OPE for super loops

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 ).

Section: Jhep10(2014)065mentioning

confidence: 99%

Section: Jhep10(2014)065mentioning

confidence: 99%

Section: Jhep10(2014)065mentioning

confidence: 99%

Section: Jhep10(2014)065mentioning

confidence: 99%

See 2 more Smart Citations

Bootstrapping an NMHV amplitude through three loops

Hippel²

2014

166

332

“…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 [15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

Hexagon functions and the three-loop remainder function

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.

Analytic result for the two-loop six-point NMHV amplitude in $ \mathcal{N} = {4} $ super Yang-Mills theory

Drummond

Henn

2012

162

246

We provide a simple analytic formula for the two-loop six-point ratio function of planar N = 4 super Yang-Mills theory. This result extends the analytic knowledge of multi-loop six-point amplitudes beyond those with maximal helicity violation. We make a natural ansatz for the symbols of the relevant functions appearing in the two-loop amplitude, and impose various consistency conditions, including symmetry, the absence of spurious poles, the correct collinear behaviour, and agreement with the operator product expansion for light-like (super) Wilson loops. This information reduces the ansatz to a small number of relatively simple functions. In order to fix these parameters uniquely, we utilize an explicit representation of the amplitude in terms of loop integrals that can be evaluated analytically in various kinematic limits. The final compact analytic result is expressed in terms of classical polylogarithms, whose arguments are rational functions of the dual conformal cross-ratios, plus precisely two functions that are not of this type. One of the functions, the loop integral Ω (2) , also plays a key role in a new representation of the remainder function R (2) 6 in the maximally helicity violating sector. Another interesting feature at two loops is the appearance of a new (parity odd) × (parity odd) sector of the amplitude, which is absent at one loop, and which is uniquely determined in a natural way in terms of the more familiar (parity even) × (parity even) part. The second non-polylogarithmic function, the loop integralΩ (2) , characterizes this sector. Both Ω (2) andΩ (2) can be expressed as one-dimensional integrals over classical polylogarithms with rational arguments.