Charged Lepton Flavor Violation in the Semi-Constrained NMSSM with Right-Handed Neutrinos

We study the \mu \to e \gamma decay in the Z_3-invariant next-to-minimal supersymmetric (SUSY) Standard Model (NMSSM) with superheavy right-handed neutrinos. We assume that the soft SUSY breaking parameters are generated at the GUT scale, not universally as in the minimal supergravity scenario but in such a way that those soft parameters which are specific to the NMSSM can differ from the soft parameters which involve only the MSSM fields while keeping the universality at the GUT scale within the soft parameters for the MSSM and right-handed neutrino fields. We call this type of boundary conditions"semi-constrained". In this model, the lepton-flavor-violating off-diagonal elements of the slepton mass matrix are induced by radiative corrections from the neutrino Yukawa couplings, just like as in the MSSM extended with the right-handed neutrinos, and these off-diagonal elements induce sizable rates of \mu \to e \gamma depending on the parameter space. Since this model has more free parameters than the MSSM, the parameter region favored from the Higgs boson mass can slightly differ from that in the MSSM. We show that there is a parameter region in which the \mu \to e \gamma decay can be observable in the near future even if the SUSY mass scale is about 4 TeV.


Introduction
It is now clear that the lepton flavor number is not a conserved symmetry because of experimental observations of neutrino oscillations [1]. In the minimal extensions of the Standard Model (SM) with the Majorana neutrino mass terms, the branching ratios for charged lepton-flavor violating (LFV) processes are extremely small since they are suppressed by at least a factor of m 2 ν /m 2 W , which makes it very difficult for near-future experiments to detect LFV signals. On the other hand, in more general extensions of the SM, which are motivated by several reasons, it is known that sizable LFV rates are predicted depending on parameter region. If LFV processes are discovered, it directly means an indirect signature of physics beyond the SM (BSM). Recently, the MEG experiment reported a new upper limit of Br(µ → eγ) < 5.7 × 10 −13 [2]. This already gives a strong constraint on models beyond the SM, and hence it is very important to keep updating these upper bounds on the LFV processes. Supersymmetry (SUSY) is still a promising candidate for physics beyond the SM [3]. Lots of efforts have been devoted to the discovery of SUSY at the LHC, but only in vain so far. The most studied model of SUSY is the minimal SUSY SM (MSSM). Even in the framework of the MSSM, there are some unsolved problems such as the µ problem. Next-to-the MSSM (NMSSM) is an extension of the MSSM with a SM-singlet Higgs chiral superfieldŜ. The NMSSM could give a hint to solve the µ problem since in this model the µ term is induced by the vacuum expectation value (VEV) of the scalar component S ofŜ. In this sense the NMSSM is a natural extension of the MSSM.
One of the difficulties in the MSSM is the Higgs boson mass. In the MSSM, the tree-level lightest Higgs boson mass is bounded from above as, and has to rely on large radiative corrections to reproduce the observed Higgs boson mass of 126 GeV [1].
The main contribution to the radiative corrections comes from the top Yukawa coupling [4][5][6], and to maximize this effects one needs a top-squark mass much larger than the top-quark mass. In the NMSSM, the lightest Higgs boson mass reads [7]: where v ∼ 174 GeV. As seen from this equation, the contribution from the new parameter λ, which is the coupling among the new singlet S and the MSSM Higgs doublets H u and H d , makes the tree-level Higgs boson mass larger, in particular for small tan β. We have to note that the mixings between S and the MSSM Higgs doublets can make a negative contribution to the lightest Higgs boson mass, and the NMSSM does not always predict a larger Higgs boson mass. We will discuss this issue in details later in this paper.
There are more than one-hundred free parameters in the MSSM. Usually, we assume an underlying scenario for SUSY breaking, and it allows us to reduce the number of free parameters. In this paper we assume the minimal supergravity (mSUGRA)-like boundary conditions that the SUSY breaking parameters m 0 , M 1/2 , A 0 are universal at the GUT scale. The parameters at the SUSY scale are obtained by evolving these parameters according to the renormalization group equations (RGE). These mSUGRA-like boundary conditions are very effective for avoiding constraints from the SUSY-induced flavor changing neutral current (FCNC) processes. This is also true for the charged LFV processes, and in the mSUGRA, also known as the constrained MSSM (cMSSM), there are essentially no charged LFV. This is similar in the case of the constrained NMSSM.
The neutrino masses are exactly zero in the framework of the SM, which clearly needs modifications in view of the observation of neutrino oscillations. One of the most natural mechanisms to explain the tiny neutrino masses is the (Type-I) seesaw mechanism [8][9][10], which we consider in this paper. The extension of the original seesaw mechanism to SUSY models is straightforward. In the MSSM extended with the right-handed neutrinos ν R , which we call the MSSM + ν R model, even if one assumes the mSUGRA-like boundary conditions at the GUT scale, off-diagonal elements in the slepton mass matrices are induced via radiative corrections from the neutrino Yukawa couplings, which can predict sizable rates for the LFV processes like µ → eγ. This mechanism also works in the NMSSM extended with the right-handed neutrinos, which we call the NMSSM + ν R model and which we consider in this paper, but since there are more free parameters than in the case of the MSSM + ν R model, the predicted LFV rates can slightly differ from those in the MSSM + ν R model in the parameter region favored from the Higgs boson mass.
The contents of this paper are as follows. In Section 2, we introduce the model we work with, and in Section 3 we explain the origin of the LFV (off-diagonal) elements of the slepton mass matrices. In Section 4, we discuss constraints on the parameters of the model. We introduce the results of numerical calculations in Section 5, and in Section 6 we summarize this paper.

Z 3 -invariant NMSSM
The NMSSM is an extension of the MSSM, and it has an extra Higgs chiral superfieldŜ which is singlet under the SM gauge group. In the Z 3 -invariant NMSSM [7], the µ term µĤ u ·Ĥ d in the superpotential of the MSSM is replaced by the term λŜĤ u ·Ĥ d , and the µ-parameter is determined from the singlet VEV s as µ eff = λs. Namely, the superpotential of the Z 3 -invariant NMSSM is given as where the dot in the termĤ u ·Ĥ d represents a product of two SU (2) doublets, and the hats on the fields stand for the superfields corresponding to the fields. We assume that the R-parity is conserved, and assign the even R-parity toŜ. The soft SUSY breaking terms are In the case of the constrained NMSSM, the gaugino masses, sfermion soft SUSY breaking masses, and the A-parameters take the values which are "universal" at the GUT scale, similarly to the case of the cMSSM: where α (α = 1, . . . , 3) labels the gauge groups of the SM, and i and j are the indices for generations, i, j = 1, . . . , 3. As for the parameters A λ , A κ and m 2 S which are specific to the NMSSM, we assume that the values of A λ and A κ at the GUT scale are not necessarily equal to A 0 , and that m 2 S at the GUT scale can be different from m 2 0 . We call the NMSSM with this class of boundary conditions the semi-constrained NMSSM.

Z 3 -invariant NMSSM extended with right-handed neutrinos
In this paper we take the simplest extension of the Z 3 -invariant NMSSM with the right-handed neutrinos, in which the (type-I) seesaw mechanism [8][9][10] is at work. The superpotential is given by where the Z 3 -charges are assigned as in Table 1 [11]. This charge assignment excludes the term (λ ν ) ijŜN c i · N c j from the superpotential 1 .
The neutrino masses in this model is where U MNS is the MNS matrix [12]. In the standard representation of the PDG, the matrix reads: where c ij = cos θ ij , s ij = sin θ ij . The mixing angles θ ij (i, j = 1, . . . , 3, i < j) describes the mixing between the mass eigenstates ν i and ν j , and the factors δ, α 21 , α 31 are complex phases, and represent the Dirac phase and the two Majorana phases, respectively. According to the latest data [1]  1 It is possible to derive the (left-handed) neutrino masses via the type-I seesaw mechanism from the Majorana masses which emerge from term (λν ) ijŜN c i ·N c j after replacing S with its VEV. In this case, since the singlet VEV S is at most O(1 − 100 TeV), the Majorana masses must be about the same order, which forces us to assume a very small neutrino Yukawa coupling (Y N ) in order to explain the tiny neutrino masses. This makes the LFV rates extremely small and hence we do not consider this scenario in this paper.
The mass-squared differences, which are also important parameters, are: 2.52 ± 0.07 (10 −3 eV 2 ) (normal mass hierarchy) 2.44 ± 0.06 (10 −3 eV 2 ) (inverted mass hierarchy) . (2.14) In this paper, we assume the normal hierarchy scenario for the neutrino masses, and take the values and, for the mixing angles, Concerning the complex phases, we take for simplicity. Another free parameters are the 3 × 3 elements of M N . Although it is known that the structure of this matrix gives an influence to the predicted LFV rates [13][14][15][16][17], in this paper we assume where M ν is a real number.

Lepton Flavor Violation
In this section we discuss charged lepton flavor violation, taking the NMSSM + ν R model as an example of new physics beyond the SM.

µ → eγ in the Standard Model with ν R
Within the SM, the neutrinos are strictly massless and lepton flavor number is exactly conserved. The experimental observations of neutrino oscillations [1], however, make it clear that we have to extend the SM in such a way that it can accommodate the neutrino masses and mixings. One of the simplest extensions is to introduce right-handed neutrinos (ν R ) which are singlet under the SM gauge group, which allows us to introduce Dirac mass terms for the neutrinos in the Lagrangian.
Once we introduce the right-handed neutrinos in the SM, in general, charged lepton flavor number is no longer conserved. This is similar to the case in the quark sector, and the mismatch between the gauge eigenstates and the mass eigenstates violates the lepton flavor number conservation. The branching ratio of µ → eγ in this model is given by [18][19][20] The suppression factor (m 2 ν,i − m 2 ν,1 ) 2 /M 4 W makes the branching ratio extremely small, and it is very difficult for near future experiments to detect µ → eγ in this model. On the contrary, in the non-minimal extensions of the SM such as the (N)MSSM+ν R , sizable LFV rates can be predicted depending on the parameter region, and this makes the LFV searches very important as a probe of new physics beyond the SM. Figure 1: The diagrams which give dominant contributions to the l i → l j γ decay in the NMSSM+ν R model.

µ → eγ in NMSSM + ν R Model
In the NMSSM+ν R model, there are two diagrams which give dominant contributions to the l i → l j γ decays (where i and j are the generation indices which run from 1 through 3 with i > j), which we show in Fig. 1. One is the diagram with the neutralino and the charged slepton in the loop, and the other is the diagram which involves the chargino and the sneutrino. In general, the amplitude T for the l i → l j γ decay can be written as where e is the positron charge, ǫ α is the polarization vector of the photon, u i and u j are the spinors for the initial-and final-state leptons, respectively. The operators P L,R stand for the chiral projection operators. The dependence of the amplitude on the models is included in the coefficients A L 2 and A R 2 , and by calculating the diagrams in Fig. 1 we can determine A L 2 and A R 2 . In the case of the MSSM+ν R model, the explicit forms of A L 2 and A R 2 are given, for example, in Refs. [21][22][23]. In the case of the NMSSM + ν R model, they are essentially the same as the MSSM + ν R model, except that there are five neutralinos, instead of four, at low energies, and we can use the expressions in Refs. [21][22][23] with small modifications. By using the formulas mentioned above, the decay branching ratio Br(l i → l j γ) can be calculated from the amplitudes to be where Γ li is the total decay width of the lepton l i . In order to have a non-vanishing LFV rate from the diagrams in Fig. 1, we must have off-diagonal elements in the slepton mass matrices. The mass matrices are given as, RL are the 3 × 3 matrices whose (i, j) elements are given as where v d ≡ v cos β. In this paper, we assume mSUGRA-like boundary conditions, in which all the SUSY breaking parameters that have flavor indices do not have flavor mixings at the GUT scale. This means that there are no off-diagonal elements in the matrices M 2 l and M 2 ν . However, off-diagonal elements in these mass matrices are induced by radiative corrections at the energy scale higher than M N , which can be seen in the RGE, where t = ln Q with Q being the renormalization scale. This directly means that both M 2 l and M 2 ν have off-diagonal elements at low energies. The size of these off-diagonal elements can be roughly estimated as [21][22][23], where i = j. As is clear from Eq. (3.11), the slepton off-diagonal elements in this model comes from the neutrino Yukawa couplings, Y N . The branching ratio can be estimated in terms of the off-diagonal elements to be [23] Br (3.12) At this moment, the most stringent experimental constraint on the µ → eγ is given by the MEG experiment and the upper limit is 5.7 × 10 −13 [1,2]. This bound will be further improved by the upgraded MEG experiment to ∼ 6 × 10 −14 [24], and this makes the experiment very important as a probe of new physics beyond the SM.

Other cLFV processes
In this paper we focus on µ → eγ in the later sections, but there are many other charged LFV processes [25]. Here we mention some of them. There are two other l i → l j γ processes, τ → µγ and τ → eγ. Their current experimental limits are Br(τ → eγ) < 3.3 × 10 −8 and Br(τ → µγ) < 4.4 × 10 −8 [1]. In the near future, these limits are expected to be improved to the level Br(τ → lγ) < 1.0 × 10 −9 at Belle-II [26]. Under the assumptions we set out at Section 2, the µ → eγ decay is more sensitive to SUSY particles, and hence we focus on µ → eγ in this paper.
Other important cLFV processes include l + i → l + j l + j l − j and µ-e conversion in nuclei. As for the former process, the branching ratio can be related to that of the l i → l j γ decay as [23] Br Br(l i → l j γ) , (3.13) and hence Br(l + i → l + j l + j l − j ) can be calculated once Br(l i → l j γ) is obtained. The current experimental limit for µ + → e + e + e − is Br(µ + → e + e + e − ) < 1.0 × 10 −12 [1], and this is expected to be improved to Br(µ + → e + e + e − ) < 1.0 × 10 −16 at the Mu3e experiment at PSI [27]. Concerning the µ-e conversion in nuclei, there is a simple relation between the conversion rate B µe (N ) and Br(µ → eγ) when the photon mediation diagram gives the dominant contribution [28], is the conversion rate normalized to the muon capture rate Γ(µ − N → capture), and R(Z) is a parameter which depends on the atomic number Z of the nucleus which captures the muon. The current limits are B µe (Ti) < 4.3 × 10 −12 , B µe (Au) < 7 × 10 −13 [1]. The near future experiments are the COMET experiment at J-PARC [29] and the µ2e experiment at FNAL [30], and the PRISM/PRIME experiment at J-PARC [31], which are expected to improve the bounds to B µe (Al) ∼ 7 × 10 −17 , B µe (Al) ∼ 6 × 10 −17 , B µe (Al) ∼ 10 −18 , respectively. Since the R(Z) factors for these experiments are R ∼ 0.0025 for Al and R ∼ 0.0040 for Ti [28], these experiments are expected to go beyond the corresponding limit of the µ → eγ decay by 3 ∼ 4 orders of magnitude, and this will be very useful to probe broader parameter region of new physics.

Constraints on the Parameters in the Model
In this section we discuss constraints on the parameters in the NMSSM + ν R model. Some of the issues below are already discussed in literature [7].

Tadpole conditions
In the NMSSM, there are three tadpole conditions. At tree-level they read: where µ eff = λs and B eff = A λ + κs. We can use these relations to determine three parameters from other parameters. For example, we can use these relations to determine µ eff , B eff and m 2 S from the other parameters. Later we will discuss which parameters we use as input.

Maximal Tree-level Higgs Mass condition
One of the advantages of the NMSSM over the MSSM is that there is a parameter region in which the lightest Higgs boson mass can be made larger than that of the MSSM. As can be seen from Eq. (1.2), in order for the Higgs boson mass to be larger, it is favorable to have large λ and small tan β. The approximate formula Eq. (1.2) is obtained by neglecting the mixings between the MSSM Higgses and the singlet Higgs in the CP-even Higgs-boson mass matrix, where v u ≡ v sin β and the lower-left components are related to the upper-right components by the condition (M 2 S,Tree ) ij = (M 2 S,Tree ) ji . If we take the mixing to the singlet Higgs into account, the lightest Higgs-boson mass reads [7]: As can be seen from this equation, the mixing to the singlet Higgs makes the tree-level lightest Higgsboson mass smaller. The λ dependence of the lightest Higgs-boson mass mainly comes from the second and third terms, and too large value of λ makes the Higgs boson very small. There are two ways to decrease the mixing with the singlet: One way is to assume a small λ ( 0.1), and the other is to tune the parameters to satisfy the relation 2 ,

Conditions from positive CP-even and CP-odd Higgs boson mass-squared
The (3, 3) element in the CP-odd Higgs-boson mass matrix is given as, At broad parameter region, the third term on the right-hand side gives the dominant contribution. Therefore, in order for the CP-odd Higgs mass-squared to be positive, we must have the condition, κsA κ 0 , (4.8) in the approximation that the first and second terms in Eq. (4.7) are negligible compared to the third term.
Another condition is that the (3, 3) element of the CP-even Higgs-boson mass-squared matrix should be positive: where we have worked in the approximation s ≫ v u , v d . This condition comes from the requirement that the singlet Higgs-boson mass-squared must be positive in the approximation that the mixing between the singlet and any of the MSSM Higgs doublets is neglected. Summing up, the condition which A κ should satisfy is −4(κs) 2 κsA κ 0 . In the numerical analysis presented in this paper, we give A κ as an input parameter at the SUSY scale.

Constraint from non-vanishing VEV of S
There is a condition on the model parameters from the requirement that the singlet Higgs S has a non-zero VEV, S ≡ s = 0. When s ≫ v u , v d , the potential for S reads: If we require that this potential has a minimum at S = s = 0, and that the value of V (S) at S = s is smaller than V (0), we obtain the condition [7],

Constraint from Perturbativity of λ
The tree-level Higgs boson mass becomes larger for larger value of λ unless we take the mixing with the singlet into account. However, there is a limit on the size of λ which comes from theoretical consideration. Namely, in order for λ not to blow up below the GUT scale, the value of λ at the SUSY scale must be smaller than ∼ 0.7 [7].

Condition from the SM-like lightest Higgs boson
In this paper, we identify the lightest CP-even Higgs boson as the Higgs boson discovered at the LHC [1]. The properties of the discovered particle such as the decay branching ratios are known to be consistent with those of the Higgs boson in the minimal SM. This means that we have to require that the lightest CP-even Higgs boson in the model we consider should not be singlet-like but like the lightest Higgs boson in the MSSM which is known to become SM-like in the decoupling limit.

Numerical Results
In this section, we give our numerical results. First, we explain how we choose independent input parameters. To maximally keep the similarity to the cMSSM, we choose tan β at the SUSY scale and m 0 , M 1/2 and A 0 at the GUT scale as input parameters. In addition, since the parameter λ directly enters in the expression for the lightest Higgsboson mass, we choose λ at the SUSY scale as input. If we further choose either κ or A λ as input, we can use the two tadpole conditions Eqs. (4.1) and (4.2) to determine µ eff (= λs) and B eff (= κs + A λ ), and then use Eq. (4.3) to fix m 2 S by using the value of A κ as an additional input. Below we consider two cases: in one case we choose κ at the SUSY scale as input, and in the other case we take A λ at the GUT scale as input. Summing up, we consider two sets of input parameters. In one case, we choose the parameters below as input, tan β , λ , κ , A κ at the SUSY scale , m 0 , M 1/2 , A 0 at the GUT scale , (5.1) which we call the case 1, and in the other case, we take the parameters below as input: tan β , λ , A κ at the SUSY scale , which we call the case 2.

Case 1
In this case, we determine the parameters s = µ eff /λ and A λ = B eff − κs by using the tadpole conditions. If we are to use Eq. (4.6), we have to tune κ to satisfy Eq. (4.6). The value of κ in this case is This equation means that for large tan β and for large λ, the κ parameter becomes too large, and then λ at the scale higher than the weak scale becomes too large to be perturbative, and eventually it blows up below the GUT scale 3 . We therefore do NOT assume Eq. (4.6) for the case 1, and assume small λ (∼ 0.1) to make the mixing of the MSSM Higgses with the singlet Higgs smaller, in order not to decrease the tree-level Higgs-boson mass.

Numerical Results
Our numerical results for Br(µ → eγ) and the Higgs boson mass in the case 1 are given in Figs. 2 (a) and (b). In the figures (a) and (b), κ at the SUSY scale is taken to be 0.09 and 0.05, respectively. The rest of the input parameters are taken to be the same in the two figures, and the input SUSY parameters are λ = 0.1, A κ = −50 (GeV) at the SUSY scale and A 0 = −500 (GeV) at the GUT scale. We take m 0 = M 1/2 , and the right-handed neutrino Majorana mass is taken to be M ν = 5.0 × 10 14 (GeV).
Also shown in Figs. 2 (a) and (b) are the contours for the lightest Higgs boson mass. From the figures, we find that smaller κ makes the Higgs boson mass smaller. We have numerically confirmed that the difference in the Higgs boson mass mainly comes from the values of κ, and the difference in the values of the other parameters like A λ are not very important for the difference in the predictions for the Higgs boson mass. This dependence of the Higgs boson mass on κ can be understood from Eq. (4.5). Namely, large κ makes the (3, 3) element of M 2 S,Tree larger and the mixing between the MSSM Higgses and the singlet Higgs, which makes a negative contribution to the lightest Higgs boson mass, smaller.
If we assume (4.6), then a large λ induces a large κ via RGEs, and λ can develop the Landau pole below the GUT scale depending on the parameters. For small tan β, the large top Yukawa coupling makes the right-hand side of the RGE for λ large, and this makes it easier for the Landau pole for λ to occur. From the figures, we find that there is a parameter region which is favored from the Higgs boson mass measurement where the predicted value of Br(µ → eγ) is within reach of the near-future experiment even if m 0 is as large as ∼ 4 TeV.
We here comment on the dependence of the Higgs boson mass on κ. In the figures, we take κ only down to 0.05. For smaller values of κ, for example, κ 0.03 for λ = 0.1, the Higgs boson mass sharply decreases for decreasing κ. This sharp κ dependence comes from the factor (λ/κ) 2 in the third term of the right-hand side of Eq. (4.5). If we take smaller value of λ, this sharp decrease of κ happens at smaller value of κ, and hence we can take smaller κ as well.

Case 2
In this case, if we are to use Eq. (4.6), the value of κ is determined to be, similarly to the case 1, Similarly to the reasoning in the case 1, the equation above implies that if tan β or λ is too large, the κ parameter at higher scale blows up and becomes non-perturbative. Therefore, if we are to use Eq. (4.6), we need small λ and small tan β, but this choice makes the Higgs boson mass very similar to the MSSM case and hence is not very interesting. We therefore do NOT use Eq. (4.6) in the case 2, either.

Numerical Results
In Figs  From the figures, we find that only in Fig. 3 (b), there is an extra Higgs-mass favored parameter space at the region where tan β and m 0 (= M 1/2 ) are both large. This difference between the two figures mainly comes from the difference in the value of κ, and the differences in the other parameters like A λ enter only indirectly through the value of κ in the prediction for the Higgs boson mass.
We now explain why the changes in the input value of A λ at the GUT scale affect the value of κ at the SUSY scale.
To do so, we first explain the dependence of κ on m 0 (= M 1/2 ) and tan β with fixed value of A λ (M GUT ). Below we will show that κ becomes smaller for larger m 0 and for larger tan β at the region tan β ≫ 1 in the parameter space shown in Figs. 3 (a) and (b). At the upper-right region of Fig. 3 (b), the value of κ becomes κ 0.03, where the Higgs boson mass decreases relatively quickly for decreasing κ, as discussed at the end of the discussion for the case 1 in this section 4 . In the parameter region shown in Fig. 3 (a), the value of κ is larger than 0.03, and hence this relatively fast decrease does not happen. Then we have to explain why κ is smaller in Fig. 3 (b). This is because A λ (M SUSY ) is larger in Fig. 3 (b) since A λ (M GUT ) is larger. The relation between A λ (M SUSY ) and κ are given by κ = (B eff − A λ )/s and hence larger A λ means smaller κ.
Let us discuss the change in κ for different m 0 (= M 1/2 ) for fixed values of tan β and A λ (M GUT ). The value of A λ at the SUSY scale is given by solving the RGE, For our sample parameters, A λ (M SUSY ) becomes larger 5 for larger m 0 (= M 1/2 ) and fixed tan β. Therefore, for a fixed value of tan β, larger m 0 (= M 1/2 ) makes κ smaller through the relation, κ = (B eff −A λ )/s. Next, we discuss the dependence of κ on tan β, fixing the values of m 0 (= M 1/2 ) and A λ at the GUT scale. Since here we are mainly interested in the difference at large tan β region, in this paragraph we assume tan β ≫ 1. For large tan β, A λ (M SUSY ) becomes larger for larger tan β since the fourth term of the right-hand side of Eq. (5.7), which involves the bottom Yukawa coupling, becomes more important. This increase in A λ (M SUSY ) for larger tan β makes κ smaller for fixed m 0 since κ = (B eff − A λ )/s. Another reason which makes κ smaller for larger tan β comes from the values of µ eff and B eff , although this effect is less important for large tan β. The values of µ eff and B eff at the SUSY scale are obtained by solving the tadpole conditions, and the solutions at the tree-level are, Both µ eff and B eff become smaller for larger tan β for our sample parameters. From the relation µ eff = λs, a smaller µ eff means a smaller s for fixed λ. From κ = (B eff − A λ )/s, the variation of κ comes from that of s (= µ eff /λ) and that of B eff . For our sample parameters, since the decrease in B eff due to increase in tan β has a larger effect on κ than that of s, κ becomes smaller for larger tan β.
As for Br(µ → eγ), also in the case 2, we find that there is a parameter region which is favored from the Higgs boson mass and in which the predicted value of Br(µ → eγ) is within reach of near-future experiment even if m 0 ∼ 4 (TeV), which has not yet been probed at the LHC.

Summary
In this paper, we have studied the cLFV in the semi-constrained NMSSM+ν R model, taking into account the recent results on the Higgs boson mass determination. We have considered the boundary conditions at the GUT scale to be MSSM-like and semi-constrained in the sense that the SUSY breaking parameters A λ , A κ , m 2 S which are specific to the NMSSM are not necessarily equal to A 0 , A 0 , m 2 0 , respectively. We have considered two cases: in one case the parameters (s, κ, m 2 S ) are determined from the tadpole conditions, which we call the case 1, while in the other case (s, A λ , m 2 S ) are determined from other input parameters, which we call the case 2.
One of the advantages of the NMSSM is that the tree-level lightest Higgs boson mass can be taken to be larger than that of the MSSM by taking a large value of λ. In addition to this effect, there is another new effect in the Higgs boson sector of the NMSSM, namely, we also have to take into account the mixing with the singlet Higgs. This mixing can decrease the Higgs boson mass depending on the parameters. In the semi-constrained scenario we have considered, we find it is difficult to realize both large λ and small mixing with the singlet at the same time. Hence in this paper we have assumed a small λ (∼ 0.1) which makes the mixing with the singlet small.
In the case 1, we have obtained the results similar to those in the MSSM + ν R model. We have also shown that the Higgs-boson-mass favored parameter region depends on the value of κ. As the case 2, we have considered the case where the κ parameter is not an input parameter but is a parameter determined from other parameters via the tadpole conditions, and we have obtained a partly different favored region from the case 1. In both cases, we have shown that in the NMSSM+ν R model there is a parameter region in which the predicted value of Br(µ → eγ) is so large that the µ → eγ decay can be observable at the near-future experiment even if the SUSY mass scale is about 4 TeV.