Multi-Regge limit of the two-loop five-point amplitudes in $$ \mathcal{N} $$ = 4 super Yang-Mills and $$ \mathcal{N} $$ = 8 supergravity

Abstract

In previous work, the two-loop five-point amplitudes in $$ \mathcal{N} $$ N = 4 super Yang-Mills theory and $$ \mathcal{N} $$ N = 8 supergravity were computed at symbol level. In this paper, we compute the full functional form. The amplitudes are assembled and simplified using the analytic expressions of the two-loop pentagon integrals in the physical scattering region. We provide the explicit functional expressions, and a numerical reference point in the scattering region. We then calculate the multi-Regge limit of both amplitudes. The result is written in terms of an explicit transcendental function basis. For certain non-planar colour structures of the $$ \mathcal{N} $$ N = 4 super Yang-Mills amplitude, we perform an independent calculation based on the BFKL effective theory. We find perfect agreement. We comment on the analytic properties of the amplitudes.

Pentagon functions for scattering of five massless particles

“…As an example, in section 6 we investigate the behavior of pentagon functions on boundaries of the physical phase space where all five momenta belong to a three-dimensional subspace, but none of the external momenta are soft or collinear. Confirming the observation of [59], we find that certain weight three and weight four pentagon functions contributing to non-planar master integrals are divergent on these boundaries.…”

“…To completely fix the solutions of DEs one needs to provide the initial values -values of the master integrals at X 0 . Building on the results of [30,59], we obtain a complete set of the initial values of all DEs at X 0 from the requirement of absence of unphysical singularities and identify a generating set of 19 algebraically-independent transcendental constants. We then employ the shuffle algebra of iterated integrals (see e.g.…”

“…The latter is a highly nontrivial task for integrals depending on many scales. In particular, some of the non-planar master integrals develop discontinuities and even divergences inside the physical region on subvarities of ∆ = 0 without collinear or soft momenta [59]; we discuss this in section 6. In this paper we follow an alternative approach which was advocated in [53].…”

“…It corresponds to the kinematics with all five momenta lying in a three-dimensional hyperplane. Crossing of the ∆ = 0 variety separating Im( 5 ) > 0 and Im( 5 ) < 0 regions is not innocuous since the master integrals could diverge there [59].…”

“…In [59], by integrating the DEs, the initial values at X 0 were transported to a point X Regge ∈ P 0 in the Regge asymptotic regime. In this regime the pentagon alphabet enormously simplifies, which leads to a more simple form of the initial values at X Regge as compared to X 0 .…”