Mixed heavy–light matching in the Universal One-Loop Effective Action

Abstract

Recently, a general result for evaluating the path integral at one loop was obtained in the form of the Universal One-Loop Effective Action. It may be used to derive effective field theory operators of dimensions up to six, by evaluating the traces of matrices in this expression, with the mass dependence encapsulated in the universal coefficients. Here we show that it can account for loops of mixed heavy-light particles in the matching procedure. Our prescription for computing these mixed contributions to the Wilson coefficients is conceptually simple. Moreover it has the advantage of maintaining the universal structure of the effective action, which we illustrate using the example of integrating out a heavy electroweak triplet scalar coupling to a light Higgs doublet. Finally we also identify new structures that were previously neglected in the universal results.

Extending the Universal One-Loop Effective Action: heavy-light coefficients

“…[27] we also identified further structures that occur in the quadratic term of the action for specific cases, for example when integrating out heavy particles coupling with "open" covariant derivatives, i.e. covariant derivatives which do not appear in commutators.…”

“…techniques exist for doing so up to one-loop level [18][19][20][21][22][23][24][25][26][27][28][29]. 2 In practice, functional methods have typically been overlooked in favour of the more traditional Feynman diagram approach, where many tools have been developed to ease otherwise complicated calculations.…”

“…[40], that functional methods did not appear to account for one-loop matching involving both heavy and light particles in the loop. This was addressed by us [27] and others [25,26,28], each demonstrating different procedures for treating mixed heavy-light matching in the path integral approach. 3 We emphasised in particular that our method also allowed for the computation of universal terms [27], that could in principle be added to the original UOLEA.…”

Integrating out heavy particles with functional methods: a simplified framework

“…[5], and fixed later on in refs. [7,8], by the use of variants of the functional approach which require additional ingredients in order to subtract the parts of the heavy-light loops which are already accounted for by the one-loop EFT contribution. Here we would like to introduce a more direct method to obtain the one-loop effective theory that builds upon the works of refs.…”

“…Although the rationale and the procedure has been well developed long ago in the literature (see for instance [1,2]), the integration at next-to-leading order in the upper theory, that is to say at one loop, is undergoing lately an intense debate [3][4][5][6][7][8] that, as we put forward in this paper, still allows for simpler alternatives. There are two techniques to obtain the Wilson coefficients of the EFT.…”