1 Introduction

Flavor violating leptonic decays of the Higgs boson

Seham Fathy***Email: p-sfathy@zewailcity.edu.eg, Tarek IbrahimEmail: tibrahim@zewailcity.edu.eg, Ahmad ItaniEmail: ahmad.it@gmail.com, Pran Nath§§§Email: nath@neu.edu

University of Science and Technology, Zewail City of Science and Technology,
6th of October City, Giza 12588, Egypt555Permanent address: Department of Physics, Faculty of Science, University of Alexandria, Alexandria, Egypt
Department of Physics, Beirut Arab University, Beirut 11-5020,Lebanon
Department of Physics, Northeastern University, Boston, MA 02115-5000, USA

Abstract

Recent data from the ATLAS and CMS detectors at the Large Hadron Collider at CERN give a hint of possible violation of flavor in the leptonic decays of the Higgs boson. In this work we analyze the flavor violating leptonic decays () within the framework of an MSSM extension with a vectorlike leptonic generation. Specifically we focus on the decay mode . The analysis is done including tree and loop contributions involving exchange of , charge and neutral higgs and leptons and mirror leptons, charginos and neutralinos and sleptons and mirror sleptons. It is found that a substantial branching ratio of , i.e., of as much a , can be achieved in this model, the size hinted by the ATLAS and CMS data. The flavor violating decays are also analyzed and found to be consistent with the current experimental limits. An analysis of the dependence of flavor violating decays on CP phases is given. The analysis is extended to include flavor decays of the heavier Higgs bosons. A confirmation of the flavor violation in Higgs boson decays with more data that is expected from LHC at TeV will be evidence of new physics beyond the standard model.

Keywords:  Flavor violation, Higgs, vector multiplet, CP phases
PACS numbers: 12.60.-i, 14.60.Fg

## 1 Introduction

Recently the ATLAS[1] and the CMS [2] Collaborations at CERN have observed some possible hints of flavor violating decays of the Higgs boson . Thus the ATLAS Collaboration finds [1]

 BR(H01→μτ)=BR((H01→μ+τ−)+BR((H01→μ−τ+)=(0.77±0.62)% (1)

while the CMS Collaboration finds [2]

 BR(H01→μτ)=BR((H01→μ+τ−)+BR((H01→μ−τ+)=(0.84+0.39−0.37)% (2)

For the and modes the experiments find a 95% CL bounds so that

 BR(H01→eμ)<0.036%, BR(H01→eτ)<0.70%. (3)

More data is expected in the near future which makes an investigation of the lepton flavor violation in Higgs decays a timely topic of investigation. Thus in the standard model there is no explanation of flavor violating leptonic decays of the Higgs boson and if they are confirmed that would be direct evidence for new physics beyond the standard model. In this work we explain the flavor violating leptonic decays of the Higgs boson in the framework of an extended MSSM with a vectorlike leptonic generation following the techniques discussed in [3, 4, 5]. Flavor changing Higgs decays are of significant theoretical interest and for some previous works see, e.g., [6]- [28].

In the analysis of this work the three leptonic generations mix with the vectorlike generation which leads to flavor violation for the Higgs interactions. The analysis is carried out at the tree (see Fig. 1) and loop level where loop diagrams involving , leptons and mirror leptons (see figs. (2) and (4)), charginos, neutralinos, sleptons and mirror sleptons (see figs. (3) and (5)), charged Higgs, neutral Higgs, sleptons and mirror sleptons (see figs. (6) and (7) are taken account of. It is shown that flavor violating decays of the Higgs of the size hinted by the ATLAS and CMS data can be achieved consistent with the Higgs boson mass constraint. The dependence of the branching ratio of the flavor violating decay and well as the dependence of the Higgs boson mass on CP phases is analyzed.

The outline of the rest of the paper is as follows. In section (2) we give a description of the extended MSSM model. In section 3 an analytic analysis of the triangle loops figs. (2) -(7) that contribute to the flavor changing processes is given. Numerical analysis is given in section 4. Here we also study the dependence of the flavor violation on CP phases. Conclusions are given in section 5. Further details of the analysis are given in the Appendix.

## 2 The Model

As mentioned in section 1 the model we use for the computation of the flavor violating leptonic decays of the Higgs boson is an extended MSSM which includes a vector like leptonic generation. As is well known vectorlike multiplets appear in a variety of unified models including string and D brane models [29, 30, 31, 32]. Many applications of these vector like multiplets exist in the literature [3, 4, 5, 33, 34, 35]. In our analysis we include one vector like matter multiplet along with the three generations of matter. We begin by defining the notation for the matter content of the model and their properties under . For the four sequential leptonic families we use the notation

 ψiL≡(νiLℓiL)∼(1,2,−12),ℓciL∼(1,1,1),νciL∼(1,1,0), (4)

where the last entry on the right hand side of each is the value of the hypercharge defined so that and we have included in our analysis the singlet field , with runs from . For the mirrors we use the notation

 χc≡(EcμLNcL)∼(1,2,12),EμL∼(1,1,−1),NL∼(1,1,0). (5)

The main difference between the leptons and the mirrors is that while the leptons have interactions type interactions with gauge bosons the mirrors have interactions. Further details of the model including the superpotential, Lagrangian, and mass matrices are given below.

As discussed above the analysis is based on the assumption that there is a vectorlike leptonic generation that lies at low scales. Including this vectorlike generation we discuss the superpotential, soft terms, the mass matrices and the particle and sparticle spectrum that enters in the analysis in this section. Thus the superpotential of the model for the lepton part is taken to be of the form

 W =−μϵij^Hi1^Hj2+ϵij[f1^Hi1^ψjL^τcL+f′1^Hj2^ψiL^νcτL+f2^Hi1^χcj^NL+f′2^Hj2^χci^EL +h1^Hi1^ψjμL^μcL+h′1^Hj2^ψiμL^νcμL+h2^Hi1^ψjeL^ecL+h′2^Hj2^ψieL^νceL+y5^Hi1^ψj4L^ℓc4L+y′5^Hj2^ψi4L^νc4L] +f3ϵij^χci^ψjL+f′3ϵij^χci^ψjμL+f4^τcL^EL+f5^νcτL^NL+f′4^μcL^EL+f′5^νcμL^NL +f′′3ϵij^χci^ψjeL+f′′4^ecL^EL+f′′5^νceL^NL +h6ϵij^χci^ψj4L+h7^ℓc4L^EL+h8^νc4L^NL, (6)

where implies superfields, stands for , stands for and stands for . Mixings of the above type can arise via non-renormalizable interactions. Consider, for example, a term such as . If and develop VEVs of size , a mixing term of the right size can be generated. We assume that the couplings in Eq.(6) are complex and we define their phases so that

 fk=|fk|eiχk,  f′k=|f′k|eiχ′k,  f′′k=|f′′k|eiχ′′k, hi=|hi|eiθhi,  h′i=|h′i|eiθ′hi,h′′i=|h′′i|eiθh′′i, (7)

where take on the appropriate values that appear in Eq.(6).

The mass terms for the neutrinos, mirror neutrinos, leptons and mirror leptons arise from the term

 L=−12∂2W∂Ai∂Ajψiψj+H.c. (8)

where and stand for generic two-component fermion and scalar fields. After spontaneous breaking of the electroweak symmetry, ( and ), we have the following set of mass terms written in the 4-component spinor notation so that

 −Lm=¯ξTR(Mf)ξL+¯ηTR(Mℓ)ηL+H.c., (9)

where the basis vectors in which the mass matrix is written is given by

 ¯ξTR=(¯ντR¯NR¯νμR¯νeR¯ν4R), ξTL=(ντLNLνμLνeLν4L) , ¯ηTR=(¯τR¯ER¯μR¯eR¯ℓ4R), ηTL=(τLELμLeLℓ4L) , (10)

and the mass matrix of neutrinos is given by

 Mf=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝f′1v2/√2f5000−f3f2v1/√2−f′3−f′′3−h60f′5h′1v2/√2000f′′50h′2v2/√200h800y′5v2/√2⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠ . (11)

We define the matrix element of the mass matrix as so that

 mN=f2v1/√2. (12)

The mass matrix is not hermitian and thus one needs bi-unitary transformations to diagonalize it. We define the bi-unitary transformation so that

 Dν†R(Mf)DνL=diag(mψ1,mψ2,mψ3,mψ4,mψ5). (13)

In are the mass eigenstates for the neutrinos, where in the limit of no mixing we identify as the light tau neutrino, as the heavier mass mirror eigen state, as the muon neutrino, as the electron neutrino and as the other heavy 4-sequential generation neutrino. A similar analysis goes to the lepton mass matrix where

 Mℓ=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝f1v1/√2f4000f3f′2v2/√2f′3f′′3h60f′4h1v1/√2000f′′40h2v1/√200h700y5v1/√2⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠ . (14)

We introduce now the mass parameter defined by the (22) element of the mass matrix above so that

 mE=f′2v2/√2. (15)

The mass squared matrices of the slepton-mirror slepton and sneutrino-mirror sneutrino come from three sources: the F term, the D term of the potential and the soft SUSY breaking terms. After spontaneous breaking of the electroweak symmetry the Lagrangian is given by

 L=LF+LD+Lsoft , (16)

where is deduced from , while the is given by

 −LD =12m2Zcos2θWcos2β{~ντL~ν∗τL−~τL~τ∗L+~νμL~ν∗μL−~μL~μ∗L+~νeL~ν∗eL−~eL~e∗L +~ER~E∗R−~NR~N∗R+~ν4L~ν∗4L−~ℓ4L~ℓ∗4L}+12m2Zsin2θWcos2β{~ντL~ν∗τL+~τL~τ∗L+~νμL~ν∗μL+~μL~μ∗L +~νeL~ν∗eL+~eL~e∗L+~ν4L~ν∗4L+~ℓ4L~ℓ∗4L −~ER~E∗R−~NR~N∗R+2~EL~E∗L−2~τR~τ∗R−2~μR~μ∗R−2~eR~e∗R−2~ℓ4R~ℓ∗4R}. (17)

For we assume the following form

 −Lsoft =~M2τL~ψi∗τL~ψiτL+~M2χ~χci∗~χci+~M2μL~ψi∗μL~ψiμL +~M2eL~ψi∗eL~ψieL+~M2ντ~νc∗τL~νcτL+~M2νμ~νc∗μL~νcμL +~M24L~ψi∗4L~ψi4L+~M2ν4~νc∗4L~νc4L+~M2νe~νc∗eL~νceL+~M2τ~τc∗L~τcL+~M2μ~μc∗L~μcL +~M2e~ec∗L~ecL+~M2E~E∗L~EL+~M2N~N∗L~NL+~M24~ℓc∗4L~ℓc4L +ϵij{f1AτHi1~ψjτL~τcL−f′1AντHi2~ψjτL~νcτL+h1AμHi1~ψjμL~μcL−h′1AνμHi2~ψjμL~νcμL +h2AeHi1~ψjeL~ecL−h′2AνeHi2~ψjeL~νceL+f2ANHi1~χcj~NL−f′2AEHi2~χcj~EL +y5A4ℓHi1~ψj4L~ℓc4L−y′5A4νHi2~ψj4L~νc4L+H.c.} . (18)

The trilinear couplings are also complex and we define their phases so that

 Ai=|Ai|eθAi. (19)

We define the scalar mass squared matrix in the basis

 (~τL,~EL,~τR,~ER,~μL,~μR,~eL,~eR,~ℓ4L,~ℓ4R). (20)

We label the matrix elements of these as where the elements of the matrix are given in [36]. We assume that all the masses are of the electroweak size so all the terms enter in the mass squared matrix. We diagonalize this hermitian mass squared matrix by the unitary transformation

 ~Dτ†M2~τ~Dτ=diag(M2~τ1,M2~τ2,M2~τ3,M2~τ4,M2~τ5,M2~τ6,M2~τ7,M2~τ8M2~τ9,M2~τ10). (21)

The mass matrix in the sneutrino sector has a similar structure. In the basis

 (~ντL,~NL,~ντR,~NR,~νμL,~νμR,~νeL,~νeR,~ν4L,~ν4R) (22)

and write the sneutrino mass matrix in the form where the elements are given in [36]. As in the charged lepton sector we assume that all the masses are of the electroweak size so all the terms enter in the mass matrix. This mass matrix can be diagonalized by the unitary transformation

 ~Dν†M2~ν~Dν=% diag(M2~ν1,M2~ν2,M2~ν3,M2~ν4,M2~ν5,M2~ν6,M2~ν7,M2~ν8,M2~ν9,M2~ν10). (23)

## 3 Analysis of flavor violating leptonic decays of the Higgs boson

Flavor changing decays of this extended MSSM model arise at both the tree level due to lepton and mirror lepton mass mixing and at the loop level. There are several diagrams that contribute to the decays. These include the exchange of the charged bosons and neutrinos and mirror neutrinos (see left panel of Fig. 2), exchange of bosons and leptons and mirror leptons (see right panel of Fig. 2), exchange of charginos, sneutrinos and mirror sneutrinos (see left panel of Fig. 3) and the exchange of neutralinos, charged sleptons and mirror charged sleptons (see right panel of Fig. 3). Additional diagrams which involve Higgs-neutrino-neutrino, Higgs-lepton-lepton, Higgs-sneutrino-sneutrino and Higgs-slepton-slepton vertices are given in Fig. 4 and Fig. 5. Other diagrams involve neutral and charged Higgs running in the loops are given in Figs. 6 and 7. So at the tree level, there is a coupling between the fields , and due to mixing given by (see section 6)

 −Leff=¯μχ31PLτH11+¯μη31PLτH22 +¯τχ13PLμH11+¯τη13PLμH22+H.c. (24)

The loop corrections produces the effective Lagrangian

 Leff=¯μδξμτPRτH11+¯μΔξμτPLτH11 +¯μδξ′μτPRτH22+¯μΔξ′μτPLτH22+H.c. (25)

This effective Lagrangian written in terms of the mass eigen states of the neutral Higgs with reads

 Leff=¯μ({−αs31i+αsi}+γ5{−αp31i+αpi})τH0i +¯τ({−αs13i+α′si}+γ5{−αp13i+α′pi})μH0i (26)

where the couplings are given by

 αskji=12√2(χkj{Yi1+iYi3sinβ}+ηkj{Yi2+iYi3cosβ} +χ∗jk{Yi1−iYi3sinβ}+η∗jk{Yi2−iYi3cosβ}) αpkji=12√2(−χkj{Yi1+iYi3sinβ}−ηkj{Yi2+iYi3cosβ} +χ∗jk{Yi1−iYi3sinβ}+η∗jk{Yi2−iYi3cosβ}) αsi=12√2({δξμτ+Δξμτ}{Yi1+iYi3sinβ}+{δξ′μτ+Δξ′μτ}{Yi2+iYi3cosβ}) αpi=12√2({δξμτ−Δξμτ}{Yi1+iYi3sinβ}+{δξ′μτ−Δξ′μτ}{Yi2+iYi3cosβ}) α′si=12√2({δξ∗μτ+Δξ∗μτ}{Yi1−iYi3sinβ}+{δξ′∗μτ+Δξ′∗μτ}{Yi2−iYi3cosβ}) α′pi=12√2({Δξ∗μτ−δξ∗μτ}{Yi1−iYi3sinβ}+{Δξ′∗μτ−δξ′∗μτ}{Yi2−iYi3cosβ}) (27)

where the matrix elements are defined by

 YM2HiggsYT=diag(m2H01,m2H02,m2H03) (28)

and and are given in Eq. (59). The decay of the neutral Higgs into an anti tau and a muon is given by

 Γi(H0i→¯τμ)=14πm3H0i√[(m2τ+m2μ−m2H0i)2−4m2τm2μ] ×{12(|−αs31i+αsi|2+|−αp31i+αpi|2)(m2H0i−m2τ−m2μ) −12(|−αs31i+αsi|2−|−αp31i+αpi|2)(2mτmμ)} Γi(H0i→¯μτ)=14πm3H0i√[(m2τ+m2μ−m2H0i)2−4m2τm2μ] ×{12(|−αs13i+α′si|2+|−αp13i+α′pi|2)(m2H0i−m2τ−m2μ) −12(|−αs13i+α′si|2−|−αp13i+α′pi|2)(2mτmμ)} (29)

We give a computation of each of the different loop contributions to , , and in the Appendix.

## 4 Numerical analysis

As discussed in the introduction, the promising Higgs boson decays for the observation of flavor violation are , i.e., . In MSSM one has three neutral Higgs bosons with being the lightest which is the observed Higgs boson. As is well known in the presence of CP phases the CP even and CP odd Higgs bosons mix [37] (for a recent analysis see [38]). Thus the mass eigenstates in general will have dependence on CP phases. We will investigate the dependence of the flavor violating decays as well as of the Higgs boson mass on the CP phases in the analysis. We also note that one may allow large CP phases consistent with the current limits on EDM constraints due the cancellation mechanism discussed in many works [39, 40, 41]. Thus the flavor violating branching ratios of into are given by

 BR(H01→¯τμ)=Γ(H01→¯τμ)Γ(H01→¯μτ)+Γ(H01→¯τμ)+∑iΓ(H01→¯fifi)+ΓH1DB BR(H01→τ¯μ)=Γ(H01→¯μτ)Γ(H01→¯μτ)+Γ(H01→¯τμ)+∑iΓ(H01→¯fifi)+ΓH1DB (30)

where stand for fermionic particles that have coupling with the Higgs boson and have a mass less than half the higgs boson mass and is the decay width into diboson states which include . Thus the computation of the branching ratios of Eq. (30) involve the decay widths

 Γi(H0i→¯ff)f=b,d,s=3g2m2f32πm2Wcos2βMi{|Yi1|2(1−4m2fM2i)3/2+|Yi3|2sin2β(1−4m2fM2i)1/2} Γi(H0i→¯ff)f=τ,μ,e=g2m2f32πm2Wcos2βMi{|Yi1|2(1−4m2fM2i)3/2+|Yi3|2sin2β(1−4m2fM2i)1/2} Γi(H0i→¯ff)f=u,c=3g2m2f32πm2Wsin2βMi{|Yi2|2(1−4m2fM2i)3/2+|Yi3|2