\Phi$. Hence, the contribution to the bending energy of the entire fibril is $$\begin{aligned}
G_{\textrm{surf}} &= \sum_{j=1}^{2L/\lambda}G_{\textrm{polar},j}\\
&=\int_0^L\frac{2\,ds}{\lambda}\frac{\varepsilon \omega^3 \ell}{12\sin^2\theta_p}
(\cos\Phi\,{\cot\theta_p}+\sin\Phi)\,\,\frac{d\Theta}{ds}, \label{eq:free0}\end{aligned}$$ where we have assumed that the bend is smooth between contacts, and converted the sum to an integral via $\sum_j \rightarrow\int ds/\lambda$.
The polar moment is given by $$\begin{aligned}
{\mathbf{P}}&=\frac{2}{\lambda}\int_{S} \,{\mathbf{r}}\, \delta\gamma({\mathbf{r}})\,\,d^2r,\\
&=\frac{2\varepsilon}{\lambda}\int_{S} \,{\mathbf{r}}\, ({\mathbf{r}}\cdot{\mathbf{\hat{{u}}}})\,\,d^2r,\\
&=\frac{\varepsilon}{\lambda}\frac{\partial}{\partial{\mathbf{\hat{{u}}}}} \int^{\frac{\omega}{2}}_{-\frac{\omega}{2}}dy\int^{y\cot\theta_{p}+\tfrac{\ell}{2}}_{y\cot\theta_{p}-\tfrac{\ell}{2}}
\left[x\cos\Phi+ y\sin\Phi\right]^2\, dx\\\,
&=\frac{\varepsilon \omega^3\ell }{6\lambda}
\left\{\left[\cos\Phi\left(\cot^2\theta_p + \left(\frac{\ell}{\omega}\right)^2\right) + \sin\Phi\cot\theta_p \right]{\mathbf{\hat{{t}}}} + \left(\sin\Phi + \cos\Phi\cot\theta_p\right){\mathbf{\hat{{n}}}}\times{\mathbf{\hat{{t}}}}\right\}.\end{aligned}$$ One component of ${\mathbf{P}}$ is parallel to the fibril direction ${\mathbf{\hat{{t}}}}$, while the other direction is perpendicular to ${\mathbf{\hat{{t}}}}$ and in the plane specified by normal vector ${\mathbf{\hat{{n}}}}$. Note that $\{{\mathbf{\hat{{t}}}},{\mathbf{\hat{{n}}}}\times{\mathbf{\hat{{t}}}},{\mathbf{\hat{{n}}}}\}$ form an orthonormal basis. Hence, $${\mathbf{P}} = P_{\parallel}{\mathbf{t}} + P_{\perp}{\mathbf{\hat{{n}}}}\times{\mathbf{\hat{{t}}}},$$ where
$$\begin{aligned}
P_{\parallel} &= \frac{\varepsilon \omega^3\ell }{6\lambda}
\left[\cos\Phi\left(\cot^2\theta_p + \left(\frac{\ell}{\omega}\right)^2\right) + \sin\Phi\cot\theta_p \right]\\
P_{\perp} &= \frac{\varepsilon \omega^3\ell }{6\lambda} \left(\sin\Phi + \cos\Phi\cot\theta_p\right).\end{aligned}$$
Comparing the definition of ${\mathbf{P}}$ with the free energy $G_{\textrm{surf}}$, we can rewrite the surface energy as $$\begin{aligned}
G_{\textrm{surf}} &=\frac{1}{\sin^2\theta_p}\int_0^Lds\,{P_{\perp}}\,\frac{d\Theta}{ds}. \end{aligned}$$ In vector form, the angular rotation is given by $\dot{\boldsymbol{\Theta}}=-{\mathbf{\hat{{n}}}}\frac{d\Theta}{ds}$ (Fig. \[fig2\]), while the component $P_{\perp}$ can be extracted via $P_{\perp}={\mathbf{\hat{{n}}}}\cdot{\mathbf{\hat{{t}}}}\times{\mathbf{P}}$. Thus, the free energy becomes $$\begin{aligned}
G_{\textrm{surf}} &=-\frac{1}{\sin^2\theta_p}\int_0^Lds\,\dot{\boldsymbol{\Theta}}\cdot{\mathbf{\hat{{t}}}}\times{\mathbf{P}},\end{aligned}$$ which corresponds to the free energy of Equations 5-6 in the main text, with $A=1/\sin^2\theta_p$.
### Induced Curvature
The total bending free energy is given by the sum of the standard bending energy and the coupling to the surface: $$\begin{aligned}
G_{bend} &= \int ds\left[\frac12 B \dot{\boldsymbol{\Theta}}^2
- \frac{1}{\sin^2\theta_p}\dot{\boldsymbol{\Theta}}\cdot{\mathbf{\hat{{t}}}}\times{\mathbf{P}} \right] \\
&= \int ds \left[ \frac12 B \kappa^2 -
\frac{\varepsilon \omega^3 \ell}{6\lambda\sin^2\theta_p}
(\cos\Phi\,{\cot\theta_p}+\sin\Phi)\kappa \right],\end{aligned}$$ where the (signed) curvature is defined by $\dot{\Theta}=\kappa{\mathbf{\hat{{n}}}}$. The bending modulus generally includes contributions from the surface, which can be calculated based on the formalism here. However, since our intent is to demonstrate the significance of the induced curvature, we do not consider such perturbations. Moreover, the main contribution to bending is usually from internal degrees of freedom that are only weakly influenced by the surface. An exception occurs for highly charged filaments. In such cases the reduction in the dielectric constant and lack of screening near a hydrophobic surface will increase the electrostatic contribution to $B$.
This bend energy is minimized by the following spontaneous curvature $\kappa_0$: $$\begin{aligned}
\kappa_0 & = \frac{\varepsilon \omega^3 \ell}{6\lambda\sin^2\theta_p\,B}
(\cos\Phi\,{\cot\theta_p}+\sin\Phi) \label{eq:spont}\\
&= \frac{\varepsilon \omega^3 \ell}{\lambda\sin^2\theta_p\,B}
\alpha(\theta_p,\Phi),\end{aligned}$$ where $ \alpha(\theta_p,\Phi)\equiv(\cos\Phi\,{\cot\theta_p}+\sin\Phi)/6$.
The sign of the induced curvature can be understood as follows. Consider $\varepsilon>0$, a helix with an opening angle of $\theta_p=\pi/4$, and a polarization direction specified by $\Phi=\pi/6$ (roughly as in Figs. \[fig1\], \[fig2\]). In this case there is a higher energy for exposing the upper right part of the parallelogram in Fig. \[fig2\] to the surface. Hence the preferred bending direction should be ‘up’ in Fig. \[fig1\] (rather than the downward shown), to allow the relatively less of the costly part of the surface interaction to attain more contact with the surface. This corresponds to a positive bend around ${\mathbf{\hat{{n}}}}$, given by $\dot{\boldsymbol{\Theta}}=\kappa_0{\mathbf{\hat{{n}}}}$ with $\kappa_0>0$ and matches the prediction in Eq. (\[eq:spont\]).
[3]{} Adamcik, J. [*et al.*]{} Understanding Amyloid Aggregation by Statistical Analysis of Atomic Force Microscopy Images. [*Nature Nanotech.*]{} [**2010**]{}, [*5*]{}, 5423–428. Pickering, S. CXCVI.Emulsions. [*J. Chem. Soc., Trans.*]{} [**1907**]{}, [*91*]{}, 2001–2021. Israelachvili, J.N. [*Intermolecular and Surface Forces*]{} Academic Press: London, 1992).
![\[Sfig1\] AFM image of fibrils at the air-water interface after $t$=60 minutes adsorption time from a $c_{\text{init}}=0.001$% w/w fibril suspension.](Sfig1.jpg){width="100.00000%"}
![\[Sfig2\] AFM image of fibrils at the air-water interface after $t$=10 minutes adsorption time from a $c_{\text{init}}\approx{0.008}$% w/w fibril suspension. Rings coexist with nematic fibril domains.](Sfig2.jpg){width="100.00000%"}
![\[Sfig4\] AFM image of fibrils at the air-water interface after $t$=60 minutes adsorption time from a $c_{\text{init}}=0.001$% w/w fibril suspension. Rings are often composed of many short fibrils.](Sfig3.jpg){width="100.00000%"}
![\[Sfig5\] AFM height and phase images of fibrils at the air-water interface immediately after sample preparation of a $c_{\text{init}}=0.001$% w/w fibril suspension. The scale bar applies to both images. Distortions in the background peptide layer are readily visible in the phase image but are rarely spherical and do not coincide spatially with fibril rings.](Sfig4.jpg){width="100.00000%"}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[![(A) Helical fibril against a surface. (B) The contact area, or interaction strip, is a parallelogram that deforms asymmetrically (C) when the fibril is bent. This leads to an excess contact area by one ‘charge’ of the polar interaction, leading to a preference for one sign of bend and thus a spontaneous curvature. A positive red ‘charge’ and a negative ‘blue’ charge corresponds to a polarization potential $\delta\gamma=\varepsilon(\cos\pi/6 + y \sin\pi/6)$, with $\varepsilon>0$. In this case the bend shown in (C) costs energy, and the preferred spontaneous curvature instead correponds to a bend $\frac{d\boldsymbol{\Theta}}{ds}=\dot{\boldsymbol{\Theta}}={\mathbf{\hat{{t}}}}\times\dot{\hat{\bf{t}}}$ which is parallel to ${\mathbf{\hat{{n}}}}$.[]{data-label="fig1"}](Sfig5ribbons.jpg "fig:"){width="15cm"}]{}
\[5.0truept\]
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[![(A) Geometry of twisted ribbon. Initial (B) and deformed (C) interaction strips, obtained by bending the fibril. The undeformed strip is shown in grey under the deformed strip. The bend causes a tilt in the two sides (right and left) depending on the change in the tangent vector across the strip, while the top and bottom sides remain parallel to each other, but rotate with respect to the undeformed strip by $\bar{\phi}=\tfrac12(\phi_L+\phi_R)$, which describes the average tilt of the individual strip.[]{data-label="fig2"}](ribbonlabellingPDO_C.pdf "fig:"){width="15cm"}]{}\
[![(A) Geometry of twisted ribbon. Initial (B) and deformed (C) interaction strips, obtained by bending the fibril. The undeformed strip is shown in grey under the deformed strip. The bend causes a tilt in the two sides (right and left) depending on the change in the tangent vector across the strip, while the top and bottom sides remain parallel to each other, but rotate with respect to the undeformed strip by $\bar{\phi}=\tfrac12(\phi_L+\phi_R)$, which describes the average tilt of the individual strip.[]{data-label="fig2"}](unbent2PDO.pdf "fig:"){width="7cm"}]{}[![(A) Geometry of twisted ribbon. Initial (B) and deformed (C) interaction strips, obtained by bending the fibril. The undeformed strip is shown in grey under the deformed strip. The bend causes a tilt in the two sides (right and left) depending on the change in the tangent vector across the strip, while the top and bottom sides remain parallel to each other, but rotate with respect to the undeformed strip by $\bar{\phi}=\tfrac12(\phi_L+\phi_R)$, which describes the average tilt of the individual strip.[]{data-label="fig2"}](bentPDO.pdf "fig:"){width="8cm"}]{}
![\[Sfig3\] Whole AFM image used for the curvature distribution analysis showing fibrils at the air-water interface after $t$=10 minutes adsorption time from a $c_{\text{init}}=0.001$% w/w fibril suspension.](Sfig5.jpg){width="100.00000%"}
|
---
abstract: |
Given an arbitrary $C^{\,0}$ flow on a manifold $M$, let $\mbox{CMin}$ be the set of its compact minimal sets, endowed with the Hausdorff metric, and $\mathcal{S}$ the subset of those that are Lyapunov stable. A topological characterization of the interior of $\mathcal{S}$, the set of Lyapunov stable compact minimal sets that are away from Lyapunov unstable ones is given, together with a description of the dynamics around it. In particular, $\mbox{int}_{H}\mathcal{S}$ is locally a Peano continuum (Peano curve) and each of its countably many connected components admits a complete geodesic metric.
This result establishes unexpected connections between the local topology of $\mbox{CMin}$ and the dynamics of the flow, providing criteria for the local detection of Lyapunov instability by merely looking at the topology of $\mbox{CMin}$. For instance, if $\mbox{CMin}$ is not locally connected at $\varLambda\in\mbox{CMin}$, then every neighbourhood of $\varLambda$ in $M$ contains Lyapunov unstable compact minimal sets (hence, if $\mbox{CMin}$ is nowhere locally connected, then every neighbourhood of each compact minimal set contains infinitely many Lyapunov unstable compact minimal sets).
author:
- pedro teixeira
---
Introduction
============
The comprehension of the dynamics around compact minimal sets plays an important role in the study of flows on manifolds. Among the concepts that are pertinent in this context, those of Lyapunov stability/instability are fundamental both in the conservative and in the dissipative settings [@LY; @BI; @M1; @M2]. Detecting the occurrence of Lyapunov unstable compact minimal sets in the neighbourhood of Lyapunov stable ones is a relevant dynamical problem, first of all because in the presence of the former, by an arbitrarily small perturbation of the phase space coordinates of a point, one may pass from stable to unstable almost periodic solutions.
The set $\mbox{CMin}$ of all compact minimal sets of a flow is naturally endowed with the Hausdorff metric, thus becoming a metric space whose “points” are the compact minimal sets $\varLambda\in\mbox{CMin}$. This is a topological invariant (topologically equivalent flows have homeomorphic $\mbox{CMin}$’s). One the other hand, examples show abundantly that completely distinct flows may also have homeomorphic $\mbox{CMin}$’s. Nevertheless, it turns out that the mere inspection of the local topology of $\mbox{CMin}$ at $\varLambda$ may reveal unexpected information about the flow dynamics around $\varLambda$. For instance, if $\mbox{CMin}$ is *not* locally connected at $\varLambda$, then every neighbourhood of $\varLambda$ in $M$ contains Lyapunov unstable compact minimal sets. The same holds if $\varLambda$ has no compact neighbourhood in $\mbox{CMin}$. This follows immediately from the fact that, on any flow, the interior of the set $\mathcal{S}$ of all Lyapunov stable compact minimal sets is an open subset of $\mbox{CMin}$ that is both locally compact and locally connected. It is this quite exceptional topological structure that furnishes criteria permitting to detect that a given $\varLambda\in\mbox{CMin}$ does *not* belong to $\mbox{int}_{H}\mathcal{S}$ or equivalently, that $\varLambda\in\mbox{cl}_{H}\mathcal{U}=\mbox{(int}_{H}\mathcal{S})^{c}$,[^1] and it is readily seen that $\varLambda\in\mbox{cl}_{H}\mathcal{U}$ iff every neighbourhood of $\varLambda$ in $M$ contains Lyapunov unstable compact minimal sets. Actually, $\mbox{int}_{H}\mathcal{S}$ is locally a Peano continuum, having the nice “pre-geometric” property of existence of a complete geodesic metric on each of its countably many connected components (Corollary , Remark ).
As a practical application, imagine that without knowing a certain flow, we are provided with a homeomorphic copy $K$ of its $\mbox{CMin}$. If, for instance, $K$ is the Cantor star,[^2] then we know immediately that every neighbourhood of each compact minimal set of the flow contains infinitely many Lyapunov unstable compact minimal sets, and this by simply observing that $K$ is not locally connected at a dense subset of its points. However, it is in general impossible, by the mere inspection of $K$, to determine *which* points of $K$ correspond (under that homeomorphism) to the Lyapunov unstable compact minimal sets we know to exist. Instead of being a limitation, this fact is actually one of the reasons that make these criteria interesting, for they are among the results that somehow escape the intrinsic bounds confining the dynamical information concerning Lyapunov stability/instability extractable from the topology of $\mbox{CMin}$, as we now explain. To give a simpler example, consider the $C^{\infty}$ flow on $\mathbb{R}^{2}$ pictured as (1) in Fig. (we suppose that all orbits outside the outer periodic orbit have that orbit as $\omega$-limit and have empty $\alpha$-limit set; the denumerably many periodic orbits are ordered by decreasing length as $\gamma_{n}$, $n\in\mathbb{N}$). Its $\mbox{CMin}$ is homeomorphic to $K=\{0\}\cup\{1/n:\, n\in\mathbb{N}\}$, the Lyapunov stable equilibrium $O$ (origin) is taken to $0$ and each Lyapunov unstable periodic orbit $\gamma_{n}$ to $1/n$. Since $K$ is not locally connected at $0$, we know that on *any* flow having its $\mbox{CMin}$ homeomorphic to $K$, every neighbourhood of the compact minimal set corresponding to $0$ (under that homeomorphism) contains Lyapunov unstable compact minimal sets. Now, it is easily seen that there is another $C^{\infty}$ flow on $\mathbb{R}^{2}$ without periodic orbits (Fig. ), whose $\mbox{CMin}$ is also homeomorphic to $K$ in such a away that $0$ is taken to the Lyapunov *unstable* equilibrium $O$ and each $1/n$ is taken to a Lyapunov *stable* equilibrium (these later equilibria are attractors i.e. asymptotically stable, see Definition ). Hence, these two flows have homeomorphic $\mbox{CMin}$’s, but under that homeomorphism, the Lyapunov unstable compact minimal sets of the first flow correspond to the Lyapunov stable ones of the second and vice versa (similar examples could be given on $\mathbb{S}^{2}$). Therefore, by the mere inspection of $K\simeq\mbox{CMin}$, it is impossible to determine which points of $K$ correspond to the Lyapunov unstable compact minimal sets that we know to occur in every neighbourhood of $O$. Observe, however, that our conclusion remains intact: in both flows every neighbourhood of the equilibrium $O$ (corresponding to $0$ under both homeomorphisms) contains Lyapunov unstable compact minimal sets $\varGamma$ (in the second flow, the only such $\varGamma$ is the equilibrium orbit $\{O\}$ itself).
It should be also mentioned that contrariwise, it is obviously impossible to detect the occurrence of Lyapunov stable compact minimal sets in a flow by merely looking at its $\mbox{CMin}$, for given any $C^{\,0\leq r\leq\infty,\omega}$ flow on a manifold $M$, there is a $C^{r}$ flow on $M\times\mathbb{S}^{1}$ with all compact minimal sets Lyapunov unstable, whose $\mbox{CMin}$ is isometric to that of the original flow on $M$. Actually, the above mentioned topological characterization of $\mbox{int}_{H}\mathcal{S}$ is quite exceptional, for among the subsets of $\mbox{CMin}$ directly related to its partition into Lyapunov stable and Lyapunov unstable compact minimal sets ($\mbox{CMin}=\mathcal{S\,}\sqcup\mathcal{\, U}$), only $\mbox{int}_{H}\mathcal{S}$ has a nice local topology (even inducing a pre-geometric structure).[^3]
This work continues the line of research initiated in [@TE] aiming to illuminate the connections between the local topology of $\mbox{CMin}$ Fig. 8.3-3 “VAK” in [@AB p.585]). Based on the local topological characterization of $\mbox{int}_{H}\mathcal{S}$ mentioned above, the following result establishes a conservative setting picture of the possible interplay between Lyapunov stability and instability in the neighbourhood of an arbitrary compact minimal set:
***(A)*** Let $\varLambda$ be a compact minimal set of a $C^{0}$ non-wandering flow on a connected manifold $M$. Then, either
1. there is a sequence of Lyapunov unstable compact minimal sets $\varLambda_{n}$ converging to $\varLambda$ in the Hausdorff metric, or
2. $\varLambda=M$ i.e. $M$ is compact and the flow is minimal, or
3. there are arbitrarily small compact, connected, invariant neighbourhoods $U$ of $\varLambda$ in $M$ such that:
1. $U$ is the union of $\mathfrak{c}=|\mathbb{R}|$ Lyapunov stable (compact) minimal sets $\varLambda_{i\in\mathbb{R}}$;
2. endowed with the Hausdorff metric, the set $\big\{\varLambda_{i\in\mathbb{R}}\big\}$ is a Peano continuum (Peano curve).
For general $C^{0}$ flows (not necessarily non-wandering), proper attractors may show up and we have the following analogue local characterization (Section , Theorem and Fig. ):
***(B)*** Let $\varLambda$ be a compact minimal set of a $C^{0}$ flow on a connected manifold $M$. Then, either
1. there is a sequence of Lyapunov unstable compact minimal sets $\varLambda_{n}$ converging to $\varLambda$ in the Hausdorff metric, or
2. *$\varLambda$* is an attractor i.e. asymptotically stable, or
3. there are arbitrarily small compact, connected, (+)invariant neighbourhoods $U$ of $\varLambda$ in $M$ such that:
1. the (compact) minimal sets contained in $U$ are all Lyapunov stable and, endowed with the Hausdorff metric, their set is a Peano continuum with $\mathfrak{c}$ elements;
2. for each $x\in U$, $\omega(x)$ is a (Lyapunov stable compact) minimal set contained in $U$ and if $x\not\in\omega(x)$, then its negative orbit leaves $U$ (and thus never returns again).
This result shows that, if a compact minimal set $\varLambda$ is away from Lyapunov unstable ones, then the dynamics around it is reasonably well understood and the set of Lyapunov stable compact minimals near $\varLambda$ has a remarkable topological structure (in relation to the Hausdorff metric $d_{H}$), exactly as in the non-wandering case (Theorem A).
Although we have assumed the phase space $M$ to be a connected manifold, these results still hold under much weaker hypothesis: it is enough to suppose that $M$ is a generalized Peano continuum i.e. a locally compact, connected and locally connected metric space (see Remark ). What seems remarkable is that *this topological structure of the phase space is actually completely inherited by each component of* ):
1. $H\mathcal{S}$ ** has countably many components $X_{i}$ (possibly none), each $X_{i}$ being a clopen generalized Peano continuum (*$H\mathcal{S}$* is endowed with the Hausdorff metric);
2. the union $X_{i}^{*}\subset M$ of the (Lyapunov stable) compact minimal sets $\varLambda\in X_{i}$ is contained in the “basin” $A_{i}$, a connected, open invariant subset of $M$, consisting of all points that have some $\varLambda\in X_{i}$ as $\omega$-limit set. Although $X_{i}^{*}$ may be noncompact, it roughly acts as an attractor with basin $A_{i}$ in the flow (see Fig. );
3. if $x\in A_{i}$ but $x\not\in\omega(x)$, then ** $\alpha(x)\subset\mbox{bd\,}A_{i}$.
This gives a fairly complete description of the dynamics within and around each $X_{i}^{*}$. If the flow is non-wandering but not minimal, then $X_{i}^{*}=A_{i}$ (Theorem ), i.e. the union of the compact minimal sets belonging to a component $X_{i}$ of $H\mathcal{S}$ is a (nonvoid) connected, open invariant set $A_{i}\subset M$ and the local density of these minimal sets is actually $\mathfrak{c}$ all over $A_{i}$: every open set $B\subset A_{i}$ intersects $\mathfrak{c}$ (Lyapunov stable) compact minimal sets $\varLambda\in X_{i}$.
The proofs explore the local topology of the phase space $M$, together with the specific dynamical constraints of the flow near $\varLambda\in H\mathcal{S}$ and the fact that these minimal sets $\varLambda$ are continua (see Remark ), to “move” the local topological structure of $M$ to the components of $H\mathcal{S}$. A kind of duality builds up between these two topologies, Lemma being a simple “show-case” of the techniques that enable this “crossing of the bridge”. Showing that $H\mathcal{S}$ is locally a Peano continuum (Corollary ) requires a fundamental result from Peano continuum theory (see the proof of Lemma ), and seems hard to establish otherwise.
This paper is organized as follows: Section introduces the general setting of the whole work and the main concepts, Lyapunov stability and hyper-stability (, and ). Examples illustrating the dynamical significance of both notions are given in and , the later section being entirely devoted to flows with all orbits periodic. Section introduces the main tools and the first dynamical consequences of the topological characterization of $H\mathcal{S}$ there obtained, criteria for the local detection of Lyapunov unstable compact minimal sets being given in . Section contains the main results, giving a reasonable global and local characterization of $H\mathcal{S}$ and of the dynamics around it. The analogue characterizations, specific to the non-wandering context, are obtained in . Section shows how global absence of Lyapunov unstable compact minimals imposes strong dynamical constraints on the flow, if the phase space is compact (these constraints vanish in the noncompact setting). Directly related to this phenomenon are the continuous decompositions of closed manifolds into closed submanifolds. The natural question of the existence of a manifold structure in the associated quotient space (endowed with the Hausdorff metric) is briefly discussed in . Finally, Section shows that the topological characterization of $H\mathcal{S}$ obtained in Corollary (Section ) is optimal in the context under consideration and ends pointing to evidence showing that intricate (generalized) Peano continua indeed appear as the $H\mathcal{S}$ sets of smooth flows on manifolds. Sections , , , and are unessential and may be skipped by any reader seeking a “straight to the core” approach. However, the first paragraph of Section and Problem in the same section are important to gain perspective of the significance of the main result (Theorem ).
Lyapunov stability versus instability
=====================================
General setting
---------------
Throughout this paper, deductions are purely topological, all results being valid for flows on much larger classes of phase spaces than those of manifolds. We now establish the general context of this work. This amounts to a minimum of hypothesis needed to deduce all the results in the paper.
[[<span style="font-variant:small-caps;">Convention</span>]{}]{}
: Except if otherwise mentioned, we will be considering an *arbitrary continuous ($C^{0}$) flow $\theta$ on a* *locally compact, connected and locally connected metric space* $M$. Such an $M$ is called a *generalized Peano continuum* (a *Peano continuum*, if compact). $M$ is *non-degenerate* if $|M|>1$, thus implying $|M|=\mathfrak{c}=|\mathbb{R}|$.
($M$ is connected ($\implies|M|\geq\mathfrak{c}$) and locally compact, hence [@KO p.269] separable ($\implies|M|\leq\mathfrak{c})$, therefore $|M|=\mathfrak{c}$).
Assuming the phase space of the flow to be a generalized Peano continuum instead of a manifold has obvious advantages, even in the differentiable setting: the results may be applied to subflows on arbitrary (e.g. non-manifold) connected, closed invariant subsets, provided these are locally connected. Obviously, if, instead, these invariant subsets are open, then only connectedness is required. On the other hand, this more general setting helps to get rid of the additional manifold structure that is irrelevant for the comprehension of the phenomena under consideration, attention becoming exclusively focused on the topological factors that are determinant to the process.
\(a) a *continuum* is a compact and connected metric space. Compact minimal sets are continua.
\(b) Peano continua (or *Peano curves*) are the continuous images of $\mathbb{D}^{1}=[-1,1]$ into Hausdorff spaces (Hahn-Mazurkiewicz Theorem).
\(c) connected manifolds[^4] (not necessarily compact or boundaryless) are generalized Peano continua and the later share important topological properties with the former: they are locally compact, *separable, arcwise connected, locally arcwise connected* [@NA p.131-132] and *admit a complete geodesic metric* (Tominaga and Tanaka [@TO], following Bing [@B1; @B2]). This implies the existence of an equivalent metric $d$ for which: (1) every two points are joined through a geodesic arc (given any $x,y\in M$, there is an isometric embedding $\varphi:[0,d(x,y)]\hookrightarrow M$ with $\varphi(0)=x$ and $\varphi(d(x,y))=y$); (2) closed balls are compact (Hopf-Rinow-Cohn-Vossen Theorem [@BU p.51]). Generalized Peano continua are at the threshold of metric geometry. It is conjectured (Busemann [@BN]), that if every geodesic in an $n$-dimensional generalized Peano continuum $M$ is locally prolongable and prolongations are unique, then $M$ is a (boundaryless) $n$-manifold. This is confirmed in dimension $n\leq4$ and (apparently) open in higher dimensions (see [@HR; @BR]). However, even $\mathbb{R}^{2}$ embeddable (generalized) Peano continua may display intricate fractal-like geometry and topology (as geodesics can have multiple prolongations, see e.g. [@CH Chap.1],[@SA Chap.9] for examples).
(*neighbourhoods and distance in $M$*) $[M,d]$ denotes the space $M$ with metric $d$. Given $x\in M$ (resp. $Y\subset M$), $\mathcal{N}_{x}$ (resp. $\mathcal{N}_{Y}$) is the set of neighbourhoods of $x$ (resp. of $Y$) in $M$. For $X,\, Y\subset M$, $d(X,Y):=\mbox{inf}\{d(x,y):\, x\in X,\, y\in Y\}$.
The classical concept of Lyapunov stability of compact invariant sets is now introduced. It is crucial to understand the much stronger dynamical constraints this notion imposes in the non-wandering setting. Given the relevance of non-wandering flows commanded by conservative dynamics, we shall deduce, throughout this paper, from the general results, the corresponding characterizations that are specific to this particularly important context.
Consider a $C^{0}$ flow on $M$ and a nonvoid, compact invariant set $K\subset M$.
(*stable / unstable*) $K$ is *(Lyapunov)* *stable* if every $U\in\mathcal{N}_{K}$ contains a positively invariant $V\in\mathcal{N}_{K}$ $(V=\mathcal{O}^{+}(V)\subset U$, Otherwise it is *unstable*. If every $U\in\mathcal{N}_{K}$ contains a negatively invariant $V\in\mathcal{N}_{K}$, we say that $K$ is (-)stable. $K$ is *bi-stable* if it is both stable and (-)stable.
*Throughout this paper, every mention to stability/instability is always in Lyapunov’s sense.*
Non-wandering setting
---------------------
The following elementary facts will be implicitly used throughout this paper: (a) a nonvoid proper subset of a connected metric space has nonvoid boundary; (b) in a locally compact metric space, compact subsets have arbitrarily small compact neighbourhoods and both closed and open sets are locally compact; (c) a (+)invariant set has (+)invariant closure and interior.
It is easily seen that in a ** non-wandering flow, a stable (or (-)stable) $K$ is always bi-stable: together, the density of recurrent points (Poincaré Recurrence Theorem)[^5] and the continuity of the flow immediately imply that $$K\mbox{ \,\emph{is unstable} }\Longleftrightarrow\, K\mbox{ \,\emph{is} }\mbox{(-)}unstable\quad\mbox{(Fig. 2.1)}$$
Therefore, in the context of *non-wandering flows*, a compact invariant set $\emptyset\neq K\subsetneq M$ is either
1. *stable $\Longleftrightarrow$* (-)*stable $\Longleftrightarrow$ bi-stable*,\
in which case $K$ *has arbitrarily small compact invariant neighbourhoods*; due to the connectedness of $M$, this implies that *for any* $U\in\mathcal{N}_{K}$, $U\setminus K$ *contains infinitely many compact minimal sets* (*Proof.* if $K$ is bi-stable and $U_{0}\in\mathcal{N}_{K}$, then $U_{0}\setminus K$ contains a compact minimal set: take a compact $U\in\mathcal{N}_{K}$, $M\neq U\subset U_{0}$. As $K$ is bi-stable, there are $V_{0},V_{1}\in\mathcal{N}_{K}$ such that $\mathcal{O}^{-}(V_{0}),\,\mathcal{O}^{+}(V_{1})\subset U$, hence $V:=\overline{\mathcal{O}(V_{0}\cap V_{1})}\subset U$. Now $V\neq M$ is a compact invariant neighbourhood of $K$ and $M$ is connected, thus $\mbox{bd\,}V\subset U_{0}$ is a nonvoid compact invariant set disjoint from $K$ and it contains at least one compact minimal set. Therefore, since there are arbitrarily small compact invariant neighbourhoods of $K$, $U_{0}\setminus K$ contains infinitely many compact minimal sets), or
2. *unstable $\Longleftrightarrow$* (-)*unstable*,\
in which case there is an $U\in\mathcal{N}_{K}$ and points $z$ arbitrarily near $K$ such that $$\mathcal{O}^{-}(z)\not\subset U\mbox{ and }\mathcal{O}^{+}(z)\not\subset U$$
i.e. points arbitrarily near $K$ escape from $U$ both in the past and future. It should be mentioned that, in case (2), Ura-Kimura-Bhatia Theorem [@BH p.114] implies that, in the absence of a pair of points $x,\, y\in K^{c}$ such that $\,\emptyset\neq\alpha(x),\,\omega(y)\subset K$, a kind of partial stability still takes place near $K$: there are points $z\in K^{c}$ whose orbits remain forever (past and future) arbitrarily near $K$ (i.e., for all $U\in\mathcal{N}_{K}$, there are orbits $\mathcal{O}(z)\subset U\setminus K$). Obviously, both phenomena may coexist in dimension $n\geq2$.
(Lyapunov) hyper-stability
--------------------------
We now introduce the central concept of this paper, Lyapunov hyper-stability. Briefly, the compact minimal sets are partitioned into Lyapunov stable and unstable ones, the hyper-stable being those that are *not* in the (Hausdorff metric) closure of the unstable. This actually amounts to having a neighbourhood in $M$ intersecting no (Lyapunov) unstable compact minimal set (see below).
($\mbox{CMin}$, $\mathcal{S}$, $\mathcal{U}$) $\mbox{CMin}$ is the set of compact minimals of the flow and $\mathcal{\mathcal{S}},\,\mathcal{U}\subset\mbox{CMin}$ the subsets of those that are, respectively, (Lyapunov) stable and unstable. For $U\subset M$, $$\mbox{CMin}(U)=\mbox{set of compact minimals contained in }U$$ $\mathcal{S}(U)$, $\mathcal{U}(U)$ are defined in an analogue way.
$\mbox{CMin}(U)$ and more generally $\mbox{Ci}(U)$, the set of nonvoid, compact invariant subsets of $U$ are naturally endowed with the Hausdorff metric $d_{H}$, $\mbox{Ci}(U)$ being compact, if $U$ is compact (see e.g. [@TE p.233]). The subscript $_{H}$ stands for Hausdorff metric concepts, e.g. $\mbox{int}_{H}X$, $\mbox{cl}_{H}X$ and $\mbox{bd}_{H}X$ are, respectively, the interior, closure and boundary of $X\subset[\mbox{CMin},\, d_{H}]$.
Let $\varLambda\in\mbox{CMin}$. The following three statements seem ordered by increasing strength:
1. every neighbourhood of $\varLambda$ in $M$ intersects some $\varGamma\in\mathcal{U}$
2. every neighbourhood of $\varLambda$ in $M$ contains some $\varGamma\in\mathcal{U}$
3. there is a sequence $\varGamma_{n}\in\mathcal{U}$ such that $\varGamma_{n}\overset{d_{H}}{\longrightarrow}\varLambda$
However, they are actually equivalent. $(1)\Rightarrow(2):$ this is obvious if $\varLambda\in\mathcal{U}$, otherwise, given $U\in\mathcal{N}_{\varLambda}$, there is a compact, (+)invariant $U\supset V\in\mathcal{N}_{\varLambda}$ (Lemma , Remark , below). By (1), $V$ intersects some $\varGamma\in\mathcal{U}$, hence $\varGamma\subset V\subset U$ (as $\varGamma=\overline{\mathcal{O}^{+}(x)}$, for each $x\in\varGamma$); $(2)\Rightarrow(3):$ for each $n\geq$1, take a $\varGamma_{n}\in\mathcal{U}$ contained in $B(\varLambda,1/n).$ Then (Lemma , below), $\varGamma_{n}\overset{d_{H}}{\longrightarrow}\varLambda$; $(3)\Rightarrow(1):$ this is obvious since $d_{H}(\varGamma_{n},\varLambda)<\epsilon\implies\varGamma_{n}\subset B(\varLambda,\epsilon)$.
(*hyper-stable*, $H\mathcal{S}$) a compact minimal set $\varLambda$ is (Lyapunov) *hyper-stable* if some neighbourhood of $\varLambda$ in $M$ intersects *no* $\varGamma\in\mathcal{U}$. For any $U\subset M$, $H\mathcal{S}(U)$ is the set of hyper-stable $\varLambda\subset U$ and $H\mathcal{S}:=H\mathcal{S}(M)=\mbox{int}_{H}\mathcal{S}.$ Therefore, $H\mathcal{S}^{c}$ is the set of compact minimals satisfying the three equivalent conditions (1) to (3), i.e. $H\mathcal{S}^{c}=\mbox{cl}_{H}\mathcal{U}$ or, equivalently, $H\mathcal{S}=\mbox{int}_{H}\mathcal{S}$.
Examples
--------
The (frictionless) mathematical pendulum, described by $\overset{.}{x}=y$, $\overset{.}{y}=-\mbox{sin}\,2\pi x$, whose phase space is the cylinder $M=\mathbb{S}^{1}\times\mathbb{R}$. Every orbit, except three (one equilibrium with two homoclinic loops) is a hyper-stable compact minimal, and they are all periodic, with the exception of the lower equilibrium point, which is a centre. $[H\mathcal{S},d_{H}]$ is homeomorphic to the (separated) union of $[0,1)$ and two copies of $\mathbb{R}$.
Simple (but instructive) models of 3-dimensional volume-preserving dynamics near a periodic orbit are given by the $C^{\infty}$ vector fields $$\begin{array}{lll}
v_{\lambda}:M=\mathbb{S}^{1}\times\mathbb{D}^{2} & \longrightarrow & \mathbb{R}^{4}=\mathbb{C}^{2}\\
\qquad\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(z_{1},z_{2}) & \longmapsto & \big(iz_{1},i\lambda(|z_{2}|^{2})z_{2}\big)
\end{array}$$ where $\lambda\in C^{\infty}([0,1],\mathbb{R})$ and $|\cdot|$ is the euclidean norm (see Fig. ). Each torus $\mathbb{S}^{1}\times\mu\mathbb{S}^{1}$, $0<\mu\leq1$, is invariant and carries either a minimal subflow ($\lambda(\mu^{2})\in\mathbb{Q}^{c}$) or foliates into periodic orbits $(\lambda(\mu^{2})\in\mathbb{Q})$. Thus, every point $z\in M$ belongs to some $\varLambda\in\mbox{CMin}$. To understand the Lyapunov’s nature of these minimal sets, three cases are particularly relevant ($\simeq$ means homeomorphic):
1. $\lambda=const.\in\mathbb{Q}$, in which case $[H\mathcal{S},d_{H}]\simeq\mathbb{D}^{2}$. This case defines the local fibre structure of Seifert fibrations [@E1 p.67];
2. $\lambda=const.\in\mathbb{Q}^{c}$, in which case $[H\mathcal{S},d_{H}]\simeq\mathbb{D}^{1}$;
3. $\lambda$ is constant in no nontrivial interval $I\subset[0,1]$, in which case $H\mathcal{S}=\emptyset$ (see below).
In general, $\lambda$ is constant on each component of a (possibly empty) open set $A\subset[0,1]$, and constant in no neighbourhood of each point $x\in A^{c}$. It follows that if $\mu^{2}\in A$, $0<\mu\leq1$, then every $\varLambda\in\mbox{CMin}$ contained in the invariant torus $\mathbb{S}^{1}\times\mu\mathbb{S}^{1}$ is hyper-stable, otherwise
- it is an unstable periodic orbit, if $\lambda(\mu^{2})\in\mathbb{Q}$ (due to the existence of a sequence of minimal tori $\mathbb{S}^{1}\times\mu_{n}\mathbb{S}^{1}$, $\mu_{n}\rightarrow\mu$, $\lambda(\mu_{n}^{2})\in\mathbb{Q}^{c}$, $d_{H}$ converging to the torus $\mathbb{S}^{1}\times\mu\mathbb{S}^{1}$ containing the periodic orbit $\varLambda$);
- $\varLambda=\mathbb{S}^{1}\times\mu\mathbb{S}^{1}$ is a stable, but not hyper-stable minimal torus, if $\lambda(\mu^{2})\in\mathbb{Q}^{c}$ (the stability of $\varLambda$ is forced by the existence of sequences of minimal tori $\mathbb{S}^{1}\times\xi_{n}\mathbb{S}^{1}$, $\xi_{n}\rightarrow\mu$, $\lambda(\xi_{n}^{2})\in\mathbb{Q}^{c}$, $d_{H}$ converging to $\varLambda$ from both sides, thus entailing the existence of arbitrarily small invariant neighbourhoods; $\varLambda$ is not hyper-stable due to the existence of unstable periodic orbits in tori $\mathbb{S}^{1}\times\zeta_{n}\mathbb{S}^{1}$, $\zeta_{n}\rightarrow\mu$, $\lambda(\zeta_{n}^{2})\in\mathbb{Q}$, and thus in every neighbourhood of $\varLambda=\mathbb{S}^{1}\times\mu\mathbb{S}^{1}$).
The core periodic orbit $\gamma=\mathbb{S}^{1}\times\{0\}$ is always stable, being hyper-stable iff $0\in A$.
Hyper-stability in flows with all orbits periodic
-------------------------------------------------
$\mbox{Per}:=\,$the set of periodic orbits of the flow.
If a flows with all orbits periodic induces a circle bundle on $M$, then $H\mathcal{S}=\mbox{Per}$ and $[H\mathcal{S},d_{H}]$ is homeomorphic to the base space of the bundle, which, however, may be a non-manifold (see Section ).
A classical example of a volume-preserving $C^{\omega}$ periodic flow[^6] inducing a nontrivial circle bundle is given by the Hopf flow ** on $\mathbb{S}^{2n+1}$ ($\mathbb{R}^{2n+2}$ identified with $\mathbb{C}^{n+1}$), $$(t,\, z)\longmapsto e^{it}z$$ The base space of the induced bundle is $\mathbb{C}P^{n}\simeq H\mathcal{S}$, the case $n=1$, $\mathbb{S}^{1}\hookrightarrow\mathbb{S}^{3}\rightarrow\mathbb{S}^{2}\simeq\mathbb{C}P^{1}$ being the original Hopf fibration [@DU p.230].
It is possible that $H\mathcal{S}=\mbox{Per}$, even if a flows with all orbits periodic does not induce a circle bundle. $C^{1}$ flows with all orbits periodic on orientable, closed 3-manifolds induce Seifert fibrations (Epstein, [@E1]), which in general are not circle bundles. This guarantees that $H\mathcal{S}=\mbox{Per}$ and also, that $[H\mathcal{S},d_{H}]$ is always a closed 2-manifold, a remarkable phenomenon unparalleled in the higher dimensions (see also Section , Example ).
However, there are $C^{\infty}$ flows with all orbits periodic on closed manifolds, for which $H\mathcal{S}\subsetneq\mbox{Per}$ i.e. $\mathcal{U}\neq\emptyset$. Sullivan [@SU] constructed a beautiful example of a such a flow (which can be made $C^{\omega}$), on a closed 5-manifold, for which there is no bound to the lengths of the orbits. This implies (Epstein [@E2 Theorem 4.3]) the existence of Lyapunov unstable orbits in the flow (take a convergent sequence $x_{n}\in\gamma_{n}$, where $\gamma_{n}$ is a sequence orbits with no bound on their lengths. Then, the orbit $\mathcal{O}(\mbox{lim}\, x_{n})$ is necessarily unstable).
Main lemmas. First consequences
===============================
Lemmas and constitute the core tools for the comprehension of the dynamics near hyper-stable compact minimals. Lemma , in particular, essentially shows how each component of $H\mathcal{S}$ inherits the local topological structure of the phase space (i.e. its local compactness and local connectedness).
Every has arbitrarily small compact, connected, (+)invariant neighbourhoods $U$ in $M$ such that *$\mbox{CMin}(U)\subset H\mathcal{S}.$*
Except for the the conclusion $\mbox{CMin}(U)\subset H\mathcal{S}$, the same holds if $\varLambda\in\mathcal{S}$.
$M$ is locally compact and $\varLambda\in H\mathcal{S}$, hence $\mbox{\ensuremath{\varLambda}}$
Suppose that
*(a)* $K,\, K\text{'}$ are nonvoid, compact invariant sets;
*(b)* *$\varLambda_{n}\in\mbox{CMin}$*;
*(c)* $\alpha(x),\,\omega(x)$ are nonvoid, compact limit sets. Then,
1. $\varLambda_{n}\overset{d_{H}}{\longrightarrow}K$ and $K'\subsetneq K$ $\Longrightarrow$ $K'$ is unstable;
2. $K'\subsetneq\omega(x)$ $\Longrightarrow$ $K'$ is unstable;
3. $K\subset\alpha(x)$ is stable $\Longrightarrow$ $x\in K=\alpha(x)$;
4. if the flow is non-wandering and $\omega(x)$ is stable, then $x\in\omega(x)$.
In each case, assume that the hypothesis hold.
\(1) take $y\in K\setminus K'$ and let $d=d(y,K')$. Given $\epsilon>0$, take $n\in\mathbb{N}$ such that $d':=d_{H}(\varLambda_{n},K)<\mbox{min}(\epsilon,d/2)$. Take $a,b\in\varLambda_{n}$ such that $$\begin{array}{c}
d(a,K')\leq d'<\epsilon\mbox{ \,\ and \,}d(b,y)\leq d'<d/2\end{array}$$ As $\varLambda_{n}$ is minimal, for some $t>0$, $$d(a^{t},b)<d/2-d(b,y)$$ thus implying that $$d(a^{t},y)\leq d(a^{t},b)+d(b,y)<d/2$$ which by its turn implies $d(a^{t},K')>d/2$. Therefore, $K'$ is unstable, as points arbitrarily near $K'$ escape $B(K',d/2)$ in positive time.
\(2) let $y\in\omega(x)\setminus K'$. Given $\epsilon_{0}>0$, take $$0<\epsilon<\mbox{min}\big(\epsilon_{0},d(y,K')/2\big)$$ Then, there are $0<t<T$ such that $d(x^{\, t},K')<\epsilon<\epsilon_{0}$ and $d(x^{T},y)<\epsilon<d(y,K')/2$, the last inequality implying $d(x^{T},K')>d(y,K')/2$, hence $K'$ is unstable as in (1).
\(3) $\emptyset\neq K\subset\alpha(x)$ and $x\not\in K$ obviously implies that $K$ is unstable, therefore $x\in K$ and $K=\alpha(x)$ (as $K$ is a closed invariant set).
\(4) $\omega(x)$ has arbitrarily small invariant neighbourhoods (Section ), hence no point outside $\omega(x)$ may have $\omega(x)$ as $\omega$-limit.
Let $U$ be a nonvoid, compact, connected, (+)invariant set such that *$\mbox{CMin}(U)\subset H\mathcal{S}$.* Then:
1. ($\omega$-limits are in $H\mathcal{S}(U)$) $x\in U\implies\emptyset\neq\omega(x)\in H\mathcal{S}(U)$;
2. ($\omega$-limits convergence) $U\ni x_{n}\rightarrow x\implies\omega(x_{n})\overset{d_{H}}{\longrightarrow}\omega(x)$;
3. (escaping orbits) $x\in U$ and $x\not\in\omega(x)\implies\mathcal{O}^{-}(x)\not\subset U$;
4. ($H\mathcal{S}(U)$ is a continuum) *$H\mathcal{S}(U)$* is $d_{H}$ compact and connected.
if $C\subset\mbox{CMin}$, then $C^{*}:=\underset{\varLambda\in C}{\cup}\varLambda\subset M$.
\(1) since $U$ is compact and (+)invariant, $\emptyset\neq\omega(x)\subset U$ is compact, for every $x\in U$ and it must be a minimal set (stable by hypothesis), otherwise there is an unstable compact minimal set $\varLambda\subsetneq\omega(x)$ (Lemma ), contradicting $\mbox{CMin}(U)\subset H\mathcal{S}$.
\(2) let $U\ni x_{n}\rightarrow x$. Since $U$ is compact and the $\omega(x_{n})$’s are nonvoid compact invariants, by Blaschke Principle ([@TE p.233]), there is a subsequence $\omega(x_{n'})\overset{d_{H}}{\longrightarrow}\varGamma$, for some $\varGamma\in\mbox{Ci}(U)$ (see Remark above). Reasoning by contradiction, suppose that $\varGamma\neq\omega(x)$. Take $z\in\varGamma\setminus\omega(x)\neq\emptyset$ ($\omega(x)$ is minimal by (1)). Then, by continuity of the flow, there are sequences $0<t_{n}<T_{n}$ such that $$d\big(x_{n}^{t_{n}},\,\omega(x)\big)\rightarrow0\mbox{ \,\ and \,}d\big(x_{n}^{T_{n}},\, z\big)\rightarrow0$$ which implies that $\omega(x)$ is unstable, contradicting (1). Hence, every subsequence $\omega(x_{n'})$ of $\omega(x_{n})$ contains a subsequence $\omega(x_{n''})\overset{d_{H}}{\longrightarrow}\omega(x)$, trivially implying that $\omega(x_{n})\overset{d_{H}}{\longrightarrow}\omega(x)$.
\(3) suppose that $x\in U$ and $x\not\in\omega(x)$. Then, $\mathcal{O}^{-}(x)\not\subset U$, otherwise $\emptyset\neq\alpha(x)\subset U$ is compact, implying that $\alpha(x)$ contains a compact minimal set $\varGamma$ and $x\not\in\varGamma$ (since $x\not\in\omega(x))$, thus implying (Lemma ) that $\varGamma\in\mbox{CMin}(U)$ is unstable, contradiction. Actually, $$\exists\, T\leq0:\, x^{(-\infty,T)}\subset U^{c}\mbox{ and }\; x^{[T,+\infty)}\subset U$$ since $U$ being compact and (+)invariant, $$-\infty<T:=\mbox{min}\big\{ t\leq0:\, x^{[t,0]}\subset U\big\}\leq0$$ is well defined and has the required properties.
(4.1) $H$$\mathcal{S}(U)$ *is $d_{H}$ compact*: given a sequence $\varLambda_{n}\in H\mathcal{S}(U)=\mbox{CMin}(U)$, take a convergent subsequence $x_{n'}\in\varLambda_{n'}$ ($U$ is compact). Then, $\varLambda_{n'}=\omega(x_{n'})$ and by (2) and (1) $$\omega(x_{n'})\overset{d_{H}}{\longrightarrow}\omega(\mbox{lim\,}x_{n'})\in H\mathcal{S}(U).$$
(4.2) $H$$\mathcal{S}(U)$ *is $d_{H}$ connected*: reasoning by contradiction assume it is not. Then, by (4.1), $H\mathcal{S}(U)$ is the union of two nonvoid, disjoint compacts $C_{0},$ $C_{1}\subset H\mathcal{S}(U)$.
*Claim.* $C_{0}^{*},\, C_{1}^{*}\subset U$ *are nonvoid, disjoint compacts*: they are obviously nonvoid. Fix $j\in\{0,1\}$. Let $x_{n}\in C_{j}^{*}$. Each $x_{n}$ belongs to some $\varLambda_{n}\in C_{j}$. Take a convergent subsequence $\varLambda_{n'}\overset{d_{H}}{\longrightarrow}\varLambda\in C_{j}$. $U$ being compact, $x_{n'}$ has a convergent subsequence $x_{n''}\rightarrow x\in U$. But
Now let $$\varOmega_{j}:=\big\{ x\in U:\,\omega(x)\in C_{j}\big\},\,\mbox{ for \,}j=0,1$$ We show that these two sets form a nontrivial partition of $U$ into closed subsets, thus getting a contradiction. Since $C_{0}\cap C_{1}=\emptyset$, $\varOmega_{0}\cap\varOmega_{1}=\emptyset$; $\varOmega_{0}\neq\emptyset\neq\varOmega_{1}$ since $C_{j}^{*}\subset\varOmega_{j}$. Also, since $C_{0},\, C_{1}\subset H\mathcal{S}(U)$ are closed, by (2), both $\varOmega_{0}$ and $\varOmega_{1}$ are closed in $U$. Finally, by (1), $U=\varOmega_{0}\sqcup\varOmega_{1}$, hence $U$ is disconnected.
Let $\mathfrak{N}$ be a neighbourhood of $\varLambda$ in $H\mathcal{S}$. By Lemmas and (below), there is
Topological detection of Lyapunov instability
---------------------------------------------
We can now establish criteria permitting to detect the presence of Lyapunov unstable compact minimal sets in arbitrarily small neighbourhoods of a given compact minimal set $\varLambda$ i.e. criteria detecting that $\varLambda\in\mbox{cl}_{H}\mathcal{U}$. The following result is an immediate consequence of Lemma (as $H\mathcal{S}^{c}=\mbox{cl}_{H}\mathcal{U}$ is a closed subset of $\mbox{CMin}$). Although trivially $1\implies$2, we list both conditions as it is often useful to have the simplest possible sufficient conditions in mind.
Let *$\varLambda\in\mbox{CMin}$.* If any of the following three conditions holds,
1. *$\mbox{CMin}$* is not locally connected at $\varLambda$;
2. $\varLambda$ has no locally connected neighbourhood in *$\mbox{CMin}$*;
3. $\varLambda$ has no compact neighbourhood in *$\mbox{CMin}$.*
From Corollary 1 we can draw some interesting conclusions: for instance, if $\mbox{CMin}$ is nowhere locally connected, ** then every neighbourhood of each compact minimal set contains infinitely many Lyapunov unstable compact minimals. This follows observing that nowhere locally connected sets have no isolated elements. Actually, we have the following stronger
[Criterion]{}
: *If any of the following two conditions holds,then every neighbourhood of each compact minimal set of the flow contains infinitely many Lyapunov unstable compact minimals:*
1. $\mbox{CMin}$ *is not locally connected at a dense subset of its points;*[^7]
2. *no point of* $\mbox{CMin}$ *has both a compact neighbourhood and a locally connected neighbourhood.*
(*attractor* / *repeller*) a nonvoid, compact invariant set $\varDelta$ is an *attractor* if it is (Lyapunov) stable and $$B^{+}(\varDelta):=\{x\in M:\,\emptyset\neq\omega(x)\subset\varDelta\}\in\mathcal{N}_{\varDelta}$$ i.e. if it is asymptotically stable. $B^{+}(\varDelta)$ is the *attraction basin* of ** $\varDelta$. $\mathcal{A}\subset H\mathcal{S}$ is the set of (compact) minimal attractors. $\varDelta$ is a *repeller* if it is an attractor in the time-reversed flow. Its *repulsion basin* is defined in the obvious way. Note that, when the phase space $M$ is compact, $M$ is always both an attractor and a repeller in any flow. This is the only possible attractor (resp. repeller) if the flow is non-wandering (as attractors are simultaneously stable and isolated, see () in Section and Definition below).
Local connectedness of $H\mathcal{S}$ (Lemma ), permits a straightforward deduction of topological-dynamical results that otherwise seem somewhat surprising. For instance, while $\mathcal{A}\subset H\mathcal{S}$*,* hyper-stable compact minimals cannot be surrounded by attractors: **
Let $\varLambda$ be a compact minimal set.
1. $\varLambda$ is an attractor (the unique one inside its basin), or
2. *$\varLambda\in\mbox{cl}_{H}\mathcal{U}$* i.e. every neighbourhood of $\varLambda$ in $M$ also contains Lyapunov unstable compact minimal sets.
Therefore, if $\varLambda\in H\mathcal{S}$ then attractors $\varGamma\neq\varLambda$ cannot occur arbitrarily near $\varLambda$.
The existence of a sequence of attractors $\mbox{\ensuremath{\varDelta}}_{n}\neq\varLambda\in\mbox{CMin}$ such that $\varDelta_{n}\subset B(\varLambda,1/n)\subset M$ thus implies the existence of another sequence $\mathcal{U}\ni\varGamma_{n}\overset{d_{H}}{\longrightarrow}\varLambda$.
Assume that $\varLambda\not\in\mbox{cl}_{H}\mathcal{U}$. By Lemma $\varLambda$ has connected neighbourhoods in $\mbox{CMin}$. We show that the only possible such neighbourhood is $\{\varLambda\}$ i.e. $\varLambda$ is $d_{H}$ isolated in $\mbox{CMin}$, hence $\varLambda$ is an attractor (see Corollary ahead). Let $\mathfrak{N}$ be a connected neighbourhood of $\varLambda$ in $\mbox{CMin}$. there is an open neighbourhood $U$ of $\varLambda$ in $M$ such that $d_{H}(\varGamma,\varLambda)<\epsilon/2$ for every compact minimal set $\varGamma\subset U$. By hypothesis, there is an attractor $Q\subset U$ and $Q$ contains at least one $\varGamma\in\mbox{CMin}$. Clearly the attraction basin $B(Q)$ is an open invariant set containing $Q$ such that $B(Q)\setminus Q$ intersects no compact minimal. Therefore, all compact minimals belonging to the connected component $\varTheta$ of $\varGamma$ in $\mbox{CMin}$ must be contained in $Q$, hence $\mbox{diam}_{H}\varTheta<\epsilon$.
$H\mathcal{S}$ is locally a Peano continuum
-------------------------------------------
One of our main goals is to prove that $H\mathcal{S}$ is locally a Peano continuum. Observe that Lemma does not prove this i.e. it does not show that each $\varLambda\in H\mathcal{S}$ has arbitrarily small compact, connected and locally connected neighbourhoods in $H\mathcal{S}$ (in relation to the Hausdorff metric $d_{H}$). While, by Lemma , $\varLambda$ has arbitrarily small compact, connected, (+)invariant neighbourhoods $U$ in $M$ such that $\mbox{CMin}(U)\subset H\mathcal{S}$, implying, by Lemma , that $H\mathcal{S}(U)$ is a continuum and thus (Lemma ) that $H\mathcal{S}$ is locally a continuum, the difficulty of following this approach lies in guarantying that $U$ is locally connected at every $x\in\mbox{bd\,}U$. We overcome this difficulty taking a more direct path: we show, through Peano Continuum Theory, that $H\mathcal{S}$ is locally a Peano continuum.
We first show that
\(1) *Peano continua are locally Peano continua*: let $X$ be a Peano continuum. Given $x\in X$ and $U\in\mathcal{N}_{x}$, take $\epsilon>0$ such that $B(x,\epsilon)\subset U$. $X$ is the union of finitely many Peano continua $X_{i}$ with $\mbox{diam\,}X_{i}<\epsilon$ [@NA p.124, 8.10]. Let $V$ be the union of the $X_{i}$’s that contain $x$, say $V=\cup_{i=1}^{^{n}}X_{i}$. $V$ is compact and connected (i.e. a continuum) and $U\supset V\in\mathcal{N}_{x}$, since $V^{c}$ is contained in a compact not containing $x$.
$V$ *is locally connected*: given $z\in V$ and a neighbourhood $W$ of $z$ in $V$, let $\beta$ be the set of the $1\leq i\leq n$ such that $z\in X_{i}$. For each $i\in\beta$ take a connected neighbourhood $C_{i}$ of $z$ in $X_{i}$ contained in $W$. Since the $X_{i}$’s are compact and finite in number, it is easily seen that $C:=\cup_{i\in\beta}C_{i}\subset W$ is a connected neighbourhood of $z$ in $V$. Therefore, besides compact and connected, $V$ is locally connected and thus a Peano continuum, hence (1) is proved.
\(2) *Generalized Peano continua are locally Peano continua*: let $X$ be a generalized Peano continuum. Since (2) coincides with (1) if $X$ is compact assume it is not. Let $X^{\propto}=X\sqcup\{\infty\}$ be a 1-point compactification of $X$ (on $X$, the metric $d'$ of $X^{\propto}$ is equivalent to the original metric $d$ of $X$). Then $X^{\propto}$ is compact, connected and locally connected at every point $x\in X=X^{\propto}\setminus\{\infty\}$ and thus also at $\infty$ [@NA p.78, 5.13]. Hence $X^{\propto}$ is a Peano continuum, therefore, by (1), $X^{\propto}$ is locally a Peano continuum and so is its open subset $X$.
$H\mathcal{S}$ is
1. locally a Peano continuum,
2. the union of countably many disjoint, clopen, generalized Peano continua.
By Lemma , $H\mathcal{S}$ is $d_{H}$ locally connected, thus each component is open. As $H\mathcal{S}$ is also $d_{H}$ locally compact, each component is actually a generalized Peano continuum. Hence, (1) follows from Lemma . It remains to show that $H\mathcal{S}$ has countably many components. $H\mathcal{S}$ is $d_{H}$ separable, since $\mbox{CMin}\supset H\mathcal{S}$ is (see [@TE p.258, Lemma 10]). The conclusion follows, since the components of $H\mathcal{S}$ are open.
The next result is valid
see [@TE p.255, Lemma 4].
From Lemma we easily deduce
see [@TE p.255, Lemma 6 and its proof].
Global and local structure of $H\mathcal{S}$ and the dynamics around it
=======================================================================
We are now in possession of all the tools needed to prove the main results of this paper, both in the global and local settings (Sections and ). If the flow is non-wandering, the inherent dynamical constraints make these characterizations assume a particularly elegant form (Section ). Section calls attention to a crucial difference between the local topologies of $H\mathcal{S}\subset\mbox{CMin}$ and $H\mathcal{S}^{*}\subset M$ (this disparity vanishes in dimensions 1 and 2).
All results in this section hold for arbitrary continuous flows on connected manifolds (possibly noncompact or with boundary, see Remark ).
Arbitrary flows
---------------
### $H\mathcal{S}$ globally
(Global Structure of $H\mathcal{S}$) Let $\theta$ be a $C^{0}$ flow on a generalized Peano continuum $M$. Endowed with the Hausdorff metric, $H\mathcal{S}$ is the union of countably many disjoint, clopen, generalized Peano continua $X_{i}$ (each admitting a complete geodesic metric). Moreover, there are disjoint, connected, open invariant sets $A_{i}\subset M$ such that:
1. $X_{i}^{*}\subset A_{i}$
2. $A_{i}=\{x\in M:\,\emptyset\neq\omega(x)\in X_{i}\subset H\mathcal{S}\}$
3. for any $x\in A_{i}$, $x\not\in\omega(x)$ implies *$\alpha(x)\subset\mbox{bd}\, A_{i}$* (possibly $\alpha(x)=\emptyset$).
(see Fig. )
**(a) if $M$ is compact and the flow is minimal, then the unique $X_{i}$ is $\{M\}$ and $A_{i}=M$;**
\(b) if $H\mathcal{S}=\emptyset$, then the collection $\{X_{i}\}$ is empty;
\(c) if $X_{i}$ contains a unique $\varLambda\neq M$, then $\varLambda$ is a proper attractor and $A_{i}$ its basin;
\(d) otherwise, $X_{i}$ is a non-degenerate generalized Peano continuum and thus contains $\mathfrak{c}$ hyper-stable compact minimals ($X_{i}$ contains nontrivial arcs of $H\mathcal{S}$);
\(e) although $X_{i}^{*}$ may be noncompact, it roughly acts as an attracting set in the flow, with basin $A_{i}$ (see Fig. );
\(f) if $A_{i}=M$ then, for every $x\in M$, either $x\in\omega(x)\in H\mathcal{S}$ or $\alpha(x)=\emptyset$ (and in the later case, $M$ is noncompact).
$H\mathcal{S}$ is countable iff it is $d_{H}$ discrete iff every $\varLambda\in H\mathcal{S}$ is an attractor.
(*of Theorem* ) Let $\{X_{i}\}$ be the components of $H\mathcal{S}$. The first conclusion of Theorem is (2) of Corollary , together with Remark .
Let $A_{i}$ be defined as in (2). Then $\varLambda\in X_{i}\implies\omega(x)=\varLambda$, for every $x\in\varLambda$, thus $\varLambda^{*}\subset A_{i}$, hence (1) $X_{i}^{*}\subset A_{i}$. As the $X_{i}$’s are disjoint, so are the $A_{i}$’s. Since points in the same orbit have the same $\omega$-limit, each $A_{i}$ is invariant. It is also open: suppose that $x\in A_{i}$ i.e. $\omega(x)\in X_{i}$. Take a neighbourhood $U$ of $\omega(x)\in H\mathcal{S}$ in $M$ as in Lemma , sufficiently small so that $H\mathcal{S}(U)\subset X_{i}$ (Lemma , using the fact that $X_{i}$ is open in $H\mathcal{S}$). By continuity of the flow, taking a sufficiently small $B\in\mathcal{N}_{x}$, $\mathcal{O}^{+}(y)\cap\mbox{int\,}U\neq\emptyset$, for every $y\in B$, thus implying (Lemma ) $\omega(y)\in H\mathcal{S}(U)\subset X_{i}$ and thus $y\in A_{i}$. Therefore $A_{i}$ is open in $M$. $A_{i}$ is connected since $X_{i}^{*}$ is connected and for every $x\in A_{i}$, $\mathcal{O}(x)\cup\omega(x)\subset A_{i}$ is connected and $\omega(x)\subset X_{i}^{*}$ ($X_{i}^{*}$ is connected: this is trivial noting that any nontrivial partition of $X_{i}^{*}$ into closed sets entails a nontrivial partition of $X_{i}$ into closed sets, as each $\varLambda\in X_{i}$ is a minimal, thus connected).
It remains to prove (3). Suppose $x\in A_{i}$ and $x\not\in\omega(x)$. As $A_{i}$ is open and invariant, $\alpha(x)\subset\overline{A_{i}}=A_{i}\sqcup\mbox{bd\,}A_{i}$. Reasoning by contradiction, suppose that $y\in\alpha(x)\cap A_{i}\neq\emptyset$. Then, $\omega(y)\subset\alpha(x)$ and $\omega(y)\in X_{i}\subset H\mathcal{S}$. Since $x\not\in\omega(y)$ (otherwise $x\in\omega(x)$, as $\omega(y)$ is minimal), $\omega(y)$ is unstable (Lemma ), contradiction. Therefore $\alpha(x)\subset\mbox{bd\,}A_{i}$. If $M$ is noncompact it is obviously possible that $\alpha(x)=\emptyset$.
### $H\mathcal{S}$ locally
(Local behaviour) Let $\varLambda$ be a compact minimal set of a $C^{0}$ flow on a generalized Peano continuum $M$. Then, either
1. *$\varLambda\in\mbox{cl}_{H}\mathcal{U}$*, or
2. $\varLambda$ is an attractor, or
3. there are arbitrarily small compact, connected, (+)invariant neighbourhoods $U$ of $\varLambda$ in $M$ such that:
1. the (compact) minimal sets contained in $U$ are all Lyapunov hyper-stable and their set $H\mathcal{S}(U)$ is a non-degenerate Peano continuum;
2. for each $x\in U$, $\omega(x)\in H\mathcal{S}(U)$ and if $x\not\in\omega(x)$, then its negative orbit leaves $U$ (and thus never returns again).
(see Fig. )
$\bullet$ (1) holds iff $\varLambda\not\in H\mathcal{S}$ i.e. if there is a sequence $\mathcal{U}\ni\varLambda_{n}\overset{d_{H}}{\longrightarrow}\varLambda$ (Section ). If $\varLambda$ is unstable, then this trivially holds since $\varLambda_{n}:=\varLambda$ is such a sequence;
$\bullet$ if $\varLambda=M$ (i.e. $M$ is compact and the flow is minimal), then (2) trivially holds;
$\bullet$ (3.a) implies that, endowed with the Hausdorff metric, $H\mathcal{S}(U)$ *is compact, nontrivially arcwise connected ($\implies|H\mathcal{S}(U)|=\mathfrak{c}),$ locally arcwise connected and admits a complete geodesic metric* (see Remark );
$\bullet$ the final conclusion in (3.b) can be written as: $$\exists\, T\leq0:\, x^{(-\infty,T)}\subset U^{c}\mbox{ and }\; x^{[T,+\infty)}\subset U$$ (see the proof of Lemma ).
$\varLambda\in\mbox{CMin}$ is *isolated (from minimals)* if there is an $U\in\mathcal{N}_{\varLambda}$ containing no compact minimal set other than $\varLambda$. This is equivalent to $\varLambda$ being $d_{H}$ isolated in $\mbox{CMin}$ (by the $d_{H}$ metric definition and Lemma ).
If $\varLambda\in\mathcal{S}$ is isolated, then it is an attractor.
If $\varLambda\in\mathcal{S}$ is isolated, then neither (1) nor (3) of Theorem can hold, since both imply the existence of compact minimals $\varGamma\neq\varLambda$ contained in every neighbourhood of $\varLambda$.
(*of Theorem* ) The three conditions are mutually exclusive.
\(A) if $\varLambda\not\in H\mathcal{S}$, then (1) holds (Section ).
\(B) suppose $\varLambda\in H\mathcal{S}$. Let $X_{i}$ be the component of $H\mathcal{S}$ to which $\varLambda$ belongs (see the proof of Theorem ). We distinguish two cases:
(B.1) if $X_{i}=\{\varLambda\}$, then $\varLambda$ is stable and $A_{i}\in\mathcal{N}_{\varLambda}$ (Theorem ) is its region of attraction, thus $\varLambda$ is an attractor and $A_{i}$ its basin. Hence (2) holds.
(B.2) otherwise, let $\mathfrak{N}$ be a Peano continuum neighbourhood of $\varLambda$ in $H\mathcal{S}$, contained in $X_{i}$ ($X_{i}$ is open in $H\mathcal{S}$ (Theorem ) and $H\mathcal{S}$ is locally a Peano continuum (Corollary )). As $X_{i}$ is non-degenerate (i.e. $|X_{i}|>1$), so is $\mathfrak{N}$. Take a sufficiently small compact, connected, (+)invariant neighbourhood $V$ of $\varLambda$ in $M$ such that $\mbox{CMin}(V)\subset\mathfrak{N}$ (Lemmas ). Observe that $\mathfrak{N}^{*}\subset M$ is (a) compact: $\mathfrak{N}$ is $d_{H}$ compact, hence given sequences $x_{n}\in\varLambda_{n}\in\mathfrak{N}$, there is a convergent subsequence $\varLambda_{n'}\overset{d_{H}}{\longrightarrow}\varLambda\in\mathfrak{N}$. We may assume that all $\varLambda_{n'}$’s are contained in some compact neighbourhood $W$ of $\varLambda$ in $M$ ($\varLambda$ is compact and $M$ is locally compact). The conclusion follows since $x_{n'}$ has a convergent subsequence $x_{n''}\rightarrow x\in W$ and necessarily $x\in\varLambda$, as $x_{n''}\in\varLambda_{n''}\overset{d_{H}}{\longrightarrow}\varLambda$; (b) connected (as $X_{i}^{*}$ in the proof of Theorem ) and (c) invariant (union of minimal sets). Then $U:=V\cup\mathfrak{N}^{*}$ is a compact, connected, (+)invariant neighbourhood of $\varLambda$ in $M$ and $\mbox{CMin}(U)=\mathfrak{N}\subset H\mathcal{S}$. Thus (3.a) is proved; (3.b) follows from Lemma and , since $\mbox{CMin}(U)\subset H\mathcal{S}$.
### Topology of $H\mathcal{S}^{*}\subset M$
It is easily seen that, as $H\mathcal{S}$, $H\mathcal{S}^{*}\subset M$ is locally compact and that $\{X_{i}^{*}\}$ are its (clopen) components ($\{X_{i}\}$ being the components of $H\mathcal{S}$). But while $H\mathcal{S}$ is locally connected (in relation to the $d_{H}$ metric), the corresponding set of points $H\mathcal{S}^{*}$ needs not to be a locally connected subset of $M$. The local topology of the minimal sets $\varLambda\in H\mathcal{S}$ plays a determinant role here. Actually, $H\mathcal{S}^{*}$ may be nowhere locally connected, even if the flow is smooth. Handel’s by product example [@HA p.166] can be transferred to $\mathbb{S}^{2}$, to yield an orientation preserving $C^{\infty}$ diffeomorphism $f:\mathbb{S}^{2}\circlearrowleft$ with only three minimal sets, two repelling fixed points (the north and south poles $\pm p$) and an attracting pseudo-circle $P$, with basin $\mathbb{S}^{2}\setminus\{\pm p\}$. The pseudo-circle is nowhere locally connected. Taking the suspension of $f$, we get a $C^{\infty}$ flow $v^{t}$ on $\mathbb{S}_{f}^{2}\simeq\mathbb{S}^{2}\times\mathbb{S}^{1}$,[^8] with a nowhere locally connected attracting minimal set $\varLambda=P_{f}$ (locally, $\varLambda$ is homeomorphic to the Cartesian product of $\mathbb{D}^{1}$ and some open subset of $P$, hence it is nowhere locally connected). In this flow, $H\mathcal{S}=\{\varLambda\}\simeq\{0\}$ and $H\mathcal{S}^{*}=\varLambda$, therefore $H\mathcal{S}^{*}$ is nowhere locally connected. Examples of $C^{\infty}$ flows in higher dimensions, with $H\mathcal{S}^{*}$ nowhere locally connected, are generated by the vector fields $$(x,y)\longmapsto\big(v(x),0\big)\mbox{ \,\,\,\ on\,\,\,\ }(\mathbb{S}^{2}\times\mathbb{S}^{1})\times\mathbb{S}^{k},\mbox{ \,\,\,}k\geq1$$ where $v$ is the original vector field on $\mathbb{S}^{2}\times\mathbb{S}^{1}$. Then, $$H\mathcal{S}=\big\{ P_{f}\times\{y\}:\, y\in\mathbb{S}^{k}\big\}\simeq\mathbb{S}^{k}\mbox{ \,\,\,\ and \,\,\,\ensuremath{H\mathcal{S}^{*}=P_{f}\times\mathbb{S}^{k}}}$$ is locally homeomorphic to the Cartesian product of $\mathbb{D}^{k+1}$ and an open subset of $P$, hence nowhere locally connected.
However, for $C^{0}$ flows on arbitrary 2-manifolds (possibly nonorientable, noncompact, with boundary), $H\mathcal{S}^{*}$ is *always* a locally connected subset of $M$. Actually, each (clopen) component $X_{i}^{*}$ of $H\mathcal{S}^{*}$ either
1. contains more than 2 equilibria, in which case it consists entirely of equilibria, hence $X_{i}^{*}\simeq X_{i}$, implying that $X_{i}^{*}$ is locally connected (see Theorem ), or
2. contains no more than 2 equilibria, in which case $X_{i}^{*}$ is a (connected) $k$-manifold ($0\leq k\leq2)$.
The key fact to establish (2) is the following result of Athanassopoulos and Strantzalos [@AT]:
A Lyapunov stable compact minimal set of a $C^{0}$ flow on an arbitrary 2-manifold is either an equilibrium orbit, or a periodic orbit, or a torus.
The reader is invited to look up for the 13 possible manifolds (up to homeomorphism) that might occur as $X_{i}^{*}$ in case (2).
Non-wandering flows
-------------------
Note that, in virtue of Poincaré Recurrence Theorem, all results below are valid, not only, for conservative flows on finite volume manifolds, but also for any flows topologically equivalent to these ones (being non-wandering is a topological equivalence invariant). This includes conjugations via homeomorphisms and time reparametrizations.
The results in this section are, in general, false if the flow is *not* non-wandering (even when $M$ is compact), the north-south flow on $\mathbb{S}^{n}$ being a trivial counter-example to all of them.
(Global structure of $H\mathcal{S}$) Given a $C^{0}$ non-wandering flow on a generalized Peano continuum $M$, either
1. $H\mathcal{S}=\emptyset$ i.e. the Lyapunov unstable compact minimal sets are $d_{H}$ dense in *$\mathcal{\mbox{CMin}},$* or
2. $H\mathcal{S}=\{M\}$ i.e. $M$ is compact and the flow is minimal, or
3. $H\mathcal{S}$ is the union of $1\leq\beta\leq\aleph_{0}=|\mathbb{N}|$ disjoint, clopen, non-degenerate generalized Peano continua $X_{i\in\beta}$, each $X_{i}^{*}\subset M$ being a (nonvoid) connected, open invariant set.
The three conditions are mutually exclusive. Assume (1) and (2) fail. Then, by Theorem , $H\mathcal{S}$ is the union of $1\leq\beta\leq\aleph_{0}$ clopen generalized Peano continua $X_{i\in\beta}$ and these are non-degenerate since $|X_{i}|=1$ implies the unique $\varLambda\in X_{i}$ is a proper attractor (as (2) fails, $\varLambda\subsetneq M$), which is impossible in the non-wandering context (as minimal attractors are both stable and isolated, see in Section ). Let $A_{i}$ be as in Theorem . Then, $x\in A_{i}$ implies $\emptyset\neq\omega(x)\in X_{i}\subset H\mathcal{S}$ and $x\in\omega(x)$, as $\omega(x)$ is stable and the flow is non-wandering (Lemma ). Therefore, $X_{i}^{*}=A_{i}$ and $X_{i}^{*}$ is as claimed.
The following result shows that any cardinal limitation on the number of compact minimal sets implies either the minimality of the flow or the $d_{H}$ denseness in $\mbox{CMin }$of the unstable compact minimal sets.
If the flow is non-wandering and *$|\mbox{CMin}|<\mathfrak{c}$*, then either
1. $H\mathcal{S}=\emptyset$ i.e. *$\mbox{cl}_{H}\mathcal{U}=\mbox{CMin}$,* or
2. $M$ is compact and the flow is minimal.
Actually, in case (1), the isolated (and thus Lyapunov unstable) compact minimal sets are $d_{H}$ dense in *$\mbox{CMin}$.*
Case (3) of Theorem implies that $|\mbox{CMin}|=\mathfrak{c}$, thus $|\mbox{CMin}|<\mathfrak{c}$, implies that either (1) or (2) holds. The last sentence is an immediate consequence of Theorem in [@TE p.248]: $|\mbox{CMin}|<\mathfrak{c}$ implies $\mathfrak{M}_{10}=\emptyset$, thus $\mathfrak{M}_{1-6}$, the set of isolated compact minimals, is $d_{H}$ dense in $\mbox{CMin}$.
Note that, in general, despite the abundance of recurrent points, compact minimal sets may be extremely scarce in non-wandering flows (e.g. the only $\varLambda\in\mbox{CMin}$ may be an equilibrium orbit). Corollary shows that this phenomenon may only occur if isolated (unstable) compact minimal sets are $d_{H}$ dense in $\mbox{CMin}$.
(Local behaviour) Let $\varLambda$ be a compact minimal set of a $C^{0}$ non-wandering flow on a generalized Peano continuum $M$. Then, either
1. *$\varLambda\in\mbox{cl}_{H}\mathcal{U}$*, or
2. $\varLambda=M$ i.e. $M$ is compact and the flow is minimal, or
3. there are arbitrarily small compact, connected, invariant neighbourhoods $U$ of $\varLambda$ such that:
1. $U$ is the union of $\mathfrak{c}$ hyper-stable compact minimal sets i.e. $U=H\mathcal{S}(U)^{*}$;
2. $H\mathcal{S}(U)$ is a (non-degenerate) Peano continuum.
(see Fig. ).
Theorem is just Theorem in the non-wandering setting: (1) coincides in both theorems. If the flow is non-wandering, then (2) of Theorem can hold iff $\varLambda=M$ since there are no proper attractors. This is (2) of Theorem . Finally, let $U\in\mathcal{N}_{\varLambda}$ be as in (3) of Theorem . By (3.b) of Theorem , $$x\in U\implies\omega(x)\in H\mathcal{S}(U)$$ As the flow is non-wandering, this implies $x\in\omega(x)\subset U$ (Lemma ). Therefore, $U=H\mathcal{S}(U)^{*}$ is invariant and (3) holds.
Global absence of Lyapunov instability
======================================
Dichotomy
---------
It is well know that transitive dynamical behaviour precludes the existence of Lyapunov stable compact minimal sets $\varLambda\subsetneq M$. At the other extreme of the spectrum are flows without Lyapunov unstable compact minimals. *If the phase space $M$ is compact*, this imposes extremely strong dynamical constraints on the flow, the following dichotomy holding:
(minimality or fragmentation into $\mathfrak{c}$ stable minimals) A $C^{0}$ flow $\theta$ without Lyapunov unstable compact minimals on a Peano continuum $M$ is non-wandering. Actually, every point is recurrent and the flow is either
1. minimal, or
2. partitions $M$ into $\mathfrak{c}$ Lyapunov hyper-stable compact minimal sets, forming a non-degenerate Peano continuum (in the Hausdorff metric). Every nonvoid open set $A\subset M$ intersects $\mathfrak{c}$ minimal sets.
Thus, in a certain sense, Lyapunov unstable compact minimal sets are a vital ingredient of dynamical complexity and diversity of flows on compact phase spaces.
As $\mathcal{U}=\emptyset$, $\mbox{CMin}=H\mathcal{S}$. Since $M$ is compact, for each $x\in M$, $\alpha(x)\neq\emptyset$ is compact and thus contains a compact minimal set $\varGamma$. Necessarily, $x\in\varGamma=\alpha(x)$, otherwise $\varGamma$ is unstable (Lemma ). Thus, every $x$ belongs to some $\varGamma\in H\mathcal{S}$ and the flow is non-wandering, with all points recurrent. Since $H\mathcal{S}^{*}=M$, if the flow is not minimal, then $|H\mathcal{S}|>1$, and by Lemma applied to $U:=M$, $H\mathcal{S}$ is $d_{H}$ compact and connected, hence by Theorem , a non-degenerate Peano continuum (thus $|H\mathcal{S}|=\mathfrak{c}$). It remains to prove the last sentence in (2). Let $x\in M$ and $\epsilon>0$. We show that $B(x,\epsilon)$ intersects $\mathfrak{c}$ minimals: $x$ belongs to some $\varLambda\in H\mathcal{S}$; as $H$$\mathcal{S}$ is a non-degenerate Peano continuum, there are $\mathfrak{c}$ distinct minimals $\varGamma_{i\in\mathbb{R}}\in H\mathcal{S}$ such that $d_{H}(\varGamma_{i},\varLambda)<\epsilon$, thus implying $\varGamma_{i}\cap B(x,\epsilon)\neq\emptyset$.
The last conclusion in (2) means that the “local density” of the hyper-stable compact minimal sets is actually $\mathfrak{c}$ all over $M$.
The same conclusion holds if the flow has no (-)unstable compact minimal sets, as the time reversed flow $\phi(t,x)=\theta(-t,x)$ contains no unstable compact minimal set and thus Theorem is valid for $\phi$, hence also for $\theta,$ as its conclusions are preserved under time-reversal.
This result, valid for arbitrary flows on compact (connected) manifolds, is, in general, false when $M$ is noncompact (trivial counter-examples include the flow on $\mathbb{R}^{n}$ generated by the vector field $v(z)=-z$ and the parallel flows $\partial/\partial x_{i}$. If $M$ is noncompact, the flow may contain no minimal sets, compact or not, see Inaba [@IN], Beniere, Meigniez [@BE]). Actually, compactness of $M$ plays the key role in the above proof. If $M$ is noncompact (i.e. a noncompact generalized Peano continuum, e.g. a noncompact manifold), then limit sets may be empty or noncompact and thus need not contain compact minimal sets. This actually implies that, in the noncompact setting, absence of Lyapunov unstable compact minimals has no analogue constraining effect on the dynamical diversity and complexity of the flow. In particular, $H\mathcal{S}$ may be empty or discrete (i.e. every $\varLambda\in H\mathcal{S}$ may be an attractor). Without entering into details, these flows may exhibit “quite freely” (on $M$ or on invariant noncompact submanifolds), an abundance of dynamical phenomena e.g. non-minimal ergodic behaviour, absence of minimal sets (compact and noncompact) etc.
Fragmentation into $\mathfrak{c}$ hyper-stable minimal submanifolds
-------------------------------------------------------------------
In virtue of Theorem , a non-minimal $C^{0}$ flow on a (connected) closed manifold $M$, displaying no unstable minimal sets, partitions $M$ into $\mathfrak{c}$ hyper-stable minimal sets $\{\varLambda_{i\in\mathbb{R}}\}$. If all these $\varLambda_{i}$ are submanifolds (necessarily closed and connected, of possibly non-fixed codimension $\geq1$), then given the nature of the examples that usually come to one’s mind, it is tempting to ask if $[\mbox{CMin},d_{H}]=[\{\varLambda_{i\in\mathbb{R}}\},d_{H}]$flows has all orbits periodic[^9]In dimension 2 this is actually straightforward, using Gutierrez Smoothing Theorem [@GU] and Poincaré Index Theorem: only the torus and the Klein bottle carry flows without equilibria. For flows with all orbits periodic on $\mathbb{T}^{2},$ $[\mbox{Per},d_{H}]\simeq\mathbb{S}^{1}$ and on $\mathbb{K}^{2}$, The 3-dimensional case was established earlier by Epstein [@E1], assuming that the flows with all orbits periodic is $C^{1}$ and $M$ orientable, with no a priori restrictions on the Lyapunov’s nature of the orbits (see Section ). However, in the higher dimensions, the landscape changes radically: there are $C^{\infty}$ periodic flows on closed $n$-manifolds (for every $n\geq4$), for which $[\mbox{CMin},d_{H}]=[\mbox{Per},d_{H}]=H\mathcal{S}$ is a *non*-manifold (see Example below).
Cannon and Daverman [@CA] constructed remarkable examples of periodic $C^{0}$ flows on $M=N\times\mathbb{S}^{1}$, $N$ any $C^{\infty}$ boundaryless $(n\geq3)$-manifold, on which every orbit is a wildly embedded $\mathbb{S}^{1}$! By construction, these periodic flows induce trivial circle bundles. As the fibres are wild in $M$, no point of the bundle’s base space $\varTheta$ has a neighbourhood homeomorphic to $\mathbb{B}^{n}$, hence $H\mathcal{S}\simeq\varTheta$ is nowhere a manifold. These flows have nowhere an $n$-cell cross section, and thus are nowhere topologically equivalent to $C^{1}$ flows, all this showing that $C^{0}$ dynamics harbours topological phenomena unparalleled in the differentiable setting (even locally).
The following construction provides examples of $C^{\infty}$ periodic flows with all orbits Lyapunov stable, on closed manifolds in all dimensions $n\geq4$, for which the space of orbits $\mbox{Per}\simeq\varTheta$ is a *non*-manifold. The construction of the underlying tangentially orientable foliations, which could hardly be simpler, was, essentially, kindly communicated to us by Professor Robert Daverman [@D5].
For $n\geq4$, let $f:\mathbb{S}^{n-1}\circlearrowleft$ act as the orthogonal reflection on the north-south axis $[-p,p]$, $p=(0,0,\ldots,0,1)$ (this is the compactification of $\mathbb{R}^{n-1}\circlearrowleft:x\mapsto-x$). The (semifree) $\mathbb{Z}/2\mathbb{Z}$ action determined by $f$ fixes $\pm p$ and the orbit space $\varTheta:=\mathbb{S}^{n-1}/f$ is homeomorphic to the topological suspension of $\mathbb{\mathbb{R}}P^{n-2}$. As $n\geq4$, $\mathbb{R}P^{n-2}\not\hookrightarrow\mathbb{R}^{n-1}$ [@MA], hence $\varTheta$ is a non-manifold (with singular points $\pm p$).
Let $F$ be the free $\mathbb{Z}/2\mathbb{Z}$ action on $\mathbb{S}^{1}\times\mathbb{S}^{n-1}$ which acts as the antipodal map on the 1st factor and as $f$ on the 2nd. Since the action is $C^{\infty},$ finite and free, the corresponding orbit space $M=(\mathbb{S}^{1}\times\mathbb{S}^{n-1})/F$ is a connected, $C^{\infty}$ closed $n$-manifold. The image of the circles $\mathbb{S}^{1}\times\{y\}$, under the quotient map $h:\mathbb{S}^{1}\times\mathbb{S}^{n-1}\rightarrow M$, are circles defining a tangentially orientable $C^{\infty}$ 1-foliation of $M$, with all leaves Lyapunov stable. Now, starting with the periodic vector field $(z_{1},z_{2})\mapsto(iz_{1},0)$ on $\mathbb{S}^{1}\times\mathbb{S}^{n-1}$, we get, via the quotient map, a $C^{\infty}$ vector field on $M$, tangent to the resulting foliation. Thus $H\mathcal{S}=\mbox{Per}$ is the space of leaves, which, by construction, is homeomorphic to $\varTheta$, a non-manifold (see above). Also, trivially, the flow is periodic with period $2\pi$ (identifying $\mathbb{S}^{1}$ with $\mathbb{R}/2\pi\mathbb{\mathbb{Z}}$), the two orbits corresponding to the image (under the quotient) of each circle $\mathbb{S}^{1}\times\{\pm p\}$ have minimal period $\pi$. All other orbits have minimal period $2\pi$.
Final remarks. Open questions
=============================
Assuming the phase space $M$ of the flow to be a (generalized) Peano continuum, the topological characterization of $H$$\mathcal{S}$ given by Theorem is optimal: it is easily seen that if a metric space $\mathfrak{M}$ is the union of countably many disjoint, clopen, generalized Peano continua, then there is a $C^{0}$ flow on a Peano continuum for which $H\mathcal{S}\simeq\mathfrak{M}$. We sketch the proof in the case $\mathfrak{M}$ is noncompact and has denumerably many components (the other cases are easier).
Let $\mathfrak{M}=\sqcup_{i\in\mathbb{N}}X_{i}$, where each $X_{i}$ is a nonvoid, clopen, generalized Peano continuum. For each $i\in\mathbb{N}$, take a $C^{0}$ flow $\theta_{i}$ on $\mathbb{D}_{i}^{1}=[-1_{i},1_{i}]\simeq\mathbb{D}^{1}$, with (exactly) three equilibria $-1_{i}$, $0_{i}$, $1_{i}$, $\{0_{i}\}$ a repeller. From each $X_{i}$ select a point $z_{i}$. Connect $z_{i}$ to $z_{i+1}$ pasting $-1_{i}$ to $z_{i}$ and $1_{i}$ to $z_{i+1}$ (the $\mathbb{D}_{i}^{1}$’s are disjoint, except that the pasting induces the identification $1_{i}\equiv-1_{i+1}$). Define the $C^{0}$ flow $\theta$ on $\mathfrak{N}=\mathfrak{M}\cup\cup_{i\in\mathbb{N}}\mathbb{D}_{i}^{1}$, which coincides with $\theta_{i}$ on $\mathbb{D}_{i}^{1}$ and has each $x\in\mathfrak{M}$ has an equilibrium. $\mathfrak{N}$ is a noncompact generalized Peano continuum and thus has an 1-point compactification $\mathfrak{N}^{\propto}=\mathfrak{N}\sqcup\{0_{\infty}\}$, which is a Peano continuum (see (2) in the proof of Lemma ). The flow $\theta$ automatically extends to a $C^{0}$ flow on $\mathfrak{N}^{\propto}$, with $0_{\infty}$ becoming an equilibrium. Now, let $\phi$ be a $C^{0}$ flow on $\mathbb{D}^{1}$ with (exactly) two equilibria, $-1$ and $1$, $\{-1\}$ an attractor. Paste $-1$ to $z_{1}$ and $1$ to $0_{\infty}$. This defines a $C^{0}$ flow on the Peano continuum $$M:=\mathfrak{N}^{\propto}\cup\mathbb{D}^{1}$$ with $\mathcal{S}=H\mathcal{S}=\big\{\{x\}:\, x\in\mathfrak{M}\big\}\simeq\mathfrak{M}$ and $\mathcal{U}=\big\{\{0_{i}\}:\, i\in\mathbb{N}\sqcup\{\infty\}\big\}$.
A more difficult question is the following:
Assuming that $\mathfrak{M}$ (see above) is finite dimensional,[^10]
when is $\mathfrak{M}$ homeomorphic to the $H\mathcal{S}$ set of some flow on a manifold?
We restrict our attention to the simpler problem:
If $K\subset\mathbb{S}^{n}$ is a Peano continuum, under which conditions is there a flow on $\mathbb{S}^{n}$ such that:
\(a) each point $x\in K$ is an equilibrium and
\(b) $H\mathcal{S}=\big\{\{x\}:x\in K\big\}\simeq K$ ?
For $n=2$, such a flow exists iff $K^{c}$ has finitely many components, and it can be made of class $C^{\infty}$ (if $\mathbb{S}^{2}$ is replaced by any compact manifold, it is easily seen that this condition remains necessary for the existence of such a flow, even of class $C^{0}$, see below). Hence the answer is positive, for example, if $K$ is homeomorphic to Wazewski’s universal dendrite ([@NA p.181],[@CH p.12]), but negative if it is homeomorphic to Sierpinski’s universal plane curve (“Sierpinski’s carpet”, [@NA p.9],[@CH p.31],[@SA p.160]). Our existence proof relies heavily on Riemann Mapping Theorem.
**Synopsis.** (*Existence*) excluding trivialities, suppose $K\subsetneq\mathbb{S}^{2}$ is a non-degenerate Peano continuum. Through topology, each component $A_{i}$ of $K^{c}$ is simply connected, hence there is a biholomorphism $\zeta_{i}:\mathbb{B}^{2}\rightarrow A_{i}$ (Riemann Mapping Theorem). We use this to put a $C^{\infty}$ vector field $v_{i}$ on each $A_{i}$, with a repelling equilibrium $O_{i}$, such that $A_{i}$ is its repulsion basin and $v_{i}$ extends to the whole $\mathbb{S}^{2}$, letting $v_{i}=0$ on $A_{i}^{c}$. Define $v=\sum v_{i}$ on $\mathbb{S}^{2}$. Clearly $v|_{A_{i}}=v_{i}$ and $\mbox{CMin}=\big\{\{x\}:\, x\in K\big\}\sqcup\big\{\{O_{i}\}\big\}_{i}$. By topology again, $\mbox{bd\,}A_{i}$ is locally connected, hence $\zeta_{i}$ extends continuously to $\zeta_{i}:\mathbb{D}^{2}\rightarrow\overline{A_{i}}$. This ensures that, for each $p\in\mbox{bd\,}A_{i}\subset K$, $\{p\}$ is a stable equilibrium orbit, actually hyper-stable, since the unstable minimals are the finitely many repellers $\{O_{i}\}$.
\(A) *Existence.* The cases $K=\emptyset$, $|K|=1$ and $K=\mathbb{S}^{2}$ are trivial. Assume that $K\subsetneq\mathbb{S}^{2}$ is non-degenerate Peano continuum such that $K^{c}$ has finitely many components $\{A_{i}\}$. Identify $\mathbb{R}^{2}$ with $\mathbb{C}$ and $\mathbb{S}^{2}$ with the Riemann sphere $\mathbb{C}\cup\{\infty\}$.
\(1) *Claim. Each $A_{i}$ is biholomorphic to* $\mathbb{B}^{2}=\mbox{int\,}\mathbb{D}^{2}$: let $\gamma$ be an $\mathbb{S}^{1}$ embedded in $A_{i}$. By Schoenflies Theorem, we may reason as if $\gamma$ is the standard equator $\mathbb{S}^{1}\times\{0\}$. Being connected and disjoint from $\gamma$, $K$ is contained in one open hemisphere, say the north one. Then, the closed south hemisphere is contained in $A_{i}$ (being a connected subset of $K^{c}$ containing $\gamma\subset A_{i}$), hence $\gamma$ is contractible to a point inside $A_{i}$. Therefore, as $A_{i}\neq\emptyset$ is open, simply connected and $|A_{i}^{c}|>1$, there is a biholomorphic map $\zeta_{i}:\mathbb{B}^{2}\rightarrow A_{i}\subset\mathbb{S}^{2}$ (Riemann Mapping Theorem).
\(2) as $K$ is Peano continuum, so is $\mbox{bd\,}A_{i}\subset K$ ([@KU]). This implies ([@PO p.18]) that $\zeta_{i}$ (uniquely) extends to a $C^{0}$ map $\zeta_{i}:\mathbb{D}^{2}\rightarrow\overline{A_{i}}$. It is easily seen that $\zeta_{i}$ maps $\mathbb{S}^{1}=\mbox{bd\,}\mathbb{D}^{2}$ onto $\mbox{bd\,}A_{i}=\overline{A_{i}}\setminus A_{i}$ (in general not injectively).
\(3) Take $\lambda\in C^{\infty}(\mathbb{R}^{2},[0,1])$ such that $\lambda^{-1}(0)=(\mathbb{B}^{2})^{c}$. Let $\upsilon$ be the vector field $\mathbb{R}^{2}\circlearrowleft:\, z\mapsto\lambda(z)z$. Transfer $\upsilon|_{\mathbb{B}^{2}}$ to $A_{i}$ via $\zeta_{i}$, getting $\upsilon_{i}=\zeta_{i_{*}}\upsilon|_{\mathbb{B}^{2}}\in\mathfrak{X}^{\infty}(A_{i})$. By Kaplan Smoothing Theorem [@KA p.157], there is $\mu_{i}\in C^{\infty}(\mathbb{S}^{2},[0,1])$ such that $\mu_{i}^{-1}(0)=A_{i}^{c}$ and $$\begin{array}{lllc}
v_{i}:\mathbb{S}^{2} & \longrightarrow & \mathbb{R}^{3}\\
\quad\;\,\, z & \longmapsto & \mu_{i}\upsilon_{i}(z) & \mbox{ on \,}A_{i}\\
\quad\;\,\, z & \longmapsto & 0 & \mbox{ on \,}A_{i}^{c}
\end{array}$$ defines a $C^{\infty}$ vector field on $\mathbb{S}^{2}$, whose restriction to $A_{i}$ is topologically equivalent to $\upsilon|_{\mathbb{B}^{2}}$ via $\zeta_{i}$. Let $v=\sum v_{i}\in\mathfrak{X}^{\infty}(\mathbb{S}^{2})$. Note that $v|_{A_{i}}=v_{i}.$ Its set of equilibria is $K\sqcup\{O_{i}\}_{i}$, where $O_{i}=\zeta_{i}(0)$. The corresponding equilibrium orbits are the only minimal sets of the flow $v^{t}$. Each $\{O_{i}\}$ is a repeller and $A_{i}$ its repulsion basin. For each $O_{i}\neq z\in A_{i}$, $\alpha(z)=\{O_{i}\}$ and $\omega(z)=\{p\}$, for some $p\in\mbox{bd\,}A_{i}$. Since $\zeta_{i}:\mathbb{S}^{1}\rightarrow\mbox{bd\,}A_{i}$ is onto, every equilibrium $p\in\mbox{bd\,}A_{i}$ is the $\omega$-limit of at least one $z\in A_{i}$ (Fig. ).
\(4) *Claim.* $H\mathcal{S}=\big\{\{x\}:\, x\in K\big\}$.
We show that each $y\in K$ has arbitrarily small (+)invariant neighbourhoods. If $y\in\mbox{int\,}K$, this is obvious since $y$ has a neighbourhood consisting of equilibria (see 3). Otherwise, given $\epsilon>0$, we get a (+)invariant neighbourhood $D_{i}\subset B(y,\epsilon)$ of $y$ in $\overline{A_{i}}$, for each component $A_{i}$ of $K^{c}$ such that $y\in\mbox{bd\,}A_{i}$. Then, as the number of components is finite, $\big(B(y,\epsilon)\cap K\big)\cup(\cup D_{i})\subset B(y,\epsilon)$ is a (+)invariant neighbourhood of $y$ in $\mathbb{S}^{2}$.
Suppose $y\in\mbox{bd\,}A_{i}$. As $\zeta_{i}:\mathbb{D}^{2}\rightarrow\overline{A_{i}}$ is $C^{0}$, $\beta_{i}=\zeta_{i}^{-1}(y)\subset\mathbb{S}^{1}$ is compact and $B_{i}=\zeta_{i}^{-1}\big(B(y,\epsilon)\cap\overline{A_{i}}\big)$ is an open neighbourhood of $\beta_{i}$ in $\mathbb{D}^{2}$. For each $x\in\beta_{i}$, take a “conic” open, (+)invariant neighbourhood $C_{x}$ of $x$ in $\mathbb{D}^{2}$, contained in $B_{i}$ ($\mathbb{D}^{2}$ is invariant under the flow $\upsilon^{t}$). Let $C_{i}$ be a finite union of $C_{x}$’s covering $\beta_{i}$. Then, $\big(B(y,\epsilon)\cap K\big)\cup\big(\cup_{y\in\mbox{bd\,}A_{i}}\zeta_{i}(C_{i})\big)$ is a (+)invariant neighbourhood of $y$ in $\mathbb{S}^{2}$, contained in $B(y,\epsilon)$. Therefore $\{y\}\in\mathcal{S}$. The only other minimals are the finitely many repellers $\{O_{i}\}$, which are necessarily away from $\mathcal{S},$ hence $H\mathcal{S}=\mathcal{S}=\big\{\{x\}:\, x\in K\big\}$.
\(B) Finally, we prove that if $K^{c}$ has infinitely many components, then there is no such flow (even of class $C^{0}$). Reasoning by contradiction, suppose there is such a flow. Let $\{A_{n}\}_{n\in\mathbb{N}}$ be the distinct components of $K^{c}$.
*Claim. Each open invariant set $A_{n}$ contains an unstable minimal set*: ** let $z\in A_{n}$. As $\mathbb{S}^{2}$ is compact, $\alpha(z)\neq\emptyset$ is compact and thus contains a minimal set $\varLambda_{n}$. Clearly, $\varLambda_{n}\subset A_{n}$, otherwise $\varLambda_{n}\subset\mbox{bd\,}A_{n}\subset K$ which implies $\varLambda_{n}=\{x\}$, for some $x\in K$ (by hypothesis , each $x\in K$ is an equilibrium) and $\{x\}$ is unstable (Lemma ), contradicting hypothesis . By hypothesis (b), $\varLambda_{n}\not\in H\mathcal{S}$ i.e. $\varLambda_{n}\in\mbox{cl}_{H}\mathcal{U}$. Take $\varGamma_{n}\in\mathcal{U}$ sufficiently $d_{H}$ near $\varLambda_{n}$ so that it is contained in $A_{n}$.
Now, by Blaschke Principle ([@TE p.223]), $\varGamma_{n}$ has a subsequence $\varGamma_{n'}$ $d_{H}$ converging to some nonvoid, compact invariant set $\varGamma\subset\mathbb{S}^{2}$. As the $A_{n}$’s are disjoint, open and invariant, $\varGamma\subset K$. But then, for each $x\in\varGamma\subset K$, $\{x\}\in\mbox{cl}_{H}\mathcal{U}=H\mathcal{S}^{c}$, contradicting $\{x\}\in H\mathcal{S}$ (hypothesis (b)): this is obvious if $\varGamma=\{x\}$, as $\mathcal{U}\ni\varGamma_{n'}\overset{d_{H}}{\longrightarrow}\varGamma$. Otherwise $\{x\}\subsetneq\varGamma$, thus $\{x\}\in\mathcal{U}$ (Lemma ).
[References]{} R. Abraham, J. Marsden, *Foundations of Mechanics*, Addison-Wesley, 2nd edition, 1987.
K. Athanassopoulos, P. Strantzalos, *On minimal sets in 2-manifolds*, J. Reine Angew. Math. 388 (1988), 206-211.
J.-C. Beniere, G. Meigniez, *Flows without minimal set*, Ergodic Th. & Dynam. Sys. 19 (1999), 21-30.
V. Berestovskii, D. Halverson, D. Repovs, *Locally G-homogeneous Busemann G-spaces,* Diff. Geom. Appl. 29 (2011), 299-318.
N. Bhatia, G. Szegö, *Stability theory of dynamical systems*, Springer-Verlag, ** Berlin, 1970.
R. H. Bing, *A convex metric for a locally connected continuum*, Bull. Amer. Math. Soc. 55 (1949), 812-819.
R. H. Bing, *Partitioning a set,* Bull. Amer. Math. Soc. 55 (1949), 1101-1110.
R. H. Bing, *Partitially continuous decompositions,* Proc. Amer. Math. Soc. 6 (1955), 124-133.
G. Birkhoff, *Dynamical systems,* American Mathematical Society Colloquium Pub. vol.IX, 1927.
D. Burago, Y. Burago, S. Ivanov, *A course in metric geometry*, Graduate studies in mathematics vol. ** 33, ** AMS*,* Providence, Rhode Island, 2001. **
H. Busemannn, *The Geometry of geodesics,* Academic Press, 1955.
J. Cannon, R. Daverman, *A totally wild flow,* Indiana J. of Math. 20(3) (1981), 371-387.
J. Charatonik, P. Krupski, P. Pyrih, *Examples in Continuum Theory,* May 2003 (unfinished electronic version available online).
R. Daverman, *Decompositions of manifolds,* Academic Press, 1986.
R. Daverman, *Decompositions into codimension one submanifolds*, Comp. Math. 55(2) (1985), 185-207.
R. Daverman, J. Walsh, *Decompositions into codimension-two submanifolds,* Trans. Amer. Math. Soc. 288(1) (1985), 273-291.
R. Daverman, *Decompositions into codimension two submanifolds: the nonorientable case,* Topol. and its Appl. 24 (1986), 71-81.
R. Daverman, private communication, 24-01-2014.
B. Dubrovin, A. Fomenko, S. Novikov, *Modern geometry - methods and applications, Part II. The geometry and topology of manifolds*, Springer Verlag, 1985.
D. Epstein, *Periodic flows on three-manifolds,* Annals of Math. 95 (1972), 68-82.
D. Epstein, *Foliations with all leaves compact,* Ann. Inst. Fourier, Grenoble 26(1) (1976), 265-282.
C. Gutierrez, *Smoothing continuous flows on two-manifolds and recurrences,* Ergod. Th. & Dynam. Sys. 6(1) (1986), 17-44.
M. Handel, *A pathological area preserving $C^{\infty}$ diffeomorphism of the plane*, Proc. Amer. Math. Soc. 86(1) (1982), 163-168.
D. Halverson, D. Repovs, *The Bing-Borsuk and the Busemann conjectures,* Math. Commun. 13 (2008), no.2, 163-184.
W. Hurewicz, H. Wallman, *Dimension Theory*, Princeton Univ. Press, 1948.
T. Inaba, *An example of a flow on a non-compact surface without minimal set*, Ergod. Th. & Dynam. Sys. 19 (1999), 31-33.
W. Kaplan, *The singularities of continuous functions and vector fields*, Michigan Math. J. 9(2) (1962), 151-159.
S. Kobayachi and K. Nomizu, *Foundations of differential geometry vol.* *I*, Interscience Publishers*,* 1963. **
K. Kuratowski, *Sur une propriété des continues Péaniens,* Fund. Math. 15 (1930), 180-184.
A. Lyapunov, *Problème générale de la stabilité du mouvement,* Ann. Fac. Sci. Toulouse 9 (1907), 203-474; (reprinted) Ann. Math. Studies 17, Princeton Univ. Press, 1947.
W. Massey, *On the imbeddability of the real projective spaces in euclidean space,* Pacific J. of Math. 9 (1959), 783-785.
J. Moser, *Stable and random motions in dynamical systems,* Annals Math. Studies 77, Princeton Univ. Press, 1973.
J. Moser, C. Siegel, *Lectures on Celestial Mechanics,* Springer, Grundlehren Bd. 187, 1971.
S. Nadler, *Continuum Theory*, ** Marcel Decker, 1992.
S. Newhouse, *Quasi-elliptic periodic points in conservative dynamical systems*, ** Amer. J. Math. 99(5) (1977), 10611087.
Ch. Pommerenke, *Boundary Behaviour of Conformal Maps*, Springer-Verlag, 1992.
H. Sagan, *Space-filling curves*, Springer-Verlag, 1994.
D. Sullivan, *A counterexample to the periodic orbit conjecture*, Publications I.H.E.S. 46(1) (1976), 5-14.
P. Teixeira, *Flows near compact invariant sets. Part I,* Fundamenta Mathematicae 223(3) (2013), 225-272.
A. Tominaga, T. Tanaka, *Convexification of locally connected generalized continua,* J. Sei. Hiroshima Univ. Ser A 19 (1955), 301-306.
E. Zehnder, *Homoclinic points near elliptic fixed points,* Comm. Pure Appl. Math. 26 (1973), 131-182.
E. Zehnder, *Hommage à Jürgen Moser*, Gaz. Math. No. 59 (1994), 57-66.
\
\
**
[^1]: $\mathcal{U}$ is the set of Lyapunov unstable compact minimals sets of the flow.
[^2]: Cantor star: union of the closed radii of $\mathbb{D}^{2}$ connecting the origin to a Cantor subset of $\mathbb{S}^{1}=\partial\mathbb{D}^{2}$.
[^3]: For instance, every $\mathbb{S}^{n\geq1}$ carries a $C^{\infty}$ flow with $\mathcal{S}$ and $\mathcal{U}$ both nowhere locally compact and nowhere locally connected (always in relation to the Hausdorff metric). It is also easily seen that every $n$-dimensional compact metric space $K$ (even if nowhere locally connected) is homeomorphic to the set $\mathcal{S}$ of all Lyapunov stable compact minimals of some $C^{\infty}$ flow on $\mathbb{R}^{2n+2}$. The same is true for $\mbox{bd}_{H}\mathcal{S}$, $\mathcal{S}\cap\mbox{bd}_{H}\mathcal{S}$, $\mbox{cl}_{H}\mathcal{S}$, $\mbox{\ensuremath{\big(}cl}_{H}\mathcal{S}\big)\setminus\mathcal{U}$, $\mathcal{U}$, $\mbox{int}_{H}\mathcal{U},$ $\mathcal{U}\cap\mbox{bd}_{H}\mathcal{U}$, $\mbox{cl}_{H}\mathcal{\mathcal{U}}$, $\mbox{\ensuremath{\big(}cl}_{H}\mathcal{\mathcal{U}}\big)\setminus\mathcal{S}$ (obviously, $\mbox{int}_{H}$, $\mbox{bd}_{H}$, $\mbox{cl}_{H}$ stand for interior boundary and closure of subsets of $\mbox{CMin }$in relation to the Hausdorff metric and $\mathcal{U}=\mathcal{S}^{c}:=\mbox{CMin\ensuremath{\setminus}}\mathcal{S}$). These examples already give an idea of how topologically arbitrary these sets may be in comparison with $\mbox{int}_{H}\mathcal{S}$.
[^4]: An $n$-*manifold* (briefly, a *manifold*) is a separable metric space, locally homeomorphic to the $n$-cell $\mathbb{D}^{n}=\{x\in\mathbb{R}^{n}:\,|x|\leq1\}$. Thus, manifolds are 2nd countable and Hausdorff (but possibly disconnected, noncompact, with boundary). As usual, *closed manifold* means compact and boundaryless. Except if otherwise mentioned, *all manifolds considered are assumed to be connected*.
[^5]: Here we are invoking the following topological version of this celebrated result: *the set of recurrent points of a non-wandering $C^{0}$ flow, on a locally compact metric space $M$, is Baire residual and thus dense in $M$* (*Proof.* for each $0\neq n\in\mathbb{Z}$, let $R_{n}=\{x\in M:\,\exists\, t\in[1,+\infty)n\,:\, d(x^{t},x)<1/|n|\}$. Then $R_{n}$ is open by continuity of the flow and dense in $M$ since the flow is non-wandering. Therefore, the set of recurrent points, $R=\underset{0\neq n\in\mathbb{Z}}{\cap}R_{n}$, is Baire residual in $M$).
[^6]: A flow $\theta$ without equilibria is *periodic* if there is a $t>0$ such that $\theta^{t}=\mbox{Id}$.
[^7]: See the Cantor star example in the introduction to this paper.
[^8]: We may actually identify these two manifolds and suppose $v^{t}$ defined on $\mathbb{S}^{2}\times\mathbb{S}^{1}$, as homeomorphic 3-manifolds are $C^{\infty}$ diffeomorphic.
[^9]: Observe that the partitions (decompositions) of $M$ into closed submanifolds we are considering are upper semicontinuous (usc) in the standard sense [@B3; @D1], as required in [@D2; @D3; @D4]. This follows immediately from Lemma . They are in fact *continuous* in the standard sense [@B3]: $d(\varLambda_{n},\,\varLambda_{0})\rightarrow0$ implies $\varLambda_{n}\overset{d_{H}}{\longrightarrow}\varLambda_{0}$ (Lemmas and ).
[^10]: Every $n$-dimensional, separable metric space embeds in $\mathbb{R}^{2n+1}$, hence in any $(2n+1)$-manifold (Menger-Nöbeling-Hurewicz Theorem, see e.g. [@HU p.60]).
|
---
abstract: 'In my talk I will present an overview of our recent work involving the use of supersymmetric quantum mechanics (SUSY-QM). I begin by discussing the mathematical underpinnings of SUSY-QM and then discuss how we have used this for developing novel theoretical and numerical approaches suitable for studying molecular systems. I will conclude by discussing our attempt to extend SUSY-QM to multiple dimensions.'
author:
- 'Eric R. Bittner'
- 'Donald J. Kouri'
title: |
\
Quantum dynamics and super-symmetric quantum mechanics.
---
A first date
============
My introduction to supersymmetric quantum mechanics (SUSY) came quite by accident. I had heard of SUSY in the context of high-energy physics where the SUSY theory postulates that for every fermion there is boson of equal mass (i.e. energy). This comes about because for every quantum Hamiltonian there is a partner Hamiltonian that has the same energy spectrum above the ground-state of the original system. In other words, above the ground state of $H_{1}$, each higher-lying eigenstate is partnered with an eigenstate of $H_{2}$. In particle physics, the $H_{1}$ “sector” is populated by bosons and the $H_{2}$ sector by fermions and SUSY predicts that the lowest lying fermion state is energetically degenerate with the first excited boson state. Evidence for SUSY has proven to be elusive and it is now believed that SUSY is a broken symmetry.
Last January (Jan-09) at a conference dedicated to Bob Wyatt, my co-author suggested that we look at SUSY as a way to develop new computational methods and approaches. Up until now, SUSY has been more of a mathematical technique that has been used more or less as a way to obtain stationary solutions to the Schrödinger equation for the variety of one-dimensional potential systems. In this paper and in my talk, I will discuss some of the work we have been doing in developing “SUSY” inspired methods for performing quantum many-body calculations and quantum scattering calculations. I shall begin with a brief overview of the SUSY theory and some of its elementary results. I shall then discuss how we have used the approach to develop both analytical and numerical solutions of the stationary Schrödinger equation. I will conclude by discussing our recent extension of SUSY to higher dimensions and for scattering theory.
Mathematical considerations
===========================
Before discussing some of our recent results, it is important to introduce briefly the mathematical formulation of SUSY quantum mechanics.
Hamiltonian formulation of SUSY
-------------------------------
In quantum theory, there is a fundamental connection between a bound state and its potential. This is simple to demonstrate by writing the Schrödinger equation for the stationary states as $$\begin{aligned}
V_{1}(x) - E_{n} = -\frac{\hbar^2}{2m}\frac{1}{\psi_{n}}\partial_x^{2}\psi_{n} = Q[\psi_{n}]\end{aligned}$$ where we recognize the right-hand side as the Bohm quantum potential which will certainly be discussed repeatedly at this conference. One of the remarkable consequences of this equation is that every stationary state of a given potential has the same functional form for its quantum potential $Q$. Thus, knowing any bound state allows a global reconstruction of the potential, $V(x)$ up to a constant energy shift.
SUSY is obtained by factoring the Schrödinger equation into the form [@Witten:1981bq; @Witten:1982nx; @Cooper:1995jt] $$\begin{aligned}
H\psi = A^{+}A \psi_{o}^{(1)} = 0 \label{susy-hamiltonian}\end{aligned}$$ using the operators $$\begin{aligned}
A = \frac{\hbar}{\sqrt{2m}} \partial_{x}+ W \,\,{\rm and}\,\,
A^{+} = -\frac{\hbar}{\sqrt{2m}}\partial_{x} + W .\end{aligned}$$ Since we can impose $A\psi_{o}^{(1)} = 0$, we can immediately write that $$\begin{aligned}
W(x) = -\frac{\hbar}{\sqrt{2m}}\partial_x \ln\psi_{o}.\end{aligned}$$ $W(x)$ is the [*superpotential*]{} which is related to the physical potential by a Riccati equation. $$\begin{aligned}
V(x) = W^{2}(x) - \frac{\hbar}{\sqrt{2m}}W'(x). \label{re}\end{aligned}$$ The SUSY factorization of the Schrödinger equation can always be applied in one-dimension.
From this point on we label the original Hamiltonian operator and its associated potential, states, and energies as $H_{1}$, $V_{1}$, $\psi_{n}^{(1)}$ and $E_{n}^{(1)}$. One can also define a partner Hamiltonian, $H_{2} = AA^{+}$ with a corresponding potential $$\begin{aligned}
V_{2} = W^{2} + \frac{\hbar}{\sqrt{2m}}W'(x).\end{aligned}$$ All of this seems rather circular and pointless until one recognizes that $V_{1}$ and its partner potential, $V_{2}$, give rise to a common set of energy eigenvalues. This principle result of SUSY can be seen by first considering an arbitrary stationary solution of $H_{1}$, $$\begin{aligned}
H_{1} \psi_{n}^{(1)} = A^{+}A\psi_{n} = E_{n}^{(1)}\psi_{n}^{(1)}.\end{aligned}$$ This implies that $(A\psi_{n}^{(1)})$ is an eigenstate of $H_{2}$ with energy $E_{n}^{(1)}$ since $$\begin{aligned}
H_{2}(A\psi_{n}^{(1)}) = AA^{+}A\psi_{n}^{(1)} = E_{n}^{(1)}(A\psi_{n}^{(1)}).\end{aligned}$$ Likewise, the Schrödinger equation involving the partner potential $H_{2}\psi_{n}^{(2)} = E_{n}^{(2)}\psi_{n}^{(2)} $ implies that $$\begin{aligned}
A^{+}AA^{+}\psi_{n}^{(2)} = H_{1}(A^{+}\psi_{n}^{(2)}) = E_{n}^{(2)}(A^{+}\psi_{n}^{(2)}).\label{chargeop}\end{aligned}$$ This (along with $E_{o}^{(1)} = 0$ ) allows one to conclude that the eigenenergies and eigenfunctions of $H_{1}$ and $H_{2}$ are related in the following way: $
E_{n+1}^{(1)} = E_{n}^{(2)},
$ $$\begin{aligned}
\psi_{n}^{(2)} = \frac{1}{\sqrt{E_{n+1}^{(1)}}} A \psi_{n+1}^{(1)},\, \,{\rm and} \,\,
\psi_{n+1}^{(1)} = \frac{1}{\sqrt{E_{n}^{(2)}}} A^{+} \psi_{n}^{(2)} \label{ops}\end{aligned}$$ for $n > 0$. [^1] Thus, the [*ground state of $H_{2}$ has the same energy as the first excited state of $H_{1}$*]{}. If this state $\psi_{o}^{(2)}$ is assumed to be node-less, then $\psi_{1}^{(1)} \propto A^{+}\psi_{o}^{(2)}$ will have a single node. We can repeat this analysis and show that $H_{2}$ is partnered with another Hamiltonian, $H_{3}$ whose ground state is isoenergetic with the first excited state of $H_{2}$ and thus isoenergetic with the second excited state of the original $H_{1}$. This hierarchy of partners persists until all of the bound states of $H_{1}$ are exhausted.
SUSY algebra
------------
We can connect the two partner Hamiltonians by constructing a matrix super-Hamiltonian operator $$\begin{aligned}
{\bf H} =
\left(
\begin{array}{cc}
H_1 & 0 \\
0 & H_2 \\
\end{array}
\right)\end{aligned}$$ and two matrix “super-charge” operators $$\begin{aligned}
{\bf Q}
=
\left(
\begin{array}{cc}
0 & 0 \\
A & 0\\
\end{array}
\right)
= A \sigma_{-} \end{aligned}$$ and $$\begin{aligned}
{\bf Q}^{+}
=
\left(
\begin{array}{cc}
0 & A^{+} \\
0 & 0\\
\end{array}
\right) = A^{+} \sigma_{+}\end{aligned}$$ where $\sigma_{\pm}$ are $2\times 2$ Pauli spin matrices. Using these we can re-write the SUSY Hamiltonian as $$\begin{aligned}
{\bf H}
=
\left(-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}} + W^{2}\right) \sigma_{o}
+W'\sigma_{z}\end{aligned}$$ The operators $\{ {\bf H}, {\bf Q}, {\bf Q}^{+}\} $ form a closed algebra (termed the Witten superalgebra) with $$\begin{aligned}
[ {\bf H}, {\bf Q} ] &=& [ {\bf H}, {\bf Q}^{+} ] = 0 \\
\{ {\bf Q}, {\bf Q} \} &=& \{ {\bf Q}^{+}, {\bf Q}^{+} \} = 0 \\
\{ {\bf Q}, {\bf Q}^{+} \} &=& {\bf H}\end{aligned}$$ The first algebraic relation is responsible for the degeneracy of the spectra of $H_{1}$ and $H_{2}$ and the supercharges transform an eigenstate of one sector into an eigenstate of the other sector.
As an example and perhaps a better connection to the physics implied by this structure, consider the case of a one-dimensional particle with an internal spin degree of freedom and with $[x,p] = i $ denoting the position and momentum of the particle. Conserved SUSY would imply that all non-diagonal coupling terms between the bosonic (coordinate) and fermionic (spin) degrees of freedom are exactly zero. This of course is equivalent to making the Born Oppenheimer approximation for a two-state system coupled to a continuous field $x(t)$. In this case, SUSY is preserved so long as $d_{t}\psi(x(t),t) = \partial_{t} \psi(x(t),t)$. SUSY is broken when $\dot x(t) \partial_{x}\psi(x(t),t) \ne 0$ which would lift the degeneracy between the states of $H_{1}$ and $H_{2}$.
Scattering in one dimension
---------------------------
The SUSY approach is not limited to bound-state problems. For a one-dimensional scattering system, it is straightforward to apply the SUSY theory to determine a relation between between the transmission and reflection coefficients of the supersymmetric partners. Asymptotically, we can assume that $W(x) \to W_{\pm}$ as $x\to \pm \infty$. In the same limit, the partner potentials become $V_{1,2} \to W_{\pm}^{2}$. For a plane wave incident from the left with energy $E$ scattering from $V_{1,2}$, we require the following asymptotic forms: $$\begin{aligned}
\lim_{x\to-\infty}\psi^{(1,2)}(k,x) &\sim &e^{ikx} + R^{(1,2)}e^{-ikx} \\
\lim_{x\to+\infty}\psi^{(1,2)}(k',x)&\sim& T^{(1,2)}e^{ik'x} \end{aligned}$$ We can derive a relation between the two scattering states by using the relation $\psi^{(1)}(k,x) = N A^{+}\psi_{2}(k',x)$. For the left-hand components ($x\to -\infty$). $$\begin{aligned}
%\psi^{(1)}(k,x\to-\infty) = N \left(-\frac{\hbar}{\sqrt{2m}}\frac{d}{dx} + W_{-}\right)\psi^{(2)}(k,x\to -\infty) \nonumber \\
e^{ikx}+R^{(1)}e^{-ikx} = N\left[
\left(-{i k } + \tilde W_{-}\right)e^{ikx}
+\left({i k } + \tilde W_{-}\right)e^{-ikx}
\right]\end{aligned}$$ where in the last line we have incorporated the $\hbar/\sqrt{2m}$ in to the normalization and wrote $\tilde W_{\pm} =W_{\pm} \sqrt{2m}/{\hbar}$. Likewise for the transmitted coefficients ($x\to +\infty$). $$\begin{aligned}
T^{(1)}e^{ik'x} = N (-ik' + \tilde W_{+} )T^{(2)}e^{ik'x}\end{aligned}$$ Eliminating the common normalization factor and using the fact that $k = \sqrt{2m(E-W_{-})}/\hbar$ and $k' = \sqrt{2m(E-W_{+})}/\hbar$ from the Schrödinger equation we can arrive at $$\begin{aligned}
R^{(1)}(k) = \frac{W_{-}+ik}{W_{-}-ik}R^{(2)}(k) \\
T^{(1)}(k) = \frac{W_{+}-ik'}{W_{-}-ik}T^{(2)}(k).\end{aligned}$$ Consequently, knowledge of the scattering states of $V_{1}$ allows one to easily construct scattering states for the partner potential.
Non-stationary states
---------------------
Finally, one can use the SUSY approach in a time-dependent context by writing $$i\hbar \partial_{t} \psi^{(1)} = H_{1} \psi^{(1)} = A^{+} A \psi^{(1)}$$ where $\psi^{(1)}$ is a non-stationary state in the first sector. If $V_{1}$ is independent of time, then the superpotential must also be independent of time and so we can write $$i\hbar A \partial_{t} \psi^{(1)} = i\hbar \partial_{t} (A \psi^{(1)}) = A A^{+} (A \psi^{(1)})$$ In other words, we have the time-dependent Schrödinger equation for the partner potential $$i\hbar \partial_{t}\psi^{(2)} = H_{2} \psi^{(2)}.$$ The two non-stationary states are partnered, $\psi^{(2)} \propto A^{+}\psi^{(1)}$. We also note that these states satisfy $$\psi^{(1)}(t) = e^{-iA^{+}At/\hbar}\psi(0)$$ and $$\psi^{(2)}(t) = e^{-iAA^{+}t/\hbar}\psi(0)$$ for some initial state $\psi(0)$. Using the charge operators we can show that $$A\psi^{(1)}(t) =
%A e^{-iA^{+}At/\hbar}\psi(0)\\
%&=&A\left(1 - \frac{it}{\hbar}A^{+}A + \cdots \right)\psi(0) \\
%&=&\left(1 - \frac{it}{\hbar}AA^{+} + \cdots \right)A\psi(0) \\
%&=&
e^{-iAA^{+}t/\hbar}(A\psi(0)).$$ As above in the scattering example, one can use the dynamics of one sector to determine the dynamics in the other sector.
The partnering scheme presents a powerful prescription for developing novel approaches for solving a wide variety of quantum mechanical problems. This allows one one use analytical or numerical solutions of one problem to determine either approximate or exact solutions to some new problem. In the sections that follow, I present some of our attempt to use SUSY in a numerical context. At the moment our numerical results are limited to one spatial dimension. As I shall discuss, extending SUSY to multiple dimensions has proven to be problematic. However, in Sec. V we present our extension using a vector-SUSY approach we are developing.
Using SUSY to obtain excitation energies and excited states
===========================================================
The SUSY hierarchy also provides a useful prescription for determining the excited states of $H_{1}$ (which may represent the physical problem of interest.) The first excited state of $H_{1} $ is isoenergetic with the ground state of $H_{2}$. Since this state is node-less, one can use either Ritz variational approaches or Monte Carlo approaches to determine this state to very high accuracy.
Two basic tools used in computational chemistry are the Quantum Monte Carlo (QMC) and the Rayleigh-Ritz variational approaches. Both approaches yield their best and most accurate results for ground state energies and wave functions. Although the variational method also gives bounds for the excited state energies as well as the ground state (the Hylleraas-Undheim theorem [@Hilleraas:1930ph]), it is well known that their accuracy is significantly lower than that of the ground state. Even more serious, the wave functions are known to converge much more slowly than the energies.
In the case of the QMC[@Hammond94; @porter:7795; @Doll87; @Needs:2001kx; @PhysRevB.71.241103; @PhysRevE.55.3664], there are additional difficulties associated with the presence of nodes in the excited state wave functions [@bouabca:114107]. While some progress has been made in dealing with this issue (e.g., the “fixed node” or “guide wave” techniques)[@Needs:2001kx; @PhysRevB.71.241103; @PhysRevE.55.3664; @bouabca:114107; @Oriols98] the computational effort required is greater and the accuracy is lower and in fact, no general solution to the difficulty has been found for reducing the computational effort and increasing the accuracy for excited state calculations in QMC to the same level as is attained for the ground state. In fact, it is very likely the presence and effects of nodes in the excited states that is largely responsible for the lower accuracy and slower convergence of excited state results in the variational method. The precise determination of nodal surfaces is expected to play a crucial role since they reflect changes in the relative phase of the wave function. Because of the ubiquitous importance of both the variational and QMC methods, solving the so-called “node problemÓ will have enormous impact on computational chemistry.
Using SUSY to improve quality of variational calculations
---------------------------------------------------------
We now turn to the proof of principle for this approach as a computational scheme to obtain improved excited state energies and wave functions in the Rayleigh-Ritz variational method. We should note that these results can be generalized to any system where a hierarchy of Hamiltonians can be generated because of the nature of the Rayleigh-Ritz scheme. In the standard approach one calculates the energies and wave functions variationally, relying on the Hylleraas-Undheim theorem for convergence[@Hilleraas:1930ph]. This, however, is unattractive for higher energy states because they require a much larger basis to converge to the same error. We stress that this is true regardless of the specific basis set used. Of course, some bases will be more efficient than others but it is generally true that for a given basis, the Rayleigh-Ritz result is less accurate for excited states. We address this situation by solving for ground states in the variational part of the problem.
To demonstrate our computational scheme, we investigate the first example system from the previous section. For the potential $$V_1(x) = x^6 + 4x^4 + x^2 - 2.$$ exact solutions are known for all states of $H_1$. We use the exact results to assess the accuracy of the variational calculations. Here we employed a $n$-point discrete variable representation (DVR) based upon the Tchebchev polynomials to compute the eigenspectra of the first and second sectors.[@light:1400; @lig92:185] In Fig. \[converge\] we show the numerical error in the first excitation energy by comparing $E_{1}^{1}(n)$ and $E_{0}^{2}(n)$ from an $n$ point DVR to the numerically “exact” value corresponding to a 100 point DVR, $$\epsilon_{1}^{1}(n) = \log_{10}|E_{1}^{1}(n)-E_{1}^{1}(exact)|.$$ Likewise, $$\epsilon_{0}^{2}(n) = \log_{10}|E_{0}^{1}(n)-E_{1}^{1}(exact)|.$$ For any given basis size, $\epsilon_{0}^{2} < \epsilon_{1}^{1}$. Moreover, over a range of $15 < n < 40$ points, the excitation energy computed using the second sector’s ground state is between 10 and 100 times more accurate than $E_{1}^{1}(n)$. This effectively reiterates our point that by using the SUSY hierarchy, one can systematically improve upon the accuracy of a given variational calculation.
![Convergence of first excitation energy $E_{1}^{1}$ for model potential $V_{1} = x^{6} + 4 x^{4} + x^{2} -2$ using a $n$-point discrete variable representation (DVR). Gray squares: $\epsilon=\log_{10}|E_{1}^{1}(n)-E_{1}^{1}(exact)|$, Black squares: $\epsilon=\log_{10}|E_{0}^{2}(n)-E_{1}^{1}(exact)|$. Dashed lines are linear fits. (From Ref[@Kouri:2009hb].)[]{data-label="converge"}](converge){width="0.5\columnwidth"}
Monte Carlo SUSY
-----------------
Having defined the basic terms of SUSY quantum mechanics, let us presume that one can determine an accurate approximation to the ground state density $\rho_{o}^{(1)}(x)$ of Hamiltonian $H_{1}$. One can then use this to determine the superpotential using the Riccati transform $$\begin{aligned}
W_{o}^{(1)} = -\frac{1}{2} \frac{\hbar}{\sqrt{2m}} \frac{\partial \ln\rho_{o}^{(1)}}{\partial x} \label{riccati}\end{aligned}$$ and the partner potential $$\begin{aligned}
V_{2} = V_{1} - \frac{\hbar^{2}}{2m} \frac{\partial^{2} \ln\rho_{o}^{(1)}}{\partial x^{2}}. \label{eq13}\end{aligned}$$ Certainly, our ability to compute the energy of the ground state of the partner potential $V_{2}$ depends on having first obtained an accurate estimate of the ground-state density associated with the original $V_{1}$.
For this we turn to an adaptive Monte Carlo-like approach developed by Maddox and Bittner.[@maddox:6465] Here, we assume we can write the trial density as a sum over $N$ Gaussian approximate functions $$\begin{aligned}
\rho_{T}(x) = \sum_{n} G_{n}(x,{\bf c}_{n}). \label{approx}\end{aligned}$$ parameterized by their amplitude, center, and width. $$\begin{aligned}
G_{n}(x,\{{\bf c}_{n}\}) = c_{no} e^{-c_{n2}(x-c_{n3})^{2}}\end{aligned}$$ This trial density then is used to compute the energy $$\begin{aligned}
E[\rho_{T}] = \langle V_{1}\rangle + \langle Q[\rho_{T}]\rangle \end{aligned}$$ where $ Q[\rho_{T}] $ is the Bohm quantum potential, $$\begin{aligned}
Q[\rho_{T} ] = -\frac{\hbar^{2}}{2m}\frac{1}{\sqrt{\rho_{T}}}\frac{\partial^{2}}{\partial x^{2}}\sqrt{\rho_{T}}.\end{aligned}$$ The energy average is computed by sampling $\rho_{T}(x)$ over a set of trial points $\{x_{i}\}$ and then moving the trial points along the conjugate gradient of $$\begin{aligned}
E(x) = V_{1}(x) + Q[\rho_{T}](x).\end{aligned}$$ After each conjugate gradient step, a new set of $\bf {c}_{n}$ coefficients are determined according to an expectation maximization criteria such that the new trial density provides the best $N$-Gaussian approximation to the actual probability distribution function sampled by the new set of trial points. The procedure is repeated until $\delta \langle E\rangle = 0$. In doing so, we simultaneously minimize the energy and optimize the trial function. Since the ground state is assumed to be node-less, we will not encounter the singularities and numerical instabilities associated with other Bohmian equations of motion based approaches. [@Bohm52a; @Holland93; @Lopreore99; @Bittner00a; @Wyatt01; @maddox:6465] Moreover, the approach has been extended to very high-dimensions and to finite temperature by Derrickson and Bittner in their studies of the structure and thermodynamics of rare gas clusters with up to 130 atoms. [@Derrickson:2006; @Derrickson:2007jo]
Test case: tunneling in a double well potential
===============================================
As a non-trivial test case, consider the tunneling of a particle between two minima of a symmetric double potential well. One can estimate the tunneling splitting using semi-classical techniques by assuming that the ground and excited states are given by the approximate form $$\begin{aligned}
\psi_{\pm} = \frac{1}{\sqrt{2}}(\phi_{o}(x) \pm \phi_{o}(-x))\end{aligned}$$ where $\phi_{o}$ is the lowest energy state in the right-hand well in the limit the wells are infinitely far apart. From this, one can easily estimate the splitting as [@Landau:1974wq] $$\begin{aligned}
\delta = 4 \frac{\hbar^{2}}{m} \phi_{o}(0)\phi_{o}'(0)\end{aligned}$$ If we assume the localized states $(\phi_{o})$ to be gaussian, then $$\begin{aligned}
\psi_{\pm} \propto \frac{1}{\sqrt{2}}(e^{-\beta(x-x_{o})^{2}}\pm e^{-\beta(x+x_{o})^{2}})\end{aligned}$$ and we can write the superpotential as $$\begin{aligned}
W = \sqrt{\frac{2}{m}}\hbar\beta \left(x - x_{o}\tanh(2 x x_{o}\beta) \right).\end{aligned}$$ From this, one can easily determine both the original potential and the partner potential as $$\begin{aligned}
V_{1,2} &=& W^{2} \pm \frac{\hbar}{\sqrt{2m}}W' \\
&=& \frac{\beta^{2} \hbar ^2}{m} \left(
2 (x-x_o \tanh (2 x x_o \beta ))^2 \right. \nonumber \\
&\pm&\left. (2 x_o^2 \text{sech}^2(2 x x_o \beta )-1\right)\end{aligned}$$ While the $V_{1}$ potential has the characteristic double minima giving rise to a tunneling doublet, the SUSY partner potential $V_{2}$ has a central dimple which in the limit of $x_{o}\rightarrow \infty$ becomes a $\delta$-function which produces an unpaired and node-less ground state. [@Cooper:1995jt] Using Eq. \[chargeop\], one obtains $\psi_{1}^{(1)} = \psi_{-} \propto A^{\dagger}\psi_{o}^{(2)}$ which now has a single node at $x = 0$.
For a computational example, we take the double well potential to be of the form $$\begin{aligned}
V_{1}(x) = a x^{4} + bx^{2} + E_{o}.\end{aligned}$$ with $a = 438.9 {\rm cm}^{-1}/(bohr^{2})$, $b = 877.8 {\rm cm}^{-1}/(bohr)^{4}$, and $E_{o} = -181.1 {\rm cm}^{-1}$ which (for $m = m_{H}$ ) gives rise to exactly two states at below the barrier separating the two minima with a tunneling splitting of 59.32 ${\rm cm}^{-1}$ as computed using a discrete variable representation (DVR) approach.[@lig85:1400] For the calculations reported here, we used $n_{p}=1000$ sample points and $N =15$ Gaussians and in the expansion of $\rho_{T}(x)$ to converge the ground state. This converged the ground state to $1:10^{-8}$ in terms of the energy. This is certainly a bit of an overkill in the number of points and number of gaussians since far fewer DVR points were required to achieve comparable accuracy (and a manifold of excited states). The numerical results, however, are encouraging since the accuracy of generic Monte Carlo evaluation would be $1/\sqrt{n_{p}} \approx 3\%$ in terms of the energy. [^2] Plots of $V_{1}$ and the converged ground state is shown in \[fig1\].
The partner potential $V_{2} = W^{2} + \hbar W'/\sqrt{2m}$, can be constructed once we know the superpotential, $W(x)$. Here, we require an accurate evaluation of the ground state density and its first two log-derivatives. The advantage of our computational scheme is that one can evaluate these analytically for a given set of coefficients. In \[fig1\]a we show the partner potential derived from the ground-state density. Where as the original $V_{1}$ potential exhibits the double well structure with minima near $x_{o} = \pm 1$ , the $V_{2}$ partner potential has a pronounced dip about $x=0$. Consequently, its ground-state should have a simple “gaussian”-like form peaked about the origin.
Once we determined an accurate representation of the partner potential, it is now a trivial matter to re-introduce the partner potential into the optimization routines. The ground state converges easily and is shown in \[fig4\]a along with its gaussians. After 1000 CG steps, the converged energy is within 0.1% of the exact tunneling splitting for this model system. Again, this is an order of magnitude better than the $1/\sqrt{n_{p}}$ error associated with a simple Monte Carlo sampling. Furthermore, \[fig4\]b shows $\psi_{1}^{(1)}\propto A^{\dagger}\psi_{0}^{(2)}$ computed using the converged $\rho_{0}^{(2)}$ density. As anticipated, it shows the proper symmetry and nodal position.
By symmetry, one expects the node to lie precisely at the origin. However, since we have not imposed any symmetry restriction or bias on our numerical method, the position of the node provides a sensitive test of the convergence of the trial density for $\rho_{0}^{(2)}$. In the example shown in Fig.\[fig3\], the location of the node oscillates about the origin and appears to converge exponentially with number of CG steps. This is remarkably good considering that this is ultimately determined by the quality of the 3rd and 4th derivatives of $\rho_{o}^{(1)}$ that appear when computing the conjugate gradient of $V_{2}$. We have tested this approach on a number of other one-dimensional bound-state problems with similar success.
[![ Location of excited state node for the last 600 CG steps. (From Ref.[@Bittner:2009jt]) []{data-label="fig3"}](Fig3b "fig:"){width="0.5\columnwidth"}]{}
Extension of SUSY to multiple dimensions.
=========================================
While SUSY-QM has also been explored for one dimensional, non-relativistic quantum mechanical problems[@baer:075024; @Eides:1984rz; @Andrianov:1985ty; @Cooper:1995jt; @Andrianov:1984nr; @PhysRevA.47.2720], thus far these studies have focused on the formal aspects and on obtaining exact, analytical solutions for the ground state for specific classes of problems. In several recent papers[@Bittner:2009jt; @Kouri:2009hb; @Kouri09; @Kouri:2009xr], we have begun exploring the SUSY-QM approach as the basis of a general computational scheme for bound state problems. Our initial studies have been restricted to one dimensional systems (for which there are, obviously, many powerful computational methods). In our first paper, we found that SUSY-QM (combined with a new periodic version of the Heisenberg-Weyl algebra) yields a robust, natural way to treat an infinite family of hindered rotors.[@Kouri:2009xr] Next we showed that the SUSY-QM leads to a general treatment of an infinite family of anharmonic oscillators, such that highly accurate excited state energies and wave functions could be obtained variationally using significantly smaller basis sets than a traditional variational approach requires[@Kouri:2009hb]. Most recently , we have considered a 1-D double well potential in which we solved for the ground state energy and wave function using a VQMC approach. Then using SUSY-QM, we (numerically) generated an auxiliary Hamiltonian whose [*nodeless*]{} ground state is iso-spectral (degenerate) with the first excited state of the original system Hamiltonian. This ground state was also easily determined by VQMC, yielding excellent accuracy for the first excited state energy. [@Bittner:2009jt] Even more significant, by using the charge operators naturally generated in the SUSY-QM approach, we also obtained excellent accuracy for the first excited state wave function. Furthermore, at no point did impose a fixed node or symmetry on the excited state wave function and our calculation only involved working with a nodeless ground state.
Of course, all this begs the question: Can this approach be generalized to higher numbers of dimensions and to more than a single particle? There has been substantial effort in the past to do just this.[@Eides:1984rz; @Andrianov:1985ty; @Cooper:1995jt; @Andrianov:1984nr; @leung:4802; @rodrigues:125023; @contreras-astorga:55; @Andrianov:1987zl; @Andrianov:1988yg; @Andrianov:1984sf; @A.A.Andrianov:1984qv; @Andrianov:1986rm; @0305-4470-35-6-305; @Andrianov:2002gf; @Das:1996sw; @SIGMA] However, to date, no such generalization has been found that is able to generate all the excited states and energies even for so simple a system as a pair of separable, 1-D harmonic oscillators (HO) or equivalently, for a separable 2-D single HO. In our most recent, unpublished work [@Kouri09], we have succeeded in obtaining such a generalization and showed that it does, in fact, yield the correct analytical results for separable and non-separable problems. In the next section, we present a succinct summary of our approach. The major question now is whether this formalism provides a basis for a robust, computational method for determining excited state energies and wave functions for large, strongly correlated systems using either QMC or variational algorithms applied solely to nodeless ground state problems.
Difficulties in extending beyond one dimension
----------------------------------------------
To move beyond one dimensional SUSY, Ioffe and coworkers have explored the use of higher-order charge operators [@A.A.Andrianov:1993bv; @Andrianov:1995ve; @0305-4470-35-6-305; @Andrianov:2002gf], and Kravchenko has explored the use of Clifford algebras[@Kravchenko]. Unfortunately, this is difficult to do in general. The reason being that the Riccati factorization of the one-dimensional Schrödinger equation does not extend easily to higher dimensions. One remedy is write the charge operators as vectors $\vec{A} = (+\vec\partial + \vec W)$ and with $\vec{A}^{+} = (-\vec\partial + \vec{W})^{\dagger}$ as the adjoint charge operator. The original Schrödinger operator is then constructed as an inner-product $$\begin{aligned}
H_{1} = \vec{A}^{+} \cdot \vec{A} .\end{aligned}$$ Working through the vector product produces the Schrödinger equation $$\begin{aligned}
H_{1}\phi = (-\nabla^{2} + W^{2} - (\vec{\nabla}\cdot \vec{W})) \phi = 0
\label{2dfactr}\end{aligned}$$ and a Riccati equation of the form $$\begin{aligned}
U(x) = W^{2} - \vec\nabla\cdot \vec{W}.\end{aligned}$$ For a 2d harmonic oscillator, we would obtain a vector superpotential of the form $$\begin{aligned}
\vec{W} =-\frac{1}{\psi_{0}^{(1)}} \vec\nabla\psi_{0}^{(1)} = \left(x, y\right) = (W_{x},W_{y})\end{aligned}$$ Let us look more closely at the $\vec\nabla\cdot \vec{W}$ part. If we use the form that $\vec{W} = - \vec{\nabla}\ln\psi$, then $-\vec{\nabla}\cdot\vec{\nabla}\ln\psi = -\nabla^{2}\ln\psi$ which for the 2D oscillator results in $\vec\nabla\cdot \vec{W} = 2$. Thus, $$\begin{aligned}
W^{2} - \vec\nabla\cdot \vec{W } = (x^{2} + y^{2}) - 2 \end{aligned}$$ which agrees with the original symmetric harmonic potential. Now, we write the scaled partner potential as $$\begin{aligned}
U_{2}=W^{2} + \vec\nabla\cdot \vec{W } =(x^{2} + y^{2}) + 2.\label{u2}\end{aligned}$$ This is equivalent to the original potential shifted by a constant amount. $$\begin{aligned}
U_{2} = U_{1} + 4.\end{aligned}$$ The ground state in this potential would be have the same energy as the states of the original potential with quantum numbers $n + m = 2$. Consequently, even with the this naïve factorization, one can in principle obtain excitation energies for higher dimensional systems, but there is no assurance that one can reproduce the entire spectrum of states.
The problem lies in the fact that neither Hamiltonian $H_{2}$ nor its associated potential $U_{2}$ is given correctly by the form implied by Eq. \[2dfactr\] and Eq. \[u2\]. Rather, the correct approach is to write the $H_{2}$ Hamiltonian as a [*tensor*]{} by taking the outer product of the charges $\overline{H}_{2} = \vec{A} \vec{A}^{+}$ rather than as a scalar $\vec{A} \cdot\vec{A}^{+}$. At first this seems unwieldy and unlikely to lead anywhere since the wave function solutions of $$\begin{aligned}
\overline{H}_{2} \vec{\psi} = E \vec{\psi}\end{aligned}$$ are now vectors rather than scalers. However, rather than adding an undue complexity to the problem, it actually simplifies matters considerably. As we demonstrate in a forthcoming paper, this tensor factorization preserves the SUSY algebraic structure and produces excitation energies for any $n-$dimensional SUSY system. Moreover, this produces a scalar $\mapsto$ tensor $\mapsto$ scalar hierarchy as one moves to higher excitations.[@Kouri09]
Vector SUSY
-----------
We now give a brief summary of our new generalization of SUSY-QM to treat higher dimensionality and more than one particle. Previous attempts generally involved introducing additional, “spin-like” degrees of freedom.[@Andrianov:1985ty; @Andrianov:1984nr; @Andrianov:1987zl; @Andrianov:1988yg; @A.A.Andrianov:1984qv; @Andrianov:1984sf; @0305-4470-35-6-305; @Andrianov:2002gf; @A.A.Andrianov:1993bv; @Dzhioev:2007; @Andrianov:1995ve] In our approach, we make use of a vectorial technique that can deal simultaneously with either higher dimensions or more than one particle. In fact, the two problems are dealt with in exactly the same manner. Therefore, for simplicity, we consider a general $n$-dimensional distinguishable particle system with orthogonal coordinates $\{x_{\mu}\}$. The Hamiltonian is given by [^3] $$\begin{aligned}
H = -\nabla^{2 } + V_{0}(x_{1},\cdots x_{n}) %= \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + V_{o}(x,y)\end{aligned}$$ and the nodeless ground state satisfies the Schrödinger equation, $$\begin{aligned}
H \psi_{0}^{(1)} = E^{(1)}_{0}\psi_{0}^{(1)}. \label{eq2}\end{aligned}$$ We now define a “vector super-potential”, $\vec{W}_{1}$, with components $$\begin{aligned}
W_{1\mu} = - \frac{\partial}{\partial x_{\mu}} \ln\psi_0^{(1)}.
%W_{1y} = - \frac{\partial}{\partial y} \ln\psi_0^{(1)}\end{aligned}$$ Then it is easily seen that the original Hamiltonian can be recast as $$\begin{aligned}
H_{1} = (-\nabla + \vec{W}_{1})\cdot (\nabla + \vec{W}_{1}) =
\vec{Q}_{1}^{+}\cdot \vec{Q}_{1}\label{eq5}\end{aligned}$$ where the $\vec{Q}_{1}$ and $\vec{Q}_{1}^{+}$ are multi-dimensional generalizations of the SUSY charge operators from Eq. \[susy-hamiltonian\]. This defines our “sector-1” (or “boson”) Hamiltonian and Eq.(\[eq2\]) can be written as [^4] $$\begin{aligned}
H_{1} \psi_{0}^{(1)} = E^{(1)}_{0}\psi_{0}^{(1)}\end{aligned}$$ One can show that the vector superpotential is related to the original (scalar) potential via: $$\begin{aligned}
V_{0} = \vec{W}_{1}\cdot \vec{W}_{1} - \nabla\cdot\vec{W}_{1}.\end{aligned}$$ The various components of the charge operators, $\vec{A}_{1}$ and $\vec{A}_{1}^{+}$ are defined by $$\begin{aligned}
A_{1\mu} = \frac{\partial}{\partial x_{\mu}} + W_{1\mu} \,\,\&\,\,\,
A_{1\mu}^{+} = -\frac{\partial}{\partial x_{\mu}} + W_{1\mu}.\end{aligned}$$ Note that since these are associated with orthogonal degrees of freedom, the charge operators can be applied either by individual components or in vector form.
Next, consider the Schrödinger equation for the first excited state of $H$. We can write this using the charge operators as $$\begin{aligned}
H_{1}\psi_{1}^{(1)} = E_{1}^{(1)}\psi_{1}^{(1)} = (\vec{A}_{1}^{+}\cdot\vec{A}_{1} + E_{0}^{(1)})\psi_{1}^{(1) }\end{aligned}$$ We apply $\vec{A}_{1}$ to Equation (9): $$\begin{aligned}
(\vec{A}_{1}\vec{A}_{1}^{+})\cdot \vec{A}_{1}\psi_{1}^{(1)} = ( E_{1}^{(1)} -E_{1}^{(0)} ) \vec{A}_{1}\psi_{1}^{(1)}\end{aligned}$$ Here we identify $(\vec{A}_{1}\vec{A}_{1}^{\dagger})$ as a new, auxiliary Hamiltonian. It is important to note that this is constructed from the outer or tensor product of the charge operators rather than from inner or dot product as used in constructing $H_{1}$. Its eigenvector, $ \vec{A}_{1}\psi_{1}^{(1)}$, is isospectral with the excited state, $\psi_{1}^{(1)}$ of $H_{1}$ ( since $E_{0}^{(1)}$ is known, determining $ ( E_{1}^{(1)} -E_{1}^{(0)} ) $ yields $
E_{1}^{(1)} $ ). We therefore define the [**tensor**]{} Hamiltonian for the second sector as $$\begin{aligned}
\tens{H}_{2} = \vec{A}_{1}\vec{A}_{1}^{\dagger} \label{eq11}\end{aligned}$$ and [**vector**]{} state function as $$\begin{aligned}
\vec\psi_{0}^{(2)} = \frac{1}{( E_{1}^{(1)} -E_{1}^{(0)} )
}\vec{A}_{1}\psi_{1}^{(1)}.\end{aligned}$$ It is easy to show that the ground state energy of $\tens{H}_{2}$ is related to the first excitation energy of the original Hamiltonianm $$\begin{aligned}
E_{0}^{(2)} = E_{1}^{(1)} -E_{1}^{(0)}. \label{eq13}\end{aligned}$$ Furthermore, the ground state of $\tens{H}_{2}$ is also [ **nodeless**]{}. This has been explicitly shown to be true for the separable 2-particle HOs considered earlier[@Kouri09]. Therefore, we propose to apply both the VQMC and the standard variational methods to determine $E_{0}^{(2)} $ and $ \vec\psi_{0}^{(2)}$. Of course, knowing the second sector ground state energy also gives us the first excited state energy of the original Hamiltonian (Eq. (\[eq13\])). Furthermore, we form the scalar product of $$\begin{aligned}
\tens{H}_{2}\cdot \vec{\psi}_{0}^{(2)} = E_{0}^{(2)}
\vec{\psi}_{0}^{(2)} \label{eq14}\end{aligned}$$ with $\vec{A}_{1}^{+}$ obtaining $$\begin{aligned}
(\vec{A}_{1}^{+}\cdot \vec{A}_{1}) \vec{A}_{1}^{+}
\vec{\psi}_{0}^{(2)} = E_{0}^{(2)} \vec{A}_{1}^{+} \cdot
\vec{\psi}_{0}^{(2)}\end{aligned}$$ Clearly, this is exactly $$\begin{aligned}
H_{1} ( \vec{A}_{1}^{+} \cdot \vec{\psi}_{0}^{(2)} ) = E_{1}^{(1)} (
\vec{A}_{1}^{+} \cdot \vec{\psi}_{0}^{(2)} )\end{aligned}$$ so we can conclude that $$\begin{aligned}
\psi_{1}^{(1)} = \frac{1}{\sqrt{E_{0}^{(2)}}} ( \vec{A}_{1}^{+} \cdot
\vec{\psi}_{0}^{(2)} )\end{aligned}$$
Thus we also obtain the excited state wave function without any significant additional computational effort. This is because applying the charge operator is much simpler than solving an eigenvalue problem (it is a strictly linear operation). Evidence from our 1-D studies indicates that the accuracy of the excited states obtained using the SUSY-QM charge operator is significantly higher, for a given basis set, than what is obtained variationally (or with QMC) from the original Hamiltonian[@Kouri09; @Kouri:2009hb]. This procedure can be continued as follows. We define a sector-2 vector super-potential with components $$\begin{aligned}
W_{2\mu} = \frac{\partial}{\partial x_{\mu}} \ln\psi_{0\mu}^{(2)} \end{aligned}$$ Then it follows that $$\begin{aligned}
\vec{A}_{2}\cdot \vec{\psi}_{0}^{(2)} = (\nabla + \vec{W}_{2})\cdot
\psi_{0}^{(2)} = 0\end{aligned}$$ so we can write $$\begin{aligned}
\tens{H}_{2} = \vec{A}_{2}^{+} \vec{A}_{2} + E_{0}^{(2)} {\bf I}\end{aligned}$$ and Eq. (\[eq14\]) is still satisfied. We form the scalar product of $ \vec{A}_{2}$ with the first excited state Schrödinger equation to obtain $$\begin{aligned}
( \vec{A}_{2} \cdot \vec{A}_{2}^{+}) \vec{A}_{2} \cdot
\vec{\psi}_{1}^{(2)} = E_{1}^{(2)} \vec{A}_{2}\cdot
\vec{\psi}_{1}^{(2)}\end{aligned}$$ Then we define the sector 3 [*scalar*]{} Hamiltonian by $$\begin{aligned}
H_{3} = \vec{A}_{2} \cdot \vec{A}_{2}^{+} + E_{0}^{(2)}\end{aligned}$$ with the ground state wave equation $$\begin{aligned}
H_{3}\psi_{0}^{(3)} = E_{0}^{(3)} \psi_{0}^{(3)}.\end{aligned}$$ It is easily seen that $E_{0}^{(3)} = E_{1}^{(2)} - E_{0}^{{2}}$. This procedure continues until all bound states of the original Hamiltonian are exhausted. It should also be clear that the sector 2 excited state wave function is obtained from the nodeless sector 3 ground state by applying $\vec{A}_{2}^{+}$ to it. Then the second excited state for sector 1 results from taking the scalar product of $
\vec{A}_{1}^{+}$ with $\vec{\psi}_{1}^{(2)}$ . The approach thus leads to an alternating sequence of scalar and tensor Hamiltonians, [*but in all cases we need only determine nodeless ground states*]{}.
There are two additional aspects of the tensor sector problem that require discussion. First we consider the validity of the Rayleigh-Ritz variational principle. It is easily seen from Eq. (\[eq11\]) that $\tens{H}_{2}$ is a Hermitian operator. Therefore, its eigenspectrum is real and its eigenvectors are complete. With these facts in hand, the proof of the variational principle follows the standard one in every detail. This is also true for the Hylleraas-Undheim theorem.
Second, the QMC method is also directly applicable to the tensor sector problem. For the example discussed above, we note that the energy is given by $$\begin{aligned}
E_{trial} = \frac{\int d\tau \vec\psi_{trial}\cdot \tens{H}_{2} \cdot
\vec{\psi}_{trial}}{\int d\tau \vec\psi_{trial}\cdot
\vec{\psi}_{trial}}
\end{aligned}$$ We next note that the integral can be expanded in terms of its components as $$\begin{aligned}
E_{trial} = \frac{\sum_{\mu\nu}\int d\tau (\psi_{\mu,trial}
H_{2,\mu\nu} \psi_{\nu,trial} )} {\sum_{\mu}\int d\tau(
\psi_{\mu,trial})^{2}}.\end{aligned}$$ It is then clear that each separate integral can be evaluated by QMC. For example, the $\mu\ne \nu$ cross term is divided and multiplied by $$\begin{aligned}
\psi_{\mu,trial}\int d\tau \psi_{\nu,trial}\psi_{\mu,trial}.\end{aligned}$$ Then the sampling is done relative to the mixed probability distribution, $$\begin{aligned}
P_{\mu\nu} = \frac{ \psi_{\mu,trial}\psi_{\nu,trial}}{\int d\tau \psi_{\mu,trial}\psi_{\nu,trial} }.\end{aligned}$$ A similar expression applies to each term in the energy expression and the evaluation would need to be performed self-consistantly.
Thus far, we have developed a formalism that appears to be suitable for extending the SUSY-QM technique to higher dimensional systems. We believe the approach we have outlined above will provide the mathematical basis for a number of potentially interesting theoretical results. Moreover, we anticipate that when combined with either variational or Monte Carlo methods, our multi-dimensional extension of SUSY-QM will facilitate the calculation of accurate excitation energies and excited state wave functions.
Outlook
=======
I presented a number of avenues we are actively pursuing with the goal of using SUSY-QM or SUSY-inspired-QM to solve problems that are difficult to solve using more conventional approaches. In addition to what I have discussed here we exploring the use of the Riccati equation to solve quantum scattering problems. It is as if one of the co-authors of this paper (DJK) has come full-circle since one of his first papers concerned solving the Hamilton-Jacobi for the action integral in quantum scattering,[@kouri:1919][^5] $$iS/\hbar = -\int_{r_{o}}^{r}W(r')dr'.$$ The integrand in this last equation is the SUSY super-potential. Furthermore, there is a connection between our work and the complex-valued quantum trajectories studied by Wyatt and Tannor and their respective co-workers.
This work was supported in part by the National Science Foundation (ERB: CHE-0712981) and the Robert A. Welch foundation (ERB: E-1337, DJK: E-0608). The authors also acknowledge Prof. M. Ioffe for comments regarding the extension to higher dimensions.
[50]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, ****, ().
, ****, ().
, , , ****, ().
, ****, ().
, , , **, vol. of ** (, , ).
, , , , ****, (), <http://link.aip.org/link/?JCP/114/7795/1>.
, , , ****, ().
, , , , , ****, ().
, ****, ().
, , , , ****, ().
, , , , ****, (pages ) (), <http://link.aip.org/link/?JCP/130/114107/1>.
, , , , , , , ****, ().
, , , ****, (), <http://link.aip.org/link/?JCP/82/1400/1>.
, in **, edited by (, , ), pp. .
, , , , (), <http://dx.doi.org/10.1021/jp905798m>.
, ****, (), <http://link.aip.org/link/?JCP/119/6465/1>.
, ****, ().
, ** (, ).
, ****, ().
, ****, ().
, , , ****, ().
, ****, (), [http://pubs.acs.org/cgi-bin/article.cgi/jpcafh/2006/110/i16/p%
df/jp055889q.pdf](http://pubs.acs.org/cgi-bin/article.cgi/jpcafh/2006/110/i16/p%
df/jp055889q.pdf).
, ****, (), , <http://pubs.acs.org/doi/abs/10.1021/jp0722657>.
, **, vol. of ** (, , ), ed.
, , , (), <http://dx.doi.org/10.1021/jp9058017>.
, , , ****, ().
, , , , ****, (pages ) (), <http://link.aps.org/abstract/PRD/v65/e075024>.
, , , , ****, ().
, , , , ****, ().
, , , ****, ().
, , , ****, ().
, .
, , , , ****, ().
, , , , , ****, (), <http://link.aip.org/link/?JMP/42/4802/1>.
, , , ****, (pages ) (), <http://link.aps.org/abstract/PRD/v58/e125023>.
, ****, (), <http://link.aip.org/link/?APC/960/55/1>.
, , , ****, ().
, ****, ().
, , , ****, ().
, , , ****, ().
, , , ****, ().
, , , ****, (), <http://stacks.iop.org/0305-4470/35/1389>.
, , , ****, ().
, ().
, , , ****, ().
, , , ****, ().
, , , ****, ().
, ****, ().
, ****, (), ISSN .
, ****, (), <http://link.aip.org/link/?JCP/43/1919/1>.
[^1]: Our notation from here on is that $\psi_{n}^{(m)}$ denotes the $n$th state associated with the $m$th partner Hamiltonian with similar notion for related quantities such as energies and superpotentials.
[^2]: In our implementation, the sampling points are only used to evaluate the requisite integrals and they themselves are adjusted along a conjugate gradient rather than by resampling. One could in principle forego this step entirely and optimize the parameters describing the gaussians directly.
[^3]: Our units are such that $\hbar^{2}/2m = 1$.
[^4]: In analogy with the original descriptions of SUSY, we refer to the partner pairs as “boson” and “fermion” sectors or less poetically as “sector-1”, “sector 2”, and so forth.
[^5]: Coincidentally, Ref. [@kouri:1919] appeared in the J. Chem. Phys. issue immediately before the birthday of the other author of this paper. There appears to be some interesting Karma at work here.
|
---
abstract: 'We demonstrate a scheme for quantum communication between the ends of an array of coupled cavities. Each cavity is doped with a single two level system (atoms or quantum dots) and the detuning of the atomic level spacing and photonic frequency is appropriately tuned to achieve photon blockade in the array. We show that in such a regime, the array can simulate a dual rail quantum state transfer protocol where the arrival of quantum information at the receiving cavity is heralded through a fluorescence measurement. Communication is also possible between any pair of cavities of a network of connected cavities.'
address:
- '$^{1}$Department of Physics and Astronomy, University College London, Gower St., London WC1E 6BT, UK'
- '$^{2}$Centre for Quantum Computation, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, CB3 0WA, UK'
- '$^{3}$Computer Science Departement, ETH Z[ü]{}rich, CH-8092 Z[ü]{}rich, Switzerland\'
author:
- 'Sougato Bose $^{1}$'
- 'Dimitris G. $^{2}$'
- 'Daniel $^{1,3}$'
title: Transfer of a Polaritonic Qubit through a Coupled Cavity Array
---
Introduction
============
Recently, the exciting possibility of coupling high Q cavities directly with each other has materialized in a variety of settings, namely fiber coupled micro-toroidal cavities [@armani-vahala03], arrays of defects in photonic band gap materials (PBGs) [@pbg; @angelakis-knight04] and microwave stripline resonators joined to each other [@supercond]. A further exciting development has been the ability to couple each such cavity to a quantum two-level system which could be atoms for micro-toroid cavities, quantum dots for defects in PBGs or superconducting qubits for microwave stripline resonators[@coupled_twolevel]. Possibilities with such systems are enormous and include the the implementation optical quantum computing [@angelakis-ekert04], the production of entangled photons [@angelakis-bose06], the realization of Mott insulating and superfluid phases and spin chain systems [@angelakis-bose06b; @hartmann-plenio06; @greentree-hollenberg06] . Such settings can also be used to verify the possibilities of distributed quantum computation involving atoms coupled to distinct cavities [@alessio] also to generate cluster states for efficient measurement based quantum computing schemes[@angelakis_kay07].
When the coupling between the cavity field and the two-level system (which we will just call atom henceforth, noting that they need not necessarily be only atoms) is very strong (in the so called strong coupling regime), each cavity-atom unit behaves as a quantum system whose excitations are combined atom-field excitations called polaritons. The nonlinearity induced by this coupling or as it is otherwise known, the photon blockade effect[@birnbaum-kimble05], forces the system to a state where maximum one excitation (polariton) per site is allowed. However, a superposition of two different polaritons, which is equivalent to a superposition of two energy levels of the cavity-atom system, is indeed allowed and naturally the question arises as to whether that can be used as a qubit. Purely atomic qubits (formed from purely atomic energy levels) in cavities have long been discussed in the literature (see references cited in [@alessio], for example), but such qubits in distinct cavities do not directly interact with each other unless mediated through light. On the other hand, a purely photonic field in a cavity is not easy to manipulate in the sense of one being able to create arbitrary superpositions of its states by an external laser. Being a mixed excitation, polaritons interact with each other as well as permit easy manipulations with external lasers in much the same manner as one would manipulate and superpose atomic energy levels. Is there any interesting form of quantum information processing that can be performed by encoding the quantum information in a superposition of polaritonic states? While an ultimate aim might be to accomplish full quantum computation with polaritonic qubits (it has been recently shown this to possible using the cluster state approach [@angelakis_kay07]), we concentrate here on a more modest aim of transferring the state of a qubit encoded in polaritonic states (a polaritonic qubit) from one end of the coupled cavity array to another.
Assume a chain of $N$ coupled cavities. We will describe the system dynamics using the operators corresponding to the localized eigenmodes (Wannier functions), $a^{\dagger}_{k}(a_{k})$. The Hamiltonian is given by $$H=\sum_{k=1}^{N}\omega_d
a^{\dagger}_{k}a_{k}+\sum_{k=1}^{N}A(a^{\dagger}_{k}a_{k+1}+H.C.).$$ and corresponds to a series quantum harmonic oscillators coupled through hopping photons. The photon frequency and hopping rate is $\omega_{d}$ and $A$ respectively and no nonlinearity is present yet. Assume now that the cavities are doped with two level systems (atoms/ quantum dots/superconducting qubits) and $|g\rangle_{k}$ and $|e\rangle_{k}$ their ground and excited states at site $k$. The Hamiltonian describing the system is the sum of three terms. $H^{free}$ the Hamiltonian for the free light and dopant parts, $H^{int}$ the Hamiltonian describing the internal coupling of the photon and dopant in a specific cavity and $H^{hop}$ for the light hopping between cavities. $$\begin{aligned}
H^{free}&&=\omega_{d}\sum_{k=1}^N a_k^\dagger a_k+\omega_{0}\sum_k|e\rangle_{k} \langle e|_{k} \\
H^{int}&&=g \sum_{k=1}^N(a_k^\dagger|g\rangle_{k}\langle e|_{k}+H.C.)\\
H^{hop}&&= A\sum_{k=1}^N(a_k^\dagger a_{k+1} +H.C)\end{aligned}$$ where g is the light atom coupling strength. The $H^{free}+H^{int}$ part of the Hamiltonian can be diagonalized in a basis of mixed photonic and atomic excitations, called [*polaritons*]{} (Fig. 1). While $|g,0\rangle_k$ is the ground state of each atom cavity system, the excited eigenstates of the $k$th cavity-atom system are given by $|n\pm\rangle_k=(|g,n\rangle_k\pm
|e,n-1\rangle_k)/\sqrt2$ with energies $E^{\pm}_{n}=n\omega_{d}\pm
g\sqrt{n}$. One can then define polariton creation operators $P_{k}^{(\pm,n)\dagger}$ by the action $P_{k}^{(\pm,n)\dagger}|g,0\rangle_k=|n\pm\rangle_k$. As we have proved elsewhere, due to the blockade effect, once a site is excited to $|1-\rangle$ or $|1+\rangle$, no further excitation is possible[@angelakis-bose06b]. In simplified terms, this is because it costs more energy to add another excitation in already filled site so the system prefers to deposit it if possible to an a nearby empty site. This effect has recently lead to the prediction of a Mott phase for polaritons in coupled cavity systems[@angelakis-bose06b]. If we restrict to the low energy dynamics of the system such that states with $n\geq 1$ are not occupied, which can be ensured through appropriate initial conditions, the Hamiltonian in becomes (in the interaction picture):
$$\begin{aligned}
H_{I}=A\sum_{k=1}^{N}P^{(-)\dagger}_{k}P^{(-)}_{k+1}+
A\sum_{k=1}^{N}P^{(+)\dagger}_{k}P^{(+)}_{k+1}+H.C. \label{hop}\end{aligned}$$
where $P_{k}^{(\pm)\dagger}=P_{k}^{(\pm,1)\dagger}$ is the polaritonic operator creating excitations to the first polaritonic manifold (Fig. 1). In deriving the above, the logic is that the terms of the type $P^{(-)\dagger}_{k}P^{(+)}_{k+1}$, which inter-convert between polaritons, are fast rotating and they vanish[@angelakis-bose06b].
We are now in a position to outline the basic idea behind the protocol. A qubit is encoded as a superposition of the polaritonic states $|1+\rangle$ and $|1-\rangle$ in the first cavity. The multi-cavity system is then allowed to evolve according to $H_I$. At the receiving cavity at the other end we then do a measurement inspired by a dual rail quantum state transfer protocol [@burgarth-bose05] which heralds the perfect reception of the qubit for one outcome of the measurement, while for the other outcome of the measurement the process is simply to be repeated once more after a time delay. Before presenting the scheme in detail, let us first present a special set of initial conditions under which $H_I$ describes the dynamics of two identical parallel uncoupled spin chains.
Suppose we are restricting our attention to a dynamics in which the initial state is obtained by the action of only one of the operators among $P_k^{(+)\dagger}$ and $P_{k}^{(-)\dagger}$ on the state $\prod_k|g,0\rangle_k$ which has all the sites in the state $|g,0\rangle$. As $P_k^{(-)\dagger}$ does not act after $P_k^{(+)\dagger}$ has acted and vice versa, under the above restricted initial conditions, the system is going to evolve only according to one of the terms in Eq.(\[hop\]) [*i.e.,*]{} only according to the first or the second term. To be more precise, if we start with a state $P_j^{(+)\dagger}\prod_k|g,0\rangle_k$ only the term $A\sum_{k=1}^{N}P^{(+)\dagger}_{k}P^{(+)}_{k+1}$ is going to be active and cause the time evolution, while if we start with the state $P_j^{(-)\dagger}\prod_k|g,0\rangle_k$ only the term $A\sum_{k=1}^{N}P^{(-)\dagger}_{k}P^{(-)}_{k+1}$ will be responsible for the time evolution. Each of the operators $P^{(+)\dagger}_{k}$ and $P^{(-)\dagger}_{k}$ individually have the same algebra as the Pauli operator $\sigma^{+}_k=\sigma^x_k+i\sigma^y_k$, which makes both the parts of the Hamiltonian individually equivalent to a $XY$ spin chain with a Hamiltonian $H_{XY}=A\sum_k(\sigma^x_k\sigma^x_{k+1}+\sigma^y_k\sigma^y_{k+1})$. The restricted set of initial states mentioned above can be mapped on to those of two parallel chains of spins labeled as chain I and chain II respectively. Let $|0\rangle$ and $|1\rangle$ be spin-up and spin-down states of a spin along the $z$ direction, $|{\bf
0}\rangle^{(I)}|{\bf 0}\rangle^{(II)}$ be a state with all spins of both chains being in the state $|0\rangle$, $|{\bf
k}\rangle^{(I)}|{\bf 0}\rangle^{(II)}$ represent the state obtained from $|{\bf 0}\rangle^{(I)}|{\bf 0}\rangle^{(II)}$ by flipping only the $k$th spin of chain $I$ and $|{\bf 0}\rangle^{(I)}|{\bf
k}\rangle^{(II)}$ represents the state obtained from $|{\bf
0}\rangle^{(I)}|{\bf 0}\rangle^{(II)}$ by flipping only the $k$th spin of chain $II$. Then, the restricted class of initial conditions for polaritonic states can be mapped on to states of the parallel spin chains as $$\begin{aligned}
|g,0\rangle_1|g,0\rangle_2....|g,0\rangle_N\rightarrow |{\bf
0}\rangle^{I}|{\bf 0}\rangle^{II},\label{map1}\\
|g,0\rangle_1..|g,0\rangle_{k-1}|1+\rangle_k|g,0\rangle_{k+1}..|g,0\rangle_{N}\rightarrow|{\bf
k}\rangle^{(I)}|{\bf 0}\rangle^{II},\\
|g,0\rangle_1..|g,0\rangle_{k-1}|1-\rangle_k|g,0\rangle_{k+1}..|g,0\rangle_{N}\rightarrow|{\bf
0}\rangle^{I}|{\bf
k}\rangle^{(II)}
\label{map3}\end{aligned}$$ Under the above mapping and under the above restrictions on state space, $H_I$ becomes equivalent to the Hamiltonian of two identical parallel XY spin chains completely decoupled from each other. Precisely such a Hamiltonian is known to permit a heralded perfect quantum state transfer from one end of a pair of parallel spin chains to the other [@burgarth-bose05], and we discuss that below.
Spin chains are capable to transmitting quantum states by natural time evolution [@bose]. However it is well known that due to the disperion on the chain [@Osborne] the fidelity of transfer is quite low except for specific engineered couplings in the spin chains [@Christandl; @Plenio] or when the receiver has access to a significant memory [@giovannettiburgath06]. The advantage of the polariton system is that we have *two parallel and identical* chains. We have recently shown how this can be made use of in a dual rail protocol [@burgarth-bose05]. The main idea of this protocol is to encode the state in a symmetric way on both chains. The sender Alice encodes a qubit $\alpha|0\rangle+\beta|1\rangle$ to be transmitted as $$|\Phi(0)\rangle=\alpha |{\bf 0}\rangle^{(I)}|{\bf 1}\rangle^{(II)}
+\beta |{\bf 1}\rangle^{(I)}|{\bf 0}\rangle^{(II)},$$ which evolves with time as $$|\Phi(t)\rangle=\sum_{j=1}^Nf_{1j}(t)(\alpha |{\bf 0}\rangle^{(I)}
|{\bf j}\rangle^{(II)} +\beta |{\bf j}\rangle^{(I)}|{\bf
0}\rangle^{(II)}), \label{phit}$$ where $f_{1j}$ is the transition amplitude of a spin flip from the $1$st to the $j$th site of a chain. Clearly, if after waiting a while Bob performs a joint parity measurement on the two spins at his (receiving) end of the chain and the parity is found to be “odd", then the state of the whole system will be projected to $\alpha |{\bf 0}\rangle^{(I)} |{\bf N}\rangle^{(II)} +\beta |{\bf
N}\rangle^{(I)}|{\bf 0}\rangle^{(II)}$, which implies the perfect reception of Alice’s state (albeit encoded in two qubits now). The protocol presented in Ref.[@burgarth-bose05] in fact suggested the use of a two qubit quantum gate at Bob’s end which measured both the parity as well as mapped the state to a single qubit state. However, here the presentation as above suffices for what follows. Physically, this protocol, which is called the dual rail protocol, allows one to perform measurements on the chain that monitor the location of the quantum information *without perturbing it*. As such it can also be used for arbitrary graphs of spins (as long as there are two identical parallel graphs) with the receiver at any node of the graph. Furthermore, for the Hamiltonian at hand (XY spin model) it is known [@multirail] that the probability of success converges exponentially fast to one if the receiver performs regular measurements. The time it takes to reach a transfer fidelity $F$ scales as $$t=0.33 A^{-1} N^{5/3} |\ln (1-F) |.$$
The difference between our current coupled cavity system and the spin chain system considered in [@burgarth-bose05] is that in our case, the two chains are effectively realized in *one* system. Therefore, it is not necessary to perform a two-qubit measurement such as a parity measurement at the receiving ends of the chain. The qubit to be transferred is encoded as $\alpha^{'}|1+\rangle_1+\beta^{'}|1-\rangle_1\equiv\alpha|e,0\rangle_1+\beta|g,1\rangle_1$. This state can be created by the sender Alice using a resonant Jaynes-Cummings interaction between the atom and the cavity field. Then the whole evolution will exactly be as in Eq.(\[phit\]) with the spin chain states have to be replaced by polaritonic states according to the mapping given in Eqs.(\[map1\])-(\[map3\]). The measurement to herald the arrival of the state at the receiving end is accomplished by a exciting (shelving) $|g,0\rangle$ repeatedly to a metastable state by an appropriate laser (which does not do anything if the atom is either in $|1\pm\rangle$). The fluorescence emitted on decay of the atom from this metastable state to $|g,0\rangle$ implies that another measurement has to be done after waiting a while. No fluorescence implies success and completion of the perfect transfer of the polaritonic qubit. Interestingly enough, the measurement at the receiving cavity need not be snapshot measurements at regular time intervals, but can also be continuous measurements under which the scheme can have very similar behavior to the case with snap-shot measurements for appropriate strength of the continuous measurement process [@kurt].
We now briefly discuss the parameter regime needed for the scheme of this paper. In order to achieve the required limit of no more than one excitation per site, the parameters should have the following values[@angelakis-bose06b]. The ratio between the internal atom-photon coupling and the hopping of photons down the chain should be $g/A=10^{2}$. We should be on resonance, $\Delta=0$, and the cavity/atomic frequencies $\omega_d,\omega_0 \sim 10^4g$ which means we should be well in the strong coupling regime. The losses should also be small, $g/max(\kappa,\gamma)\sim 10^3$, where $\kappa$ and $\gamma$ are cavity and atom/other qubit decay rates. These values are expected to be feasible in both toroidal microcavity systems with atoms and stripline microwave resonators coupled to superconducting qubits [@coupled_twolevel], so that the above states are essentially unaffected by decay for a time $10/A$ ($10$ns for the toroidal case and $100$ns for microwave stripline resonators type of implementations).
We conclude with a brief discussion about the positive features of the scheme and situations in which the scheme might be practically relevant. The scheme combines the best aspects of both atomic and photonic qubits as far as communication is concerned. The atomic content of the polaritonic state enables the manipulation to create the initial state and measure the received state of the cavity-atom systems with external laser fields, while the photonic component enables its hopping from cavity to cavity thereby enabling transfer. Unlike quantum communication schemes where an atomic qubit first has to be mapped to the photonic state in the transmitting cavity and be mapped back to an atomic state in the receiving cavity by external lasers, here the polaritonic qubit simply has to be created. Once created, it will hop by itself though the array of cavities without the need of further external control or manipulation.
In what situations might such a scheme have some practical utility? One case is when Alice “knows" the quantum state she has to transmit to Bob. She can easily prepare it as a polaritonic state in her cavity and then let Bob receive it through the natural hopping of the polaritons. Another situation is when a multiple number of cavities are connected with each other through an arbitrary graph. The protocol of Ref.[@burgarth-bose05] still works fine in this situation with Alice’s qubit being receivable in any of the cavities simply by doing the receiving fluorescence measurements in that cavity.
We acknowledge the hospitality of Quantum Information group in NUS Singapore, and the Kavli Institute for Theoretical Physics where discussions between DA and SB took place during joint visits. This work was supported in part by the QIP IRC (GR/S82176/01), the European Union through the Integrated Projects QAP (IST-3-015848), SCALA (CT-015714) and SECOQC., and an Advanced Research Fellowship from EPSRC.
[99]{}
D. K. Armani, T. J. Kippenberg, S.M. Spillane & K. J. Vahala. [*Nature*]{} [**421**]{}, 925 (2003).
D.G. Angelakis, E. Paspalakis and P.L. Knight, [*Contemp. Phys.*]{} [**45**]{}, 303 (2004).
A. Yariv, Y. Xu, R. K. Lee and A. Scherer. Opt. Lett. [**24**]{}, 711 (1999); M. Bayindir, B. Temelkuran and E. Ozbay. [*Phys. Rev. Lett.,*]{} [**84**]{}, 2140 (2000); J. Vuckovic, M. Loncar, H. Mabuchi. & Scherer A. [*Phys. Rev. E*]{}, [**65**]{}, 016608 (2001);
Barbosa Alt H., Graef H.-D.C. et al., [*Phys. Rev. Lett.*]{} [**81**]{} (1998) 4847; A. Blais, R-S. Huang, et al., [*Phys. Rev. A*]{} [**69**]{}, 062320 (2004).
G. S. Solomon, M. Pelton, and Y. Yamamoto Phys. Rev. Lett. [**86**]{}, 3903 (2001); A. Badolato et al, [*Science*]{} [**308**]{} 1158 (2005); A. Wallraff, D. I. Schuster, et al., [*Nature*]{} [**431**]{}, 162-167 (2004); T. Aoki et al. quant-ph/0606033; W.T.M. Irvine et al., Phys. Rev. Lett. [**96**]{}, 057405 (2006).
D.G. Angelakis, M.Santos, V. Yannopapas and A. Ekert, quantum-ph/0410189, Phys. Lett. A. 362, 377 (2007).
D. G. Angelakis and S. Bose. J. Opt. Soc. Am. B [**24**]{}, 266-269 (2007).
D. G. Angelakis, M. Santos and S. Bose. quant-ph/0606157.
M. J. Hartmann, F. G. S. L. Brandao and M. B. Plenio, Nature Physics [**2**]{}, 849 (2006).
A. Greentree et al., Nature Physics [**2**]{}, 856 (2006).
A. Serafini, S. Mancini and S. Bose, Phys. Rev. Lett. [**96**]{}, 010503 (2006).
D.G. Angelakis and A. Kay, quant-ph/0702133.
K. M. Birnbaum, A. Boca et al. [*Nature*]{} [**436**]{}, 87 (2005); A. Imamoglu, H. Schmidt, et al. [*Phys. Rev. Lett.*]{} [**79**]{} 1467 (1997).
D. Burgarth and S. Bose, Phys. Rev. A [**71**]{}, 052315 (2005).
S. Bose, Phys. Rev. Lett **91**, 207901(2003).
T. J. Osborne and N. Linden, Phys. Rev. A [**69**]{}, 052315 (2004). M. Christandl, N. Datta, A. Ekert and A. J. Landahl, Phys. Rev. Lett. **92**, 187902 (2004).
M. B. Plenio and F. L. Semiao, New J. Phys. 7, [**73**]{} (2005).
V. Giovannetti and D. Burgarth, Phys. Rev. Lett. [**96**]{}, 030501 (2006).
D. Burgarth, V. Giovannetti and S. Bose, J. Phys. A: Math. Gen. [**38**]{}, 6793 (2005).
K. Shizume, K. Jacobs, D. Burgarth & S. Bose, arXiv.org eprint: quant-ph/0702029 (2007).
|
---
abstract: 'Inspired by recent discussions of inverse magnetic catalysis in the literature, we examine the effects of a uniform external magnetic field on the chiral phase transition in quenched ladder QED at nonzero chemical potential. In particular, we study the behaviour of the effective potential as the strength of the magnetic field is varied while the chemical potential is held constant. For a certain range of the magnetic field, the effective potential develops a local maximum. Inverse magnetic catalysis is observed at this maximum, whereas the usual magnetic catalysis is observed at the true minimum of the effective potential.'
author:
- 'W.-C. Syu, D.-S. Lee'
- 'C. N. Leung'
title: External magnetic fields and the chiral phase transition in QED at nonzero chemical potential
---
The effect of external fields on the symmetry properties of the vacuum has been extensively studied in the past decades. Under the quenched, ladder approximation of QED, chiral symmetry is dynamically broken at weak gauge couplings when a uniform magnetic field is present [@GMS; @LNA]. The dynamically generated fermion mass is obtained that increases with growing magnetic field strength. The phenomenon is referred to as magnetic catalysis of chiral symmetry breaking. A subsequent analysis [@LW] shows that an improved truncation beyond the quenched ladder approximation produces a gauge independent dynamical fermion mass within the lowest Landau level approximation, provided that it is obtained from the fermion self-energy evaluated on shell.\
Recent lattice studies of hot QCD in an external (electro)magnetic field found that the critical temperature of the chiral phase transition decreases with increasing magnetic field [@BBE], contrary to the expectation from magnetic catalysis. This is known as inverse magnetic catalysis. Although there are several proposed ideas for explaining this unexpected behaviour [@FH], what causes this anomalous phenomenon remains an open question. Recent field theoretic studies did not arrive at a definite conclusion about how the critical chemical potential for the chiral phase transition vary with the strength of the external magnetic field [@chempot].\
Previous works on quenched ladder QED in an external magnetic field show that chiral symmetry is restored above a certain critical value of temperature [@LLN1; @GS; @LLN2] as well as critical chemical potential [@LLN2]. Since the critical temperature and chemical potential are measured in units of $\sqrt{|eH|}$, where $H$ is the magnetic field, it suggests that their values will increase with increasing magnetic field and the system does not exhibit behaviour of inverse magnetic catalysis. Prompted by the current interest in inverse magnetic catalysis, we reexamine these earlier works more carefully to determine the effects of the magnetic field on the chiral phase transition. We shall focus on the effects on the critical chemical potential in this paper.\
We employ the effective potential approach of Ref. [@LMNS], with modifications relevant to studying the chiral dynamics in the case of nonzero chemical potential. The detailed profile of the effective potential and the location of its extrema will enable one to construct the phase diagram of the chiral dynamics and understand the nature of the phase transition.\
We begin by constructing the effective potential for chiral dynamics in terms of the expectation value of composite local fields, $\sigma(x)=\langle 0| \bar{\psi}(x)\psi(x)|0\rangle$ and $\pi(x)=\langle 0|\bar{\psi}(x) i\gamma_5\psi(x) |0 \rangle$. To do so, consider the generating functional $$\begin{aligned}
Z[J_{\sigma}, J_{\pi}] &\equiv& \exp{\left(iW[J_{\sigma}, J_{\pi}]\right)} \nonumber \\
&=& \int {\cal D} \psi(x) {\cal D} \bar{\psi}(x) {\cal D} A_{\mu}(x) \cdot
\nonumber \\
&&\exp{\left( i\int d^4 x\left[{\cal L}+J_{\sigma}(x)\bar{\psi}(x)
\psi(x)+J_{\pi}(x)\bar{\psi}(x)i\gamma_5\psi(x)\right] \right)}\, ,
%\nonumber \\
\label{gen-fun}\end{aligned}$$ where ${\cal L}$ is the Lagrangian density of massless QED in a uniform external magnetic field pointing in the $z$ direction. The expectation values of the composite fields can be obtained by taking the usual variation of the generating functional with respect to the sources: $$\frac{\delta W}{\delta J_{\sigma}(x)}=\sigma(x),\qquad \frac{\delta
W}{\delta J_{\pi}(x)}=\pi(x)\, .
\label{del-W}$$ By inverting the expressions (\[del-W\]) to write $\sigma$ and $\pi$ as a function of the sources, the effective action can be obtained through the Lengedre transformation $$\Gamma[\sigma, \pi] = W[J_{\sigma}, J_{\pi}]-\int d^4 x
\left[J_{\sigma}(x)\sigma(x) +J_{\pi}(x)\pi(x)\right] \, ,
\label{eff-act}$$ from which $$\frac{\delta \Gamma}{\delta \sigma(x)}=-J_{\sigma}(x),\qquad
\frac{\delta \Gamma}{\delta \pi(x)}=-J_{\pi}(x).
\label{del-G}$$ For spacetime independent fields, $\sigma_0$ and $\pi_0$ are given by the corresponding constant sources $j_{\sigma}$ and $j_{\pi}$, respectively. The effective potential is found to be $$V[\sigma_0,\pi_0]=-\frac{1}{\Omega} \Gamma [\sigma_0, \pi_0] \, ,
\label{effect_pot}$$ where $\Omega$ is the spacetime volume. The presence of chiral symmetry renders the effective action/potential a function of $\rho=(\sigma^2+\pi^2)^{1/2}$ only. It is thus sufficient and convenient to simply consider, e.g., the case $\pi=0$ and $\sigma \neq 0$. The complete functional form of the effective potential can be found via substituting $\sigma_0=\rho$. In terms of the spacetime independent generating functional, denoted by $w[j]=W[j]/\Omega$, where we have simplified the notation by setting $j_{\sigma}=j$, the effective potential now becomes $$V[\rho]= j \rho -w[j] \, ,
\label{V_fun}$$ where $$w[j]=\int \rho \,\, d j \, .
\label{w_fun}$$\
Applying the above to the case of nonzero chemical potential and using the quenched ladder approximation that takes into account contributions from the lowest Landau level only, we find $$\begin{aligned}
j &~\simeq~& -m_{\mu} + \frac{\alpha}{2 \pi} |eH| \, m_{\mu} \int_{-\infty}^\infty
dq_3 \int_0^\infty d\hat{q}_\perp^2 ~
\frac{{\rm e}^{-\hat{q}_\perp^2}}{Q_1 Q_2} \nonumber \\
& & ~~~~\cdot \left[f_+ (Q_1,Q_2) +
\theta (Q_1 - \mu ) f_-(Q_1,Q_2)+ \theta( \mu - Q_1) f_- (Q_1,-Q_2)
\right) \, ,
\label{gapu}\end{aligned}$$ where $\hat{q}_\perp^2 \equiv (q_1^2 + q_2^2)/(2|eH|)$, $~Q_1^2 \equiv q_3^2 + m^2_{\mu}$, $~Q_2^2 \equiv q_3^2 + 2
|eH| \hat{q}_\perp^2$, $\alpha$ is the fine structure constant, and $m_{\mu}$ is the infrared dynamical fermion mass which is a function of the chemical potential $\mu$ ($\mu \equiv |\mu| > 0$) and the external source $j$ [@LLN1; @LLN2]. The functions $f_{\pm}$ are defined as $$f_{\pm} (Q_1,Q_2)=\frac{1}{Q_1 + Q_2 \pm \mu } \, .$$ The chiral condensate, $\rho= \langle \bar{\psi} \psi \rangle_{\mu}$, is found to be $$\rho~\simeq~-~\frac{\vert eH
\vert}{\pi^2} m_\mu \left[\theta(m_\mu-\mu) \int_0^{\sqrt{\vert
eH \vert}} \frac{dq_3}{\sqrt{q_3^2+m_\mu^2}} + \theta(\mu-m_\mu)
\int_{\sqrt{\mu^2-m_\mu^2}}^{\sqrt{\vert eH \vert}} \frac{dq_3}
{\sqrt{q_3^2+m_\mu^2}} \right]_.
\label{rhoeq}
\label{ccmu}$$\
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
\[figur\]
The effective potential $V(\rho)$ is evaluated numerically for various values of the magnetic field in units of the chemical potential $\mu$. Some sample results are shown in Fig.1. It is found that, for $\sqrt{|eH|} \leq \mu$, the effective potential has only one minimum, which is located at $\rho=0$, corresponding to the chirally symmetric phase. See the first graph in Fig.1. For $\sqrt{|eH|}> \mu$, the second and third graphs in Fig.1 show that, as the magnetic field is increased, the effective potential starts to develop two additional local extrema, a local minimum and a local maximum, at nonzero values of $\rho$. The global minimum at $\rho=0$ preserves chiral symmetry. As the magnetic field reaches a certain strength, the two minima become degenerate and a first-order phase transition is about to occur (see the fourth graph in Fig.1). When the field strength is above this critical value, the global minimum of $V(\rho)$ is shifted to a nonzero value of $\rho$ and chiral symmetry is spontaneously broken. In particular, the expectation value $\rho$ at the global minimum increases with the increase in the magnetic field, consistent with the expectation of magnetic catalysis (see the last two graphs in Fig.1). On the contrary, as the strength of the magnetic field increases, the expectation value at the local maximum, the unstable state, increases and then shifts toward $\rho=0$. This gives rise to the anomalous inverse magnetic catalysis effect and is shown in Fig.2. In Fig.3, the corresponding dynamical mass for the ground state and the unstable state is also shown. Again the unstable state exhibits features of inverse magnetic catalysis. These figures show again that the magnetic field must exceed a certain critical value for spontaneous chiral symmetry breaking to take place when the chemical potential is not zero.\
 \[figur\]
![The dynamical fermion mass (in units of $\mu$) as a function of the magnetic field for $\alpha=\pi/10$. The red curve corresponds to the global minimum of the effective potential and behaves as expected from magnetic catalysis, whereas the blue curve for the local maximum of the effective potential exhibits inverse magnetic catalysis behaviour. For comparison, the approximate result for $m_{\mu 0 ; {\rm unstable}}$ from Eq. (\[m\_unstable\]) is shown by the dotted curve.](fig3.pdf "fig:") \[figur\]
To better understand the numerical results above, let us examine the behaviour of the effective potential more closely. For $m_{\mu} < \mu$, the behavior of the effective potential can be found by first obtaining the corresponding approximate expression of (\[gapu\]) for small $m_{\mu}$, $$j ~\simeq~ c_1 ~m_{\mu} + c_3~ m_{\mu}^3 + {\cal O}(m_{\mu}^5)\, ,
\label{j_smallm}$$ where the constants $c_1$ and $c_3$ depend on the chemical potential and the magnetic field. Specifically, $$\begin{aligned}
c_1 (H,\mu) &=& -1 + \frac{\alpha}{2 \pi} |eH| \,
\int_{-\infty}^\infty
dq_3 \int_0^\infty d\hat{q}_\perp^2 ~
\frac{{\rm e}^{-\hat{q}_\perp^2}}{q_3 Q_2} \cdot \nonumber\\
&& \quad\quad \left[ f_+ (q_3,Q_2) + \theta(q_3 - \mu ) f_- (q_3,Q_2)
+ \theta(\mu - q_3) f_- (q_3,-Q_2) \right] \, ,
\nonumber \\
c_3 (H,\mu) &=& \frac{\alpha}{ \pi} \frac{|eH|}{\mu^2}
\, \int_0^\infty d\hat{q}_\perp^2 ~
\frac{{\rm e}^{-\hat{q}_\perp^2}}{2 \vert e H \vert \hat{q}_{\perp}^2
+ \mu^2}\, .\end{aligned}$$ A nontrivial approximate solution for the gap equation can be obtained by setting the source $j$ equal to zero, yielding $$m_{\mu 0; {\rm unstable}} \simeq \sqrt{- c_1 /{c_3}} \, .
\label{m_unstable}$$ Note that $c_3$ is always positive and $c_1$ is negative for some range of the parameters. It can be seen from Fig.3 that this solution, which is valid for $m_{\mu} < \mu$, corresponds to the unstable local maximum of the effective potential. We also plotted in Fig.3 this approximate result for $m_{\mu 0; {\rm unstable}}$ to compare with the exact numerical result. A reasonably good agreement is found. The decrease of $m_{\mu 0 ; {\rm unstable}}$ with increasing magnetic field can be understood by observing that, for a larger value of the magnetic field, $c_1$ is less negative whereas $c_3$ is larger.\
Turning now to the behaviour of the chiral condensate $\rho$ for small $m_\mu$, we find from Eq. (\[rhoeq\]) that $$\rho \simeq -\frac{\vert e H \vert}{\pi^2} m_{\mu} \ln \left[ \left(
\sqrt{\vert e H \vert}+\sqrt{\vert e H \vert +m_{\mu}^2} \right)/
\left(\mu + \sqrt{\mu^2-m^2_{\mu}} \right) \right] \, .
\label{rho_smallm}$$ Using the approximate expressions for $j$ and $\rho$ above, one finds from Eq. (\[w\_fun\]) that, for $m_\mu < \mu$, $$\begin{aligned}
w_<&=&\int_0^{m_\mu} \rho (m)\, \frac{dj}{dm} \, dm
\nonumber\\
&\simeq& - \frac{|eH|}{2 \pi^2} \ln \left(\sqrt{|eH|}/{\mu} \right)
(c_1 m_\mu^2 + \frac{3 c_3}{2} m_\mu^4)
- \frac{c_1}{16 \pi^2}
\left[1 + \frac{|eH|}{\mu^2} \right] m_{\mu}^4 + {\cal O}(m^6_{\mu})\, .
\label{w_fun_m=0}\end{aligned}$$ It follows from Eq. (\[V\_fun\]) that the corresponding effective potential is $$\begin{aligned}
V_< &\simeq& - \, \frac{c_1}{ 2
\pi^2} \vert e H \vert \ln \left(\sqrt{|eH|}/{\mu} \right) m_{\mu}^2
-\left[\frac{3 c_1}{16 \pi^2}
\left(1 + \frac{|eH|}{\mu^2} \right) \right. \nonumber\\
&& \left. \quad\quad\quad\quad \quad\quad\quad\quad +\frac{c_3}{ 4\pi^2}
\vert eH \vert \ln \left(\sqrt{|eH|}/{\mu} \right) \right] m_{\mu}^4
+ {\cal O}(m^6_{\mu})\, .
\label{v_fun_m=0}\end{aligned}$$ This approximation is compared with the numerical result for the full effective potential in Fig.4. It is seen to give a reliable description for $m_{\mu} < 0.5 \mu$. We check that, for sufficiently large magnetic field, the term proportional to $c_3$ in the coefficient of the $m_\mu^4$ dominates, thus reproducing the solution (\[m\_unstable\]). The result of (\[v\_fun\_m=0\]) also indicates that the unstable local maximum of the effective potential starts to develop when $\sqrt{|eH|}>\mu$ and $c_1<0$, leading to a positive coefficient for the $m_{\mu}^2$ term and negative coefficient for the $m_{\mu}^4$ term, and then disappears when $c_1$ turns positive as the magnetic field exceeds certain critical value, consistent with our numerical finding in Fig.1.\
 \[figur\]
We perform a similar analysis for the case $m_\mu > \mu$. Specifically we examine the behaviour of the effective potential in the neighbourhood of $m_\mu = m_{\mu 0}$, the ground-state solution of the gap equation. We find approximate expressions for $$j \simeq b_1 ~(m_{\mu}-m_{\mu 0})+ b_2 ~(m_{\mu}-m_{\mu 0})^2 +
{\cal O}(m_{\mu}-m_{\mu 0})^3
\label{j_largem}$$ and $$\rho \simeq -\frac{\vert e H \vert}{\pi^2} m_{\mu} \ln
\left[ ( \sqrt{\vert e H\vert}+\sqrt{\vert e H \vert
+m_{\mu}^2})/{m_{\mu}} \right] \, ,
\label{rho_largem}$$ from which we obtain the effective potential in the form $$V_> \simeq v_0 \,+ v_2 \, (m_{\mu}-m_{\mu 0})^2 + {\cal O}
(m_{\mu}-m_{\mu 0})^3 \, .
\label{v_fun_largem}$$ The detailed expressions for the coefficients $b_1$, $b_2$, $v_0$, and $v_2$ are not very illuminating. Instead, we show the result in Fig.4 and compare it with the full effective potential. The agreement is quite good and the global minimum of the effective potential is correctly produced. As shown in Figs.2 and 3, this true vacuum behaves like what one would expect from magnetic catalysis as the magnetic field is varied.\
In summary, we have reexamined the effect of an external magnetic field on the chiral phase transition in QED at a finite chemical potential through the effective potential of order parameter fields relevant to the chiral dynamics. We observe differing behaviour between the true vacuum (global minimum of the effective potential) and the unstable local maximum of the effective potential. While both are solutions to the gap equation, the true vacuum behaves according to magnetic catalysis while the solution at the local maximum behaves according to inverse magnetic catalysis. Since the chiral phase transition is governed by the vacuum solution, we conclude that quenched ladder QED at nonzero chemical potential exhibits characteristics of magnetic catalysis. It is useful and interesting to also include thermal fluctuations in this study. Work along this direction is in progress.\
The work of DSL was supported in part by the National Science Council of Taiwan. Part of the work of CNL was carried out while he was visiting the Academia Sinica and National Dong Hwa University in Taiwan. He thanks these institutions for their support and hospitality.
V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy, Phys. Rev. D [**52**]{}, 4747 (1995); Nucl. Phys. [**B462**]{}, 249 (1996). C. N. Leung, Y. J. Ng, and A. W. Ackley, Phys. Rev. D [**54**]{}, 4181 (1996).
C. N. Leung and S.-Y. Wang, Nucl. Phys. B [**747**]{}, 266 (2006); Annals Phys. [**322**]{}, 701 (2007).
G.S. Bali [*et al*]{}, JHEP [**1202**]{}, 044 (2012); PoS [**LATTICE2011**]{}, 192 (2011); Phys. Rev. D [**86**]{}, 071502 (2012); G. S. Bali, PoS [**ConfinementX**]{}, 198 (2012).
K. Fukushima and Y. Hidaka, Phys. Rev. Lett. [**110**]{}, 031601 (2013); T. Kojo and N. Su, Phys. Lett. B [**720**]{}, 192 (2013); F. Bruckmann, G. Endrodi and T. G. Kovacs, JHEP [**1304**]{}, 112 (2013); J. Chao, P. Chu and M. Huang, arXiv:1305.1100 \[hep-ph\].
F. Preis, A. Rebhan and A. Schmitt, JHEP 1103, 033 (2011); AIP Conf. Proc. 1492, 264 (2012); A.F. Garcia and M.B. Pinto, arXiv:1307.2451 \[hep-ph\]; P.G. Allen and N.N. Scoccola, arXiv:1307.4070 \[hep-ph\].
D.-S. Lee, C. N. Leung and Y. J. Ng, Phys. Rev. D [**55**]{}, 6504 (1997).
V. P. Gusynin and I. A. Shovkovy, Phys. Rev. D [**56**]{}, 5251 (1997).
D.-S. Lee, C. N. Leung and Y. J. Ng, Phys. Rev. D [**57**]{}, 5224 (1998).
D.-S. Lee, P. N. McGraw, Y. J. Ng, and I. A. Shovkovy, Rev. D [**59**]{}, 085008 (1999).
|
---
abstract: 'In addition to producing a strong gravitational signal, a short gamma-ray burst (GRB), and a compact remnant, neutron star mergers eject significant masses at significant kinetic energies. This mass ejection takes place via dynamical mass ejection and a GRB jet but other processes have also been suggested: a shock-breakout material, a cocoon resulting from the interaction of the jet with other ejecta, and viscous and neutrino driven winds from the central remnant or the accretion disk. The different components of the ejected masses include up to a few percent of a solar mass, some of which is ejected at relativistic velocities. The interaction of these ejecta with the surrounding interstellar medium will produce a long lasting radio flare, in a similar way to GRB afterglows or to radio supernovae. The relative strength of the different signals depends strongly on the viewing angle. An observer along the jet axis or close to it will detect a strong signal at a few dozen days from the radio afterglow (or the orphan radio afterglow) produced by the highly relativistic GRB jet. For a generic observer at larger viewing angles, the dynamical ejecta, whose contribution peaks a year or so after the event, will generally dominate. Depending on the observed frequency and the external density, other components may also give rise to a significant contribution. We also compare these estimates with the radio signature of the short GRB 130603B. The radio flare from the dynamical ejecta might be detectable with the EVLA and the LOFAR for the higher range of external densities $n\gtrsim 0.5{\rm cm^{-3}}$.'
date: 8 January 2015
title: 'Mass ejection from neutron star mergers: different components and expected radio signals'
---
\[firstpage\]
gravitational waves$-$binaries:close$-$stars:neutron$-$gamma-ray burst:general
Introduction {#sec:Introduction}
============
A binary neutron star (ns$^2$) merger is one of the most promising targets of ground-based gravitational-wave (GW) interferometers, such as Advanced LIGO, Advanced Virgo, and KAGRA [@ligo2014; @virgo2011CQG_v2; @kagra2013PRD]. The expected event rate of ns$^2$ mergers is $0.4$ – $400 {\rm~yr^{-1}}$ [@abadie2010CQG]. Most of these events will be just above or just below the detection threshold. Observations of an electromagnetic counterpart will confirm the validity of these GW signals, increasing significantly the potential detection rate and detection confidence [@kochanek1993ApJ]. In addition, an electromagnetic counterpart will enable the localization of the sources and the identification of their host galaxies and their redshifts, enhancing significantly the potential information from this event. Finally any electromagnetic counterpart will provide invaluable information on the physics of the merger process.
[lcccccccc]{} & Mass \[$M_{\odot}$\] & Kinetic Energy $E$ \[erg\] &Fiducial $E$ \[erg\]& Average $\beta \Gamma$& Fiducial $\beta \Gamma$& Average $Y_{e}$ & Reference\
Dynamical ejecta[^1] & $10^{-4}$ – $10^{-2}$ & $10^{49}$ – $10^{51}$& $5\cdot 10^{50}$& $0.1$ – $0.3$ & 0.2& $0.01$ – $0.4$& \[1\]\
GRB jet & $\lesssim 10^{-8}$ & $10^{47}$ – $10^{50.5}$ & $10^{48}$ & $> 30$& –&– & \[2\]\
Cocoon & $10^{-6}$ – $10^{-4}$ & $10^{47}$ – $10^{50.5}$ & $10^{48}$ & $0.2$ – $10$ & $0.3$ &– & \[3\]\
Shock breakout &$10^{-6}$ – $10^{-4}$ & $10^{47}$ – $10^{49.5}$ & $10^{48.5}$ & $1$ & $1$ &– & \[4\]\
Wind[^2] & $10^{-4}$ – $5\cdot 10^{-2}$ &$10^{47}$ – $10^{50}$ & $10^{50}$ &$0.03$ – $0.1$ & $ 0.07$& $0.2$ – $0.4$ & \[5\]\
\
\[tab1\]
The detection horizon distance will extend up to a few hundred Mpc. The size of the GW-sky localization error box will depend on the number of detectors used and between a few tens and thousands Milky-Way size galaxies will reside within this error box (see e.g., @nissanke2011ApJ; @fairhurst2011CQG). Follow-up observations will be a challenging task, even for a search limited to these galaxies. Clearly, a good understanding of the expected electromagnetic signals is essential to detect an electromagnetic counterpart [@metzger2012ApJ; @nissanke2013ApJ; @kanner2013ApJ; @kasliwal2014ApJ; @bartos2014MNRAS].
Ns$^2$ mergers have been recognized as the possible progenitors of short gamma-ray bursts (GRBs) (@eichler1989Nature; @nakar2007) and short GRBs and their afterglows are one of the most attractive electromagnetic counterparts of GW events. However, GRBs and their early afterglows are believed to be highly beamed with a half-opening angle $\theta_{j}\approx 10^{\circ}$ [@fong2014ApJ]. This results in about $5\%$ probability[^3] to detect a short GRB in coincidence with the GW signal (@shutz2011CQG; @nissanke2011ApJ; @seto2015MNRAS). Even if the viewing angle is larger than the jet opening angle, off-axis afterglows, called orphan afterglows, can be observed at late times when the relativistic jet slows down and its emission is less beamed. As the frequency of the peak flux of afterglows decreases with time, at sufficiently small viewing angles the off-axis afterglows in the optical to radio bands can be a good potential candidate of electromagnetic counterparts to GW events (@vaneerten2011ApJ; @metzger2012ApJ).
In addition to the GRB beamed emission and its late more isotropic orphan afterglow, electromagnetic waves will be emitted quasi isotropically at different stages from material that is ejected during the merger. Most notable one is a macronova (also called kilonova), an optical–infrared transient driven by the radioactive decay of the heavy nuclei synthesized in the ejecta (@li1998ApJ; @metzger2010MNRAS; @kasen2013ApJ; @barnes2013ApJ,; @tanaka2013ApJ; @grossman2014MNRAS). Recently, the [*Hubble Space Telescope*]{} detected a near infrared bump at 9 days after the [*Swift*]{} short GRB 130603B (@tanvir2013Nature; @berger2013ApJ), which is consistent with the theoretical expectation of macronovae. While the identification hinges on a single data point if correct this is the first observational evidence for a significant mass ejection with a high velocity from a ns$^2$ merger.
Synchrotron radiation of electrons accelerated in shocks formed between the (mildly) relativistic ejecta and the interstellar medium (ISM) is a second electromagnetic counterpart [@nakar2011Nature]. This emission can last up to a few years and peaks in the radio band. All the ejected material will contribute to this emission, but different components with different velocities will contribute at different timescales, at different frequencies, and at different intensities. The rise time and the peak flux depend on the density of the ISM surrounding the merger but for modest densities the radio signals can be observed up to the detector horizon [@piran2013MNRAS].
Recent studies have shown that mass ejection from mergers is driven by several different processes. The most robust one, that appears in numerical merger simulations, is the dynamical mass ejection. If mergers are accompanied by GRBs then clearly relativistic GRB jets are another component. Other mass ejection mechanisms that have been proposed are merger shock-breakout material, viscous/neutrino/magnetically driven winds, and a possible cocoon that forms when the jet propagates within the other components of the ejecta.
The different components have different masses and kinetic energies. Their characteristics depend on the nature of the progenitors, in particular on their relative sizes, on the nature of the merger remnant, which could be either a black hole or a massive neutron star (MNS), as well on the, unknown yet, neutron star matter equation of state (see, e.g., @hotokezaka2013PRDa [@bauswein2013ApJa]). The different components will interact with each other and these interactions will affect their dynamics [@bucciantini2012MNRAS; @nagakura2014ApJ; @murguia2014ApJ], possibly producing electromagnetic signatures (e.g., @bucciantini2012MNRAS [@zhang2013ApJ; @metzger2014MNRASmag; @nakamura2014ApJ; @rezzolla2014; @ciolfi2014; @kisaka2014]).
In this paper we examine the long lasting radio emission arising from the different components of the ejecta. The structure of the paper is as follows: We summarize in Sec. \[sec:Components\] the properties of the different components of the ejecta. In Sec. \[sec:radio\] we calculate the expected long-lasting radio flares produced by the interaction of the different components of the ejecta with the ISM. In Sec. \[sec:grb\] we compare these estimates with the radio signature of the short GRB 130603B. Finally, in Sec. \[sec:conc\], we summarize our results and their possible implications on the detection of radio signals accompanying mergers.
Different component of ejecta and their properties {#sec:Components}
==================================================
{width="85mm"} {width="85mm"}
As material is ejected in different processes the different components will have different masses, kinetic energies, velocities, and electron fractions. The first three quantities determine the radio flare signals while all four are important for macronova estimate. Table \[tab1\] summarizes the values of these quantities as taken from the recent literature. The properties of the different components of the ejecta are also shown in Fig. \[fig1\]. The left panel of the figure depicts the possible range of the kinetic energy, $E$, and the four velocity, $\Gamma \beta$. Here $\Gamma$ is a Lorentz factor and $\beta$ is a velocity in units of the speed of light $c$. Also shown in the figure are the deceleration timescales due to the interaction with the ISM, which are discussed later. This timescale gives the characteristic peak time of the radio flares from each component. The right panel of the figure shows schematically the expected morphology of the ejecta.
In the following, we briefly describe the properties of the different components. In each case we focus on the total mass, energy, and the corresponding velocities. We also mention the expected distribution of energy as a function of velocity, which is essential in order to estimate the radio flares from these components. For completeness we also mention the electron fraction $Y_e$. This is not needed for the radio estimate but it is a critical quantity that determines the composition of the ejected material as well as the heating rate that is essential for macronova estimates.
The dynamical ejecta
--------------------
Gravitational and hydrodynamical interactions produce the dynamical ejecta. In many senses it is the easiest to calculate and as such it is the most robust element. It was investigated using Newtonian simulations (e.g., ) and using general relativistic simulations (e.g., ). According to these numerical simulations, the mass and kinetic energy of the dynamical ejecta are expected to be in the range $10^{-4}\lesssim M_{\rm ej} \lesssim 10^{-2}M_{\odot}$ and $10^{49}\lesssim E \lesssim 10^{51}$ erg, respectively. The median value of $E$ in the general relativistic simulations is a few times $10^{50}$ erg. The properties of the dynamical ejecta are as follows.
[*The tidal ejecta.*]{} A fraction of the material obtains sufficient angular momentum and is ejected via tidal interaction due to non-axisymmetry of the gravitational forces. This matter is ejected even before the two stars collide with each other and it lasts as long as the gravitational field is not axisymmetric (about $10$ ms after the merger in the case that the remnant is a MNS). This tidal component is mostly ejected into the equatorial plane of the binary within an angle about $20^{\circ}$ (see e.g., Fig. 17 in @hotokezaka2013PRDa).
The electron fraction of the dynamical ejecta and the resulting nucleosynthesis have been studied in the literature (e.g. @goriely2011ApJ [@korobkin2012MNRAS; @wanajo2014ApJ]). The tidally ejected material has initially a low electron fraction $Y_{e}\ll 0.1$ as this matter does not suffer from shock heating and neutrino irradiation [@wanajo2014ApJ]. This is particularly important concerning the possibility that this is the source of heavy (high atomic number) $r$-process nuclides, but it is not so relevant for our discussion that is concerned mostly with the radio flare. This fraction can increase by electron neutrino absorption or by positron absorption. The tidal component ejected at late times has higher $Y_{e}$ values.
[*The shocked component.*]{} A shock is formed at the interface of the merging neutron stars. The shock sweeps up the material in the envelope of the merging neutron stars. Furthermore, a shock is continuously produced around the envelope of a remnant MNS as long as the MNS has radial oscillation. As a result, a fraction of the shocked material obtains sufficient energy and is ejected from the system. Recent general relativistic simulations show that this component can dominate over the tidal component in the case of a nearly equal mass binary (e.g., @hotokezaka2013PRDa [@bauswein2013ApJa]). The shocked component is ejected even in the direction of the rotation axis of the binary. The average electron fraction of the shocked components is relatively large compared with that of the tidal ejecta [@wanajo2014ApJ]. It may be as large as $Y_{e}\sim 0.2$ – $0.4$ and it will result in a different nucleosynthesis signature.
We take the velocity distribution of the dynamical ejecta from the result of a numerical relativity simulation of [@hotokezaka2013PRDa] for a $1.4$–$1.4M_{\odot}$ ns$^2$ merger for the case of APR4 equation of state. The energy distribution of this model can be approximately described as $E(\geq \beta)\propto \beta^{-0.5}$ with a cut off at $\beta \simeq 0.4$ and an average velocity is $\beta\simeq 0.2$, where $E(\geq \beta)$ is the kinetic energy with a velocity larger than $\beta$. Note that it is not clear whether the cut off at $\beta \simeq 0.4$ is physical or that it arises just because it is difficult to resolve such a small amount of fast material in the numerical simulations. For our fiducial model, we use a total kinetic energy of $5 \times 10^{50}~{\rm erg}$.
[*The relativistic shock-breakout component.*]{} When the shock breaks out from the neutron star surface to the ISM, it is accelerated and a fraction of the shocked component can have a relativistic velocity with $\beta \Gamma \gtrsim 1$. @kyutoku2014MNRAS showed analytically that the kinetic energy of the relativistic ejecta can be $\sim 10^{47}$ – $10^{49.5}~{\rm erg}$. More recently, @metzger2015MNRAS found that there is a mildly relativistic component with $\beta\gtrsim 0.8$ in a merger simulation of [@bauswein2013ApJa]. This fast component is likely resulted from the acceleration of a shock emerging from the neutron star surface. They found that the mass and kinetic energy of the fast component with $\beta \gtrsim 0.8$ are $\sim 10^{-5}M_{\odot}$ and $\sim 5\times 10^{48}$erg, respectively. Because of this large velocity, the radio signature of this component would be different from the slower material. We denote this component as a “shock-breakout material” and we consider it separately from the sub-relativistic dynamical ejecta.
Here, we assume that the kinetic energy distribution of this component is a simple power-law, $E(\geq \beta \Gamma)=10^{48.5}(\beta \Gamma)^{-\alpha}~{\rm erg}$ as the fiducial model. The value of $\alpha$ varies from $1.1$ for $\beta \Gamma \gg 1$ to $5.2$ for $\beta \Gamma \ll 1$ [@kyutoku2014MNRAS; @tan2001ApJ]. We set $\alpha$ to be $3$, which is valid around $\beta \Gamma\sim 1$ and we take into account only the fast component with $\beta \Gamma\geq 1$.
The ultra-relativistic jet
--------------------------
If ns$^2$ mergers are progenitors of short GRBs, they involve relativistic jets. Assuming the kinetic energy of the jet as the gamma-ray energy of the prompt emission, the energy of the jet can be estimated from the observed GRBs. The minimal and maximal values of the observed isotropic-equivalent gamma-ray energy for non-Collapsar short GRBs are $2\times 10^{49}$ erg and $4\times 10^{52}$ erg, respectively [@wanderman2014]. Taking into account the average value of the measured jet-half opening angles of $\theta_{j} \approx 10^{\circ}$ [@fong2014ApJ], the kinetic energy of a relativistic jet is in the range of $10^{47}$ – $3\times 10^{50}~{\rm erg}$. The luminosity function is rather steep and there are more weak GRBs than strong ones. Hence we consider here a fiducial GRB jet with a kinetic energy of $10^{48}~{\rm erg}$ and a jet-half opening angle of $10^{\circ}$.
The wind from the merger remnant
--------------------------------
Some of the debris of the neutron stars form an accretion disk that surrounds the central remnant. The mass of this accretion disk is estimated to be in the range of $10^{-3} \lesssim M_{\rm disk} \lesssim 0.3~M_{\odot}$ (see e.g., @shibata2006PRD [@rezzolla2010CQG; @hotokezaka2013PRDc]). This accretion disk produces an outflow driven by viscous and neutrino heating. The properties of this outflow depend on the central object as follows.
[*A black hole with an accretion disk.*]{} The wind from an accretion disk surrounding a black hole has been explored, in the context of mergers, by [@fernandez2013MNRAS; @just2014; @fernandez2015MNRAS]. The disk is extremely dense and the accretion rate is huge. Initially it is opaque even for neutrinos. After $\sim 0.1$ – $1$ s from the onset of the merger, the density and temperature of the accretion disk decrease and neutrino-cooling becomes inefficient. As a result, a fraction of the material is ejected isotropically due to the viscous heating in the accretion disk. The amount of ejected material is about $5$ – $20\%$ of the initial disk mass depending on the $\alpha$-viscosity parameter and on the spin parameter of the black hole. Increasing these parameters, the fraction of the ejected mass to the initial disk mass increases. The average velocity of the ejecta is $0.03 \lesssim \beta \lesssim 0.05$ and the expected kinetic energy of the outflow is in the range of $10^{47} \lesssim E \lesssim 10^{50}$ erg. The average electron fraction is $Y_{e}\sim 0.2$ – $0.3$.
[*A neutron star with an accretion disk.*]{} The wind from a neutron star with an accretion disk can be divided into three parts; a neutrino-driven wind from the remnant neutron star itself, a neutrino driven wind from the accretion disk, and a viscous driven wind from the accretion disk [@dessart2009ApJ; @metzger2014MNRAS; @perego2014MNRAS]. The neutrino-driven wind expands into relatively high latitudes and it has a larger velocity $\beta \lesssim 0.1$ and a higher electron fraction $Y_{e}\sim 0.4$ than those of the viscous-driven wind. The amount of material ejected by the neutrino-driven wind depends on the lifetime of the central neutron star. [@perego2014MNRAS] showed that the ejected mass is more than $3\times 10^{-3}M_{\odot}$ with a velocity $\beta\sim 0.06$ – $0.9$ at 100 ms after the merger in the case of an initial disk mass of $0.17M_{\odot}$. [@metzger2014MNRAS] also showed that the amount of the ejected mass and average velocity are $M_{\rm ej}\sim 10^{-3}M_{\odot}$ and $\beta \sim 0.05$ at $100$ ms with an initial disk mass of $0.03M_{\odot}$. When a MNS does not collapse into a black hole, about $20\%$ of the initial disk mass may be ejected as neutrino driven or viscous driven winds [@metzger2014MNRAS].
In the following, we take the kinetic energy of $10^{50}$ erg with a single velocity $0.07c$ for the wind from the merger remnant as the fiducial model. Note that these values correspond to the most optimistic case. As shown later even in this case, the expected radio signals are very weak.
The cocoon
----------
The interaction of a GRB jet with the pre-ejected material such as the dynamical ejecta or the wind along the rotation axis would produce a hot cocoon surrounding the jet [@nagakura2014ApJ; @murguia2014ApJ; @rezzolla2014]. After the jet emerges from the expanding ejecta, the cocoon will break out from the surface of the ejecta and will expand nearly spherically. Assuming that the material inside the cone of a jet-half opening angle $\theta_{j}$ is shocked by the jet and forms a cocoon and the deposited energy into the cocoon is $E_{c}$, the Lorentz factor of the cocoon can be estimated as $$\begin{aligned}
\Gamma \approx 1+0.05\left(\frac{M_{\rm ej}(\theta_{j})}{10^{-5}M_{\odot}}\right)^{-1}
\left(\frac{E_{c}}{10^{48}{\rm~erg}}\right),\label{Lc}\end{aligned}$$ where $M_{\rm ej}(\theta_{j})$ is the ejecta mass within $\theta_{j}$. As the jet crossing time is comparable with the duration of a short GRB and the jet energy deposited in the cocoon will be comparable to the jet energy, we expect that the cocoon energy will be similar to the GRB jet energy. For our fiducial value, we take a kinetic energy of $10^{48}$ erg with a single velocity of $0.3c$ and the cocoon is sub-relativistic. Note that it becomes relativistic at energies higher than $10^{49}$ erg.
The radio signature {#sec:radio}
===================
The various components of ejecta interact first with each other and ultimately with the ISM. This last interaction produces a long-lived blast wave. This shock that propagates into the ISM will enhance magnetic fields and accelerate electrons that will emit synchrotron radiation. The process is similar to GRB afterglows and to radio emission from some early supernova remnants. In this section, we explore the synchrotron radiation from a merger taking into account the various components of ejecta. Except for the relativistic jet we consider all components of the ejecta as spherically symmetric. We discuss the implications of this approximation in Sec. \[sec:results\].
We assume that the ISM is homogeneous and characterized by an external density $n$. The ejecta slows down with the deceleration timescale given by $$\begin{aligned}
t_{\rm dec} = \left(\frac{3E}{4\pi m_{p}c^{5}n \Gamma_{0} (\Gamma_{0}-1)\beta_{0}^{3}}\right)^{1/3},\label{tdec1}\end{aligned}$$ where $\Gamma_{0}$ and $\beta_{0}$ are the initial Lorentz factor and the corresponding initial velocity of the ejecta, $m_{p}$ is the proton mass. The values of $t_{\rm dec}$ for the different components of ejecta are shown in the left panel of Fig. \[fig1\]. For a mildly or sub-relativistic outflow the deceleration timescale characterizes the observed peak time. For an ultra-relativistic beamed jet with a viewing angle $\theta_{\rm obs}>\theta_{j}$, we have an orphan afterglow. Namely, we do not see the highly beamed burst and early afterglow. But we see the late afterglow when it slows down and its less beamed emission includes our line of sight. As this happens when $\Gamma \sim \theta_{\rm obs}^{-1}$, the peak time in the source frame is around $t_{\rm dec}$ given by Eq. (\[tdec1\]) with $\Gamma \sim \theta_{\rm obs}^{-1}$ instead of the initial Lorentz factor. Note that for a relativistic outflow the observer time is different from the time in source frame and it is smaller by a factor of $\Gamma^{-2}$. Note that for an observed GRB the peak in the radio arises when the observed frequency equals the typical synchrotron frequency.
The deceleration timescale (see the left panel of Fig. \[fig1\]) suggests three types of the radio flares. First, the ultra-relativistic jet produces at early times the radio afterglow, that can be seen only by observers along the jet axis or close to it. Second, the mildly relativistic components, including the cocoon, the shock-breakout material, and the jet for an observer away from its axis produce radio flares with a timescale of a few dozen days. Finally, the sub-relativistic dynamical ejecta produces a late-time radio flare with a timescale of a few years.
Ultra-relativistic beamed jet {#sec:jetafterglow}
-----------------------------
The Blandford-Mckee self-similar solution describes the jet dynamics, in the relativistic regime after the energy of the ISM swept up by the jet becomes comparable to the energy of the jet itself. Once the Lorentz factor of the jet decreases to $\Gamma \sim \theta_{j}^{-1}$, the jet expands laterally and approaches a quasi- spherical shape. To describe the evolution during this sideway expansion phase we adopt a semi-analytic formula for a homogeneous jet given by [@granot2012MNRAS][^4], which shows a good agreement with the results of a numerical simulation by [@decolle2012ApJ]. The observed signal depends strongly on the viewing angle and we consider five different viewing angles $\theta_{\rm obs} = (0^{\circ},~30^{\circ},~45^{\circ},~60^{\circ},~90^{\circ})$.
To calculate the synchrotron radio emission we assume, as common in GRB afterglows and in radio supernovae modeling (see e.g., @sari1998ApJ), that the shock generates magnetic fields and accelerates electrons to a power law distribution $N(\gamma)\propto \gamma^{-p}$, where $\gamma$ is the Lorentz factor of an accelerated electron. The value of $p$ is estimated as $p \approx 2.1$ – $2.5$ in late GRB afterglows and afterglows of low luminosity GRBs and as $p\approx 2.5$ – $3$ in typical radio supernovae [@chevalier1998ApJ]. We assume $p=2.5$. The total energy of the electrons and the magnetic field intensity are characterized by equipartition parameters: $\epsilon_{e}$ and $\epsilon_{B}$ that are the conversion efficiency from the internal energy of the shock into the energy of the accelerated electrons and magnetic fields, respectively. We set these parameters as $\epsilon_{e}=\epsilon_{B}=0.1$. These values are consistent with those evaluated from late radio afterglows in long GRBs [@frail2000ApJ; @frail2005ApJ]. For our purposes the radio emission is always below the cooling frequency hence the system has only two characteristic frequencies, the synchrotron frequency of the “typical" electron and the self absorption frequency. We implement the effect of the synchrotron-self absorption following [@granot1999ApJb; @rybicki1979]. Once we determine the local emissivity we integrate over the intensity of each line of sight with an equal arrival time (see e.g., @sari1998ApJb [@granot1999ApJ])[^5].
Mildly and sub-relativistic isotropic components. {#sec:mild}
-------------------------------------------------
We briefly discuss the simple analytic estimates of the radio signals for a mildly and sub-relativistic ejecta (see @piran2013MNRAS for details). The hydrodynamics of a mildly and sub-relativistic blast wave with a kinetic energy $E$ and an initial velocity $\beta_{0}$ expanding into a homogeneous ISM with an external density $n$ can be approximately described by $\beta=\beta_{0}$ until the deceleration time $t_{\rm dec}$. The dynamics approaches to the Sedov-Taylor self-similar solution after $t_{\rm dec}$.
The synchrotron emission is slow cooling and it is strongly suppressed by the self absorption below the self absorption frequency: $$\begin{aligned}
\nu_{a}(t) =
\left\{
\begin{array}{ll}
\nu_{a{\rm ,dec}} \left( \frac{t}{t_{\rm dec}}\right)^{\frac{2}{p+4}}
&~~(t\leq t_{\rm dec}),\\
\nu_{a{\rm ,dec}}\left( \frac{t}{t_{\rm dec}}\right)^{-\frac{3p-2}{p+4}}
&~~(t > t_{\rm dec}),\\
\end{array}
\right.\end{aligned}$$ where $$\begin{aligned}
\nu_{a{\rm ,dec}} = 1~{\rm GHz}~E_{49}^{\frac{2}{3(p+4)}}
n^{\frac{3p+14}{6(p+4)}}
\epsilon_{B,-1}^{\frac{2+p}{2(p+4)}}
\epsilon_{e,-1}^{\frac{2(p-1)}{p+4}}
\beta_{0}^{\frac{15p-10}{3(p+4)}}.\end{aligned}$$ These expressions are valid for $\nu_{a}>\nu_{m}$, where $\nu_{m}$ is the synchrotron frequency of electrons with the minimum Lorentz factor. Here and elsewhere, $Q_{x}$ denotes the value of $Q/10^{x}$ in cgs units.
For $\nu>\nu_{a}$, the peak flux and the peak time can be estimated as $$\begin{aligned}
F_{{\rm peak},\nu>\nu_{a}(t_{\rm dec})}\approx 0.8~{\rm mJy}~E_{49}n^{\frac{p+1}{4}}\epsilon_{B,-1}^{\frac{p+1}{4}}
\epsilon_{e,-1}^{p-1}\beta_{0}^{\frac{5p-7}{2}}\label{f1} \\ \nonumber
\times \left(\frac{D_{L}}{200~{\rm Mpc}}\right)^{-2}
\left(\frac{\nu}{1.4~{\rm GHz}}\right)^{-\frac{p-1}{2}},\end{aligned}$$ and $$\begin{aligned}
t_{\nu > \nu_{a}(t_{\rm dec})}=t_{\rm dec} \approx 40~{\rm day}~E_{49}^{\frac{1}{3}}n^{-\frac{1}{3}}\beta_{0}^{-\frac{5}{3}},\label{t1}\end{aligned}$$ where $D_{L}$ is the luminosity distance to the source and we approximate the Lorentz factor as $\Gamma_{0}-1\approx \beta_{0}^{2}$ in Eq. (\[t1\]). The peak flux and its time depend sensitively on the external density, the kinetic energy, and the initial velocity of the ejecta in the optically thin regime.
For $\nu<\nu_{a}$ at $t_{\rm dec}$, the peak flux and peak timescale are $$\begin{aligned}
F_{{\rm peak},\nu<\nu_{a}}\approx 0.1~{\rm mJy}~E_{49}^{\frac{4}{5}}n^{\frac{1}{5}}\epsilon_{B,-1}^{\frac{1}{5}}
\epsilon_{e,-1}^{\frac{3}{5}}\label{f2} \\
\times \left(\frac{D_{L}}{200~{\rm Mpc}}\right)^{-2}
\left(\frac{\nu}{150~{\rm MHz}}\right)^{\frac{6}{5}},\nonumber\end{aligned}$$ and $$\begin{aligned}
t_{\nu < \nu_{a}(t_{\rm dec})}\approx 200~{\rm day}~E_{49}^{\frac{5}{11}}n^{\frac{7}{22}}\epsilon_{B,-1}^{\frac{9}{22}}\label{t2}
\epsilon_{e,-1}^{\frac{6}{11}}\left(\frac{\nu}{150~{\rm MHz}}\right)^{\frac{13}{11}}.\\ \nonumber\end{aligned}$$ In the optically thick regime, the peak flux and its timescale depend weakly on the external density and they are independent of the initial velocity of the ejecta. The dependence on the energy is also weaker than in the optically thin case.
As the velocity distribution is not uniform, we estimate the emission from each shell of matter and combined the results. For a given distribution of energies as a function of velocity, we divide the outflow into shells. An external ISM mass, $M(R)$, swept up at a radius $R$ can be associated with each shell such that this mass slows down the shells: $$\begin{aligned}
M(R)(c\beta \Gamma)^{2} = E(\geq \beta \Gamma).
\label{MR}\end{aligned}$$ Once we solve the implicit Eq. (\[MR\]), we determine the observed light curves for each shell. We then combine the contributions of the different shells to obtain the total light curve. In the non-relativistic limit, the ejecta dynamics described by Eq. (\[MR\]) is consistent with the self-similar solution derived by [@chevalier1982ApJ] up to a factor of order unity. In the relativistic limit and the case of $E(\geq \beta \Gamma)={\rm const}$, it agrees with the Blandford-Mackee self-similar solution again up to a factor of order unity.
Numerical result {#sec:results}
----------------
{width="85mm"} {width="85mm"}\
{width="85mm"} {width="85mm"}\
{width="85mm"} {width="85mm"}
{width="85mm"} {width="85mm"}\
{width="85mm"} {width="85mm"}\
{width="85mm"} {width="85mm"}
Figure \[fig2\] depicts the resulting radio light curves of the different components for our fiducial model (see Table \[tab1\] for the fiducial parameters). We examine three different values of the external density $n=0.01$ – $1~{\rm cm^{-3}}$ and we present the light curves for two observed frequencies $150$ MHz (left panels) and $1.4$ GHz (right panels) corresponding to the LOFAR and the EVLA of radio telescopes. We set the luminosity distance of the source to be $200$ Mpc, which is roughly the sky averaged horizon distance of the advanced GW detectors for ns$^2$ mergers.
{width="85mm"} {width="85mm"}\
{width="85mm"} {width="85mm"}\
{width="85mm"} {width="85mm"}
The ultra-relativistic jet always arrives first. This on-axis emission of the jet, the GRB radio afterglow, is the strongest at 1.4 GHz for low external densities $n\lesssim 0.1~{\rm cm^{-3}}$. At 150 MHz this GRB afterglow as well as the other relativistic component, the shock-breakout material, is strongly suppressed by self-absorption and it is much weaker. For a generic observer, the off-axis orphan afterglow at viewing angles of 60$^\circ$ and even at 45$^\circ$ is always subdominant compared to the shock-breakout material and the dynamical ejecta. The mildly relativistic component, that arises from the shock-breakout material, peaks later at around 20–100 days depending on the observed frequency and external density. Finally the sub-relativistic dynamical ejecta arises at late times (typically 1000 days). It is always the brightest at 150 MHz and it is also brightest at 1.4GHz for higher external densities. For our fiducial parameters, that are based on a weak GRB, the radio emission from the sub-relativistic cocoon is always negligible.
As mentioned earlier, at early times, synchrotron self-absorption strongly suppresses the radio flux at $150$ MHz. As a result, the peak flux is only $F_{\nu }\sim 0.01~{\rm mJy}$ for the relativistic components such as the shock-breakout material and the off-axis GRB jet for all the cases. As expected from Eq. (\[f2\]), in this case the peak flux depends only on the kinetic energy among the parameters of ejecta. Indeed, the dynamical ejecta is the brightest as $F_{\nu}\sim 1~{\rm mJy}$ for $n\gtrsim 0.1~{\rm cm^{-3}}$ and its peak time is $\sim 1000$ days. For very low densities $n \lesssim~0.01 {\rm cm^{-3}}$, the on-axis GRB afterglow is comparable to the dynamical ejecta flare, peaking at about 20 days with $F_{\nu}\sim 0.1 ~{\rm mJy}$.
At $1.4$ GHz, there are the early and late-time radio flares. The relativistic components such as the GRB afterglows and the shock-breakout material contribute the flare at early times as expected from Eqs. (\[f1\]) and (\[t1\]). For low densities $n\lesssim 0.1~{\rm cm^{-3}}$, the GRB afterglows within a viewing angle $\theta_{\rm obs}\sim 30^{\circ}$ is the brightest at 1.4 GHz as $F_{\nu}\sim 0.1$ – $0.5$ mJy. Note that the off-axis GRB afterglows are very faint for large viewing angle $\theta_{\rm obs}\gtrsim 60^{\circ}$ compared with the shock-breakout material and the dynamical ejecta. At this stage the originally beamed jet has already slowed down and its emission is already quasi spherical because of its low Lorentz factor (this is independent of the question how much did the jet physically expand sideways). As the jet energy is smaller than those of the other components, its radiation is weaker. The dynamical component dominates at late times $t\gtrsim 100$ days and has a relatively flat light curve.
As mentioned earlier, our fiducial GRB was a typical, low luminosity one. Figure \[fig3\] shows the radio light curves for the case of a strong GRB with a jet energy of $10^{49}$ erg (corresponding to an isotropic equivalent energy of $\sim 10^{51}$ erg). The energy of the cocoon is also larger as this should be comparable to the jet energy. We use $E_{c}=10^{49}$ erg and a corresponding Lorentz factor of $\Gamma = 1.5$ is obtained from Eq. (\[Lc\]). Now, for this GRB, the on-axis GRB afterglow is the brightest at all densities at 1.4 GHz and for very low densities at 150 MHz. The cocoon is much brighter than that of the weak GRB and its peak flux at 1.4 GHz is comparable to that of the dynamical ejecta and to the off-axis orphan afterglow for $\theta_{\rm obs}=45^{\circ}$.
Unfortunately, there are numerous uncertainties in our estimated light curves. One of the strongest sources of the uncertainties arises from lack of precise estimates of the mass and energy of the different components. To demonstrate the possible variability of the light curves with these unknown parameters, we present in Fig. \[fig4\] the dependence of the radio flares on the kinetic energy of the different components. Here we show the light curves of the fiducial model (thick curves) and those with kinetic energies larger and smaller by a factor of 3 (thin curves) than those of the fiducial model. Above (below) the self-absorption frequency, the amplitude of the light curves scales as $\propto E$ ($E^{4/5}$) and the timescale of the light curves behaves as $\propto E^{1/3}$ ($E^{5/11}$) as expected (see Sec. \[sec:mild\]).
While the overall peak luminosity and peak flux depend mostly on the global properties of the outflow component, the details depend also on the spatial distribution and on the velocity distribution. For instance, the light curve of an off-axis jet rises steeply because of the collimation of the jet and relativistic beaming effect. The detailed shapes of this rise will depend on the angular structure of the jet. The light curves of the spherical components rise slowly compared to that of off-axis afterglows. The slope of the light curve depends on the velocity distribution $E(\geq \Gamma \beta)\propto (\Gamma \beta)^{-\alpha}$. The rise of light curves will be shallower for lower values of $\alpha$.
Examination of the numerical simulations reveals that the mildly and sub-relativistic components that we have examined do not satisfy indeed the spherical symmetry assumption that we have made here. [@margalit2015] have estimated the effects of a-sphericity on the emission, focusing on the dynamical ejecta. They found that, for a given total mass, energy, and external density, a-sphericity typically delays the peak emission and reduces the peak flux. This can be understood intuitively as follows. If more mass and energy are concentrated in one direction the matter propagating in that direction will slow down later. This longer deceleration time results in a longer and weaker radio flare compared with the isotropic one. Note that, however, as the outflow is only mildly relativistic, even from a highly a-spherical ejecta, the emission will be roughly isotropic and viewing angle effects will be small. It is worth noting that the effect of a-sphericity is more relevant for black hole neutron star mergers, which can result in highly a-spherical mass ejection (see e.g., @kyutoku2013PRD [@foucart2013PRD]).
The radio signature of the short GRB 130603B {#sec:grb}
============================================
![ Radio signatures of the short GRB 130603B and light curves at 6.7 GHz (red curves), 1.4 GHz (green curves), and 150 MHz (blue curves). The red dotted curve denotes the GRB radio afterglow from a jet with $E=8\times 10^{48}$ erg, $\theta_{j}=4^{\circ}$, $p=2.3$, $\epsilon_{e}=0.2$, $\epsilon_{B}=8\times 10^{-3}$, and $n=1.0~{\rm cm^{-3}}$. The solid and dashed curves denote the expected radio light curves from a dynamical ejecta with an external density $n=1.0~{\rm cm^{-3}}$ and $0.5~{\rm cm^{-3}}$, respectively. For the dynamical ejecta, the kinetic energy is assumed to be $8\times 10^{50}$ erg and the other microphysics parameters are the same as those in Sec. \[sec:radio\]. The filled squares and the open triangles show the observed data points and the upper limits at 6.7 GHz obtained with the VLA [@fong2014ApJ]. The blue shaded region shows the expected sensitivity of the EVLA at 1.4 GHz.[]{data-label="fig5"}](130603B_v5.eps){width="85mm"}
The short GRB 130603B had an associated macronova candidate [@berger2013ApJ; @tanvir2013Nature]. While the macronova identification is based only on one observed data point in the $H$-band at about $7$ days (in source frame) after the burst, this is the first, even though rather weak, evidence of the significant mass ejection from a ns$^{2}$. Using this data point, one can estimate, from the observed luminosity of the macronova, the minimal ejecta mass as $M_{\rm ej}\approx 0.02(\epsilon_{\rm th}/0.5)~M_{\odot}$ [@hotokezaka2013ApJ; @piran2014], where $\epsilon_{\rm th}$ is the conversion efficiency from the total energy generated by radioactive decay into the thermal energy of the ejecta. The velocity can be also estimated as $v\gtrsim 0.1c$ from the condition that radiation can diffuse out from the ejecta with a mass $M_{\rm ej}\gtrsim 0.02M_{\sun}$ and an opacity of $10~{\rm cm^{2}/g}$ [@kasen2013ApJ; @tanaka2013ApJ] at about $7$ days after the burst. Assuming that the ejecta mass and average velocity is $M_{\rm ej}\sim 0.02M_{\odot}$ and $v\sim 0.2c$, the estimated kinetic energy is about $10^{51}$ erg.
Figure \[fig5\] shows the expected radio flares from the dynamical ejecta at 150MHz, 1.4 GHz, and 6.7 GHz as well as the observed early radio afterglow of GRB 130603B at 6.7 GHz by the VLA [@fong2014ApJ]. Here we also show a GRB afterglow light curve which is obtained with parameters $E=8\times 10^{48}$ erg, $\theta_{j}=4^{\circ}$, $p=2.3$, $\epsilon_{e}=0.2$, $\epsilon_{B}=8\times 10^{-3}$, and $n=1.0~{\rm cm^{-3}}$. The light curve is consistent with the observed data points and upper limits[^6] within a factor of 2. It is worth emphasizing that the parameters can change by orders of magnitude and still fit the data. For instance, the external density lies in the range $n\approx 5\times 10^{-3}$ – $30~{\rm cm^{-3}}$ [@fong2014ApJ]. For modeling the radio flare from the dynamical ejecta, we assume an external density to be $1.0$ and $0.5~{\rm cm^{-3}}$ and we use an ejecta mass $M=0.02M_{\odot}$ and a kinetic energy $E=8\times 10^{50}$ erg. The predicted dynamical ejecta light curves at 6.7 GHz are well below the upper limits $F_{\nu}\lesssim 30~\mu{\rm Jy}$ at $\sim 80$ days. However, later observations may detect a signal. Specifically the peak flux at 1.4 GHz can be as high as $F_{\nu}\approx 20~\mu{\rm Jy}$ (depending on the external density). The expected sensitivity of the EVLA at 1.4 GHz is also shown in the figure. For the higher range of external densities $n\gtrsim 0.5~{\rm cm^{-3}}$, the radio flare might be detectable with the EVLA. The signal at 150 MHz can be $F_{\nu}\approx 30~\mu{\rm Jy}$, which might be also detectable with the LOFAR. A positive detection of a varying radio signal will confirm the identification of this event as a ns$^2$ merger and will establish the observed infrared bump as a macronova.
Conclusion and discussion {#sec:conc}
=========================
A ns$^2$ merger ejects a significant amount of mass in several different components: a dynamical ejecta, a shock-breakout material, a wind from a black hole/neutron star surrounded by an accretion disk, and a relativistic jet, producing a GRB. As a result of the interaction of the relativistic jet with the earlier and slower ejecta along the rotational axis, a cocoon is expected to be formed and expands nearly spherically. Among the different components of the ejecta, the dynamical ejecta, which is also the most robustly found in numerical simulations, has the largest amount of kinetic energy up to $E\sim 10^{51}$ erg. This is comparable to the “isotropic equivalent” energy of the highly beamed GRB jet. Somewhat surprisingly the beamed GRB jet is among the least energetic. Overall we have three important distinct components, the dynamical ejecta, which is mostly sub-relativistic, the mildly relativistic shock-breakout material and cocoon [^7], and the ultra-relativistic beamed jet.
We have calculated the expected radio signals produced via synchrotron emission from electrons accelerated in the shocks formed between the different components of the ejecta and the ISM. This would be a low frequency (as opposed to the X-ray or optical GRB afterglow or the optical–infrared macronova) electromagnetic counterpart of the GW event. This process is similar to GRB afterglows and radio emission of some early supernova remnants. In contrast with the high frequency counterparts, this emission lasts much longer and may even peak a few years after the merger. We focused on the expected radio flux at two frequencies of 150 MHz and 1.4 GHz. We found that there are three types of the radio flares: (i) The ultra-relativistic jet produces the earliest bright radio flare with a timescale of $\sim 10$ days for an observer along the jet axis or close to it. (ii) A radio flare with a timescale of a few dozen days is produced by the mildly relativistic components such as the shock-breakout material, the off-axis jet, and the cocoon. The latter flare from the cocoon is significant only when the cocoon is energetic enough so that it has a relativistic velocity. (iii) Finally the sub-relativistic dynamical ejecta produces a radio flare with a timescale of a few years. At 150 MHz, however, the radio flare is strongly suppressed by synchrotron self absorption for $n\gtrsim 0.01~{\rm cm^{-3}}$ until a few months. Hence the earlier signals are much weaker at low frequencies. Depending on the external density and the distance to the source, these radio flares could be detectable as GW counterparts. Such a detection can reveal the nature of ns$^{2}$ mergers.
Although the modeling of radio emission from mergers contains uncertain parameters such as the kinetic energy of the ejecta and the external density, it is worth to estimate these values from the nature of the short GRB 130603B. The detection of a macronova candidate associated with this event allows us to estimate the ejecta mass $M_{\rm ej}\approx 0.02(\epsilon_{\rm th}/0.5)~M_{\odot}$. Assuming the velocity of the ejecta is $0.2c$, the estimated kinetic energy is about $10^{51}$ erg. The detection of the afterglows implies that the external density is in the range of $n\approx 5\times 10^{-3}$ – $30~{\rm cm^{-3}}$ [@fong2014ApJ]. The radio afterglow of GRB 130603B, that arose from the relativistic jet, decayed quickly and it was below $30~\mu {\rm Jy}$ at $\sim 4$ days with a similar upper limit at 80 days. However it is still possible to observe the late-long lasting radio signal arising from the dynamical ejecta. This signal could be as high as $20~\mu{\rm Jy}$ at 1.4 GHz depending mostly on the external density. For the higher range of external densities $n\gtrsim 0.5~{\rm cm^{-3}}$, this would be detectable at 1.4 GHz with the EVLA. At 150 MHz, the expected flux is about $30~\mu{\rm Jy}$, which depends weakly on the external density and peaks at late times. This flux might be detectable with the LOFAR. While a detection is uncertain, a positive radio signal will confirm the identification of this event as a ns$^2$ merger and will establish the observed infrared bump as a macronova.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank J. Granot, E. Nakar, L. Nava, S. Nissanke, R. Sari, and R. Shen for fruitful discussions. This research was supported by an ERC advanced grant (GRBs) and by the I-CORE Program of the Planning and Budgeting Committee and The Israel Science Foundation (grant No 1829/12).
J., [Abbott]{} B. P., [Abbott]{} R., [Abernathy]{} M., [Accadia]{} T., [Acernese]{} F., [Adams]{} C., [Adhikari]{} R., [Ajith]{} P., [Allen]{} B., et al. 2010, Classical and Quantum Gravity, 27, 173001
T., et al. 2011, Classical and Quantum Gravity, 28, 114002
Y., [Michimura]{} Y., [Somiya]{} K., [Ando]{} M., [Miyakawa]{} O., [Sekiguchi]{} T., [Tatsumi]{} D., [Yamamoto]{} H., 2013, [Phys. Rev. D. ]{}, 88, 043007
J., [Kasen]{} D., 2013, [ApJ]{}, 775, 18
I., [Veres]{} P., [Nieto]{} D., [Connaughton]{} V., [Humensky]{} B., [Hurley]{} K., [M[á]{}rka]{} S., [M[é]{}sz[á]{}ros]{} P., [Mukherjee]{} R., [O’Brien]{} P., [Osborne]{} J. P., 2014, [MNRAS]{}, 443, 738
A., [Goriely]{} S., [Janka]{} H.-T., 2013, [ApJ]{}, 773, 78
E., [Fong]{} W., [Chornock]{} R., 2013, [ApJL]{}, 774, L23
N., [Metzger]{} B. D., [Thompson]{} T. A., [Quataert]{} E., 2012, [MNRAS]{}, 419, 1537
R. A., 1982, [ApJ]{}, 258, 790
R. A., 1998, [ApJ]{}, 499, 810
R., [Siegel]{} D. M., 2014, ArXiv e-prints
M. B., [Benz]{} W., [Piran]{} T., [Thielemann]{} F. K., 1994, [ApJ]{}, 431, 742
F., [Ramirez-Ruiz]{} E., [Granot]{} J., [Lopez-Camara]{} D., 2012, [ApJ]{}, 751, 57
L., [Ott]{} C. D., [Burrows]{} A., [Rosswog]{} S., [Livne]{} E., 2009, [ApJ]{}, 690, 1681
D., [Livio]{} M., [Piran]{} T., [Schramm]{} D. N., 1989, [Nature]{}, 340, 126
S., 2011, Classical and Quantum Gravity, 28, 105021
R., [Kasen]{} D., [Metzger]{} B. D., [Quataert]{} E., 2015, [MNRAS]{}, 446, 750
R., [Metzger]{} B. D., 2013, [MNRAS]{}
W., [Berger]{} E., [Metzger]{} B. D., [Margutti]{} R., [Chornock]{} R., [Migliori]{} G., [Foley]{} R. J., [Zauderer]{} B. A., [Lunnan]{} R., [Laskar]{} T., [Desch]{} S. J., [Meech]{} K. J., [Sonnett]{} S., [Dickey]{} C., [Hedlund]{} A., [Harding]{} P., 2014, [ApJ]{}, 780, 118
F., [Deaton]{} M. B., [Duez]{} M. D., [Kidder]{} L. E., [MacDonald]{} I., [Ott]{} C. D., [Pfeiffer]{} H. P., [Scheel]{} M. A., [Szilagyi]{} B., [Teukolsky]{} S. A., 2013, [Phys. Rev. D. ]{}, 87, 084006
D. A., [Soderberg]{} A. M., [Kulkarni]{} S. R., [Berger]{} E., [Yost]{} S., [Fox]{} D. W., [Harrison]{} F. A., 2005, [ApJ]{}, 619, 994
D. A., [Waxman]{} E., [Kulkarni]{} S. R., 2000, [ApJ]{}, 537, 191
S., [Bauswein]{} A., [Janka]{} H.-T., 2011, [ApJL]{}, 738, L32
J., [Piran]{} T., 2012, [MNRAS]{}, 421, 570
J., [Piran]{} T., [Sari]{} R., 1999a, [ApJ]{}, 513, 679
J., [Piran]{} T., [Sari]{} R., 1999b, [ApJ]{}, 527, 236
D., [Korobkin]{} O., [Rosswog]{} S., [Piran]{} T., 2014, [MNRAS]{}, 439, 757
K., [Kiuchi]{} K., [Kyutoku]{} K., [Muranushi]{} T., [Sekiguchi]{} Y.-i., [Shibata]{} M., [Taniguchi]{} K., 2013, [Phys. Rev. D. ]{}, 88, 044026
K., [Kiuchi]{} K., [Kyutoku]{} K., [Okawa]{} H., [Sekiguchi]{} Y.-i., [Shibata]{} M., [Taniguchi]{} K., 2013, [Phys. Rev. D. ]{}, 87, 024001
K., [Kyutoku]{} K., [Tanaka]{} M., [Kiuchi]{} K., [Sekiguchi]{} Y., [Shibata]{} M., [Wanajo]{} S., 2013, [ApJL]{}, 778, L16
O., [Bauswein]{} A., [Ardevol Pulpillo]{} R., [Goriely]{} S., [Janka]{} H.-T., 2014, ArXiv e-prints
J., [Baker]{} J., [Blackburn]{} L., [Camp]{} J., [Mooley]{} K., [Mushotzky]{} R., [Ptak]{} A., 2013, [ApJ]{}, 774, 63
D., [Badnell]{} N. R., [Barnes]{} J., 2013, [ApJ]{}, 774, 25
M. M., [Nissanke]{} S., 2014, [ApJL]{}, 789, L5
S., [Ioka]{} K., [Takami]{} H., 2014, ArXiv e-prints
C. S., [Piran]{} T., 1993, [ApJL]{}, 417, L17
O., [Rosswog]{} S., [Arcones]{} A., [Winteler]{} C., 2012, [MNRAS]{}, 426, 1940
Kyutoku K., Ioka K., Shibata M., 2013, Phys. Rev. D, 88, 041503
K., [Ioka]{} K., [Shibata]{} M., 2014, [MNRAS]{}, 437, L6
L.-X., [Paczy[ń]{}ski]{} B., 1998, [ApJL]{}, 507, L59
B., [Piran]{} T., 2015, in preparation
B. D., [Bauswein]{} A., [Goriely]{} S., [Kasen]{} D., 2015, [MNRAS]{}, 446, 1115
B. D., [Berger]{} E., 2012, [ApJ]{}, 746, 48
B. D., [Fern[á]{}ndez]{} R., 2014, [MNRAS]{}, 441, 3444
B. D., [Mart[í]{}nez-Pinedo]{} G., [Darbha]{} S., [Quataert]{} E., [Arcones]{} A., [Kasen]{} D., [Thomas]{} R., [Nugent]{} P., [Panov]{} I. V., [Zinner]{} N. T., 2010, [MNRAS]{}, 406, 2650
B. D., [Piro]{} A. L., 2014, [MNRAS]{}, 439, 3916
A., [Montes]{} G., [Ramirez-Ruiz]{} E., [De Colle]{} F., [Lee]{} W. H., 2014, [ApJL]{}, 788, L8
H., [Hotokezaka]{} K., [Sekiguchi]{} Y., [Shibata]{} M., [Ioka]{} K., 2014, [ApJL]{}, 784, L28
T., [Kashiyama]{} K., [Nakauchi]{} D., [Suwa]{} Y., [Sakamoto]{} T., [Kawai]{} N., 2014, [ApJ]{}, 796, 13
E., 2007, [Phys. Rep.]{}, 442, 166
E., [Piran]{} T., 2011, [Nature]{}, 478, 82
S., [Kasliwal]{} M., [Georgieva]{} A., 2013, [ApJ]{}, 767, 124
S., [Sievers]{} J., [Dalal]{} N., [Holz]{} D., 2011, [ApJ]{}, 739, 99
R., [Janka]{} H.-T., [Marek]{} A., 2007, [A&A]{}, 467, 395
A., [Rosswog]{} S., [Cabez[ó]{}n]{} R. M., [Korobkin]{} O., [K[ä]{}ppeli]{} R., [Arcones]{} A., [Liebend[ö]{}rfer]{} M., 2014, [MNRAS]{}, 443, 3134
T., [Korobkin]{} O., [Rosswog]{} S., 2014, ArXiv e-prints
T., [Nakar]{} E., [Rosswog]{} S., 2013, [MNRAS]{}, 430, 2121
L., [Baiotti]{} L., [Giacomazzo]{} B., [Link]{} D., [Font]{} J. A., 2010, Classical and Quantum Gravity, 27, 114105
L., [Kumar]{} P., 2014, ArXiv e-prints
S., 2013, Royal Society of London Philosophical Transactions Series A, 371, 20272
S., [Liebend[ö]{}rfer]{} M., [Thielemann]{} F.-K., [Davies]{} M. B., [Benz]{} W., [Piran]{} T., 1999, [A&A]{}, 341, 499
M., [Janka]{} H.-T., [Takahashi]{} K., [Schaefer]{} G., 1997, [A&A]{}, 319, 122
G. B., [Lightman]{} A. P., 1979, [Radiative processes in astrophysics]{}
R., 1998, [ApJL]{}, 494, L49
R., [Piran]{} T., [Narayan]{} R., 1998, [ApJL]{}, 497, L17
B. F., 2011, Classical and Quantum Gravity, 28, 125023
N., 2015, [MNRAS]{}, 446, 2887
M., [Taniguchi]{} K., 2006, [Phys. Rev. D. ]{}, 73, 064027
J. C., [Matzner]{} C. D., [McKee]{} C. F., 2001, [ApJ]{}, 551, 946
M., [Hotokezaka]{} K., 2013, [ApJ]{}, 775, 113
N. R., [Levan]{} A. J., [Fruchter]{} A. S., [Hjorth]{} J., [Hounsell]{} R. A., [Wiersema]{} K., [Tunnicliffe]{} R. L., 2013, [Nature]{}, 500, 547
2014, ArXiv e-prints
H. J., [MacFadyen]{} A. I., 2011, [ApJL]{}, 733, L37
S., [Sekiguchi]{} Y., [Nishimura]{} N., [Kiuchi]{} K., [Kyutoku]{} K., [Shibata]{} M., 2014, [ApJL]{}, 789, L39
D., [Piran]{} T., 2014, ArXiv e-prints
B., 2013, [ApJL]{}, 763, L22
[^1]: This component is composed by the tidal tail and shocked component. The main difference between them is the value of $Y_{e}$. The tidal component can have a lower $Y_{e}$.
[^2]: This component includes two cases depending on whether the remnant is a black hole or a neutron star. In the former the wind is just from the surrounding disk while in the latter it arises from the neutron star as well. The wind from the remnant neutron star has a higher value of $\beta \Gamma$ and $Y_{e}$. Note that the fiducial value used is an optimistic one.
[^3]: Note that these estimates take into account that the GW horizon is larger in the direction of the GRB hence the chance of coincidence with a short GRB is larger than the beaming fraction.
[^4]: We adopt the conical model of [@granot2012MNRAS]. The difference of the afterglow flux between their different models is a factor $2\sim 3$ during the side-expansion phase.
[^5]: We do not use the afterglow library of [@vaneerten2011ApJ] which is incorrect below the absorption frequency. Above the absorption frequency our light curves are consistent with those of [@vaneerten2011ApJ].
[^6]: The light curve that we obtain is also consistent with the observed data in other frequencies except for the late time excess in the $H$-band (the macronova candidate) and in the X-ray band.
[^7]: Note that the velocity of the cocoon depends on the energy deposited into the cocoon. The cocoon will be relativistic when the deposited energy is larger than about $10^{49}$ erg.
|
= -1.5cm = -1.5cm =.8cm
[Decay Constant of Pseudoscalar Meson in the Heavy Mass Limit]{} $$$$ $$$$
[V.V. Andreev\
Gomel State University, Physics Department, Gomel, 246699,Belarus.\
E-mail: andreev@gsu.unibel.by\
]{} $$$$ [Published in Proceedings 7 Annual Seminar “Nonlinear Phenomena In Complex System” (NPCS’98)\
( Minsk, Belarus, 1998 )\
.]{}
$$$$ $$$$
[Abstract]{}
[The leptonic decay constant of the pseudoscalar mesons a calculated by use of the relativistic constituent quark model constructed on the point form of Poincare-covariant quantum mechanics. We discuss the role relativistic corrections for decay constants of pseudoscalar mesons with heavy quarks. We consider the heavy mass limit of decay constant for two-particle system with equal masses.]{}
$$$$ [V.Andreev\
Gomel State University, Physics Department,\
Gomel, 246699,Belarus.\
E-mail: andreev@gsu.unibel.by\
]{}
**Introduction**
================
According to the structure of the current interaction, electroweak decays of hadrons can be divided into following classes: leptonic decays,in which the quarks of the decaying hadron annihilate each other and only leptons appear in the final state; semileptonic decays, in which both leptons and hadrons appear in the final state; photons decays,in which the final state consists of photons only; radiative transitions between hadrons, in which hadrons and photon are caused by hadron decays. non-leptonic decays,in which the final state consists of hadrons only. Over the last decade, a lot of information on hadron decays has been collected in experiments at $e^{+}e^{-}$ and hadron colliders. This has led to a rather detailed knowledge of the flavour sector of the Standard Model and many of the parameters associated with it. In this work we investigate the decay constant of the mesons with spinor quarks in the heavy mass limit.
$q\overline{q}$ bound state in the point form of RQM
====================================================
There are three forms of the dynamics in the relativistic quantum mechanics (RQM), called instant ,point,light-front forms [@Di1]. In this work we use point form of the RQM [@Gross]. The description in the point form implies that the generators of the Poincare group ${\hat M}^{\mu\nu }$ are the same as for noninteracting particles and bound systems. Interaction terms can be present only in the four-momentum operators ${\hat P}_\mu $, but the four-velocities of bound and free-particle systems are equal.
The momenta $\vec p_1$, $\vec p_2$ of the quarks with the masses $m_1$ and $
m_2$ of relativistic system can be transformed to the total $\vec P$ and relative momenta $\vec k$ to facilitate the separation of the center mass motion: $$\vec P_{12} =\vec p_1+\vec p_2,$$ $$\overrightarrow{k} =\overrightarrow{p_1}+\frac{\overrightarrow{P_{12}}}{M_0 }
\left( \frac{\left( \overrightarrow{P_{12}}\overrightarrow{p_1}\right) }{
\omega _{M_0}\left( \overrightarrow{P_{12}}\right) +M_0}+\omega _{m_1}\left(
\overrightarrow{p_1}\right) \right) , \label{veck}$$ where $M_0=\omega _{m_1}\left( \overrightarrow{k}\right) +\omega
_{m_2}\left( \overrightarrow{k}\right) ,\omega _{m_1}\left( \overrightarrow{
p_1}\right) =\sqrt{\overrightarrow{p_1}^2+m_1^2}$.
The solution of the eigenvalue problem will lead to eigenfunction of the form $$_0\left\langle \vec V_{12}\mu ,\left[ J\hskip 2ptk\right] ,(l s)\right.
\left| \vec V\mu ,\left[ J\hskip 2ptM\right] \right\rangle =$$ $$=\delta_{J J^{\prime }} \delta _{\mu \mu^{\prime }} \delta (\vec V-\vec
V_{12}) \Psi ^{J\mu }\left( k\hskip 2pt l\hskip 2pt s\hskip 2pt; M \right)$$ with the velocities of bound system $\vec V={\vec P}/{M}$ and noninteracting system $\vec V_{12}={\vec P_{12}}/{M_0}$. The function $\Psi ^{J\mu }\left(
k \hskip 2ptl\hskip 2pts\hskip 2pt;M\right) $ satisfies in the point form a following equation [@Pol1]: $$\sum_{l^{\prime }s^{\prime }}\int\limits_0^\infty <k\hskip 2ptl\hskip
2pts\parallel W^J\parallel k^{\prime }\hskip 2ptl^{\prime }\hskip
2pts^{\prime }>\Psi ^J(k^{\prime }\hskip 2ptl^{\prime }\hskip 2pts^{\prime
}; \hskip 2ptM)k^{\prime 2}dk^{\prime }+$$ $$+k^2\Psi ^J(k\hskip 2ptl\hskip 2pts;M)=\eta \Psi ^J(k\hskip 2ptl\hskip 2pt
s;M) \label{maineq}$$ with reduced matrix element of operator $\hat W$.
In the point form the meson state is defined by as state of on-shell quark and antiquark with the wave function $\Psi ^{J\mu }\left( k\hskip 2pt l
\hskip 2pt s\hskip 2pt;M\right) $ $$\left| \overrightarrow{P}\mu \hskip 2pt\left[ JM\right] \right\rangle =\sqrt{
\frac M{\omega _M\left( \overrightarrow{P}\right) }}*$$ $$\ast \sum_{ls\lambda _1\lambda _2}\int d^3k\sqrt{\frac{\omega _{m_1}\left(
\overrightarrow{p_1}\right) \omega _{m_2}\left( \overrightarrow{p_2}\right)
}{\omega _{m_1}\left( \overrightarrow{k}\right) \omega _{m_2}\left(
\overrightarrow{k}\right) }}\Psi ^{J\mu }\left( kls;M\right)$$ $$\sum_{m\lambda }\sum_{\nu _1\nu _2}\left\langle s_1\nu _1,s_2\nu _2\right|
\left. s\lambda \right\rangle \left\langle lm,s\lambda \right| \left. J\mu
\right\rangle Y_{lm}\left( \theta ,\phi \right)$$ $$D_{\lambda _1\nu _1}^{1/2}\left( \overrightarrow{n}\left( p_1,P\right)
\right) D_{\lambda _2\nu _2}^{1/2}\left( \overrightarrow{n}\left(
p_2,P\right) \right) \left| p_1\lambda _1\right\rangle \left| p_2\lambda
_2\right\rangle \label{state}$$ where $\left\langle s_1\nu _1,s_2\nu _2\right| \left. s\lambda \right\rangle
$, $\left\langle lm,s\lambda \right| \left. J\mu \right\rangle $ are Clebsh-Gordan coefficients of $SU(2)$-group, $Y_{lm}(\theta ,\phi )$ - spherical harmonic with spherical angle of $\vec k$. Also, in Eq.(\[state\] ) $D^{1/2}\left( \overrightarrow{n}\right) =1-i\left( \overrightarrow{n}
\overrightarrow{\sigma }\right) /\sqrt{1+\overrightarrow{n}^2}$is $D$ -function of Wigner rotation, which determined by vector-parameter $
\overrightarrow{n(}p_1,p_2)=\overrightarrow{u_1}\times \overrightarrow{u_2}
/(1-\left( \overrightarrow{u_1}\overrightarrow{u_2}\right) )$ with $
\overrightarrow{u}=\overrightarrow{p}/\left( \omega _m\left( \overrightarrow{
p}\right) +m\right) $.
**Leptonic decay constant**
============================
The leptonic decay constant for pseudoscalar meson is defined by $$\left\langle 0\left| \hat J^\mu \left( 0\right) \right| \overrightarrow{P}
,M\right\rangle =i\left( 1/2\pi \right) ^{3/2}\frac 1{\sqrt{2\omega _M\left(
\overrightarrow{P}\right) }}P^\mu f_p, \label{deconst}$$ where $\hat J^\mu (0)$ is the operator axial-vector part of the charged weak current. Using Eq.(\[state\]) and Eq.(\[deconst\]) we found in the point form dynamics, that [@And1] $$f_p=\frac{N_c}{\pi \sqrt{2}}\int_0^\infty dk\hskip 2ptk^2\sqrt{\frac{
M_0^2-(m_1-m_2)^2}{\omega _{m_1}\left( \overrightarrow{k}\right) \omega
_{m_2}\left( \overrightarrow{k}\right) }}*$$ $$\ast \frac{\left( m_1+m_2\right) }{M_0^{3/2}}\Psi \left( k,M\right) ,
\label{dec1}$$ where $N_c$-number of colors, $m_1$ and $m_2$ are the respective masses of the two quarks. The wave function for pseudoscalar meson have the normalization $$\int_0^\infty dk\hskip 2ptk^2\hskip 2ptN_c\left| \Psi \left( k,M\right)
\right| ^2=1.$$ When $m_1=m_2=m_Q$, the leptonic decay constant is defined by
$$f_p=\frac{2N_cm_Q}\pi \int_0^\infty \frac{dk\hskip 2ptk^2\Psi \left(
k,M\right) }{\omega _{m_Q}^{3/2}\left( \overrightarrow{k}\right) }.
\label{dec1d}$$
The equation for the bound $q\bar q$ states (\[maineq\]) in the RQM is relativistic equation with effective potential $W$ . However, it is hard problem to obtain wave function $\Psi (k,M)$ as solution of this equation. Therefore, we use simple model wave function depending on length scale parameter $1/\beta $:
$$\Psi (k\hskip 2pt,M)\equiv \Psi (k\hskip 2pt,\beta )=2/(\sqrt{N_c}\beta
^{3/2}\pi ^{1/4})exp(-\frac{k^2}{2\beta ^2}) . \label{wv1}$$
Using the equations (\[dec1d\]) and (\[wv1\]), one can see that
$$\begin{aligned}
f_\pi &=&\frac{\sqrt{N_c}\beta }{\pi ^{5/4}\Gamma \left( -\frac 14\right) W}
\nonumber \\
&&(2^{3/4}\Gamma \left( -\frac 14\right) \Gamma \left( \frac 34\right) {
_1F_1 }\left( \frac 34;\frac 14;\frac 1{2W^2}\right) - \nonumber \\
&&\frac{2\sqrt{\pi }}{W^{3/2}}\Gamma \left( -\frac 34\right) {_1F_1}\left(
\frac 32;\frac 74;\frac 1{2W^2}\right) ), \label{dec1a}\end{aligned}$$
with $W=\beta /m_Q$, hypergeometric function $_1F_1(a;b;z)$ and $\Gamma (z)$ -Gamma function.
We now consider the heavy mass limit of (\[dec1\]). This limit is defined as $m_1$, $m_2\longrightarrow \infty $ with $V=P/M$ fixed. The starting point in the construction of the effective theory with the heavy quarks (HQET) is the observation that a heavy quark bound inside a hadron moves more or less with the hadron’s velocity $V$. Its momentum can be written as $$p_Q=m_QV+\widetilde{k}, \label{hq1}$$ where the components of the so-called residual momentum $\widetilde{k}$ are much smaller than $m_Q$. Interactions of the heavy quark with light degrees of freedom change the residual momentum by an amount of order $\widetilde{k}
\sim \Lambda _{QCD}\simeq 1/R_{Hadron}$, but the corresponding changes in the heavy-quark velocity vanish as $\Lambda _{QCD}/m_Q\longrightarrow 0$. In the system of the center mass we are obtained, that relative momentum $
\overrightarrow{k}$ (\[veck\]) and residual momentum$\overrightarrow{
\widetilde{k}}$ are equal and therefore, the heavy mass limit in the point form is given by $$\left| \overrightarrow{k}\right| \leq \Lambda _{QCD}\ll m_Q. \label{hlim}$$ The nonrelativistic variant furnishes the following relationship for leptonic decay constant (\[dec1\]):
$$\begin{aligned}
f_{nonrel} &=&\frac{2N_cm_Q}\pi \int_0^{\Lambda _{QCD}}dk \hskip 2pt k^2
\nonumber \\
&&(1-\frac{3 k^2}{4 m_Q^2}+\frac{21 k^4}{32 m_Q^4})\Psi _{nonrel}\left(
k,M\right) , \label{dec2}\end{aligned}$$
where $\Psi _{nonrel}\left( k,M\right) $ have the normalization $$\int_0^{\Lambda _{QCD}}dk\hskip 2pt k^2\hskip 2pt N_c\left| \Psi
_{nonrel}\left( k,M\right) \right| ^2=1.$$ The parameter $\beta $ can be estimate from mean square radius (MSR) $
\left\langle r^2\right\rangle _{nonrel}=1/\Lambda _{QCD}^2$ of the meson with the heavy quarks. In the nonrelativistic approximation the MSR is $
\left\langle r^2\right\rangle _{nonrel}=3/8\beta ^2$ and we obtain the relationship between $\beta $ and $\Lambda _{QCD}$:
$$\Lambda _{QCD}=\sqrt{\frac 83}\beta . \label{lb}$$
The nonrelativistic wave function can be choose the form:
$$\Psi _{nonrel}\left( k,M\right) \sim \Psi \left( k,M\right) \sim exp(-\frac{
k^2}{2\beta ^2}), \label{nonfun}$$
since the model wave function (\[wv1\]) has not the small parameter $
\Lambda _{QCD}/m_Q$ (or $W=\beta /m_Q$).Using (\[dec2\]),(\[lb\]) and ( \[nonfun\]) we obtain the following result for $f_{nonrel}$:
$$f_{nonrel}\approx \sqrt{N_c}\beta \sqrt{W}(0.72 -0.73 W^2+1.09 W^4).
\label{dec1c}$$
Let us discuss in brief the role of relativistic corrections in leptonic decay constant of pseudoscalar meson with heavy quarks. This effect can be extracted easily. Using asymptotic limit for Kummer’s function ($
1/W\longrightarrow \infty $)we found, that decay constant (\[dec1a\]) can be written as
$$\begin{aligned}
f_p &\approx &\frac{\sqrt{N_c}\beta \sqrt{W}}{\pi ^{3/4}16\sqrt{2}}\left(
32-72 W^2+315 W^4\right) \nonumber \\
&\approx &\sqrt{N_c}\beta \sqrt{W}(0.60 -1.35 W^2+5.90 W^4). \label{dec1b}\end{aligned}$$
Comparison of series (\[dec1b\]) and (\[dec1c\]), that the factors at addenda of these series differ, especially second and third term of a series. Just, these addenda also give corrections to the effective theory of heavy quarks. If we let’s assume, that the parameter $\Lambda _{QCD}=a\beta $ with $a=\sqrt{2}$, the first terms of series practically coincides for two variants $$f_{nonrel}\approx \sqrt{N_c}\beta \sqrt{W}(0.60-0.47 W^2+0.55 W^4),
\label{de1ca}$$ but the second and third terms nevertheless essentially differ. Practically we compare two approaches of an evaluation of relativistic corrections for the effective theory of heavy quarks: the first approach follows from an exact solution of a problem with a consequent passage to the limit of heavy quarks; the second approach is based on an approximate solution of a problem; In the second approach, and such approach, as we see requires cutting relative momentum by magnitude $\Lambda _{QCD}$ in a quark model of a meson, relativistic corrections has the smaller value just because of cutting. Such divergence can be reduced by introduction of a parameter $\mu $ , which $\gg \Lambda _{QCD}$. However, it can be defined a value only using exact calculation. Therefore use of exact expressions for observable magnitudies is represented preferable to us, as, the numerical integration both approximate relations, and exact expressions has an identical order of complexity.
**ACKNOWLEDGMENTS**
===================
This work was supported by grant [**N. F96-326 (17.02.97**]{}) from the Foundation for Fundamental Research of Republic Belarus.
[9]{} P.A.M. Dirac Rev.Mod.Phys., [**21**]{}, p.392 (1949)
F.Gross Phys.Rev.,[** 186**]{}, p.1448 (1969); Phys. Rev. D, [**10**]{}, p.223 (1974). F.M.Lev ”Forms of relativistic dynamics, current operators and deep inelastic lepton-nucleon scattering” hep-ph/9505373.
F. Coester and W.N. Polyzou, Phys.Rev. D, [**26**]{},1348, (1982).
W.N. Polyzou, Annals of Physics [**193**]{}, p.367 (1989). B.D.Keister, W.N.Polyzou in Advances in Nuclear Physics, edited J.W.Negele and E.Vogt (Plenum, New York, 1991).
V.V.Andreev Proc.3-d Annual Seminar NPCS’94. Editors V.I.Kuvshinov & G.Krylov,Minsk 1995. P.141-145.
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.