Abstract
We review the systematics of Mandelstam cut contributions to planar scattering amplitudes in the multi-Regge limit. Isolating the relevant cut terms, we explain how the BFKL expansion can be used to construct the perturbative n-point multi-Regge limit amplitude in certain kinematic regions from a finite number of basic building blocks. At three loops and at leading logarithmic order, two building blocks are required. Their symbols are extracted from the known three-loop six-point and seven-point symbols for general kinematics. The new seven-point building block is constructed in terms of single-valued multiple polylogarithms to the extent it can be determined using the symbol as well as further symmetry and consistency constraints. Beyond the leading logarithmic order, the subleading and sub-subleading terms require two and one further building block, respectively. The latter could either be reconstructed from further perturbative data, or from BFKL integrals involving yet-unknown corrections to the central emission block.
Highlights
We review the systematics of Mandelstam cut contributions to planar scattering amplitudes in the multi-Regge limit
In the expansion around large logarithms, they admit a reconstruction of perturbative amplitudes to any multiplicity, once the BFKL building blocks are known
It should be emphasized that the identity has a two-fold meaning: on the one hand, it holds at the level of the complete remainder function’s symbol. It holds at the level of full functions once one restricts the remainder function to its simplest cut contribution as in (3.5), neglecting the Regge pole terms as well as higher Regge cut contributions such as the ones in the last line of (2.23)
Summary
The cut diagram stands for all contributions from two-Reggeon bound state exchange in the t5 channel This picture generalizes to higher multiplicities: the planar n-point multi-Regge. Passing to the multi-Regge limit, and stripping off the universal absolute value, the BDS amplitude reduces to a region-dependent phase factor From the latter, one can separate off a conformally invariant, infrared finite part exp(iδnρ), which again is regiondependent, and contains the finite part of the one-loop Regge cut terms [11, 27]. Just like the pole terms of the four-point and five-point amplitudes, they enjoy the virtue of Regge factorization, in the following sense: the multi-Reggeon bound states that propagate in the intermediate t-channels are governed by the BFKL [1,2,3] and BKP [4, 5] equations The solutions to these equations are most naturally expressed in terms of their SL(2, C) representation labels (n, ν). At a given loop order, the coefficient of each monomial in log(εk) is a function of the kinematics that exclusively depends on the complex anharmonic ratios wk (2.7)
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