Single-Valued Motivic Periods and Multiple Zeta Values

106

159

We present a recursive method to calculate the α -expansion of disk integrals arising in tree-level scattering of open strings which resembles the approach of Berends and Giele to gluon amplitudes. Following an earlier interpretation of disk integrals as doubly partial amplitudes of an effective theory of scalars dubbed as Z-theory, we pinpoint the equation of motion of Z-theory from the Berends-Giele recursion for its tree amplitudes. A computer implementation of this method including explicit results for the recursion up to order α 7 is made available on the website http://repo.or.cz/BGap.git.

Section: Jhep01(2017)031mentioning

confidence: 99%

Non-abelian Z-theory: Berends-Giele recursion for the α ′-expansion of disk integrals

Mafra

Schlotterer

2017

106

159

“…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 [29][30][31]. 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.…”

Section: Jhep12(2013)049mentioning

confidence: 99%

“…It was argued in ref. [30,31,35] that it is consistent to define, ∆(ζ 2n ) = ζ 2n ⊗ 1 and ∆(π) = π ⊗ 1 . We choose F (1, 1, 1) = 0 for all functions except for the special case Ω (2) (1, 1, 1) = −6ζ 4 .…”

Section: Jhep12(2013)049mentioning

confidence: 99%

Hexagon functions and the three-loop remainder function

Dixon

Drummond

Hippel

et al. 2013

165

398

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 ζ 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.

“…The functions f s k describe the {n − 1, 1} component of a coproduct ∆ associated with a Hopf algebra for iterated integrals [73][74][75][76]. Similarly, each f s k can be differentiated,…”

Section: Jhep10(2014)065mentioning

confidence: 99%

Bootstrapping an NMHV amplitude through three loops

Dixon

Hippel²

2014

166

332

Abstract:We extend the hexagon function bootstrap to the next-to-maximally-helicityviolating (NMHV) configuration for six-point scattering in planar N = 4 super-Yang-Mills theory at three loops. Constraints from theQ differential equation, from the operator product expansion (OPE) for Wilson loops with operator insertions, and from multi-Regge factorization, lead to a unique answer for the three-loop ratio function. The three-loop result also predicts additional terms in the OPE expansion, as well as the behavior of NMHV amplitudes in the multi-Regge limit at one higher logarithmic accuracy (NNLL) than was used as input. Both predictions are in agreement with recent results from the flux-tube approach. We also study the multi-particle factorization of multi-loop amplitudes for the first time. We find that the function controlling this factorization is purely logarithmic through three loops. We show that a function U , which is closely related to the parity-even part of the ratio function V , is remarkably simple; only five of the nine possible final entries in its symbol are non-vanishing. We study the analytic and numerical behavior of both the parity-even and parity-odd parts of the ratio function on simple lines traversing the space of cross ratios (u, v, w), as well as on a few two-dimensional planes. Finally, we present an empirical formula for V in terms of elements of the coproduct of the six-gluon MHV remainder function R 6 at one higher loop, which works through three loops for V (four loops for R 6 ).