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57ed2c4b42fd49fd8f9a0c58ed6af2c21f036ca2 | subsection | 38 | 47 | Results | Inserting these
expression into Eq (REF ) for a quick, qualitative
estimate, we find \bar{T}_g\sim {v_{ij}v_{ij}}\equiv v_s^2 for \gamma _0\gg \gamma _1\bar{T}_g, and \bar{T}_g\sim v_s for \gamma _0\ll \gamma _1\bar{T}_g. The first
regime is essentially elastic, because the relaxation
term, u_{ij}/\tau \sim u_{ij}\bar{... | {
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557e0760d12577971b54542a6a8b2f10716f3e61 | subsection | 39 | 47 | The Hypoplastic Regime | Hypoplasticity , a modern,
well-verified, yet comparatively simple theory of soil
mechanics, models solid dynamics as realistically as the
best of the elasto-plastic theories. Its starting point
is the rate-independent constitutive relation,\partial _t{\sigma }_{ij}
=H_{ijk\ell }\,\,v_{k\ell }+\Lambda _{ij}\sqrt{v^0_{i... | {
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f1048115fbfb0f430f6e4eb4d582f5359f482495 | subsection | 40 | 47 | The Hypoplastic Regime | (Rate-independence means \partial _t{\sigma }_{ij} is linearly proportional to the magnitude of the
velocity, such that the change in stress remains the same
for given displacement irrespective how fast the change
is applied, a well verified observation in both the
elastic and hypoplastic regime.) Great efforts are
inv... | {
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ca7d56f5f9e411cd5f25cc824a51d8b064331960 | subsection | 41 | 47 | The Hypoplastic Regime | It is then
quite easy to evaluate \partial _t \sigma _{ij}
employing Eqs (,),{\partial _t}\sigma _{ij}=(1-\alpha ){\partial _t}
\pi _{ij}=(1-\alpha )M_{ijk\ell } {\partial _t} u_{k\ell }=\\
{\textstyle (1-\alpha ) M_{ijk\ell } [(1-\alpha )v_{k\ell }-
u^0_{k\ell }/\tau -\delta _{k\ell }u_{mm}/{\tau _1}]}.Clearly, given ... | {
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8d9c445974ac6b5550d5ea5140fff0c63a20a888 | subsection | 42 | 47 | The Hypoplastic Regime | But |\Lambda _{ij}|/
|H_{ijk\ell }| \sim |u_{k\ell }^0|\cdot 114/(1-\alpha ) is
of order unity for |u_{ij}| around 10^{-3}.Although the functions of Eqs (REF ,)
appear rather different from that of
Eqs (REF ,), the stress-strain
increments are quite similar, as the comparison
in shows. Moreover, the residual
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20789535f512183bb0e3cdf789cb57a8ae423cac | subsection | 43 | 47 | Conclusion | The success of Granular Elasticity, the theory
we employ to account for static stress distribution in
granular media, is mainly due to the fact that the
information on the plastic strain is quite irrelevant
there. This is no longer true in granular dynamics,
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db1aaecacbe437ab5a6d8a56d5c18105f9ca4422 | subsection | 44 | 47 | Equilibrium Conditions | First, noting \pi _{ij}{\rm d}u_{ij}=\pi _{ij}{\rm d}\nabla _jU_i because \pi _{ij} is symmetric, we write
the energy density per unit volume (dropping the
subscript of w_0 in this section) as{\rm d}w=T{\rm d}s+\mu {\rm d}\rho -\pi _{ij}{\rm d}\nabla _jU_i.Next, we vary the energy \int w{\rm d}V for given
entropy \int ... | {
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} | 10.1007/s10035-009-0137-3 | 0807.1883 | Granular Solid Hydrodynamics | [
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e4d6fdfd305dd08ed7be2a5adad7bd47ab5dbecb | subsection | 45 | 47 | Equilibrium Conditions | Since
\nabla _j\pi _{ij}\sim \partial ^2_x[U_x(x_0)-U_x(0)]=0, the one-dimensional strain
is u_{\ell \ell }=\partial _x U_x=(U_x(x_0)-U_x(0))/x_0
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dd9f18b978692ac9ff079be98386e247db275769 | subsection | 46 | 47 | Equilibrium Conditions | But if Eq (REF ) is
implemented, turning Eq (REF ) to{\rm d}e&=&T{\rm d}\sigma -(P_T+\pi _{\ell \ell }/3){\rm d}v-(\pi ^0_{ij}/\rho ){\rm d}\nabla _jU_i,\\ &=& T{\rm d}\sigma -\rho ^{-1}(P_T\,\delta _{ij}+\pi _{ij})\,{\rm d}\nabla _jU_i,the equilibrium conditions are altered to become\nabla _iT=0,\quad \nabla _i(P_T+\p... | {
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821daa999db777d6df41a7eb8dd16a07b0983792 | abstract | 0 | 14 | Abstract | Atomic hydrogen escaping from the extrasolar giant planet HD209458b provides
the largest observational signature ever detected for an extrasolar planet
atmosphere. In fact, the upper atmosphere of this planet is evaporating.
Observational evidences and interpretations coming from various models are
reviewed. Implicatio... | {
"cite_spans": []
} | 0807.1885 | Evaporation of extrasolar planets | [
"David Ehrenreich"
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c2896eec39f3f88321be93f0d0e99be0055136fe | subsection | 1 | 14 | Introduction | Extrasolar planets are now commonly found around main
sequence stars, usually by measuring the variations of the stellar velocity
projected on the line of sight. If the inclination of the planetary system is
close to 90^{\rm o}, the planet is seen from Earth passing in front of its host
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f6bb20576e6beaf9669ea12807844d4f770322e9 | subsection | 2 | 14 | Exosphere and atmospheric escape: some definitions | All the notions addressed in this chapter are developed by
Chamberlain & Hunten (1987), where the reader is referred for more details. | {
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00ae6bd07d539c544cf73d2694adaca0f66c4d07 | subsection | 3 | 14 | Collisionless atmospheres | A planetary atmosphere can be roughly split in two parts:
the bottom part is the homosphere, where atmospheric constituants are mixed by
various processes (convection in the troposphere, eddy diffusion, molecular
diffusion, collisions, etc.) resulting in a more-or-less uniform composition
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94883ad580e6d5cc09197ec1fa6459481a5fca26 | subsection | 4 | 14 | Escape processes | Being a boundary layer between a relatively dense gas
envelope and space vacuum, the exosphere is where atmospheric mass loss
processes, or escape processes, take place. The simplest, yet usually not very
efficient, escape mechanism is the thermal escape (Jeans 1925). It assumes that
particles in the exosphere have a M... | {
"cite_spans": []
} | 0807.1885 | Evaporation of extrasolar planets | [
"David Ehrenreich"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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b60ccff4ae80f02fbbc3a6f5fbd9978ab899ff67 | subsection | 5 | 14 | Escape processes | At this
point, the best description of atmospheric escape is not that of Jeans, nor the
limit to the escape flux is the transport to the exobase; the description is
rather hydrodynamic and the energetic input to the upper atmosphere represents
the bottleneck to the escape flux.Furthermore, close-in giant extrasolar pla... | {
"cite_spans": []
} | 0807.1885 | Evaporation of extrasolar planets | [
"David Ehrenreich"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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69d592b7e7f3d704f23a464361f2ab1e39c53df4 | subsection | 6 | 14 | Roche lobe | The gravitational potential of a rotating circularized binary
system, such as a star-planet system, expresses (Hameury 2000)\phi = -\frac{\mathrm {G}m_p}{|x-r_p|} - \frac{\mathrm {G}m_\star }{|x-r_\star |} -
\frac{1}{2} |\omega \times x|^2,where x is the distance from the centre of mass and \omega is the system
angular... | {
"cite_spans": []
} | 0807.1885 | Evaporation of extrasolar planets | [
"David Ehrenreich"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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663b1d90c8e71bc7b720febfec170cbe478c875a | subsection | 7 | 14 | Detection of the escaping exosphere of HD 209458b from resolved data | The transit of HD 209458b was originally observed by Vidal-Madjar (2003) with the Space Telescope Imaging Spectrograph (STIS; Woodgate 1998) on board the Hubble Space Telescope (HST) to search for
H i in the planet atmosphere. Since H i is the lightest possible
component of the planet, it could be detected high in the ... | {
"cite_spans": []
} | 0807.1885 | Evaporation of extrasolar planets | [
"David Ehrenreich"
] | [
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45550570965c40668c292d38f817a5cc8445913f | subsection | 8 | 14 | Constraints on the evaporation of HD 209458b from unresolved data | The detection of HD 209458b extended exosphere was first confirmed by
Vidal-Madjar (2004) with low-resolution STIS spectra gathered during 4
transits of the planet. A low-resolution grism was employed to probe a large
spectral domain in the ultraviolet (from 120 to 170 nm) and test whether other
atomic elements could b... | {
"cite_spans": []
} | 0807.1885 | Evaporation of extrasolar planets | [
"David Ehrenreich"
] | [
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] | 2,008 | en | Physics | [
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c89890aa00c7796293e8cf4c1bc05352d7a03d05 | subsection | 9 | 14 | Are there other cases of evaporation? | The atmospheric evaporation phenomenon was first observed in 2003 when the
number of transiting planets around bright stars (V\lesssim 10) was minute.
The situation drastically changed in 2004, when several `bright' transits were
spotted, in particular, HD 149026b (V = 8.15; Sato 2005) and
HD 189733b (V = 7.67; Bouchy ... | {
"cite_spans": []
} | 0807.1885 | Evaporation of extrasolar planets | [
"David Ehrenreich"
] | [
"astro-ph"
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b85981f4053be40459dde027808ed24f163440e3 | subsection | 10 | 14 | Interpreting the observations | Models of extended exospheres have early been crafted by observers to justify
the search for large spectroscopic signatures (Rauer 2000; Moutou 2001) and, later, to provide an interpretation of the 15\% absorption in the
Ly\alpha stellar line during the transit (Vidal-Madjar 2003). These last
authors used a particle si... | {
"cite_spans": []
} | 0807.1885 | Evaporation of extrasolar planets | [
"David Ehrenreich"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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c8ec873397cb753c2da76a3928ef11fd5483c0a2 | subsection | 11 | 14 | Understanding the dynamics of an evaporating atmosphere | Numerical models of atmospheres enduring hydrodynamic escape were initially
built to explain the loss of hydrogen from the primitive terrestrial and
cytherean atmospheres (Watson, Donahue & Walker 1981; Kasting & Pollack 1983;
Chassefière 1996). The discovery of HD 209458b evaporation sparked new
modelling insights, wh... | {
"cite_spans": []
} | 0807.1885 | Evaporation of extrasolar planets | [
"David Ehrenreich"
] | [
"astro-ph"
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f8dbe49213bc77c7351684bce292669e50ae9908 | subsection | 12 | 14 | Atmospheric evaporation in the frame of comparative exoplanetology | Although a lot of detailed models has been developed to understand the physics
of evaporation, it appears that the resulting escape rate does not depend on
the details of the models but only on the assumed input energy used to escape
the potential well of the planet. As a consequence, an energy diagram was
introduced b... | {
"cite_spans": []
} | 0807.1885 | Evaporation of extrasolar planets | [
"David Ehrenreich"
] | [
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07a7f180ab6fb17006997ec0a8c5f0a3d3e94183 | subsection | 13 | 14 | Summary | A planetary transit in the ultraviolet produces remarkably large spectroscopic
signatures compared to what is observed in the visible. So far, the
observational results obtained on planet HD 209458b have been obtained from
two different instruments and strengthened by independent data analysis. The
evaporation of close... | {
"cite_spans": []
} | 0807.1885 | Evaporation of extrasolar planets | [
"David Ehrenreich"
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79831928cc21532873eb2014dd040627452a972a | abstract | 0 | 7 | Abstract | Cold Fermionic atoms with three different hyperfine states confined in
optical lattices show pronounced Atomic Density Waves (ADWs). These ADWs are
pinned due to the confining potential that traps the atoms in the optical
lattice and can be considered a crystal of strongly bound trions. We show that
the crystalline pha... | {
"cite_spans": []
} | 10.1103/PhysRevA.80.013616 | 0807.1886 | Crystallization of trions in SU(3) cold atom gases trapped in optical
lattices | [
"Rafael A. Molina",
"Jorge Dukelsky",
"Peter Schmitteckert"
] | [
"cond-mat.str-el",
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2a19af0d910c471da035f703a237c3a20dd8887d | subsection | 1 | 7 | Body | Crystallization of trions in SU(3) cold atom gases trapped in
optical latticesRafael A. Molina^{1,2}
Jorge Dukelsky^1
Peter Schmitteckert^{3}^1Instituto de Estructura de la Materia - CSIC, Serrano 123,
28006 Madrid, Spain^2Institut für Theorie der Kondensierten Materie,
Universität Karlsruhe, 76128 Karlsruhe, Germany^3... | {
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"e... | 10.1103/PhysRevA.80.013616 | 0807.1886 | Crystallization of trions in SU(3) cold atom gases trapped in optical
lattices | [
"Rafael A. Molina",
"Jorge Dukelsky",
"Peter Schmitteckert"
] | [
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b6bbd432a1c32fa7c1ab213cb691b3f596b0aaf4 | subsection | 2 | 7 | Body | There is a
competition between the superfluid trionic behavior and ADW order with the former phase dominating for generic
fillings, and the latter probably appearing at half-filling.The main goal of this letter is to study the emergence of a trionic crystalline phase in three color atoms loaded
in optical lattices. We ... | {
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"Rafael A. Molina",
"Jorge Dukelsky",
"Peter Schmitteckert"
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2ba0f6c0e1b2ab636eff277247f2c193bae25016 | subsection | 3 | 7 | Body | For the number of pairs we defineIn the non-polarized we will be interested in the total number of pairs at each site i,The local correlation that counts the number of trions isFor non-interacting SU(3) fermions
P_{i\alpha \beta }=\left< n_{i\alpha }\right> \left< n_{i\beta }\right>-\left< n_{ir} \right> \left< n_{ig} ... | {
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"Rafael A. Molina",
"Jorge Dukelsky",
"Peter Schmitteckert"
] | [
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e7245302a2c816599f4fe96a16d950a9d98c2bee | abstract | 0 | 9 | Abstract | We introduce a minimal Agent Based Model with two classes of agents,
fundamentalists (stabilizing) and chartists (destabilizing) and we focus on the
essential features which can generate the stylized facts. This leads to a
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7303ee5c189d76936b16732fb769b23b5733f1fb | subsection | 1 | 9 | Introduction | The Random Walk (RW) represents the simplest modelization for the behavior of financial time series. It was in fact introduced by Louis Bachelier in 1900 to model the dynamics of prices. Recently, the availability of large amounts of data has been accompanied by a systematic study of the empirical properties of price ... | {
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9b90dc547a9eaff0eb5b4d200da394ac97accf29 | subsection | 2 | 9 | Introduction | Finally we would like to stress that the general perspective of the present model is to have the maximum simplification in order to identify the specific origin of the SF. The model can be easily generalized to higher degrees of realism in systematic way . | {
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86884e43c34c1a6a378462cd195824ae73ac2957 | subsection | 3 | 9 | Simplest Agent Based Model with Fundamentalists and Chartists | We consider N agents divided in N_f fundamentalists and N_c chartists (N=N_f+N_c). For the moment we keep N fixed but later, in the study of self-organization, N will be a fluctuating variable. An important point is to have a description of chartists as simple as possible and with the minimal number of parameters. In t... | {
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0c52b39d19b3a06e87832cf55e6ed41e5e070885 | subsection | 4 | 9 | Simplest Agent Based Model with Fundamentalists and Chartists | It is clear therefore that the intermittent region corresponds to finite size effects which disappear for N very large.]For small price fluctuations the use of linear or log returns are essentially equivalent and in this paper
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35fd4fc0dbd23316edcd9a097c6add83bf8652eb | subsection | 5 | 9 | Simplest Agent Based Model with Fundamentalists and Chartists | This means that specific effects in the real data and in the model can affect their values, which should not be considered as fundamental test for the models.
In Fig. REF we report the Stylized Facts corresponding to the intermittent case (N=500).
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d88250316c8561e3d153642bdbd36a31e321aa79 | subsection | 6 | 9 | Nonstationarity and the Microscopic Origin of Stylized Facts | The simplicity of the model permits a detailed interpretation of the origin and nature of the stylized facts. The long time correlation of volatility and its behavior in terms of bursts depends essentially on two elements. The first is that there is a certain probability to have an appreciable number of chartists and t... | {
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b2058253f043012cc335e2ee8d202c9a21eea1bb | subsection | 7 | 9 | Self-Organization of Stylized Facts | Finally we can consider the problem of the self-organization of the system in the intermittent state. In fact, also in our model, the SF are obtained only for specific values of the parameters and a basic question is to understand why the systems chooses to stabilize in this special situation. The key point in this res... | {
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f18bf9f0f4b8034d73623c1d3c3487b7a5e37064 | subsection | 8 | 9 | Conclusions | In summary the model we have introduced was motivated by the objective of having the maximum mathematical simplification that is still able to reproduce the stylized facts. This has permitted a detailed interpretation of their nature and origin of SF and it is a crucial point in order to consider the phenomenon of the ... | {
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280a1374d60ca743d5b9450e26f126cf083bd1ea | abstract | 0 | 114 | Abstract | We review, in a self-contained and pedagogical manner, recent developments
and techniques for the evaluation of the scattering amplitudes of planar N=4
SYM theory at both weak and strong coupling. Special emphasis is placed on the
newly discovered connection between a special class of amplitudes and the
expectation val... | {
"cite_spans": []
} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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efc3e644a3fd87a3097baf2b6191a20b31c01d4c | subsection | 1 | 114 | Introduction | There are two approaches to understanding and solving {\cal N}=4
Yang-Mills (SYM): on the one hand, being a conformal field theory,
it is uniquely specified by the spectrum of (anomalous) dimensions
of gauge-invariant operators and their three-point correlation
functions, while, on the other hand, like any other quantu... | {
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{
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"doi": "",
"end": 1079,
"openalex_id": "",
"raw": "N. Beisert and M. Staudacher, “Long-range PSU(2,2|4) Bethe ansaetze for gauge theory and strings,” Nucl. Phys. B 727, 1 (2005) [arXiv:hep-th/0504190].",
"source_ref_id": "2909eab541fec559bee6... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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1f435c3bbf1047850f3cee5f266c0cd93266e4f3 | subsection | 2 | 114 | Introduction | The four- and five-gluon amplitudes,
which are currently known to all orders in perturbation theory (up to
a set of undetermined constants), provide a proof of principle in this
direction.Here we review, in a self-contained and pedagogical manner, some of
the recent developments and techniques for the evaluation of the... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 471,
"openalex_id": "",
"raw": "L. J. Dixon, “Gluon scattering in N=4 super-Yang-Mills theory from weak to strong coupling,” arXiv:0803.2475 [hep-th].",
"source_ref_id": "f349ebb8ffc4817f519c121f4dcac69625e5439a",
"sta... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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9045a546e852fba9e2f87c868136c341aa6c3937 | subsection | 3 | 114 | Introduction | While not yet clear how to extend the calculations beyond
leading order, this regularization scheme has the advantage of
being analogous to dimensional regularization as used in gauge
theory calculations and thus of allowing a direct comparison of
results.We carry out the calculation of the four-gluon scattering amplit... | {
"cite_spans": []
} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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975c85600453eb38e15e7a8ac8b93c796bc8ff33 | subsection | 4 | 114 | Introduction | Its origins and full
implications remain to be uncovered; in section we
collect some open questions whose answers may lead to an improved
understanding of the deep and powerful structures governing the
dynamics of {\cal N}=4 super-Yang-Mills theory and perhaps other
four-dimensional gauge theories. | {
"cite_spans": []
} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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638ccdc0b89402e40c8853e5aa46e0a5daabfe1e | subsection | 5 | 114 | Scattering amplitudes at weak coupling | On-shell scattering amplitudes are perhaps the most basic
quantities computed in any quantum field theory. The standard
textbook approaches proceed to relate them through the LSZ
reduction to Green's functions which are in turn
computed in terms of Feynman diagrams. Each diagram evaluated
separately is typically more c... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 2166,
"openalex_id": "",
"raw": "F. A. Berends and W. T. Giele, “Recursive Calculations for Processes with n Gluons,” Nucl. Phys. B 306, 759 (1988).",
"source_ref_id": "35ebaa51f3ecf9591ada2b1e9ffc193fac906b77",
"start... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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571dc8b1b038dddacb09b81b8e9d5e6954f5fe09 | subsection | 6 | 114 | Organization, presentation and general properties | A good notation as well as an efficient organization of the
calculation and result are indispensable ingredients for the
calculation of scattering amplitudes, whether with Feynman diagrams or
by other means. They are provided, respectively, by the spinor
helicity method (for massless theories) and by color ordering, wh... | {
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"doi": "",
"end": 515,
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"raw": "L. J. Dixon, “Calculating scattering amplitudes efficiently,” arXiv:hep-ph/9601359.",
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"start": 455
}
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} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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ab305fedb4e32e4a2cc2627f35f19a86c320473e | subsection | 7 | 114 | Spinor helicity and color ordering | In a massless theory, solutions of the chiral Dirac equation
provide an excellent parametrization of momenta and polarization
vectors which allows, among other things, the construction of
physical polarization vectors without fixing noncovariant gauges.
The main observation is that the sum over polarizations of a
direc... | {
"cite_spans": []
} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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162516971379a679b80694d9958800e436bc1fd8 | subsection | 8 | 114 | Spinor helicity and color ordering | Moreover, in the frame in which the vector field
propagates along a specified axis, they should take the standard form
of circular polarization vectors.A solution to these constraints can be constructed in terms of an
arbitrary null (reference) vector \xi
(\xi _\mu \sigma ^\mu _{\alpha {\dot{\alpha }}}=\xi _\alpha {\t... | {
"cite_spans": []
} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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43c7da5964c8a70cb296bf76646cecd57530a292 | subsection | 9 | 114 | Spinor helicity and color ordering | For an SU(N) gauge
theory with gauge group generators denoted by T^a, it is possible to
show that any L-loop amplitude may be decomposed as follows:A^{(L)}=N^L\sum _{\rho \in S_n/\mathbb {Z}_n}
{\rm Tr}[T^{a_{\rho (1)}}\dots T^{a_{\rho (n)}}]A^{(L)}(k_{\rho (1)}\dots k_{\rho (n)},N) +{\rm multi{\scriptstyle -}traces}wh... | {
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"arxiv_id": "",
"doi": "",
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"raw": "Z. Bern and D. A. Kosower, “Color Decomposition Of One Loop Amplitudes In Gauge Theories,” Nucl. Phys. B 362, 389 (1991).",
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... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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78a21aa612fa812f280cb4714ae47b28d371115c | subsection | 10 | 114 | General properties of color ordered amplitudes | The general properties of color-ordered amplitudes follow from their
construction in terms of Feynman diagrams (or string diagrams). The
results of other constructions must obey the same properties. Some of
them – such as the analytic structure – impose powerful constraints
and in some cases uniquely determine the (tre... | {
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... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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e9e8b2680cfc3400d132355ce68b682cee13c10f | subsection | 11 | 114 | General properties of color ordered amplitudes | For a given gauge theory, the l-loop splitting amplitudes {\rm Split}^{(l)}_{-h}((n-1)^{h_{n-1}},n^{h_n}) are universal functions
of the helicities of the collinear particles,
the helicity of the external leg of the resulting amplitude and of the
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z=\frac{\xi \cdot k_{n-1}}{\xi \cdot (k_{... | {
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"sta... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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af7ae187961be3c4d439588590d2e1bb343c4579 | subsection | 12 | 114 | General properties of color ordered amplitudes | \longrightarrow \\
\sum _{h_p = \pm }\Big [
\!\!\!\!\!\!\!\!\!\!\!\!\! && A^{\rm tree}_{m+1}(1,\ldots ,m,k^{h_k}) \frac{i}{k_{1,m}^2} A^{\rm 1\,loop}_{n-m+1}((-k)^{-h_k},m+1,\ldots ,n)
\cr &+&\!\!\!A^{\rm 1\,loop}_{m+1}(1,\ldots ,m,k^{h_k})
\frac{i}{k_{1,m}^2} A^{\rm tree}_{n-m+1}((-k)^{-{h_k}},m+1,\ldots ,n)
\vphantom... | {
"cite_spans": []
} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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970bf832e6287ab8ca9384b5e09866d1afa72136 | subsection | 13 | 114 | Some simple examples | Besides color ordering, scattering amplitudes can be organized
following the number of negative helicity gluons. One can easily
see that the amplitude with only positive helicity gluons as well
as the amplitude with a single negative helicity gluons vanish
identically at tree level in any gauge theory.
This is realized... | {
"cite_spans": []
} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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dcedc14b1153c25e82f50e6d059df8d6b92d8151 | subsection | 14 | 114 | Some simple examples | Similarly, using the vanishing of \langle 0|\lambda ^{a+}
g^- g^+\dots g^+|0\rangle and making judicious choices for the
reference spinor leads to the vanishing of the amplitude with a
single negative helicity gluon To
spell out the details, we use an {\cal N}=1 part of the
{\cal N}=4 supersymmetry algebra and denote b... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 970,
"openalex_id": "",
"raw": "M. T. Grisaru and H. N. Pendleton, “Some Properties Of Scattering Amplitudes In Supersymmetric Theories,” Nucl. Phys. B 124 (1977) 81.",
"source_ref_id": "f4e6355ec2b9168d02779d9f352cb0dea34ff... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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0... |
7a1a40501fa7ee25787a298c5a53eaa06ef5d766 | subsection | 15 | 114 | Some simple examples | It thus follows that, to
any loop order L,A_{MHV}^{(L)}=A^{\rm tree}_{MHV}\,{\cal M}^{(L)}(s_{i,i+1}, s_{i\dots i+2},\dots )~~,where {\cal M}^{(L)}(s_{i,i+1}, s_{i\dots i+2},\dots ) is a cyclicly symmetric
function of momenta and s_{i...j}=(k_i+k_{i+1}+...+k_j)^2. This
factorization of the tree-level amplitude also hol... | {
"cite_spans": []
} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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44b3aba67f7ebffc374575c44f0f0b9f035acf73 | subsection | 16 | 114 | Soft/Collinear factorization | A general feature of massless gauge theories in four dimensions is the
existence of infrared singularities.Ultraviolet divergences
may of course be present as well; as previously mentioned, our focus
is {\cal N}=4 SYM theory, which is free of such divergences. Unlike
ultraviolet divergences they cannot be renormalized ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 529,
"openalex_id": "",
"raw": "R. Akhoury, “Mass Divergences Of Wide Angle Scattering Amplitudes,” Phys. Rev. D 19, 1250 (1979);",
"source_ref_id": "e12f1d0a1963ddd80080f1d2d208329ef6e09a9a",
"start": 434
},
{... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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f9c87f997dc041fcb065a45c71e3092f7f761ab7 | subsection | 17 | 114 | Soft/Collinear factorization | Finally, h_n(k,
\frac{Q}{\mu },\alpha _s(\mu ),\epsilon ) contains
the effects of highly virtual fields and is finite as
\epsilon \rightarrow 0. The jet and soft functions can be
independently defined in terms of specific matrix elements.The factorization scale Q is arbitrary (within some physical
limits); it is simply... | {
"cite_spans": []
} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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4fb8d3fc7e283160d11c55394b671bd79dfc22b4 | subsection | 18 | 114 | Soft/Collinear factorization | They read\frac{d}{d\ln Q^2}{\cal M}^{[gg\rightarrow 1]}{\left(
\frac{Q^2}{\mu ^2},\lambda ,\epsilon \right)}=\frac{1}{2}\left[K(\epsilon ,
\lambda )+G\left(\frac{Q^2}{\mu ^2},\lambda ,\epsilon \right)\right]
{\cal M}^{[gg\rightarrow 1]}{\left(
\frac{Q^2}{\mu ^2},\lambda ,\epsilon \right)}~,~~where the function K contai... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 620,
"openalex_id": "",
"raw": "A. H. Mueller, “On The Asymptotic Behavior Of The Sudakov Form-Factor,” Phys. Rev. D 20, 2037 (1979);",
"source_ref_id": "fd79d58c6aa376efb3231960a80f97fa4e253372",
"start": 359
},
... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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... |
e2cc798b6a7b09ab71b2f7f15489c5de3fd3833e | subsection | 19 | 114 | Soft/Collinear factorization | The solution for K and G may then be used to
reconstruct the Sudakov form factor (REF ) which, in
turn, leads to the following expression for the factorized amplitude
:{\cal M}_n&=& \exp \left[-\frac{1}{8}\sum _{l=1}^\infty a^l \left(\frac{\gamma _K^{(l)}}{(l\epsilon )^2}
+\frac{2{\cal G}_0^{(l)}}{l\epsilon }\right)\su... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 567,
"openalex_id": "",
"raw": "Z. Bern, L. J. Dixon and V. A. Smirnov, “Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond,” Phys. Rev. D 72, 085001 (2005) [arXiv:hep-th/0505205... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
29806,
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c32d2d6ca725a276e1db5871e024e9c246fd874b | subsection | 20 | 114 | Soft/Collinear factorization | The leading
term in the strong coupling expansion of G was computed in
:f(\lambda )&=&\frac{\sqrt{\lambda }}{\pi }\left(1-\frac{3\ln 2}{\sqrt{\lambda }}-\frac{\rm K}{\lambda }+\cdots \right) \,,
\hspace{28.45274pt}\lambda \rightarrow \infty \,,\\
G(\lambda )&=& (1 - \ln 2) { \sqrt{\lambda } \over 8\pi } + \cdots \,,
\h... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 465,
"openalex_id": "",
"raw": "L. F. Alday and J. M. Maldacena, “Gluon scattering amplitudes at strong coupling,” JHEP 0706, 064 (2007) [arXiv:0705.0303 [hep-th]].",
"source_ref_id": "d6cbe432fa71a63c9bfec171d91c197adf561ce... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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d2394161628d2fb56e0a988d2d644d32108c16b1 | subsection | 21 | 114 | Loop amplitudes; generalized unitarity-based method | Having discussed general properties of scattering amplitudes, we
now proceed to describe methods for their construction at loop
level. The goal will be to use only on-shell information for this
purpose. we will be assuming (quite accurately) that tree-level
amplitudes are known. As we will see, the fact that Feynman
di... | {
"cite_spans": []
} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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d7d8a476b4f589f263b82625570591d98241f4eb | subsection | 22 | 114 | Loop amplitudes; generalized unitarity-based method | The first
string scattering amplitudes at one-loop were evaluated through such a
method , .While perfectly valid, such an approach does not make use of recent
sophisticated techniques for evaluating Feynman integrals: identities,
modern reduction techniques, differential equations, reduction to
master integrals, etc. A... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 91,
"openalex_id": "",
"raw": "K. Kikkawa, B. Sakita and M. A. Virasoro, “Feynman-like diagrams compatible with duality. I: Planar diagrams,” Phys. Rev. 184, 1701 (1969).",
"source_ref_id": "02177e998ae89fa70a5bca69b5ce92d5e... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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c90a68b6c948b04bfaabc10eee7d8fb9a4c283c0 | subsection | 23 | 114 | Loop amplitudes; generalized unitarity-based method | We will organize
the calculation in terms of regular, two-particle cuts reinterpreted in
the spirit of generalized unitarity-based method. There are two cuts
– in the s and in the t-channels. Depending upon the external
helicity configuration either one or both cuts are of non-singlet
type, with the complete {\cal N}=4... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 2228,
"openalex_id": "",
"raw": "Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, “One loop n point gauge theory amplitudes, unitarity and collinear limits,” Nucl. Phys. B 425, 217 (1994) [arXiv:hep-ph/9403226].",
"sour... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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d7a1cf9fc6450cb7f22870f855971a89683f0d74 | subsection | 24 | 114 | Loop amplitudes; generalized unitarity-based method | It was shown
in that in a supersymmetric theory this rational
contributions are absent and that in such theories one-loop
amplitudes are constructible using four-dimensional cuts.
[Figure: Box,triangle and bubble scalar integrals. The clusters at each cornerare constructed from color-adjacent external legs. If more tha... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 180,
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"raw": "Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, “One loop n point gauge theory amplitudes, unitarity and collinear limits,” Nucl. Phys. B 425, 217 (1994) [arXiv:hep-ph/9403226].",
"sourc... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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351f7d4c42b28c9f5aaa4147d068ab7cc1425ffd | subsection | 25 | 114 | Loop amplitudes; generalized unitarity-based method | For an arbitrary number of external legs
(larger than four), the result initially obtained in
(which can be reproduced using quadruple cuts and
complex momenta) reads:{\cal M}_{n=2m+1}^{(1)}&=&-\frac{1}{2}\sum _{r=2}^{m-1}\sum _{i=1}^n
(t_{i-1}^{[r+1]}t_i^{[r+1]}-t_{i}^{[r]}t_{i+r+1}^{[n-r-2]})\;I_{4;r;i}^{2me}
-\frac... | {
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"raw": "Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, “One loop n point gauge theory amplitudes, unitarity and collinear limits,” Nucl. Phys. B 425, 217 (1994) [arXiv:hep-ph/9403226].",
"sourc... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
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aabbf6d82b2b9aa0ae5949c9870b2f67d0357e18 | subsection | 26 | 114 | Calculations at higher loops | Higher loop calculations in {\cal N}=4 SYM enjoy similar
simplifications, though to a lesser extent. An analog of the 1-loop
integral basis is not available, in the sense that the members of all
proposed bases are in fact functionally dependent integrals
Notable examples are the two-loop four-point integral basis
with ... | {
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"raw": "V. A. Smirnov and O. L. Veretin, “Analytical results for dimensionally regularized massless on-shell double boxes with arbitrary indices and numerators,” Nucl. Phys. B 566, 469 (2000) [arXiv:hep-ph/... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
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ee2bc8835818695184733a47438234de39c30e43 | subsection | 27 | 114 | Calculations at higher loops | This rule is
illustrated in figure
REF .
[Figure: The rung rule.]For higher multiplicity amplitudes the rung rule is less effective
and it is necessary to explicitly evaluate the relevant iterated
cuts. The strategy discussed in this section can be used to
compute quite high loop amplitudes in {\cal N}=4 SYM. In the
ne... | {
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"raw": "Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, “One-loop self-dual and N = 4 superYang-Mills,” Phys. Lett. B 394, 105 (1997) [arXiv:hep-th/9611127].",
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Correspondence | [
"Luis F. Alday",
"Radu Roiban"
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15d7cf731db00e8c52166c576533eb650b0aef6a | subsection | 28 | 114 | Calculations at higher loops | The odd part
of the two-loop six-point amplitude was constructed only through this
method . | {
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af56ebc43134fb42af6ec7d40e637d3deba090f0 | subsection | 29 | 114 | Some explicit higher loop results at
low multiplicity | Using generalized unitarity, a number of higher loop amplitudes have
been explicitly computed and their properties analyzed. Due to the
increase in the number of kinematic invariants with the number of
external particles, the complexity of the analysis increases as the
number of external legs is increased.
Here we disc... | {
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f59c15432d6dcdccc9a8e57680acdf1fad3708dd | subsection | 30 | 114 | Some explicit higher loop results at
low multiplicity | The product of the
first two tree-level amplitudes may be easily reorganized following
equation (REF ) to beA_4^{\rm tree}{\scriptstyle (l_2, k_1^-, k_2^-, l_1)}
A_4^{\rm tree}{\scriptstyle (-l_1, -l_4, -l_3, -l_2)}
=i s_{12}(k_2-l_4)^2 A{\scriptstyle (-l_3,1^-,2^-,-l_4)}
\frac{1}{(l_2-k_1)^2(l_2+l_3)^2}~~.Further appl... | {
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} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
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49e822aebacf475b1f928c1a1428201ab9fb045b | subsection | 31 | 114 | Some explicit higher loop results at
low multiplicity | Thus, one finds that they imply that the two-loop
four-gluon amplitude in {\cal N}=4 SYM is (for any choice of helicity
assignment) given by {\cal M}^{(2)}_4{\scriptstyle (k_1,k_2,k_3,k_4)}
=-\frac{1}{4}
s_{12}s_{23}\left\lbrace ~s_{12} \hbox{~\mbox{\includegraphics [scale=.25]{sch_2lp}}}\right.~+\;s_{23}
\hbox{~\mbox{... | {
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608f476ca413b366d1a2b0e036f6ac364c3782d4 | subsection | 32 | 114 | A possible integral basis at higher loops;
Conformal integrals | {\cal N}=4 SYM is a conformal field theory at the quantum level;
conformal invariance may be observed in correlation functions of
operators of definite (anomalous) dimension. In the context of the
AdS/CFT correspondence this symmetry is related to the existence of an
exact SO(2,4) isometry of the anti-de-Sitter space. ... | {
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988bf4c4be6cf44786834726c50dbb51e3fd04c0 | subsection | 33 | 114 | A possible integral basis at higher loops;
Conformal integrals | Up to numerator factors, the
relevant integral isI_a=\int d^4 x_5 \frac{1}{x_{51}^2x_{52}^2 x_{53}^2 x_{54}^2}~~;each of the propagators present is denoted by a solid line in figure
REF a. As mentioned, under inversion this
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\frac{(x_5^2x_1^2)(x_5^2x_2^2)... | {
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"Radu Roiban"
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86d99939bb583a99cdf010481d083166b0007425 | subsection | 34 | 114 | A possible integral basis at higher loops;
Conformal integrals | Figure
REF ) and the even part of the two-loop
six-point amplitude which we shall review shortly.It is not clear what is the underlying reason for the appearance of
dual conformal invariance at weak coupling. It is moreover not clear
whether its appearance persists to all loop orders (perhaps up to
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"Luis F. Alday",
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a79b1f4384195ced94cfc0fd8e50a81ff31a223b | subsection | 35 | 114 | The BDS conjecture and potential departures form it | In section REF we discussed, following
, , higher loop corrections to the
four-gluon amplitude. The direct evaluation of the integrals in
the two-loop four-gluon amplitude reveals
a surprising structure: up to terms of order \epsilon ,{\cal M}_4^{(2)}(\epsilon )&=& \frac{1}{2}\left({\cal M}_4^{(1)}(\epsilon )\right)^2 ... | {
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cd6b9d2d636a073428eca59144c4b32cffbfbaed | subsection | 36 | 114 | The BDS conjecture and potential departures form it | Assuming that the same
is true for the splitting amplitude, Bern, Dixon and Smirnov
suggested that, to all loop orders, the rescaled n-point
MHV amplitude is given by{\cal M}_n=
\exp \left[\sum _{l=1}^\infty \, a^l f^{(l)}(\epsilon ){\cal M}^{(1)}_n(l\epsilon )+C^{(l)}
+{\cal O}(\epsilon )\right]where the coefficients... | {
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"raw": "Z. Bern, L. J. Dixon and V. A. Smirnov, “Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond,” Phys. Rev. D 72, 085001 (2005) [arXiv:hep-th/0505205... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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af6b5daf72dbac138cfe2bcd1c949c57d19915f1 | subsection | 37 | 114 | The BDS conjecture and potential departures form it | Then, taking the collinear limit of the
five-point amplitude immediately yields (REF ).The infrared poles are apparent in the equation (REF )
and, using equation (REF ), may be readily
isolated together with the associated dependence on the two-particle
invariants:{\rm Div}_n =
-\sum _{i=1}^n \Biggl [
\frac{1}{8 \epsil... | {
"cite_spans": []
} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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4181f4f119547caa5828a1bebe1180ab71dc203f | subsection | 38 | 114 | The BDS conjecture and potential departures form it | For the even case (n=2m) these quantities are given byD_{2m,i} &=&
-\sum _{r=2}^{m-2}
{\rm Li}_2 \Biggl ( 1- \frac{t_i^{[r]} t_{i-1}^{[r+2]}}{t_i^{[r+1]}t_{i-1}^{[r+1]}} \Biggr )
- \frac{1}{2} {\rm Li}_2 \Biggl ( 1- \frac{t_{i}^{[m-1]} t_{i-1}^{[m+1]}}{t_i^{[m]} t_{i-1}^{[m]}}
\Biggr )
\,,
\\
L_{2m,i} &=&
\frac{1}{4}
\... | {
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"raw": "Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, “One loop n point gauge theory amplitudes, unitarity and collinear limits,” Nucl. Phys. B 425, 217 (1994) [arXiv:hep-ph/9403226].",
"sourc... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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72219788ca0d5f988a3db61fbad791bf126b8b19 | subsection | 39 | 114 | The BDS conjecture and potential departures form it | Explicit
calculations show that, at least at
two-loop order, all parity-odd terms in the six-point amplitude
exponentiate following the BDS ansatz.u_1=
\frac{s_{12}s_{45}}{s_{123}s_{345}}~~~~
u_2=
\frac{s_{23}s_{56}}{s_{234}s_{123}}~~~~
u_3=
\frac{s_{34}s_{61}}{s_{345}s_{234}}~~.The number of such cross-ratios – i.e. x... | {
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"raw": "F. Cachazo, M. Spradlin and A. Volovich, “Leading Singularities of the Two-Loop Six-Particle MHV Amplitude,” arXiv:0805.4832 [hep-th].",
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Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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ec52ea3629ed5560a2310bfb32de5d9155d8b852 | subsection | 40 | 114 | The six-point MHV amplitude at two-loops and the BDS ansatz | As previously mentioned, the ABDK/BDS ansatz was constructed based on
explicit calculations of four gluon amplitudes at two- and three-loop
orders as well as of the collinear splitting amplitudes at two-loop
order and was subsequently tested through the calculation of the
five-point amplitude at two-loops. Assuming tha... | {
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Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
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9f108cb472e60a4eb5b8254d4f9144eb70570168 | subsection | 41 | 114 | The six-point MHV amplitude at two-loops and the BDS ansatz | As before, \mu _p and \mu _q denote the(-2\epsilon )-dimensional part of the loop momenta.]The four-dimensional cut-constructible parity-even part of the
amplitude is given entirely in terms of pseudo-conformal integrals
;M_6^{(2),D=4}(\epsilon ) &=& \frac{1}{16} \sum _{12~{\rm perms.}}
\Bigg [
\frac{1}{4} c_1 I^{(1)}(... | {
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Correspondence | [
"Luis F. Alday",
"Radu Roiban"
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b152066350b526fdd0e90cc1a0dff2cdc91e4db7 | subsection | 42 | 114 | The six-point MHV amplitude at two-loops and the BDS ansatz | Their relative coefficients are 0,\pm 1,\pm 2 and \pm 4, which
represents a chance of patterns from the four- and five-point
amplitudes where they were only 0 and \pm 1. It is currently
unclear what is the origin of this change.The remaining parity-even part of the amplitude, which may be
determined by performing gener... | {
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Correspondence | [
"Luis F. Alday",
"Radu Roiban"
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8809430ee77bd8695acfaf96e76d76f0b3721d27 | subsection | 43 | 114 | The six-point MHV amplitude at two-loops and the BDS ansatz | For this
purpose it is instructive to identify the origin of the remainder
function within the arguments that led to this ansatz. In short,
the full structure of collinear limits for n-point amplitudes with
n\ge 6 is somewhat more involved. One may consider limits in
which more than two particles are simultaneously col... | {
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Correspondence | [
"Luis F. Alday",
"Radu Roiban"
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a3a1ee31f3537c292c827347f716c7257b3c72a1 | subsection | 44 | 114 | Scattering amplitudes at strong coupling through the
AdS/CFT duality | The AdS/CFT correspondence , , provides the only direct access to the strong coupling
regime of the {\cal N}=4 SYM; it relates four-dimensional {\cal N}=4 SYM theory and type IIB string theory on AdS_5
\times S^5 space through the identification of string states and
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"raw": "J. M. Maldacena, “The large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [arXiv:hep-th/9711200].",
"so... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
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83a56d6b4d8853061f07577877d6e3fd2e85133e | subsection | 45 | 114 | The general construction | In the presence of an open string sector on the string theory side of
the AdS/CFT correspondence, the calculation of open string scattering
amplitudes in the Poincaré patch is, in principle, conceptually
straightforward: one simply computes the transition amplitude between
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"Luis F. Alday",
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42090f6254d173af798a3734f282196f32989676 | subsection | 46 | 114 | The general construction | Thus, when described in terms of vertex operators placed on the
regulator D3-brane, the scattering process occurs at arbitrarily high
(proper) momenta and fixed angle.In flat space this regime was studied by Gross, Mende and Manes
, . The key result of their
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5bb846f7db7523969250f95ede49958f8e8e01b4 | subsection | 47 | 114 | The general construction | From (REF ) it
follows that the curvature expansion is a series in
\frac{1}{\sqrt{\lambda }} which should therefore reproduce the
strong coupling expansion of scattering amplitudes.To summarize this (long) argument, the leading term in the strong
coupling expansion of a scattering amplitude of states in {\cal N}=4
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] | [
"hep-th"
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65280e54d3c4af4fdf0a3038fb307d4d3c176ee1 | subsection | 48 | 114 | The general construction | The transformation (REF ) has a
similar effect: the zero-mode of the field y corresponding to the
momentum k^\mu (and described by a local vertex operator) is
replaced by a “winding” mode of the field x implying that the
difference between the two endpoints of the string obeys\Delta x^\mu =2 \pi k^\mu ~~,as may be seen... | {
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"doi": "",
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"raw": "S. J. Rey and J. T. Yee, “Macroscopic strings as heavy quarks in large N gauge theory and anti-de Sitter supergravity,” Eur. Phys. J. C 22 (2001) 379 [arXiv:hep-th/9803001].",
"source_ref_id"... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
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47342e0aa0053701d9776055db4bdcc6e8ace491 | subsection | 49 | 114 | The general construction | Thus, one finds one light-like polygon for each
color-ordered amplitude. The corresponding color factor may be
reconstructed from the order of momenta of particles.In the following, we will show in detail how this construction works
for the scattering of four gluons and compare our results with field
theory expectation... | {
"cite_spans": []
} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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57f731e797aa818df9524112ae991f5bba7ef7e0 | subsection | 50 | 114 | Four gluon scattering | The simplest scattering process involves four particles and
is characterized by the usual Mandelstam invariantss=-(k_1+k_2)^2
~,~~~~
t=-(k_2+k_3)^2The discussion in the previous section suggests that, in the strongly
coupled {\cal N}=4 SYM theory, the amplitude for this process is
governed by the minimal surface ending... | {
"cite_spans": []
} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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608c5e48985484a8e18084c8810b00cbd0092829 | subsection | 51 | 114 | The single cusp solution | As a warm up exercise let us discuss the solution near the cusp
where two of the light-like lines meet. This problem was
originally considered in and it will
prove useful for generating the solution relevant for the
four-gluon scattering. The surface can be embedded into an AdS_3
subspace of AdS_5ds^2=\frac{-dx_0^2+dx_... | {
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"doi": "",
"end": 239,
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"raw": "M. Kruczenski, “A note on twist two operators in N = 4 SYM and Wilson loops in Minkowski signature,” JHEP 0212 (2002) 024 [arXiv:hep-th/0210115].",
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Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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c905bfea7b81dcc87ee57a2677d91e1c5504a4f1 | subsection | 52 | 114 | Four cusps solution | The four cusps solution is closely related to the single cusp
solution discussed above. The relevant solution of the Nambu Goto
action can be embedded in a AdS_4 subspace of AdS_5,
parametrized by (r,x_0,x_1,x_2,x_3=0). It is moreover convenient
to fix reparametrization invariance by choosing
(\sigma _1,\sigma _0)=(x_1... | {
"cite_spans": []
} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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08e31004cd5d738344eae168b274f264d3207be6 | subsection | 53 | 114 | Four cusps solution | The Poincaré coordinates ({\bf x},r) are but a particular solution
of the constraint equation (REF ):X^\mu &=&\frac{x^\mu }{r},~~~~~~\mu =0,...,3 \\
X_{-1}+X_4&=&\frac{1}{r},\hspace{21.68121pt}X_{-1}-X_4=\frac{r^2+x_\mu x^\mu }{r}~~.Direct use of the solution (REF ) implies that, in the
embedding coordinates, the
minim... | {
"cite_spans": []
} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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ab38f3ad70d597ff47c825d419e2552f93b12d46 | subsection | 54 | 114 | Four cusps solution | The
solution can be conveniently written asr={a \over \cosh u_1 \cosh u_2+b \sinh u_1
\sinh u_2},~~~~ x_0= {a \sqrt{1+b^2} \sinh u_1 \sinh u_2 \over \cosh u_1 \cosh u_2+b \sinh u_1 \sinh u_2} \\ x_1={a \sinh u_1
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x_2={a \cosh u_1 \sinh u_2 \over \cosh u_1 \... | {
"cite_spans": []
} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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6920097de5bebb094ad845ecb1ead22b25e03870 | subsection | 55 | 114 | Dimensional regularization at strong coupling | Gauge theory amplitudes are regularized by considering the theory
in D=4-2\epsilon dimensions. More precisely (see discussion in
section 2), one starts with {\cal N}=1 in ten dimensions and
then dimensionally reduces to 4-2\epsilon dimensions. For
integer 2\epsilon this is precisely the low energy theory living
on a Dp... | {
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{
"arxiv_id": "",
"doi": "",
"end": 1158,
"openalex_id": "",
"raw": "M. J. Duff, R. R. Khuri and J. X. Lu, “String solitons,” Phys. Rept. 259, 213 (1995) [arXiv:hep-th/9412184].",
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... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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7a714e468ce167dba93183bf94b5ad59587f9dad | subsection | 56 | 114 | Dimensional regularization at strong coupling | The evaluation of integrals leads to S \approx \sqrt{\lambda } \frac{\mu ^\epsilon }{a^\epsilon } ~ _2
F_1\left( \frac{1}{2},-\frac{\epsilon }{2},\frac{1-\epsilon }{2};b^2
\right)~~.Finally, expanding in powers of \epsilon yields the final answer{\cal A}&=&e^{i S}=\exp \left[i
S_{div}+\frac{\sqrt{\lambda }}{8\pi }\left... | {
"cite_spans": [
{
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"doi": "",
"end": 182,
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"raw": "L. F. Alday and J. M. Maldacena, “Gluon scattering amplitudes at strong coupling,” JHEP 0706, 064 (2007) [arXiv:0705.0303 [hep-th]].",
"source_ref_id": "d6cbe432fa71a63c9bfec171d91c197adf561ce... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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7b6ffb0ad9f65b01bafb2e192b79167846abccd0 | subsection | 57 | 114 | Dimensional regularization at strong coupling | Thus, similarly to the weak coupling result of
, , ,
the cusp anomaly equals the large spin limit of the anomalous
dimension of twist-2 operators to all orders in the 1/\sqrt{\lambda }
expansion.f(\lambda )=\frac{\sqrt{\lambda }}{\pi }~~.Moreover, from (REF ) one could extract the strong coupling
behavior of the functi... | {
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"raw": "G. P. Korchemsky, “Asymptotics of the Altarelli-Parisi-Lipatov Evolution Kernels of Parton Distributions,” Mod. Phys. Lett. A 4, 1257 (1989).",
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Correspondence | [
"Luis F. Alday",
"Radu Roiban"
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"hep-th"
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6fcf8c61a5d99317aaae94987f00e5e7ee418925 | subsection | 58 | 114 | Radial Cut-off | A more common regularization scheme for computing minimal areas in
AdS is to introduce a cut-off in the radial direction. The correct
procedure would be to impose the boundary conditions at some small
r=r_c. It turns out, however, that in order to compute the finite
piece as r_c \rightarrow 0 it suffices to use the ori... | {
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"raw": "J. M. Drummond, G. P. Korchemsky and E. Sokatchev, “Conformal properties of four-gluon planar amplitudes and Wilson loops,” Nucl. Phys. B 795 (2008) 385 [arXiv:0707.0243 [hep-th]].",
"source_... | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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4edc1c21c400aade31c3edb67c1c782f2f59bd4a | subsection | 59 | 114 | Structure of infrared poles at strong coupling | Even if the relevant solutions for minimal surfaces for the cases
n>4 are presently unknown, the IR structure of amplitudes at strong
coupling for the general case of n-point amplitudes can easily be
understood.Given the
cusp formed by a pair of neighboring gluons with momenta k_i and
k_{i+1} we associate the kinematic... | {
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Correspondence | [
"Luis F. Alday",
"Radu Roiban"
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"hep-th"
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0.0438537597... |
fcc9be799da383fced316f56a97ae8d5f75a22fc | subsection | 60 | 114 | Structure of infrared poles at strong coupling | For instance, a
simple example is that of a cut-off that takes the value
r_c(x_i) at the ith cusp and varies linearly between cusp and
cusp, in this caseE_{edge}^i
={\sqrt{\lambda } \over 4 \pi } \log ^2{r_c(x_i)
\over r_c(x_{i+1})} | {
"cite_spans": []
} | 10.1016/j.physrep.2008.08.002 | 0807.1889 | Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
Correspondence | [
"Luis F. Alday",
"Radu Roiban"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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