Abstract
Abstract We present an all-orders formula for the six-point amplitude of planar maximally supersymmetric $ \mathcal{N}=4 $ Yang-Mills theory in the leading-logarithmic approximation of multi-Regge kinematics. In the MHV helicity configuration, our results agree with an integral formula of Lipatov and Prygarin through at least 14 loops. A differential equation linking the MHV and NMHV helicity configurations has a natural action in the space of functions relevant to this problem — the single-valued harmonic polylogarithms introduced by Brown. These functions depend on a single complex variable and its conjugate, w and w * , which are quadratically related to the original kinematic variables. We investigate the all-orders formula in the near-collinear limit, which is approached as |w| → 0. Up to power-suppressed terms, the resulting expansion may be organized by powers of log |w|. The leading term of this expansion agrees with the all-orders double-leading-logarithmic approximation of Bartels, Lipatov, and Prygarin. The explicit form for the sub-leading powers of log |w| is given in terms of modified Bessel functions.
Highlights
A differential equation linking the maximally-helicity violating (MHV) and NMHV helicity configurations has a natural action in the space of functions relevant to this problem — the single-valued harmonic polylogarithms introduced by Brown
We studied the six-point amplitude of planar N = 4 super-Yang-Mills theory in the leading-logarithmic approximation of multi-Regge kinematics
The remainder function assumes a simple form, which we exposed to all loop orders in terms of the single-valued harmonic polylogarithms introduced by Brown
Summary
We consider the six-gluon scattering process g3g6 → g1g5g4g2 where the momenta are taken to be outgoing and the gluons are labeled cyclically in the clockwise direction. To obtain a non-vanishing result, we must consider a physical region in which one of the cross ratios acquires a phase [28] One such region corresponds to the 2 → 4 scattering process described above. An ansatz for the result may be expanded around |w| = 0 and matched term-by-term to the truncated double sum The latter method requires knowledge of the complete set of functions that might arise in this context. A simpler method is to make use of the following differential equation, which may be deduced by comparing the two expressions (2.16) and (2.19), w∗ In principle, solving this equation requires the difficult step of fixing the constants of integration in such a way that single-valuedness is preserved. As discussed in ref. [42], this step becomes trivial when working in the space of SVHPLs, which are the subject of the section
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