The last of the simple remainders

TL;DR

This paper computes two-loop form factors of half-BPS operators in N=4 super Yang-Mills for any n, revealing simplified expressions using symbols and analyzing their infrared and collinear behavior.

Contribution

It introduces a method to derive simplified expressions for n-point two-loop form factors of half-BPS operators using symbols, applicable for all n > 2.

Findings

Infrared divergences exponentiate as in amplitudes.

Compact remainder functions are obtained for all n.

Behavior in collinear and soft limits is analyzed, showing deviations from amplitude patterns.

Abstract

We compute the n-point two-loop form factors of the half-BPS operators Tr(phi_{AB}^n) in N=4 super Yang-Mills for arbitrary n >2 using generalised unitarity and symbols. These form factors are minimal in the sense that the n^{th} power of the scalar field in the operator requires the presence of at least n on-shell legs. Infrared divergences are shown to exponentiate as for amplitudes, reproducing the known cusp and collinear anomalous dimensions at two loops. We define appropriate infrared-finite remainder functions and compute them analytically for all n. The results obtained by using the known expressions of the integral functions involve complicated combinations of Goncharov multiple polylogarithms, but we show that much simpler expressions can in fact be derived using the symbol of transcendental functions. For n=3 we find a very compact remainder function expressed in terms of classical polylogarithms only. For arbitrary n>3 we are able to write all the remainder functions in terms of a single compact building block, expressed as a sum of classical polylogarithms augmented by two multiple polylogarithms. The decomposition of the symbol into specific components is crucial in order to single out a natural combination of multiple polylogarithms. Finally, we analyse in detail the behaviour of these minimal form factors in collinear and soft limits, which deviates from the usual behaviour of amplitudes and non-minimal form factors.