Hexagonalization of correlation functions II: two-particle contributions

Abstract

In this work, we compute one-loop planar five-point functions in N = 4 superYang-Mills using integrability. As in the previous work, we decompose the correlation functions into hexagon form factors and glue them using the weight factors which depend on the cross-ratios. The main new ingredient in the computation, as compared to the four-point functions studied in the previous paper, is the two-particle mirror contribution. We develop techniques to evaluate it and find agreement with the perturbative results in all the cases we analyzed. In addition, we consider next-to-extremal four-point functions, which are known to be protected, and show that the sum of one-particle and two-particle contributions at one loop adds up to zero as expected. The tools developed in this work would be useful for computing higher-particle contributions which would be relevant for more complicated quantities such as higher-loop corrections and non-planar correlators.

The octagon as a determinant

“…The correlation function for four such operators can be computed non-perturbatively by the hexagonalisation method designed first in [7] for the computation of the three-point functions and adjusted for the four-point functions in [8][9][10]. The sum over virtual states prescribed by the hexagonalisation simplifies for heavy fields (large K) and particular choices of the polarisations because some of the channels of propagation of the virtual particles are suppressed.…”

“…4 The characters are expressed in terms of the variables φ, ϕ, θ parametrizing the cross sections in the coordinate and in the flavour space. As suggested in [8,10] and elaborated upon in [2], one should consider two ways of dressing the mirror basis with Z markers and then take the average between the two choices. The matrix part takes the form of a sum of two terms…”

“…where χ ± a are the characters of the antisymmetric representations of psu(2|2), [8,10] (cf eq. (47) of [8])…”

“…In this paper we address the computation of the octagon form factor which appears as a building block for the evaluation of a class of four-point correlation functions of single-trace half-BPS operators in N = 4 planar SYM [2]. We are doing so using integrability-inspired techniques [3][4][5][6], and in particular the recently proposed geometric decomposition of correlation functions into hexagons [7][8][9][10][11][12]. The hexagons are form factors for non-local operators creating curvature defects.…”

“…In this sum the dependence on the polarisations is carried by the weight factors X n [8,10], to be recalled in the following, the coefficients c j 1 ...jn are rational numbers, and the (conveniently normalised) ladder integrals are given, for |z| < 1, by…”