Higgs flavor violation as a signal to discriminate models

We consider the Higgs Lepton Flavor Violating process h → τμ, in which CMS found a 2.5 σ excess of events, from a model independent perspective, and find that it is difficult to generate this operator without also obtaining a sizeable Wilson coefficient for the dipole operators responsible for tau radiative decay, constrained by BABAR to BR(τ → μγ) < 4.4 × 10−8. We then survey a set of representative models for new physics, to determine which ones are capable of evading this problem. We conclude that, should this measurement persist as a signal, type-III Two Higgs Doublet Models and Higgs portal-like models are favored, while SUSY and Composite Higgs models are unlikely to explain it.

We will approach the problem, in section 2, from a model independent perspective, using an effective theory and including dimension six and eight dipole operators, which have been mostly neglected in previous works on LFV Higgs decays. We will argue that the inclusion of these operators changes how the bounds from low energy data can be translated to constraints on LFV Higgs couplings, making them in general more restrictive. We will also show that, in many realistic and simple UV continuations, dipole operators are generated with sizeable Wilson coefficients, and it is natural that in those cases the h → τ µ branching ratio will be much smaller than predicted by analyses that neglected these operators.
In section 3 we will explore some representative classes of models to determine which ones generate which operators, and what are the relations between the Wilson coefficients. That classification will allow us to have a general view of the problem, separating models that predict branching ratios much smaller than the current experimental reach from those that are actually favored in case the 2.5 σ excess, observed by CMS in the h → τ µ channel, gets more significant with increasing luminosity.

Effective Field Theory
In the Effective Field Theory (EFT) approach, we consider the most general Lagrangian compatible with the SM local symmetries and containing no new degrees of freedom below a set scale Λ: where L SM is the renormalizable SM Lagrangian, and L (i) contains the higher dimensional operators (of dimension i) generated by integrating out heavy fields above the scale Λ. A complete survey of the operators generated at dimension five and six can be found in [20].
We will be interested only in the leptonic sector, for which the SM Lagrangian is:

2)
1 Which can be ambiguously read as either a fluctuation to set upper limits to the branching ratio, or as a signal and a measurement of the same branching ratio. We give more details on section 2.1.
where L and E are the lepton weak doublet and singlet fields (triplets in flavor space, indexes are suppressed), H is the Higgs doublet, D µ is the appropriate covariant derivative and y 4 is a matrix in flavor space. In a phenomenological approach, LFV Higgs interactions are mediated at lowest order by the dimension six operator: 2 where c 6H is a matrix in flavor space and we define the operatorÔ 6H ≡LHH † HE.
In principle, the couplings for different chirality combinations may differ (e.g. c τ L µ R 6H = c µ L τ R 6H ) but, since we will be always looking at initial and final states containing both combinations, we consider a single coupling c 6H , understood to be the average value c 6H ≡ c 2 LR + c 2 RL , where LR and RL are the chiralities of heavier and lighter fermions, in this order. This operator, in conjunction with the renormalizable Yukawa couplings (y 4 ) generates the effective flavor violating Yukawas, given in the mass basis by: In our analysis we assume that the diagonal entries of this matrix are close to their SM values (i.e., Y ii m i /v).
In the presence of a non-diagonal Y , a one-loop contribution to radiative lepton decays l i → l j γ is generated (see figure 1a). IfÔ 6H is the only higher dimensional operator relevant for Higgs flavor physics, Y ij is constrained by the experimental bounds shown in table 1. This diagram however must contain an insertion of the lepton Yukawa coupling, which greatly suppresses the radiative decay, allowing the LFV Higgs couplings Y ij to be sizeable while still respecting the limits in table 1.
Once a large enough Y ij is allowed, one may study the Higgs LFV decay h → l i l j generated by it (see figure 1b). In the case of h → µe, the strong bound of µ → eγ (table 1) puts this process at a rate beyond the reach of the LHC: BR(h → µe) 10 −8 [3]. The authors of [3] claim that the non-diagonal Yukawas for h → τ e and h → τ µ are still allowed to be big enough to produce a branching ratio at an observable level. Recently,

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Process Upper bound on BR µ → eγ < 5.7 × 10 −13 [21] τ → eγ < 3.3 × 10 −8 [22] τ → µγ < 4.4 × 10 −8 [22] h → τ µ < 1.57 × 10 −2 [6]  CMS has released results on the direct search for h → τ µ and there is an excess of events of a 2.5 σ significance in the channel [6]. That could be read as a statistical fluctuation and be used to put a bound of BR(h → τ µ) < 1.57% at 95% confidence level, or as a signal, which then leads to BR(h → τ µ) = 0.89 +0.40 −0.37 % [6], only with more data will the situation become clear. The possibility of confirming this as a signal is specially interesting because, as we will see in the following sections, the conclusion of [3] does not hold if there are other sources of flavor violation (besidesÔ 6H ), and a confirmation would disfavor models with such sources. We will focus on this process for the remainder of this text.

Interplay with radiative LFV
The conclusion of [3], that Higgs LFV decays may be observed in the tau sector, holds if the only contribution to the tau radiative decay comes from the Higgs one-loop diagram (figure 1a). In general, however, other sources LFV above the scale Λ may generate the dipole operators [20,23]: where F αβ is the electromagnetic field strength, e is the electric charge and the dipole operators are defined asÔ 6γ ≡LHσ αβ EF αβ andÔ 8γ ≡LHH † Hσ αβ EF αβ . Since these operators contribute at tree level to the flavor violating radiative decays they cannot be immediately dismissed. In the literature, one usually ignoresÔ 8γ on account of its larger suppression with the scale Λ [5,[24][25][26]. However, in cases whereÔ 6γ is not present (i.e., it is independently suppressed or unrelated to LFV, see the next section for examples), it may become the leading contribution to LFV radiative decays. Thus, in general, the bounds from table 1 will apply to a combination of c 6H Λ 2 , c 6γ Λ 2 and c 8γ Λ 4 , generally making the restrictions on c 6H Λ 2 stronger, unless we restrict the analysis to specific UV theories in which c 6γ Λ 2 and c 8γ Λ 4 are suppressed in relation to c 6H Λ 2 , either by having the dipole operators be generated a higher loop orders or by different physics at higher scales (making Λ in eq. (2.5) different and bigger than Λ in eq. (2.3)).
It turns out that it is quite hard to find simple UV completions that generate c 6H c 6γ and c 6H c 8γ at a given scale Λ. For the sake of illustration, lets first assume thatÔ 6H is generated by the UV theory at loop level (one loop or more). If there is charge flowing through the loop, one can JHEP11(2015)074  easily add an emitted photon, generatingÔ 8γ with c 8γ c 6H (see figure 2a). The caveat here is that one could propose UV theories in which there is no charge going through the loop at leading order, in this case c 8γ would be suppressed by loop factors. We provide one example in section 3.4. One could also obtainÔ 6γ fromÔ 8γ by closing a loop with two external Higgs lines or by simply removing them. In the first case c 6γ will be a loop factor smaller than c 8γ , and in the second case the relation between c 6γ and the other two couplings depends on the specific couplings of the Higgs with the particles going around the loop, making it effectively independent.
On the other hand, ifÔ 6H is present at tree level in the UV theory, one cannot simply attach a photon and getÔ 8γ because the diagram must vanish by Ward Identities. But now, one can obtainÔ 6γ by closing two of the Higgs legs in a loop and attaching a photon to that loop (see figure 2b). In this case, c 6γ will be smaller than c 6H by a loop factor, but since it contributes to the radiative decay at tree level while c 6H contributes only at loop level, the factors cancel and the dipole operator must be taken into account.
We see that one may connect the dipole operators toÔ 6H and consequently the Wilson coefficients c 6γ , c 8γ are in general not independent from c 6H . Of course, it is always possible to imagine that contributions from additional degrees of freedom to the dipole operators, as well as (approximate or exact) flavor symmetries, unrelated toÔ 6H , may greatly suppress the values of c 6γ , c 8γ with respect to c 6H , allowing the latter to be generated at lower scales and to have observable effects. However, a survey of the recent literature on the subject [7][8][9][10][11][12][13][14][17][18][19] reveals that in many well motivated UV completions, such as SUSY and Composite Higgs scenarios, there is a great correlation between the sizes of c 6γ , c 8γ and c 6H , as we will discuss in the next section.

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If the theory that UV completes the SM is a gauge theory, it is possible to show that the radiative dipole operators are always generated at one-loop level [27]. This follows directly from imposing the QED Ward identities in the radiative decay invariant amplitude. Such a consideration motivates extracting a universal 1/(16π 2 ) factor from the Wilson coefficients c 6γ and c 8γ . Here we do not do this, because while these operators are always generated at loop level, the Wilson coefficient c 6H may be generated at tree or loop level, depending on the specific UV theory. As such, we will at first assume that c 6H , c 6γ and c 8γ are generated at the same order, implying that any loop factors are the same in both, to obtain our bounds. Later, when we discuss specific models, we will note the cases where c 6H is generated at tree level, or more generally at a lower loop order than the dipole operators, and hence is less suppressed by one or more loop factors.
Assuming that c 6γ , c 8γ and c 6H are of the same order, we may compare the contributions to the τ → µγ decay rate generated by each of the three higher dimensional operators. In the following expressions we neglect interference between channels, since our goal is to find out in what regime only one of the Wilson coefficients is sizeable and the other two are small enough to be neglected. The contribution ofÔ 6H is shown in figure 1a and the contributions ofÔ 6γ andÔ 8γ are obtained by substituting H by v where: We note that in the processes involving lepton radiative decay one must include a certain class of two loop diagrams, the so-called Barr-Zee diagrams [3,28]. These diagrams, while of higher loop order, are less chirally suppressed, and become parametrically more important than the diagram in figure 1a and involve the same non-diagonal Yukawa coupling. The Barr-Zee contribution is denoted by c 2-loop in eqs. (2.8) and (2.10). We also note that we are assuming that the one loop and two loop contributions have a positive relative sign i.e., there is no destructive interference between these diagrams. A cancelation may be achieved if the top Yukawa (that enters in c 2-loop ) has a negative sign with respect to the top mass, y T = −m T √ 2/v. In this case, the numerical value of our bounds changes, but qualitatively our results remain unchanged.

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Figure 3: τ → µγ branching ratio (including the Barr-Zee diagrams) from various operators as a function of the new physics scale Λ. The continuous black, dashed blue and dotted red lines are respectively the branching ratios generated byÔ 6γ ,Ô 8γ andÔ 6H . Here, we consider c 6γ ∼ c 8γ ∼ c 6H ∼ 1, and interference effects are neglected.
In figure 3, we plot the branching ratio BR(τ → µγ) calculated using eqs. (2.6) to (2.8) as a function of the UV scale Λ. We see that, for similarly sized Wilson coefficients,Ô 6γ dominates completely the amplitude, generating the strongest bound on Λ, while even the highly suppressed dimension eight operatorÔ 8γ becomes important for a cutoff below approximately 25 TeV and c 6γ ∼ c 8γ ∼ c 6H ∼ 1. In generalÔ 8γ will be less important than O 6H for: In the case whereÔ 8γ is generated at one higher loop order thanÔ 6H , there is an extra factor of ∼ 1/(16π 2 ) in the definition of the operator, as discussed before. In this case the crossover happens at: It should not be so surprising that a dimension eight operator may play a role in such a case, since the diagram of figure 1a that generates the dipole operator at one loop has an additional suppression due to the tau Yukawa insertion. In fact, it is for this same reason that we must include the Barr-Zee two loop diagrams when computing the contribution of the operator of eq. (2.3) to the tau decay rate. Hence,Ô 8γ cannot be neglected in many cases of interest.

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The relation between the allowed BR(τ → µγ) and BR(h → τ µ) can be seen in figure 4, where we use BR(τ → µγ) to fix c 6γ , c 8γ and c 6H in three different scenarios, subsequently using c 6H to obtain BR(h → τ µ). The three scenarios we consider are: (I) when onlyÔ 6H contributes to both decays, with the dipole operators suppressed: c 6γ = c 8γ = 0. This scenario reproduces the results of [3]; (II) the scenario shown if figure 2a, whereÔ 6H is generated at loop level leading to c 8γ ∼ c 6H and the dimension six dipole is suppressed (we set c 6γ = 0). In this case the cut-off scale becomes important becauseÔ 8γ is suppressed by an additional 1 Λ 2 . We show two extreme cases, Λ = 2 TeV (close to the current bounds) and Λ = 50 TeV (which brings it close to the first scenario); (III) the third scenario is shown in figure 2b, with a tree generatedÔ 6H leading to c 6γ ∼ c 6H 16π 2 and a suppressedÔ 8γ (c 8γ = 0).
One can clearly see that, barring contrived cancelations, 3 the presence of dipole operators strongly restrict the FV Higgs decay. Even the dimension eight dipole will be important unless we are willing to accept high cut-offs. It should also be stressed that these bounds become even stricter if we allow the suppressed operators in each scenario to have coefficients different from zero, so the curves in figure 4 can be seen as upper limits in each case.
Once we have estimated the size of the various contributions to the LFV decay process, it becomes a relevant question to ask whether there are any interesting models in which the connection between the operators generating radiative and Higgs mediated LFV decay is broken, allowing the Higgs LFV decay to be seen at the LHC. One such example is a type-III two Higgs doublet model, explored in [14], while another one, explored recently in [17], is provided by a scalar gauge singlet model, which may be a viable DM candidate. We discuss both scenarios as well as the SUSY and Composite Higgs ones in the next section, keeping attention to the relative size of the higher dimentional LFV operators produced by each model and whether or not they may generate the h → τ µ process at an observable level for the LHC.
3 Results for specific models

Composite Higgs
In models where the EWSB happens due to the dynamics of a new strong sector, such as Randall Sundrum (RS) models [29] and their discrete versions (Quiver or N -site models [30][31][32]), heavy vector-like leptons may mix linearly with the chiral SM leptons, in the so-called partial compositeness scenario. In these models, the Higgs boson is a light composite bound state and does not couple directly (i.e. in the flavor basis) to the SM leptons. Following JHEP11(2015)074 In this class of models,Ô 6H is generated at tree level by integrating out the heavy fermions, as per the diagram of figure 5. It is essential in obtaining this operator that there be non-zero "wrong" chirality couplingsỸ of the Higgs to the composite sector. Such an operator is naturally obtained when considering the brane localized (pure composite) Higgs as a limit of a bulk localized Higgs [18]. In the low momenta limit, the diagram on the left of figure 5 is given by: where u L and u R are chiral projections of the Dirac Spinor (there is a similiar expression with L and R exchanged, in what follows we show only the LR case to avoid duplication).

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Upon closing two of the external Higgs legs and adding an external photon, one gen-eratesÔ 6γ , the diagrams are shown in figure 6.
In the low external momenta limit we have: where we see that the Wilson coefficients c 6γ and c 6H are proportional (as expected from figure 2b). In fact, besides the predicted 1/(4π) 2 loop suppression, we get only a −11/18 JHEP11(2015)074 −0.6 proportionality factor between them. Because of this strong correlation, the stringent bounds on the radiative decay τ → µγ forces the Higgs decay h → τ µ to be well below observational level. The authors of [7] obtain BR(h → τ µ) < 8.6 × 10 −6 , which is much more restrictive than the current bounds obtained by direct searches: BR(h → τ µ) < 1.57 × 10 −2 [6].
In a more complete model, such as the RS model explored in [33], there are additional contributions to c 6γ and c 8γ , due to composite gauge bosons (electroweak Kaluza-Klein (KK) modes). In this case, the main contribution is due to the first charged resonance W , which sets the strongest bound on the KK scale, of Λ 5 TeV for Y,Ỹ of order one. Barring cancellations with the typically smaller contributions from the neutral sector (Z and h), one may use such a bound on the KK scale to obtain the expected value of the h → τ µ branching ratio to be of order m h /(8πΛ 4 ) ∼ 10 −14 , for c 6H 1.
Both these cases consider the Higgs to be a bound state of the strong sector that is accidentally light. A well motivated proposal to explain the lightness of the Higgs with respect to the other resonances is considering it to be a pseudo Nambu-Goldstone Boson (pNGB). In this case, one assumes the strong sector has a large global symmetry G, spontaneously broken down to H, such that the NGBs parametrizing the G/H coset contain the Higgs doublet, conveniently represented by the non-linear sigma model field Σ = exp (ih/f ), where h are the components of the doublet H and f is the Higgs decay constant. The symmetry G is then broken by the Yukawa couplings and by gauging a subgroup H ⊃ SU(2) L × U(1) Y , giving the Higgs, at the one loop level, a small mass compared to the strong sector resonances.
The requirement that the Higgs couplings must respect a global symmetry of the strong sector, in which the SM gauge symmetry is embedded, modifies the effective Lagrangian, such thatÔ 6H must be aligned to the dimension four Yukawa coupling, suppressing flavor violation [34]. 4 Following [34], this alignment can be seen by promoting the strong sector operators Ψ,Ψ to full representations of G. Then, the Yukawa couplings λ l , λ e of eq. (3.1) are made formally invariant under G by embedding the elementary fields in incomplete G representations, where the additional degrees of freedom are spurions. In place of the direct couplings to H, the strong sector must couple G invariantly to Σ.
Global invariance then forces the couplings to the doublet H to take the schematic form where P (Σ) is a polynomial in Σ, projected over the elementary fields. As long as the elementary fields are coupled to a single representation (e.g. the 5 of SO(5) in the MCH5 model [36], in which case P (Σ) = ΣΣ T → sin(h/f ) cos(h/f )), then all the higher dimensional non-derivative operators in H are aligned and can be simultaneously diagonalized.
In this case the leading contribution comes from the kinetic mixing generated by integrating out the heavy leptons, which is typically smaller than (2.3) and leads to even lower

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Higgs LFV decay rates. One may then conclude that the observation of the process h → τ µ at a rate near the current experimental sensitivity would disfavour this class of models.

2HDM
In models with two electroweak Higgs doublets, the Yukawa Lagrangian for the charged leptons is written as where Φ 1 and Φ 2 are the Higgs doublets and y 1 , y 2 their corresponding Yukawa matrices. Neglecting CP violation for simplicity, the neutral CP even scalar mass eigenstates are related to those in the flavor basis by We consider the case where the lightest eigenstate h is the 125 GeV Higgs, current limits on 2HDM parameter space can be found in [38,39] and references therein. If there is a parity symmetry distinguishing the two doublets, (e.g. type II 2HDM), one may forbid the couplings of Φ 2 to the charged leptons at tree level. This case is closely related to the SUSY models considered in the following section and will be discussed there.
In the general case where there is no such parity, it is generally impossible to diagonalize simultaneously y 1 , y 2 and the Higgs mass matrix, then we see that a Higgs LFV operator is already present in the renormalizable Lagrangian in the Higgs mass basis. In order to connect this result with our effective theory framework, one must integrate out the other physical scalars (assumed heavier than the 125 GeV Higgs). Since, by eq. (3.8) the physical scalar h lives in both doublets, the contributions coming from Φ 2 proportional to y 2 cos(α) to the Yukawa matrix do not decouple for big m H and are not suppressed by the new physics scale, being in general stronger than the higher dimensional operators.
Recently, the authors of [14] have shown that in such a case one may obtain the h → τ µ process at an observable level, despite the limits from radiative decays. Numerous other authors have also obtained observable BR(h → τ µ) [40][41][42][43] in 2HDMs. The reason being that the Higgs LFV is generated by a renormalizable operator, at tree level, while the tau radiative decay is loop suppressed. Furthermore, in certain regions of parameter space, a cancellation between the heavy scalar and pseudoscalar becomes possible, leading to additional suppression of the radiative decay. For a deeper analysis on the flavor structure of 2HDMs see [44].

SUSY
In supersymmetric models such as the Minimal Supersymmetric Standard Model (MSSM), anomaly cancellation and holomorphy force one to include two Higgs doublets of opposite Hypercharge, that give mass separately to up type and down type fermions. This corresponds then to a type II 2HDM, at least at tree level.
At one-loop order, however, one may generate Yukawa couplings with the "wrong" Higgs boson, as in the diagrams of figure 7. Including the analogous diagrams from both

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Higgs doublets, the effective Lagrangian of charged leptons and Higgs bosons may be parametrized as: where ∆ 1 , ∆ 2 are loop induced couplings, E, L are the singlet and doublet lepton fields, H 1 and H 2 the down and up type Higgs doublets and tan(β) = v 2 /v 1 is the ratio of their VEVs.
On the second line we rearranged the Lagrangian so that it is easy to see that the first term is proportional to the charged lepton mass matrix and is diagonal in the mass basis, while the second term is in general flavor violating. The implications for flavor transitions of the non-holomorphic couplings ∆ 2 have been extensively studied in the literature, see for instance [45,46]. The source of flavor violation is found in the soft SUSY breaking trilinear terms and slepton masses: Rotating to the Higgs mass basis, √ 2Re(H 1 ) = v 1 + H cos α − h sin α and √ 2Re(H 2 ) = v 2 + H sin α + h cos α, the τ µ coupling to lightest Higgs eigenstate h is given by: plus the analogous contribution with exchanged chiralities. We see from this expression that these diagrams become important in the large tan β limit. In this limit, then, we expect the diagrams obtained by adding a photon to the loops in figure 7, and exchanging the Higgs for its VEV, to dominate the τ → µγ amplitude. 5 In this case, the radiative lepton decay is generated with a Wilson coefficient that is directly proportional to the Higgs LFV one, and we can find a correlation between c 6γ and the coefficient of the renormalizable interaction c 4H . While both are generated at one loop, it is clear that since the Higgs flavor violating interaction in dimension four, it is non-decoupling, so that one may always suppress c 6γ by going to a high SUSY breaking scale, obtaining an effective 2HDM. One then recovers the discussion of the previous section. In terms of effective operators, this is very similar to scenario (II) in figure 4 (the dashed lines), replacing c 6H → c 4H and c 8γ → c 6γ . In order to illustrate the correlation, consider the diagram of figure 7a with a wino exchange. Explicitly, one finds: (c) Figure 7: Diagrams leading to coupling between charged leptons and the "wrong" Higgs doublet (i.e., the one not responsible for their mass at tree level). The slepton, higgsino, wino and bino fields are denotedl ,H 0,± 1,2 ,W 0,± andB 0 , respectively, and the cross denotes a mass insertion. Chirality labels are suppressed for simplictity.
where M 2 and µ are the wino and higgsino soft mass parameters, andm L2 ,m L3 the slepton masses. The slepton mass basis is defined byμ L = cos θ Ll2 −sin θ Ll3 ,τ L = sin θ Ll2 +cos θ Ll3 . The loop function I 1 is given by: Adding a photon to the slepton line of figure 7a and replacing the Higgs by its VEV, one can extract c 6γ and show that: , (3.14) where Λ is the dominant scale in I 2 (the bigger of the three masses) and: In figure 8, we plot contours for the ratio c 4H /c 6γ with tan β = 40, and the pseudoscalar mass M A = 200 GeV varying M 2 andm L for µ = 250 GeV on figure 8a, and varying µ andm L for M 2 = 250 GeV on figure 8b. We see that for this region of parameter space, the ratio is order one, indicating a strong correlation. SinceÔ 6γ contributes to τ → µγ at tree level, c 6γ will be strictly bounded. The relation between c 4H and c 6γ then implies BR(h → τ µ) to be very small. This constraint is relaxed if we increase the overall SUSY scale, making c 6γ /Λ 2 small and taking us back to a 2HDM scenario. Of course, in order to derive rigorous bounds on the Wilson coefficients a thorough analysis of all contributions JHEP11(2015)074 to h → τ µ and τ → µγ needs to be performed. For that we refer the reader to [10,47], for instance, where the authors obtain BR(h → τ µ) 10 −4 from BR(τ → µγ) < 3.1 × 10 −7 (which is an older constraint, superseded by the one shown in table 1). A R-Parity violating scenario was explored in [11] with BR(h → τ µ) 10 −5 as a constraint.
This puts the LFV Higgs decays at least two orders of magnitude below the current experimental sensitivity, and disfavors this class of models in case the current excess turns out to be signal.

Higgs portal
One class of models where radiative lepton decay may be small enough to satisfy bounds and still allow observable Higgs LFV decays is given by the Higgs portal models [48][49][50][51][52][53], where a scalar singlet odd under a Z 2 symmetry is added to the SM. The coupling terms of this field to the Higgs and leptons, as well as its self couplings, are given by the effective Lagrangian: where S is the singlet field, ξ,ỹ are coupling constants (ỹ is a matrix in flavor space), and Λ is a scale associated with some heavier sector (e.g., vector-like leptons or an additional scalar weak doublet) which generates the effective interaction of S with SM fermions, not to be confused with the mass of S, m S . The flavor violating decays are generated by thẽ y couplings, which may be misaligned with respect to the SM Yukawas. The mass of S is given at tree level by At one loop order, there are additional contributions from λ S andỹ which we neglect.
In this class of models, the Higgs LFV interaction is generated at one loop order by the diagram of figure 9a, which becomesÔ 6H if we also integrate out S.
The radiative lepton flavor violation is generated only at two-loop order by the sunset diagram of figure 9b. BothÔ 6γ andÔ 8γ are obtained in the large m S limit,Ô 8γ is directly obtained from figure 9b, whileÔ 6γ is obtained by closing a third loop with two of the three Higgs lines in that diagram, making this operator even more suppressed.
We note that, while from a low energy point of view, m S is the cutoff scale that determines the suppression of the irrelevant operators, an explicit calculation from the diagrams of figures 9a and 9b shows that c 6H and c 8γ depend on m S only logarithmically, becoming important for a larger gap between m S and the scale Λ, see below. 6 Because the radiative decays are generated at two loops, they are naturally suppressed with respect to the Higgs LFV rates. Furthermore, since S is uncharged, it is impossible to add a photon to the loop of figure 9a, and the Wilson coefficients c 6H and c 8γ are uncorrelated. In this case, we find that the conclusions of [3] apply.
We estimate the size of the contributions to h → τ µ and τ → µγ to be, respectively: where the result for c 8γ is only a leading order estimate. Using eqs. that the gap between m S and Λ, as well as the size of the couplings ξ andỹ τ µ , be small enough to mantain perturbative control. Explicitly, we use the perturbativity contraints In the second equation, E = 2.718 . . . is the Euler number. We note that no stringent bound onỹ τ µ is implied by c 8γ , because of the tau Yukawa suppression. Imposing these constraints, we obtain the plots of figure 10.
One sees that, satisfying the constraints, it is possible to generate a Higgs LFV decay rate of order BR(h → τ µ) ∼ 10 −2 , observable at the LHC, while the radiative tau decay is well under control, with a rate of order BR(τ → µγ) < 1.5 × 10 −17 , where the maximum value is obtained for couplings saturating the perturbativity bound. However, if S was stable (being the lightest Z 2 odd particle), its thermal relic abundance would have to be below or equal to the observed Dark Matter abundance, Ω S ≤ Ω DM = 0.227, implying ξ ≤ 3 × 10 −4 (m S /1 GeV) [54,55], which is far too small to allow c 6H to be sizeable. One way to avoid this is to extend the scalar sector to obey a larger global symmetry than Z 2 . In this case, only the lightest scalar must satisfy the DM constraints, while its heavier partners may have larger couplings to the Higgs, of order ξ 10 and so allow the h → τ µ rate to be generated at an observable level.

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This class of models then allows one to evade the bound on τ → µγ and still generate h → τ µ at a rate near the current experimental sensitivity. A signal of this decay would then be indicative that this kind of model may be playing a role.
Detailed phenomenology of the scalar sector and the viability of the lightest scalar as a DM candidate will be presented elsewhere, however, see [17].

Conclusions
We have examined the possibility of a new physics contribution to the Higgs LFV process h → τ µ, which has been recently constrained by CMS to BR(h → τ µ) < 1.57% at 95% or that may be rather seen as a signal, with BR(h → τ µ) = 0.89 +0. 40 −0.37 % [6]. From a model independent perspective, in order to obtain a signal big enough to be detected or restricted by the LHC (BR(h → τ µ) ∼ 10 −2 ) one must consider models in which the Higgs LFV operatorÔ 6H is sizeable but does not violate the important constraint from radiative lepton decay, BR(τ → µγ) < 4.4 × 10 −8 [22]. This usually requires the contribution of the dipole operatorsÔ 6γ ,Ô 8γ to the tau decay rate to be negligible, which implies that they are generated at a higher loop order thanÔ 6H by the UV continuation of the SM. The dipole operators, speciallyÔ 8γ , have been generally neglected in model independent approaches to LFV Higgs decays. We show that the contributions of these operators can be important and thus put strong bounds onÔ 6H .
Looking at specific classes of models found in the literature, we see that typically the Wilson coefficients ofÔ 6γ ,Ô 8γ are correlated withÔ 6H , and make it difficult to avoid the radiative decay bound and still get Higgs LFV at an observable level. This is the case of SUSY, whereÔ 6H is generated at one-loop order, and Composite Higgs models, in which it is obtained at tree level. In both these cases, one may add a photon toÔ 6H and thus getÔ 6γ with a similarly sized Wilson coefficient. For this reason, these kinds of models are disfavored by the data, should the measured rate for h → τ µ be confirmed as a signal.
On the other hand, there are certain kinds of models in which one may avoid this conclusion. Examples are a type III Two Higgs Doublet Model, where Higgs LFV is produced by a renormalizable operator, and models with an extended, gauge singlet scalar sector, in which case the tau radiative decay only appears viaÔ 8γ , generated at two loops, and is negligible. These options for extending the Standard Model may be favored, should this signal persist.
More generally, any model aiming to explain the data should have a mechanism to suppress τ → µγ independently from Higgs LFV, by guaranteeing that the dipole operators are generated at a higher loop order thanÔ 6H .
As this work was being prepared for publication, ATLAS collaboration released the analysis on its search for h → τ µ decays [56], with results compatible with those of CMS. ATLAS constraint on the decay is BR(h → τ µ) < 1.85% at 95% and the best fit to the data is given by BR(h → τ µ) = (0.77 ± 0.62) %. These results have little impact on the analysis of our paper and no updating of constraints was needed (in table 1 and figure 4), since the CMS constraints are stronger.