scholarly journals Scattering Amplitudes, the AdS/CFT Correspondence, Minimal Surfaces, and Integrability

2010 ◽  
Vol 2010 ◽  
pp. 1-31
Author(s):  
Luis F. Alday
Keyword(s):  
Scattering Amplitudes ◽  
Integral Equations ◽  
Strong Coupling ◽  
Minimal Surfaces ◽  
Four Dimensions ◽  
Detailed Exposition

We focus on the computation of scattering amplitudes of planar maximally supersymmetric Yang-Mill in four dimensions at strong coupling by means of the AdS/CFT correspondence and explain how the problem boils down to the computation of minimal surfaces in AdS in the first part of this paper. In the second part of this review we explain how integrability allows to give a solution to the problem in terms of a set of integral equations. The intention of the review is to give a pedagogical, rather than very detailed, exposition.

scholarly journals GLUON SCATTERING AMPLITUDES FROM GAUGE/STRING DUALITY AND INTEGRABILITY

2013 ◽  
Vol 21 ◽  
pp. 1-21
Author(s):  
YUJI SATOH
Keyword(s):  
Perturbation Theory ◽  
Scattering Amplitudes ◽  
String Duality ◽  
Bethe Ansatz ◽  
Strong Coupling ◽  
Minimal Surfaces ◽  
Wilson Loops ◽  
Thermodynamic Bethe Ansatz ◽  
Yang Mills ◽  
Sine Gordon Model

We discuss gluon scattering amplitudes/null-polygonal Wilson loops of [Formula: see text] super Yang-Mills theory at strong coupling based on the gauge/string duality and its underlying integrability. We focus on the amplitudes/Wilson loops corresponding to the minimal surfaces in AdS3, which are described by the thermodynamic Bethe ansatz equations of the homogeneous sine-Gordon model. Using conformal perturbation theory and an interesting relation between the g-function (boundary entropy) and the T-function, we derive analytic expansions around the limit where the Wilson loops become regular-polygonal. We also compare our analytic results with those at two loops, to find that the rescaled remainder functions are close to each other for all multi-point amplitudes.

scholarly journals QQ-system and non-linear integral equations for scattering amplitudes at strong coupling

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Davide Fioravanti ◽  
Marco Rossi ◽  
Hongfei Shu
Keyword(s):  
Scattering Amplitudes ◽  
Integral Equations ◽  
Bethe Ansatz ◽  
Strong Coupling ◽  
Strong Coupling Limit ◽  
Wilson Loops ◽  
Functional Relations ◽  
Non Linear ◽  
Linear Integral ◽  
Linear Integral Equations

Abstract We provide the two fundamental sets of functional relations which describe the strong coupling limit in AdS3 of scattering amplitudes in $$ \mathcal{N} $$ N = 4 SYM dual to Wilson loops (possibly extended by a non-zero twist l): the basic QQ-system and the derived TQ-system. We use the TQ relations and the knowledge of the main properties of the Q-function (eigenvalue of some Q-operator) to write the Bethe Ansatz equations, viz. a set of (‘complex’) non-linear-integral equations, whose solutions give exact values to the strong coupling amplitudes/Wilson loops. Moreover, they have some advantages with respect to the (‘real’) non-linear-integral equations of Thermodynamic Bethe Ansatz and still reproduce, both analytically and numerically, the findings coming from the latter. In any case, these new functional and integral equations give a larger perspective on the topic also applicable to the realm of $$ \mathcal{N} $$ N = 2 SYM BPS spectra.

scholarly journals The Wilson-loop d log representation for Feynman integrals

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Song He ◽  
Zhenjie Li ◽  
Yichao Tang ◽  
Qinglin Yang
Keyword(s):  
Scattering Amplitudes ◽  
Differential Equations ◽  
Integral Equations ◽  
Wilson Loop ◽  
Boundary Term ◽  
Feynman Integrals ◽  
Four Dimensions ◽  
Loop Integral ◽  
Six Dimensions

Abstract We introduce and study a so-called Wilson-loop d log representation of certain Feynman integrals for scattering amplitudes in $$ \mathcal{N} $$ N = 4 SYM and beyond, which makes their evaluation completely straightforward. Such a representation was motivated by the dual Wilson loop picture, and it can also be derived by partial Feynman parametrization of loop integrals. We first introduce it for the simplest one-loop examples, the chiral pentagon in four dimensions and the three-mass-easy hexagon in six dimensions, which are represented by two- and three-fold d log integrals that are nicely related to each other. For multi-loop examples, we write the L-loop generalized penta-ladders as 2(L − 1)-fold d log integrals of some one-loop integral, so that once the latter is known, the integration can be performed in a systematic way. In particular, we write the eight-point penta-ladder as a 2L-fold d log integral whose symbol can be computed without performing any integration; we also obtain the last entries and the symbol alphabet of these integrals. Similarly we study the symbol of the seven-point double-penta-ladder, which is represented by a 2(L − 1)-fold integral of a hexagon; the latter can be written as a two-fold d log integral plus a boundary term. We comment on the relation of our representation to differential equations and resumming the ladders by solving certain integral equations.

scholarly journals Two-loop rational terms in Yang-Mills theories

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Jean-Nicolas Lang ◽  
Stefano Pozzorini ◽  
Hantian Zhang ◽  
Max F. Zoller
Keyword(s):  
Scattering Amplitudes ◽  
Gauge Theories ◽  
A Posteriori ◽  
Four Dimensions ◽  
Yang Mills ◽  
General Properties ◽  
Finite Set ◽  
Numerical Tools ◽  
Loop Amplitudes

Abstract Scattering amplitudes in D dimensions involve particular terms that originate from the interplay of UV poles with the (D − 4)-dimensional parts of loop numerators. Such contributions can be controlled through a finite set of process-independent rational counterterms, which make it possible to compute loop amplitudes with numerical tools that construct the loop numerators in four dimensions. Building on a recent study [1] of the general properties of two-loop rational counterterms, in this paper we investigate their dependence on the choice of renormalisation scheme. We identify a nontrivial form of scheme dependence, which originates from the interplay of mass and field renormalisation with the (D−4)-dimensional parts of loop numerators, and we show that it can be controlled through a new kind of one-loop counterterms. This guarantees that the two-loop rational counterterms for a given renormalisable theory can be derived once and for all in terms of generic renormalisation constants, which can be adapted a posteriori to any scheme. Using this approach, we present the first calculation of the full set of two-loop rational counterterms in Yang-Mills theories. The results are applicable to SU(N) and U(1) gauge theories coupled to nf fermions with arbitrary masses.

Multiple scattering of plane harmonic elastic waves in an infinite solid by an arbitrary configuration of obstacles

1969 ◽  
Vol 66 (2) ◽  
pp. 469-480 ◽  
Author(s):  
P. J. Barratt
Keyword(s):  
Scattering Amplitudes ◽  
Integral Equations ◽  
Multiple Scattering ◽  
Elastic Waves ◽  
Iterative Procedure ◽  
Elastic Solid ◽  
Single Scattering ◽  
Far Field ◽  
Rigid Spheres ◽  
Arbitrary Configuration

AbstractThe multiple scattering of plane harmonic P and S waves in an infinite elastic solid by arbitrary configurations of obstacles is considered. Integral equations relating the far-field multiple scattering amplitudes to the corresponding single scattering functions are obtained and asymptotic solutions are found by an iterative procedure. The scattering of a plane harmonic P wave by two identical rigid spheres is investigated.

On the integral equations for three-body scattering amplitudes

1971 ◽  
Vol 1 (25) ◽  
pp. 1057-1060 ◽  
Author(s):  
V. Vanzani ◽  
G. Cattapan
Keyword(s):  
Scattering Amplitudes ◽  
Integral Equations ◽  
Three Body
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