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.
Highlights
Dimensional regularisation [2] is the most widely used method to regularise the ultraviolet (UV) and infrared (IR) singularities of scattering amplitudes in quantum-field theory
The most widely used tools for the automated calculation of one-loop amplitudes are based on numerical algorithms that build the numerators of loop integrands in Dn = 4 dimensions, while the remnant parts are reconstructed by means of rational counterterms
All two-loop contributions stemming from the interplay of UV poles with the (D−4)-dimensional parts of loop numerators can be reconstructed through insertions of the well known one-loop rational counterterms δR1 into one-loop amplitudes and insertions of two-loop rational counterterms δR2 into tree amplitudes
Summary
Dimensional regularisation [2] is the most widely used method to regularise the ultraviolet (UV) and infrared (IR) singularities of scattering amplitudes in quantum-field theory. To this end, we consider a finite multiplicative renormalisation of couplings, masses and fields, and we study its interplay with the projection of loop numerators to Dn = 4 dimensions. These two operations do not commute at two loops, but the effect of their commutator can be encoded in a new set of scheme- and process-independent one-loop counterterms δK1 This allows us to derive the general formulas (4.62)–(4.64), which describe the scheme dependence of δR2 counterterms as the result of the multiplicative renormalisation of the known δR1 counterterms plus a nontrivial part that can be written as a combination of one-loop renormalisation constants and δK1 counterterms. For convenience of the reader, in appendix B we have collected all relevant UV renormalisation constants for the case of the MS scheme
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
