Abstract
We evaluate the master integrals for the two-loop non-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 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. In particular our results, together with the planar master integrals recently computed, represent the complete set of functions needed for the evaluation of the photonic two-loop virtual next-to-next-to-leading order QED corrections to μe → μe and e+e− → μ+μ−.
Highlights
Exploited to constrain non-standard eeμμ interactions [6], and the current estimates suggest that the knowledge of the NNLO QED differential cross section is needed, as QED itself produces an asymmetry starting at NLO
We adopt the method of differential equations and the Magnus exponential to determine a canonical set of integrals, expressed as a Taylor series around four spacetime dimensions, with coefficients written as combination of generalised polylogarithms
We complete the task of determining all functions required by the NNLO QED virtual photonic corrections to μe scattering, by evaluating the two-loop integrals coming from non-planar four-point Feynman diagrams
Summary
The general solution of the system of DEQs in terms of GPLs, which is obtained from the integration of eq (3.4), must be complemented by a suitable set of boundary conditions. We can generate the DEQs for the analogous triangle integrals with p24 = 0, and u = 0 and an off-shell leg p21, solve them by using as an integration base-point the regular point p21 = 0, and extract the boundary values of I11 and I12 by means of eq (3.18) The details of this computation are reported in appendix A. As the numerical evaluation of those integrals is challenging, we identified an alternative set of independent MIs that are quasi finite [29] in d = 6 The latter have been computed semi-numerically by means of an in-house algorithm: starting from the Feynman parametrisation of the integrals, we carried out as many analytic integration as possible, until we reached a form where the left over multivariate integral could be numerically evaluated by means of Gauss quadrature.
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