Abstract
We present the calculation of the master integrals needed for the two-loop QCDxEW corrections to $ q + \bar{q} \to l^- + l^+$ and $ q + \bar{q}' \to l^- + \overline{\nu} \, , $ for massless external particles. We treat W and Z bosons as degenerate in mass. We identify three types of diagrams, according to the presence of massive internal lines: the no-mass type, the one-mass type, and the two-mass type, where all massive propagators, when occurring, contain the same mass value. We find a basis of 49 master integrals and evaluate them with the method of the differential equations. The Magnus exponential is employed to choose a set of master integrals that obeys a canonical system of differential equations. Boundary conditions are found either by matching the solutions onto simpler integrals in special kinematic configurations, or by requiring the regularity of the solution at pseudo-thresholds. The canonical master integrals are finally given as Taylor series around d=4 space-time dimensions, up to order four, with coefficients given in terms of iterated integrals, respectively up to weight four.
Highlights
Drell-Yan processes constitute the SM background in searches of New Physics, involving for instance new vector boson resonances, Z and W, originating from GUT extensions of the SM
We presented the calculation of the master integrals (MIs) needed for the virtual QCD×EW two-loop corrections to the Drell-Yan scattering processes, q + q → l− + l+, q + q → l− + ν, for massless external particles
We identified a basis of 49 MIs and evaluated them with the method of the differential equations
Summary
In which the line integral of the product of k matrix-valued 1-forms dA is understood in the sense of Chen’s iterated integrals [117] (see [118] and the pedagogical lectures [119]) and γ is a piecewise-smooth path γ : [0, 1] t → γ(t) = (γ1(t), γ2(t)) , Such that γ(0) = x0 and γ(1) = x. In the limit x → x0, the line shrinks to a point and all the path integrals in eq (3.7) vanish, so that I( , x) → I( , x0), i.e. the integration constants have a natural interpretation as initial values, and the path-ordered exponential as evolution operator. Which emphasizes that the iterated integrals in (3.16) are in general functionals of the path γ
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