Flavor Violating Higgs Couplings in Minimal Flavor Violation

Motivated by the rencent LHC data on the lepton-flavor violating (LFV) decays $h\to \ell_1 \ell_2$ and $B_{s,d}\to \ell_1 \ell_2$, we study the Higgs-mediated flavor-changing neutral current (FCNC) interactions in the effective field theory (EFT) approach without and with the minimal flavor violation (MFV) hypothesis, and concentrate on the later. After considering the $B$ and $K$ physics data, the various LFV processes, and the LHC Higgs data, severe constraints on the Higgs FCNC couplings are derived, which are dominated by the LHC Higgs data, the $B_s - \bar B_s$ mixing, and the $\mu \to e \gamma$ decay. In the general case and the MFV framework, allowed ranges of various observables are obtained, such as $\mathcal B (B_s\to \ell_1 \ell_2)$, $\mathcal B (h\to \ell_1 \ell_2)$, $\mathcal B (h\to q_1 q_2)$, and the branching ratio of $\mu \to e$ conversion in Al. Future prospects of searching for the Higgs FCNC interactions at the low-energy experiments and the LHC are discussed.


Introduction
The Higgs boson has been discovered at the LHC [1,2], with a mass of 125 GeV and properties consistent with the standard model (SM) predictions. Precision measurements on the Higgs couplings with the SM particles will be one of the most important tasks for the LHC Run II and its high-luminosity upgrade. A significant deviation from the SM expectations in Higgs phenomenology would be an indicator of new physics (NP) [3,4].
Besides precision measurements of the Higgs boson at high-energy colliders, it is equally important to search for low-energy processes that are rare or forbidden in the SM [5]. In the SM, the flavor-changing neutral current (FCNC) Yukawa interactions are forbidden at the tree level. However, the Higgs-mediated FCNCs generally appear at the tree level in models beyond the SM [6][7][8][9]. These Higgs-mediated couplings can generate the processes which are forbidden in the SM, or enhance some rare decays. In this respect, the leptonflavor violating (LFV) decays provide excellent probes for such FCNC interactions, such as the µ → eγ, B s,d → 1 2 and h → 1 2 decays ( 1,2 = e, µ, τ ) and can be probed by the LHC and other low energy experiments.

Higgs FCNC
In the SM, the Yukawa interactions with quarks are described by the following Lagrangian in the interaction basis, (2.1) where Q L denotes the left-handed quark doublet, d R the right-handed down-type quarks, u R the right-handed up-type quarks, H the Higgs doublet, andH ≡ iσ 2 H * . The Yukawa coupling matrices Y u,d are 3 × 3 complex matrices in flavor space.
In the SM, the Higgs doublet develops a non-zero vacuum expectation value H = v/ √ 2 which breaks electroweak weak symmetry down to U(1) em , the charged Higgs fields H ± and the imaginary part of the neutral components are "eaten" by W ± and Z bosons and left a physical neutral Higgs h. Working in the basis of quark mass eigenstates, the above Lagrangian gives a flavor conserving Higgs-fermion coupling of the form m ff f (1 + h/v).
When going beyond the SM, the above simple flavor conserving couplings will be modified. Considering the BSM effects in the EFT approach, these Higgs Yukawa interactions can be affected by dim-6 operators at the tree level. There are several different bases to choose for writing down the operators. We will work in the Warsaw basis in ref. [28]. There exist only three operators relevant to our analysis to the lowest order. They are given by (2.2) where the doublet/singlet Q L , u R , d R , and the couplings C uH,dH, H are in flavour space, and their flavour indices are omitted. It is also noted that these operators can arise in many BSM scenarios, such as Extra Dimensions [29,30], Composite Higgs [31,32], models with vector-like quarks [33], Two-Higgs Doublet Models [34,35], and other extended Higgs sectors [36]. In a genuine EFT approach, analysis of low-energy flavor transitions, such as B s −B s mixing, should include not only the operators listed in eq. (2.2), but also other dim-6 operators which can affect these flavor transitions at the tree level. In the Warsaw basis, these additional operators are 4-fermion operators, such as (Q L γ µ Q L )(Q L γ µ Q L ), dipole operators, such as (Q L σ µν d R )HB µν , and operators involving ( . It is noted that all these operators do not contribute to the Higgs FCNC Yukawa couplings at the tree level. Therefore, in this work, we don't consider these operators and their effects to the low-energy FCNC processes are assumed to be negligible compared to the contributions from the Higgs FCNC Yukawa couplings. Our numerical results in section 4 are valid only under this assumption. We refer to refs. [38] and [39,40] for general EFT analysis of B-meson FCNC processes and µ → e transitions, respectively. The assumption that the BSM contributions to the low-energy flavour transitions are dominated by the effect of the modified Higgs Yukawa couplings in the form of eq. (2.2) has limitation in EFT analysis. For example, in the type-II 2HDM, a tree-level matching gives the effective Lagrangian in the scalar potential, Λ denotes the mass scale of m A , m H ± and m H 0 , η u = 1, and η d = − tan 2 β [34]. Therefore, in the processes involving four down-type fermions, e.g.,ssdd, the condition for the effect of modified Yukawa couplings dominates over the one of four-fermion operator is Z 6 v 2 /m 2 h tan β, which requires a large Z 6 or small tan β. However, these parameter space are disfavored by the combined constraints of the Electro-Weak Precision Test, the current LHC Higgs data, and flavour physics. The situation in the Composite Higgs models is different. In the Composite Higgs scenario in ref. [37], a Naive Dimension Analysis (NDA) gives that the Wilson coefficients of the low-energy ∆F = 2 FCNC fourfermion operators from the Higgs, Z and heavy vector exchange have the power counting C Higgs : Here y * and g * denote the typical coupling of the composite Higgs to the other strong states and the typical coupling of the heavy resonances of the strong sector, respectively. In the parameter region y 6 * (v 4 /m 2 h Λ 2 ) g 2 * , the contribution from the Higgs exchange is dominant. The operators above can contribute to the fermion mass terms in dim-4 L SM after the symmetry breaking H † H → 1/2v 2 . The Yukawa couplings of h to fermions are given by with the definitionȲ where Λ denotes some NP scale.
In the mass-eigenstate basisȲ f becomes diagonal, but the Higgs Yukawa interactions [41] and induces FCNC interactions. We write them as Where f denotes (u, c, t), (d, s, b) or (e, µ, τ ). Y L and Y R are 3 × 3 complex matrices in flavor space and connect to each other by the relation 2m i /v and the vacuum expectation value v = 246 GeV. Now Y u plays the role of Y u . Similarly for d and sectors. Here we have used dim-6 operators to show how to parametrize the general form of a Higgs to fermions couplings. This should apply to more general cases.
In the literature, the following basis for the Higgs Yukawa interactions is also widely used Here, Y andȲ are 3 × 3 Hermitian matrices. This form is related to eq. (2.5) by Y R,L = Y ± iȲ . It is noted that real Y ij L,R do not imply real Y ij orȲ ij , and vice versa.

Higgs FCNC in MFV
In the SM, the Yukawa interactions in eq. (2.1) violate the global flavor symmetry

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This flavor symmetry can be recovered by formally promoting the Yukawa matrices to spurions fields, which transform as The coefficients C uH,dH in eq. (2.3) are in general independent from the tree level defined Yukawa couplings Y u,d . To have more definitive framework for our later analysis, we will work under the assumption of MFV hypothesis. This hypothesis implies that [27], all CP and flavor violating sources come from Y u,d and the effective Lagrangian is invariant under the flavor symmetry group G QF . Therefore C uH,dH can be written in the following form in order to have the right transformation properties under G QF , (2.9) The function f u,d (A, B) can be expanded in an infinite series of the form f u,d (A, B) ≡ ξ u,d ijk··· A i B j A k · · · with ξ u,d ijk··· to be real since no new CP violating source should be introduced other than that already contained in Y u,d . Using the Cayley-Hamilton identity, f (A, B) can be generally resumed into 17 terms [42,43], Due to the resummation, the coefficients κ i can receive some contributions from combinations of A and B and become complex generally. However, it can be shown that Imκ i ∝ |Tr(A 2 BAB 2 )| 1 and therefore these tiny imaginary parts can be neglected in our numerical analysis [42][43][44][45][46]. Since the spurion B is highly suppressed by the small down-type quark Yukawa couplings, terms with B are neglected and we obtain [47] (2.10) The coefficients u 0,1,2 and d 0,1,2 are free complex parameters but have negligible imaginary components [42][43][44][45][46].
For the down-type quarks, the Yukawa interactions with the dim-6 operator O dH after the EW symmetry breaking read with the definitionȲ Using the MFV hypothesis in eq. (2.9) and the approximation in eq. (2.10),

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With the redefinition in eq. (2.12) (2.14) Finally, we obtain the Yukawa interactions for down-type quarks in the mass eigenstate Due to the large hierarchy in the diagonal matrix λ u , theˆ d 1 andˆ d 2 terms have almost the same structure. Therefore, we will use the following approximation in the numerical analysis We have checked that the numerical differences due to this approximation are negligible.
Similarly, the Yukawa interactions for up-type quarks in the MFV are obtained with the definitionˆ u i = (v 2 /Λ 2 ) u i . Due to the large hierarchy in the diagonal matrix λ u and λ t ≈ 1, we take the approximationˆ u We have checked that the numerical differences due to this approximation are negligible.
In the MFV, the FCNC in the up sector is negligibly small. For the lepton sector, definition of MFV depends on the underlying mechanism responsible for neutrino masses and is not unique [48][49][50][51]. Here, we adopt the approach in ref. [47], which is based on type-I seesaw mechanism. Then, the basic building block spurion similar to A in the quark sector, reads in the mass eigenstate where U denotes the Pontecorvo-Maki-Nakagawa-Sakata matrix,m ν the diagonal neutrino mass matrix diag(m 1 , m 2 , m 3 ) and M mass of the right-handed neutrinos. Matrix O is generally complex orthogonal, satisfying OO T = 1 [52]. Then, after neglecting small B terms, the Yukawa interactions for charged lepton reads

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In summary, the Yukawa couplings in the MFV framework can be written as in the basis of eq. (2.5), (2.21) All the above Yukawa matrices are Hermitian in the MFV framework.

Relevant processes
In this section we consider possible processes which can constrain the Higgs FCNC couplings to fermions. We find the most relevant processes are B s −B s , B d −B d and K 0 −K 0 mixing, B s,d → 1 2 decays, the leptonic decays i → j γ and µ → e conversion in nuclei, and Higgs production and decay at the LHC, which are investigated in detail in this section.

Neutral B and K meson mixing
Including the Higgs FCNC contributions, the effective Hamiltonian for B s −B s mixing can be written as [53] H ∆B=2 where the operators relevant to our study are with α and β color indices. V ij denote the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements. The SM contributes to only the O VLL 1 operator, whose Wilson coefficients C VLL 1 can be found in ref. [54]. The other operators can be generated by tree-level Higgs FCNC exchange, whose Wilson coefficients read [41] C SLL,NP The contribution from H ∆B=2 eff to the transition matrix element of B s −B s mixing is given by [53], where recent lattice calculations of the hadronic matrix elements O i can be found in refs. [55,56]. Then the mass difference and CP violation phase read ∆m s = 2|M s 12 | , and φ s = arg M s 12 . (3.5)

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In the case of complex Yukawa couplings, φ s can deviate from the SM prediction, i.e., can affect the CP violation in the B s → J/ψφ decay [57], as well as A ∆Γ in the B s → µ + µ − decay as in eq. (3.14). In the basis in eq. (2.6), it can be seen that the mass difference ∆m s depends only on Y 2 sb andȲ 2 sb , but not Y sbȲsb . In addition, we follow ref. [53] to perform renormalization group evolution of the NP operators O SLL 1 , It is found that including RG effects of the NP operators enhances the NP contributions by about a factor of 2.

B s → 1 2 decay
In this subsection, we consider the B s → µ + µ − decay as an example to recapitulate the theoretical framework of the B s → 1 2 processes. Within the Higgs FCNC effects, the effective Hamiltonian of the B s → µ + µ − decay reads [54] where α e is the fine structure constant, and s 2 W ≡ sin 2 θ W with θ W being the weak mixing angle. The operators O In the framework we are working with, the Wilson coefficient C A contains only the SM contribution, and its explicit expression up to the NLO QCD corrections can be found in refs. [58][59][60]. Recently, corrections at the NLO EW [61] and NNLO QCD [62] have been completed, with the numerical value approximated by [63] where m P t denotes the top-quark pole mass. In the SM, the Wilson coefficients C SM S and C SM P can be induced by the Higgs-penguin diagrams but are highly suppressed. Their expressions can be found in refs. [64,65]. As a very good approximation, we can safely take With the Higgs-mediated FCNC interactions in the effective Lagrangian, eq. (2.5), the scalar and pseudoscalar Wilson coefficients read with the common factor

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For the effective Hamiltonian eq. (3.6), the branching ratio of B s → µ + µ − reads [64,65] where m Bs , τ Bs and f Bs denote the mass, lifetime and decay constant of the B s meson, respectively. The amplitudes P and S are defined as From these expressions and using the basis in eq. (2.6), it can be seen that the branching ratio of B s → µ + µ − only depends onȲ sb Y µµ andȲ sbȲµµ . Due to the B s -B s oscillations, the measured branching ratio of B s → µ + µ − should be the time-integrated one [66]: with [67] (3.14) Here, Γ L s (Γ H s ) denote the decay widths of the light (heavy) B s mass eigenstates. ϕ P and ϕ S are the phases associated with P and S, respectively. The CP phase φ NP s comes from B s -B s mixing and has been defined in eq. (3.5). In the SM, A SM ∆Γ = 1.

Leptonic decays i → j γ
Considering the Higgs FCNC interactions, the effective Lagrangian for the i → j γ decays are given by [5] where m i denotes the mass of the lepton i and F µν the photon field strength tensor. Then, the decay rate of i → j γ is given by [5] The Wilson coefficients c L and c R receive contributions from the one-loop penguin diagrams. Their analytical expressions read [5] At the two-loop level, there are also comparable contributions from the Barr-Zee type diagrams. Here, we use the numerical results in ref. [5].
which are obtained from a full two-loop analytical calculations [68]. Here, Y tt L and Y tt R are assumed to be real.

µ → e conversion in nuclei
The Higgs FCNC interactions could induce µ → e conversion in nuclei. The relevant effective Lagrangian reads [5] For the vector operators, the leading contributions arise from one-loop penguin diagrams, and the corresponding Wilson coefficients read [5] g with the loop function Here, Q q is the electric charge of quark q. q 2 denotes square of the moment exchange and takes the value JHEP02(2019)007 of −m 2 µ , which corresponds to the limit of an infinitely heavy nucleus. The coupling g q RV can be obtained from g q LV with the replacement Y → Y † . Using these Wilson coefficients, the rate of µ → e conversion in a nuclei N can be written as [69] Here,g (n,p) L/RS,L/RV denote the couplings to proton and neutron and can be evaluated from the quark-level ones where the summation runs over all quark flavors q ∈ {u, d, s, c, b, t}, and the nucleon matrix elements f (q,p) ≡ p|m qq q|p /m p are numerically [70,71] f where Γ capt. N denotes the muon capture rate, and numerically Γ capt. Au = 1.307 × 10 7 s −1 and Γ capt. Al = 7.054 × 10 5 s −1 [72].

Numerical analysis
In this section, we proceed to present our numerical analysis for the Higgs FCNC couplings in the general case and in the MFV framework presented in section 2.1 and section 2.2, respectively. SM predictions and the current experimental data for various processes discussed in the previous sections.
To constrain the Higgs FCNC couplings, we impose the experimental constraints in the same way as in refs. [41,73]; i.e., for each point in the parameter space, if the difference between the corresponding theoretical prediction and experimental data is less than 1.96 σ (1.65 σ) error bar, which is calculated by adding the theoretical and experimental errors in quadrature, this point is regarded as allowed at 95% CL (90% CL). Since the main theoretical uncertainties arise from hadronic input parameters, which are common to both the SM and the Higgs FCNC contributions, the relative theoretical uncertainty is assumed to be constant over the whole parameter space. As discussed in section 2.1, the numerical results in this section are valid only under the assumption that BSM effects to the lowenergy flavor transitions are dominated by the effective operators listed in eq. (2.2).

Analysis within general Higgs FCNC
In our previous paper [41], the Higgs LFV interactions in eq. (2.5) have already been studied in detail. Here, we focus on the couplings Y eµ L,R and Y eτ L,R . These two couplings could induce h → eµ and h → eτ decay, respectively. The current Higgs data give the following bounds at 95% CL. When obtaining these bounds, the contributions of Y eµ L,R and Y eτ L,R to the Higgs total width have been included.
The FCNC couplings Y sb andȲ sb are constrained by B s −B s mixing. In the case of real Y sb andȲ sb , their allowed regions obtained from ∆m s are shown in figure 1. There are two allowed regions. The one near the origin corresponds to the case where the Higgs FCNC effects are destructive with the SM contribution. In the other region, the Higgsmediated FCNC interactions dominate over the SM contribution. Another bound on these two parameters comes from the h → sb decay. Although there is no upper limits on this process currently, we consider the bound B(h → new) < 34% at 95%CL, which is obtained from a global fit of the LHC run I data and applies to modifications of the decays into SM particles that are not directly measured by the LHC [81]. However, this constraint is much weaker than the one from B s −B s mixing, as shown in figure 1. Furthermore, assuming a SM-like hµµ coupling, B(B s → µ + µ − ) also provides a constraint onȲ sb . Such constraint is comparable with the one from B s −B s mixing, as can be seen in figure 1. In the case of complex Y sb andȲ sb , situation becomes quite different. Since the contributions of Y sb andȲ sb to ∆m s can cancel out each other, B s −B s mixing can't provide upper limits on |Y sb | and |Ȳ sb |. In this case, the upper limits are given by B(B s → µ + µ − ) with the assumption of a SM-like hµµ coupling and are weaker than the ones in the case of real couplings. Finally, the combined constraints on the complex couplings Y sb andȲ sb result in the following prediction Γ(h → sb) < 0.17 MeV, at 95% CL. For the B s → 1 2 decays, using the analytical expressions in section 3, we can obtain the following numerical expression where the SM Higgs total width Γ SM h ≈ 4.07 MeV [82] is assumed. In the case of complex Yukawa couplings, the combined bounds on Y sb andȲ sb discussed above and the LHC bounds on h → i j result in following upper limits at 95% CL. For the branching ratio of B s → eµ decay, our predicted upper limit is three times lower than the current LHCb bound B(B s → eµ) < 6.3 × 10 −9 [10]. The Higgs FCNC couplings can also affect the LFV processes in the lepton sector, such as the µ → eγ decay. However, their dominated contributions arise at loop level and involve several Yukawa couplings. These processes can't provide model-independent bounds on one or two particular Yukawa couplings except assuming some special hierarchy among the Higgs FCNC couplings Y i,j L,R , as in ref. [5].

Analysis in the MFV framework
The Higgs FCNC couplings in the MFV framework have been discussed in detail in section 2.2. In the following numerical analysis, without loss of generality, we take the NP scale Λ = v, such thatˆ u,d, 0,1,2 = u,d, 0,1,2 , 1 and the right-handed neutrinos' mass M = 10 15 GeV. For the MFV in the lepton sector, we consider the simplest possibility that the orthogonal matrix O in eq. (2.19) is real. Since mass ordering of light neutrinos is not yet established, both the normal ordering (NO), where m 1 < m 2 < m 3 , and the inverted ordering (IO), where m 3 < m 1 < m 2 , are included in our analysis. In the NO (IO) case, we take m 1(3) = 0. Finally, the Higgs Yukawa couplings in the MFV framework are determined by the following 6 real parameters which correspond to the up-type quark, down-type quark and lepton sectors, respectively. In the following, the constraints on these parameters will be discussed in detail. The parameters ( u 0 , d 0 , 0 ) control the Higgs flavor-conserving couplings to up-type quarks, down-type quarks and leptons, respectively. They are constrained by the Higgs production and decays processes at the LHC. We perform a global fit for these three parameters with the Lilith package [83], which is used to take into account the Higgs data measured by LHC Run I [81] and Tevatron [84]. Although the flavor-changing parameters d 1 and 1,2 can also affect the Higgs signal strengths, they are strongly bounded by

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other processes, as discussed in the following. Therefore, their contributions can be safely neglected in the global fit. The allowed regions of ( u 0 , d 0 , 0 ) at 90% CL are shown in figure 2. Our global fit shows that O(30%) deviations from the SM values are allowed for the flavor-conserving couplings in the MFV framework.
The flavor-changing couplings for down-type quarks are determined by the parameter d 1 . Constraints on this coupling come from B s −B s , B d −B d and K 0 −K 0 mixing. Since hadronic uncertainties in K 0 −K 0 mixing are relatively large [85,86], we adopt the conservative treatment in ref. [86]; i.e., the Higgs FCNC effects to ∆m K are allowed within 50% range of ∆m exp K , and | K | is allowed to vary within a 20% symmetric range. Since the current experimental data of the B and K mixing are in good agreement with the SM prediction, we obtain the strong bound on the MFV parameter at 95% CL. This bound is dominated by ∆m s in B s −B s mixing. Since the Yukawa couplings Y sd L,R in the MFV framework are suppressed by s or d quark mass as in eq. (2.16), K 0 −K 0 mixing can't provide strong constraint. Using this bound, the predicted upper limits for various Higgs FCNC decays are obtained at 95% CL. These channels suffer from a huge QCD background which makes measurement of these decays very challenging at the LHC, and even at the ILC. At a centre-of-mass energy 500 GeV with an integrated luminosity of 4000 fb −1 , ILC can provide a discovery sensitivity of 0.5% for B(h → bj), with j representing a light quark [87], which is still about one order of magnitude larger than these upper limits listed above. The parameters ( 1 , 2 ) control the flavor-changing couplings for charged leptons. They should be bounded by the LFV processes. However, as discussed in section 3, the quark Yukawa couplings also appear in some leptonic processes, e.g., top quark Yukawa couplings are involved in the two-loop diagrams of µ → eγ and all the quark Yukawa couplings affect µ → e conversion in nuclei at the tree level. Generally, all relevant parameters in the LFV processes are ( u 0 , d 0 , 0 , 1 , 2 ). We don't include the MFV parameter d 1 , since its effect is highly suppressed in the LFV processes. When deriving the bounds on these parameters and studying their effects, it's useful to separate from the effects of the quark Yukawa couplings. Therefore, we consider the following two scenarios in the discussion of the LFV processes.
Scenario I : − 0.5 < 0,1,2 < +0.5, Scenario II : − 1.0 < 0,1,2 < +1.0, To constrain the MFV parameters, we consider various LFV processes including h → i j , i → j k¯ l , i → j γ, µ → e conversion in nuclei, leptonic EDM, and anomalous magnetic moment. The previously obtained bounds on ( u 0 , d 0 , 0 ) from the Higgs data have been also included. After combining all these constraints, the allowed parameter space of ( u 0 , d 0 , 0 , 1 , 2 ) are obtained for scenario I and II in the NO and IO cases, which are plotted in the ( 1 , 2 ) plane in figure 3. All the correlations are showing in figure 6 in appendix A, from which one can see the correlations between the flavor-conserving couplings ( u 0 , d 0 , 0 ) and the flavor-changing couplings ( 1 , 2 ) are weak. It is found that the most strong constraints on the MFV parameters come from the branching ratio of µ → eγ decay. Our detailed numerical analysis shows that the µ → eγ decay in the allowed parameter space is dominated by the two-loop contribution c 2−loop R in eq. (3.19), which is proportional to the couplings Y eµ R and Y tt L . Due to the values of the PMNS matrix in the IO case, the contributions from 1 and 2 can strongly cancel out each other in the Yukawa couplings Y eµ R . It makes the allowed ranges of 1 and 2 in the IO case are much wider than the one in the NO case but have larger fine-tuning.
For comparison, the bounds from µ → e conversion in Au are shown in figure 4, which are much weaker than ones from the µ → eγ decay. In the future Mu2e experiment, the sensitivity for the branching ratio of µ → e conversion is expected to be improved by 4 orders of magnitude compared to the current SINDRUM II bound, which corresponds to 7 × 10 −17 in Al at 90% CL [16]. The allowed parameter space corresponding to the future sensitivity at the Mu2e experiment are shown in figure 3. It can be seen that the expected bounds at the Mu2e experiment are much more stringent than the ones obtained from the current measurements on µ → eγ decay. In the near future, with three-year run, the MEG II experiment can reach a sensitivity of 6 × 10 −14 at 90% CL for B(µ → eγ) [88]. However, the corresponding bounds on the MFV parameters are much weaker than the ones expected at the Mu2e experiment.  Since the experimental sensitivity to the LFV processes µ → eγ and µ → e conversion in nuclei will be greatly improved in the near future, we show the correlations between B(µ → eγ) and B(µ Al → e Al) in figure 5, which are obtained in the allowed parameter space corresponding to figure 3. It can be seen that, the correlations in the NO and IO cases are almost the same. To understand this, we should notice that the Higgs FCNC effects on these two processes are dominated by the contributions of c 2−loop R and g q LS in the allowed parameter space in both the NO and IO cases. In the scenario I, from their definitions in eq. (3.19) and (3.22), they are proportional to the Yukawa coupling Y eµ R , which makes the branching ratios of both the two processes are proportional to |Y eµ R | 2 . Therefore, although Y eµ R depends on ( 1 , 2 ) differently in the NO and IO cases, the correlation between B(µ Al → e Al) and B(µ → eγ) is very strong and does no depend on the ordering of the light neutrinos' masses, as shown by the thin red regions in figure 5. In the scenario II, the contributions c 2−loop R and g q LS are also proportional to Y tt L and Y qq R , respectively. In the MFV framework, the flavor-conserving couplings Y tt L and Y qq R mainly depend on the parameters ( u 0 , d 0 ) and their dependence are the same between in the NO and IO cases. These flavor-conserving couplings make the correlation between B(µ Al → e Al) and B(µ → eγ) much weaker than the one in the scenario I in both the NO and IO cases, as shown by the wide green regions in figure 5. Considering that the bounds on the flavorconserving couplings will be largely improved by the future LHC data, the correlation in the scenario II is expected to become much stronger and approach the one in the scenario I. For the anomalous magnetic moment a µ , current data show about 3σ deviation from the SM prediction [74,89]. In the MFV framework, explanation for this anomaly needs large LFV parameters 1 and 2 , which is ruled out by the µ → eγ decay.
Using the combined bounds obtained in the previous sections, the upper limits on various LFV B s and Higgs decays are obtained for the scenario I and II and in the NO and IO cases, which are shown in table 3. For the h → µτ decay, the upper limits in the MFV are about two orders of magnitude lower than the current LHC bounds, which make searches for this channel challenging at the LHC. For the other LFV decays, since the upper bounds on their branching ratios are lower than the current LHC bounds by several orders of magnitude, they are very difficult to be measured at the LHC. For the B s → µ + µ − decay in both the NO and IO cases, it is found that its branching ratio can't deviate from the SM prediction by more than 1%.

Conclusions
Motivated by the recent LHC searches for the LFV decays B s → i j and h → i j , we study the tree-level Higgs FCNC interactions in the EFT approach. With and without the MFV hypothesis, we investigate the Higgs FCNC effects on the B s −B s , B d −B d and K 0 −K 0 mixing, the lepton FCNC processes i → j γ, i → j k¯ l , µ → e conversion in nuclei, the LHC Higgs data, and etc, and derive the bounds on the Higgs FCNC couplings.
In the general case, the two LFV decays B s → 1 2 and h → 1 2 are related to each other by the following expression assuming the SM Higgs total width. After deriving the bounds onȲ sb from B s −B s mixing and B s → µ + µ − , predictions on various Higgs and B s FCNC decays are obtained, such as B(B s → eµ) < 2.1 × 10 −9 , B(h → sb) < 4.1 × 10 −2 , at 95% CL, where the SM Higgs total width is assumed.

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In the MFV hypothesis, strong constraints on the free parameters ( u 0 , d 0 , d 1 , 0 , 1 , 2 ) are derived. We find that the bounds on ( u 0 , d 0 , 0 ) are dominated by the LHC Higgs data, at 90% CL, where the SM Higgs total width is assumed. For the B s → µ + µ − decay, its branching ratio can't deviate from the SM prediction by more than 1%. For the various B s → 1 2 and h → 1 2 decays, since the upper limits of their branching ratios are much lower than the current LHC bounds, searches for these LFV processes are very challenging at the LHC. However, with the improved measurements at the future MEG II and Mu2e experiments, searches for the LFV Higgs couplings in the µ → eγ decay and µ → e conversion in Al are very promising. In the MFV, the branching ratios of these two processes are strongly correlated to each other. Our bounds and correlations for the various processes can be used to obtain valuable information about the Higgs FCNC couplings from future measurements at the LHC and the low-energy experiments.

A Allowed parameter space in scenario II
In section 4, allowed parameter space of ( u 0 , d 0 , 0 , 1 , 2 ) in scenario II are derived. The results in ( u 0 , d 0 ) and ( u 0 , 0 ), and ( 1 , 2 ) plane have been shown in figure 2 and figure 3, respectively. Here, we show the results in the other planes in figure 6. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.