Multi-particle finite-volume effects for hexagon tessellations

Abstract

Correlation functions of gauge-invariant composite operators in N = 4 super Yang-Mills theory can be computed by integrability using triangulations. The elementary tile in this process is the hexagon, which should be glued by appropriately inserting resolutions of the identity involving virtual ("mirror") magnons. We consider this problem for five-point functions of protected operators. At one-loop in the 't Hooft coupling, it is necessary to glue three adjacent tiles which involves two virtual magnons scattering among each other. We show that the result can be simplified by using an adapted mirror rotation and employing appropriate summation techniques. The mirror-particle contributions then yield hyperlogarithms of weight two. Finally, we use these results to investigate braiding prescriptions introduced in earlier work on the problem.

Loop equation and exact soft anomalous dimension in $$ \mathcal{N} $$ = 4 super Yang-Mills

“…(2.47) Interestingly this integral resembles multiparticle integrals [31][32][33][34][35][36][37][38][39][40][41] in the hexagon approach to the correlation functions and we will use this connection in our upcoming paper [30] to analyze the defect CFT correlators on the higher-rank Wilson loop.…”

Decagon at two loops

“…Note that the bar over χ means conjugation of all the indices of the fields φ 1 ↔ φ 2 and ψ 1 ↔ ψ 2 . The foremost sum in the expression (2.18) is over the dressings, in other words, the naive basis of the bound states has to be modified by inserting some Z-markers in two different ways denoted plus and minus (the factor of half is because one has to take the average, see [41,42] for a recent discussion about Z-markers). The rule for the plus dressing is the following…”

“…The only known complet analytically evaluation of a two-particle contribution is the simpler case of the tree-level propagator contribution for fishnet theories appearing in [46]. It should be possible to extend these tecniques to more complicated integrals or greatly simplifies the integrand, we hope to address these questions in the future and some recent progresses in evaluating these integrals analytically can be found in [41,42]. In this work, we only generate power series from integrability and fit the results with a basis of two-loop integrals that we evaluate as described in the appendix B.…”

“…So, in this work, we have computed the integrability contributions for five points and then reduced the result to four by identifying a pair of points. This can be improved by a deeper understanding of the integrability integrand and by performing some analytical integrations, see [41,42] for interesting progress in this direction.…”

Three-point functions in $$ \mathcal{N} $$ = 4 SYM at finite Nc and background independence

“…The integrability approach tells us how single-trace correlation functions depend on the 't Hooft coupling λ = N c g 2 YM . However, only the non-extremal correlation functions have been studied, because the non-extremality is related to the so-called bridge length (the number of Wick contractions between a pair of operators), which suppresses the complicated wrapping corrections to the asymptotic formula [13][14][15][16][17].…”