New differential equations for on-shell loop integrals

2013

We introduce and solve an infinite class of loop integrals which generalises the well-known ladder series. The integrals are described in terms of single-valued polylogarithmic functions which satisfy certain differential equations. The combination of the differential equations and single-valued behaviour allow us to explicitly construct the polylogarithms recursively. For this class of integrals the symbol may be read off from the integrand in a particularly simple way. We give an explicit formula for the simplest generalisation of the ladder series. We also relate the generalised ladder integrals to a class of vacuum diagrams which includes both the wheels and the zigzags.

“…They were used in [19] to prove relations between different integrals. Similar equations for on-shell integrals were derived and studied in [20].…”

Section: Jhep02(2013)092mentioning

confidence: 92%

Generalised ladders and single-valued polylogs

2013

“…The symbol of Ω (2) can be deduced [34] from the differential equations it satisfies [75,76]. There are only three distinct final entries of the symbol of Ω (2) (u, v, w), namely…”

Section: Jhep12(2013)049mentioning

confidence: 99%

“…(4.45) because in this case Ω (2) vanishes at the lower endpoint, Ω (2) (1, 0, 0) = 0 [34,75]. Continuing onward, we construct the remaining functions of the hexagon basis in an iterative fashion, using the above methods.…”

Section: Jhep12(2013)049mentioning

confidence: 99%

Hexagon functions and the three-loop remainder function

Dixon

Hippel

et al. 2013

Self Cite

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 final entry should be expressible in terms of only six letters rather than all nine. This constraint comes from a supersymmetric formulation of the polygonal Wilson loop [63] and also from examining the differential equations obeyed by one-loop [64][65][66] and multi-loop integrals [22,23] related to N = 4 super-Yang-Mills scattering amplitudes. The final-entry constraint on the symbol corresponds to a differential constraint we shall impose at function level, which also has a simple description in terms of the coproduct of the function.…”

Section: Jhep06(2014)116mentioning

confidence: 99%

The four-loop remainder function and multi-Regge behavior at NNLLA in planar $ \mathcal{N} $ = 4 super-Yang-Mills theory

Dixon

Duhr

et al. 2014

Self Cite

147

252

We present the four-loop remainder function for six-gluon scattering with maximal helicity violation in planar N = 4 super-Yang-Mills theory, as an analytic function of three dual-conformal cross ratios. The function is constructed entirely from its analytic properties, without ever inspecting any multi-loop integrand. We employ the same approach used at three loops, writing an ansatz in terms of hexagon functions, and fixing coefficients in the ansatz using the multi-Regge limit and the operator product expansion in the near-collinear limit. We express the result in terms of multiple polylogarithms, and in terms of the coproduct for the associated Hopf algebra. From the remainder function, we extract the BFKL eigenvalue at next-to-next-to-leading logarithmic accuracy (NNLLA), and the impact factor at N 3 LLA. We plot the remainder function along various lines and on one surface, studying ratios of successive loop orders. As seen previously through three loops, these ratios are surprisingly constant over large regions in the space of cross ratios, and they are not far from the value expected at asymptotically large orders of perturbation theory.