Two-loop rational terms in Yang-Mills theories

Abstract

Scattering amplitudes in D dimensions involve particular terms that originate from the interplay of UV poles with the (D − 4)-dimensional parts of loop numerators. Such contributions can be controlled through a finite set of process-independent rational counterterms, which make it possible to compute loop amplitudes with numerical tools that construct the loop numerators in four dimensions. Building on a recent study [1] of the general properties of two-loop rational counterterms, in this paper we investigate their dependence on the choice of renormalisation scheme. We identify a nontrivial form of scheme dependence, which originates from the interplay of mass and field renormalisation with the (D−4)-dimensional parts of loop numerators, and we show that it can be controlled through a new kind of one-loop counterterms. This guarantees that the two-loop rational counterterms for a given renormalisable theory can be derived once and for all in terms of generic renormalisation constants, which can be adapted a posteriori to any scheme. Using this approach, we present the first calculation of the full set of two-loop rational counterterms in Yang-Mills theories. The results are applicable to SU(N) and U(1) gauge theories coupled to nf fermions with arbitrary masses.

Two-loop rational terms for spontaneously broken theories

“…One-loop rational counterterms δR 1 do not depend on the choice of UV renormalisation scheme, while two-loop rational counterterms δR 2 are scheme dependent [29]. This dependence involves a trivial contribution, which corresponds to the naive renormalisation of δR 1 counterterms, plus a non-trivial scheme dependence, which is due to the fact that the multiplicative renormalisation of UV subdivergences does not commute with the projection of loop numerators to D n = 4 dimensions.…”

“…This method was a key ingredient for the automation of one-loop calculations, and the foundations for its extension to two loops have been established in [28,29], where it was shown that renormalised two-loop amplitudes can be constructed through a modified version of the R-operation for loop amplitudes with D n = 4 dimensional integrand numerators. In this approach, the usual one-loop and two-loop counterterms for the subtraction of UV poles are supplemented by corresponding rational counterterms, which reconstruct the contributions that arise from the interplay of UV poles with the (D − 4)-dimensional parts of loop numerators.…”

“…Within a given theoretical model, each potentially UV-divergent 1PI vertex function gives rise to a rational counterterm, which needs to be determined only once and for all. A general method for the determination of two-loop rational counterterms has been presented in [28,29].…”

Calculational Techniques in Particle Theory

“…In the same spirit, but from a different point of view, interesting results have also recently been obtained on extending recursion techniques (based on Feynman diagrams!) to construct two-loop integrands efficiently starting from lower loop amplitudes [16][17][18]. These developments are very promising, and they have made it possible to compute various non-trivial amplitudes, in particular in the very special all-plus configuration [19,20].…”

May the four be with you: novel IR-subtraction methods to tackle NNLO calculations

“…We would like to remark that the method of pulling out the UV behaviour of the amplitude can be traced back to the original papers regarding the BPHZ theorem[1,[76][77][78][79], used in[86][87][88] and has recently been reconsidered in[89] 13. We point out that this procedure of extracting the UV behaviour of multi-loop scattering amplitudes, directly from the Feynman propagators, can be compared with the procedure described in Sect.…”