Abstract
The multi-Regge limit of scattering amplitudes in strongly-coupled mathcal{N} = 4 super Yang-Mills is described by the large mass limit of a set of thermodynamic Bethe ansatz (TBA) equations. A non-trivial remainder function arises in this setup in certain kinematical regions due to excitations of the TBA equations which appear during the analytic continuation into these kinematical regions. So far, these analytic continuations were carried out on a case-by-case basis for the six- and seven-gluon remainder function. In this note, we show that the set of possible excitations appearing in any analytic continuation in the multi-Regge limit for any number of particles is rather constrained. In particular, we show that the BFKL eigenvalue of any possible Reggeon bound state is a multiple of the two-Reggeon BFKL eigenvalue appearing in the six-gluon case.
Highlights
About scattering amplitudes with more external gluons, at least in general kinematics, where only the two-loop symbol is known for any number of gluons [17,18,19], and the remainder function of the eight- and nine-gluon amplitude was determined up to two loops in [20]
The appearance of BFKL eigenvalues and impact factors is closely linked to the concept of so-called Mandelstam regions, which are kinematical regions reached by analytic continuations of the scattering amplitude in the kinematic variables
We will not specify the paths of analytic continuation for the auxiliary parameters explicitly, as we argue that, as long as the endpoint of the analytic continuation corresponds to a Mandelstam region, the possible BFKL eigenvalues governing the remainder function can be determined without knowing the explicit path chosen for the auxiliary parameters
Summary
The dependence of the remainder function on the cross ratios is described by three terms,. To explain the other two terms in eq (2.3), we introduce the auxiliary parameters ms =: |ms|eiφs and Cs, which are connected with the cross ratios (2.2) as described below. We introduce 3n − 15 functions Ya,s(θ), which depend on the auxiliary parameters and a complex parameter θ These Y-functions satisfy the non-linear integral equations log Ya,s(θ) = −|ma,s| cosh θ + Ca,s +. Aper = Aper(ms) is a polynomial in the auxiliary parameters ms, which we spell out for particular amplitudes in later sections This procedure allows to us to calculate the remainder function for given values of the auxiliary parameters ms and Cs. we still need to connect the auxiliary parameters with the cross ratios which are used to describe the kinematics. To calculate the remainder function in practice, we would need to specify the values of the cross ratios we are interested in and try to find those values of the auxiliary parameters that reproduce the behavior of the cross ratios via eq (2.9)
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