Two-loop and n-loop vertex corrections for eikonal diagrams with massive partons

We present the ultraviolet poles and finite terms of two-loop vertex corrections for diagrams with massive partons in the eikonal approximation. We discover and prove that the results for a set of the corrections generalize to n-loop eikonal diagrams. These results will enhance theoretical understanding and allow greater calculational accuracy for the many partonic processes where the eikonal approximation is applied. Perturbation theory for partonic processes entails the calculation of increasingly complicated loop diagrams as one moves to higher orders. At present, many QCD processes and some beyond the Standard Model processes have been calculated to next-to-leading order (NLO) in the strong coupling αS, and they require the evaluation of one-loop diagrams. Higher-order calculations involve diagrams at two-loops and higher [1, 2]. Beyond the use of standard fixed-order calculations in perturbation theory, it is possible to evaluate higher-order contributions to partonic cross sections, that are dominant in certain regions of phase space, using a panoply of resummation methods and related techniques. The eikonal approximation has been an extremely useful tool in these other approaches to perturbative calculations. The eikonal approximation is valid for emission of soft gluons from partons in the hard scattering and leads to a simplified form of the Feynman rules. When the gluon momentum goes to zero, the usual Feynman rules for the quark propagator and quark-gluon vertex in the diagram in Figure 1 simplify as follows: ū(p) (−igsT c F ) γ μ i(p/+ k/+m) (p+ k)2 −m2 + iǫ → ū(p) gsT c F γ μ p/+m 2p · k + iǫ = ū(p) gsT c F v v · k + iǫ (1) with v a dimensionless vector, p ∝ v, g s = 4παs, and T c F the generators of SU(3) in the fundamental representation.