Electroweak Corrections to Higgs to $\gamma\gamma$ and $W^+W^-$ in the SMEFT

Higgs decays to gauge boson pairs are a crucial ingredient in the study of Higgs properties, with the decay $H\rightarrow\gamma\gamma$ being particularly sensitive to new physics effects. Assuming all potential new physics occurs at energies much above the weak scale, deviations from Standard Model predictions can be parameterized in terms of the coefficients of an effective field theory (SMEFT). When experimental limits on the SMEFT coefficients reach an accuracy of a few percent, predictions must be done beyond the lowest order in the SMEFT in order to match theory and experimental accuracy. This paper completes a program of computing the one-loop electroweak SMEFT corrections to $H\rightarrow VV^\prime$, $V=W^\pm,Z,\gamma$. The calculation of the real contribution to $H\rightarrow W^+W^-\gamma$ is performed by mapping two-loop amplitudes to the $3-$ body phase space.


I. INTRODUCTION
The LHC Higgs program is entering an era of precision measurements that requires a program of higher order theoretical calculations .The need for precise calculations is driven by: (1) the non-discovery of new particles that implies that the scale of Beyond the Standard Model (BSM) physics must typically be much higher than ∼ 1 T eV and (2) the anticipated precision in the Higgs measurements at the high luminosity LHC [1,2].In order to study deviations of Higgs properties from the SM predictions, a consistent theoretical framework is needed so that the accuracy of theoretical calculations is comparable to that of the measurements.
The Standard Model (SM) QCD and electroweak contributions to Higgs production and decay are known to at least NLO for all relevant processes and provide a framework for comparison [3].In the LHC Run-1, deviations of Higgs measurements from SM predictions were typically expressed in terms of limits on coupling constant modifiers [4].This κ approach rescales all Higgs couplings by constant factors and is not sensitive to kinematic distributions.As measurements approach the level of 5 − 10% accuracy, however, it becomes necessary to include electroweak corrections to the predictions, which in turn necessitates the use of effective field theory techniques, since electroweak corrections typically cannot be incorporated into a simple rescaling of the Higgs couplings.
The use of effective field theories for studying Higgs production and decay is well established [5][6][7].The SM effective field theory (SMEFT) assumes that the Higgs is an SU (2) L doublet and parameterizes new physics through an expansion in higher dimensional operators, where the SU (3) × SU (2) L × U (1) Y invariant dimension-k operators are constructed from SM fields and all of the BSM physics effects reside in the coefficient functions, C k i .If the scale Λ >> v, then it suffices to truncate the expansion at dimension-6.We need predictions to NLO QCD and EW accuracy in the SMEFT so that the theoretical predictions have roughly the same uncertainties as the experimental results.For processes with strong interactions, many NLO QCD results in the SMEFT exist, particularly in the top-Higgs sector [8].Electroweak corrections in the SMEFT [9] are available for only a handful of processes: H → bb [10,11], H → γγ [9,[12][13][14], H → Zγ [15] and Z → f f [16].Here, (from [19]).For brevity we suppress fermion chiral indices L, R. I = 1, 2, 3 is an SU (2) index, p, r are flavor indices, and we complete the program of computing the on-shell decays H → V V , (V = Z, W ± , γ), at one-loop in the SMEFT.Previously, we presented one-loop SMEFT results for H → Zγ and for the (unphysical) on-shell decay H → ZZ and [15].
In this paper, we present the one-loop SMEFT results for H → γγ and the on-shell process H → W + W − .The result for the decay H → γγ follows from the results of Ref. [15] and we compare with the results of Refs.[12][13][14].We consider two different renormalization schemes in order to assess the numerical significance of the scheme dependence.Our result contains the full (constant plus logarithmic terms) SMEFT result for the renormalization of G F1 .
Our one-loop H → W + W − result is an intermediate step on the way to the physical process H → W + W − → 4 fermions.The calculation of the real contributions from H → W + W − γ is performed using a mapping of the 3-body phase space to 2-loop amplitudes, which is of technical interest [17].
Section II reviews the one-loop electroweak renormalization for H → V V decays, Section III has results for H → W + W − , and Section IV contains the one-loop results for H → γγ.
Conclusions are contained in Section V.

II. BASICS
We use the Warsaw basis [18,19] where the relevant operators for the one-loop contributions to the decays H → V V are given in Table I and the Feynman rules and conventions in R ξ gauge are taken from Ref. [20].For simplicity, we assume a diagonal flavor structure for the coefficients C, i.e.C i p,r , where p, r are flavor indices.Furthermore, we assume The Higgs Lagrangian is, where φ is the usual Higgs doublet: and v is the vacuum expectation value (vev) defined as the minimum of the potential, The Higgs kinetic terms in the resulting Lagrangian are not canonically normalized due to O φ and O φD .As a consequence we need to shift the fields, The physical mass of the Higgs to The SMEFT interactions also cause the gauge field kinetic energies to have non-canonical normalizations and following Ref.[20], we define "barred" fields and couplings, such that W µ g 2 = W µ g 2 and B µ g 1 = B µ g 1 .The "barred" fields defined in this way have properly normalized kinetic energy terms.The masses of the W and Z fields to Dimension-6 4-fermion operators give contributions to the decay of the µ, changing the relation between the vev, v, and the Fermi constant G µ .Considering only contributions that interfere with the SM amplitude, we obtain the tree level result, where we assume the C i are flavor universal.The tadpole counterterms are defined such that they cancel completely the tadpole graphs [22].This condition identifies the renormalized vacuum as the minimum of the renormalized scalar potential at each order of perturbation theory.
Since the SMEFT theory is only renormalizable order by order in the (v 2 /Λ 2 ) expansion, we drop all terms proportional to (v 2 /Λ 2 ) a with a > 1.The one-loop SMEFT calculations contain both tree level and one-loop contributions from the dimension-6 operators, along with the full electroweak one-loop SM amplitudes.
We use a modified on shell (OS) scheme, where the SM parameters are OS quantities.
Since the coefficients of the dimension-6 operators are not physical observables, we treat them as M S parameters, so the renormalized coefficients are C(µ) = C 0 − poles, where C 0 are the bare quantities.The poles of the coefficients C 0 are found from the renormalization group evolution of the coefficients computed in the unbroken phase of the theory in Refs.[21,23,24], where µ is the renormalization scale, γ ij is the one-loop anomalous dimension, and ˆ −1 ≡ −1 − γ E + log(4π).At one-loop, tree level parameters (denoted with the subscript 0 in this section) must be renormalized.The renormalized SM masses are defined by, where Π V V (M 2 V ) is the one-loop correction to the 2-point function for Z or W computed on-shell.The gauge boson 2-point functions in the SMEFT can be found analytically in Refs.[9,25].
The one-loop relation between the vev and the Fermi constant is, where v 0 is the unrenormalized minimum of the potential and ∆r is obtained from the oneloop corrections to µ decay.Analytic expressions for ∆r in both the SM and the SMEFT at dimension-6 are given in Ref. [15].
The calculation proceeds in the same way as Ref. [15].We obtain the relevant amplitudes using FeynArts [26] with a model file generated by FeynRules [27] with the Feynman rules presented in [20].Then we use FeynCalc [28,29] to manipulate and reduce the integrals and LoopTools [30] for the numerical evaluation.
We consider 2 renormalization schemes.For the W + W − calculation, we choose the G µ scheme, where we take the physical input parameters to be 2 We then follow the same procedure as in Ref. [15].In our discussion, we term this the "G µ , M W , M Z scheme".
For the decay H → γγ, we consider the effects of explicitly pulling out an overall factor of α from the amplitudes, that is we calculate where F is a function of the bare parameters v 0 , M 0,W , M 0,Z , that we renormalize as described before and express in terms of G µ , M W and M Z .The on-shell renormalization of the overall factor α is extracted from the renormalization of the γ ll vertex and we take the physical parameter α = 1 137.035999139(31) .
We term this the "α, G µ , M W , M Z scheme".
The tree level decay width for H → W + W − receives contributions from the rescaling of the Higgs field (Eq.5), the SMEFT contributions fo G µ (Eq.9) and the direct interaction of O φW .For M H = 200 GeV , the numerical result SMEFT tree level result in GeV is, We have retained terms of O(C 2 i ) in Eq. 16 although the numerical coefficients are suppressed relative to those of the O(C i ) terms.The usual tree level scaling factor is defined to At NLO in the SMEFT, the decay width receives contributions from the one-loop virtual diagrams, including the renormalization terms discussed in the previous section, and from real photon emission.These contributions are separately IR divergent and we regulate them with a photon mass.The SM rate including all electroweak corrections is well known, both for the on-shell decay H → W + W − [31] and the off-shell decays, H →4 fermions [32].The off-shell effects are known to be significant for the physical M H = 125 GeV Higgs and the extension of our calculation to include the off-shell effects is clearly a needed step.The SM electroweak corrections are of order ∼ 6% for our reference Higgs mass, M H = 200 GeV .
The calculation of the virtual contribution in the SMEFT follows the identical procedure as for H → ZZ, with the exception of the introduction of a finite photon mass.The renormalization prescription is described in the previous section.
The IR divergences from the virtual diagrams are cancelled by real photon emission contributions, H → W W γ. Due to the complex Lorentz structures of the SMEFT vertices, the calculation of the width H → W W γ through direct integration of the phase space is extremely intricate.In order to calculate the real corrections we used the method developed in [17], where the integration over the phase space is replaced with a loop integration.This is possible after we recognize that the Cutkosky rules allow us to replace the delta functions inside the phase space integrations with propagators: After making this replacement, we can treat the momenta of the outgoing particles as internal loop momenta, and the integration over the phase space becomes an integration over the loop momenta.This allow us to use the IBP relations to reduce the loop integrals in terms of Master Integrals (MI).The methodology of this approach is described in Ref. [17].
In the specific case of H → W W γ, the integrals obtained are 2-point 2-loop integrals, for which a generic basis of MI is known [33,34].The reduction was done using FIRE [35].Since many 2-point 2-loop MI are known analytically, and the rest can be calculated numerically with high precision, for example using TSIL [36], we evaluate the MI directly and take the imaginary part of the result.An important caveat is that after the reduction to MI, we have to select only the MI that still have a physical W W γ cut, while we put to zero those that have lost one or more of the propagators generated by eq. ( 18).An interesting consequence of this is that if an integral can be cut in more than one way it is necessary to add a counterterm to cancel the extra imaginary part.As an example of this proceedure, see Fig. 1.
We have verified analytically that the IR divergences proportional to the photon mass cancel using this technique.
The total width is then the sum of the virtual and real contributions, and is given for (1) where the coefficients are evaluated at the scale Λ and X Λ = log(Λ 2 /M 2 Z ).(The terms in the square brackets occur at tree level.) We define the (on-shell) scaling factor at one-loop for M H = 200 GeV and Λ = 1 T eV , The change in the coefficients of operators that appear at tree level (in the square brackets in Eq. 20) is typically a few percent, while a few of the operators that first appear at one-loop have sizable coefficients and could potentially be probed in H → W + W − decays.

IV. H → γγ
As a by-product of our calculation of H → ZZ and H → Zγ [15], we obtain the SMEFT result for H → γ µ (p 1 )γ ν (p 2 ) at one-loop.Gauge invariance requires that the one-loop amplitude take the form, where we have broken up the coefficient into the tree level SMEFT piece, F 0 SM EF T , the one-loop SM piece, F 1 SM , and the one loop SMEFT contribution, F 1 SM EF T .Initially, we take M W , M Z and G µ as input parameters.At tree level, there is only the SMEFT contribution, where The analytic one-loop SM result can be found in many places [37] and we write the results numerically.The SMEFT logarithms contributing to F 1 SM EF T can be obtained from the anomalous dimensions given in Ref. [21] and are written in terms of The full 1-loop SMEFT result for H → γγ is extracted from the calculation of Ref. [15] (for the input parameters of Eq. 14).Our complete result is, • G µ , M W , M Z Scheme: φl (Λ) + 0.02258C ll (Λ) .
The coefficients are given in GeV and are evaluated at the scale Λ.This is the appropriate scale for matching with high-scale UV complete models [38][39][40][41].Note the dependence at one-loop on coefficients that do not appear at tree level, leading to the interesting possibility of obtaining limits on previously unconstrained coefficients.
We recalculate the result using α, G µ , M Z , and M W as inputs, as described in Sec.II.
We also note that in both schemes, typically the coefficients of the logarithms are of similar sizes to the constant pieces and that the differences between the schemes are small in most cases.The loop corrections to C φW , C φB and C φW B are on the order of 1 − 2%, relative to the tree level results.
Our results can be compared with the limits from ATLAS [4,42] and CMS [4,43], AT LAS, Run − 2 : µ γγ = 0.99 ± 0.15 AT LAS, Run − 1 : µ γγ = 1.14 ± 0.27 CM S, Run − 2 : µ γγ = 1.18 ± 0.17 We make the simplifying assumption that there are no cancellations between terms and require that no single contribution saturate the experimental bound.This is probably a poor assumption, since in any specific model, there are relations between the SMEFT coefficients [39,40,44].When the complete set of one-loop SMEFT predictions to Higgs decay is known, it will be possible to do a global fit incorporating these effects.In Fig. 2 we show the bounds on the coefficients that occur at tree level.The argument of the logarithms is evaluated at Λ = 1 T eV .The solid lines are the contributions in the G µ , M Z , M W scheme and the dotted lines are the G µ , M Z , M W , α scheme.Requiring that 0 < µ γγ < 1.28, we find for Λ = 1 The coefficients can be evolved to low scales, µ ∼ M Z , using the anomalous dimension matrix, where the anomalous dimension matrices can be found in Refs.[21] and the analogous numerical result to Eq. 28, but with the coefficients evaluated at a low scale, can be found in Ref. [14].
On the LHS of Fig. 3 we show the contributions to µ γγ from C uB and C uW .These coefficients first appear at loop level and it is interesting that H → γγ has the potential to place limits on them.We find (for Λ = 1 TeV), | C uB (Λ) | < 0.14 (H → γγ limit) The contribution from C W is shown on the LHS of Fig. 3.This operator is particularly interesting because it contributes to W + W − pair production [45,46].Translating the tree level results of Ref. [47] into our notation, we have for Λ = 1 T eV , C W < 0.08 (W + W − limit) .
From Fig. 3, the limit on C W from H → γγ assuming that C W is the only non-zero coefficient is | C W |< 0.3, significantly weaker than the limit from gauge boson pair production.

FIG. 1 :
FIG.1: Example of the calculation of W W γ phase space.From the reduction we obtain the central integral.There are four possible ways to cut it: two over W W and two over W W γ. Since we are interested in calculating the integral with a single physical cut over W W γ (left integral), we need to subtract a counterterm (right integral).

FIG. 2 : 2 − 5 . 2 0
FIG.2: Contributions to µ γγ when one operator at a time is varied, setting the remaining operators to 0. The coefficients are evaluated at the scale Λ = 1 T eV .

FIG. 3 :
FIG.3: Contributions to µ γγ when one operator at a time is varied, setting the remaining operators to 0. The coefficients are evaluated at the scale Λ = 1 T eV .On the LHS, the solid and dotted lines are indistinguishable.

TABLE I :
Dimension-6 operators relevant for the one-loop contributions to