Exact result in N=4 SYM theory: Generalised double-logarithmic equation

TL;DR

This paper derives a generalized double-logarithmic equation from seven-loop anomalous dimensions in planar N=4 SYM theory, revealing insights into scattering amplitude asymptotics and pole structures across all perturbative orders.

Contribution

It introduces a new generalized double-logarithmic equation that includes subleading logarithms and can be expressed as rational functions, advancing understanding of anomalous dimension poles.

Findings

Resummation of the equation as rational functions with simple denominators

Identification of pole structures in anomalous dimensions at all orders

Extension of the equation near BFKL-related values

Abstract

We present the new results for the generalised double-logarithmic equation, obtained from the analytical continuation of the seven-loop anomalous dimension of twist-2 operators in the planar N=4 SYM theory. The double-logarithmic equation is related to the special asymptotic of the scattering amplitudes, when the large logarithms of the energy of scattering particles are appeared and should be summed in all order of perturbative theory. These large logarithms correspond to the poles of the analytically continued anomalous dimension. The generalised double-logarithmic equation includes the subleading logarithms. We have found, that the expansion of the generalised double-logarithmic equation can be ressumed in the form of rational functions with simple denominator. The solution of the generalised double-logarithmic equation provides a lot of information about the poles of the analytically continued anomalous dimension in all orders of perturbative theory. We have found also the generalised double-logarithmic equation for the analytically continued anomalous dimension near the value, which is related with BFKL-equation.