An operator product expansion for polygonal null Wilson loops

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%

Bootstrapping an NMHV amplitude through three loops

Dixon

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

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.

“…Furthermore, in the limit where the spin S, and as a consequence ρ 0 tends to infinity, the tips of the string touch the boundary writing light-like segments [17,11]. Consequently, we have seen that the analytic continuation of the large spin, S/ √ λ → ∞, string solution (2.1) produces the square Wilson loop of [10] which describes the scattering of four gluons at strong coupling.…”

Section: -Point Function Of Twist 2 Operatorsmentioning

confidence: 99%

“…We can now analytically continue the world-sheet time τ → iτ while at the same time we perform an analytic continuation to the embedding coordinates of the form [11]:…”

Section: -Point Function Of Twist 2 Operatorsmentioning

confidence: 99%

See 1 more Smart Citation

Two and three-point correlators of operators dual to folded string solutions at strong coupling

Georgiou

2011

A particular analytic continuation of classical string solutions having a single AdS 5 spin is considered. These solutions describe strings tunnelling from the boundary to the boundary of AdS 5 . We use the Legendre transform of the dimensionally regularised action of these solutions to evaluate the 2-point functions of the dual operators, holographically. Subsequently, we evaluate the structure coefficient of correlators involving two operators with spin S and a BPS state, at strong coupling. Our expressions are valid for any value of the AdS spin S/ √ λ and can be applied both at the case of long and short strings. For long strings and at leading order, the structure coefficient is independent of the spin S for twist J operators, while it scales as 1/ log l S √ λ for the case of operators with two equal angular momenta in S 5 . For short strings, the structure coefficient is proportional to the energy E of the string. Finally, we comment on the possibility of relating the strong coupling 3-point function coefficient of three large spin twist 2 operators to the action of the 6-gluon scattering amplitude.