Combinatorial aspects in the one-loop renormalization of higher derivative theories

Abstract

An efficient way to calculate one-loop counterterms within the Feynman diagrammatic approach and dimensional regularization is to expand the propagators in the integrands of the Feynman integrals around vanishing external momentum. In this way, a generic one-loop diagram is reduced to a sum of vacuum diagrams. The logarithmically divergent part can be extracted by power counting arguments. In case of higher derivative theories, the standard implementation of this procedure on a computer algebra system can become quickly inefficient due to a high proliferation of terms coming from the intermediate replacement of high-rank tensor-integrals with symmetrized product of metric tensors. In this note we present a simple combinatorial solution to this problem which makes the implementation much more efficient. This method is especially relevant in the renormalization of higher derivative theories, but might as well be integrated as a standard routine in existing computer algebra programs designed to automatize Feynman diagrammatic calculations.

One-loop renormalization in Galileon effective field theory

“…This generalizes the previously known results for the off-shell 2-point function [35] and the on-shell 4-point function [39]. The actual calculation has been performed with the recently suggested combinatorial algorithm designed to efficiently extract the one-loop divergences in higher derivative field theories [41]. In view of the large number of terms, which involve O(10 2 ) terms for the 4-point function and O(10 4 ) terms for the 5-point function, we have provided the explicit results for the UV divergences in an ancillary file.…”

“…. , 5e by the general combinatorial algorithm introduced in [41]. There are however certain diagrams, for which it is obvious that the associated integrals identically vanish.…”

“…In [41], a combinatorial formula was derived by which the remaining logarithmically divergent integrals can be easily evaluated…”

“…The coefficient C k corresponds to the combinatorial weight of the kth partition and was derived explicitly in [41],…”

“…We derive the off-shell one-loop divergences for the Galileon in flat spacetime within the traditional momentum space Feynman diagrammatic approach. Since the derivation of JCAP01(2020)014 the one-loop counterterms in the MS scheme only requires the extraction of the ultraviolet divergent (UV) part of the 1PI one-loop diagrams, we make use of the recently proposed combinatorial algorithm, specifically designed for an efficient evaluation of one-loop UV divergences in higher derivative theories [41]. We discuss the hierarchy of loop induced operators in an effective field theory expansion and comment on various resummation schemes.…”