On the exotic Higgs decays in effective field theory

We discuss exotic Higgs decays in an effective field theory where the Standard Model is extended by dimension-6 operators. We review and update the status of two-body lepton- and quark-flavor-violating decays involving the Higgs boson. We also comment on the possibility of observing three-body flavor-violating Higgs decays in this context.


Introduction
If new particles beyond the standard model (SM) are much heavier than 100 GeV, physics at the weak scale can be described by an effective field theory (EFT) with the SM Lagrangian perturbed by higher-dimensional operators. The latter encode, in a model-independent way, possible effects of new heavy particles at energies well below the new physics scale . The EFT framework allows for a systematic expansion of these effects in operator dimensions or, equivalently, a e-mail: hermes.belusca@th.u-psud.fr b e-mail: adam.falkowski@th.u-psud.fr in powers of 1/ . The leading effects are expected from operators of dimension 6, as their coefficients are suppressed by 1/ 2 . The first classification of dimension-6 operators was performed in Ref. [1]. For one generation of fermions, a complete non-redundant set (henceforth referred to as a basis) was identified and explicitly written down in Ref. [2]. Reference [3] extended this to three generations of fermions, in which case a basis is characterized by 2499 independent parameters.
In this paper we are interested in the subset of these operators that lead to exotic decays of the 125 GeV Higgs boson. By "exotic" we mean decays that are forbidden in the SM or predicted to occur with an extremely suppressed branching fraction. More specifically, we are interested in decays that violate the lepton flavor or quark flavor. Leptonflavor-violating (LFV) processes are completely forbidden in the SM in the limit of zero neutrino masses. Quark-flavorviolating (QFV) Higgs decays as a flavor-changing neutral current process are forbidden in the SM at tree level. Theoretical studies of exotic Higgs decays have a long history; see e.g. Refs. [4][5][6][7][8][9][10][11] and [12] for a review. Most of these papers assume new light degrees of freedom, in which case the EFT approach described here is not adequate. On the other hand, Refs. [13,14] recently studied the possibility of LFV and QFV two-body decays of the 125 GeV Higgs within the EFT framework. Such decays can arise in the presence of Yukawa-type dimension-6 operators [15]. These papers demonstrated that LFV Higgs decays to τ ± μ ∓ and τ ± e ∓ with the branching fraction as large as 10 % are allowed by current indirect constraints. At the same time, the LHC is currently sensitive to branching fractions of order 1 % [16]. This corresponds to probing the scale suppressing the corresponding dimension-6 operators at the level of ∼ 10 TeV.
The goal for this paper is to extend this study to a full set of dimension-6 operators. Apart from the Yukawa-type operators, exotic Higgs decays can arise in the presence of vertextype ∼ H † D μ Hψγ μ ψ and dipole-type ∼ Hψσ μν ψ F μν operators. We systematically discuss these operators and the new Higgs decay channels that they imply. The structure of the dimension-6 Lagrangian then implies certain relations between these Higgs couplings, as well as relations between single-Higgs interactions and Lagrangian terms without a Higgs that affect precision observables. We give the limits on each of the couplings from precision tests of the SM. That information can be explored to place limits on the allowed magnitude of the Higgs couplings. We will discuss the maximum exotic Higgs branching fraction that these limits permit.
The paper is organized as follows. In Sect. 2 we define our notation and introduce the dimension-6 Lagrangian with LFV and QFV interactions in the Higgs basis. In Sect. 3 we review and update the results of Refs. [13,14] concerning two-body exotic Higgs decays. In Sect. 4 we study the possibility of LFV and QFV Higgs decays mediated by vertex-and dipole-type operators, respectively. Obviously, studying the full parameter space of the dimension-6 Lagrangian would be an extremely difficult task. To deal with the degeneracies among the parameters, one simplifying assumption we make throughout this paper is that the flavor-diagonal Higgs couplings are not significantly affected by higher-dimensional operators. Furthermore, we will assume that there are no large fine-tuned cancellations between different parameters so as to satisfy constraints from precision experiments. In such a constrained framework, we discuss the limits on the LFV and QFV Higgs couplings from various precision measurements. Given these constraints, we discuss the implications for the rate of exotic Higgs decays at the LHC.

Exotic Higgs couplings from dimension-6 Lagrangian
We consider an effective theory where the SM is extended by dimension-6 operators: We assume the SM electroweak symmetry is linearly realized. This implies L eff contains local operators invariant under the SU (3) × SU (2) × U (1) symmetry; in particular, the Higgs boson h enters the Lagrangian only through gauge invariant interactions of the Higgs doublet H . The SM Lagrangian in our notation takes the form The gauge couplings of SU (3) × SU (2) × U (1) are denoted by g S , g L , g Y , respectively; we also define the electromagnetic coupling e = g L g Y / g 2 L + g 2 Y , and the weak angle where v ≈ 246.2 GeV. We also defineH i = i j H * j . After electroweak symmetry breaking, the gauge mass eigenstates are defined as The fermions q L = (u L , V † CKM d L ) and L = (ν L , e L ) are doublets of the SU (2) gauge group. All fermions are three-component vectors in the generation space. We work in the basis where the fermions are mass eigenstates, thus The Higgs boson interactions following from Eq. (2), do not contain any LFV nor QFV couplings. We move to describe the effect of dimension-6 operators. In Eq. (1) we choose to normalize them by the electroweak scale v, while the new physics scale is absorbed into the coefficients c i ∼ v 2 / 2 of these operators in the Lagrangian. A complete non-redundant L D=6 for three generations of fermions was explicitly written down in Ref. [3]. Here we work at the level of Higgs boson couplings with other SM mass eigenstates, as in [17,18]. In this language, the Lagrangian is defined by a set of couplings [δy f ], [δg V f ], and [d V f ], which are in general 3 × 3 matrices with non-diagonal elements for all fermion species f . A subset of these interactions violates lepton flavor and introduces tree-level flavor changing neutral currents for quark-flavor violation.
The first group is related to corrections to the SM Higgs Yukawa couplings in Eq. (3): These couplings arise from dimension-6 operators of the form The second group is related to the contact interactions between the Higgs boson, fermions, and the massive SU (2) vector bosons: where [δg V f ] are Hermitian matrices. These couplings arise from dimension-6 operators of the form H † D μ Hf γ μ f . The gauge symmetry of the dimension-6 Lagrangian implies δg Furthermore, it implies that the Higgs boson enters via (1+h/v) 2 . Therefore, the strength of the Higgs contact interactions of this form is correlated with vertex corrections to the W and Z boson interactions with fermions.
Finally, we also consider the dipole-type Higgs interactions: where σ μν = ı 2 [γ μ , γ ν ], and [d V f ] are general 3 × 3 matrices. These couplings are absent in the SM at the tree level, but they arise from dimension-6 operators of the form Hf σ μν f V μν . The gauge symmetry of the dimension-6 Lagrangian implies that the W boson dipole couplings are related to those of the Z boson and the photon: Again, it also dictates that the Higgs boson enters via (1 + h/v). Therefore, the strength of this type of Higgs interactions is correlated with the strength of dipole interactions of the SM fermions and gauge bosons.
In Eqs. (4) and (6) we isolated the factor √ m f i m f j in the Yukawa and dipole interactions. This is done for convenience, and we do not assume any particular pattern of [δy f ] i j and [d V f ] i j . The Yukawa and dipole interactions are distinguished by the fact that they violate chirality (they allow for transitions of left-handed fermions into right-handed ones and vice versa), much like the fermion mass terms in the SM. Any model addressing the flavor problem and generat-ing these parameters in the low-energy EFT is expected to exhibit some sort of chiral suppression. Exactly this pattern will arise from models following the minimal flavor-violation paradigm, where all sources of flavor violation are proportional to the SM Yukawa matrices. Although, more generally, the chiral suppression does not have to be proportional to the fermion masses, isolating the mass factor leads to a more transparent picture for natural values of these parameters. For the Yukawa interactions, the off-diagonal couplings can be more readily compared to the diagonal ones which, in this normalization, are just equal to 1 in the SM limit.
In the rest of this paper, we discuss LFV and QFV exotic Higgs decays induced by the operators in Eqs. (4), (5), and (6). As mentioned before, we assume that the flavor-diagonal Higgs couplings are not significantly affected by higherdimensional operators, 1 and that there are no large fine-tuned cancellations between different parameters so as to satisfy constraints from precision experiments. In such a constrained framework, we discuss the limits on the LFV and QFV Higgs couplings from various precision measurements. With these assumptions, we give the limits on the couplings from precision experiments and discuss the maximum exotic Higgs branching fractions allowed.

Two-body Higgs decays
In this section we discuss two-body flavor-violating decays involving the Higgs boson. Such processes are generated via the Yukawa couplings in Eq. (4). The important point is that the [δy f ] i j are free parameters from the EFT point of view, and can take any value within the EFT validity range.

Lepton-flavor-violating decays
No experimental dedicated searches have been done so far for h → μe and h → τ e. For h → τ μ, the 95 % CL upper limit on the branching ratio was set by CMS [20] and ATLAS [21]: The CMS search shows a 2.4σ excess over the expected null background, 2 Br(h → τ μ) = 0.84 +0.39 −0.37 %, while the "excess" in ATLAS is only 1σ , Br(h → τ μ) = 0.77 +0.62 −0.62 %. A naive combination of the ATLAS and CMS results yields In terms of the parameters in Eq. (4), the branching ratio can be written as where we assumed the h → τ τ decay is not significantly affected by new physics. Using m μ = 105.7 MeV, m τ = 1.78 GeV, Br(h → τ τ ) = 6.3 % from the SM value, we obtain the best fit value and the 95 % CL bound on the EFT parameters: The strongest constraints on the LFV Higgs couplings come from 2 → 1 γ decays [13,14]. In the SM, such processes are completely forbidden in the limit of zero neutrino masses, but they can be generated in the presence of D=6 operators. In the EFT with LFV Yukawa couplings, they occur at one-loop level. The amplitude for the process is parametrized as and the decay width is given by (in the approximation m 1 m 2 ): Evaluating the one-loop diagrams we find the following results: • μ → eγ : • τ → eγ : • τ → μγ : Above, we kept only the contributions from diagrams with the τ lepton in the internal fermion line. Other contributions are suppressed by m μ /m τ or m e /m τ and can be neglected, unless there is a huge hierarchy between different off-diagonal elements of [δy f ]. Such hierarchy is not expected for EFT arising as low-energy approximation of specific models where the flavor problem is addressed. Our results agree with Refs. [13,14]. It was pointed out in the literature [13,14,23,24] that certain two-loop corrections, the so-called Barr-Zee diagrams with a W or a top loop, may give comparable contributions as the one-loop diagrams computed above. Their analytical forms can be found in Appendix A.2 of [14], which were adapted from the μ → eγ formulas of Chang et al. [25] and Leigh et al. [26]. It turns out that Barr-Zee contributions are proportional to √ m i m j δy i j C, where C is common for all the processes. Numerically, one finds • μ → eγ : • τ → eγ : • τ → μγ : Indeed, the two-loop contributions turn out to be dominant, for τ → μγ and τ → eγ by approximately a factor of 4. For μ → eγ the ratio of two-and one-loop contributions depends on the ratios of the different off-diagonal Yukawa couplings.
Limits on the off-diagonal Yukawa couplings from their oneloop contribution to 2 → 3 1 decays are weaker [29]. Finally, motivated by the constraints discussed above, we write the LFV Higgs branching fractions as We can immediately see that the indirect constraints allow for a sizable branching fraction of h → τ e, and h → τ μ decays. In particular, the percent-level branching fraction for h → τ μ, hinted at by the CMS excess, can be addressed in the EFT context without any tension with τ → μγ bounds. However, one should note that the μ → eγ constraint does not allow Br(h → τ μ) and Br(h → τ e) to be simultaneously large. Observing both of these decays at the LHC would thus signify a breakdown of the EFT approach. On the other hand, Br(h → μe) is constrained to be small by the μ → eγ constraint, so as to be unobservable in practice.

Flavor-changing top quark decays
Dimension-6 operators may also violate flavor in the quark sector. In the SM, quark flavor is not conserved due to offdiagonal CKM matrix elements, but flavor-changing neutral currents are forbidden at tree level. Therefore, the quarkflavor-violating processes involving the Higgs boson are suppressed by a loop factor, and in addition suppressed by the GIM mechanism. On the other hand, the couplings in Eq. (4) may lead to flavor-changing neutral currents at tree level.
From the experimental point of view, the most interesting of these processes are the ones involving the top quark. ATLAS and CMS have performed direct searches for Higgs-mediated flavor-changing neutral currents in top quark decays: t → hq, q = c, u. Due to loop and GIM suppression, the branching fractions for these decays in the SM are prohibitively small. However, in models beyond the SM with new sources of flavor violation these decays are often enhanced to a level that may be observable at the LHC; see e.g. [43].
In the limit of massless charm or up quarks, the tree-level decay width is given by the formula: This translates to the branching fractions: where we used t ≈ 1.35 GeV.
The current 95 % upper limits on the branching fractions for these decays are given in Table 2. Using these, we find the following constraints on the off-diagonal Higgs Yukawa couplings:  [44] t → uh 4.5 × 10 −3 (ATLAS) [44] t → q(= c + u)h 7.9 × 10 −3 (ATLAS) [45] t → ch 5.6 × 10 −3 (CMS) [46] Much as for LFV Higgs decays to tau leptons, the current indirect constraints on δy qt and δy tq do not forbid the t → hq branching fraction to be close to the current LHC limits. While the relative phase between δy qt and δy tq is severely constrained by neutron electric dipole moment searches [47], the absolute values (which enter into the t → hq widths) are allowed to be large. One should also mention that D-meson oscillations place more severe constraints on the products δy ut δy tc and δy tu δy ct ; see [47]. Therefore, in the EFT context, it is impossible for both t → hc and t → hu branching fractions to be close to the current experimental limits.

Three-body decays
In the previous section we discussed two-body exotic decays induced by dimension-6 operators of the Yukawa type. We concluded that indirect constraints on the LFV and QFV Higgs Yukawa couplings to fermions are consistent with the branching fractions of h → τ μ and h → τ e decays that are readily observable at the LHC. In fact, the best limits on the relevant couplings currently come from the LHC. This agrees with the conclusions from previous literature [14]. In this section we extend this discussion to three-body exotic Higgs decays and other operators appearing at the dimension-6 level in the EFT Lagrangian.

h → W bq
We begin with the h → t * q → W bq decays. These decays are mediated by the same Yukawa couplings as the ones leading to the t → hc/u decays, and they are constrained by ATLAS and CMS searches as in Eq. (25): where we summed over the W + and W − modes. Note that Higgs decays with O(10 −4 ) branching fractions have already been seen in LHC Run-1 in the h → Z Z → 4 channel. Thus, if t → hq decays are observed at the LHC close to the current limit, it should be possible to also observe the h → W bq decays in the future (although the tt background will be a challenge in this case).
The same decay can also be mediated by dipole-type couplings in Eq. (6). Implementing the relevant vertices in Feyn-Rules [48,49] and calculating the decay width numerically in aMC@NLO [50] one finds where we sum over W ± decay modes. Barring fine-tuning, in the EFT approach the W-boson dipole couplings d W q are related to the analogous dipole couplings of the photon, Given that, the parameters d W d entering the expressions above are subject to stringent constraints from B-physics, such as the measurement of the b → sγ branching fraction, and thus their contribution to the Higgs decays is completely negligible. As for the d W u parameters, they are less severely constrained by the limits on Br(t → qγ ) set by the CMS experiment [51], and we estimate This implies that the maximum branching fraction for dipole-mediated h → W bq decays at most ∼ 10 −6 , which is lower than the maximum Yukawa-mediated rate, and probably too low to be observable.

h → 1 2 γ
We move to the dipole-type operators in Eq. (6). In the lepton sector, these may lead to h → 1 2 γ decays, where the presence of a hard photon in the decay would allow experiments to distinguish it from h → mediated by Yukawa couplings.
As discussed before, the strength of Higgs dipole-type interactions is fixed by the strength of the corresponding dipole interaction between fermions and a gauge boson. Therefore the constraint on the Higgs coupling will come from dipole-mediated 1 → 2 γ decays. In the limit where the leptons are massless, the width of the latter is given by where we summed over the + 1 − 2 and − 1 + 2 decay modes. Using the experimental results from Table 1, we get the following constraints on the dipole couplings: The dipole-mediated h → 1 2 γ decay width is given by where we summed over the + Unlike for Yukawa-mediated two-body decays, this time the decays with τ in the final states are constrained to be extremely rare. As long as the EFT framework is adequate for describing Higgs decays, there is no prospect of observing the dipole-mediated LFV decays at the LHC or the future 100 TeV collider [52].

h → 1 2 Z
Another process that can be generated by dipole-type interactions in Eq. (6) is h → 1 2 Z . To calculate the width, we have implemented the (1 + h/v)¯ 1 σ μν 2 V μν vertex in FeynRules, and calculated the decay width numerically in aMC@NLO. We obtain the branching fractions: where we summed over the +  Table 3. The dipole mediated partial decay width is given by where we summed over the + 1 − 2 and − 1 + 2 decay modes. This results in the following constraints on the dipole couplings: Stronger constraints on these couplings are obtained through their loop contributions to radiative lepton decays [56,57]. At one loop one finds Using the experimental results from Table 1, and assuming no cancellations between the tree-level d Ae and the one-loop contribution from Z dipole, we get the following constraints on d Ze : This translates to the constraints on the branching fractions: Br(h → τ μZ ) ≤ 1.9 × 10 −11 . The suppression is slightly smaller than for the decays with a photon in the final state, however, observing decays with this low branching fraction is impossible at the LHC or at the 100 TeV collider. The same process (though with a different helicity structure for the final-state fermions) can also be generated by vertex-type couplings in Eq. (5). Implementing the relevant vertices in FeynRules and calculating the decay width numerically in aMC@NLO one finds Again, the off-diagonal vertex corrections are constrained by LFV Z boson decays. The decay width is Then the experimental constraints in Table 3 imply: Again stronger constraints arise through one-loop contributions to LFV lepton decays, for which the experimental limits are collected in Table 4. Assuming no cancellations with tree-level contributions of 4-fermion operators, one obtains the bounds [29] [ This translates to the following bounds on the branching fractions: The bounds are somewhat weaker than for the dipolemediated Higgs decays, however, the suppression is still too much for any realistic prospects of experimental detection.

t → hqV
Finally, we consider flavor-violating three-body decays of the top quark mediated by dipole-type operators: t → hqV , where V is a photon or a gluon. We have implemented thē tσ μν cV μν and htσ μν cV μν vertices in FeynRules, and calculated the decay width numerically in aMC@NLO. We find The current best constraints on Br(t → qγ ) come from searches for anomalous top production at the LHC. For the dipole couplings to photons the strongest limits come from the CMS experiment [51]. They translate to the following limits on the branching fractions: For t → uγ , even stronger limits can be placed due to the dipole contributions to the neutron electric dipole moment [60], though these constraints do no apply when the dipole couplings are parity conserving. For the dipole couplings to the gluon, the strongest limits come from the ATLAS experiment [61]: Br(t → ug) ≤ 4.0 × 10 −5 , Limits on the flavor-violating dipole top couplings from tγ production at the LHC [62] are currently weaker. The experimental bounds in Eqs. (44) and (45)  Br(t → cgh) ≤ 7.5 × 10 −12 .
As in the case of h → + 1 − 2 γ decays, the branching fraction for t → qγ /gh may be larger than the limits in Eq. (46) if the process is mediated by off-diagonal Yukawa couplings and the photon or gluon is emitted from the final state quark. Although the top production cross section is larger than that of the Higgs boson (O(1) nb at the 14 TeV LHC, and a factor of 30 larger at 100 TeV, which corresponds to O(10 11 ) tt pairs in the future 100 TeV collider [52]), the limits in Eq. (46) leave little room for observing the dipole-mediated top quark decays. A more realistic probe of flavor-violating dipole-type Higgs interactions may be offered by the production processes pp → ht j and pp → htγ . 4

Conclusions
In this paper we discussed the possibility of observing at the LHC exotic Higgs decays that violate lepton or quark flavor. Our study was done in the context of an EFT which describes the effective interactions of the Higgs boson with other SM particles after heavy particles from beyond the SM have been integrated out. In this context, the possibility of a significant rate of two-body decays such as h → μτ , h → eτ , and t → ch was pointed out in the previous literature. Our analysis confirms and updates these conclusions.
We also studied the possibility of exotic three-body decays involving the Higgs boson. Here, our conclusions are largely negative. The existing precision constraints imply that the rate of such three-body processes must be prohibitively small and cannot be observed in colliders in the foreseeable future. This is an important and robust conclusion that can derived in the EFT framework. Conversely, if such three-body processes are observed, this would signal a breakdown of the EFT description we used regarding Higgs decay processes. Such a breakdown would be a harbinger of new light degrees of freedom, or a non-linear realization of electroweak symmetry.