An explanation of one loop induced $h\to \mu \tau$ decay

We discuss a possibility to explain the excess of $h\to\mu\tau$ at one-loop level. We introduce three generation of vector-like lepton doublet $L'$ and two singlet scalars $S_{1,2}$ which are odd under $Z_2$, while all the standard model fields are even under this discrete symmetry. We show that $S_1$ can be a good dark matter candidate. We show that we can explain the dark matter relic abundance, large part of the discrepancy of muon $g-2$ between experiments and the standard model predictions, as well as the $h\to\mu\tau$ excess of $\sim 1$ %, while evading constraints from experiments of dark matter direct detection and charged lepton flavor violating processes. We also consider prospects of production of $S_2$ at LHC with energy $\sqrt{s}=14$ TeV.


I. INTRODUCTION
The mechanism of the electroweak symmetry breaking is being further understood by the discovery of the Standard Model(SM)-like Higgs boson with mass around 125 GeV at the ATLAS [1] and CMS [2] experiments. Furthermore, we would obtain some insight on physics beyond the SM by exploring nature of the Higgs boson such as its decay channels and scalar potential.
The CMS [3] and ATLAS [4] collaborations reported the results on their search for rare Higgs decay h → µτ with the dataset obtained at the LHC 8 TeV. An excess of the events was observed by CMS, with a significance of 2.4σ, where the best fit value of branching ratio is BR(h → µτ ) = (0.84 +0. 39 −0.37 )%. ATLAS' best fit value is BR(h → µτ ) = (0.77 ± 0.62)%, consistent with but less significant than CMS. Since the lepton flavor violating Higgs decay is highly suppressed in the SM, their findings are an intriguing hint indicating new physics (NP) which induces lepton flavor violation in the charged lepton sector, although we need more data to get conclusive evidence for NP. Actually, inspired by the excess, new physics effects in h → µτ decay have been studied in . Earlier works on the flavor violating Higgs decay can be found in [42][43][44][45][46][47][48][49][50][51][52].
In this paper, we investigate lepton flavor violating effect which is mediated by an exotic lepton doublet L ′ and inert singlet scalars S which are odd under discrete symmetry Z 2 . The interaction termLL ′ S allowed by both the SM gauge and the Z 2 symmetries often appears in radiative seesaw models, providing active neutrino masses. The neutral components of L ′ or S can be also good dark matter(DM) candidates. In addition lepton flavor violating (LFV) Higgs decay can be induced at one-loop level with the interaction. Thus this interaction provides interesting effects connecting active neutrino masses, dark matter, and lepton flavor violating Higgs decays. Focusing on the interaction, we explore h → µτ , charged lepton flavor violations, anomalous magnetic moment of muon and relic density of bosonic dark matter candidate, considering a specific model as an example. Then we search for the parameter region which explains the excess of h → µτ observed by CMS with sizable muon magnetic moment and observed relic density of DM, taking into account the constraints from flavor violating lepton decays. Furthermore we discuss possible signature of our scenario which could be tested at the LHC. This paper is organized as follows. In Sec. II, we show our model, including LFVs, muon Lepton Fields Scalar Fields is the quantum number of the electric charge.
anomalous magnetic moment, and LFV Higgs decay. In Sec. III, we carry out numerical analysis including bosonic DM candidate to explain relic density and direct detection in a specific case. We conclude and discuss in Sec. IV.

II. MODEL SETUP
In this section, we explain our model. The particle contents and their charges are shown in Table I. We add three iso-spin doublet vector-like exotic fermions L ′ with hypercharge −N/2, and an isospin singlet scalar S m with (−1 + N)/2 hypercharge to the SM, where they are odd under Z 2 , N(≥ 1) is an odd integer and m(≡ N −1 2 ) is the electric charge of S. Then we define the exotic lepton as We assume that only the SM Higgs Φ have vacuum expectation value (VEV), which is The relevant Lagrangian and Higgs potential under these symmetries are given by where the first term of L Y can generates the SM charged-lepton masses m ℓ ≡ y ℓ v/ √ 2 after the spontaneous electroweak symmetry breaking by VEV of Φ. We assume all the coefficients 8   TABLE II: Summary of ℓ b → ℓ a γ process and the lower bound of experimental data [54].
are real and positive for simplicity. The scalar fields can be parameterized as where v ≃ 246 GeV is VEV of the Higgs doublet, and w ± and z are Goldstone bosons which are absorbed by the longitudinal component of W and Z boson, respectively. Inserting the tadpole condition; ∂V/∂φ| v = 0, the SM Higgs mass is given by √ 2λ Φ v. The mass eigenstate S ′m of the exotic scalar has mass

A. Lepton Flavor Violations and Muon anomalous magnetic moment
First, let us consider the LFV decays in the charged lepton sector, which impose constraints on the h → µτ anomaly. They are summarized in Table II. The processes Then their branching ratios BR(ℓ b → ℓ a γ) are defined by where α em is the fine structure constant, GeV −2 is the Fermi constant. The Wilson coefficients (a R ) ab are obtained to be while the chirality-flipped ones are suppressed by small mass ratios: a L = a R m a /m b . Here Our formula of the muon anomalous magnetic moment (muon g −2) is also given in terms of a L/R by where the lower index 2 of a R is muon eigenstate.

B. h → µτ excess
In our case, the excess of h → µτ can be generated at one-loop level as the leading contribution. Its Feynman diagram is shown in Fig. 1.
The resultant decay rate formulas are expressed as where µ hSS ≡ λ ΦS v/2 is the strength of the trilinear hS ±m S ∓m interaction. Then the branching ratio reads GeV is the total decay width of the SM Higgs boson at 125.5 GeV. III.

THE CASE m = 0
Let us first consider briefly the case m = 0 before we discuss more constrained model with m = 0. DM candidate does not exist in this case. Nonzero m, however, can enhance the muon g − 2 as well as the LFVs. Thus one can obtain the sizable value of muon g − 2 which can as large as O(10 −9 ), while one can assume that the Yukawa coupling matrix (y L ) is diagonal or at least one of m S m , m Ψ m+1 's are very large to evade the constraints of LFVs.
The model has all the ingredients to generate Majorana neutrino mass matrix via radiative seesaw mechanism, which, however, is not straightforward due to the Dirac nature of L ′ [53].
In this sense, radiative neutrino models with (at least) two-loop diagrams are favored for nonzero m. Another difficulty is decays of S m , which is charged scalar for m = 0, into the SM fields is not possible. Some more additional fields need to be introduced in order to evade this problem for each m, and the detailed phenomenology depends on the implemented models. Thus we do not discuss the case of nonzero m further.
We will focus on the special case of m = 0 because it includes a DM candidate S m ≡ S 0 in the boson sector, which can possibly solve the above mentioned problems of m = 0 case.
Notice here that the neutral component of the SU(2) L -doublet L ′ fermion cannot be DM due to the interaction with the SM neutral gauge boson Z that is ruled out by the direct detection search.
We redefine the exotic fields as in (III.23) can be rewritten as where the corresponding trilinear coupling µ hS ±m S ∓m appearing on Eq. (II.12) is rewritten The formulae for LFVs, muon g − 2, and the excess of h → µτ are obtained simply by putting m = 0 in Eqs. (II.7), (II.8), and (II.12). Now we discuss the property of a DM candidate S R/I in the next subsection.

A. Dark Matter Candidate
Before the analysis of the DM candidate let us make some assumptions for simplicity as Relic density: The thermal averaged annihilation cross section comes from the processes [55,56], f SM and V being the SM fermions and gauge bosons, respectively. The Feynman diagrams are shown in Fig. 2. It can be calculated as where s, t, u are the Mandelstam variables; p 1 , p 2 (k 1 , k 2 ) are four-momenta of the initial The observed relic density reported by Planck suggest that Ωh 2 ≈ 0.12 [57]. In terms of the model parameters the expansion coefficients are where we would not show the explicit forms of b eff and d eff , because they are too complicated.
Direct detection: The DM-nucleon scattering is induced by the SM Higgs exchanging process in our model, which is calculated in non-relativistic limit. The dominant tree-level diagram is obtained by crossing Fig. 2  Ref. [58]. Firstly we obtain the following effective Lagrangian by integrating out h for non-relativistic momentum transfer, where q and m q represent the corresponding quark fields and the quark masses respectively, the sum is over all quark flavors, and we neglected higher dimensional operators.
The coefficient C hSS determines the effective interaction between the quarks and X. The corresponding value in our model is Then the effective X-nucleon (N) interaction can be written down by where the effective coupling constant f N is given by Note that the quark mass m q is absorbed into the definition of quark mass fraction The heavy quark contributions are replaced by the gluon contributions by calculating the From the scale anomaly, the trace of the stress energy tensor is written as [59] θ Here we use the DM-neutron (n) scattering cross section to consider constraints from direct detection where that of DM-proton case is almost same for Higgs portal interaction. Then the spin independent scattering cross section of the X with neutron through the SM Higgs(φ) portal process is obtained to be [58] σ We scanned the above regions of parameters randomly to obtain the allowed range of h → µτ branching ratio and muon g − 2, imposing the constraint from dark matter relic density and direct detection experiments. The results are shown in Fig. 3. As we can see in the figure, we can accommodate the excess observed by CMS. In this case, the maximum value of the muon g − 2 is around (3 × 10 −10 ), which is smaller than the current discrepancy O(10 −9 ) [62]. The relic density is O(0.001 − 0.01), which is also smaller than the current measurement 0.12 as can be seen in Fig. 3. It is mainly due to the direct detection bound. In order to obtain enough excess of h → µτ , one has to increase the value of trilinear coupling To explain DM relic abundance we extend the model as minimally as possible. One of the minimal extension to solve this issue is to introduce another gauge singlet boson having the same charge with S, which we denote by S 2 . Then all the terms of Eq. (III.2) remain in the same form with only the number of terms doubled. For our convenience let us rename two singlet bosons (S 1 , S 2 ). Then the Lagrangian is simply obtained by replacing, e.g., where µ hSαS β ≡ λ ΦSαS β v/2 and we neglect µ hS 1 S 2 for simplicity. We assume M S 2 > M S 1 so that S 1 still remains as a DM candidate. In this case, µ hS 2 S 2 can play a crucial role in generating the excess h → µτ , while it need not contribute to the interaction of the direct detection searches. We take the same regions given Eq. then we can obtain the following observables: BR(µ → eγ) ≈ 2.5 × 10 −13 , BR(τ → eγ) ≈ 8.6 × 10 −12 , BR(µ → µγ) ≈ 1.3 × 10 −8 .

(III.26)
Collider phenomenology of our scenario: Now we discuss the signature of our scenario at the LHC. Here we focus on the interaction µ hS 2 S 2 h|S 2 | 2 to produce S 2 via gluon fusion, gg → h → S 2 S 2 , since the coupling constant of the interaction hS 2 S 2 is required to be large as O(100) GeV in obtaining sizable h → µτ branching ratio. The produced S 2 mainly decays into S 1 h through the interaction like vhS 1 S 2 in scalar potential where we assume L ′ is heavier than S 2 for simplicity. It suggests that the S 2 can be measured at the LHC where the signature will be two SM Higgs boson with missing transverse energy. Then the production cross section is numerically estimated with CalcHEP [64] using CTEQ6L PDF [65] by implementing relevant interactions in the code. In Fig. 5, we show the production cross section of pp → S 2 S 2 as a function of S 2 mass adopting some values of µ hS 2 S 2 and collision energy of √ s = 14 TeV. Here the cross section is at the leading order and it will be larger when we consider K-factor. We find that the cross section can be sizable when µ hS 2 S 2 is large and m S 2 is around 100 GeV. Note that the cross section becomes significantly large when the m S 2 close to m h /2 due to resonant enhancement since the process is SM Higgs boson exchanging s-channel, although the resonant point is below our parameter region.
Thus some parameter space of our scenario can be tested by exploring hh E T signal at the LHC. In Table III, we also show the number of expected events at the LHC 14 TeV for several values of m S 2 and µ hS 2 S 2 with luminosity of 100 fb −1 as a reference. Moreover the study of exotic lepton production will be also interesting. The detailed simulation study is beyond the scope of this paper and it is left as future study.

IV. CONCLUSIONS AND DISCUSSIONS
We studied a possibility to explain the excess of h → µτ and muon g − 2 in a model with a dark matter candidate. At first, we provided a simple set up with generic hypercharge  assignments, in which we formulated the lepton flavor violations, muon g − 2, and the branching ratio of h → µτ . Then we moved on to the specific case where single DM candidate can be included. We found the sizable excess of h → µτ has been obtained. However the relic density and muon g − 2 cannot be explained due to the stringent constraint from the direct detection via Higgs portal.
We extended the model as minimally as possible so that we can explain the relic density and the muon g − 2 as well as h → µτ . We introduced another gauge singlet boson S 2 having the same quantum numbers with S 1 , and we solved all the issues. At the end, we have discussed the signature of our scenario at the LHC, focusing on the interaction µ hS 2 S 2 h|S 2 | 2 to produce S 2 via gluon fusion, gg → h → S 2 S 2 . This is because the coupling constant of the interaction hS 2 S 2 is required to be large as O(100) GeV in obtaining sizable h → µτ branching ratio. The produced S 2 mainly decays into S 1 h through the coupling vhS 1 S 2 in scalar potential. It suggests that the S 2 can be searched for at the LHC where the signature is two SM Higgs boson with missing transverse energy. We found that the cross section can be sizable when when m S 2 is around 100 GeV for √ s = 14 TeV pp collision.
It is worth to mention possible application of our model to the other sectors such as neutrinos. Since our set up is very simple, several applications to the neutrino sector could be possible. Let us just briefly comment on two possibilities. First, if we introduce a gauge singlet Majorana fermion with Z 2 odd charge, we can explain the neutrino masses and mixings at the one-loop level, and the fermion can be a good DM candidate. Second possibility is to introduce a SU(2) triplet boson with nonzero VEV. In this case, the neutrino masses and the mixings are induced through the type-II seesaw mechanism.The neutral component of the SU(2) L doublet exotic lepton can be a DM candidate since sizable mass splitting between right-handed and left handed neutral fermions can be obtained to evade the strong bound from direct detection experiments [66]. However since there is no new source of the muon g − 2 for both cases, we need some extensions such as we have mainly discussed in our paper.