Abstract
Abstract We present the three-loop remainder function, which describes the scattering of six gluons in the maximally-helicity-violating configuration in planar $ \mathcal{N} $ = 4 super-Yang-Mills theory, as a function of the three dual conformal cross ratios. The result can be expressed in terms of multiple Goncharov polylogarithms. We also employ a more restricted class of hexagon functions which have the correct branch cuts and certain other restrictions on their symbols. We classify all the hexagon functions through transcendental weight five, using the coproduct for their Hopf algebra iteratively, which amounts to a set of first-order differential equations. The three-loop remainder function is a particular weight-six hexagon function, whose symbol was determined previously. The differential equations can be integrated numerically for generic values of the cross ratios, or analytically in certain kinematic limits, including the near-collinear and multi-Regge limits. These limits allow us to impose constraints from the operator product expansion and multi-Regge factorization directly at the function level, and thereby to fix uniquely a set of Riemann ζ valued constants that could not be fixed at the level of the symbol. The near-collinear limits agree precisely with recent predictions by Basso, Sever and Vieira based on integrability. The multi-Regge limits agree with the factorization formula of Fadin and Lipatov, and determine three constants entering the impact factor at this order. We plot the three-loop remainder function for various slices of the Euclidean region of positive cross ratios, and compare it to the two-loop one. For large ranges of the cross ratios, the ratio of the three-loop to the two-loop remainder function is relatively constant, and close to −7.
Highlights
For roughly half a century we have known that many physical properties of scattering amplitudes in quantum field theories are encoded in different kinds of analytic behavior in various regions of the kinematical phase space
We present the three-loop remainder function, which describes the scattering of six gluons in the maximally-helicity-violating configuration in planar N = 4 superYang-Mills theory, as a function of the three dual conformal cross ratios
There exists a different kind of bootstrap program, whereby correlation functions can be determined by imposing consistency with the operator product expansion (OPE), crossing symmetry, and unitarity [3, 4]
Summary
For roughly half a century we have known that many physical properties of scattering amplitudes in quantum field theories are encoded in different kinds of analytic behavior in various regions of the kinematical phase space. The application of integrability to the pentagon-transition decomposition of Wilson loops provides, through the OPE, all-loop-order boundary-value information for the problem of determining Wilson loops (or scattering amplitudes) at generic nonzero (interior) values of the cross ratios We will use this information in the six-point case to uniquely determine the three-loop remainder function, not just at symbol level, but at function level as well. In the case of the six-point MHV remainder function at L loops, we require the symbol to be that of a weight-2L parity-even function with full S3 permutation symmetry among the cross ratios. Knowing all such pure functions at weight five will enable us to promote the weight-five quantities S(Reup) and S(Reypu) to well-defined functions, subject to ζ-valued ambiguities that we will fix using physical criteria
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