The $\mu\to e\gamma$ decay in an EW-scale non-sterile RH neutrino model

We study in this research the phenomenology of $\mu\to e\gamma$ decay in a scenario of the class of extended models with non-sterile right-handed (RH) neutrino at electroweak (EW) scale proposed by P.Q. Hung. Field content of the standard model (SM) is enlarged by introducing for each SM fermion a corresponding mirror partner with the same quantum numbers beside opposite chirality. Light neutrino masses are generated via the type-I see-saw mechanism and it is also proved to be relevant with low energy within the EW scale of the RH neutrino masses. We introduce the model and derive branching ratio of the $\mu\to e\gamma$ decay at one-loop approximation with the participation of W gauge boson, neutral and singly charged Higgs scalars. After that we set constraints on relevant parameters and predict the sensitivities of the decay channel under the present and future experiments.


Introduction
The electroweak-scale right-handed neutrino (EW-scale ν R ) model is an extended version of the standard model (SM) was proposed for the first time in [1]. The fermion contents are doubled by introducing a mirror partner for each of the SM particle. Thus, correspond to a normal fermionic component is a mirror one with opposite chirality. Obey the mirror symmetry, right-handed mirror neutrinos and leptons, for example, are combined to form a doublets of the model's gauge symmetry SU(2) × U(1) Y . Other particles in the mirror sector are arranged in the similar way.
With left-handed and right-handed neutrinos respectively introduced in the normal and mirror sectors, that acquires enough conditions for the type I see-saw mechanism operating to give masses for the light active neutrinos [2,3,4,5]. In contrast to the normal type I seesaw, in which Dirac mass matrix is generated at electroweak scale, therefore right-handed neutrino masses, in general, should be extremely heavy to ensure the SM active neutrino masses as small as experiments have identified, Dirac mass matrix in the current model is given by a new Higgs singlet apart from the mechanism of SM mass production. It is proved that if the Higgs singlet's VEV is at relevant scale, heavy Majorana mass matrix is about hundred GeV, thus at the EW scale [1].
Apparently, mirror partners of the SM matter particles introduced in this model, which might impact on various physical phenomena, have to confront with experimental high precise measurements of the EW processes. The effects of extra chiral doublets have been carefully examined in [6]. The research shows that there is still large free parameter space after being constrained by the EW precision data. One of the exciting reasons, which has been demonstrated, is the partial cancellation of the contributions from the mirror fermions by those of the physical scalars, especially the SU(2) triplets. An updated version has been introduced after the discovery of the 125 GeV SM-like scalar [7,8], which has opened up a new stage for elementary particle physics, particularly model building for physics beyond the SM. Differ from the old version, an additional Higgs doublet has been introduced to give masses for mirror quarks and charged leptons (mirror sector), and the original one for those of the normal sector. Two candidates are found out to have signals in agreement with ATLAS and CMS observations [9].
Inspire of compelling evidence for lepton flavour violations (LFV) in neutrino oscillations, all efforts looking for those in the charged lepton sector have given negative results so far. In fact, it is demonstrated that minimal extensions of the SM with massive neutrinos, the rates of LFV processes involving the charged leptons are so extremely tiny, that unobservable in practice. For instance, the µ → eγ decay branching ratio is evaluated to be about 10 −55 using the currently known neutrino oscillation data [10,11,12]. For the reason, the decay is considered as one of the most important channels to look for signals of physics beyond the SM. On the experimental aspect, the best upper limit implies from the non-observation of the muon decay µ → eγ given by MEG in 2016 and it has been recently upgraded to work at sensitivity BR(µ + → e + γ) < 6.0 × 10 −14 [14,15].
On the theoretical aspect, a large number of researches have been interested in the same topic in various scenarios of physics beyond the SM [10,11,12,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]. Among them, the µ → eγ branching ratios and related consequences in the considering schemes of EW-scale ν R models are discussed for three different versions, which are the original [25], the updated with light of the 125 GeV SM-like Higgs scalar discovery [26] and an extension with A 4 discrete symmetry [27], respectively.
Although the phenomenology of µ → eγ decay in the three mentioned above EW-scale ν R scenarios has been considered, these researches have only taken into account contributions of one-loop diagrams, which are formed by light neutral scalar and the mirror charged leptons. The theoretically predicted branching ratio confronted with experimental upper bound will set upper limits on the magnitudes of Yukawa couplings involving the Higgs singlet. Following that light singlet vacuum expectation value v s is identified to have right Dirac mass magnitude for the see-saw mechanism working properly with hundred GeV Majorana righthanded neutrino masses to generate sub-eV scale of those for the light active neutrinos. However it is only part of the whole story, at one-loop approximation there are still contributions from diagrams with W − gauge boson, other neutral and singly charged scalars. These contributions might be sizable and therefore required careful considerations to set constraints on the involving interaction strengths, or evaluate their observed possibilities with the current and future experiments.
As aim of this research, we discuss the phenomenology of µ → eγ decay in the scenario of EW-scale ν R model with two Higgs doublets, the extended version to accommodate with the 125 GeV SM-like scalar detection [9,26]. The process is considered upto one-loop approximation with participation of light physical scalar and other particles, which have not been studied in the previous researches. The paper is divided into 4 main sections as follows. Beside this section for introduction, in Sect. 2, we briefly introduce the model and write down the relevant LFV vertexes, which contribute to our process of interest. In Sect. 3, after introducing the form-factors with explicitly algebraic expressions, one will derive the decay branching ratio and perform numerical analysis. Conclusion is given in the Sect. 4.

2
A review of the model

The model content
That has been mentioned in an earlier part, this research works on an extended version of the EW-scale ν R model, which is constructed based on the symmetric group SU(2) × With this choice, in contrast to the SM, right-handed neutrinos are components of a SU(2)× U(1) Y doublets, therefore they are non-sterile and take part in the weak interaction.
Two SM-like Higgs doublets are introduced to give masses to the charged and mirror fermions, respectively. The Higgs doublet, which is denoted as Φ 2 = (φ + 2 , φ 0 2 ), couples to the SM fermions to produce their masses, while the another, called Φ 2M = (φ + 2M , φ 0 2M ), is responsible for mass generations of the matter partner particles in the mirror sector. The mechanisms of lepton mass generations, including both charged leptons and neutrinos, will be detail presented in a later part of the research.
The heavy right-handed neutrinos, in this model, are introduced naturally in the mirror sector, therefore light active neutrino masses are commonly expected to be generated by the type I seesaw mechanism. The right-hand neutrino mass matrix is provided by a complex Higgs triplet with Y = 2χ If only one triplet, e.g.χ, is introduced in the model, the tree-level result ρ = 1, which is precisely measured by experiment, will be spoiled out, then one might obtain ρ = 2, instead. However, it is also proven in [31] that, if one has two triplets with relevant hyper-charges, when combined, to form (3,3) representation under the global SU(2) L ⊗ SU(2) R symmetry, the custodial SU(2) symmetry is preserved and ρ = 1. Thus accompanying with the tripletχ, it is required to add a real Higgs triplet with Y = 0, called (ξ + , ξ 0 , ξ − ). Finally, we introduce SM singlet Higgs φ S , which is responsible for creating Dirac mass matrix and connecting between matter and mirror sectors.
In this considered model, two Higgs doublets Φ 2 and Φ 2M are used to couple to SM and mirror fermions, respectively. To prevent unexpected couplings, a global symmetries U(1) SM × U(1) M F are imposed. Transformations of matter and Higgs fields are defined as following: Before writing down the Yukawa couplings, we should keep in mind that the global symmetry defined above only allows Φ 2 to couple to SM fermions, while Φ 2M will couple to the mirror ones. Apparently, terms involved φ S should contain both SM and mirror fermion fields. Two unit lepton number violation term, which stands in need for acquiring Majorna mass for the operation of the see-saw type I, can only be constructed without the absence ofχ. Detail expressions of the Yukawa couplings are: where σ 2 is the second Pauli matrix,

Symmetry breaking and mass generations
We will discuss next the mechanism of mass generations for matter particles in this model, especially for leptons including both charged leptons and neutrinos which are involved in the later discussions of the research, when the symmetry is spontaneously breaking. For further discussions, we suppose that Higgs fields develop their vacuum expectation values (VEV) as following: Charged lepton mass matrix obtained from eq. (4) can be expressed as where The matrix shown in eq. (7) can be diagonalized to give eigenvalues in the mass basis and mixing matrix for charged lepton. Without losing physical reality and is easier for calculation to obtain algebraic expression of the mixing matrix, let us assume that m ℓM ≫ m ℓ and m ℓM , m ℓ ≫ m D ℓ . The assumption allows us to approximately block diagonalized M ℓ in the same way usually done for the see-saw type I neutrino mass matrix, then one has where Suppose that normal and mirror charged lepton matrices are written in formm ℓR † , where m d ℓ and m d ℓM are diagonal, we have the relation between the gauge states and physical states as following Similarly, we have the quark mass matrix Discussion about quark sector will not be carried on any further, because they do not involve in the LFV decays being considered in this research.
Denote the heavy Majorana mass matrix as M R = g M v M , one easily obtains the full neutrino mass matrix, which has canonical form of the type-I see-saw mechanism Block diagonalizing the matrix, one could rewrite it as the following, for M R ≫ m D ν , where Differ from eq. (8), where charged lepton mass matrix M ℓ = M † ℓ is Hermitic thus being transformed by an unitary and its conjugated matrices, the complex symmetric M ν = M T ν is rotated by an unitary and its orthogonal ones (see eq. (16) ). Light neutrino mass matrixm ν is experimentally constrained to be smaller than 1 eV , if (g 2 ℓs /g M ) ∼ O(1) and v S ∼ O(10 5 eV ), Majorana mass of the right-handed neutrino M R could be at order of the electroweak scale. In fact, g ℓs is constrained by some rare processes (for instant µ → eγ decay, as we will see latter), which might lead to constraint (g 2 ℓs /g M ) ≪ 1. In this case v S would require to be at order GeV to ensure neutrino mass generation operating at the electroweak scale. Suppose that light and heavy (mirror) neutrino mass matrices are written respectively as To guarantee ρ = 1, as mentioned before, the Higgs potential should have a global symmetry SU(2) L ⊗ SU(2) R , and it is broken down to the custodial SU (2) when Higgs fields gain their VEVs. Before the symmetry broken, two triplets combine to form (3,3) representation and the doublets also maintain the (2, 2) structures under the global symmetry. The detail expressions are: From the above equations, one have the proper vacuum alignment for breaking gauge sym- Thus, the VEVs of real components of Φ 2 , Φ 2M and and v M respectively, they satisfy the condition where v ≈ 246 GeV . For further discussion, the following definitions are used After gauge symmetry is spontaneously broken, the L-R global symmetry of the Higgs potential is also broken down to the custodial SU(2) D . Seventeen degrees of freedom of the two Higgs triplets (one real and one complex) and two Higgs doublets are rearranged into physical Higgs bosons, beside three of the Nambu-Goldstone bosons are absorbed to give masses for W's and Z. Among the physical bosons, those which have degenerate masses, are grouped in the same physical scalar multiplets of the global custodial symmetry as following: The above discussion about physical scalars does not depends on the specific case of Higgs potential, but works for any one of that which possesses SU(2) L ⊗ SU(2) R global symmetry, including the cases have been detail considered in [9]. The physical Higgs bosons, which are generated after the process of gauge symmetry breaking, should have masses at electroweak scale, thus in range of hundred to few hundred GeVs. The scalars arranged in the same multiplets (not singlet) have same masses, while three singlets H 0 1 , H 0 1M , H 0′ 1 are not physical states, in general. These states are linear combinations of mass eigenstates, which are denoted respectively asH 0 1 ,H 0 2 ,H 0 3 . Relations between physical and gauge states are expressed as H 0 Note that the SM Higgs scalar discovered by LHC with mass 125-GeV is one of the three mentioned above mass states.
Final physical scalar to be mentioned in this research is the light singlet φ 0 s , which originates from the degree of freedom of gauge singlet Higss φ S . As a singlet, φ S does not break the gauge symmetry and its VEV v S is expected to be much lower than the electroweak scale in order to give tiny masses for the light active neutrinos. It is reasonable to take φ 0 s mass at the same order as v S .

The LFV vertexes
It is known that LFV vertex does not present in the SM at tree-level, because charged lepton mass matrix and the matrix of Yukawa couplings are diagonal at the same time, and vector gauge bosons interact only with the left-handed components of the matter fields. In this considered scenario, vector fields interact not only with the left-handed SM fermions but also with the right-handed components of the mirror sector. Moreover, LFV interactions occur at tree-level for both the charged currents and Yukawa couplings. Lagrangian involving charged currents in this model can be written as where ψ SM and ψ M stand for the SM and mirror fermionic fields in the gauge basis, respectively.
The relevant Yukawa couplings between the leptons and mirror lepton with scalars, which contribute to the phenomenology of µ → eγ decay, in the gauge basis are listed in the  suppose that charged lepton and mirror charged lepton mixing matrices are real (thus all involved complex phase are ignored) and U ℓL = U ℓR = U ℓ , U M ℓL = U M ℓR = U M ℓ . Using the relations described in eqs. (11), (18), one easily obtains the main vertexes that contribute to our process of interest in the mass eigenstate basis. After ignoring terms, which are proportional to the second order of R ν(ℓ) , the detail expressions are listed in table 2 and eqs. from (29) to (34): Here the notations U P M N S = U † ℓ U ν , which is the famous neutrino mixing PMNS matrix, have been used. For simplicity, we have also neglected the complex phases in U ℓ and U M ℓ .

Form-factors and µ → e + γ branching ratio
Before introducing the µ → e+γ branching ratio, the loop integral factors must be calculated.
In this research, we take into account the effective charged lepton flavor changing operators arising at one-loop level with the participation of physical Higgs scalars (which include the single and neutral charges, heavy and light ones) and W gauges boson. The final result could be written as Here A L,R are the form factors: where where we have defined λ k = m 2 k /M 2 Wµ(H Q ) .
Note that the monotonic functions G γ (x) and R γ (x), which are defined for x variable varying in the interval [0, +∞), have been introduced in some researches so far, for instance [17,28]. At some specific points, such as x = 0, 1 and x tends to infinity, G γ (x) obtains the limited values as −1/2, −3/8 and 0, respectively. Similarly, we also have R γ (x → 0) = 4, R γ (x → 1) = 3/2 and R γ (x → ∞) = −1/2. Compare with the form-factors for Higgs scalar one-loop have been used in some previous publications [19,28], G Q H (x) has better expression 1 , and R Q H (x), as far as my knowledge, has not been given so far.
The branching ratio of µ → e + γ decay is easily obtained as where α em = 1/137 is the fine-structure constant.

Numerical analysis of the µ → e + γ decay
We perform in this section the numerical analysis of µ → e + γ branching ratio using the current experimental data and expected sensitivity of the future experiments. Apparently, taking into account at the same time all the possible contributions to the process would not a good strategy to give detail understanding the role of each kind of the diagrams. Thus, we separately consider the contributions of the one-loops diagrams with virtual W gauge boson, neutral and singly charged Higgs scalars to the ratio. For simplicity in further numerical discussion in the later part, we suppose that three heavy neutrinos possess equal masses denoted as m M χ . Similarly, three mirror charged lepton masses are m M ℓ .
The LFV vertexes involving neutrinos and W boson taken upto R ν(ℓ) first order are the three firsts listed in the table 2. In fact, the contribution of light neutrinos to the µ → e + γ decay are extremely small, which could be easily seen from eq. (36). For light neutrino masses are 1 G Q H (x) introduced the current research is valid for any x in the interval [0, +∞), while the previous calculations are applied only for infinitesimal λ k . When x tends to zero, functions G Q H (x) tends to 1/6 − Q/4, that is consistent to the results obtained in [19,28]. at sub-eV order or less, and M W = 80 GeV, λ k = m 2 k /M 2 W ≈ 0, that leads to (see eq. (36)) due to the unitarity of PMNS matrix. Moreover, contribution from the interference term in A R is strongly suppressed by the factor m k /m µ ∼ 10 −1 eV/10 2 MeV ∼ 10 −9 .
Constraint on the interaction strength between heavy neutrinos and W gauge boson by the lepton flavour violation decay µ → eγ is given in figure 1. The blue and red lines correspond to the constraints obtained using whether the current upper bound or the designed sensitivity of the future experiment, respectively. The results, apparently, are less meaningful comparing with the limits, which could be directly derived from what we have already known about light neutrino masses. One havẽ thus |R ν | 2 ∼ 10 −12 for M R ∼ 100 GeV that is at least 7 (6) orders smaller than the constraints obtained from figure  (36), (37). The interference term is no longer suppressed but dominated over the first term due to the large masses of accompanying particles with the physical scalar, which are heavy neutrinos or new charged leptons depending on kind of charge carried by the scalar. For heavy neutrino mass m χ M L and new charged lepton mass m M ℓ about hundreds GeV, the ratio m k /m µ ∼ 100 GeV/100 MeV ∼ 10 3 , that is also the dominated factor of the second term in comparison with the first term.
We show in figure 2 constraint on the Yukawa coupling only if the first term in (36), (37) are taken into account for singly charged scalar cases. The stringency obtained in this case on the magnitude of the couplings is almost the same as the previous consideration of W gauge boson and heavy neutrino couplings, and thus does not provide any new meaning. Our study shows that similar results are obtained for the case of neutral Higgs scalars.
Contributions of the first terms in the expressions of the form-factors A L and A R are less important, that is because they are strongly suppressed by the second terms as explained in a previous paragraph. The figure 3 describes the upper constraints on the relevant Yukawa couplings as functions of Higgs scalar mass (left-panel) and heavy neutrino mass (rightpanel), which are varied from about hundred GeV to several hundreds GeV. The constraints are about six orders more stringent than the values of those obtained from previous figure. That, one again, proves the dominated contribution of the interference term and is also consistent with the illustration given somewhere above. The left plot shows lines, which have shapes of monotonically increasing functions, therefore the most stringent constraints on The results read:   (see table 2) will decide how sensitive they are with the µ → eγ decay experiments. The involved couplings depend on lager number of new parameters, where most of them are unknown, thus it is too difficult if not want to say impossible to make a detail analysis. In this research, we try to roughly make estimation on the sensitivities of the µ → eγ decay with the present and future experiments, using the known data and supposing that the model is functioning at the electroweak scale.
Let us make a numerical estimation. As have been explained in the earlier part that R ν ∼ 10 −5 1GeV M R ∼ 10 −6 . In the same way, we also have R ℓ ∼ 10 −5 1GeV M R M 2 ℓM ∼ 10 −6 . It is reasonable to estimateR ℓ(ν) = U † ℓ R ℓ(ν) U M ℓ ∼ 10 −6 , at the same order as R ℓ and R ν , since basis transformation matrices U ℓ and U M ℓ are normalized. Furthermore, we take heavy neutrino and mirror charged lepton masses about 100 GeV. The Yukawa coupling depended factor of the branching ratio in eq. (42) reads: with the future expected upper bound.
Carry on similar estimation as the previous part, the results are: Thus |g ℓs | upper limit is estimated with current experimental data to be at order 10 −5 , it might be improved little more if next generation of µ → eγ experiment would not probe any signal. Moretheless the constraint is still in the same scale.
As one can see in the right-panel of figure 5, constraints on the Yukawa couplings become less and less stringent as mirror charged lepton mass goes up. However, the shape does not effect strongly on |g ℓs | upper bound. Let us make a simple evaluation. For m M ℓ = 500 GeV, which corresponds to the highest or the end point of each line, one has 2|(Y L

Conclusion
In this research, we have performed a numerical analysis for the µ → eγ decay in a scenario of the EW-scale non-sterile right-handed neutrino model, which is accommodated with the 125 GeV SM-like scalar discovery. The decay is suggested to occur at one-loop diagrams, formed by neutrinos (heavy and light) accompanying with W-boson or singly charged scalars, and light or heavy neutral scalars with charged leptons. It has been shown that the contribution provided by neutrino and W-boson loop channels give trivial upper constraint, which is about six orders less stringent than the limit of that directly derived from neutrino mass currently known data. For the case particles running inside the loops are light scalar and mirror charged leptons, upper bound for the Yukawa couplings |g ℓs | roughly obtained after comparing theoretical prediction and experimental results is some number of order 10 −5 .
Consequently that brings the singlet vacuum expectation value upto magnitude of few GeVs or few ten GeVs, if the RH neutrino mass is managed within the electroweak scale. The research has also demonstrated that the branching ratio of µ → eγ decay might be large enough to reach expected sensitivity, Br(µ → eγ) ≤ 6.0 × 10 −14 , of the upgraded MEG experiment, if one of the two particles running in the loop is the heavy scalar with neutral or single charge. For instance, as one of the most promising possibilities, magnitude of the branching ratios might be within the detectable range with c M ≤ 0.03 for the case neutral scalar H 0 3 participating in the process.