The Higgs width in the SMEFT

We calculate the total and partial inclusive Higgs widths at leading order in the Standard Model Effective Field Theory (SMEFT). We report results incorporating SMEFT corrections for two and four body Higgs decays through vector currents in this limit. The narrow width approximation is avoided and all phase space integrals are directly evaluated. We explain why the narrow width approximation fails more significantly in the SMEFT compared to the SM, despite the narrowness of the observed $\rm SU(2) \times U(1)$ bosons in both theories. Our results are presented in a manner that allows various input parameter schemes to be used, and they allow the inclusive branching ratios and decay widths of the Higgs to be numerically determined without a Monte Carlo generation of phase space for each Wilson coefficient value chosen.


Introduction
In the Standard Model (SM) the width of the Higgs is small (∼ 4 MeV) compared to the Higgs mass of m h ∼ 125 GeV. The width is known to high accuracy in terms of the parameters of the SM, and this makes it interesting to study perturbations due to physics beyond the SM on the total and partial widths. Although difficult to directly measure, the Higgs width is essential to inferring the full set of partial widths from the observed branching ratios -which match well with SM predictions for the Higgs at the ∼ 10% level. Precise knowledge of the Higgs width is a key requirement to accurately interpreting experimental results on Higgs decays now and in the future. This remains true when the SM is extended into the Standard Model Effective Field theory (SMEFT). The SMEFT is defined under the assumptions that: physics beyond the SM is present at scales Λ > 2 H † H =v T , no light (m v T ) hidden states are lurking in the particle spectrum with couplings to the SM, and a SU(2) L scalar doublet with hypercharge y h = 1/2 is present in the low energy limit defining the EFT. 1 The SMEFT extends the SM with operators Q The operators Q are the Wilson coefficients. In this work we use the non-redundant L (6) Warsaw basis [2]. This basis removed some residual redundancies (see also [3,4]) in the over-complete basis of Ref. [5]. We often use the notationC = Cv 2 T /Λ 2 for dimensionless rescaled Wilson coefficients. A key strength of a SMEFT analysis of experimental data is that it represents a consistent general low energy (or infrared -IR) limit of physics beyond the SM, so long as its defining assumptions are satisfied, and all operators at each order in the power counting of the theory are retained. This is the approach we adopt in this paper. A further strength of the SMEFT is that it addresses a key challenge to the program of studying the Higgs precisely to look for deviations in its properties as a sign of physics beyond the SM. The difficulty of directly 1 More precisely the direct meaning of this standard assumption is that the local operators are analytic functions of the field H in the SMEFT. The analyticity of the local contact operators making up the SMEFT is a basic feature of this theory. This basic EFT point was discussed in the recent SMEFT review [1]. measuring the Higgs width experimentally model independently in the LHC environment is well known. For some related results see Refs. [6][7][8][9][10]. This fact is also relevant when considering successor machines for a future precision Higgs phenomenology program. It is important to stress that the perturbations to the Higgs width are systematically calculable and of a limited form in the SMEFT, when the assumptions of this theoretical framework are adopted. Due to this, even when the Higgs width is difficult to directly measure, it is possible to bound it indirectly due to calculating directly its allowed perturbations in the SMEFT.
In this paper, we report a consistent calculation of the width of the Higgs to order 1/Λ 2 for a set of two and four body decays (through vector currents) in the SMEFT. 2 Our results are presented in a semi-analytic fashion, with inclusive phase space integrals explicitly evaluated and reported. Our results allow the total inclusive width, partial widths and branching ratios to be determined as a function of the Wilson coefficients without a Monte Carlo generator being run. This allows the Wilson coefficient space of the SMEFT to be sampled efficiently in global studies of the properties of the Higgs, and combined with other particle physics experimental results. We believe this is of some value going forward in the LHC experimental program.
A key observation feeding into the important impact of the calculation reported here is the relative success of the narrow width approximation in the SM and the SMEFT. The narrow width approximation in the SM relies on the fact that SM interactions are of limited mass dimension (d ≤ 4) for its numerical adequacy in predicting many experimental results (this is the case as renormalizability leads to hγγ and hγZ effective vertices being one loop effects). In the SMEFT, the presence of interaction terms of mass dimension d > 4 leads to a more serious breakdown of the narrow width approximation, primarily due to neglected interference effects using this approximation. This is despite the fact that the SU(2) × U(1) gauge bosons remain narrow, with Γ/M 1. In this work we incorporate off-shell effects neglected in the narrow width approximation, and a consistent set of interference effects present in the SMEFT at LO for the processes we calculate, to address this issue.
The outline of this paper is as follows. In Section 3 we define how the {α ew ,m Z ,Ĝ F ,M h } and {m W ,m Z ,Ĝ F ,M h } electroweak parameter input schemes are related to Lagrangian parameters. In Section 4 we define some common Lagrangian parameter shifts, including vertex corrections, widths, and shifts to the propagators as combinations of Wilson coefficients. In Section 5 we define the consistent leading order results for the SMEFT corrections to a critical set of two and four body decays of the Higgs. This includes an extensive discussion of the results for four body Higgs decays, and the required determination of the phase space integrations over four body phase space. In Section 6 we discuss the numerical results and quantify the impact of different contributions, with special attention to the terms that are usually neglected when using the narrow width approximation for the W, Z bosons. Finally in Section 7 we conclude. 2 Four body decays where a vector is emitted off the fermion pair produced by the Higgs is considered beyond the scope of this work. These results, as well as a set of other interference effects that are also omitted here, will be included in a follow up work.

SM and SMEFT theoretical conventions
The SM Lagrangian [11][12][13] notation is fixed to be The chiral projectors have the convention ψ L/R = P L/R ψ where P R = (1 + γ 5 ) /2, and the gauge covariant derivative is defined with a positive sign convention The normalization of τ I is such that tr[τ I τ J ] = 2 δ IJ . Our conventions are consistent with Ref. [1], and we refer the reader to this work for more notational details. We use the notation k α ij = (k i + k j ) α and k 2 ij = (k i + k j ) 2 for the Lorentz invariant four vector and its square, with final state spinor pairs produced from the decay of a vector boson. For example, in the massless fermions limit, pairs (ū(k i ), v(k j )), (ū(k k ), v(k l )) can be produced by vectors carrying four momentum k 2 ij = 2k i · k j , k 2 kl = 2k k · k l .

Input schemes and analytical results
For the {α ew ,m Z ,Ĝ F } input parameter scheme many of these results are summarized in Ref. [1], which in turn is based on [14][15][16][17][18][19]. The corresponding results in the {m W ,m Z ,Ĝ F } input parameter scheme largely descend from Ref. [20]. Here we collect and complete the theoretical results used for a self contained presentation, and to define a consistent LO set of analytic results of the SMEFT. These results are then used to consistently define the Higgs width with leading SMEFT corrections. Our notation follows the "hat-bar" convention of Refs. [1,15,16]. Lagrangian parameters directly determined from the measured input parameters are defined as having hat superscripts. Lagrangian parameters in the canonically normalized SMEFT Lagrangian are indicated with bar superscripts. The differences between these parameters come about due to the SMEFT perturbations of the SM. With this convention, a leading order shift in a SM -3 -Lagrangian parameter (P ) due to the SMEFT, when such a parameter is determined from an input parameter set, is given by (3.1) Note the sign convention applied to these shift definitions, and that in the SM limit (C i → 0) hatted and bar quantities coincide and the SM inference from experimental results (at tree level) is recovered. The implementation of this convention has some historical legacies. δG F is dimensionless while G F has mass dimension minus two requiring a further dimensionful rescaling from a naive implementation of this convention. In unitary gauge, the Higgs doublet is expanded as to obtain a canonical normalization. Here H † H has been defined to include corrections due to L (6) so thatv T ≡ (1 + 3C Hv 2 /8λΛ 2 )v where 2 H † H SM ≡v. Below, we include cross terms in theoretical predictions, where higher order SM perturbative corrections interfere with the L (6) corrections. We note that the total contribution to S matrix elements is gauge invariant order by order in the SMEFT power counting expansion; i.e. the A SM amplitude contributing to an S matrix element through A SM × A (6) /Λ 2 is separately gauge invariant, as is A (6) /Λ 2 alone. The gauge fields are redefined into script fields to canonically normalize the SMEFT, including L (6) corrections, as The modified coupling constants are simultaneously redefined g 3 = g 3 1 +C HG , g 2 = g 2 1 +C HW , g 1 = g 1 1 +C HB , (3.4) so that the products g 3 G A µ = g 3 G A µ , etc. are unchanged. The rotated script field eigenstate basis for {W 3 µ , B µ } in the SMEFT to L (6) is given by [14,15] The L SM + L (6) rotation angle is tanθ =ḡ (3.6) The mass eigenstate fields of the SM {Z µ , A µ } are defined via the C HW B → 0 and {cos θ, sin θ} → {cos θ, sin θ} limit of Eq. (3.5) where c θ = cos θ = g 2 / g 2 1 + g 2 2 , s θ = sin θ = g 1 / g 2 1 + g 2 2 . The relation between the mass eigenstate fields in L SM and L SM + L (6) is explicitly [21] Z µ = Z µ 1 + s 2 θC HB + c 2 θC HW + sθcθC HW B , These expressions hold in both input parameter schemes using notation defined in the following section. L (8) corrections to this formalism where recently reported in Ref. [22] In addition to the electroweak input parameters we discuss below in detail, we also require experimental inputs to fix {m t , α s , m c , m b , m τ , V ij CKM , ∆α (5) had , · · · }. Barred mass parameters are generally defined to be the pole masses in L SM EF T , including L (6) corrections. For recent discussion and results on CKM parameters in the SMEFT from an input parameter perspective, see Refs. [21,23]. Note that we generally neglect terms in the SMEFT corrections to SM results relatively suppressed by small quark masses.
Hl ee The U(3) 5 limit used here treats the two flavour contractions of Q ll as independent [24] (C ll mnop δ mn δ op + C ll mnop δ mp δ no )(l m γ µ l n )(l o γ µ l p ). (3.13) We also define corrections to the Z and h mass parameters even though the corresponding input parameterM Z ,M h fix the location of the propagator pole, i.e. by definition a pole in a resonance scan is such that δM 2 We define shifts to the Z, h masses as a convenient shorthand notation for common combinations of Lagrangian parameters in L SMEFT . We are then faced with a notational conundrum, as the natural notational choice in each case is zero by definition. We overcome this challenge with a slight modification of notation compared to Ref. [1] by defining 14) where the lowercase m takes on an meaning distinguishing it from the uppercase M Z,h resonance pole mass, whose shift vanishes by definition. One should note this notational refinement when comparing to past works. See Ref. [1] for more details.

Preliminaries: some common parameter shifts
For each input parameter scheme, the expression for a physical observable depends (in part) on the shift in the usual SM Lagrangian parameters through the formulae in the previous two -6 -sections. Here we give a common set of such shifts. The δP are a useful short hand notation that can be used at times in a specific gauge, but do not span, and are not equivalent to, a complete and well defined gauge independent operator basis for L (6) in the SMEFT. The remaining SMEFT corrections to physical observables appear through the direct dependence on the operators in calculated amplitudes, and through the expansion of the W pole mass in the {α ew ,M Z ,Ĝ F ,M h } input parameter scheme.

Effective
In either input parameter scheme we can define the A µ effective couplings as where Q ψ = {2/3, −1/3, −1} for ψ = {u, d, e}. As class seven operators in the Warsaw basis are of the form H † i ← → D µ Hψγ µ ψ and the Higgs is uncharged under U(1) em , further flavour nonuniversal contact operator contributions due to expanding out these operators are not present. Chirality flipping dipole operators generate effective couplings of the photon field to U(1) em charged fermions at L 6 . However, as these contributions interfere with the SM amplitudes proportional to quark masses, even if the Wilson coefficient is not assumed proportional to the Yukawa matrix to impose a controlled breaking of flavour symmetry; these contributions are neglected.

Effective W µψ
L γ µ ψ L and W µ hψ L γ µ ψ L couplings In the case of the W effective couplings we define In the SM is the CKM matrix, U = U (e, L) † U (ν, L) the PMNS matrix, with U (ψ, L/R) the rotation matrix between the weak and mass eigenstates. For flavour diagonal components Hψ rr Hq pr Hq pr The off diagonal terms trivially follow. Again, we note that the left and right handed δg W,ψ L,R SMEFT corrections are the sum and difference of the vector and axial W shifts respectively.

Effective hψψ couplings
The pole masses of quarks and leptons inferred from experimental results can define input parametersm ψ . These inputs also determine the Yukawa couplings through the definition In the SM g SM hψ pp =Ŷ ψ pp / √ 2, and in the SMEFT [15] δg hψ pr =Ŷ ψ pr

Massive boson propagator and width shifts
For a consistent treatment of the SMEFT corrections to the SM, the propagators need to be expanded up to linear order in the Wilson coefficients, when a massive vector boson mediates an experimental measurement [19]. For a massive boson B = {Z, W, h} we define The propagator in unitary gauge is then Note that as we calculate in the massless limit of the final state fermions, the longitudinal term ∝ k ν k µ vanishes in this limit. The shift in the propagator is given by which can be directly used if V CKM , U P M N S phases are neglected and one considers a CP conserving set of Wilson coefficients in L (6) . In a near on-shell region of phase space The width shift should be included when studying experimental results in any scheme for a consistent SMEFT analysis. This can be done by expanding in the correction to the width, linearizing the dependence on the SMEFT correction in the final result for an observable. This procedure is difficult to directly carry out interfacing SMEFTsim [21] with MadGraph5 due to the implementation of widths in MadGraph5. Our results present this result for inclusive quantities in a semi-analytic form, determining this correction using direct numerical integration. This makes the dependence on the total width clear in the case of inclusive quantities, and at least clearer when considering non-inclusive quantities.
The decay of a Higgs boson to four fermion final states occurs through physical phase space where some of the intermediate off shell propagators are necessarily off-shell. The width of the unstable SM Bosons remains parametrically < m B in the SMEFT by definition, as the local contact operators are a perturbative correction. As such, one can still expand in the small ratios Γ B /M B , δΓ B /M B the modified propagator, finding At tree level corrections to Z partial and total widths due to L (6) are N ψ C depends on the SU(3) c representation of ψ. Off diagonal corrections due to local contact operators are neglected, as they interfere with SM contributions that have a significant numerical suppression. This reasoning is used in part to define a "pole parameter" set of SMEFT Wilson coefficients in Ref. [21], and our results are consistent with this reasoning. 3 Similarly corrections due to four fermion operators modify the inference of a partial Z width from an experimental cross section, with an intermediate Z boson. We also neglect these corrections as they are kinematically suppressed beyond the power counting suppression.

δΓ W
At tree level corrections to W partial and total widths due to L (6) are [19] V ψ corresponds to the CKM (ψ = q) or Hermitian conjugate of the PMNS (ψ = ) matrix.

Corrections to the partial and total Higgs decay widths
The total and partial Higgs width is also corrected in the SMEFT as follows.

δΓ h→ψψ
The decays to ψ = {u, c, d, s, b, e, µ, τ } are each modified as (5.1) 3 The neglect of flavour violating effects for the h interactions with fermions also follows from the structure of flavour changing effects in the SM.

δΓ h→AA
The leading order SM prediction of Γ(h → AA) is [25][26][27] The correction in the SMEFT due to L (6) is given by whereC AA =C HW /ĝ 2 2 +C HB /ĝ 2 1 −C HW B /ĝ 1ĝ2 . Here the first line indicates the scheme dependent linear expansion in SMEFT corrections feeding into the SM loop diagrams. The second term indicates a shift due to a possible shift in the W mass, while the fermion mass inputs are assumed to be pole masses. One loop calculations of this process in the SMEFT were reported in Refs. [28][29][30][31]. Such corrections are important, but they the same order (in the SMEFT expansion and the loop expansion) as the scheme dependent corrections to the SM results, which introduces scheme dependence that is only removed once a full one loop improvement of SMEFT predictions is obtained. We consistently drop such loop suppressed SMEFT effects in this work and only retain the contribution from the third line of Eq. (5.6).

Four fermion decays δΓ h→ψψψψ
The largest partial widths that remain are due to h → V V →ψψψψ, through combinations of vector bosons V = {W ± , Z, A}. These calculations, when the intermediate gauge bosons are allowed to be off-shell, have been developed for the SM in Refs. [7,[35][36][37][38][39]. Here we extend this approach to the SMEFT, avoiding an on-shell assumption and narrow width approximation to ensure the consistency of the SMEFT corrections included in a leading order analysis.
To define these corrections, it is useful to introduce the notation and J V ν ψ pr a (k 2 ij ) when the gauge boson V coupling to the current producing the final states labeled with ψ p,r a (of flavours p, r) carries four momentum squared k 2 ij . The definition of the propagator has assumed a width prescription that is consistent with the implementation of widths for unstable states in MadGraph5. 4 Further notation is defined as follows for example The flavour, colour and spin sums (denoted s ) in each case are restricted to the allowed final states. Note that we are not using a convention that repeated indicies are always summed. 5 At times the momentum dependence of the T are suppressed. Determining the partial width from the expressions that follow is defined aŝ for a 4 body phase space element with k i denoting the momentum of each final state spinor. In the Appendix we transform this four body phase space into a form where Lorentz invariants are integrated over in the phase space volume, to allow direct numerical evaluations. A convenient trace product to define for a compact presentation is The diagrams for pure charged current (CC) interference effects are shown in Fig. 1 We label h → W W →ψ 1 ψ 1ψ 2 ψ 2 as h → F 1 (ψ rs 1 , ψ tu 2 ). In the SM the corresponding leading order result is Here k ij,kl are the momentum carried by the W propagators associated with the spinor pairs (ū(k i ), v(k j )), (ū(k k ), v(k l )). The couplings in this expression areĝ 4 2ḡ 4 2 as theĝ 2 couplings are 4 A generalization of the results to a different width prescription and the complex mass scheme is clearly also of interest, but is beyond the scope of this work. 5 At times an Einstein summation convention is in place, particularly for flavour indicies for brevity of notation. The presence of a summation or not is believed to be clear from the physics in each case.
. Figure 1. Charged current contributions to h → 4ψ. The SM diagram also represents O(v 2 T /Λ 2 ) operator insertions perturbing the SM prediction through two charged current exchanges.
defined to be those that couple the vector to J W , while the remaining dependence descends from the hW 2 vertex.
The partial SM decay width at leading order is constructed from the one "kinematic number" We extract this number, and similar numbers below using various techniques to cross check results. When considering SMEFT corrections, novel kinematic numbers result from the novel populations of phase space due to the presence of local contact operators of mass dimension d > 4. An extended set of such kinematic numbers is required for describing four fermion Higgs decays in the SMEFT. Such corrections are a key difference from the κ formalism developed in Refs. [40][41][42][43][44][45][46][47].
We determine a kinematic number in a process using the Vegas algorithm and CUBA numerical integration package [48] primarily. These results are also cross checked in Mad-Graph5 [49] from leading order SM results. We have also (when possible) cross checked these results with an independent evaluation using the RAMBO algorithm [50] to directly determine the phase space integrals. In some cases, for phase space integrals that are highly singular, or in the presence of multiple poles, the Vegas numerical approach was considered an essential step to obtain a reliable determination of sufficient numerical accuracy. Some details on these approaches are given in the Appendix.
The SM result for the pure charged current partial width is The L (6) SMEFT corrections can be classified by the phase space integrations they multiply.
The partial width corrections that simply perturb the SM prediction proportional to N WW 19) in the limit that we neglect phases in the CKM and PMNS matrices. The generalization to the case where SM phases are not neglected is via The remaining corrections lead to non-SM phase space integrations due to the local contact operators present in the SMEFT, and are given by 2M h δΓ h→F 1 (ψ rs Here we have neglected interference effects due to SM phases in the propagator correction, and final state fermion masses. The relevant inclusive phase space integrals can be evaluated to be The shift in this inclusive partial decay width (δΓ h→F 1 (ψ rs ) SM ) can then be defined as (5.25) 6 Note the hat notation on the predicted observable is again used here.
-16 - The first term in this expression can be obtained from rescaling the SM result, which is consistent with the approach in the κ formalism. When the population of phase space in the SM and the SMEFT due to an interaction is the same, the ratio of kinematic numbers is one. The ratios of the kinematic numbers in the remaining terms give some intuition as to how the κ formalism fails due to the decay kinematics being able to differ in the SMEFT, compared to the SM. When measuring a decay channel, assumptions on SM like kinematics to define event rate acceptances is expected to require the introduction of further correction factors when the ratio of the kinematic numbers is very far from one. The acceptance correction will be strongly dependent on the detailed experimental signal definition and is not determined in the calculation reported here. Our results are intended to define and determine the theoretical inclusive total width and branching ratios in the SMEFT.  In the SM, this LO result is The second term in the expression above is complex. A relative sign in the two terms is due to Fermi statistics, and there are relative numerical factors due to counting Wick contractions.
Here k ij,kl,jk,li are the momentum carried by the Z propagators associated with the momenta of the final state spinors pairs (ū(k i ), v(k j )), (ū(k k ), v(k l )), (v(k j ), u(k k )), (v(k l ), u(k i )).
It is useful to expand these results out explicitly obtaining Integrating over phase space we extract the kinematic numbers, one finds With these results the corresponding SM inclusive partial widths are constructed as . A subset of the SMEFT L (6) corrections to this partial width directly follow as whereC ZZ = (c 2 θC HW +s 2 θC HB +cθ sθC HW B ). This set of δΓ(h → F 2 (ψ rr a , ψ ss b )) perturbations numerically reduce to (neglecting fermion masses) In the SM, the amplitudes with V V = {Z A, AA} are loop suppressed. This is not the case in the SMEFT in general [51]. This leads to a more significant breakdown of the narrow width approximation in the SMEFT. We include the tree level effects of these processes due to L (6) interfering with the SM process through ZZ for a consistent LO SMEFT analysis. The corresponding diagrams are shown in Fig. 3 and we define 7 Here we slightly abuse notation definingΓ ZZ   Zê 3v2 Here the labeled momentum P Z is generated in the effective hAZ vertex associated with C HAZ . This interference effect in the SMEFT with the SM neutral current mediated Higgs decay is given by The kinematic numbers can be approximated as We also define the following expression for interference with AA with the SM neutral currents This result contributes to a partial width as where the numerical results can be approximated as (5.43)

IR behavior when interfering with tree level photon exchange
The numerical evaluation of the four body phase space integrations in the cases with intermediate photons are more challenging than the remaining numerical evaluations. All the kinematic numbers are extracted with a direct numerical evaluation with the Vegas Monte Carlo integration algorithm and the CUBA package [48] and cross-checked both with the RAMBO phase space generator and with the numbers extracted from massless simulations in MadGraph5 with SMEFTsim. The numerical integration in Vegas was evaluated using both massless and massive phase space boundaries and validated with two different phase space variable sets and numerical methodologies. In the case of extracting an interference effect with a double photon pole, i.e. the results giving N ZZAA 1 , N ZZAA 2 , RAMBO did not converge with sufficient numerical accuracy to afford a cross-check of results and the Mad-Graph5 simulation was found to be subject to significant numerical uncertainties in the massless fermions case. Retaining fermion masses overcomes the latter issue, and allowed us to confirm the Vegas results.
The reason these results are numerically challenging to determine is due to the IR behavior of the corresponding phase space in the massless fermion limit. The phase space volume in part is where the photon invariant masses are κ 2 12 , κ 2 34 . A logarithmic dependence on the final state fermion masses results when integrating the phase space. We believe this is due to soft and collinear emissions of the final state fermions. For example, consider the massless fermion limit. The boundaries of the phase space volume are defined by with free parameters c i .A constant term was also included and determined in the fit. This expression should not be understood to imply that the massless limit is formally divergent, as cancellations can occur between the logarithmic and polylogarithmic terms shown. The massless limit is show in Fig.4, and is empirically found to be finite in our numerical fit. These fermion mass effects are numerically small enough to be neglected in the LO analysis included here, so long as an appropriate theoretical error is included in the corresponding theoretical predictions. In the case of the decay through AZ the IR limit is sufficiently regulated by the mass of the Z to further soften the logarithmic behavior.
It is important to note the interplay of these regions of phase space, where fermion masses regulate IR behavior in this manner, also coincide with the final state photon being reconstructed in the detector, not the experimental case where the photon has converted to two distinct final state fermions. As such, the regulation of phase space is practically cut off by detector effects and the signal definition, in addition to fermion masses, when this particular decay is studied experimentally. We stress that the results in Sections 5.2, 5.4 are not a double counting even in this collinear limit. The interference effects in each case are with distinct processes, at tree or the loop level in the SM. There is also a contribution due to the interference of the charged and neutral currents, where V = {Z, A} in this subsection. The corresponding diagrams are shown in Fig. 5. In the SM, as the couplings to ZA, AA are loop suppressed, and the LO expression is given by 8 leading to the SM result (5.50) 8 The presence of a minus sign again follows from Fermi statistics, see Ref. [7]. Figure 5. Neutral-charged current interference contributions to h → 4ψ. The SM diagrams also represent O(v 2 T /Λ 2 ) operator insertions perturbing the SM prediction with the same pole structure as the SM. with the kinematic number The SMEFT corrections to charged-neutral current interference are defined as δΓ h→F 5 (ψ ss a ,ψ rr b ) and is given by , and also

Numerical results and analysis of the contributions to h → 4f
Taking into account all of these results, the total Higgs width combining these decays is given by u,c,d,s, b,e,µ,τ } δΓ h→ψψ + δΓ h→AA + δΓ h→ZA + δΓ h→gg +  Table 1. Note that the fermion masses M b,c,τ were used for the h →f f channels but were neglected in the h → 4f estimates. The CKM matrix is always taken to be the unit matrix, thereby omitting flavor changing channels. Finally, the top quark mass is relevant for the numerical evaluation of the SM Higgs couplings to gg, Zγ, γγ (see Sec. 5.2 -5.4).
The SM predictions for 2-body decays (Table 7) are provided by the LHC Higgs Cross Section Working Group [52,53]. The SM predictions for the 4f channels (Table 8)   It is interesting to examine the impact of different contributions to the final result and in particular of contributions that were previously neglected, to our knowledge, in the estimate of SMEFT corrections to h → 4f . In the SM, these decays are well-described in a narrow-width approximation for the W, Z bosons, that gives Γ N C,nw.
for channels proceeding through NC, and analogously for charged currents 9 . The same approach is usually generalized to the SMEFT case, leading to estimates of the form The implementation of the narrow-width approximation in this context is not unique, as there is some arbitrariness in the choice of the contributions included in each term. However, the following classes of terms are often omitted in this approach:

Diagrams with intermediate off-shell photons.
Contributions containing the Zγ interaction are compatible with the narrow-width assumption for NC, and could therefore be included, while γγ-mediated diagrams are always missed in this approximation.
2. Interference terms between NC and CC contributions, that are not compatible with the amplitude factorization into (h →ψψV ) × (V →ψψ).
3. Interference terms between ZZ diagrams with different current contractions in channels with 2 indistinguishable fermion pairs (ψ a ψ aψ a ψ a vsψ a ψ aψ a ψ a ).

Propagator corrections for the off-shell boson.
In the following we isolate and quantify the impact of each of these terms.

Photon-mediated diagrams
As mentioned previously, due to its coupling to Zγ and γγ the Higgs boson can decay to 4 fermions via electromagnetic currents, in addition to the weak ones. In the SM this effect is negligible due to the hZγ, hγγ effective couplings being loop suppressed (this is essentially an accidental suppression due to the d ≤ 4 operator mass dimensions of the SM, for a related discussion see Ref. [51]). In the SMEFT, in contrast, these interactions formally arise at treelevel together with the leading corrections to the hZZ, hW W couplings. This is the prime reason that the narrow width approximation fails more dramatically in the SMEFT compared to the SM. The calculation presented in this work includes for the first time the interference terms 1 Γ SM,tree  which are proportional to either C AZ or C AA and therefore affect the dependence on C HW , C HB , C HW B . Table 2 shows the numerical contribution of these diagrams to the coefficients a (S) i in the linearized SMEFT expressions compared to the contributions from W W and ZZ diagrams 10 .
It is immediate to see that the photon contribution to these quantities is significant, especially for the Zγ terms that exceed in absolute value the ZZ, W W contributions in most channels. In several cases, the Zγ contribution flips the overall sign in the C i dependence compared to the one when only including ZZ, W W currents. The photon effect is largest for channels with NC only, and involving the up quark, due to a color factor and electromagnetic charge enhancement. Channels allowing both NC and CC decay are largely dominated by the CC diagrams, so both ZZ and photon contributions are suppressed.

NC-CC interference terms
The channels h → + −ν ν, h →ūudd with 4 fermions of the same generation admit both CC and NC diagrams. When assuming narrow W or Z bosons, one usually sums over the 4 configurations in which either a Z or a W is nearly on-shell. By construction this calculation neglects interference terms between diagrams mediated by W and Z.  Tables 3, 4 show a breakdown of the contributions to the quantities a (S) i for the relevant operators O i from different interference terms in the squared amplitude, obtained with the full computation. The first three rows report the contribution from ZZ and W W mediated diagrams, while the last three rows indicate the contributions from contact interactions HV f f . All numbers are normalized to the corresponding SM tree level partial width, that contains both ZZ, W W and interference terms. This table omits the contributions from the W, Z propagator corrections as well as contributions from photon diagrams.
As the SM partial width for both channels is dominated by the W W diagram, corrections to the latter are generally more important than corrections to the ZZ topology. The -30 -interference between the two gives significant contributions to the dependence on C HW B and on C (1) Hl in the leptonic case. The latter is due to an accidental cancellation between the corresponding charged lepton and neutrino corrections in ZZ diagrams that does not occur in the interference with W currents.

Interference between NC diagrams with different current contractions
Decays with 2 pairs of identical particles in final state admit 2 independent neutral-current contractions, depicted in Figure 6 for the ZZ and contact-term cases. The same contractions are allowed for photon mediated diagrams.
In the squared amplitude, the "direct" products A 1234 A 1234, † , A 1432 A 1432, † are related by relabeling of the final states and give therefore identical results, while the "crossed" interference A 1234 A 1432, † provides an independent contribution, that is neglected in the narrow width approximation.
In the complete calculation, the "crossed" interference terms are found to be most relevant in the h → + − + − channel, particularly for diagrams involving the photon. Table 5 shows a comparison of the contribution to a (eeee) i from direct and crossed amplitude products for Z, γ and contact diagrams independently.
For the remaining channels h →νννν,ūuūu,dddd we find that the size of A 1234 A 1432, † contributions is generally smaller, ranging between a few % and 20 % of the corresponding "direct" contribution. The different behavior is due to two numerical effects: on one hand, all "crossed" contributions in the quarks case are suppressed by a factor N c = 3 compared to the "direct" ones. In addition, the photon contributions are further reduced by factors of |Q q | < |Q e | = 1.

Propagator corrections to the off-shell boson
Finally, the complete calculation allows to extract the exact dependence on the W, Z propagator corrections. In the narrow V width approximation, neglecting for simplicity the off-shell boson's contribution, one would just have (see also Eq. (6.8)): Once all the contributions are taken into account, the coefficient of δΓ V /Γ SM V in this expression generally deviates from −1. Table 6 shows the values obtained in this work. We use the  {M W ,M Z ,Ĝ F ,M h } input scheme, so the only relevant corrections are due to shifts in the width of W, Z, as δM Z = δM W = 0.
For completeness, we also report here the numerical expression of the width shifts in terms of Wilson coefficients, in the same scheme: Hl +C ll . (6.12) 6.5 Summary of the impact of various contributions to h → 4f In this section we have examined the impact of various classes of terms in the squared amplitude to the final SMEFT calculation for h → 4f , and in particular those that are usually omitted in narrow W, Z-width calculations. We find that the largest among the latter contributions are those from photon-mediated diagrams. These have a very significant impact on the determination of the dependence on the Wilson coefficients C HW , C HB , C HW B in the h → 4f partial widths. This effect can be a few times larger in absolute value compared to the contribution from ZZ, W W diagrams only and is most relevant for channels proceeding via NC.
The accurate estimate of the corrections due to W, Z propagator shifts is also found to be important, as it leads to a O(20 − 30)% difference in the dependence on δΓ V /Γ SM V with respect to the naive narrow-width estimate.
The interference among NC and CC diagrams, when present, is found to affect significantly the C HW B dependence, as well as that on C HB and C

A Tables of numerical results
In this appendix we report tables that summarize the SM partial width and relative SMEFT corrections for all the Higgs decay channels considered in this work.
We parameterize each partial width as in Eq. (6.5), with the SM result taken to be the current best estimate, as provided by the LHC Higgs Cross Section Working Group [52,53]. The SMEFT corrections are tabulated reporting the values of the coefficients a    Table 8. Partial SM Higgs decay width and coefficients a

B.1 Analytic results
Integrating four body phase space is a formally solved problem. Executing such integrations in the SMEFT still presents technical challenges. Our interest in the four body phase space volume is to describe the decays of the form h →ψψψψ. When directly numerically integrating this phase space volume, we use the approach in Ref. [60], which relies on Ref. [61]. It is helpful to transform the phase space integral to an integration over the set of independent Lorentz invariants κ ij , the scalar product of the two four vectors k i and k j , instead of angular variables which are not Lorentz invariant. There are five independent invariants of the form {κ 12 , κ 13 , κ 14 , κ 23 , κ 24 , κ 34 } that are present in four body decays, subject to the momentum conservation condition When an index is repeated, we use the convention that κ 2 i = κ ii Although closely related in the massless limit the notation κ ij and k 2 ij are distinct. The massless limit relationship between the quantities is k 2 ij = 2 κ ij . The phase space volume in these variables [61] is The momentum configuration is physical so long as the matrix M 4 has one positive and three negative eigenvalues [60,61]. Imposing this condition on the momentum is aided by performing a Gram-Schmidt diagonalization of the momentum vectors. The basis vectors of the Lorentz space of the κ ij can be chosen to be independent. This is easily done by imposing the condition that one vector is time-like and three are space-like. Then the physical momentum configurations defining the phase space are defined by the simultaneous set of conditions κ 2 1 > 0, κ 2 1 κ 2 2 − κ 2 12 < 0, κ 2 1 κ 2 2 κ 2 3 − κ 2 1 κ 2 23 − κ 2 2 κ 2 13 − κ 2 3 κ 2 12 + 2 κ 12 κ 23 κ 13 < 0, (B.4) Retaining final state masses is numerically required when the double photon pole is present in some interference cases. The conditions above can be directly imposed on a numerical integration over the κ ij variable set in this case, modifying the allowed phase space volume further.