The seven-gluon amplitude in multi-Regge kinematics beyond leading logarithmic accuracy

Abstract

Abstract:We present an all-loop dispersion integral, well-defined to arbitrary logarithmic accuracy, describing the multi-Regge limit of the 2 → 5 amplitude in planar N = 4 super Yang-Mills theory. It follows from factorization, dual conformal symmetry and consistency with soft limits, and specifically holds in the region where the energies of all produced particles have been analytically continued. After promoting the known symbol of the 2-loop N -particle MHV amplitude in this region to a function, we specialize to N = 7, and extract from it the next-to-leading order (NLO) correction to the BFKL central emission vertex, namely the building block of the dispersion integral that had not yet appeared in the wellstudied six-gluon case. As an application of our results, we explicitly compute the sevengluon amplitude at next-to-leading logarithmic accuracy through 5 loops for the MHV case, and through 3 and 4 loops for the two independent NMHV helicity configurations, respectively.

Cluster adjacency and the four-loop NMHV heptagon

“…In this limit amplitudes develop large logarithms in some of the kinematic variables (1.4), and so at each loop order they reduce to a polynomial of these logarithms, the highest order of which corresponds to the leading logarithmic approximation or LLA, with an obvious generalisation to the (next-to) k -leading logarithmic approximation or N k LLA. The analysis of our four-loop amplitude provides a check of the consistency of our result with the expected structure of the Fourier-Mellin representation described in [35,37] at LLA and NLLA. It then also provides new predictions at the next two logarithmic orders.…”

“…In this final subsection, we will compare our findings for the 4-loop NMHV heptagon in MRK with independent results obtained for the latter to LLA [35] and NLLA [37], based on the Balitsky-Fadin-Kuraev-Lipatov (BFKL) approach [60,61,62]. We will also discuss our new predictions for the amplitude in question up to N 3 LLA.…”

“…In this section, we will consider the multi-Regge limit of our n = 7, 4-loop NMHV symbol, with a two-fold aim: First, to check our calculation against independent results available for the amplitude in this limit up to NLLA [35,37]. And second, to obtain new predictions up to N 3 LLA, which we hope will play an important role in further elucidating the perturbative structure of the heptagon in the limit, and guide its finite-coupling determination, similarly to the hexagon case.…”

“…Let us start by reviewing what has been previously known for the heptagon in the limit. Building on earlier work at LLA, in [37] an all-loop dispersion integral was presented, yielding the 2 → 5 amplitude in MRK to arbitrary logarithmic accuracy. It reads,…”

Six-Gluon amplitudes in planar $$ \mathcal{N} $$ = 4 super-Yang-Mills theory at six and seven loops

“…Because R 6 (u, v, w) is totally 10 The original analyses focused on amplitudes with a one-loop leading OPE contribution. More generally, it can be shown that if the leading OPE contribution an amplitude receives is at k loops, the highest degree of logarithmic divergence is L − k [116], with k = 0 being of course maximal. symmetric, and there are only two totally symmetric quadratic polynomials, we find that that are products of just two odd zeta values (for example, (ζ 3 ) 2 in c (4) 2 ) all have vanishing coefficients.…”