Abstract
Abstract We argue that the natural functions for describing the multi-Regge limit of six-gluon scattering in planar $ \mathcal{N}=4 $ super Yang-Mills theory are the single-valued harmonic polylogarithmic functions introduced by Brown. These functions depend on a single complex variable and its conjugate, (w, w ∗). Using these functions, and formulas due to Fadin, Lipatov and Prygarin, we determine the six-gluon MHV remainder function in the leading-logarithmic approximation (LLA) in this limit through ten loops, and the next-to-LLA (NLLA) terms through nine loops. In separate work, we have determined the symbol of the four-loop remainder function for general kinematics, up to 113 constants. Taking its multi-Regge limit and matching to our four-loop LLA and NLLA results, we fix all but one of the constants that survive in this limit. The multi-Regge limit factorizes in the variables (ν, n) which are related to (w, w ∗) by a Fourier-Mellin transform. We can transform the single-valued harmonic polylogarithms to functions of (ν, n) that incorporate harmonic sums, systematically through transcendental weight six. Combining this information with the four-loop results, we determine the eigenvalues of the BFKL kernel in the adjoint representation to NNLLA accuracy, and the MHV product of impact factors to N3LLA accuracy, up to constants representing beyond-the-symbol terms and the one symbol-level constant. Remarkably, only derivatives of the polygamma function enter these results. Finally, the LLA approximation to the six-gluon NMHV amplitude is evaluated through ten loops.
Highlights
Enormous progress has taken place recently in unraveling the properties of relativistic scattering amplitudes in four-dimensional gauge theories and gravity
We argue that the natural functions for describing the multi-Regge limit of six-gluon scattering in planar N = 4 super Yang-Mills theory are the single-valued harmonic polylogarithmic functions introduced by Brown
We find that the single-valued HPLs (SVHPLs) describe the multi-Regge limit of the one remaining helicity configuration for six-gluon scattering in N = 4 super-Yang-Mills theory, namely the next-to-maximally-helicity violating (MHV) (NMHV) configuration with three negative and three positive gluon helicities
Summary
Enormous progress has taken place recently in unraveling the properties of relativistic scattering amplitudes in four-dimensional gauge theories and gravity. We extract the remaining four-loop coefficient functions, g1(4), h(04) and g0(4), introducing some additional beyond-the-symbol parameters at this stage We use this information to determine the NNLLA BFKL eigenvalue and the N3LLA MHV impact factor, up to these parameters. We include the expressions for the NNLL BFKL eigenvalue and impact factor and the N3LL impact factor in terms of the building blocks in the variables (ν, n) constructed in section 6, as well as a dictionary between these building blocks and the SVHPLs up to weight five
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