Abstract
We compute the two-loop four-point form factor of a length-3 half-BPS operator in planar N=4 SYM theory, which belongs to the class of two-loop five-point scattering observables with one off-shell color-singlet leg. A new bootstrapping strategy is developed to obtain this result by starting with an ansatz expanded in terms of master integrals and then solving the master coefficients via various physical constraints. We find that consistency conditions of infrared divergences and collinear limits, together with the cancellation of spurious poles, can fix a significant part of the ansatz. The remaining degrees of freedom can be fixed by one simple type of two-double unitarity cut. Full analytic results in terms of both symbol and Goncharov polylogarithms are provided.
Highlights
Introduction.—The past two decades have seen tremendous progress in our understanding of scattering amplitudes in quantum field theories (QFTs), where the study of the maximally supersymmetric N 1⁄4 4 super-Yang-Mills (SYM) theory has been beneficial, see, e.g., [1,2]
Two-loop corrections for 2 → 3 processes have been at the frontier of amplitude computations which have been under intense study in the last couple of years: based on the advancements of integral computations [3,4,5,6,7,8], a number of amplitudes have been obtained in compact analytic form in both supersymmetric and nonsupersymmetric theories, including all massless cases [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24] and a two-loop five-point amplitude with one massive vector boson [25]
In this work we present an analytic computation of a two-loop four-point form factor in planar N 1⁄4 4 SYM theory, which may be understood as a supersymmetric version of the two-loop Higgs-plus-four-parton scattering, see, e.g., [26]
Summary
We point out that similar ideas have been developed for computing amplitudes [30,31,32,33,34,35,36,37,38,39,40,41,42] and form factors [26,43] based on the symbol techniques [44]. Gð22Þ 1⁄4 Gð12Þjðp1↔p3Þ; ð9Þ c1;i and c2;i are not independent Since both the form factor and integral basis have degree 4, the coefficients ca;i are expected to be pure rational numbers independent of dimensional regularization parameter ε. The integrals IUdBTox2c are of OðεÞ order, so they are irrelevant if one is only interested in getting the ε0 order of the form factor The coefficients of these masters can be fixed by the single type of two-double cuts shown by Fig. 3(d), given by the product of three tree building blocks: F ð30ÞAð40Þ;MHVAð50Þ;MHV. It is a linear combination of two masters used in [7,8] as ð22Þ in which the UT numerators are indicated
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