Lepton Flavor Violating Decays of Neutral Higgses in Extended Mirror Fermion Model

We perform the one-loop induced charged lepton flavor violating decays of the neutral Higgses in an extended mirror fermion model with non-sterile electroweak-scale right-handed neutrinos and a horizontal $A_4$ symmetry in the lepton sector. We demonstrate that for the 125 GeV scalar $h$ there is tension between the recent LHC result ${\cal B}(h \to \tau \mu) \sim $ 1% and the stringent limits on the rare processes $\mu \to e \gamma$ and $\tau \to (\mu$ or $e) \gamma$ from low energy experiments.


I. MOTIVATION
As is well known, lepton and baryon number are accidental global symmetries in the fundamental Lagrangian of Standard Model (SM). Processes like µ → eγ, p → eγ, etc that violating either one (or both) of these two quantum numbers are thus strictly forbidden in the perturbation calculations of SM. Experimental limits for these processes are indeed very stringent. For example, from Particle Data Group [1], we have the following bounds and τ (p → e + γ) > 670 × 10 30 years .
Search for lepton flavor violating (LFV) Higgs decay h → τ µ at hadron colliders was proposed some time ago [2]. Recently both ATLAS [3] and CMS [4]    On the other hand, we have stringent limits for LFV radiative decays like µ → eγ in Eq. (1) as well as both at 90% CL from the low energy data of BaBar experiment [5].
Over the years, many authors had studied the flavor changing neutral current Higgs decays h → f i f j in both the SM and its various extensions. For a recent updated calculation on h → q i q j in the SM we refer the readers to [6] and references therein. For earlier calculations for the leptonic case with large Majorana neutrino masses, see for example [7,8]. Recently large flux of works on new physics implications for the LHC result Eq. (3) is easily noticed .
In [40], an up-to-date analysis of a previous calculation [41] of µ → eγ in a class of mirror fermion models with non-sterile electroweak scale right-handed neutrinos [42] was presented for an extension of the models with a horizontal A 4 symmetry in the lepton sector [43]. It was demonstrated in [40] that although there exists parameter space relevant to electroweak physics to accommodate the muon magnetic dipole moment anomaly ∆a µ = 288(63)(49) × 10 −11 [1], the current low energy limit Eq. (1) on the branching ratio B(µ → eγ) from MEG experiment [44] has disfavored those regions of parameter space.
In this work, we present the calculation of LFV decay of the neutral Higgses in an extended mirror fermion model. In Section 2, we briefly review the extended model and show the relevant interactions that may lead to the LFV decays of the neutral Higgses in the model. In Section 3, we present our calculation. Numerical results are given in Section 4. We conclude in Section 5. Detailed formulas for the loop amplitudes are given in the Appendix.

II. THE MODEL AND ITS RELEVANT INTERACTIONS
The other two 1-particle reducible diagrams corresponding to the wave function renormalization of the external fermion lines are not shown.
In the original mirror fermion model [42], while the gauge group is the same as SM, every left-handed (right-handed) SM fermion has a right-handed (left-handed) mirror partner, and the scalar sector consists of one SM Higgs doublet Φ, one singlet φ 0S and two triplets ξ andχá la Georgi-Machacek [45,46]. One peculiar feature of the model is that the right-handed neutrinos are non-sterile.
where l Li and l Ri are SM leptons, l M Rm and l M Lm are mirror leptons (i, m are generation indices); U L k im and U R k im are the coupling coefficients given by where the matrix elements for the four matrices M k (k = 0, 1, 2, 3) are listed in Table I and M k jn can be obtained from M k jn with the following substitutions for the Yukawa couplings g 0S → g 0S and g 1S → g 1S [40]; U PMNS is the usual neutrino mixing matrix defined as and its mirror and right-handed counter-parts U M PMNS , U PMNS and U M PMNS are defined analogously as and where U l R and U l M L are the unitary matrices relating the gauge eigenstates (fields with superscripts 0) and the mass eigenstates and where ω ≡ exp(i2π/3) entered in the multiplication rules of A 4 . The matrix in Eq. (15) was first discussed by Cabibbo and also by Wolfenstein in the context of CP violation in three generations of neutrino oscillations [48].
The second Yukawa interaction is for the couplings of neutral Higgses with the SM fermion pairs and the mirror fermion pairs. It was shown in [47] that the physical neutral Higgs states ( H 1 , H 2 , H 3 ) 1 are in general mixture of the unphysical neutral where H 0 1 and H 0 1M are the neutral components of the SM Higgs and mirror Higgs doublets respectively, and H 0 1 is linear combination of the neutral components in the Georgi-Machacek triplets. The couplings of the physical Higgs H a with a pair of SM fermions f and a pair of mirror fermions f M are given by [47] where g is the SU ( with and v M are the VEVs of the Higgs doublet, mirror Higgs doublet and Georgi-Machacek triplets respectively. For the original mirror model [42], one can simply set H 1 → H 0 1 ≡ h, O 11 /s 2 and O 12 /s 2M → 1, and drop all other terms with a = 1 in Eq. (17).

III. THE CALCULATION
The matrix element for the process H a (q) → l i (p) + l j (p ) (Fig. 1) can be written where P L,R = (1 ∓ γ 5 )/2 are the chiral projection operators. In terms of scalar and pseudoscalar couplings the above amplitude can be rewritten as where The partial decay width is given by where λ(x, y, z) = x 2 + y 2 + z 2 − 2(xy + yz + zx). The one-loop induced coefficients A aij and B aij are related to C aij L and C aij R according to Eq. (23). The formulas for the latter are given in the Appendix.
We now comment on the divergent cancellation in the calculation. For the original mirror model [42] in which there is only one Higgs doublet with Yukawa couplings to the SM fermions and to the mirror fermions that are differ only by the corresponding fermion masses, the divergence in the one-loop diagram in Fig. (1) will cancel with those in the two 1-particle reducible diagrams associated with wave function renormalization. On the other hand, for the extended model [47] these divergences do not cancel each other. Recall that in the extended model, besides the SM Higgs doublet an additional mirror Higgs doublet was introduced. Both Higgs doublets can then couple to the SM fermions and may lead to LFV decay of the Higgses at tree level. In [42], a global U (1) × U (1) symmetry was employed such that the SM Higgs doublet only couples to the SM fermions, while the mirror Higgs doublet only couples to the mirror fermions. Hence there will be no tree level LFV vertices for the SM Higgs decays into SM fermions. However this global symmetry is broken by a term in the scalar potential. This term also provide the Higgs mixings in Eq. (16) that eventually responsible to LFV decays of the Higgses in the extended model. Due to renormalizability, the presence of this symmetry breaking term in the scalar potential forces one to reintroduce the Yukawa terms that are forbidden by the symmetry.
Hence tree level LFV decays of the Higgses are generally present in the extended model. According to the general analysis in [39] such tree level LFV couplings are constrained to be quite small by low energy data. For our purpose, we will assume these tree level LFV couplings are vanishing small and the main reason for their existence is to provide counter terms to absorb the divergences in the calculation in the extended model. The results of C aij L,R should then be regarded as renormalized quantities.
The amplitude for l i → l j γ in the extended model can be found in [40].

IV. NUMERICAL ANALYSIS
We will focus on the case of lightest neutral Higgs H 1 → τ µ with H 1 identified as the 125 GeV Higgs, and adopt the following strategy which has been used in [40] for the numerical analysis of µ → eγ: • Two scenarios were specified according to the following forms of the three unknown mixing matrices: for the neutrino masses with normal and inverted hierarchies respectively. The Majorana phases have been ignored in the analyses. For each scenario, we consider these two possible solutions for the U PMNS . Due to the small differences between these two solutions, we expect our results are not too sensitive to the neutrino mass hierarchies.
• All Yukawa couplings g 0S , g 1S , g 0S and g 1S are assumed to be real. For simplicity, we will assume g 0S = g 0S , g 1S = g 1S and study the following 6 cases: (a) g 0S = 0, g 1S = 0. The A 4 triplet terms are switched off.
(b) g 1S = 10 −2 × g 0S . The A 4 triplet couplings are merely one percent of the singlet ones.
(d) g 1S = 0.5 × g 0S . The A 4 triplet couplings are one half of the singlet ones.
(e) g 1S = g 0S . Both A 4 singlet and triplet terms have the same weight.
• For the masses of the singlet scalars φ kS , we take with a fixed common mass M S = 10 MeV. As long as m φ kS m l M m , our results will not be affected much by this assumption.
• For the masses of the mirror lepton l M m , we take m l M m = M mirror + δ m with δ 1 = 0, δ 2 = 10 GeV, δ 3 = 20 GeV and vary the common mass M mirror .
• As shown in [47], the 125 GeV scalar resonance h discovered at the LHC Hence the 125 GeV Higgs identified as H 1 is an impostor in this scenario; it is mainly composed of the two neutral components in the Georgi- Machacek triplets.
In Fig. (2), we plot the contours of the branching ratios B(h → τ µ) = 0.84% By studying in details of all the plots in these two figures, we can deduce the following results: • The bumps at M mirror ∼ 200 GeV at all the plots in these two figures are due to large cancellation in the amplitudes between the two one-particle reducible (wave function renormalization) diagrams and the irreducible one-loop diagram shown in Fig. (1). As a result, the Yukawa couplings have to be considerable larger in the contour lines of fixed branching ratios of the processes.    • For the two processes τ → µγ (blue lines) and τ → eγ (green lines) in all these plots, the solid and dotted lines are coincide to each other while the dashed and dot-dashed lines are very close together. Thus there are essentially no differences between the normal and inverted mass hierarchies in both Scenarios 1 and 2 in these two processes. However, for the process µ → eγ (black lines), only the solid and dotted lines are coincide to each other. Thus there are some differences between normal and inverted mass hierarchies in Scenario 2 but not in Scenario 1 for this process, in particular for cases (a)-(d) in which • For h → τ µ (red lines), the solid (dashed) and dotted (dot-dashed) lines are either very close (in Fig. (2) for Dr. Jekyll scenario) or mostly coincide (in  from LHC and the low energy limits of B(τ → (µ, e)γ) and B(µ → eγ), in particular the latter one. Before we depart, we comment on the possible collider signals for the mirror fermions [49]. Mirror leptons if not too heavy can be produced at the LHC via electroweak processes [42], e.g.
The mirror lepton decays as l M R → l L + φ S or l M R → ν R + W −( * ) for m l M R > m ν R plus the conjugate processes, while the right-handed neutrino can decay as ν R → ν L + φ S or ν R → l M R + W +( * ) for m ν R > m l M R followed by l M R → l L + φ S . If kinematics allowed, the scalar singlet φ S can decay into lepton pair as well through mixings; otherwise they would appear as missing energies like neutrinos. Thus the signals at the LHC or future 100 TeV SPPC would be multiple lepton pairs plus missing energies. In the case where the right-handed neutrinos are Majorana fermions, we would have same sign dilepton plus missing energies. Assuming l M R → l L + φ S is the dominant mode and the Yukawa couplings are small enough, the decay length of the mirror lepton could be as large as a few millimeter [49]. Thus the mirror lepton may lead to a displaced vertex and decay outside the beam pipe. These leptonic final states may have been discarded by the current algorithms adopted by the LHC experiments.
It is therefore quite important for the experimentalists to devise new algorithms to search for these mirror fermions that may decay outside the beam pipe.
The scale of new physics may be hidden in the lepton flavor violating processes like h → τ (µ, e), τ → (µ, e)γ, µ → eγ, µ → eee, µ-e conversion etc. Ongoing and future experiments at high energy and high intensity frontiers could shed light in the mirror fermion model that may responsible to these lepton flavor violating processes.

ACKNOWLEDGMENTS
We would like to thank P. Q. Hung for useful discussions. This work was supported in part by the Ministry of Science and Technology (MoST) of Taiwan under grant numbers 104-2112-M-001-001-MY3.

APPENDIX
The dimensionless coefficients C aij L and C aij R defined in Eq. (22) are given by C aij R can be obtained from C aij L simply by substituting U L ↔ U R , namely