Standard Model Effective Field Theory: Integrating out a Generic Scalar

We consider renormalisable models extended in the scalar sector by a generic scalar field in addition to the standard model Higgs boson field, and work out the effective theory for the latter in the decoupling limit. We match the full theory onto the effective theory at tree and one-loop levels, and concentrate on dimension-6 operators of the Higgs and electroweak gauge fields induced from such matching. The Wilson coefficients of these dimension-6 operators from tree-level matching are further improved by renormalisation group running. For specific $SU(2)_L$ representations of the scalar field, some"accidental"couplings with the Higgs field are allowed and can lead to dimension-6 operators at tree and/or one-loop level. Otherwise, two types of interaction terms are identified to have only one-loop contributions, for the Wilson coefficients of which we have obtained a general formula. Using the obtained results, we analyse constraints from electroweak oblique parameters and the Higgs data on several phenomenological models.


I. INTRODUCTION
After its discovery, the 125-GeV Higgs boson has been studied and found to be consistent with the standard model (SM) expectation as we know at present. This observation suggests that any new physics degrees of freedom that directly couple with the SM-like Higgs boson should reside at a sufficiently high mass scale or be very weakly coupled with the SM particles so that they do not affect its properties significantly. To study Higgs physics in this case, as in the case of Fermi theory, it is useful and satisfactory to work with an effective field theory (EFT) with higher dimensional operators of the SM fields organised in inverse powers of the new physics scale Λ. Using the EFT approach, we can learn about possible types of new interactions at low energies. By accumulating sufficient clues, a complete model of the new physics may be constructed.
As a start, we assume that new physics does not violate known gauge and Lorentz symmetries in the SM so that the higher dimensional operators obtained by integrating out the heavy degrees of freedom also satisfy the same symmetries. There is only one dimension-5 operator (for one family of fermions) consistent with this, i.e., the Weinberg operator that gives rise to Majorana mass for neutrinos [1]. This operator violates the lepton number by two units. In the case of dimension-6 operators, the original attempt to compile a complete basis [2] was later found to be redundant [3][4][5], leaving 64 independent operators (also for one family of fermions) [6] with five of them violating either baryon or lepton number [1,7,8].
For weakly interacting renormalizable gauge theories that are perturbatively decoupling, the dimension-6 operators can be classified into potentially tree-generated and loop-generated ones [9,10]. Note that it has been stressed with explicit examples that the classification of higher dimensional operators into tree and loop ones within the EFT can be ambiguous [11].
A good discussion and comparison of different operator bases of popular choices [2,12,13] can be found in Ref. [14].
There are some attractive motivations to consider models with an extended scalar sector.
For example, new scalar bosons in these models may facilitate a strong first-order phase transition for successful electroweak baryogenesis, provide Majorana mass for neutrinos, and/or have a connection with a hidden sector that houses dark matter candidates. Even though it may not be possible to directly probe this sector due to the heavy masses of new scalar bosons and/or their feeble interactions with SM particles, they can nevertheless leave imprints in some electroweak precision observables.
In this paper, we analyse the EFT of the SM-like Higgs boson for a wide class of weakly coupled renormalisable new physics models extended by one type of scalar field(s) 1 and respecting CP symmetry. It can be shown that only a few types of interactions contribute to the Higgs dimension-6 operators. Two of them are µH † HS and µ H † τ a H T a , where µ is a dimensionful quantity, H is the SM Higgs doublet, S is a singlet field, T a form a triplet field, and τ a are the SU(2) generators. These interactions only arise for specific representations of the new scalar and lead to dimension-6 operators when matching onto the EFT at tree level. Another two are ( where t a are the SU(2) operators appropriate for the new scalar field Φ. These interactions are more generic and, after the heavy scalar fields are integrated out, give rise to dimension-6 operators at one-loop level. We work out the effective operators and the associated Wilson coefficients for an arbitrary new scalar field Φ(m, n, Y ), where m and n denote its multiplicities under SU(3) C and SU(2) L , respectively, and Y is its hypercharge. Phenomenological results of a few benchmark models are studied in this framework. This paper is organised as follows. In Section II, we define our framework of UV-complete models whose scalar sector is augmented from the SM by one new scalar field, and list dimension-6 operators composed of the SM Higgs and electroweak gauge fields that are of interest to us. For the new scalar field of a generic representation, the dimension-6 operators are induced only from one-loop matching due to two specific types of quartic interactions in the UV theory. We also identify accidental interaction terms for specific representations of the new scalar field that can lead to the dimension-6 operators already from tree-level matching. We first concentrate on the accidental interactions in Section III, and work out the Wilson coefficients of induced operators from tree-level matching for these specific scalar representations. These Wilson coefficients are further improved by renormalisation group running. Section IV discusses the matching of the full theory onto the effective theory at oneloop level for the new scalar of a general representation in the SM gauge group. In Section V, we work out the results for a few benchmark models commonly considered in the literature.
Using the results, we show numerically how the model parameters are constrained by current 1 Multiple new scalar fields are allowed provided they have a common mass scale, as will be seen in the Zee-Babu model and the Georgi-Machacek model analysed in Sections V-A and V-C. and future electroweak precision observables and SM Higgs data. Section VI summarises our findings.

II. EFFECTIVE OPERATORS AND WILSON COEFFICIENTS
In the following, we will consider the renormalizable model having a scalar sector extended with a generic scalar field that couples to the SM Higgs field, and match it at tree and oneloop levels onto an effective theory with operators up to dimension-6. Moreover, we will use the renormalization group equations (RGE's) to evolve the Wilson coefficients obtained from the tree-level matching from the new physics scale down to the electroweak scale, thereby capturing the leading-log loop corrections to them, and combine with those from the direct one-loop matching. Throughout this paper, we will use H and Φ to denote the SM Higgs field and the generic new scalar field, respectively.

Symbol Operator expression
Symbol Operator expression In Table I, we list ten CP-even dimension-6 operators composed of only the Higgs and electroweak gauge boson fields that are relevant for the electroweak precision and Higgs observables. In the table, D µ denotes the SM covariant derivative; G a µν , W a µν and B µν are respectively the field strength tensors of the SU(3) C , SU(2) L and U(1) Y groups with the associated gauge couplings g s , g, and g ′ ; and A ← → The Wilson coefficient corresponding to the operator O i will be denoted by c i and have mass dimension Assuming that the new scalar field Φ is a complex scalar, the kinetic and interaction terms relevant to our discussions are where L acc denotes the "accidental" part to be detailed below. For a real scalar field, terms quadratic in the new scalar should include an extra factor of 1/2 and Φ † is identified as Φ.
In Eq. (1), τ a = σ a /2 are the SU(2) generators for the fundamental representation and t a are those for a generic representation, and the parameter M sets the new physics scale that is assumed to be much higher than the electroweak scale. Note that other terms such as the quartic interactions of the Higgs and the new scalar fields have been omitted, since they are irrelevant to the dimension-6 operators. We will assume λ, λ ′ > 0 and the other quartic terms to be positive-definite as well to ensure that the potential is bounded from below.
The accidental part L acc in Eq. (1) contains terms that are allowed only for specific representations of Φ and lead to dimension-6 operators from tree-level and/or one-loop matching. It has two types of interactions. The first one involves dimension-3 operators, and the only possibilities are: The second one involves dimension-4 operators that are only possible when Φ is an SU(2) L doublet. For example, Note that the above term is allowed because Y Φ = Y H = 1/2. The effects of the λ ′′ terms have been discussed, for example, in Refs. [15,16]. The λ ′′′ terms only lead to the operator O 6 of no interest to our analysis. New Yukawa terms involving Φ will also arise; yet they are irrelevant for our discussions.

III. TREE-LEVEL MATCHING AND RGE IMPROVEMENT
For the dimension-3 interaction terms in Eq.
(2), one can integrate out the new scalar field from the UV-complete theory by solving the associated equation of motion and plugging it back into the original Lagrangian 2 . This gives rise to the following dimension-6 operators and some renormalization corrections for |H| 4 in the EFT: Note that the operator O T will lead to corrections of the oblique T parameter, given by with the fine-structure constant α EM = 1/128 and v ≃ 246 GeV. Therefore, the T parameter measured to a high precision imposes a stringent constraint on the triplet models in Eq. (4), as the corresponding Wilson coefficients are not loop suppressed. Using the measured electroweak ρ parameter [18], one can obtain an upper bound on |µ/M 2 | to be 12.0 × 10 −5 and 3.4 × 10 −5 GeV −1 at 95% confidence level (CL) for the real and complex triplet cases, respectively. These bounds can be translated into the corresponding bounds on the vacuum expectation values of the triplet field in the models. By combining the real and complex triplet fields with a common M parameter, corresponding to the Georgi-Machacek (GM) model, one gets a cancellation for the operator O T so that c T = 0. That is, the GM model has only nonzero c H and c R at tree level. We note in passing that the GM model also has other contributions from one-loop matching, which is to be discussed in Section V-C.
In addition to the above-mentioned contributions directly from tree-level matching, it is also possible to have additional corrections through RGE running of the other Wilson coefficients from the new physics scale M to the electroweak scale, characterized by the W boson mass M W (or sometimes the Higgs mass M h = 125 GeV is used). Here we will focus on the electroweak oblique corrections, of which T has been discussed above and Note that with only tree-level matching and no RGE running, S would be zero in the above-mentioned models. The oblique U parameter is not considered here because it first arises from a dimension-8 operator. The anomalous dimensions for the RGE's are given in Ref. [19][20][21][22][23][24]. It is well-known that the RGE anomalous dimensions have basis dependence. Redundant operators would be radiatively generated even if one starts with an irreducible complete basis. Using such a basis requires one to make use of equations of motion and/or field redefinitions to project the redundant Wilson coefficients generated in the running. Since the redundant operator O R already arises from tree-level matching in our analysis, we choose to work with a redundant basis containing O R whose anomalous dimensions are given by [24]: where we have kept only the tree-level generated Wilson coefficients while leaving the other contributions in the "· · · " parts. Gauge independence of the running of these Wilson coefficients are checked and discussed at length in Ref. [24]. It is readily seen that the list of models that can lead to dimension-6 operators from tree-level matching is rather limited [15,25]. So is the list of models that can have oblique corrections induced at the tree-level matching with RGE improvements. With the assump-tion of vanishing tree-level T (but still allowing corrections from RGE), only two models are left: the real singlet model and the GM model. In Fig. 1, we show the corrections to the S and T parameters from RGE-improved tree-level matching for the two models in red (n ≥ 2) of Φ can give rise to the λ ′ term. 3 We use the normalisation that the electric charge of a particle Q = I 3 + Y with I 3 being its third weak isospin component.
After identifying the interaction terms, we then implement the CDE for the Coleman-Weinberg potential, as detailed in Refs. [15,25].
where thus vanish for SU(2) L singlets. 4 We note in passing that tree-level perturbative unitarity of the SU(2) L interaction alone imposes a limit that the representation of a complex (real) scalar cannot be larger than an octet (nonet) [26]. 4 In Ref. [15], it is pointed out that there are additional universal contributions to the pure gauge dimension- and O 3G defined in the reference. The last two will not affect the electroweak and Higgs physics, while the first three are usually small in effect because they are proportional to the SM gauge couplings. For completeness, we also quote the general result here and include them in the following fits:  In this section, we explicitly work out the Wilson coefficients of relevant dimension-6

V. A FEW PHENOMENOLOGICAL EXAMPLES
operators for a few well-motivated models whose scalar sector is extended with one new scalar field or multiple scalar fields of a common type, and discuss how the models are constrained by the electroweak precision data and the Higgs data, both for existing data and future expected measurements. We have already seen that most such models have effective dimension-6 operators starting only at the one-loop level. For current measurements we refer to the U = 0 oblique parameter measurements of Ref. [29] and the ATLAS (CMS) Higgs data with 4.5 (5.1) fb −1 integrated luminosity at √ s = 7 TeV and 20.3 (19.7) fb −1 integrated luminosity at √ s = 8 TeV [27,28], as listed in Table II. We do not include tri-gauge boson precision measurements in our fitting. For future expected sensitivities, we take the most aggressive oblique parameter measurements expected from the Tera Z experiment [30] and the projected Higgs data from Table 4 of Ref. [31].

A. Zee-Babu model
The Zee-Babu model [32][33][34] is one of the simplest phenomenological models that lead to dimension-6 operators only from the one-loop matching. The model has one singly-charged and one doubly-charged singlet scalar fields without a color charge. Assuming a common mass parameter M for both of them and noticing that these fields cannot have the λ ′ term in Eq. (1), we find that only the O BB and O H operators are induced at one-loop level:

B. Two-Higgs doublet model
As a second example for loop-induced dimension-6 operators, we consider the two-Higgs doublet model (2HDM). We follow the notation and convention of Ref. [38], except that each of our λ's is twice bigger, and write down the scalar potential: where we have left out the λ 6,7 terms that are forbidden by the Z 2 symmetry Φ 1 → Φ 1 and Φ 2 → −Φ 2 . As usual, we define the angle β in terms of tan β = v 2 /v 1 , the ratio of the vacuum expectation values of Φ 2 and Φ 1 , and the angle α as the rotation from the above basis to the mass eigenbasis for the CP-even neutral Higgs bosons. In the decoupling limit, cos(β − α) → 0 and the SM-like Higgs boson is much lighter than the other Higgs bosons.
In this limit, the above potential is turned into a form consistent with Eqs. (1) and (3) after the basis rotation of angle β. We then determine At this point, it is useful to make a comparison with results already existing in the literature, as we use a different parameterisation from others. Using the identity τ a ij τ a kl = We therefore make the identification λ = λ 1 + 1 2 λ 2 , λ ′ = 2λ 2 , where λ 1,2 here are those used in Ref. [15]. Without the λ ′′ terms in Eq. (3), our most generic result corresponds to the "Simpler 2HDM theory" in Ref. [15]. to the 2HDM except that the λ's in Eq. (9) should be replaced by either 1 8 (g 2 ± g ′2 ) or − 1 4 g 2 at tree level. Therefore, in the decoupling limit of the MSSM, the relevant interaction coefficients λ and λ ′ are O(0.5). As seen in Fig. 2, the oblique parameters are not very constraining even in the case of future lepton colliders. In certain models such as the inert 2HDM where β = 0, λ = 2λ 3 + λ 4 and λ ′ = 4λ 4 , which are not constrained by the SM Higgs coupling/mass and potentially large. In this case, the precision measurements alone can probe a much larger parameter region.

C. Georgi-Machacek model
Our third example is the GM model, which is special in that it is the only model that has contributions from both RGE-improved tree-level matching and the one-loop matching.
In addition to the SM Higgs doublet, the scalar sector of the GM model has a Y = 1 complex triplet scalar field X ≡ (χ ++ , χ + , χ 0 / √ 2) T and a Y = 0 real triplet scalar field Ξ ≡ (ξ + , ξ 0 , ξ − ) T with the scalar potential where H C = iσ 2 H * , terms have been written in accordance with Eqs. (1) and (2), and the "· · · " part contains the irrelevant ones. Note that the one-loop matching is not simply a sum of separate Y = 1 complex and Y = 0 real triplet contributions, as the last term mixes the real and complex fields.
Corrections to the oblique parameters are   The models in Groups II and III of Table III have  For nontrivial representations of SU(2) L in the other groups of Table III, they involve the additional coupling λ ′ . In this case, the T parameter, which scales like (λ ′ /M) 2 , plays an important role in restricting the parameter space. Therefore, the bounds are usually for λ ′ /M instead of λ ′ /M 2 from the S parameter.

VI. SUMMARY
Although it is widely believed that the standard model (SM) is at best a good effective theory at low energies, the fact that the observed 125-GeV Higgs boson has properties very close to that in the SM suggests that the new physics scale is high and the new degrees of freedom are likely to be in the decoupling limit. Therefore, it is useful to work out an effective field theory (EFT) in terms of operators up to dimension 6 and composed of only the SM fields.
In this paper, we have analysed the EFT of the SM Higgs field for a wide class of weakly coupled renormalisable new physics models extended by one type of scalar fields and respecting CP symmetry, concentrating on the dimension-6 operators that have corrections to the electroweak oblique parameters and current Higgs observables. We have shown that for the new scalar field of specific representations (SU(2) L singlet, doublet, and triplet), there are "accidental" interactions between the scalar and the SM Higgs fields that lead to dimension-6 operators at both tree and one-loop level. For the scalar field of a general representation under the SM gauge groups, we have pointed out that there are only two generic quartic interactions that will lead to dimension-6 operators only at one-loop level.
We work out the Wilson coefficients associated with these operators for the general case in terms of the new physics parameters.
Using the existing LEP oblique parameter measurements and LHC Higgs data, we study the current constraints on the parameters of several benchmark models. The same is also done for the projected results expected in the future experiment. Although indirect, comparing the higher dimensional operators in the effective field theory with precision measurements is always a useful probe and complementary to the direct search method.