Comments on all-loop constraints for scattering amplitudes and Feynman integrals

Abstract We comment on the status of “Steinmann-like” constraints, i.e. all-loop constraints on consecutive entries of the symbol of scattering amplitudes and Feynman integrals in planar $$ \mathcal{N} $$ N = 4 super-Yang-Mills, which have been crucial for the recent progress of the bootstrap program. Based on physical discontinuities and Steinmann relations, we first summarize all possible double discontinuities (or first-two-entries) for (the symbol of) amplitudes and integrals in terms of dilogarithms, generalizing well-known results for n = 6, 7 to all multiplicities. As our main result, we find that extended-Steinmann relations hold for all finite integrals that we have checked, including various ladder integrals, generic double-pentagon integrals, as well as finite components of two-loop NMHV amplitudes for any n; with suitable normalization such as minimal subtraction, they hold for n = 8 MHV amplitudes at three loops. We find interesting cancellation between contributions from rational and algebraic letters, and for the former we have also tested cluster-adjacency conditions using the so-called Sklyanin brackets. Finally, we propose a list of possible last-two-entries for MHV amplitudes up to 9 points derived from $$ \overline{Q} $$ Q ¯ equations, which can be used to reduce the space of functions for higher-point MHV amplitudes.

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SURPRISING SYMMETRY OF SCATTERING AMPLITUDES IN GAUGE THEORIES

Modern Physics Letters A ◽

10.1142/s021773230900108x ◽

2009 ◽

Vol 24 (35n37) ◽

pp. 2868-2881 ◽

Cited By ~ 1

Author(s):

G. P. KORCHEMSKY

Keyword(s):

Scattering Amplitudes ◽

String Duality ◽

Recent Progress ◽

Gauge Theories ◽

Strongly Coupled ◽

Superconformal Symmetry ◽

Yang Mills ◽

Arbitrary Coupling

I will review a recent progress in computing scattering amplitudes in strongly coupled gauge theories — a fascinating subject which has been recently boosted by the formulation of the gauge/string duality in maximally supersymmetric Yang–Mills theory. In addition to the conventional symmetry of the underlying Lagrangian, the scattering amplitudes in this theory exhibit a new, dual superconformal symmetry. This symmetry is powerful enough to completely determine the scattering amplitudes for arbitrary coupling in a suitably defined limit.

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SURPRISING SYMMETRY OF SCATTERING AMPLITUDES IN GAUGE THEORIES

International Journal of Modern Physics A ◽

10.1142/s0217751x10048664 ◽

2010 ◽

Vol 25 (02n03) ◽

pp. 351-366

Author(s):

G. P. KORCHEMSKY

Keyword(s):

Scattering Amplitudes ◽

String Duality ◽

Recent Progress ◽

Gauge Theories ◽

Strongly Coupled ◽

Superconformal Symmetry ◽

Yang Mills ◽

Arbitrary Coupling

I review a recent progress in computing scattering amplitudes in strongly coupled gauge theories - a fascinating subject which has been recently boosted by the formulation of the gauge/string duality in maximally supersymmetric Yang-Mills theory. In addition to the conventional symmetry of the underlying Lagrangian, the scattering amplitudes in this theory exhibit a new, dual superconformal symmetry. This symmetry is powerful enough to completely determine the scattering amplitudes for arbitrary coupling in a suitably defined limit.

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Bootstrapping octagons in reduced kinematics from A2 cluster algebras

Journal of High Energy Physics ◽

10.1007/jhep10(2021)084 ◽

2021 ◽

Vol 2021 (10) ◽

Author(s):

Song He ◽

Zhenjie Li ◽

Yichao Tang ◽

Qinglin Yang

Keyword(s):

Scattering Amplitudes ◽

Minkowski Spacetime ◽

Dimensional Subspace ◽

Cluster Algebras ◽

Wilson Loops ◽

Two Dimensional ◽

Feynman Integrals ◽

Yang Mills ◽

Two Sectors ◽

Special Case

Abstract Multi-loop scattering amplitudes/null polygonal Wilson loops in $$ \mathcal{N} $$ N = 4 super-Yang-Mills are known to simplify significantly in reduced kinematics, where external legs/edges lie in an 1 + 1 dimensional subspace of Minkowski spacetime (or boundary of the AdS3 subspace). Since the edges of a 2n-gon with even and odd labels go along two different null directions, the kinematics is reduced to two copies of G(2, n)/T ∼ An−3. In the simplest octagon case, we conjecture that all loop amplitudes and Feynman integrals are given in terms of two overlapping A2 functions (a special case of two-dimensional harmonic polylogarithms): in addition to the letters v, 1 + v, w, 1 + w of A1 × A1, there are two letters v − w, 1 − vw mixing the two sectors but they never appear together in the same term; these are the reduced version of four-mass-box algebraic letters. Evidence supporting our conjecture includes all known octagon amplitudes as well as new computations of multi-loop integrals in reduced kinematics. By leveraging this alphabet and conditions on first and last entries, we initiate a bootstrap program in reduced kinematics: within the remarkably simple space of overlapping A2 functions, we easily obtain octagon amplitudes up to two-loop NMHV and three-loop MHV. We also briefly comment on the generalization to 2n-gons in terms of A2 functions and beyond.

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Algebraic singularities of scattering amplitudes from tropical geometry

Journal of High Energy Physics ◽

10.1007/jhep04(2021)002 ◽

2021 ◽

Vol 2021 (4) ◽

Author(s):

James Drummond ◽

Jack Foster ◽

Ömer Gürdoğan ◽

Chrysostomos Kalousios

Keyword(s):

Scattering Amplitudes ◽

Natural Language ◽

Tropical Geometry ◽

Cluster Algebras ◽

Finite Alphabet ◽

Square Root ◽

Finite Sets ◽

Yang Mills ◽

Square Roots ◽

Algebraic Singularities

Abstract We address the appearance of algebraic singularities in the symbol alphabet of scattering amplitudes in the context of planar $$ \mathcal{N} $$ N = 4 super Yang-Mills theory. We argue that connections between cluster algebras and tropical geometry provide a natural language for postulating a finite alphabet for scattering amplitudes beyond six and seven points where the corresponding Grassmannian cluster algebras are finite. As well as generating natural finite sets of letters, the tropical fans we discuss provide letters containing square roots. Remarkably, the minimal fan we consider provides all the square root letters recently discovered in an explicit two-loop eight-point NMHV calculation.

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Two-loop rational terms in Yang-Mills theories

Journal of High Energy Physics ◽

10.1007/jhep10(2020)016 ◽

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.

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Celestial superamplitude in $$ \mathcal{N} $$ = 4 SYM theory

Journal of High Energy Physics ◽

10.1007/jhep08(2021)031 ◽

2021 ◽

Vol 2021 (8) ◽

Author(s):

Hongliang Jiang

Keyword(s):

Field Theory ◽

Conformal Field Theory ◽

Scattering Amplitudes ◽

Momentum Space ◽

Ward Identity ◽

Yang Mills ◽

Conformal Field ◽

New Perspective ◽

Celestial Sphere

Abstract Celestial amplitude is a new reformulation of momentum space scattering amplitudes and offers a promising way for flat holography. In this paper, we study the celestial amplitudes in $$ \mathcal{N} $$ N = 4 Super-Yang-Mills (SYM) theory aiming at understanding the role of superconformal symmetry in celestial holography. We first construct the superconformal generators acting on the celestial superfield which assembles all the on-shell fields in the multiplet together in terms of celestial variables and Grassmann parameters. These generators satisfy the superconformal algebra of $$ \mathcal{N} $$ N = 4 SYM theory. We also compute the three-point and four-point celestial super-amplitudes explicitly. They can be identified as the conformal correlation functions of the celestial superfields living at the celestial sphere. We further study the soft and collinear limits which give rise to the super-Ward identity and super-OPE on the celestial sphere, respectively. Our results initiate a new perspective of understanding the well-studied $$ \mathcal{N} $$ N = 4 SYM amplitudes via 2D celestial conformal field theory.

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