Planar Amplitudes in Maximally Supersymmetric Yang-Mills Theory

Abstract

The collinear factorization properties of two-loop scattering amplitudes in dimensionally-regulated N = 4 super-Yang-Mills theory suggest that, in the planar ('t Hooft) limit, higher-loop contributions can be expressed entirely in terms of one-loop amplitudes. We demonstrate this relation explicitly for the two-loop four-point amplitude and, based on the collinear limits, conjecture an analogous relation for n-point amplitudes. The simplicity of the relation is consistent with intuition based on the AdS/CFT correspondence that the form of the large-Nc L-loop amplitudes should be simple enough to allow a resummation to all orders.PACS numbers: 11.15. Bt, 11.25.Db, 11.25.Tq, 11.30.Pb, 11.55.Bq, 12.38.Bx Four-dimensional quantum field theories are extremely intricate, and generically have complicated perturbative expansions in addition to non-perturbative contributions to physical quantities. Gauge theories are interesting in that numerous cancellations occur. This renders perturbative computations more tractable, and their results simpler, than one might otherwise expect. The Maldacena conjecture [1] implies that a special gauge theory is simpler yet: the 't Hooft (planar) limit of maximally supersymmetric four-dimensional gauge theory, or N = 4 super-Yang-Mills theory (MSYM). The conjecture states that the strong coupling limit of this conformal field theory (CFT) is dual to weakly-coupled gravity in fivedimensional anti-de Sitter (AdS) space. The AdS/CFT correspondence is remarkable in taking a seemingly intractable strong coupling problem in gauge theory and relating it to a weakly-coupled gravity theory, which can be evaluated perturbatively. There have been multiple quantitative tests of this correspondence, using observables protected by supersymmetry (see e.g. ref.[2]). Because of the different domains of validity of coupling expansions on the gauge and gravity sides, quantitative comparisons involving unprotected quantities rely at present on an additional expansion parameter, such as in the large-J ("spin") limit of BMN operators [3,4].In this latter context, the AdS/CFT correspondence can be used to motivate a search for patterns in the perturbative expansion of planar MSYM. Intuitively, observables in the strongly coupled limit of this theory should be relatively simple because of the weakly-coupled gravity interpretation. Yet infinite orders in the perturbative expansion, as well as non-perturbative effects, contribute to the strong coupling limit. How might such a complicated expansion organize itself into a simple result? For quantities protected by supersymmetry, nonrenormalization theorems, or zeros in the perturbative series, are one possibility. Another possibility, for unprotected quantities, is some iterative perturbative structure allowing for a resummation. There have been some hints of an iterative structure developing in the correlation functions of gauge-invariant composite operators [5], but the exact structure, if it exists, is not yet clear.Amplitudes for scattering of on-shell (mass...

Simplifying multiloop integrands and ultraviolet divergences of gauge theory and gravity amplitudes

“…= istu M tree 4 and changing the gauge coupling to the gravitational coupling, we have 2) where N i (k j , l j ) are the gauge-theory numerator factors given in appendix B. In contrast to the sYM amplitudes, potential snail contributions from graphs 83-85 vanish identically, as expected from our heuristic argument in section II D, and confirmed by an analysis of the unitarity cuts.…”

“…Ignoring the snail graph contributions in N = 4 sYM theory would lead to incorrect values for the UV divergences. 2 Since we are interested in this paper in the coefficient of the UV divergences, these snail graphs must be included.…”

“…It may soon be possible to completely determine all planar scattering amplitudes in this theory, for all values of the coupling, going far beyond the (now thoroughly understood) cases of four and five external gluons [2]. Much of this progress has been surveyed recently [3].…”

Analytic Form of the Two-Loop Planar Five-Gluon All-Plus-Helicity Amplitude in QCD

“…Their expression was initially conjectured in [25], tested numerically in [26], and proven in [27] from a Ward identity for dual conformal symmetry. Our calculation is the first direct analytical one, and we find complete agreement with the above references.…”

Comments on gluon 6-point scattering amplitudes in 𝒩 = 4 SYM at strong coupling

“…Using this prescription, [1] computed the 4-point gluon scattering amplitude at strong coupling in N = 4 at large N , and compared it with the conjectured exact result of Bern, Dixon, and Smirnov (BDS) [8] (see also [9]). The BDS conjecture states that the planar contributions to scattering amplitudes of N = 4 SYM have an iterative structure (at least, for MHV amplitudes): the higher-loop amplitudes are determined by the one-loop amplitude and some functions of the coupling constant.…”