Analytic result for the one-loop scalar pentagon integral with massless propagators

Abstract

The method of dimensional recurrences proposed by one of the authors [1,2] is applied to the evaluation of the pentagon-type scalar integral with on-shell external legs and massless internal lines. For the first time, an analytic result valid for arbitrary space-time dimension d and five arbitrary kinematic variables is presented. An explicit expression in terms of the Appell hypergeometric function F_3 and the Gauss hypergeometric function_2F_1, both admitting one-fold integral representations, is given. In the case when one kinematic variable vanishes, the integral reduces to a combination of Gauss hypergeometric functions_2F_1. For the case when one scalar invariant is large compared to the others, the asymptotic values of the integral in terms of Gauss hypergeometric functions_2F_1 are presented in d = 2 - 2 epsilon, 4 - 2 epsilon, and 6 - 2 epsilon dimensions. For multi-Regge kinematics, the asymptotic value of the integral in d = 4 - 2 epsilon dimensions is given in terms of the Appell function F_3 and the Gauss hypergeometric function_2F_1.

21 pages, 1 figure; discussion extended, several equations, appendix and references added, references updated; accepted for publication in Nucl. Phys. B

References in corpus (12)

• Conformal properties of four-gluon planar amplitudes and Wilson loops
• MHV Amplitudes in N=4 Super Yang-Mills and Wilson Loops
• Hexagon Wilson loop = six-gluon MHV amplitude
• The Two-Loop Six-Gluon MHV Amplitude in Maximally Supersymmetric Yang-Mills Theory
• NNLO corrections to 3-jet observables in electron-positron annihilation
• Space-time dimensionality D as complex variable: calculating loop integrals using dimensional recurrence relation and analytical properties with respect to D
• The infrared structure of e+ e- --> 3 jets at NNLO reloaded
• Hypergeometric representation of the two-loop equal mass sunrise diagram
• The one-loop pentagon to higher orders in epsilon
• New relationships between Feynman integrals
• Functional equations for one-loop master integrals for heavy-quark production and Bhabha scattering
• One-loop derivation of the Wilson polygon - MHV amplitude duality

Cited by in corpus (17)

• HYPERDIRE: HYPERgeometric functions DIfferential REduction: MATHEMATICA based packages for differential reduction of generalized hypergeometric functions pFq, F1,F2,F3,F4
• HYPERgeometric functions DIfferential REduction: MATHEMATICA based packages for differential reduction of generalized hypergeometric functions: Horn hypergeometric functions of two variables
• Are Soft Theorems Renormalized?
• From positive geometries to a coaction on hypergeometric functions
• One-loop Amplitudes in Six-Dimensional (1,1) Theories from Generalised Unitarity
• NumExp: Numerical epsilon expansion of hypergeometric functions
• The One-Loop One-Mass Hexagon Integral in D=6 Dimensions
• HYPERgeometric functions DIfferential REduction: Mathematica-based packages for the differential reduction of generalizedhypergeometric functions: Fc hypergeometric function of three variables
• Symbols of One-Loop Integrals From Mixed Tate Motives
• One-loop pentagon integral in $d$ dimensions from differential equations in $\epsilon$-form
• Pentagon integrals to arbitrary order in the dimensional regulator
• The Epsilon Expansion of Feynman Diagrams via Hypergeometric Functions and Differential Reduction
• Multiple Mellin-Barnes integrals with straight contours
• q-derivatives of multivariable q-hypergeometric function with respect to their parameters
• One-loop N = 4 super Yang-Mills scattering amplitudes in d dimensions, relation to open strings and polygonal Wilson loops
• One-loop central-emission vertex for two gluons in $\mathcal{N}=4$ super Yang-Mills theory
• A Probabilistic Angle on One Loop Scalar Integrals