Master integrals for the NNLO virtual corrections to μe scattering in QED: the planar graphs

Abstract

We evaluate the master integrals for the two-loop, planar box-diagrams contributing to the elastic scattering of muons and electrons at next-to-next-to leading-order in QED. We adopt the method of differential equations and the Magnus exponential series to determine a canonical set of integrals, finally expressed as a Taylor series around four space-time dimensions, with coefficients written as combination of generalised polylogarithms. The electron is treated as massless, while we retain full dependence on the muon mass. The considered integrals are also relevant for crossing-related processes, such as di-muon production at e + e − -colliders, as well as for the QCD corrections to top-pair production at hadron colliders.

Double-Higgs boson production in the high-energy limit: planar master integrals

“…However, this limit cannot been taken since the calculation is performed in the Euclidean region with the assumption m 2 H < s < 0 and the results are expressed in terms of multiple polylogarithms, which can not easily be analytically continued into other regions. Similar arguments apply to other recent calculations such as [21] or [22]; analytic results have been obtained in terms of multiple polylogarithms which can in principle be evaluated numerically, but are very unwieldy. This is a another reason why we have decided to perform an expansion in the high energy limit.…”

NNLO QED contribution to the µe → µe elastic scattering

“…The complete amplitude gets contributions from ∼ 10 4 Feynman integrals with maximum rank equal to 4. AID and IBPs reduction with respectively Aida and Kira have decreased significantly the number of integrals [19,20], providing N MI = 120 with max rank equal to 2 (Fig. 3).…”

Two-loop non-planar master integrals for top-pair production in the quark-annihilation channel

“…where 0 n indicates a list of n weights, all equal to 0. On the other hand, an analytic expression for the coefficients B qq and C qq is not yet available, since they depend on certain Feynman integrals families whose analytic solution have been obtained just recently [24,26,28,29]. In the following we consider two non-planar integral families that contribute to B qq and C qq which have been computed in [28].…”

“…In this context, we report on the analytic computation of certain master integrals that are needed to evaluate the two color coefficients in the quark-annihilation channel which are not yet known analytically. Indeed, while part of the relevant master integrals have already been computed in other works [18,19,20,13,14,21,22,23,24] (see also the Loopedia database [25]), the nonplanar master integrals that are described by the two non-planar integral families shown in figure 1 have been computed just recently [26,27,28,29]. In the present work, we summarise the results obtained for the master integrals of topology A [28], which have not been considered analytically so far, and we carry out an independent calculation for the master integrals of Topology B, originally evaluated in [26,27].…”