Lepton flavor violations in SUSY models for muon $g-2$ with right-handed neutrinos

We consider supersymmetric (SUSY) models for the muon $g-2$ anomaly without flavor violating masses at the tree-level. The models can avoid LHC constraints and the vacuum stability constraint in the stau-Higgs potential. Although large flavor violating processes are not induced within the framework of minimal SUSY standard model, once we adopt a seesaw model, sizable lepton flavor violating (LFV) processes such as $\mu \to e \gamma$ and $\mu \to e$ conversion are induced. These LFV processes will be observed at future experiments such as MEG-II, COMET and Mu2e if right-handed neutrinos are heavier than $10^9$ GeV motivated by the successful leptogenesis. This conclusion is somewhat model independent since Higgs doublets are required to have large soft SUSY breaking masses, leading to flavor violations in a slepton sector via neutrino Yukawa interactions.


Introduction
Higgs couples to chiral multiplets of LH leptons and RH neutrinos through flavor violating neutrino Yukawa couplings. Consequently, lepton flavor violating (LFV) processes such as µ → eγ and µ → e conversion become non-negligible and detectable at future LFV experiments when the RH neutrinos are heavier than O(10 9 ) GeV. 4 In fact, the RH neutrinos heavier than O(10 9 ) GeV are motivated by the successful thermal leptogenesis [43,44] for the baryon asymmetry of the universe.

SUSY models for muon g − 2
We introduce the three different models for the muon g −2 without flavor violating masses at the tree-level. All of the models include the direct couplings between the Higgs fields and a SUSY breaking field Z, which are needed to avoid the vacuum stability constraint in the stau-Higgs potential.

CP-safe gaugino mediation model (model A)
We first consider a gaugino mediation model given in Ref. [23], which respects the shift symmetry of the SUSY breaking field Z: Z → Z + iR with R being a real constant. With the shift symmetry, dangerous CP violating phases are suppressed [45]. We refer to this model as the model A. The Kähler potential is given by where x = Z + Z † , f (x) is an arbitrary function of x and Here, Φ I is a matter multiplet; the matter multiplets include three generations of leptons, quarks and RH neutrinos; g u (x) and g d (x) are arbitrary functions of x; we have omitted gauge interactions and taken M P = 1, where M P is the reduced Planck mass. We assume the above Kähler potential is defined at the GUT scale. With the Kähler potential, all the sfermions are massless at the tree-level, which is a very important assumption to solve the SUSY flavor problem. The sfermions masses are dominantly generated from gaugino masses through radiative corrections (gaugino mediation). The superpotential is where and Here, we have taken M N , Y d and Y e to be diagonal by the field redefinitions ofN i , Q i ,D i , L i andĒ i without loss of generality. 5 The Yukawa coupling, Y u , is given by with V CKM being the CKM matrix. The mass parameter C is a constant term. We take µ and C to be real by U (1) R rotation and field redefinitions of H u and H d . The cosmological constant vanishes under the following condition: The SUSY is broken at the minimum of vanishing cosmological constant [46] and F -term of Z is given by where m 3/2 = e K /2 C is a gravitino mass and Note that F Z is a real number since n and m 3/2 are both real. The canonically normalized kinetic terms for φ I , H u and H d are obtained by the following field redefinitions: Accordingly, the parameters in the superpotential are rescaled as [47] µ → e − K /2 1 + g u 1 − f /3 The soft SUSY breaking masses for H u and H d are where with c u , c d , d u and d d being real numbers. A-terms and the Higgs B-term are where we have no CP violating phase. The gaugino masses are generated by the coupling between Z and field strength superfields in a way consistent with a grand unified theory. Here, we consider SU (5) × SU (3) H × U (1) H product group unification [48,49], which solves the doublet triplet splitting problem in a simple way. The relevant couplings are where g 5 , g 3H and g 1H are gauge couplings of SU (5) where M 1 , M 2 and M 3 are the bino, wino and gluino masses, respectively; N is a real constant depending on the U (1) H charge of the GUT breaking Higgs field; we have rescaled the gauge couplings as g −2 a → g −2 a + 2k a Z . We note that the gaugino masses are non-universal at the GUT scale, which is important to explain the muon g − 2 while avoiding LHC constraints on colored SUSY particles [50,51].
Apart from the RH neutrino masses, the free parameters in this model are m 2 Hu , m 2 H d , A u , B µ , M 1 , M 2 and M 3 , or more conveniently, we choose where m A is a CP-odd Higgs mass. Here, µ, m A and tan β are defined at the EWSB scale while M 1 , M 2 and M 3 are given at the GUT scale, M GUT .
In order to avoid the vacuum stability constraint in the stau-Higgs potential, we consider a small µ case. Then, the chargino contribution to the muon g − 2 is dominant. The small µ is achieved by taking m 2 Hu at the GUT scale to be large and positive. As we will see, this m 2 Hu induces flavor violating slepton masses through the Yukawa interactions in Eq. (4).

Higgs mediation model with bino-wino coannihilation (model B)
Next, we consider a Higgs mediation model presented in Ref. [24], focusing on the bino-wino coannihilation region [52], where the bino and wino masses are quasi degenerated at the EWSB scale. We refer to this model as the model B. In this model, the Higgs soft masses are assumed to be tachyonic and large as O(10) TeV. These Higgs soft masses lead to natural spitting of sfermion masses through radiative corrections [13,14]: third generation sfermions become much heavier than first/second generation sfermions without inducing too large flavor violating masses [17]. Then, the vacuum stability constraint in the stau-Higgs potential is easily avoided due to the heavy staus as discussed in Sec. 3. The Kähler potential is given by where f (Z, Z † ) is a function of Z and Z † and Here, c h is assumed to be positive and κ d = 1. The concrete models justifying these assumptions are given in Ref. [26]. In this model, we do not consider the shift symmetry of Z. However, to construct a model with the shift symmetry is not difficult. The superpotential is given by By assuming Z 0, the SUSY breaking F -term is obtained as where | F Z | 2 = 3m 2 3/2 . Here, we take a canonically normalized kinetic term for Z. From Eqs. (19) and (20), we obtain The gaugino masses are generated from the couplings between Z and field strength superfields in Eq. (15), and they are non-universal at the GUT scale. This allows us to explain the correct relic abundance of dark matter through the bino-wino coannihilation, avoiding experimental constraints. The free parameters in this model are which are given at M GUT . In the following analysis, we take µ > 0.

Higgs mediation model with bino-slepton coannihilation (model C)
Lastly, we consider the Higgs mediation model focusing on the bino-slepton coannihilation region [53], where the masses of the bino, selectron and smuon are quasi degenerated at the EWSB scale. We refer to this model as the model C although the Lagrangian is completely same as that of the model B. The only differences are as follows: the wino mass is larger and the Higgs soft masses are smaller compared to the model B. The free parameters in this model are same as those in the model B. We also take µ > 0.
3 Lepton flavor violations and muon g − 2 In this section, we calculate the LFV processes in the model A, B and C, focusing on parameter regions consistent with the muon g − 2 experiment. The experimental value of the muon g − 2 [1] is deviated from a SM prediction (see [5] and references therein) with a significance of 3.7 σ level: This deviation is explained only when the smuon(s) and electroweak gauginos are light as O(100) GeV together with a large tan β of O (10). In this case, the vacuum stability constraint in the stau-Higgs potential becomes important: if µ tan β is too large, the EWSB minimum decays into the charge breaking minimum with a life time shorter than the age of the universe. The constraint is shown in Ref. [21] as where ∆ τ is a radiative correction to the tau Yukawa coupling [54], and mL 3 (mẼ 3 ) is the mass of LH (RH) stau. The normalization factor η τ ≈ 1 slightly depends on tan β [20]. Clearly, this constraint is avoided when µ is small or staus are much heavier than the smuons, which requires large soft SUSY breaking masses for the Higgs doublets.
Let us firstly consider the small µ case, where the chargino diagram dominantly contributes to the muon g − 2. The µ parameter is determined by the EWSB condition, which is given by where . . . denotes higher order terms of 1/ tan n β (n ≥ 4); m Hu (M GUT ) and m H d (M GUT ) are soft SUSY breaking masses for H u and H d , respectively; ∆m 2 Hu and ∆m 2 H d are radiative corrections. To explain the Higgs boson mass of 125 GeV, we need a large stop mass mt or a large trilinear coupling A t [55][56][57][58][59]. In this case, ∆m 2 Hu ∼ (m 2 t or A 2 t ), is inevitably large. Therefore, the small µ is only achieved with m 2 Hu (M GUT ) ∼ (m 2 t or A 2 t ). Numerically, we find m 2 Hu (M GUT ) ∼ 4 TeV 2 to be consistent with µ ∼ 100 GeV.
For the large µ case, where the neutralino diagram dominantly contributes to the muon g − 2, we need the large stau masses to avoid the constraint in Eq. (25). One possibility is to make the staus heavy by hand as explored in Refs. [60][61][62]. However, in this case, the constraint from µ → eγ is too severe unless we assume a special structure of the lepton Yukawa couplings [61]. Alternatively, we can make the staus heavy using the Higgs-loop effects [13,14], without inducing LFV in the framework of MSSM. Here, the Higgs soft masses are assumed to be large as m H u,d = O(10) TeV and tachyonic. Then, the staus become heavy as ∼ 10 TeV by a Higgs loop at the one-loop level due to the large tau-Yukawa coupling while the selectrons and smuons remain light as O(100) GeV. 6 The generated stau masses are estimated as and where Hu is used and M SUSY is a SUSY particle mass scale. We note that the condition, is not too large: if the D-term contribution is too large, the smuon and slectron become tachyonic. See appendix A for renormalization group equations for the slepton masses.
We have shown that, in order to avoid the vacuum stability constraint in Eq. (25), |m 2 Hu (M GUT )| needs to be large. This feature is somewhat model independent. Then, off-diagonal elements of the slepton mass matrix are induced through the neutrino Yukawa interactions in Eq. (4), which are estimated as [40] (see also appendix A) As for the diagonal element, (m 2 L ) 33 is given by the sum of Eq. (27) and (29) with i = j = 3. 7 The neutrino Yukawa coupling Y ν is parameterized as [63] where R is a complex orthogonal matrix, α 1 and α 2 are Majorana phases and V PMNS is the PMNS matrix. In the following numerical calculation, we take R to be a real orthogonal matrix, α 1 = α 2 = 0 for simplicity. The neutrino mass differences, the mixing angles and the Dirac phase are taken from PDG [64]. The neutrino parameter dependence of LFV appears in the In figure 1, we check the m ν 1 dependence of the LFV coupling parameter, i=1 m ν i < 0.12 eV [64]. We observe that the larger m ν 1 induces the larger LFV. Hereafter we take m ν 1 = 0 for a conservative estimate.   In the dark (light) green regions, the muon g − 2 is explained at 1σ level (2σ level). The purple shaded regions are excluded due to too large Br(µ → eγ). The stau becomes the LSP in the gray shaded region.
Once we obtain the non-negligible off-diagonal elements of the slepton mass matrix, the LFV processes, such as µ → eγ and µ → e conversion, are induced [65]. In what follows, we estimate the sizes of the LFV processes for each models, and discuss impacts on future experiments. We employ SuSpect 2.43 [66] to calculate the spectrum for SUSY particles. Combining the output of SuSpect 2.43 and the general formulae given in Ref. [65], we estimate the sizes of 6 The selectron and smuon masses are dominantly generated by two-loop diagrams involving the Higgs doublets and one-loop diagrams involving gauginos. 7 The equation (29) is also valid for i = j. However, it is subdominant.  Let us first focus on model A, where the vacuum stability constraint is avoided with the small µ. Figure 2 shows the sizes of the muon g−2 and LFV processes for M N 1 = M N 2 = M N 3 (= M N ). In this case, the LFV processes become independent of R (Y † ν Y ν is independent of R). As for   In the gray shaded region, there should be severe constraints from LHC because the stau becomes the lightest SUSY particle (LSP) and long-lived [80,81]. The muon g − 2 is explained at 1σ level (2σ level) in the dark (light) green region. We see that the model A can explain muon g − 2 if we take M 2 , µ O(100 GeV). On the other hand, our model simultaneously predicts sizable LFV processes as we discussed above. The current limit on the LFV processes is shown by the purple shaded region, which is given by MEG experiment [82]. We observe that the MEG experiment excludes the parameter region for muon g − 2 when M N = 10 11 GeV. Future sensitivities on the relevant LFV processes, on the other hand, are shown by the colored lines. The region below these lines can be tested by future LFV experiments. The purple line corresponds to the future sensitivity of µ → eγ, Br(µ → eγ) ≈ 5 × 10 −14 , at MEG-II [83]. The red and dashed red lines  are the future sensitivities of µ − e conversion in Al at COMET Phase-I [84] and COMET phase II [85], which correspond to R Al (µ → e) ≈ 7 × 10 −15 and ≈ 3 × 10 −17 respectively. Mu2e [86] gives similar sensitivity as COMET phase II. The blue line is the future sensitivity of µ → e conversion in Ti, R Ti (µ → e) ≈ 2 × 10 −19 , at PRISM/PRIME [87]. We find that the future LFV experiments can investigate the parameter region for muon g − 2 if the RH neutrinos are heavier than 10 8 GeV.
In table 1, we show the typical mass spectrum in this model. Here we fix the RH neutrino masses as M N 1 = M N 2 = M N 3 = 10 9 GeV. We note, however, that the SUSY mass spectrum is almost insensitive to the masses of the RH neutrinos. It should be reminded that, if we relax the degeneracy of the RH neutrino masses, the size of LFV depends on the structure of a matrix R which cannot be determined by observables. Let us estimate the R dependence of LFV by taking R randomly. The results are given by figures 3, 4 and 5 which show the size of Br(µ → eγ), R Ti (µ → e) and R Al (µ → e) as a function of the mass of the lightest RH neutrino, respectively. We consider the two cases; i) the case where the RH neutrinos are almost degenerate, namely M N 1 : M N 2 : M N 3 = 1 : 2 : 3 and ii) the mass spectrum for the RH neutrinos is hierarchical, namely M N 1 : M N 2 : M N 3 = 1 : 10 : 100. In both cases, we fix M 2 = 250 GeV and µ = 260 GeV and the other parameters are same as figure 2. We see that the degeneracy of the RH neutrino masses reduces the dependence of R. In these figures, we also show the current limits and the future sensitivities on the LFV. The purple and dashed purple lines in figure 3 are the current limit by MEG [82] and the future sensitivity at MEG-II [83]. The red and dashed red lines in figure 4 show the future sensitivities of R Al (µ → e) which are same as figure 2. The blue and dashed blue lines in figure 3 are the current upper limit by SINDRUM II, R Ti (µ → e) ≈ 4.4 × 10 −12 , [88] and the future sensitivity at MEG-II [83].
We next consider the model B. In this model, µ tan β is large but the vacuum stability constraint is avoided thanks to the heavy staus. Figure 6 shows the contours of the muon g − 2 and LFV in the (m 2 Hu , M 2 ) plane. We also show the contours of the mass of the lighter selectron and smuon, mẽ and mμ, as the black and black-dotted lines. Here we take M 3 = −4 TeV, tan β = 40, m 2 H d = m 2 Hu , M N 1 = M N 2 = M N 3 , and M 1 is fixed as we obtain the correct DM relic density. We need to avoid the gray shaded region because slepton becomes LSP in the region. The color notation for the muon g − 2 and LFV is same with figure 2. The blue shaded region in the bottom-right figure is the current exclusion limit given by SINDRUM II, R Ti (µ → e) ≈ 4.3 × 10 −12 , [88]. We see that the model B can explain muon g − 2, and the favorable parameter region can be tested by future LFV experiments if the RH neutrino are heavier than 10 7 GeV. The typical mass spectrum in this model is summarized in the left- Finally, we discuss the model C. In this model, the vacuum stability constraint is also avoided thanks to the heavy staus. In figure 10, we take the same parameters as figure 6, but we focus on the different (m 2 Hu , M 2 ) region where the correct DM relic density can be realized by bino-slepton coannihilation process. The gray shaded region should be avoided because selectron becomes lighter than 100 GeV in the lower gray shaded region, while the DM abundance becomes larger than the observed value in the upper gray shaded region. We observe that the model C can explain muon g − 2 keeping the consistency with the current LFV measurements, and the future LFV experiments can test the parameter region where M N 1 > 10 8 GeV. We show the typical mass spectrum of this model in the right-handed side of the table 2. Figures 11, 12, and 13 show the R dependence of the LFV in the model C. We here take m 2 Hu = m 2 H d = −10 8 GeV 2 and M 2 = 2 TeV. The other parameters are taken as same as figure 10.

Conclusion
In this paper, we have shown that, in SUSY models explaining the muon g − 2 anomaly, µ → eγ and µ → e conversion are very likely to be observed at the future experiments if the RH neutrinos are heavier than 10 9 GeV, motivated by the successful thermal leptogenesis. We have observed BR(µ → eγ) 10 −14 − 10 −12 , R Al (µ → e) 10 −17 − 10 −14 , and R Ti (µ → e) 10 −17 − 10 −14 for the case with the RH neutrinos heavier than 10 9 GeV. The LFVs originate from the slepton mass mixing, which is induced by the neutrino Yukawa interactions together with the large soft SUSY breaking mass for the up-type Higgs. We have confirmed that the degeneracy of the RH neutrino masses reduces the uncertainty in the relationship between the neutrino Yukawa couplings and the RH and the SM neutrino masses. Since the large soft SUSY breaking masses for the Higgs doublets seem to be inevitable to avoid the vacuum stability constraint in the stau-Higgs potential, this conclusion is somewhat model independent provided that the scale of SUSY breaking mediation is high enough.

Acknowledgments
This work is supported by JSPS KAKENHI Grant Numbers JP16H06492 (N. Y.), JP16H06490, JP18H05542 and JP19K14701 (R. N.). The work of R. N. was supported by the University of Padua through the "New Theoretical Tools to Look at the Invisible Universe" project and by Istituto Nazionale di Fisica Nucleare (INFN) through the "Theoretical Astroparticle Physics" (TAsP) project.

A Beta-functions for the slepton masses
Here, we show one-loop beta-functions for the slepton masses.
where t = ln Q r with Q r being the renormalization scale. Here, Y e , Y ν , m 2 L , m 2Ẽ , m 2Q , m 2 U and m 2D are 3 × 3 matrices.