Dimensional recurrence relations: an easy way to evaluate higher orders of expansion in $\epsilon$

Abstract

Applications of a method recently suggested by one of the authors (R.L.) are presented. This method is based on the use of dimensional recurrence relations and analytic properties of Feynman integrals as functions of the parameter of dimensional regularization, $d$. The method was used to obtain analytical expressions for two missing constants in the $\epsilon$-expansion of the most complicated master integrals contributing to the three-loop massless quark and gluon form factors and thereby present the form factors in a completely analytic form. To illustrate its power we present, at transcendentality weight seven, the next order of the $\epsilon$-expansion of one of the corresponding most complicated master integrals. As a further application, we present three previously unknown terms of the expansion in $\epsilon$ of the three-loop non-planar massless propagator diagram. Only multiple $\zeta$ values at integer points are present in our result.

Talk given at the International Workshop `Loops and Legs in Quantum Field Theory' (April 25--30, 2010, Wörlitz, Germany).

References in corpus (14)

• Algorithm FIRE -- Feynman Integral REduction
• The Multiple Zeta Value Data Mine
• Quark and gluon form factors to three loops
• Three-loop static potential
• Feynman Integral Evaluation by a Sector decomposiTion Approach (FIESTA)
• Static QCD potential at three-loop order
• Space-time dimensionality D as complex variable: calculating loop integrals using dimensional recurrence relation and analytical properties with respect to D
• Resolution of singularities for multi-loop integrals
• The massless higher-loop two-point function
• Fermionic contributions to the three-loop static potential
• Group structure of the integration-by-part identities and its application to the reduction of multiloop integrals
• Nine-Propagator Master Integrals for Massless Three-Loop Form Factors
• Master Integrals for Fermionic Contributions to Massless Three-Loop Form Factors
• Calculation of Massless Feynman Integrals using Harmonic Sums

Cited by in corpus (21)

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• Master Integrals for Four-Loop Massless Propagators up to Transcendentality Weight Twelve
• On the Integrand-Reduction Method for Two-Loop Scattering Amplitudes
• The three-loop unpolarized and polarized non-singlet anomalous dimensions from off shell operator matrix elements
• The three-loop form factor in N=4 super Yang-Mills
• The three-loop polarized singlet anomalous dimensions from off-shell operator matrix elements
• The massless three-loop Wilson coefficients for the deep-inelastic structure functions $F_2,F_L,x{F}_3$ and $g_1$
• On the Computation of Form Factors in Massless QCD with Finite Master Integrals
• Analytic Epsilon Expansions of Master Integrals Corresponding to Massless Three-Loop Form Factors and Three-Loop g-2 up to Four-Loop Transcendentality Weight
• Introducing SummerTime: a package for high-precision computation of sums appearing in DRA method
• On the analytic computation of massless propagators in dimensional regularization
• The Dimensional Recurrence and Analyticity Method for Multicomponent Master Integrals: Using Unitarity Cuts to Construct Homogeneous Solutions
• Numerical Multi-Loop Calculations via Finite Integrals and One-Mass EW-QCD Drell-Yan Master Integrals
• Application of the DRA method to the calculation of the four-loop QED-type tadpoles
• Konishi Form Factor at Three Loop in $\mathcal{N}=4$ SYM
• Planar master integrals for four-loop form factors
• On Epsilon Expansions of Four-loop Non-planar Massless Propagator Diagrams
• DRA method: Powerful tool for the calculation of the loop integrals
• One-loop N = 4 super Yang-Mills scattering amplitudes in d dimensions, relation to open strings and polygonal Wilson loops
• Meromorphic solutions of recurrence relations and DRA method for multicomponent master integrals
• Symmetric reduction of high-multiplicity one-loop integrals and maximal cuts