# One-loop effective lagrangians after matching

## Abstract

We discuss the limitations of the covariant derivative expansion prescription advocated to compute the one-loop Standard Model (SM) effective lagrangian when the heavy fields couple linearly to the SM. In particular, one-loop contributions resulting from the exchange of both heavy and light fields must be explicitly taken into account through matching because the proposed functional approach alone does not account for them. We review a simple case with a heavy scalar singlet of charge \(-1\) to illustrate the argument. As two other examples where this matching is needed and this functional method gives a vanishing result, up to renormalization of the heavy sector parameters, we re-evaluate the one-loop corrections to the T-parameter due to a heavy scalar triplet with vanishing hypercharge coupling to the Brout–Englert–Higgs boson and to a heavy vector-like quark singlet of charged 2 / 3 mixing with the top quark, respectively. In all cases we make use of a new code for matching fundamental and effective theories in models with arbitrary heavy field additions.

## 1 Introduction

^{1}In this scenario one must use an effective lagrangian approach to study the low energy effects of possible heavy new resonances beyond the LHC reach:

*n*the dimension of the local operators \(\mathcal {O}^{(n)}_i\) entering in \(\mathcal {L}_n = \sum _i \alpha ^{(n)}_i \mathcal {O}^{(n)}_i\), with \(\alpha ^{(n)}_i\) the corresponding Wilson coefficients. The classification of all operators \(\mathcal {O}^{(6)}_i\) of dimension 6 in \(\mathcal {L}_6\) parameterizing the SM extensions in a model independent way was put forward some time ago [18].

^{2}The coefficients \(\alpha ^{(6)}_i\), which are expected to gather the largest low-energy contributions of the heavy particles, do depend on the particular SM extension considered. As already noticed, the picture emerging from the LHC searches has boosted the revival of the phenomenological interest in the theoretical prediction of the coefficients of the effective lagrangian up to dimension 6 and up to one-loop order, to cope with the expected experimental precision. With this purpose, the procedure to evaluate the contributions of new (heavy) physics to this order has been revised in Ref. [27] (see [28, 29] for related previous works), providing the one-loop corrections for any SM addition with no

*linear*couplings to the SM (light) fields. In this work the evaluation of the one-loop contribution of a generic heavy sector is reduced to an algebraic problem, getting rid of the difficulties associated to the handling of the loop integrals. This is achieved by the clever use of functional methods using the so called covariant derivative expansion (CDE). Its results readily apply to supersymmetric models with R-parity [30, 31, 32, 33], to models in which the heavy sector does not mix linearly with the SM [34] and, in general, to models with a (discrete) symmetry forbidding such linear terms [35, 36]. This work has also been generalized to extend its range of applicability to the case of non-degenerate heavy field masses [37]. However, as already emphasized, although it has been claimed that the method applies in general, it does not fully account for all quantum corrections when the SM addition involves heavy fields coupling linearly to the light (SM) fields. Since in this case there are one-loop corrections resulting from the exchange of both heavy and light fields within the loops which are not included in the algebraic result, which only accounts for the one-loop diagrams exchanging heavy particles alone.

^{3}These contributions can be taken care of, however, performing a full matching with the proper local operators, as argued time ago in Refs. [38, 39].

^{4}

*l*by integrating out a heavy field

*h*,

*M*is the

*h*mass and

*U*is the pertinent function of the light fields

*l*, is solved by iteration (first equation below) also making use of an asymptotic expansion for the non-local operator \(\mathcal{O}^{-1} = - (D^2 + M^2 + U[l])^{-1}\) (second equation)

^{5}:

*N*, in which case the linear term is not eliminated but suppressed to the power \(M^{-2N}\). In practice, one redefines

^{6}One may argue that in the limit \(N \rightarrow \infty \) this coupling goes to zero, but then the expansion of \(\mathcal{O}^{-1}\) is asymptotic and the resulting theory and physical predictions of both limits (integrating to arbitrary momenta for

*N*finite and taking \(N \rightarrow \infty \) afterwards, or \(N \rightarrow \infty \) and integrating to arbitrary momenta) are different. In summary, one can use Eq. (6) keeping track of the linear term suppressed to the corresponding order, or use the quadratic contributions obtained by the CDE with the subsequent matching as indicated in Refs. [38, 39]. We must insist again at this point when using the former approach that although the linear coupling is removed up to order \(M^{-2N}\), it contributes to order \(M^{-2}\) at one loop, as we will explicitly show in the example below.

In the following section we work out an explicit example, reviewing a simple SM extension studied in full detail in Refs. [42, 43], the addition to the SM of one extra heavy charged scalar singlet *h* of mass \(M\ (={\Lambda })\). We want to elaborate on the fact that the purely functional methods used in the CDE to compute the one-loop effective lagrangian (see Ref. [27]) require further matching, as pointed out in Refs. [38, 39, 42, 43]. The one-loop effective action computed with the proposed functional method is entirely governed by the terms in the full lagrangian which are quadratic in the heavy fields. Furthermore, light fields are kept constant through the calculation. Such contributions correspond, diagrammatically, to one-loop diagrams in which only heavy particles circulate in the loop. In contrast, the diagrammatic calculation of the one-loop effective lagrangian by matching the fundamental and effective theories includes those contributions plus those in which both heavy and light particles circulate in the loop (these diagrams depend on the linear couplings of the heavy fields to the SM). Hence, the latter contributions do have to be taken into account but the CDE with its present formulation does not incorporate them. (See footnote 4.)

As another example of physical interest where further matching is required after using the CDE recipe, we discuss in Section 3 the proper one-loop matching for the T-parameter in two other SM extensions with a heavy sector coupling linearly to the SM. In one case the extended model has an extra heavy scalar triplet coupling linearly to the BEH boson, and in the other one the SM is extended with one extra heavy vector-like quark singlet of charge 2/3 mixing with the top quark. In both cases the CDE alone gives a vanishing one-loop contribution (or gives a contribution that can be reabsorbed in the renormalization of the mass of the heavy fields), in contrast with the straightforward diagrammatic computation. For this calculation we make use of MatchMaker [44], a new automated tool for evaluating tree-level and one-loop matching conditions for arbitrary UV completions into effective lagrangians. The result for the examples worked out below agrees with that obtained in Refs. [45, 46] for the SM unbroken phase in the scalar triplet case, and with the result in Refs. [47, 48] for the vector-like quark singlet addition. Section 4 is devoted to a summary. Technical details on the comparison with previous results in the literature are relegated to an appendix.

## 2 Extending the SM with a heavy charged scalar singlet

Let us assume the existence of a heavy scalar singlet *h* of hypercharge \(-1\) and of mass *M*, much larger than the electroweak scale, as in Refs. [42, 43]. Following it, we review in this section the discussion of the need of further matching of the effective field theory (EFT) obtained by the CDE integration of the heavy field with the fundamental theory to one loop, if both must describe the same physics at this order. However, it is not necessary in our case to go through the complete calculation of the one-loop effective lagrangian, which is already worked out in detail in Refs. [42, 43], but it is enough to show that the CDE does not account for a definite physical contribution at this order and, hence, that it must be added through matching.

^{7}

*a*. Besides, since \(\tilde{\ell } = i \tau _2 \ell ^c\), \(f_{ab}\) is antisymmetric in the flavor indices.

In order to substantiate our point with this example we will identify first a physical amplitude for which the predictions in the fundamental theory and in the EFT obtained applying the CDE prescription are different. This means that the EFT mimicking the fundamental theory must be completed with the required local operators as shown in Refs. [42, 43]. We will then show by analogy that this is needed because the implicit field redefinition used in this case when decomposing the heavy field as its classical counterpart plus its fluctuation corresponds to a non-allowed transformation, for the classical field definition involves a non-local operator which renders it to a different theory. This is made apparent observing that successive heavy field redefinitions in the fundamental theory with local transformations suppressing the linear coupling of the heavy field to SM fields up to order \(M^{-2N}\) give the same physical results till the limit \(N \rightarrow \infty \) is taken. Then no linear heavy field coupling to light fields is present at all and the heavy field redefinition involves an infinite sum of terms expanding the non-local operator in Eq. (5), then requiring further matching.

### 2.1 Matching the EFT to the fundamental theory

*l*) effective lagrangian (at the renormalization scale \(\mu = M\) and omitting flavor indices)where dimensional regularization with \(d = 4 - 2\epsilon \) is used and \(\Delta = 1/\epsilon -\gamma _E + \ln 4 \pi \). \(g^\prime \) and \(B_{\mu \nu }\) stand for the hypercharge coupling and field strength, respectively. The first line in Eq. (9) renormalizes the SM lagrangian while the last two are part of the dimension-6 effective lagrangian. What matters to us, however, is that all the terms but the last one correspond to diagrams with only

*h*running in the loop, whereas the last operator corresponds to a diagram with both heavy (

*h*) and light (\(\ell \)) particles running in the loop. Hence, the latter contribution, which is also proportional to the linear

*h*coupling in the full lagrangian in Eq. (8) (proportional to

*f*), is missing in the functional formalism. Therefore, it has to be computed through matching with the fundamental theory. In the effective theory there is no such one-loop contribution to the dimension-6 operator Open image in new window. We will then focus on this term.

*l*) effective lagrangian

Let us now discuss by analogy what happens when we redefine the heavy field in the fundamental theory by successive shifts, Eq. (6), corresponding to keeping only a finite number of terms *N* in the expansion of \(\mathcal{O}^{-1}\) in Eq. (5). The transformation has unit Jacobian but involves a non-local operator in the limit \(N \rightarrow \infty \).

### 2.2 Heavy field redefinition at leading order

*h*, named

*H*after redefining it, equals to first order

*H*momentum.

### 2.3 Heavy field redefinition at next to leading order

## 3 SM extensions with heavy scalars and fermions

The issue raised in the previous section also applies to any SM extension with heavy fields coupling linearly to the light fields. In the following we provide the tree-level and one-loop matching conditions relevant for the calculation of the T-parameter [49] in two of these SM extensions. In both cases the one-loop contribution to the T-parameter entirely arises from terms linear in the heavy fields. Hence, these contributions are missing in the one-loop effective lagrangian obtained by functional methods only.^{8}

### 3.1 SM extension with a heavy scalar triplet

### 3.2 SM extension with a heavy vector-like quark singlet

*T*in the (3, 1, 2 / 3) representation of the SM gauge group. In this case the T-parameter is only generated at one-loop order, which was originally computed in [47] (see also [48, 51] for extensions to vector-like quarks in arbitrary representations). The lagrangian involving the heavy field readswith \(T = T_L + T_R\), and

*L*and

*R*stand for left- and right-handed fermions, respectively.

In this case, since the tree-level contribution vanishes, wave function renormalization gives no further contributions at one loop. Previous calculations of the T-parameter in this model have been performed at the electroweak scale. In order to compare with our calculation, we have to run the Wilson coefficients down to the top quark mass and integrate out the top quark with the anomalous couplings induced by the heavy fermion. We present the details of this computation in Appendix A, showing the agreement with previous results.

## 4 Conclusions

The LHC picture of nature seems to confirm a significant gap between the SM (light fields) and the new layer of physics (heavy fields). This makes the use of EFT compulsory in order to describe (bound) possible small deviations from the SM predictions in the high energy tail of the experimental distributions. Although an EFT with SM symmetries and light fields and arbitrary dimension-6 operators built with them must be in general enough to describe such a scenario (neglecting in this context neutrino masses and the new physics associated to them), it is mandatory to recognize the relations among the different Wilson coefficients of these operators to identify the particular new physics realized in nature. With this purpose, different calculations of the one-loop contributions to the Wilson coefficients of the dimension-6 operators for different SM extensions have been made available using the CDE [27]. SM additions with linear couplings to light fields are treated in the same way as those without them, but in the former case the general results miss extra contributions, which must be added by further matching with the specific fundamental theory [38, 39]. Hence, although there are many phenomenologically relevant SM extensions without such linear terms, as supersymmetric theories with unbroken R-parity or models with universal extra dimensions, and in general theories with a discrete symmetry requiring interactions with only an even number of heavy fields, also many phenomenologically relevant theories include heavy fields with linear couplings to the SM, and they demand further treatment.

This problem and its solution were pointed out some time ago [38, 39], and definite examples have been also worked out in detail [42, 43]. The CDE does not include those linear couplings in loops, in contrast with the fundamental theory. What means that the corresponding contributions must be added through matching. (See footnote 4.) In this paper we elaborate on this issue. Noticing first in a simple case with a heavy charged scalar singlet that the problem arises when we perform the non-local heavy field redefinition implicit in this functional treatment of theories with linear couplings of heavy fields to the SM. The fundamental theory (lagrangian) transformed by a local heavy field redefinition expressible as a series with a finite number of terms (local operators) *N* in general gives the same physical predictions, till the infinite limit \(N \rightarrow \infty \) is taken and the series becomes the asymptotic expansion of a non-local operator with a finite radius of convergence. Then the physical predictions, as well as the resulting theory, are in general different, up to the proper matching.

We have also discussed the beyond the SM contributions to the T-parameter in two other SM extensions with linear couplings of the heavy sector to the light fields, providing the missing pieces in the CDE. They result from the addition of a heavy scalar triplet with vanishing hypercharge and of a heavy vector-like quark of charge 2/3, respectively. As a matter of fact, the CDE gives in both cases a contribution that is either vanishing or can be reabsorbed in the physical definition of the heavy field mass. We obtain perfect agreement with previous calculations in both cases, with Refs. [45, 46] in the scalar triplet case and with Ref. [47] in the vector-like quark one. At any rate, all SM extensions with linear couplings between the heavy and light sectors can require such an extra matching, which can be in general of phenomenological interest (sizable), too.

For our explicit calculations we have made use of the new code MatchMaker [44], aimed at automated calculation of tree-level and one-loop matching conditions in arbitrary extensions of the SM. Details of the code and its use will be presented elsewhere [44].

## Footnotes

- 1.
- 2.
- 3.
There can be one-loop contributions proportional to the linear couplings due to the running of heavy particles alone, which are fully accounted for in the CDE method, see below.

- 4.
- 5.
\(\mathcal{O}^{-1}\) has a finite radius of convergence and the series expansion is only valid for small values of the momenta and of the light fields.

- 6.
In general, there can be further contributions to Eq. (7) due to higher-order terms in

*h*in the lagrangian but they result in higher-order contributions in \(M^{-1}\) at one loop. - 7.
We have not explicitly written an allowed quartic term in the heavy field, \(\alpha |h|^4\), which plays no relevant rôle in our discussion.

- 8.
There is a contribution in our first example proportional to the linear couplings that arise from loops involving only heavy particles and therefore, it is correctly accounted for in the CDE [27]. This term can be reabsorbed by a renormalization of the heavy particle mass as we discuss below.

## Notes

### Acknowledgments

We thank useful discussions with C. Anastasiou, J. R. Espinosa, and A. Lazopoulos, and comments and a careful reading of the manuscript by M. Pérez-Victoria and A. Santamaría. We also thank B. Henning, X. Lu, and H. Murayama for useful correspondence regarding their work. This work has been supported in part by the European Commission through the contract PITN-GA-2012-316704 (HIGGSTOOLS), by the Ministry of Economy and Competitiveness (MINECO), under grant number FPA2013-47836-C3-1/2-P (fondos FEDER), and by the Junta de Andalucía grants FQM 101 and FQM 6552.

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