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

# An explanation of one loop induced h→μτ decay

Seungwon Baek School of Physics, KIAS, Seoul 130-722, Korea    Takaaki Nomura School of Physics, KIAS, Seoul 130-722, Korea    Hiroshi Okada Physics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan 300
July 12, 2019
###### Abstract

We discuss a possibility to explain the excess of at one-loop level. We introduce three generation of vector-like lepton doublet and two singlet scalars which are odd under , while all the standard model fields are even under this discrete symmetry. We show that 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 between experiments and the standard model predictions, as well as the excess of %, while evading constraints from experiments of dark matter direct detection and charged lepton flavor violating processes. We also consider prospects of production of at LHC with energy TeV.

preprint: KIAS-P16030

## 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 Chatrchyan:2012xdj () and CMS Aad:2012tfa () 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 Khachatryan:2015kon () and ATLAS Aad:2015gha () collaborations reported the results on their search for rare Higgs decay with the dataset obtained at the LHC 8 TeV. An excess of the events was observed by CMS, with a significance of 2.4, where the best fit value of branching ratio is . ATLAS’ best fit value is , 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 decay have been studied in Campos:2014zaa (); Sierra:2014nqa (); Lee:2014rba (); Heeck:2014qea (); Crivellin:2015mga (); Dorsner:2015mja (); Omura:2015nja (); Crivellin:2015lwa (); Das:2015zwa (); Bishara:2015cha (); Varzielas:2015joa (); He:2015rqa (); Chiang:2015cba (); Altmannshofer:2015esa (); Cheung:2015yga (); Arganda:2015naa (); Botella:2015hoa (); Baek:2015mea (); Huang:2015vpt (); Baek:2015fma (); Arganda:2015uca (); Aloni:2015wvn (); Benbrik:2015evd (); Omura:2015xcg (); Zhang:2015csm (); Hue:2015fbb (); Bizot:2015qqo (); Han:2016bvl (); Chang:2016ave (); Chen:2016lsr (); Alvarado:2016par (); Banerjee:2016foh (); Hayreter:2016kyv (); Huitu:2016pwk (); Chakraborty:2016gff (); Lami:2016mjf (); Thuc:2016qva (). Earlier works on the flavor violating Higgs decay can be found in Arganda:2004bz (); Blankenburg:2012ex (); Arhrib:2012mg (); Harnik:2012pb (); Dery:2013rta (); Arana-Catania:2013xma (); Arroyo:2013tna (); Celis:2013xja (); Falkowski:2013jya (); Arganda:2014dta (); Dery:2014kxa ().

In this paper, we investigate lepton flavor violating effect which is mediated by an exotic lepton doublet and inert singlet scalars which are odd under discrete symmetry . The interaction term allowed by both the SM gauge and the symmetries often appears in radiative seesaw models, providing active neutrino masses. The neutral components of or 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 , 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 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 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 1. We add three iso-spin doublet vector-like exotic fermions with hypercharge , and an isospin singlet scalar with hypercharge to the SM, where they are odd under , is an odd integer and is the electric charge of . Then we define the exotic lepton as

 L′≡[Ψ−m,Ψ−m−1]T. (II.1)

We assume that only the SM Higgs have vacuum expectation value (VEV), which is symbolized by .

The relevant Lagrangian and Higgs potential under these symmetries are given by

 −LY =(yℓ)ij¯LLiΦeRj+(yL)ij¯LLiL′RjSm+(ML)ij¯L′LiL′Rj+h.c., V =m2ΦΦ†Φ+m2S|Sm|2+λΦ|Φ†Φ|2+λS|Sm|4+λΦS|Φ|2|Sm|2 (II.2)

where the first term of can generates the SM charged-lepton masses after the spontaneous electroweak symmetry breaking by VEV of . We assume all the coefficients are real and positive for simplicity. The scalar fields can be parameterized as

 Φ=[w+v+h+iz√2], (II.3)

where GeV is VEV of the Higgs doublet, and and are Goldstone bosons which are absorbed by the longitudinal component of and boson, respectively. Inserting the tadpole condition; , the SM Higgs mass is given by . The mass eigenstate of the exotic scalar has mass

 mS′=m2S+λΦSv22. (II.4)

### ii.1 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 anomaly. They are summarized in Table 2. The processes arise from one-loop diagrams through the term . Then their branching ratios are defined by

 BR(ℓb→ℓaγ)=48π3αemCbG2Fm2b(|(aR)ab|2+|(aL)ab|2), (II.5)

where is the fine structure constant, for (), GeV is the Fermi constant. The Wilson coefficients are obtained to be

 (aR)ab =−mb(4π)23∑i=1(y†L)ai(yL)ib[(m+1)F[mSm,MΨm+1]+mF[MΨm+1,mSm]], (II.6) F[ma,mb] ≡2m6a+3m4am2b+12m4am2bln[mbma]−6m2am4b+m6b12(m2a−m2b)4, (II.7)

while the chirality-flipped ones are suppressed by small mass ratios: . Here .

Our formula of the muon anomalous magnetic moment (muon ) is also given in terms of by

 Δaμ≈−mμ(aR)22, (II.8)

where the lower index of is muon eigenstate.

### ii.2 h→μτ excess

In our case, the excess of 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

 Γ(h→μτ)=|¯M|28πm2h√(mh+mμ)2−m2τ2mh(mh−mμ)2−m2τ2mh, (II.9) |¯M|2=3∑i=1|(y†L)2i(yL)i3μhSS|2(4π)4[(m2h−m2μ−m2τ)(m2μF2L+m2τF2R)−4m2μm2τFLFR], (II.10) FL=∫δ(x+y+z−1)y2dxdydz(z2−z)m2μ+(x2−y)m2τ−xz(m2h−m2μ−m2τ)+xM2Ψm+1+(y+z)m2Sm, (II.11) FR=∫δ(x+y+z−1)z2dxdydz(z2−z)m2μ+(x2−y)m2τ−xz(m2h−m2μ−m2τ)+xM2Ψm+1+(y+z)m2Sm, (II.12)

where is the strength of the trilinear interaction. Then the branching ratio reads

 BR(h→μτ)≈Γ(h→μτ)Γ(h→μτ)+Γ(h), (II.13)

where 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 before we discuss more constrained model with . DM candidate does not exist in this case. Nonzero , however, can enhance the muon as well as the LFVs. Thus one can obtain the sizable value of muon which can as large as , while one can assume that the Yukawa coupling matrix () is diagonal or at least one of ’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  scoto (). In this sense, radiative neutrino models with (at least) two-loop diagrams are favored for nonzero . Another difficulty is decays of , which is charged scalar for , into the SM fields is not possible. Some more additional fields need to be introduced in order to evade this problem for each , and the detailed phenomenology depends on the implemented models. Thus we do not discuss the case of nonzero further.

We will focus on the special case of because it includes a DM candidate in the boson sector, which can possibly solve the above mentioned problems of case. Notice here that the neutral component of the SU(2)-doublet fermion cannot be DM due to the interaction with the SM neutral gauge boson that is ruled out by the direct detection search.

We redefine the exotic fields as , . The Lagrangian in (III.23) can be rewritten as

 −LY =(yℓ)ij¯LLiΦeRj+(yL)ij¯LLiL′RjS+(ML)ij¯L′LiL′Rj+h.c., (III.1) V =m2ΦΦ†Φ+(m2S1S2+h.c.)+m2S2|S|2+λΦ|Φ†Φ|2 +4∑i=0[λiSi(S∗)4−i+h.c.]+(λΦS1|Φ|2S2+h.c.)+λΦS2|Φ|2|S|2, (III.2)

where the corresponding trilinear coupling appearing on Eq. (II.12) is rewritten by and . The formulae for LFVs, muon , and the excess of are obtained simply by putting in Eqs. (II.7), (II.8), and (II.12). Now we discuss the property of a DM candidate in the next subsection.

### iii.1 Dark Matter Candidate

Before the analysis of the DM candidate let us make some assumptions for simplicity as follows: (), , therefore .

Relic density: The thermal averaged annihilation cross section comes from the processes , , , and -exchanging  Griest:1990kh (); Edsjo:1997bg (), and being the SM fermions and gauge bosons, respectively. The Feynman diagrams are shown in Fig. 2. It can be calculated as

 σvrel≈∑f=h,fSM,ℓ,V∫π0sinθdθ|¯M|216πs√1−4m2fs, (III.3)

where

 |¯M|2≈|¯M(2X→2h)|2+∑fSM=(t,b)|¯M(2X→fSM¯fSM)|2 +∑ℓ=(ℓ,νL)|¯M(2X→ℓ¯ℓ)|2+∑V=(Z,W±)|¯M(2X→VV∗)|2, (III.4) |¯M(2X→2h)|2≈λ2ΦS2∣∣ ∣∣1+3v2λΦ2(s−m2h)+v24[1t−M2X+1u−M2X]∣∣ ∣∣2, (III.5) |¯M(2X→fSM¯fSM)|2≈48μ2hSSm2fSM(s−m2h)2v2(s2−2m2fSM), (III.6) |¯M(2X→VV∗)|2≈4λΦS2μ2hSSm4V(s−m2h)2v2(2+(s/2−m2V)2m4V), (III.7) |¯M(2X→ℓ¯ℓ)|2≈82−3∑a,b∑i=1−3|(yL)i,b|2|(yL)i,a|2× [4(p1⋅k1t+p2⋅k1u)(p1⋅k2t+p2⋅k2u)−sM2X(1t2+1u2)−2s(p1⋅p2tu)], (III.8)

where are the Mandelstam variables; are four-momenta of the initial (final) states; ; among all the SM fermions in heavy quarks such as top quark or bottom quark dominate; is the SM vector gauge bosons. We neglect the masses of the SM leptons () in the final states. Notice here that the mode is wave dominant. To include its effect we retain terms up to the in expansion for all the modes. Then the relic density of DM is finally obtained from

 Ωh2≈1.07×109g1/2∗Mpl[GeV]∫∞xf(aeffx2+6beffx3+60deffx4), (III.9)

where is the total number of effective relativistic degrees of freedom at the time of freeze-out, is the Planck mass, , and , and are coefficients in the expansion of the annihilation cross section:

 σvrel≈aeff+beffv2rel+deffv4rel. (III.10)

The observed relic density reported by Planck suggest that  Ade:2013zuv (). In terms of the model parameters the expansion coefficients are

 aeff ≈3μ2hSS∑f=b,tm2f(M2X−m2f)2πM2Xv2(m2h−4M2X)2 ⎷1−m2fM2X +λ2ΦS256πM2X∣∣ ∣∣2+v2(1m2h−2M2X−3λΦm2h−4M2X)∣∣ ∣∣2 ⎷1−m2hM2X +3μ2hSS∑V=W,Z16πM2Xv2(m2h−4M2X)2(2m4V+(2M2X−m2V)2) ⎷1−m2VM2X, (III.11)

where we would not show the explicit forms of and , 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 (b) which gives (III.6). However, the leptons in in the crossed diagram does not contribute to the direct detection process because there is no valence leptons inside nucleons. Although heavy quark contributions to the parton distribution function of nucleon are suppressed, they can make contribution via Higgs-gluon-gluon triangle diagram. Here we estimate the DM-nucleon scattering cross section following Ref. Cline:2013gha (). Firstly we obtain the following effective Lagrangian by integrating out for non-relativistic momentum transfer,

 Leff=∑qChSSmqm2hX2¯qq, (III.12)

where and 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 determines the effective interaction between the quarks and . The corresponding value in our model is

 ChSS=μhSSv. (III.13)

Then the effective -nucleon () interaction can be written down by

 LNeff=fNChSSmNm2hX2¯NN (III.14)

where the effective coupling constant is given by

 fN=∑qfNq=∑qmqmN⟨N|¯qq|N⟩. (III.15)

Note that the quark mass is absorbed into the definition of quark mass fraction

 fNq≡mqmN⟨N|¯qq|N⟩. (III.16)

The heavy quark contributions are replaced by the gluon contributions by calculating the triangle diagram

 ∑q=c,b,tfNq=1mN∑q=c,b,t⟨N|(−αs12πmqGaμνGaμν)|N⟩. (III.17)

From the scale anomaly, the trace of the stress energy tensor is written as Shifman:1978zn ()

 θμμ=mN¯NN=∑qmq¯qq−7αs8πGaμνGaμν. (III.18)

From (III.17) and (III.18) we finally obtain

 ∑q=c,b,tfNq=29(1−∑q=u,d,sfNq), (III.19)

which results in

 fN=29+79∑q=u,d,sfNq. (III.20)

Here we use the DM-neutron () 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 with neutron through the SM Higgs() portal process is obtained to be Cline:2013gha ()

 σSI(Xn→Xn)×(ρXρDM) =1πμ2nXM2Xm2nC2hSSf2nm4h×(ρXρDM)≃μ2hSSf2nπv2m4nm4hM2X(Ωh20.12) ≈5.29×10−43(μhSSMX)2(Ωh20.12) [cm2], (III.21)

where is the neutron mass, we approximated , and are current density of and total density of DM, and (with ) represents the sum of the contributions of partons to the mass fraction of neutron  Belanger:2013oya (). The scattering cross section imposes a strong constraint on the parameter space relevant to the DM. The constraint from the LUX experiment is the strongest at present with less than (10) cm for DM mass about (10) GeV Akerib:2013tjd (). Notice here that the experimental bound on the direct detection is obtained by assuming that one of the DM components occupies all of the DM components (i.e., ) in the current universe. Otherwise we should multiply the factor for the scattering cross section as can be seen in Eq. (III.21). It suggests that the upper bound from direct detection is relaxed when our is subcomponent of DM, because .

Numerical analysis:

Now that all of the analytical formulae are derived, we perform numerical analysis and explore the allowed region. We scan the parameters in the ranges:

 MX∈[100GeV,500GeV],μhSS∈[50GeV,500GeV],ML(=MEi=MNi)∈[MX, 1TeV], (yL)ℓ,m∈[−0.01,0.01], (ℓ,m)=((1,1),(2,1),(3,1)),(yL)i,j∈[−√4π,4π], (i,j)≠(ℓ,m), (III.22)

where denotes DM mass. The ranges are chosen to satisfy perturbativity of , the bound from the charged lepton flavor violation, and also electroweak scale new particles are assumed. The result does not change much even if we enlarge the ranges. Here we assume Yukawa couplings are small when it has index corresponding to electron in order to satisfy constraint from .

We scanned the above regions of parameters randomly to obtain the allowed range of branching ratio and muon , 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 is around , which is smaller than the current discrepancy  bennett (). The relic density is , 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 , one has to increase the value of trilinear coupling . On the other hand, the direct detection bound suggests , if cm. To evade the constraint from LUX the DM mass scale should be above 100 TeV, which conflicts with relic density and excess. This leads to the conclusion that cannot be main source of the relic DM.

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 , which we denote by . 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 . Then the Lagrangian is simply obtained by replacing, e.g., , , etc. Explicitly the new terms include

 −LY ⊃∑α=1,2⎛⎝(yαL)ij¯LLiL′RjSα−∑β=1,2λΦSαSβ|Φ|2SαSβ+h.c.⎞⎠, (III.23)

where and we neglect for simplicity. We assume so that still remains as a DM candidate. In this case, can play a crucial role in generating the excess , while it need not contribute to the interaction of the direct detection searches. We take the same regions given Eq. (III.22) as our new input parameters except the following,

 MS2∈[1110GeV,1TeV],μhS1S1∈[0.01GeV,0.1GeV], (III.24)

and in this case. We show the results in Fig. 4, in which relic density is within the current observational value. The maximum value of the muon is around , which can explain the discrepancy  discrepancy1 () at the 2 level. To obtain muon , and are preferred. For simplicity we assume the mass difference ratio to evade the coannihilation regime. The trilinear coupling can be decreased by three orders magnitude below the original value of one model to satisfy the direct detection experiments without affecting other observables, , , DM relic density. In this case the DM relic density is achieved dominantly by channel. Here we provide the typical parameter set as follows:

 MX≈146 GeV,ML(=MEi=MNi)≈(663,980,460)[TeV], MS2≈332[TeV],μhS1S1≈0.079[GeV],μhS2S2≈23[GeV], (yL)ℓ,m≈⎡⎢⎣−0.0076−1.00.16−0.0076−0.83−2.8−0.00630.38−2.4⎤⎥⎦, (yL)i,j≈⎡⎢⎣−0.0060−0.89−3.0−0.0062−0.25−3.30.02.5−0.38⎤⎥⎦, (III.25)

then we can obtain the following observables:

 (g−2)μ≈1.8×10−10,Ωh2≈0.12,BR(h→μτ)≈0.48%,σSI≈4.4≈10−50 [cm2], 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 to produce via gluon fusion, , since the coupling constant of the interaction is required to be large as GeV in obtaining sizable branching ratio. The produced mainly decays into through the interaction like in scalar potential where we assume is heavier than for simplicity. It suggests that the 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 Belyaev:2012qa () using CTEQ6L PDF Nadolsky:2008zw () by implementing relevant interactions in the code. In Fig. 5, we show the production cross section of as a function of mass adopting some values of and collision energy of 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 is large and is around 100 GeV. Note that the cross section becomes significantly large when the close to 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 signal at the LHC. In Table 3, we also show the number of expected events at the LHC 14 TeV for several values of and with luminosity of 100 fb 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 and muon 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 , and the branching ratio of . Then we moved on to the specific case where single DM candidate can be included. We found the sizable excess of has been obtained. However the relic density and muon 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 as well as . We introduced another gauge singlet boson having the same quantum numbers with , and we solved all the issues. At the end, we have discussed the signature of our scenario at the LHC, focusing on the interaction to produce via gluon fusion, . This is because the coupling constant of the interaction is required to be large as GeV in obtaining sizable branching ratio. The produced mainly decays into through the coupling in scalar potential. It suggests that the 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 is around 100 GeV for TeV 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 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) 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 Arina:2012aj (). However since there is no new source of the muon for both cases, we need some extensions such as we have mainly discussed in our paper.

## Acknowledgments

H. O. is sincerely grateful for all the KIAS members, Korean cordial persons, foods, culture, weather, and all the other things. This work is supported in part by National Research Foundation of Korea (NRF) Research Grant NRF-2015R1A2A1A05001869 (SB).

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