Hexagon functions and the three-loop remainder function

Open Access 0 Simons Center for Geometry and Physics, Stony Brook University , Stony Brook NY 11794, U.S.A 1 CERN, Geneva 23, Switzerland 2 LAPTH, CNRS et Universite de Savoie , F-74941 Annecy-le-Vieux Cedex, France 3 SLAC National Accelerator Laboratory, Stanford University , Stanford, CA 94309, U.S.A 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. 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 multiRegge factorization directly at the function level, and thereby to fix uniquely a set of Riemann valued constants that could not be fixed at the level of the symbol. The nearcollinear 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. Contents 1 Introduction 2 Extra-pure functions and the symbol of R6(3) 3 Hexagon functions as multiple polylogarithms 3.1 Symbols 3.2 Multiple polylogarithms 3.3 The coproduct bootstrap 3.4 Constructing the hexagon functions 4 Integral representations 4.1 General setup 4.2 Constructing the hexagon functions 4.3 Constructing the three-loop remainder function 5 Collinear limits 5.1 Expanding in the near-collinear limit 5.2 Examples 5.3 Fixing most of the parameters 5.4 Comparison to flux tube OPE results 6 Multi-Regge limits 6.1 Method for taking the MRK limit 6.2 Examples 6.3 Fixing d1, d2, and 7 Final formula for R6(3) and its quantitative behavior 7.1 The line (u, u, 1) 7.2 The line (1, 1, w) 7.3 The line (u, u, u) 7.4 Planes of constant w 7.5 The plane u + v w = 1 7.6 The plane u + v + w = 1 7.7 The plane u = v 7.8 The plane u + v = 1 8 Conclusions A Multiple polylogarithms and the coproduct A.1 Multiple polylogarithms A.2 The Hopf algebra of multiple polylogarithms C Coproduct of Rep Introduction 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 idea that the amplitudes of a theory can be reconstructed (or bootstrapped) from basic physical principles such as unitarity, by exploiting the link to the analytic behavior, became known as the Analytic S-Matrix program (see e.g. ref. [1]). In the narrow resonance approximation, crossing symmetry duality led to the Veneziano formula [2] for tree-level scattering amplitudes in string theory. In conformal field theories, 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]. This program was most successful in two-dimensional conformal field theories, for which conformal symmetry actually extends to an infinite-dimensional Virasoro symmetry [5]. However, the basic idea can be applied in any dimension and recent progress has been made in applying the program to conformal field theories in three and four dimensions [68]. In recent years, the scattering amplitudes of the planar N = 4 super-Yang-Mills theory have been seen to exhibit remarkable properties. In particular, the amplitudes exhibit dual conformal symmetry and a duality to light-like polygonal Wilson loops [914]. The dual description and its associated conformal symmetry mean that CFT techniques can be applied to calculating scattering amplitudes. In particular, the idea of imposing consistency with the OPE applies. However, since the dual observables are non-local Wilson loop operators, a different OPE, involving the near-collinear limit of two sides of the light-like polygon, has to be employed [1518]. Dual conformal symmetry implies that the amplitudes involving four or five particles are fixed, because there are no invariant cross ratios that can be formed from a five-sided light-like polygon [1921]. The four- and five-point amplitudes are governed by the BDS ansatz [22]. The amplitudes not determined by dual conformal symmetry begin at six points. When the external gluons are in the maximally-helicity-violating (MHV) configuration, such amplitudes can be expressed in terms of the BDS ansatz, which contains all of the infrared divergences and transforms anomalously under dual conformal invariance, and a so-called remainder function [23, 24], which only depends on dual-conformally-invariant cross ratios. In the case of non-MHV amplitudes, one can define the ratio function [25], which depends on the cross ratios as well as dual superconformal invariants. For six external gluons, the remainder and ratio functions are described in terms of functions of three dual conformal cross ratios. At low orders in perturbation theory, these latter functions can be expressed in terms of multiple polylogarithms. In general, multiple polylogarithms are functions of many variables that can be defined as iterated integrals over rational kernels. A particularly useful feature of such functions is that they can be classified according to their symbols [2628], elements of the n-fold tensor product of the algebra of rational functions. The integer n is referred to as the transcendental weight or degree. The symbol can be defined iteratively in terms of the total derivative of the function, or alternatively, in terms of the maximally iterated coproduct by using the Hopf structure conjecturally satisfied by multiple polylogarithms [2931]. Complicated functional identities among polylogarithms become simple algebraic relations satisfied by their symbols, making the symbol a very useful tool in the study of polylogarithmic functions. The symbol can miss terms in the function that are pro (...truncated)