Muon-to-Electron Conversion in Mirror Fermion Model with Electroweak Scale Non-Sterile Right-handed Neutrinos

The muon-to-electron conversion in nuclei like aluminum, titanium and gold is studied in the context of a class of mirror fermion model with non-sterile right-handed neutrinos having mass at the electroweak scale. At the limit of zero momentum transfer and large mirror lepton masses, we derive a simple formula to relate the conversion rate with the on-shell radiative decay rate of muon into electron. Current experimental limits (SINDRUM II) and projected sensitivities (Mu2e, COMET and PRISM) for the muon-to-electron conversion rates in various nuclei and latest limit from MEG for the radiative decay rate of muon into electron are used to put constraints on the parameter space of the model. Depending on the nuclei targets used in different experiments, for the mirror lepton mass in the range of 100 to 800 GeV, the sensitivities of the new Yukawa couplings one can probe in the near future are in the range of one tenth to one hundred-thousandth, depending on the mixing scenarios in the model.


I. INTRODUCTION
As is well known, lepton flavor is an accidental conserved quantity in Standard Model (SM) with strictly massless neutrinos. For example, a muon never decays radiatively into an electron plus a photon and neutrinos do not oscillate in SM.
However various experiments have now established firmly that neutrinos do oscillate from one flavor to another. The common wisdom, motivated by the physics of K −K oscillation in the kaon system, is to give tiny masses with small mass differences to the various light neutrino species. Radiative decay of the muon into electron is then possible but with an unobservable rate highly suppressed with the minuscule neutrino masses [1,2]. Searches for lepton flavor violating rare processes in high intensity experiments are thus important for new physics beyond the SM.
The most updated limit on B(µ → eγ) is from MEG experiment [3] B(µ → eγ) ≤ 4.2 × 10 −13 (90% C.L.) (MEG 2016) , and its projected improvement [4] is Recent data from T2K experiment [5] agrees well with the global analysis of neutrino oscillation data [6][7][8], suggesting that the normal neutrino mass hierarchy (NH) with a CP violating Dirac phase δ CP ∼ 3π/2 is slightly preferred. The best fit result for the central values of the PMNS matrix elements in the normal neutrino mass hierarchy can be extracted from [ For the µ − e conversion in nuclei, the present experimental upper limits on the branching ratios were obtained by SINDRUM II experiment [9,10] for the targets titanium and gold, B(µ − + Au → e − + Au) < 7 × 10 −13 (90% C.L.) .
A positive signal of any of the above processes (or any process with charged lepton flavor violation (CLFV)) at the current or projected sensitivities of various high intensity experiments would be a clear indication of new physics as well, just like neutrino oscillations. Given the fact that no new physics has showed up yet at the high energy frontier of the Large Hadron Collider (LHC), it is not a surprise that many recent works have been focused on new physics implication of CLFV in the high intensity frontier. For a review on this topics and its possible connection with the muon anomaly, see [16] and references therein.
In a recent work [17], we updated a previous calculation [18] for the radiative process µ → eγ in the mirror fermion model with electroweak scale non-sterile righthanded neutrinos [19] to an extended version [20] where a horizontal A 4 symmetry in the lepton sector was imposed. In this work we extend this previous analysis [17] to the µ − e conversion in nuclei, in particular for aluminum, gold and titanium.
This paper is organized as follows. In Sec. II, after presenting some highlights of the crucial features of the extended mirror fermion model, the calculation of µ − e conversion in the model is presented. In Sec. III, we derive a simple relation between the µ − e conversion rate and the radiative decay rate of µ → eγ in the limit of zero momentum transfer and large mirror lepton masses. Numerical results are shown in Sec. IV. We summarize In Sec. V. In Appendix A, we briefly review the effective Lagrangian [21,22] for describing µ − e conversion; and in Appendix B, we collect some useful formulas used in Sec. III.

II. MIRROR FERMION MODEL CALCULATION
In this section, we first provide some highlights for the original mirror fermion model [19] and its A 4 extension [20]. Then we compute the effective coupling constants induced at one loop level in the extended model for the µ − e conversion.
A. Brief Review of Mirror Fermion Model

Motivation
The motivation of introducing mirror fermions in [19]  of a Higgs triplet with hypercharge Y = 2 with mass at the electroweak scale, rather than the grand unification scale in the usual scheme. Tiny Dirac masses can also be given via small VEVs of Higgs singlets with Y = 0. This is the electroweak scale see-saw mechanism in mirror fermion model which is testable at the LHC [28,29].
The original model in [19] has been shown to be consistent with electroweak precision test data [30] as well as the 125 GeV Higgs data from the LHC with an additional mirror Higgs doublet [31]. In [20], the original model was extended with a horizontal A 4 symmetry imposed in the lepton sector to address various issues of lepton mixings. We briefly review this A 4 extension in the next subsection.

Particle Content and Its A 4 Assignments
The particle content of leptons and bosons of the model are shown in Table I.
The fields l M Ri and e M Li are the mirrors of the SM lepton doublet l Li and singlet e Ri respectively for the i-th generation. For the scalars, Φ M is the mirror Higgs doublet of Φ introduced in [31]; ξ andχ are the Georgi-Machacek triplets [32,33]; and φ 0S and φ S are singlets introduced in [20]. The A 4 assignments of these particles are listed at the second row in Table I. Note that the scalar SU(2) singlet φ S is a A 4 triplet with its three components shown explicitly in the Table. The singlet scalars φ 0S , φ S are the only fields connecting the SM fermions and their mirror counterparts. Recall that the tetrahedron symmetry group A 4 has four irreducible representations 1, 1 , 1 , and 3 with the following multiplication rule 1 : 3 × 3 = 3 1 (23, 31, 12) + 3 2 (32, 13, 21) + 1(11 + 22 + 33) + 1 (11 + ω 2 22 + ω33) + 1 (11 + ω22 + ω 2 33) where ω = e 2πi/3 . In the gauge eigenbasis (fields with superscript 0), one can write down the following A 4 invariant Yukawa couplings, As shown in [20], after the scalar singlets develop VEVs with v 0 = φ 0S and v i = φ iS , one obtains the neutrino mass matrix from the first line of (9) Hermiticity of the M Dirac The above form of M Dirac ν can be diagonalized by unitary transformation, i.e.
where ω is the same as in the multiplication rules of A 4 given in (8). The matrix U CW in (12) was first discussed by Cabibbo [34] and also by Wolfenstein [35] in the context of CP violation in three generations of neutrino oscillations. In recent years, advocating A 4 symmetry in the lepton sector was mainly due to Ma [36]. and g 0S and g 1S are complex Yukawa couplings. M k can be obtained from M k with the following substitutions g 0S → g 0S and g 1S → g 1S .

Mixings
Let U l L,R and U l M R,L be the unitary matrices relating the gauge eigenstates and the mass eigenstates (fields without superscripts 0) defined as Following [17], we express the Yukawa couplings in (9) as follows The coupling coefficients U L k im and U R k im are given by where the matrix elements for the four auxiliary matrices M k (k = 0, 1, 2, 3) are listed in Table II, 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 ; 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

B. Photon Contributions and the Monopole and Dipole Form Factors
In this work we will focus on the contributions from the photon exchange as shown in the Feynman Diagrams of Fig. 1. We also compute the contributions from the Z-exchange but since they are suppressed by m 2 µ /m 2 Z we will not present them here. The invariant amplitude for µ − (p) → e − (p )γ * (q) with an off-shell photon can be parametrized as where Γ µ γ (q) has the following Lorentz and gauge invariant decomposition The monopole form factors f E0 , f M 0 and the dipole form factors f M 1 , f E1 can be obtained by generalizing our previous on-shell calculation of µ → eγ in the same model [17] to the case of off-shell photon γ * . From the Feynman diagrams of Fig. 1, we obtain the following expressions for the monopole form factors, and for the dipole form factors. Here, we have defined where m k denotes the mass of scalar singlet φ kS for k = 0, 1, 2, 3 and M m the mass of mirror lepton l M m for m = 1, 2, 3. At q 2 = 0, we have f E0,M 0 (0) = 0 as one would expect. Thus the following reduced monopole form factorsf E0,M 0 with an explicit factor of q 2 extracted from f E0,M 0 are often defined in the literature, For small q 2 , one can setf E0,M 0 (q 2 ) ≈f E0,M 0 (0) with The explicit factor of q 2 in (26) will cancel the 1/q 2 of the photon propagator in Fig. 1. This leads to four-fermion vector-vector interaction and hence the reduced monopole form factors will contribute to the effective coupling C (q) V (R,L) in the effective Lagrangian of (48) in Appendix A. We will discuss more about these four-fermion interactions in the next subsection.
At q 2 = 0, the contributions from the magnetic and electric dipole terms of (22) to the amplitude M γ in (21) can be reproduced by the following effective Lagrangian Comparing (28) C. Four-Fermion Coupling Constants C The amplitude for µ(p)q(k) → e(p )q(k ) from the monopole form factors of the photon exchange in Fig. 1 can be obtained as (30) where q = p − p = k − k, and f E0 , f M 0 are given in (23). The q µ term in (30) can be dropped due to quark current conservation. As mentioned earlier, the 1/q 2 of the photon propagator will be cancelled from a factor of q 2 in f E0,M 0 . Thus in terms of the reduced form factorsf E0,M 0 of (26), the amplitude M γ can be rewritten as wheref E0,M 0 are defined in (27) for small q 2 . At q 2 = 0, this amplitude can be reproduced by the following Fermi interaction By matching (32) Note that we have the relation C We also note that for the photon contributions, only C Since the momentum transfer q 2 is expected to be quite small in the µ − e conversion process in nuclei, we can make a Taylor expansion for the various form factors deduced in the previous section around q 2 = 0. Thus for small q 2 , we have Here r km = m 2 k /M 2 m and the expressions for the Feynman parameterization integrals I, J and I i0 (i = 1, 2, · · · , 5) can be found in Appendix B.
From (49) in Appendix B, the conversion rate (for γ exchange) is given by where C DR,DL is given by (29), andC (p) V R,V L are given by (50) and (52) in Appendix A. To obtain (36), we have used the following result valid for the neutron, Using the above approximate form factors (34) and (35) for small q 2 , we can derive and summing over the contributions from light quarks, we havẽ Dropping the q 2 terms in C DR,DL and keeping only those terms up to O(1/M 2 m ) iñ C (p) V L,V R , we obtain for the conversion rate where Thus, one obtains Note that since C km L,R is scaled by 1/M m , the second and the third terms in (43) are suppressed by 1/M m and 1/M 2 m respectively, as compared with the first term. If one drops these two suppressed terms further, one obtains a simple relation Thus, where Γ µ is the total decay width of the muon.

IV. NUMERICAL ANALYSIS
In our analysis, we adapt the same assumptions for the parameter space as was done in [17]. We summarize them as follows.
• For the mass parameters, we take the masses of the singlet scalars φ kS to be where the common mass M S is set to be 10 MeV; and for the mirror lepton masses, we set where δ 1 = 0, δ 2 = 10 GeV, δ 3 = 20 GeV and the common mass M mirror is varied in the range of 100 − 800 GeV. Our results are insensitive to these choices as long as m k /M m 1.
• Note that the relations g 2S = (g 1S ) * and g 2S = (g 1S ) * hold due to the hermiticity of the neutrino Dirac mass matrix. However, all the Yukawa couplings g 0S , g 1S , g 2S , g 0S , g 1S , and g 2S are assumed to be real.
• Out of the four mixing matrices, only the one U PMNS associated with the lefthanded SM fermions are known. Following [17], we will consider two scenarios below: where U CW is given by (12). For the PMNS mixing matrix, we will use the best fit result in (3). In the two scenarios that we are studying, our results do not depend sensitively on the mass hierarchies.
• We will study the following two cases for the Yukawa couplings.
Several comments are in order here.
• We have studied in some details the effects of different settings of couplings on our results. Generally, we observe that as one varies the A 4 triplet coupling g 1S from 10 −2 g 0S to g 0S (from Figs. 2 to 5) the contour plots for log 10 B(µ − e conversion) are shifted to the left. The A 4 triplet is playing a significant role in putting constraints on the parameter space for the CLFV processes, such as µ → eγ and µ − e conversion in the model. • For the sensitivity of the two scenarios, we find that -Generally, Scenario 2 is less stringent constraint than Scenario 1.
-In particular, when the A 4 singlet couplings are dominated, Scenario 2 is less stringent than Scenario 1 by at least two order of magnitude (10 −3 vs. 10 −1 ), regarding the constraint on the couplings (as shown in Figs. 2 and 3). This is due to the fact that in Scenario 2, the three unknown unitary mixing matrices are now departure from U PMNS which allows for larger effects since the amplitudes involve products of both the couplings and the elements of mixing matrices. decay vertices with decay lengths larger than 1 mm or so [28]. Although unrelated to the present analysis, a similar remark can be made for the search for mirror quarks [29].
It would be interesting to extend this study to the electric dipole moment of the electron and neutron, which requires the new Yukawa couplings to take on complex values instead of real ones as assumed in the present analysis. This work is in progress and will be presented elsewhere [37].
Effective Lagrangian is a powerful technique to analyze low energy processes like µ → e conversion in nuclei since the momentum transfer q 2 is typically of the order O(m 2 µ ) m 2 N for nucleus N . The most general CLFV effective Lagrangian which contributes to the µ − e conversion in nuclei has been studied by various groups [21,22,38]. At the scale Λ where the heavy particles (including particles beyond the SM as well as the heavy top, bottom and charm quarks) being integrated out, the relevant terms for the model we are studying are Here m µ is the muon mass; P L,R = (1 ∓ γ 5 )/2, σ µν = i [γ µ , γ ν ] /2; F αβ is the electromagnetic field strength; finally, C D(L,R) and C (q) V (L,R) are dimensionless coupling constants depending on specific LFV model. In the specific mirror model calculation, we will be focusing on the photon and Z boson exchange diagrams which contribute only to the magnetic and electric dipole moment operators as well as the vector and axial vector lepton bilinears.
To determine the conversion rate, the above effective Lagrangian (48) is needed to scale down to the nuclear scale where the hadronic matrix elements N |qγ µ q|N , N |F αβ F αβ |N are evaluated. In addition, the muon and electron wave functions may be significantly deviated from plane wave due to distortion by the coulomb potential of the nuclei. For high Z nuclei, relativistic corrections to their wave functions are important as well. The formula for the conversion rate is given by [22,38]  V (R,L) are defined as [38] C (p) where f (q) V p and f The dimensionless quantities D and V (p,n) in (49) are the overlap integrals of the relativistic wave functions of muon and electron in the electric field of the nucleus weighted by appropriate combinations of proton and neutron densities [22]. Their values for the four nuclei aluminum, titanium, gold and lead are listed in Table III for reference. The µ − e conversion branching ratio is defined as where Γ conv is given by (49) and Γ capt is the standard model muon capture rate. The SM capture rates for aluminum, titanium and gold have been determined experimentally [39] and they are listed in Table IV for