On one-loop corrections in the standard model effective field theory; the $\Gamma(h \rightarrow \gamma \, \gamma)$ case

We calculate one loop contributions to $\Gamma(h \rightarrow \gamma \, \gamma)$ from higher dimensional operators, in the Standard Model Effective Field Theory (SMEFT). Some technical challenges related to determining Electroweak one loop"finite terms"are discussed and overcome. Although we restrict our attention to $\Gamma(h \rightarrow \gamma \, \gamma)$, several developments we report have broad implications. Firstly, the running of the vacuum expectation value modifies the $\log(\mu)$ dependence of processes in a manner that is not captured in some past SMEFT Renormalization Group (RG) calculations. Secondly, higher dimensional operators can source ghost interactions in $R_\xi$ gauges due to a modified gauge fixing procedure. Lastly, higher dimensional operators can contribute with pure finite terms at one loop in a manner that is not anticipated in a RG analysis. These results cast recent speculation on the nature of one loop corrections in the SMEFT in an entirely new light.


Introduction
The properties of the Standard Model (SM) Higgs Boson are consistent with the measured properties of the 0 + particle discovered in 2012. The discovery of this state was the main accomplishment of the first run of the LHC. After a two year shutdown, the beams of the LHC have started circulating once again and within a month, collisions will start providing data from Run II. The effects of beyond the Standard Model (BSM) physics will be robustly searched for in Run II, in searches for new resonances and by studying the properties of the discovered 0 + state ever more precisely. It is important to systematically improve the theoretical frameworks that allow consistent interpretations of the increasingly precise measurements of the properties of the 0 + state in tandem with the increasing experimental precision. In this paper, we advance this effort by determining the complete one loop contribution to Γ(h → γ γ) due to a set of higher dimensional operators, including finite terms, in the linearly realized Standard Model Effective Field Theory (SMEFT). The linear SMEFT assumes that the low energy limit of any BSM physics is adequately described when two requirements are fulfilled; the observed 0 + scalar is embedded in the Higgs doublet, and Higher dimensional operators, denoted O i and built out of the SU(3) × SU(2) × U(1) invariant SM fields, are added to the renormalizable SM interactions. In the remainder of this paper, all statements are to be understood to apply to the linear SMEFT.
The operator with lowest dimension to add to the SM, suppressed by a cut off scale, is of dimension five. 1 A complete classification of the dimension six operators is given in Refs. [1,2], the latter of which finds that there are 59 (+ h.c) independent operators, assuming baryon number conservation, after eliminating redundant operators. Recently, the dimension seven operators have also been classified in Ref. [3].
In this paper, we calculate a set of one loop contributions from dimension six operators to Γ(h → γ γ). The one loop improvement of Γ(h → γ γ) is necessary for a precise study of this process in the SMEFT. Such a calculation accounts for the scale dependence of the operators, which is required to interpret deviations at the scale µ ≃ m h in terms of an underlying new physics model, matched onto at the scale µ ≃ Λ ≫ m h .
The first step in such an analysis is to determine the Renormalization Group (RG) running of the O i . 2 For the basis in Ref. [2], the required results were systematically determined in Refs. [4][5][6][7][8]. Most of the log(µ) dependence that is present in the physical impact of the O i at one loop can be directly determined in the unbroken phase of the theory (i.e. calculating when H † H = 0). We demonstrate how to determine the remaining log(µ) dependence.
The main focus of this paper is, however, on the calculation of the non log finite terms in Γ(h → γ γ). The effects of the O i decouple as Λ ≫ m h . As such, even when all the Λ enhanced log terms are known, they are only modestly enhanced compared to the remaining finite terms. For example, for Λ ∼ 3 TeV, the Λ enhanced log terms are ∼ 6, and larger numerical factors can occur in one loop diagrams as pure "finite terms". For this reason, it is essential to go beyond a minimal RG analysis when precisely interpreting future experimental bounds on Γ(h → γ γ).
Using the broken phase of the theory, (i.e. calculating when H † H = v 2 ) to determine one loop finite terms, is somewhat technically demanding. We show how these technical challenges are overcome in the case of Γ(h → γ γ), and advance the full characterization of this process at one loop in the SMEFT. Many of our results have broad implications and inform ongoing efforts to systematically develop the SMEFT to next to leading order (NLO). 3 The summary of the most important developments are: • We explicitly demonstrate how the counterterms determined in a one loop RG analysis can be incorporated into full one loop calculations in the SMEFT, see Section 2.1.
• In some cases, we find that the one loop renormalization of the theory does not directly give the full log(µ) dependence relevant to the one loop contribution of the O i to a physical process. When calculating assuming H † H = 0, one can identify all the operator counter terms. However, the running of the vacuum expectation value (vev), or, equivalently, the running of H † H , introduces further log(µ) dependence, see Section 3.2.
• We discuss how the Background Field Method (BFM) implementing R ξ gauge fixing is generalized into the case of the SMEFT, and how this leads to the sourcing of ghosts in interactions proportional to O i Wilson coefficients, see Appendix A.
• We show how a full one loop calculation can yield a "pure finite term" contribution to a process proportional to an O i Wilson coefficient, in a manner that is unanticipated in a RG analysis. This result makes clear that recent speculation on the structure of the one loop RG in the SMEFT is not speculation on the full one loop structure of the SMEFT, see Section 3.2.
The outline of the paper is as follows. In Section 2 we introduce the Effective Lagrangian, from which we calculate the finite contributions to the Γ(h → γγ) decay. In this section, we also explain the renormalization scheme and we show the cancellation of the divergent contributions explicitly. In Section 3, the finite results are presented, including both the direct contributions from the one loop diagrams, as well as the contributions entering through the definition of the vev and through renormalization conditions defining the external fields and couplings. Subtleties regarding extra contributions through redefinitions of gauge fields and mixing angles, leading to ghost contributions, are also discussed in this section. In Section 4 some phenomenological implications are presented. We conclude in Section 5.

Calculational framework
The one loop improvement of Γ(h → γ γ), due to the O i of the SMEFT, is given in part by the Effective Lagrangian The operator notation used here follows that in Ref. [2], and the operators in the second line of Eqn. 2.1 have Hermitian conjugates. The indicies r, s are flavour indicies. The bare operators considered in detail in this paper are normalized as The SM fields and couplings are also bare on the right hand side of Eqn. 2.2, but the (0) labels are suppressed. σ a are the Pauli matrices for weak isospin. The g i are the SM gauge couplings. The gauge coupling normalization of the operators in Eqn. 2.1 is chosen so that in the unbroken phase, the background field method can be more directly used to determine the counterterms. 4 The notationḡ i is used for the canonically normalized couplings in Ref. [7]. Although we canonically normalize the theory as in Ref. [7], the bar notation is suppressed in this paper. Also note that the Wilson coefficients are dimensionfull as they include a factor of 1/Λ 2 . We choose to consider in detail the subset of operators in Eqn. 2.2, as these operators illustrate the basic issues involved with determining the Electroweak finite terms in the SMEFT. In particular, the operator O HW B , which leads to a redefinition of the mixing angles and Z, A fields in the SM, generates most of the technical challenges introduced in the SMEFT. 5 Of course, a full one loop improvement, including all O i allowed at one loop, is eventually required. Further, the effect of redefining the input parameters of the SM prediction of Γ(h → γ γ) is required for a complete one loop treatment of this process in the SMEFT. Our results are a first step in this direction, when considering finite terms.

Operator counterterms
The operators are renormalized through introducing a renormalization matrix Z ij to cancel the extra operator induced divergences, such that and we restrict our attention to i, j = HB, HW, HW B in this paper. Here, the superscript (r) is used to denote a renormalized operator. With the normalization in Eqn. 2.2, the one loop operator counterterms are This normalization of the operators by the gauge couplings was adopted in Ref. [4], which is closely related to this work. The normalization chosen, however, differs by a factor of −2 from that used in Ref. [4]. Conversely, the complete renormalization results of Ref. [5][6][7] do not introduce gauge coupling factors into the normalization of the operators. 5 The Wilson coefficient of this operator corresponds to the S parameter, and the effects that we will discuss are present in any operator basis. In some alternative operator basis, the S parameter is related to a sum of operators, which enhances the challenges involved in developing the SMEFT to NLO. in the basis of the operators given by O i = (O HB , O HW , O HW B ). Our notation is such that while the Higgs potential is defined so that v ∼ 246 GeV and m 2 h = 2 λ v 2 . The renormalized interactions are introduced as (2.7) The operators are multiplied by a factor N i that includes the corrections from the counterterm matrix Z and absorbs coupling factors. The N 's are given by Although the subtraction required due to the introduction of the O i has been performed to render the theory finite, the subtractions to renormalize the SM interactions are still required. This is indicated by the inclusion of Z SM in the above expressions. The exact forms of the required Z SM and Z i,j depend on the normalization and scheme chosen. 6

Background field method
The renormalization scheme we use is to define the one loop finite terms in Γ(h → γ γ) in a manner that is consistent with implementing the MS scheme, while simultaneously utilizing the Background Field method (BFM). In the BFM [16][17][18], fields are split into classical and quantum components, and a gauge fixing term is added that maintains the gauge invariance of the classical background fields, while breaking the gauge invariance of the quantum fields. We use R ξ gauge with background gauge fixing, with φ ± , φ 0 , the Goldstone bosons defined through the convention The one loop vev (v + δv) is defined as the classical background field expectation value for which the one point h function vanishes. This induces finite (and gauge dependent) terms into the definition of the vev, which are discussed further in Section 3. The requirement to include these terms is indicated by the introduction of δv in the above expression, which is formally of one loop order. Technical simplifications result from the use of the BFM. One can choose a gauge in the quantum calculation to one's advantage. Also, the counterterms accounting for the renormalization of the SM gauge fields and couplings cancel for a specific choice of operator normalization. This follows from the unbroken Ward identities of the theory, when using the BFM. The Ward identities result in the following relations among the SM counterterms [19],67 The gauge fixing is undertaken in the BFM in a manner discussed in Ref. [19,20]. 7 The renormalized fields (where F = A, W, Z and S = h, φ 0 , φ ± ) and couplings (denoted c = e, g 2 , g 1 ) are defined in terms of bare fields and couplings via The factor µ −ǫ/2 is included in the coupling relation to render the renormalized coupling dimensionless [21]. The vev is also renormalized with the inclusion of The remaining SM counterterms required to render the theory finite are well known. The Higgs wavefunction renormalization in the background field method with the gauge fixing in Eqn. A.3 is given by 8 The sum of the divergent terms in the vev renormalization constant Z v and δv also needs to be fixed. A simple approach to determine this divergence that avoids operator mixing complications is to consider the operator H † HG µ ν G µ ν . The counterterm associated with this operator is [4] Considering the one loop h → gg decay with the diagrams in Fig. 1, this fixes ( (2.14) The left hand side of Eqn. 2.14 reflects an ambiguity that remains in Electroweak perturbation theory when using the MS scheme at one loop in this way. This requires the renormalization condition to fix δv, as discussed above. We choose to adopt the vev defining condition described in Ref. [19], which fixes the finite terms of the vev as in Section 3. The renormalizations of the vev and the Higgs field are seen to satisfy the following relation, This is in fact expected in the BFM, see Ref. [24].

The Effective Lagrangian
Expanding the O i into the mass eigenstate fields, one finds the renormalized effective interactions relevant to the one loop improvement of Γ(h → γ γ). 9 In all of the L ef f terms, other than some forms given in the first line of Eqn. 2.16, the contribution to Γ(h → γ γ) decay is itself a one loop contribution. As such, the relevant L ef f for the one loop Γ(h → γ γ) result simplifies to O(v) in the following way Each N i in L ef f now contains dependence on the counterterms Z g 1 , Z g 2 , etc. implicitly. In the same way, the fields and couplings of lines two through six in L ef f should be multiplied by their respective renormalization constants. However, these corrections are two loop order and neglected. This expression is also simplified using the relations given in Eqn. 2.10. All of the operator Wilson coefficients in this expression are evaluated at the scale µ = Λ, which allows a direct interpretation of any measured deviation in terms of the underlying BSM physics sector. The combination of Wilson coefficients given by corresponds to the effective Wilson coefficient for h → γ γ at tree level and one scale. This effective Wilson coefficient, however, does not physically correspond to this process in a scale independent fashion in the SMEFT, see Section 3.2.
In addition to the terms in Eqn. 2.16, contributions proportional to v 2 are present in L ef f . These terms are not expected to contribute with divergent terms to the amplitude before renormalization, as the O i counterterm subtraction is proportional to v. 10 We find by explicit calculation that this is true.
However, finite terms could exist at one loop in Γ(h → γ γ) due to v 2 terms in L ef f . We find by explicit calculation that such terms do contribute at one loop. Terms of order v 2 come about due to the direct expansion of the operators present and from the canonical normalization of the SMEFT. Interestingly, terms of this form involve ghost fields (denoted u ± ), which come about due to the effects of canonically normalizing the SMEFT. The relevant (2.18)

Cancelation of the divergent terms
In this section, we explicitly demonstrate that the counterterm subtractions render the theory finite and cancel the ξ gauge dependence of the divergent terms while doing so. The required one loop diagrams explicitly calculated are shown in Figs. 2, 3. To calculate the diagrams in the BFM, the Feynman rules of the theory, determined using the results of Ref. [19], are used. Our gauge coupling conventions differ from those chosen in Ref. [19], so we summarize our conventions in Appendix A. It is convenient to introduce the notation to simplify results. The divergent contributions of the diagrams shown in Fig. 2 are as follows iA a = C ǫ C γ γ (−4 λ − (2.28) The Here we are suppressing the dependence on the gauge parameter that exactly cancels against the gauge dependence in the √ Z h , √ Z v + δv/v counterterms in A j . The gauge independent divergent contributions exactly cancel the insertion of the gauge independent counterterms for the O i in A j , which gives establishing that the process is rendered finite as expected. This is another non-trivial check of the mixing results reported in Ref. [4]. The remaining divergences from iA b,k−o in Fig. 3 due to L ef f (v 2 ) must vanish as argued, which occurs via the intermediate results Figure 3. Diagrams contributing to H → γγ decay due to L ef f (v 2 ). The divergent terms exactly cancel in this class of contributions, as expected. u ± are ghost fields.

Finite terms
The finite terms in the calculation come about from expanding the results of the diagrams in Figs. 2,3 to O(ǫ 0 ), as well as from finite one loop terms defined via renormalization conditions in the MS scheme for the fields and couplings entering at tree level in the calculation. The latter correspond to the vacuum expectation value (δv), the renormalization conditions fixing the external two point functions for the Higgs (δR h ) and the photon (δR A ) fields, and the definition of the electric coupling e, which fixes δR e . 11 These BSM physics contributions enter the S matrix element as Note that δv appears explicitly in A j . We choose to use a scheme [19,27] that utilizes the BFM, where the finite on shell renormalization conditions are defined as follows: 12 • The tree level vacuum expectation value of the Higgs is defined by the potential The one loop correction (δv) to the vacuum expectation value is fixed by the condition that the one point function of the Higgs field vanishes to one loop order, including δv 11 The definition of these so-called R factors is well explained in Refs. [27,28]. 12 In calculating the renormalization conditions with the BFM, the two and three point functions and all external fields are treated as classical. An alternative scheme where systematically the two and three point functions are treated as having mixed classical and quantum external states, combined with a suitable modification of the results due to Figs. 2,3, can also be consistent.
Setting T = 0 defines the vev at one loop and fixes the finite terms of δv. Gauge dependence is thus present in the one loop definition of the vacuum expectation value of the Higgs. The remaining contributions to Γ(h → γ γ) from the sum of all other "tadpole" diagrams and insertions of δv vanish due to this chosen renormalization condition. These contributions are not shown in Figs. 2,3.
• The one loop correction to the electric charge e (0) = (1 + δR e ) e (r) is fixed by the on shell renormalization condition defining the electric charge as the e e γ coupling in the Thompson limit. Fixing all corrections to this vertex to vanish on shell at zero momentum transfer fixes the finite terms given by δR e . The Ward identities of the theory also fix where δR A is the wavefunction renormalization finite term fixed via the on shell renormalization condition of the photon field. Due to the above relation, no dependence on the photon two point function appears in the S matrix element at one loop.
• The finite terms present in the wavefunction renormalization of the Higgs is defined via the on shell renormalization condition where p is the momentum of the Higgs field. Here Π hh (p 2 ) is the two point function of the Higgs calculated using the BFM, which is reported in Appendix B.
Combining these results gives the BSM contributions to the S matrix element relevant for Γ(h → γ γ), due to the set of operators, we have considered. Using the notation , we find the result for this amplitude to be remarkably compact, 13 This equation is the main result of this paper.

Gauge independence of the results
The results in Eqn. 3.6 is gauge independent. Our use of the BFM allows us to choose a gauge for the quantum fields (we use the convenient choice ξ = 1), while maintaining the explicit gauge invariance of the result. We have also verified the gauge independence of the results with extensive checks of the calculation, which include the following: • The gauge independence of the coefficient of C γ γ is due to a cancelation of the gauge dependence in δR h , δv and the one loop contributions of diagrams a) and b) in Fig. 2 proportional to C γ γ . We have explicitly verified this gauge independence numerically using the CUBA library for numerical integration [33].
• The gauge independence of the remaining results proportional to C HW and C HW B in Eqn. 3.6 due to Figs. 2,3 has also been checked numerically. Moreover, the results in Feynman R ξ gauge and unitary gauge have been compared analytically, finding exact agreement. 14

Implications of the results for the one loop structure of the SMEFT
The results in Eqn. 3.6 are presented with terms grouped together in the following manner. The first line is the tree level result and the modification of this result due to the on shell renormalization conditions at one loop. The second line corresponds to log enhanced terms with µ ≡ Λ. These results are consistent with the results reported in Refs. [4,34]. However, the log(µ) dependence proportional to C γ γ receives additional contributions from R h and δv.
For ξ = 1, the relevant terms of this form are Here we have normalized all log(µ 2 ) terms to the scale m 2 h . The scale dependence of the RG results in Ref. [4] for C γ γ is obtained by adding Dependence on C HW is present at one loop, which is not absorbed into the dependence on C γ γ and is not hinted at in the one loop structure of the RG. Nevertheless, such terms do exist. We refer to this class of terms as one loop "pure finite" terms, as no log(µ) dependence is present for this class of contributions.
The form of the result in Eqn. 3.6 is exactly what one expects on general grounds. Although a particular combination of the operators O HB , O HW and O HW B corresponds to Γ(h → γ γ) at tree level, there is no sense in which this particular combination of operators is preserved at one loop in the SMEFT. The general expectation is that all three Wilson 14 One aspect of the gauge independence that is interesting is the generation of the λ dependence for one loop corrections to CHW B in unitary gauge. This dependence is due to diagram (b) in R ξ gauges, but is given by diagram (k) in unitary gauge, due to the appearance of a factor of m 2 h in a numerator in the one loop finite terms.
coefficients will appear as three independent parameters contributing to Γ(h → γ γ) at one loop. This is exactly what is found in the explicit calculation of this process.
Finally, we note the effect of properly accounting for the class of effects due to L ef f (v 2 ) in the results. These contributions do not contribute with divergences to be subtracted by the O i counterterms and so are expected to be pure finite terms. This can be understood due to the scaling in v that these terms are proportional to. This is exactly what is found in the explicit calculation. Again, studies of only the O i RG of the SMEFT are blind to such contributions. These contributions do not vanish at one loop when considering finite terms, and are the source of the e 4 terms in Eqn. 3.6.
For these reasons, the structure of the RG is not a good proxy for the full one loop structure of the SMEFT.

Phenomenology
When considering the numerical effect of the Wilson coefficients of the O i on measured processes, there are two broad perspectives, one can adopt. The SMEFT can be studied and experimentally constrained, while considering the SMEFT as a real and consistent theoretical formalism. This approach is a "bottom up" EFT point of view, and does not impose a UV bias or prejudice as to the size of the Wilson coefficients of various operators. Within this approach, consistent organizational schemes to power counting operators can be constructed. For example, a popular scheme is given by the Naive Dimensional Analysis (NDA) approach laid out in Ref. [35]. The NDA scheme is incomplete in some scenarios, but it can be consistently extended, see Refs. [36,37]. 15 Within this bottom up EFT perspective, it is interesting to note that Ref. [39] has recently advanced an elegant operator level understanding of the approximate holomorphic structure of the SMEFT discussed in detail in Ref. [40]. These works are concerned with statements on the SMEFT without invoking UV bias. The need to calculate finite terms that need not have the structure of the RG, which is emphasized and explicitly demonstrated in this paper, is consistent with the discussions in Refs. [39,40].
As an alternative to the bottom up approach, some literature employs UV dependent assumptions to classify operators. For some discussion along these lines see Refs. [41,42]. We will not discuss UV bias in any great detail in this paper.

Numerical results
The perturbative results reported in this paper are one loop corrections to the effect of the chosen O i on the process Γ(h → γ γ). The size of the O i Wilson coefficients is UV dependent. We treat the Wilson coefficients as free parameters, to be constrained by experiments, multiplying the naive power counting suppression O(v 2 /Λ 2 ). Currently the measured signal strength for γγ is given by ATLAS [43] as µ γγ = 1.17 ± 0.27, (4.1) for m h = 125.4 ± 0.4 GeV, while CMS reports [44] µ γγ = 1.14 +0.26 −0.23 , for m h = 124.70 ± 0.34 GeV. The inferred results for the Higgs mass from these experiments are consistent [45]. There is no evidence at this time for a deviation from the expectation in the SM. The modified signal strength due to the chosen O i Wilson coefficients can be written as The expression for A SM is well known from the literature, see for example Refs. [46][47][48], and is given by The corrections that we have computed are comparable in size to the RG log terms calculated in Refs. [4,34]. The results of the analyses of Refs. [4,34] are gauge independent. We compare our results to Ref. [4], which includes terms that were not calculated in Ref. [34]. For the coefficient of C HW B , the ratio of the full one loop terms, to the results of Ref. [4] is given by so that for TeV cut off scales, the RG log terms are larger by only a factor of ∼ 6 (for Λ ∼ 1 TeV). For the coefficient of C HW not already included in C γ γ , there are no RG terms. The pure finite term is given in Eqn. 3.6. The ratio of the size of this term to the full one loop C HW B term, not already included in C γ γ , is approximately which makes clear that the pure finite terms are not negligible in favour of an RG analysis for cut off scales in the TeV range. If perturbative corrections are incorporated in an analysis of future µ γγ constraints, an RG analysis is simply insufficient.
Comparing the size of the one loop corrections included in our results that were neglected in past results, it is clear that the appearance of the scale dependent term in δv is particularly problematic. For the top quark Yukawa, this is a dominant scale dependent log that is not captured in past RG analyses of the SMEFT. However, the interpretation of the numerical impact of this correction on µ γ γ is more subtle. Naively using v + δv as a numerical value of the vev extracted from a measured G F in µ − → e − +ν e + ν µ would assign a gauge dependent quantity a numerical value. It is required that µ − → e − +ν e + ν µ be calculated at one loop in the SMEFT 16 to avoid this gauge dependence and allow the measurement to be used as a direct input into the prediction of µ γ γ . This is beyond the scope of this work, but when doing so it is expected that the large log in Eqn. 4.7 will be absorbed in the numerical value of the vev in the SMEFT. Finally, we note that there is some interest in the literature on the question of a one loop contribution from an operator that is not loop suppressed in a UV matching, leading at one loop to a contribution to µ γ γ through the RG. This can occur in general, but does not occur in some cases if all UV physics is weakly coupled and renormalizable, see Refs. [11,[50][51][52] for some coherent related discussion. Again we note that when considering the physical impact of such a scenario, studying the RG is only a part of a full one loop correction to a physical amplitude.

Conclusions
In this paper we have calculated one loop finite terms for the contribution of the operators O HB , O HW B and O HW to the decay Γ(h → γ γ) in the SMEFT. The terms calculated are not small in general compared to terms that have been calculated in past works using the RG of the SMEFT operators. If experimental bounds on Γ(h → γ γ) are to be studied to next to leading order in the SMEFT, these terms should not be neglected. In developing this result at one loop in the SMEFT, we have also uncovered a number of interesting subtleties. These have broad implications when developing the SMEFT to next to leading order.
generator. Here T a = σ a /2, with σ a being the Pauli matrices. The sign convention in the covariant derivative fixes the sign conventions in the Yang-Mills part of the Lagrangian to be The conventions used here are consistent with Refs. [7,21]. We use the BFM, and the background fields are introduced with a hat superscript. Quantum fields are denoted without the hat superscript. The gauge fixing is given by From the gauge fixing term, choosing ξ B = ξ W , one directly finds This result is consistent (up to sign conventions) with the result in Ref. [19]. From this, the Feynman rules of the theory are straightforward to derive.

A.1 Digression on gauge fixing
Gauge fixing in the SMEFT has some interesting subtleties compared to gauge fixing in the SM. Fundamentally, the number of degrees of freedom being fixed are the same, and in this sense standard t'Hooft R ξ gauge fixing [53,53] can be imposed. However, the relationship between the SU(2) L × U(1) Y gauge fields W and B and the mass eigenstate fields is modified in the SMEFT. In this sense, it is clearly legitimate to consider more carefully the question -What exactly is being gauged in the SMEFT?
The fact that the relation between the W and B fields and the physical propagating mass eigenstate fields changes order by order in the power counting of the theory, as do the mixing angles, has a number of interesting consequences. These contributions can physically contribute to amplitudes in pure finite terms at one loop, as we have shown. As such, the interesting consequences of gauge fixing in the SMEFT are worthy of some comment.
For example, the ghost Lagrangian can be derived from Eqn. A.3 adding a standard Faddeev-Popov term [54] to the Lagrangian, where Here α, β are summed over the physical mass eigenstate fields, and δG α /δθ β is the variation of the gauge fixing terms under the infinitesimal quantum gauge fixing transformations. The gauge transformations in Ref. [19] can be used to derive the ghost Lagrangian, including the effect of redefining the mixing angles and states with SMEFT corrections. In this manner, at O(v 2 /Λ 2 ), we observe that the Wilson coefficient C HW B sources ghost interactions. Another interesting consequence of gauge fixing in the SMEFT is the presence of the interaction terms where in this case the gauge fixing has not imposed the relation ξ B = ξ W . In the SM, the choice ξ B = ξ W is usually made in t'Hooft gauge fixing, and as a consequence A-Z mixing is not present at tree level. The same choice in the SMEFT results in tree level A-Z mixing. This is due to the redefinition of the mass eigenstate fields of the theory at O(v 2 /Λ 2 ) in the SMEFT. These field redefinitions cause these terms to result from the gauge fixing procedure.
The presence of such unanticipated interaction terms when generalising the SM into the SMEFT emphasizes the importance of not obscuring the distinction between a basis of gauge invariant operators, and the Effective Lagrangian. The severe challenges of implementing a calculation of the form presented here, if the distinction is lost, should be manifest. This is a very serious limitation to the gauge dependent approach to the SMEFT discussed in Refs. [25,26,55], which is not well suited for long term use, as studies in the SMEFT are already being systematically improved beyond tree level.
Conversely, avoiding some of the gauge dependence challenges in the SMEFT is a feature in favour of the covariant derivative expansion discussed in Refs. [11,56,57] and by direct use of a well defined non-redundant basis, such as the Warsaw basis of Ref. [2].

B Higgs self energy
The finite part of the Higgs self energy will define the R factors of the Higgs field and mass renormalization. In Feynman gauge, setting ξ = 1, the finite part of the Higgs self energy amplitude is given by and we note that although we present this result with ξ = 1, it was determined in the background field method with classical external Higgs fields assumed for the two point function.