Single-mass-scale diagrams: construction of a basis for the ε-expansion

Abstract

Exploring the idea of Broadhurst on the "sixth root of unity" we present an ansatz for construction of a basis of transcendental numbers for the ε-expansion of single mass scale diagrams with two particle massive cut. As example, several new two-and three-loop master integrals are calculated.

High-precision ϵ-expansions of massive four-loop vacuum bubbles

“…Some results are already available in the literature: two-loop singlescale self-mass master integrals and three-loop vacuum master integrals were evaluated analytically respectively in Refs. [12,13,14,15] and Ref. [16], for all possible combinations of masses 0 and m; among the numerous applications, for example, three-loop vacuum integrals were used for calculating the relation [11] between the MS and the on-shell quark mass at order α 3 s .…”

lsjk—a C++ library for arbitrary-precision numeric evaluation of the generalized log-sine functions

“…The special interest in the calculation of the multiloop Feynman diagrams [7,8,10,19] is related to the value of the generalized log-sine functions at the "sixth root of unity", θ =…”

“…The argument x is supposed to satisfy 0 < x < π, parameter k should be 0 ≤ k ≤ 9 7 . lsjk can be used as any regular library, e.g.…”

“…More examples of the one-, two-and threeloop Feynman diagrams whose ε-expansion contains the generalized log-sine functions can be found in [8][9][10][11][12][13]. The generalized log-sine functions are also related to the derivatives of the generalized hypergeometric functions with respect to their parameters [13,14] and to the multiple Euler-Zagier sums [7,8]. A collection of the known analytical properties of these functions can be found in Appendix A of [8] (see also [15]).…”

Three-loop electroweak correction to the rho parameter in the large Higgs mass limit

“…In the second step, the counterterms δ (1) m and δ (2) m are replaced with (37)-(38) and the master integrals are expanded in ε, using results from refs. [24,25]. In the expansions, singular terms of order 1/ε j , j = 1, 2, 3, 4, appear and cancel each other.…”