Feynman integrals via hyperlogarithms

Abstract

This talk summarizes recent developments in the evaluation of Feynman integrals using hyperlogarithms. We discuss extensions of the original method, new results that were obtained with this approach and point out current problems and future directions. Loops and Legs in Quantum Field Theory

Non-abelian Z-theory: Berends-Giele recursion for the α ′-expansion of disk integrals

“…[50,[66][67][68][69][70] and references therein. Moreover, their symbolic computation have been recently implemented in computer programs [71,72].…”

“…The same kind of shuffle operations including G(0, 0; z) = 1 2 G(0; z) 2 allows to reduce cases with multiple terminal labels 0 such as G(A, a n−2 , 0, 0; z) with a n−2 = 0 to convergent polylogarithms and polynomials in G(0; z) [69]. Analogous statements based on a regularization prescription for G(z; z) can be made for upper-endpoint divergences in integrals like G(z, z, .…”

Renormalization group functions of $\phi^4$ theory in the MS-scheme to six loops

“…Graphs which admit such a good order are called linearly reducible and can be computed, order by order in ε, with the algorithm described in [11,21,22] and implemented in [10,49,51]. In [48] it was found that all massless propagators with ≤ 4 loops are indeed linearly reducible, and the same method was even applied to some 6-loop p-integrals [49,50]. The remaining challenge to the straightforward application of parametric integration in the linearly reducible case is the presence of subdivergences.…”

“…Instead, in this paper we present a simple new approach based on parametric integration using hyperlogarithms [21,22,49] which allows us to compute the counterterms of all 6-loop 2-and 4-point graphs analytically, with the sole exception of a single graph which is, however, well-known [20,56,58]. We use the Maple TM implementation HyperInt [51] of this method, which was presented at the preceding conference [50].…”

“…We use the Maple TM implementation HyperInt [51] of this method, which was presented at the preceding conference [50]. 1 Our new ingredient is an efficient and general procedure to generate convergent integrands for integrals which initially have subdivergences.…”