Abstract
The advent of efficient numerical algorithms for the construction of one-loop amplitudes has played a crucial role in the automation of NLO calculations, and the development of similar algorithms at two loops is a natural strategy for NNLO automation. Within a numerical framework the numerator of loop integrals is usually constructed in four dimensions, and the missing rational terms, which arise from the interplay of the (D − 4)-dimensional parts of the loop numerator with 1/(D − 4) poles in D dimensions, are reconstructed separately. At one loop, such rational terms arise only from UV divergences and can be restored through process-independent local counterterms. In this paper we investigate the behaviour of rational terms of UV origin at two loops. The main result is a general formula that combines the subtraction of UV poles with the reconstruction of the associated rational parts at two loops. This formula has the same structure as the R-operation, and all poles and rational parts are described through a finite set of process-independent local counterterms. We also present a general formula for the calculation of all relevant two-loop rational counterterms in any renormalisable theory based on one-scale tadpole integrals. As a first application, we derive the full set of two-loop rational counterterms for QED in the Rξ -gauge.
Highlights
Contributions stemming from the interplay of 1/ε poles with the (D − 4)-dimensional parts of loop integrands
When the loop numerator is restricted to four dimensions, the contributions associated with its (D − 4)-dimensional counterpart, referred to as N, need to be reconstructed with a different technique
In order to open the door to the usage of two-loop numerical algorithms in Dn = 4 numerator dimensions, in this paper we have presented a general analysis of rational N -contributions at two loops
Summary
For the regularisation of UV divergences in this paper we use the ’t Hooft-Veltman scheme [1], where external states are four-dimensional, while loop momenta as well as the metric tensors and Dirac matrices inside the loops live in. In Dn = D dimensions, all relevant ingredients of loop numerators will be decomposed into four-dimensional parts and (D − 4)-dimensional remnants. Contractions of Lorentz vectors in D dimensions are decomposed as. For the integration measure in loop-momentum space we use the shorthand dq = μ2ε dDq (2π)D. where μ is the scale of dimensional regularisation and will be identified with the renormalisation scale. (2.4) will be typically written as qμ = qμ + qμ This leads to contractions of objects that carry different kinds of indices and have to be understood as follows, AμBμ = AμBμ , AμBμ = 0.
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