Origin of the Six-Gluon Amplitude in Planar N = 4 Supersymmetric Yang-Mills Theory

112

We study a special class of four-point correlation functions of infinitely heavy half-BPS operators in planar N = 4 SYM which admit factorization into a product of two octagon form factors. We demonstrate that these functions satisfy a system of nonlinear integro-differential equations which are powerful enough to fully determine their dependence on the 't Hooft coupling and two cross ratios. At weak coupling, solution to these equations yields a known series representation of the octagon in terms of ladder integrals. At strong coupling, we develop a systematic expansion of the octagon in the inverse powers of the coupling constant and calculate accompanying expansion coefficients analytically. We examine the strong coupling expansion of the correlation function in various kinematical regions and observe a perfect agreement both with the expected asymptotic behavior dictated by the OPE and with results of numerical evaluation. We find that, surprisingly enough, the strong coupling expansion is Borel summable. Applying the Borel-Padé summation method, we show that the strong coupling expansion correctly describes the correlation function over a wide region of the 't Hooft coupling.

Section: Jhep07(2020)219supporting

confidence: 72%

Octagon at finite coupling

Belitsky

Korchemsky

2020

112

“…Another example of an interesting kinematic region is the "origin", which for the sixpoint case entails taking all three cross-ratios to zero. In this case, perturbative data [56] and all-orders arguments [68] show that the logarithm of the MHV amplitude depends only quadratically on logarithms of the cross ratios. Beyond one loop, the quadratic dependence gets multiplied by transcendental constants (in this case, Riemann zeta values), which can all be expressed in terms of a "tilted" version [68] of the Beisert-Eden-Staudacher (BES) kernel controlling the cusp anomalous dimension at finite coupling [69].…”

Section: Jhep10(2020)031 Introductionmentioning

confidence: 89%

“…We define the Collinear-Origin (CO) surface, which interpolates between the heptagon origin and the soft/collinear limits. The heptagon origin is the seven-point analog of the hexagon origin [68], where as many of the cross ratios u i go to zero as possible. (The u i at six and seven points all contain two-particle invariants in their numerator, and so the origin can be defined by maximizing how many two-particle invariants vanish.)…”

Section: Boundary Of Integration -The Collinear-origin Surfacementioning

confidence: 99%

See 1 more Smart Citation

Lifting heptagon symbols to functions

Dixon

Liu

2020

Self Cite

Seven-point amplitudes in planar $$ \mathcal{N} $$ N = 4 super-Yang-Mills theory have previously been constructed through four loops using the Steinmann cluster bootstrap, but only at the level of the symbol. We promote these symbols to actual functions, by specifying their first derivatives and boundary conditions on a particular two-dimensional surface. To do this, we impose branch-cut conditions and construct the entire heptagon function space through weight six. We plot the amplitudes on a few lines in the bulk Euclidean region, and explore the properties of the heptagon function space under the coaction associated with multiple polylogarithms.

“…Fredholm determinants often provide a representation for several physical quantities that go well beyond the g-functions. Examples include two-point functions in some 2d integrable models [44], the S 3 partition functions of supersymmetric gauge theories [45][46][47][48][49], nonperturbative formulation of topological string [50], certain correlation functions and amplitudes in N = 4 SYM [51][52][53][54][55][56][57][58], N = 2 supersymmetric index in two dimensions [59], and the partition function of 2d polymers [60,61]. In the last two cases, it was pointed out by Zamolodchikov [61] that such determinants are unexpectedly related to the solution of a set of integral equations reminiscent of the Thermodynamic Bethe Ansatz.…”

Section: Tracy-widom Tbamentioning

confidence: 99%

Functional equations and separation of variables for exact g-function

Caetano

Komatsu

2020

The g-function is a measure of degrees of freedom associated to a boundary of two-dimensional quantum field theories. In integrable theories, it can be computed exactly in a form of the Fredholm determinant, but it is often hard to evaluate numerically. In this paper, we derive functional equations — or equivalently integral equations of the thermodynamic Bethe ansatz (TBA) type — which directly compute the g-function in the simplest integrable theory; the sinh-Gordon theory at the self-dual point. The derivation is based on the classic result by Tracy and Widom on the relation between Fredholm determinants and TBA, which was used also in the context of topological string. We demonstrate the efficiency of our formulation through the numerical computation and compare the results in the UV limit with the Liouville CFT. As a side result, we present multiple integrals of Q-functions which we conjecture to describe a universal part of the g-function, and discuss its implication to integrable spin chains.