Renormalized $\epsilon$-finite master integrals and their virtues: the three-loop self energy case

Abstract

Loop diagram calculations typically rely on reduction to a finite set of master integrals in $4-2\epsilon$ dimensions. It has been shown that for any problem, the masters can be chosen so that their coefficients are finite as $\epsilon \to 0$. I propose a definition of renormalized $\epsilon$-finite master integrals, which incorporate ultraviolet divergence subtractions in a specific way. A key advantage of this choice is that in expressions for physical observables, expansions to positive powers in $\epsilon$ are never needed. As an example, I provide the subtractions for general three-loop self-energy integrals. The differential equations method is used to compute numerically the renormalized $\epsilon$-finite master integrals for arbitrary external momentum invariant, in special cases with internal masses equal to a single scale or zero. These include the ones necessary for the three-loop QCD corrections to the self-energies of the W, Z, and Higgs bosons. In principle, the same method should provide for numerical computation of general three-loop self energies with any masses.

33 pages. v2 includes one additional even threshold topology

References in corpus (13)

• Algorithm FIRE -- Feynman Integral REduction
• FIRE5: a C++ implementation of Feynman Integral REduction
• Lectures on differential equations for Feynman integrals
• Presenting LiteRed: a tool for the Loop InTEgrals REDuction
• Adler Function, Bjorken Sum Rule, and the Crewther Relation to Order alpha_s^4 in a General Gauge Theory
• Reducing differential equations for multiloop master integrals
• Gauss hypergeometric function: reduction, epsilon-expansion for integer/half-integer parameters and Feynman diagrams
• Effective potential at three loops
• Pole mass of the W boson at two-loop order in the pure MS-bar scheme
• Standard and epsilon-finite Master Integrals for the rho-Parameter
• $\epsilon$-regular basis for non-polylogarithmic multiloop integrals and total cross section of the process $e^{+}{e}^{-}\to 2(Q\bar{Q})$
• Differential Reduction Algorithms for Hypergeometric Functions Applied to Feynman Diagram Calculation
• TVID: Three-loop Vacuum Integrals from Dispersion relations