The decay $h\to \gamma\gamma$ in the Standard-Model Effective Field Theory

Assuming that new physics effects are parametrized by the Standard-Model Effective Field Theory (SMEFT) written in a complete basis of up to dimension-6 operators, we calculate the CP-conserving one-loop amplitude for the decay $h\to \gamma\gamma$ in general $R_\xi$-gauges. We employ a simple renormalisation scheme that is hybrid between on-shell SM-like renormalised parameters and running $\overline{\mathrm{MS}}$ Wilson coefficients. The resulting amplitude is then finite, renormalisation scale invariant, independent of the gauge choice ($\xi$) and respects SM Ward identities. Remarkably, the $S$-matrix amplitude calculation resembles very closely the one usually known from renormalisable theories and can be automatised to a high degree. We use this gauge invariant amplitude and recent LHC data to check upon sensitivity to various Wilson coefficients entering from a more complete theory at the matching energy scale. We present a closed expression for the ratio $\mathcal{R}_{h\to \gamma\gamma}$, of the Beyond the SM versus the SM contributions as appeared in LHC $h\to \gamma\gamma$ searches. The most important contributions arise at tree level from the operators $Q_{\varphi B}, Q_{\varphi W}, Q_{\varphi WB}$, and at one-loop level from the dipole operators $Q_{uB},Q_{uW}$. Our calculation shows also that, for operators that appear at tree level in SMEFT, one-loop corrections can modify their contributions by less than 10%. Wilson coefficients corresponding to these five operators are bounded from current LHC $h\to \gamma\gamma$ data -- in some cases an order of magnitude stronger than from other searches. Finally, we correct results that appeared previously in the literature.


Introduction
The discovery of the Higgs boson [1][2][3] in year 2012 was made possible mainly because of its decay into two photons [4,5]. The current outcome for this decay channel from LHC (Run-2) with center-of-mass energy √ s = 13 TeV, integrated luminosity of 36.1 fb −1 and Higgs boson mass, M h = 125.09 ± 0.24 GeV is summarised as the ratio between the experimentally measured value (which may include contributions from new physics scenarios) relative to the Standard-Model (SM) predicted value [6,7] R h→γγ = Γ(EXP, h → γγ) Γ(SM, h → γγ) . (1.1) The most recent measurements are presented by ATLAS [8] and CMS [9] experiments of LHC, ATLAS: R h→γγ = 0.99 +0.15 −0.14 , CMS: R h→γγ = 1.18 +0.17 −0.14 , (1.2) and are consistent with the SM prediction, with the error margin expected to be reduced in the near future.
If we consider the SM as a complete theory of electroweak (EW) and strong interactions up to the Planck scale, with no other scale involved in between, then the decay amplitude h → γγ arises purely from dimension d ≤ 4 (renormalisable) interactions. In this case the amplitude is finite, calculable and, since all relevant parameters are experimentally known, it is a certain prediction of the SM. It is this prediction entering the denominator in eq. (1.1). If however, there is New Physics beyond the SM already at a scale Λ which is above, but not far from, the EW scale, say Λ ∼ O(1 − 10) TeV, then its effects can be parametrized by the presence of effective operators with dimension d > 4 at scale Λ. These operators together with various parameters (or Wilson coefficients) will then run down to the EW scale and feed the on-shell scattering S-matrix amplitude together with d ≤ 4 interactions.
All dimension d ≤ 6 effective operators among SM particles that obey the SM gauge symmetry have been classified in refs. [10,11]. The SM augmented with these effective operators -remnants of unknown heavy particles' decoupling [12] -is called the SM Effective Field Theory, or for a short SMEFT. The quantization of SMEFT has recently been undertaken in ref. [13] in linear R ξ -gauges with explicit proof of BRST symmetry and where all relevant primitive interaction vertices have been collected.
Within SM, numerous calculations for the h → γγ amplitude exist. The first calculation was performed in ref. [6] in the limit of light Higgs mass (M h ≪ M W ), using dimensional regularisation in the 't Hooft-Feynman gauge. Since then, there are other works completing this calculation in linear and non-linear gauges [7,14,15], with different regularisation schemes [16][17][18][19][20][21][22]. To our knowledge the complete SM one-loop h → γγ amplitude in linear R ξ -gauges is performed in ref. [23].
In SMEFT 1 there is already a number of papers that calculate the h → γγ amplitude [26][27][28][29] 2,3 The current, state of the art calculation, has been presented by Hartmann and Trott in refs. [33,34]. The analysis was carried out using the Background Field Method (BFM) [35] 4 consistent with minimal subtraction renormalisation scheme (MS) and included all relevant (CP-conserving) dimension d ≤ 6 operators in calculating finite, non-log parts of the diagrams. Our work here is complementary but incorporates some additional features of importance: 1 For a recent review see, ref. [24] and for pedagogical lectures ref. [25]. 2 For earlier attempts see, refs. [30,31]. 3 Also, recently, the one-loop calculation for h → ZZ and h → Zγ decay in SMEFT has appeared in ref. [32]. 4 For a more recent approach on BFM-SMEFT see ref. [36].
• a simple calculational treatment in linear R ξ -gauges based on Feynman rules of ref. [13], • an analytical proof of gauge invariance (independence on the gauge choice ξ-parameter(s)) of the S-matrix element, • a simple renormalisation framework which leads to a finite and renormalisation scale invariant amplitude, • a compact semi-analytical expression highlighting the effect of new operators in the ratio R h→γγ and corresponding bounds on Wilson coefficients, • a field content of simple, perturbative, high energy models valid at the energy scale Λ, which, under gentle assumptions, can affect the ratio R h→γγ .
There are quite a few papers addressing a global fit to the Higgs data from LHC Run-1 and Run-2 in the SMEFT framework [37][38][39]. Our work provides a simple semi-analytic one-loop formula for the ratio R h→γγ in eq. (1.1) that can be used by these (usually tree level) fits or by analogous experimental analysis at LHC for Higgs boson searches. Our paper is organised as follows. In section 2 we list operators contributing to the decay h → γγ in SMEFT. Next, in section 3 we develop, in a pedagogical fashion, the renormalisation scheme for calculating the h → γγ amplitude. In section 4 we give analytical expressions for all types of SM and SMEFT contributions to the h → γγ amplitude and to the ratio R h→γγ . Semianalytical prediction for R h→γγ , depending on the running Wilson coefficients and renormalisation scale µ, are collected in section 5, and supplied with a discussion on numerical constraints of these coefficients. We conclude in section 6. Finally, in Appendix A we collect analytical expressions for the relevant one-loop self-energies and, relevant to h → γγ, three-point one-loop corrections in general R ξ -gauges.

Relevant Operators
In EFT, an effect from the decoupling of heavy particles with masses of order Λ is captured by the running parameters of the low energy theory influenced by higher dimensional operators added to SM renormalisable Lagrangian L (4) SM . The full effective Lagrangian we consider here can be expressed as, where Q X denotes dimension-6 operators that do not involve fermion fields, while Q f denotes operators that contain fermion fields. All Wilson coefficients should be rescaled by Λ 2 , for example C X → C X /Λ 2 . We shall restore 1/Λ 2 only in section 4 and thereafter. The prime in C ′ f , denotes a coefficient in flavour ("Warsaw") basis of ref. [11] while we use unprimed coefficients in fermion mass basis defined in ref. [13].
The operators involved in the calculation of decay h → γγ are collected in Table 1. They can easily be identified when drawing the Feynman diagrams for h → γγ looking at the primitive vertices listed in ref. [13]. There are 8 classes of such operators X 3 , ϕ 6 , ϕ 4 D 2 , ψ 2 ϕ 3 , X 2 ϕ 2 , ψ 2 Xϕ, ψ 2 ϕ 2 D, ψ 4 where X represents a gauge field strength tensor, ϕ the Higgs doublet, D a covariant derivative and ψ a generic fermion field. Not counting flavour multiplicities and hermi- ϕl present themselves indirectly through the translation of the renormalised vacuum expectation value (vev) into the well measured Fermi coupling constant, cf. eq. (3.5). The notation is the same as in refs. [11,13]. For brevity we suppress fermion chiral indices L, R.
tian conjugation, in general, there are 16+2 CP-conserving operators. 5 Actually, not all operators in Table 1 contribute in the final result for the h → γγ amplitude. The operator Q ϕ cancels out completely after adding all contributions. This leaves 17 CP-conserving operators (or Wilson coefficients) relevant to the h → γγ amplitude.
Another classification of various d = 6 operators can be devised alongside with their strength [40,41]. The division is between operators that are potentially tree level generated (PTG operators) and those that are loop generated (LG operators) by the more fundamental theory at high energies (UV-theory) under the assumption that the latter is perturbatively decoupled. Under this classification operators relevant for h → γγ amplitude are arranged as follows: LG operators are suppressed by 1/(4π) 2 factors for each loop and may be thought to be sub-dominant corrections with respect to PTG operators. Relevant to h → γγ, PTG and LG classes of operators are listed in Table 2 (also for later convenience in section 5.3). On the other hand, a perturbative decoupling of the UV-theory may not necessarily be the case that Nature chooses. In this work, although we do not assume any distinction amongst the d = 6 operators involved in h → γγ amplitude, we shall be referring to Table 2 as our analysis progresses.

PTG
LG

Parameter initialisation in SMEFT
There is a set of very well measured quantities, to which we rely upon, in relating our calculation for R h→γγ to the LHC data. This set of experimental values is [42]  We identify these input values with the ones obtained in SMEFT consistent with the given accuracy of up to 1/Λ 2 expansion terms. Consequently, following ref. [13], for the gauge and Higgs boson masses at tree level, it is enough to set M W , M Z and M h , respectively, equal to where λ is the Higgs quartic coupling,ḡ ′ ,ḡ are, respectively, the U (1) Y and SU (2) L gauge couplings (redefined to obtain canonical form of the gauge kinetic terms, see ref. [13]) and the C-coefficients correspond to operators defined in Table 1. Moreover, the fine structure constant is identified through the Thomson limit (Q 2 = 0) as α EM =ē 2 /4π whereē is given at tree level bȳ Similarly, the experimental values for lepton and quark masses, taken as pole masses from ref. [42], are equal to eqs. (3.27) and (3.29) of ref. [13]. The Fermi coupling constant G F , is identified through the muon decay process. In addition to the W -boson exchange which is modified in SMEFT by the PMNS matrix that is (now) a nonunitary matrix containing the operator Q (3) ϕl , G F is also affected by dipole operators e.g., Q eW or by new diagrams with Z-or Higgs-boson exchange. However, the expression for G F is simplified by making the approximation of zero neutrino masses and also by assuming that for any generic C 1 and C 2 coefficients entering the muon-decay amplitude and m l being a charged lepton mass. Only then we identify the Fermi coupling constant of eq. (3.1), within tree level in SMEFT, as All Wilson coefficients entering in eq. (3.5) are real since they are diagonal elements of Hermitian matrices. In fact, and as a side test of the approximations assumed in eq. (3.4), we have checked that, at tree level in SMEFT, the full S-matrix element for the process µ − → e −ν e ν µ is gauge invariant independently of lepton-number conservation. The formula (3.5) agrees with the corresponding one from refs. [24,32].

Renormalisation framework
We ultimately want to bring the expression for the amplitude A(h → γγ), into a form that contains only renormalised parameters that are most closely related to observable quantities, the relevant ones given in eq. (3.1). At tree level in SMEFT, the hγγ-vertex appears only in association with the unrenormalised (bare) Wilson coefficients, C ϕB 0 , C ϕW 0 and C ϕW B 0 and these are multiplied by the bare vev parameter 6 v 0 (in what follows bare parameters are always denoted with a subscript zero). In order to set the stage, let us for example consider from Table 1 the d = 6, CP-invariant operator of the form X 2 ϕ 2 , where ϕ is the scalar Higgs doublet and B µν is the U (1) Y -hypercharge gauge field strength tensor. All fields and coupling constants are unrenormalised quantities in this expression. In what follows, and in order to keep the expressions as simple as possible, we keep working with unrenormalised fields i.e., no usual field redefinition is performed. This is justified, because we are interested in calculating only an S-matrix amplitude rather than a Green function. 7 After Spontaneous Symmetry Breaking (SSB) in SMEFT (see ref. [13] for details), the expression in eq. (3.6) contains the following term describing the interaction of the Higgs field and two "photons", where h is the Higgs field. We split these bare quantities into renormalised parameters v, C ϕB and counterterms, δv, δC ϕB respectively, as We follow the steps of a simple on-shell renormalisation scheme, first described in SM by Sirlin [45], and introduce new unrenormalised fields A µ and Z µ through the linear combinations with c ≡ cos θ W and s ≡ sin θ W defined as a ratio of the physical masses of W and Z bosons, like Therefore, the Lagrangian term for the considered operator, Q ϕB , describing (part of) the hγγ interaction, reads, Note that the vev counterterm arises from pure SM contributions because it multiplies C ϕB , while δC ϕB cancels infinities that arise only from pure SMEFT diagrams i.e., in general, diagrams proportional to other C-coefficients, not necessarily only C ϕB . Besides operator Q ϕB , counterterms for operators Q ϕW and Q ϕW B need to be added, too. Because all these three operators are proportional to the Higgs bilinear combination, ϕ † ϕ, they all contain the vev counterterm as a universal contribution to h → γγ amplitude. The contributions discussed so far are depicted and explained in Fig. 1. By making use of the Feynman rules of ref. [13], their sum is written in momentum space, as One-loop, 1PI vertex contributions proportional to C ϕB , C ϕW and C ϕW B are denoted (up to pre-factors) with Γ ϕB , Γ ϕW and Γ ϕW B in the first three lines of the above equation. The SM contribution, Γ SM , is just the SM-famous result of ref. [6] but with the SM parameters replaced by the SMEFT ones (that is why "barred" Γ), taken from refs. [13,46]. Furthermore, there are additional one-loop corrections, Γ X , proportional to Wilson coefficients C X , like for instance C W , which are collected in the last line, last term of eq. (3.13).
There are additional diagrams participating in the h → γγ amputated amplitude. These are shown in Fig. 2. The first two classes of diagrams are the Higgs tadpole and its counterterm Figure 2: Tadpole and Zγ self-energy contributions with their associated counterterms. Crosses denote SM counterterms and the black boxes indicate pure d = 6 operator insertions. contributions. These two diagrams do not enter in our renormalised amplitude because, following the renormalisation scheme of ref. [47], the Higgs tadpole counterterm is adjusted to cancel the 1PI Higgs tadpole diagrams. This guarantees that the vev is unchanged to one-loop order. The last two diagrams in Fig. 2 represent the Zγ-self energy at p 2 = 0, A Zγ (0), plus its counterterm, δm 2 Zγ . The expression for the counterterm δm 2 Zγ (given below) is gauge invariant independently of the renormalisation condition for the Higgs tadpole. This is practically very useful for proving the gauge invariance of the h → γγ amplitude.
Finally, as usual, by multiplying the amputated graph with the LSZ-factors [48] (see for instance section 7.2 of textbook [49]) for the external Higgs and photon fields, we arrive at the following S-matrix amplitude: Eq. (3.15) is our master formula for the renormalised amplitude A µν (h → γγ). The definitions for the various self-energies 8 are stated in Fig. 3 and where Π γγ (p 2 ) is regular at p 2 = 0. All self-energies in eq. (3.15) should arise purely from SM diagrams because we are including terms up to 1/Λ 2 in SMEFT. As noted earlier, the SM counterterm, δm 2 Zγ , is gauge invariant and is given by [45]: (3.17) The quantity δv/v is not gauge invariant. Following standard on-shell renormalisation conditions of refs. [45,47], we write where the counterterm δg of the SU (2) L gauge coupling is gauge invariant and reads as Here δe is the electromagnetic charge renormalisation counterterm which is also gauge invariant. This is given by eq. (26) of ref. [45] δe where µ is the renormalisation scale parameter and ǫ ≡ 4−d. Leptonic and hadronic contributions, Π lept γγ (0) and Π had γγ (0), to the photon vacuum polarisation are gauge invariant and the infinite part in the squared brackets should be gauge invariant too. The hadronic contribution from light quarks, Π had γγ (0), is in principle non-calculable due to strong interaction at zero momenta. A dispersive or other non-perturbative methods should be in order. There is no such problem of course with Π lept γγ (0). SM vector boson self-energy contributions can be found in ref. [50]. The Higgs self-energy contribution can be found in refs. [34,47]. These results have been obtained in the particular case of the 't Hooft-Feynman gauge where ξ = 1. Thanks to the set of SMEFT Feynman Rules in general R ξ -gauges [13], we present in Appendix A all contributions needed in eq. (3.15) with the explicit ξ-dependence. This is necessary for checking the gauge invariance of the amplitude. Finally, the counterterms δC ϕB , δC ϕW and δC ϕW B can be read from refs. [27,33,34,46,51,52] where they have been calculated again in 't Hooft-Feynman (ξ = 1) gauge. However, in MS renormalisation scheme and at one-loop, cancellation of infinities should be independent on the gauge choice as we confirm below.

ξ-independence
Knowing the gauge invariant and non-invariant parts of various contributions, as described above, is particularly useful for proving the ξ-independence of the amplitude. We first prove gauge invariance by means of ξ-independence for the infinite parts proportional to ξ W or ξ Z . We find that the combination of δv/v and Π ′ HH (M 2 h ) in eq. (3.15) is ξ-independent. For the C ϕB contribution in eq. (3.15), the ξ W -dependent terms inside Π γγ (0) and A Zγ (0) cancel among each other, as they should since the infinite part of Γ ϕB is ξ-independent by itself. For contributions proportional to C ϕW (C ϕW B ), the ξ W cancellations take place throughout the self-energy contributions and Γ ϕW (Γ ϕW B ). Furthermore, diagrams proportional to C X with X = ϕB, ϕW, ϕW B, contributing to the last term of eq. (3.15), are gauge invariant on their own. Of course Γ SM is finite and gauge invariant as it is known from a direct calculation in R ξ -gauges with dimensional regularisation [23]. 9 We then prove analytically the cancellation of all ξ-dependent finite parts. This was done by first performing a maximal reduction on the related Passarino-Veltman functions [53] and then analytically checking for ξ-dependence among the parametric integrals. This is a highly non-trivial check of the validity of our calculation because the gauge parameter ξ appears everywhere in both the SM and SMEFT contributions which are directly related to the h → γγ amplitude. Moreover, this should be also considered as a direct proof for the validity of the expressions for vertices given in ref. [13] in general R ξ -gauges. Most importantly, the ξ-cancellation shows that the amplitude A µν (h → γγ) given in eq. (3.15) is gauge invariant as it should be. Needless to say, this is a very encouraging indication towards the correctness of our final result.
As an additional non-trivial check of our calculation, we have also proved gauge invariance for our amplitude before adopting any renormalisation scheme. We confirm that the regularised but yet unrenormalised S-matrix amplitude for h → γγ, written in terms of bare parameters, is gauge invariant.

MS scheme for Wilson coefficients
All renormalised coefficients, say C, and the counterterms, δC, in eq. (3.15), can be readily written in terms of the MS-scheme running C-coefficients as where µ is the renormalisation (or subtraction) scale that lays somewhere between the EW scale and the scale Λ, while δC is a counterterm that subtracts only terms proportional to in the loop corrections for the Wilson C-coefficients. In MS scheme and at one-loop, these counterterms are independent of the choice of the gauge fixing and can be read directly from refs. [46,51,52] to be , (3.25) where Γ u,d,e is our notation [11,13] for the usual Yukawa couplings in SM, and using Table 4 from ref. [13] the coefficients C ′ f are rotated to the fermion mass basis (denoted now as unprimed ones)

The amplitude
The remaining part of A µν (h → γγ) in eq. (3.15) is, at one-loop and up to 1/Λ 2 terms, renormalisation scale invariant: the renormalisation group running ofC(µ) coefficients cancels the explicit µ-dependence within various contributions in the RHS of eq. (3.15). Therefore, the amplitude, to be squared in finding the h → γγ decay width, is where (3.28) The subscript "finite" in the final parenthesis means that infinities proportional to E have been subtracted from all contributions in eq. Although allC(µ) coefficients in eq. (3.28) are MS parameters, the weak mixing angle θ W and the vev v that appear explicitly to multiply Wilson coefficients are defined in terms of physical quantities through eqs. (3.11) and (3.5) [see also eq. (4.16) below]. This is a virtue of our hybrid renormalisation scheme: SM on-shell parameters appear together with MS SMEFT parameters (Wilson coefficients) in the renormalised amplitude. This scheme can easily be applied to every process at one-loop in SMEFT.
From now on, all Wilson coefficients should be considered as running MS quantities, C ≡C(µ). We remove the "bar" over the MS-coefficients letting the argument to denote, or to implicitly imply, the difference.

Anatomy of the effective amplitude
In this section we present explicit expressions for the SM contribution, and, contributions proportional to all Wilson coefficients entering the h → γγ amplitude in eq. (3.28), and in Table 1. These coefficients are taken to be real. For clarity, we reinstate explicitly 1/Λ 2 factors in the expressions appeared in this and subsequent sections, so they are no longer incorporated into the definition of C's. Our EFT expansion stops at the order 1/Λ 2 and is one-loop at the -expansion. In our conventions, we denote electromagnetic fermion charges and the third component of particle weak isospin as and The colour factors are N c,e = 1 and N c,u = N c,d = 3. It is useful to note, when reading the expressions below, that the actual dimensionless EFT expansion parameter is 1 G F Λ 2 . To get a quantitative feeling of its numerical magnitude and to compare with standard loop expansion in the EW gauge couplings, we simply note that it is 1 G F M 2 W ∼ 4π, while for Λ = 1 TeV one has 1 G F Λ 2 ∼ 1 4π , for Λ = 10 TeV one has 1 G F Λ 2 ∼ α EM 4π and, finally, for Λ = 100 TeV one has

SM and C ϕW B , C ϕl(3) , C ll
The famous "SM" contributions from W and fermion triangle loops are represented by the penultimate term in eq. (3.28). This is with and Here Q f and m f are the fermion charge (in the units of proton charge), and mass, respectively, N c,f is the colour factor for fermions (3 for quarks, 1 for leptons) and The result is of course finite and is governed by a single function f (r), which reads It is useful for order of magnitude calculations to state that A 1 (r W ) ≈ 8.33, A 1/2 (r t ) ≈ 1.38 and I γγ ≈ −6.56 with a negligible imaginary part. The expression given in eq. (4.2) is not exactly the SM contribution for it is written in terms of SMEFT parameters and not in terms of measurable quantities like those listed in eq. (3.1). We therefore rewrite eq. (4.2) in terms of physical quantities using the expression forē from eq. (3.3) and G F from eq. (3.5) that bring in the new coefficients C ϕW B and C ϕl(3) Note that the piece before the square brackets on the RHS is the SM contribution to amplitude [up to a Lorentz factor in eq. (3.27)], as it would be calculated in the absence of any higher order operators. Inside the square brackets there are contributions from SMEFT i.e., running Wilson coefficients evaluated at a scale µ. Hence, the precise determination of the R h→γγ in eq. (1.1) is where the SM decay width reads, in accordance with standard refs. [23,54,55], as with I γγ given in eq. (4.3). The SMEFT contributions of eq. (4.8) are encoded in a part of δR h→γγ of eq. (4.9), in terms of measurable quantities s, c and G F , as where c 2 = 1 − s 2 = M 2 W /M 2 Z . Following our EFT expansion assumption, in obtaining eq. (4.11), corrections of O(1/Λ 4 ) have been consistently ignored.
4.2 C ϕD , C ϕ , C ϕ A direct calculation shows that the contribution from operators C ϕ and C ϕD is simply where Z h is the field redefinition factor for making the kinetic term of the Higgs field canonical in going from SM to SMEFT (see eq.(3.5) of ref. [13]) and iA SM is the full SM contribution to h → γγ amplitude. There is an explanation for this result based on the quantization of SMEFT presented in ref. [13]. In unitary gauge these operators appear in Higgs boson vertices (hW W and hf f ) with exactly the same Lorentz structure as in the corresponding SM vertices. On the other hand, in "renormalisable" gauges these operators appear in a complicated way e.g., there are contributions from Goldstone bosons hG 0 G 0 that have a non-trivial, non-SM Lorentz structure [13] and eq. (4.12) is not easily seen without performing the actual calculation. However, the result should be independent on the gauge choice as we explicitly confirm. We can view eq. (4.12) in a different way starting from the SM amplitude and perform the redefinition H = Z −1 h h on the single external Higgs boson leg.
As we already mentioned in section 2, the coefficient C ϕ does not contribute explicitly to the h → γγ amplitude in unitary gauge. Although there are apparent non-trivial contributions from it to vertices in R ξ -gauges, once again, gauge invariance implies that the amplitude is explicitly independent of C ϕ . Again, we explicitly verify this situation as well.
In summary, the contribution of operators discussed in this subsection to the ratio (4.9) reads trivially, up to ∼ 1/Λ 2 terms, as The relevant diagrams for these operators contain a fermion circulating in the loop. They contribute a ξ-independent piece in the last term of eq. (3.28) which takes the form (4.14) The contribution runs over all charged fermions f = e, u, d with their generation flavours denoted as i = 1, 2, 3, i.e., u 1 = u, u 2 = c, u 3 = t etc. The electromagnetic charges Q f and colour factors N c,f , are given in and below eq. (4.1). The function f (r) is defined in eq. (4.7). Turning all parameters into measurable ones in eq. (4.14) we obtain for the R h→γγ ratio of eq. (4.9) with A 1/2 (r) being a function defined in eq. (4.4) and I γγ defined in eq. (4.3). The function inside the square parenthesis peaks at the charm mass and as we shall see below [cf. eq. (5.1)] this is the most important contribution in δR h→γγ . All operators we have examined thus far are of PTG type. These operators create only finite contributions in the h → γγ amplitude. On contrary, operators that will be examined next will need to be renormalised.
The amplitude in eq. (3.28) contains contributions from Q ϕB , Q ϕW , Q ϕW B operators 11 appearing already at tree level in SMEFT. These are collected in the first three lines of eq. (3.28), but still contain the renormalised vev v. This parameter needs to be turned into Fermi coupling constant, G F , that is a measurable quantity with experimental value given in eq. (3.1). We only need the SM one loop corrections to ∆r, which appear through the expression .
Note that ∆r is a gauge invariant quantity and its form can be found in ref. [45]. This is consistent with our remark in section 3 that the pre-factors of C ϕB , C ϕW , C ϕW B in eq. (3.28) are respectively gauge invariant quantities and therefore the whole amplitude is gauge invariant. We then use eq. (3.5) to order 1/Λ 2 i.e., setḠ F → G F in eq. (4.16) and apply the result in eq. (3.28). We find that ∆r nicely cancels out when using an alternative expression for δv/v derived in ref. [47] in Feynman gauge ξ = 1, where the parameter E is given in ref. [47] The quantity A W W (0) is presented in ref. [50] in 't Hooft-Feynman gauge and is recalculated here for completeness in eq. (A.13). By putting eqs. (4.16) and (4.17) in eq. (3.28) we obtain the relevant finite contributions from operators Q ϕB , Q ϕW , Q ϕW B , to the physical amplitude A h→γγ This expression takes this particular form only in ξ = 1 gauge and replaces the first three lines in eq. (3.28). It is important for the reader to notice, that numerically big corrections from ∆r have been cancelled out in eq. (4.19). The quantities Γ ϕV , V = B, W, W B are fairly lengthy and are given in the Appendix A together with the self-energies, all in general R ξ -gauges. Nevertheless, following our tactic here, we can write down a clear formula for the relevant corrections to the ratio R (4) h→γγ in eq.(4.9), as (recall that tan 20) where I ϕB , I ϕW , I ϕW B represent the expressions in corresponding squared brackets of eq. (4.19).

C W
The contribution from W -loops gives rise to terms proportional to C W in eq. (3.28). The relevant expression is ξ-independent, and is written as where E is the infinite piece [see eq. (3.22)] formed as usual in dimensional regularisation, of course removed from eq. (3.28). The integral function B is where the functions f (r), J 2 (r) are given in eqs. (4.7) and (A.11), respectively, and µ is the renormalisation scale. The contribution from the operator Q W in the ratio (4.9) is with I γγ defined in eq. (4.3).
4.6 C eB , C eW , C uB , C uW , C dB , C dW These are again contributions from operators affecting fermion loops and, as such, they are ξindependent. They are, however, infinite since they involve dipole operators (as one can easily see from ref. [13] there is an extra momentum in the numerator of their corresponding Feynman rules expressions). We obtain the following contribution in the last term of eq. (3.28): where the function D(r f i ) is defined as Here again f stands for a fermion type, f = e, u, d, and i = 1, 2, 3 runs over its flavour eigenstates. The relevant contribution from the operators Q f B and Q f W to the ratio R h→γγ of eq. (4.9) is Functions I γγ , f (r) and J 2 (r) are defined in eqs. (4.3), (4.7) and (A.11), respectively. The expression δR (6) h→γγ in eq. (4.27) has few interesting features. It is proportional to the mass of the fermion circulated in the loop and also proportional to O(1) loop functions ratio. Comparing δR (6) h→γγ , which arises from LG operators, with, for example, δR (3) h→γγ of eq. (4.15) which arises from PTG operators and recall Table 2, we see that there is a huge enhancement of the former by a factor of O(10) in particular for the top-quark. Hence, for the top quark in the loop and for µ = M W , this is the biggest correction from all one-loop contributions in SMEFT as we shall see shortly in section 5.

Semi-numerical expression for the ratio R h→γγ
In this section, we sum all contributions to R h→γγ found in section 4, leaving as unknowns, the renormalisation group running Wilson coefficients, C = C(µ), the renormalisation scale µ divided by the W -boson mass and the energy scale Λ. Everything we have discussed so far is within the perturbative renormalisation framework explained in section 3. For EFT expansion to be valid, this means that the maximum value of a generic coefficient, C/Λ 2 , is at most O(1). Experimentally, it is suggested from eq. (1.2) that the corrections to δR h→γγ should be at most 15%. Being conservative, and in order to display all "important" contributions from operators in δR h→γγ , we present below semi-numerical results for δR h→γγ that are up to 1% × C/Λ 2 .
With the energy scale Λ written in TeV units, we obtain (in Warsaw basis) 12 : where the ellipses denote contributions from the operators Q in Table 1 that are less than 1% × C/Λ 2 . Terms in the first three parentheses arise from finite loop contributions, δR (1,2,3) h→γγ in eqs. (4.11), (4.13) and (4.15), while all the rest arise from "infinite" diagrams; for these the renormalisation scale µ appears explicitly. All coefficients are running quantities, C = C(µ), and δR h→γγ should be RGE invariant up to one-loop and up to 1/Λ 2 expansion terms. This can be checked numerically already from the explicit µ-dependence in eq. (5.1) and the β-functions for the C-coefficients calculated in refs. [46,51,52]. 13 Furthermore, we remark that in eq. (5.1) and for µ = 1 TeV, the logarithmic parts are of the same order of magnitude as the finite, constant, parts. Interestingly, for the coefficients in the last three lines of eq. (5.1), the two parts constructively interfere, while for the rest of coefficients they partially cancel.
At the end of the day, only five operators in eq. (5.1) can be bounded by the LHC experimental measurement (1.2) of the ratio R h→γγ . Taking µ = M W , we find All bounded coefficients above are associated with LG operators in Table 2 in a perturbative decoupled UV-theory. Eq. (5.2) seems to be consistent with this observation and Λ ≈ 1 TeV. On the other hand, assuming |C ϕV | (|C uB,uW 33 |) ≃ 1 we obtain Λ 10 (3) TeV, outside but close to the near-future LHC region. Other operators in eq. (5.1) may contribute at most 15% only when C = 1 and Λ = 1 TeV so their effects are less likely to be observed at present in LHC searches for the h → γγ process.
Operators Q ϕB , Q ϕW and Q ϕW B contribute already at tree level in SMEFT and this explains the large value of their coefficients in eq. (5.1). As our calculation shows, taking also into account one-loop corrections, modify their respective tree level contributions to the ratio δR h→γγ by 1.3% for C ϕB , by 7.5% for C ϕW B and by 8.7% for C ϕW at the renormalisation scale µ = M W , in agreement with the commonly expected magnitude of the SM-like electroweak one-loop corrections. What is surprising however, is the large loop contribution of dipole operators Q 33 uB,uW . This is basically due to the largeness of the top-quark mass and other features already noted in the discussion below eq. (4.26).

Other constraints
In the section above, we found that the dominant coefficients in R h→γγ are those given in eq. (5.2). These coefficients maybe also bounded by observables other than h → γγ. It has been noted in refs. [59,60] that the coefficient C ϕW B contributes directly to the electroweak S-parameter, one of the parameters that fits Z-pole observables. Its contribution reads With ∆S ∈ [−0.06, 0.07] [39] we obtain |C ϕW B | Λ 2 0.005 TeV −2 which is of the same order of magnitude as the upper bound we find here in eq. (5.2) from h → γγ measurement. The coefficients C ϕW and C ϕB are constrained by LHC Higgs data (giving upper limits on deviations from the SM predictions) or electroweak fits to EW observables. The respective bounds, as they read from refs. [39,61], are also about the same order of magnitude as in eq. (5.2).
The other two operators in eq. (5.2), Q 33 uB and Q 33 uW , are constrained from thettZ production and the latter also by the single top production measurements at LHC. Bounds quoted in ref. [62] are |C uB 33 |/Λ 2 7.1 TeV −2 and |C uW 33 |/Λ 2 2.5 TeV −2 . Here, bounds from h → γγ derived in eq. (5.2) are more than an order of magnitude stronger.
Restrictions to all other coefficients appeared in eq. (5.1) can be found in various articles in the literature. For example, following ref. [39], Q ϕD contributes to the T -electroweak parameter and the corresponding bound is, |C ϕD |/Λ 2 0.03 TeV −2 . This makes its contribution in h → γγ negligible. However, the coefficients C ϕ and C W are not really constrained by fitting the LHC Higgs data. It is obvious from eq. (5.1) that these two coefficients can give O(10)% contributions to R h→γγ only when one is in the vicinity of EFT validity.

h → γγ relevant UV-models
The question we want to address here is related to possible UV-field theories connected with the Wilson coefficients of eq. (5.1) contributing to the h → γγ amplitude. A possible UV-theory, valid in and above the neighbourhood of the energy scale Λ, contains heavy (w.r.t. the EW scale) fields, which upon their integration out result in a subset of SMEFT operators. Following an analysis based on ref. [63], under the assumption that at energies much smaller than all masses M of the extra particles (M ∼ Λ) the theory is well described by SMEFT, power counting rules result, interestingly, in a limited number of allowed heavy fields with definite quantum numbers and spins 0, 1/2 and 1. Other assumptions include that a candidate UV-theory must be invariant under the linearly realised SM-gauge group, is non-anomalous, and it must contain a multiplet with the SM Higgs field in representation (SU (3 We divide the Wilson coefficients appeared in eq. (5.1) into PTG and LG operators [40] as in Table 2. Then, following the tables in Appendix C of ref. [63], we check which coefficients can originate from integrating out fields with certain quantum numbers. Our results are shown in Tables 3 and 4. There are 5 spin-0 scalars, 13 Weyl fermions with vector-like masses, and 5 spin-1 gauge bosons, that can possibly appear in a UV-theory and affect the h → γγ amplitude through eq. (5.1). Remarkably, the LG coefficients in Table 4 are only a small subset of the PTG ones shown in Table 3. In addition, the C W -coefficient is absent from both Tables 3 and 4.  Tables 3 and 4, which in connection with Appendix D of ref. [63] relate the Wilson coefficient to the actual couplings of heavy fields, can be used to put bounds on the latter. We illustrate it by presenting an example. Imagine a triplet scalar, Ξ(1, 3) 0 , that is directly found or implied by an experiment with mass M in the TeV-range. According to Tables 3 and 4, at low energies there are contributions from "integrating out" Ξ in PTG coefficients C ϕ , C ϕD , C uϕ , C dϕ , C eϕ and in a LG coefficient C ϕW B . From eq. (5.1) we obtain that C ϕ and C ϕW B are multiplied by the biggest pre-factors and therefore play more important role in h → γγ amplitude. The UV-Lagrangian which originates these, is [63] where τ I are Pauli matrices and f is an energy scale with Λ 4πf . From Appendix D of ref. [63] we identify Potentially Tree Generated (PTG) Operators involved in h → γγ Spin Field Spin-1 Table 3: Dictionary for possible UV-completions with fields that, upon their "integration out", lead to PTG operators affecting the h → γγ amplitude in eq. (5.1). Flavour indices are suppressed. The field notation follows ref. [63]. and using our eq. (5.1) we find (5.6) If κ takes on its maximal value thenκ/f 0.004 TeV −1 . Of course one can advance a similar analysis in every case of an observable, not necessarily h → γγ, that is needed to be explained by a subset of fields affecting eq. (5.1).

Comparison with literature
As we mentioned in the introduction, the calculation for h → γγ in SMEFT was first performed several years ago in refs. [33,34] and to our knowledge these are the only complete studies prior to ours here. Our check shows that there are two, numerically important differences. First, all corresponding δR h→γγ in ref. [33] are smaller by exactly a factor of four. We think that this is due to a mistake in eq. (26) of ref. [33][arXiv v3]. Second, our eq. (4.15) is not in agreement with the corresponding expression of ref. [33]. We believe there is a Yukawa coupling missing for each generation and flavour in the corresponding expression of ref. [33]. Up to the aforementioned differences, we found agreement with δR (1,2,3,5,6) h→γγ . As far as δR (4) h→γγ is concerned, a direct comparison of our formulae in eq. (4.19) with the corresponding one in ref. [34] is very difficult. Checking individually quantities appearing in both works, for example, δv/v or Π ′ HH , is meaningless since the calculations in refs. [33,34] were performed in background field gauges while ours in linear R ξ -gauges. Comparing numerically the correction, δR (4) h→γγ , appearing in our eq. (5.1) with a corresponding ratio based on refs. [33,34], we find, upon fixing the factor of four mentioned above, a maximal difference of 5% for µ = M W , originating from what multiplies the coefficient C ϕB .

Conclusions
In our analysis we have calculated the one-loop decay width of the h → γγ process in the SM extended by all CP-conserving gauge invariant operators up to dimension-6 in Warsaw basis. We performed the calculations using the general R ξ -gauges and a hybrid renormalisation scheme, where we assumed the on-shell conditions for the SM parameters and MS subtraction for the running Wilson coefficients of the higher order operators. We explicitly checked the gauge ξ-parameter cancellation, which provides the very strict test of correctness of our calculations. In addition, we also explicitly proven that at the one-loop and 1/Λ 2 order, the calculated amplitude is independent of the renormalisation scale µ. Our work is complementary to previous analyses [33,34] of this process using the Background Field Method and comparisons of our results with theirs were made whenever possible. Our master formula for the S-matrix amplitude is given by eqs. (3.27) and (3.28).
We give a complete set of analytical formulae for all classes of SM and SMEFT contributions to h → γγ decay rate, normalised to the SM result as in published LHC searches [see eq. (4.9)]. We also present them in a form of simple and compact semi-analytical expressions depending only on running Wilson coefficients and renormalisation scale µ. Eq. (5.1) summarises all dominant contributions. Such formula can be readily used as additional constraint in experimental or theoretical analyses considering other observables in SMEFT.
We show that numerically largest corrections to the SM prediction can arise from Q ϕB , Q ϕW and Q ϕW B operators, contributing already at the tree level, and from Q 33 uB , Q 33 uW operators arising at the loop level. Only Wilson coefficients of these operators can be meaningfully constrained using the current precision of the LHC measurements for the h → γγ decay width. In some cases, like C uB 33 and C uW 33 , such constraints are already stronger than those from other measurements, in this case for instance from top-quark LHC-physics. Furthermore, we consider possible UV-field theoretic model completions at an energy scale nearby and above Λ. After integration out of all possible heavy fields, tabulated in Tables 3 and 4, we list all possible SMEFT operators that originate and affect the h → γγ amplitude in eq. (5.1).
A general look of our SMEFT calculational framework does not differ from common frameworks calculating electroweak one-loop corrections, like in the renormalisable SM for example. Our work can easily be automatised although we performed as many manual calculations we could for comparisons and cross checks. For example, one can use the SMEFT Feynman rules, given also in a Mathematica code, from ref. [13], and existed codes to calculate Feynman diagrams, employ a "traditional" renormalisation prescription from 80's described also here and, checking gauge invariance at every step, present a concise form of an amplitude in a useful semi-numeric form, as in eq. (5.1). It is worth for pursuing this SMEFT framework further. and the light quark contribution needed in eq. (4.21) is Π had γγ (M 2 Z ) =ḡ 2 g ′2 12π 2 (ḡ 2 +ḡ ′2 ) q N c Q 2 q E − log m 2 q µ 2 + 1 + r qZ 2 J 2 (r qZ ) + r qZ + 5 3 .