Muon \(g2\) in SplitFamily SUSY in light of LHC run II
Abstract
The SplitFamily supersymmetry is a model in which the sfermion masses of the first two generations are in \({\mathcal {O}}(100{}{1000})\,\hbox {GeV}\) while that of the third one is in \({\mathcal {O}}(10)\) TeV. With such a hierarchical spectrum, the deviation of the muon \(g2\) and the observed Higgs boson mass are explained simultaneously. In this paper, we revisit the SplitFamily SUSY model in light of the updated LHC constraints. We also study the flavor changing neutral current problems in the model. As we will show, the problems do not lead to stringent constraints when the CabibboKobayashiMaskawa matrix is the only source of the flavor mixing. We also study how large flavor mixing in the supersymmetry breaking parameters is allowed.
1 Introduction
The standard model (SM) of particle physics is now complete by the discovery of the Higgs boson with a mass around \(125\,\hbox {GeV}\) [1, 2, 3, 4, 5]. In the minimal supersymmetric SM (MSSM) (see [6] and references therein), the measured Higgs boson mass can be explained when the masses of the superpartners of the top quark (the stop) are in \({\mathcal {O}}(10{}100)\) TeV [7, 8, 9, 10, 11]. No evidence of the super particles at the LHC experiments also suggests that their masses are in a multiTeV range .
In [23], the SplitFamily supersymmetry (SUSY) model has been proposed to explain the observed Higgs boson and the muon \(g2\) deviation simultaneously. There, the sfermions of the first two generations are in \({\mathcal {O}}(100{}1000)\,\hbox {GeV}\) while that of the third one is in \({\mathcal {O}}(10)\) TeV.^{2} Such a hierarchical SUSY spectrum is motivated by the Yukawa hierarchy.
In this paper, we revisit the SplitFamily SUSY model in light of the updated LHC constraints.^{3} As we will see, almost the entire region which explains the muon \(g2\) within \(2\sigma \) is excluded for the universal gaugino mass. We also show that the collider constraints can be evaded for the nonuniversal gauino masses while explaining the muon \(g2\).
We also study the FCNC problems in the SplitFamily SUSY model. As we will explain, the precise construction of the model requires careful treatment of the family basis, which generically leads to sizable SUSY contributions to the flavor changing neutral currents (FCNC). To see such effects, we first discuss the case where the Cabibbo–Kobayashi–Maskawa (CKM) matrix is the only source of the flavor mixing. As we will see, the FCNC constraints are not so stringent in that case. We also discuss how large flavor mixing in the supersymmetry breaking parameters are allowed.
The organization of this paper is as follows. In Sect. 2, we review the SplitFamily SUSY model. In Sect. 3, we update the constraints from the collider experiments. In Sect. 4, we discuss the FCNC constraints on the model. In Sect. 5, we discuss the successful bottomtau unification as a bonus feature of the model. The final section is devoted to our conclusion.
2 SplitFamily SUSY model
The precise construction of the SplitFamily model requires careful treatment of the family basis. If the soft masses are universal, for example, the general Yukawa couplings in the superpotential do not lead to the SUSY FCNC contributions, since the soft breaking parameters are proportional to the unit matrix in any family basis. On the other hand, for the nonuniversal soft masses, the general Yukawa couplings result in the nonzero SUSY contributions to the flavor mixing. Since we have assumed the rather light sfermions, those mixing leads to unacceptably large FCNC processes.

The CKM matrix is the only source for the flavor mixing (the minimal mixing scenario)

Small flavor mixing comes from the supersymmetry breaking parameters (the small mixing scenario)
2.1 Minimal mixing scenario
2.2 Small mixing scenario
3 Phenomenology of SplitFamily SUSY model
In this section, we update the favored region for the muon \(g2\) and the Higgs boson mass in Ref. [23] in light of the current LHC data. The SUSY contributions to the FCNC are discussed in the next section.
3.1 Parameter choice at input scale
3.2 SUSY contributions To Muon \(g2\)
Here, we show the parameter space to explain the muon \(g2\). In the MSSM, the relevant one loop contributions to the muon \(g2\) come from the diagrams with the smuon/neutralino or the muontype sneutrino/chargino loops [51]. In our scenario, the oneloop binosmuon diagram dominates the SUSY contributions to the muon \(g2\), the SUSY contribution is proportional to the \(\mu \) parameter. The Higgsino contributions are, on the other hand, suppressed due to their heaviness.
In our analysis, we use the package SPheno4.0.3^{9} to calculate the low energy spectrum from the input parameter at the GUT scale. In the code, the renormalization group equations are solved including the flavor mixing parameters, which are relevant for analyses in the next section. We also use the package FeynHiggs2.14.3 [52, 53, 54, 55, 56] to compute the muon \(g2\) and the Higgs mass from the low energy spectrum obtained by SPheno.^{10}
3.2.1 Universal Gaugino mass at the GUT scale
3.2.2 Nonuniversal gaugino masses at the GUT scale
For the nonuniversal gaugino masses, we also study the parameter space explaining the muon \(g2\) on the \(m_0{}M_2\) plane. In Fig. 2, we plot the orange (yellow) shaded regions predicting the observed muon \(g2\) within \(1\sigma ~(2\sigma )\). The gray shaded regions are excluded by the tachyonic masses of the sleptons. Here, the ratio of the gaugino masses are fixed to be \(M_1=1.725\times M_2\) (\(M_1=1.74\times M_2\)) and \(M_3=2.6\,\mathrm{TeV}\) (\(2.5\,\mathrm{TeV}\)) for \(m_3=11\) TeV (10 TeV) at the GUT scale, respectively. The motivations of these choices will be explained in Sect. 3.3.2. Again, the SM Higgs boson mass is consistent with the observed Higgs boson mass in the favored parameter space. As in the case of the universal gaugino mass, the muon \(g2\) is explained by slightly larger \(m_0\) and \(M_2\) for the larger \(m_3\) when we compare both the figures.
3.3 Collider constraints
A sample point in the universal gaugino mass case. \(m_{\mathrm{gluino}}\), \(m_{{\tilde{Q}}}\), \(m_{{\tilde{e}}_L}~(m_{\tilde{\mu }_L})\), \(m_{{\tilde{e}}_R}~(m_{\tilde{\mu }_R})\), \(m_{\tilde{\chi }^1_0}\), \(m_{\tilde{\chi }^2_0}\), \(m_{\tilde{\chi }_1^\pm }\) denote the masses of the gluino, the lightest squark, the lightest almost left handed selectron (smuon), the lightest almost righthanded selectron (smuon), the lightest neutralino, the next to the lightest neutralino, the lightest chargino, respectively. It should be noted that Rparity violation is required for the decay of the LSP before the BigBang Nucleosynthesis (BBN). Then, we also need to introduce a dark matter candidate in the model (see the next section)
\(m_0,~m_3\)  \(0\,\mathrm{GeV},~12\,\mathrm{TeV}\) 
\(M_{1/2}\)  \(1650\,\mathrm{GeV}\) 
\(\mathrm{tan}\beta \)  50 
\(m_{\mathrm{Higgs}}\)  \(124.6\,\mathrm{GeV}\) 
\(a_{\mu }\)  \(1.48\times 10^{9}\) 
LSP  Charged slepton 
\({\varDelta } M_K\)  \(3.0\times 10^{21}\,\mathrm{GeV}\) 
\(\epsilon _K\)  \(2.6\times 10^{7}\) 
\( {\varDelta } M_D\)  \(1.5\times 10^{17}\,\mathrm{GeV}\) 
\(\mu \)  \(9751~\mathrm{GeV}\) 
\(m_{\mathrm{gluino}}\)  \(3619\,\mathrm{GeV}\) 
\(m_{{\tilde{Q}}}\)  \(2730 \,\mathrm{GeV}\) 
\(m_{{\tilde{e}}_L}(m_{\tilde{\mu }_L})\)  \(486\,\mathrm{GeV}\) 
\(m_{{\tilde{e}}_R}(m_{\tilde{\mu }_R})\)  \(942\,\mathrm{GeV}\) 
\(m_{\chi _0^1}\)  \(766\,\mathrm{GeV}\) 
\(m_{\chi _0^2}\)  \(1360\,\mathrm{GeV}\) 
\(m_{\chi _1^\pm }\)  \(1360\,\mathrm{GeV}\) 
3.3.1 Universal gaugino mass at the GUT scale
The bino is the lightest SUSY particle (LSP) on the right sides of the black dashed lines in Fig. 3. In this case, the parameter space with the gluino and the squarks lighter than about 2.6 TeV is excluded by the search for multijets plus missing transverse momentum at ATLAS 13 TeV using 36 \(\hbox {fb}^{1}\) [60]. We show the excluded regions as the blue shaded ones in Fig. 3, which correspond to the \(95\%\) CL limits.
A charged slepton is the LSP on the left side of the black dashed line in Fig. 3. Here, we assume that the LSP has a short lifetime by Rparity violation so that the scenario is consistent with cosmology. However, the size of the Rparity violation is limited from above not to wash out the baryon asymmetry made by baryogenesis (such as thermal leptogenesis). As a result, the charged slepton LSP is expected to be stable inside the detectors (See e.g. [61]). The heavy stable charged particles searches in [62] put upper limits on the production cross section of the SUSY particles, which is converted to the constraints on the mass parameters by using the crosssection given in [63].
In Fig. 3, the blue shaded regions on the left of the black dashed lines are excluded by the constraints on the heavy stable charged particle (95% CL).^{12} In the figure, almost entire region favored by the muon \(g2\) is excluded except for a tiny region near \((m_0,\,M_{1/2})=(0\,\mathrm{GeV},\,1.7\,\mathrm{TeV})\) for \(m_3=12\,\mathrm{TeV}\) in the case of the universal gaugino mass.^{13} In Table 1, we show a sample spectrum which evades the constraints while explains the muon \(g2\) within \(2\sigma \) in the charged slepton LSP scenario.
In the analysis for the charged slepton LSP, the electroweakino productions are the dominant SUSY production modes, where we use the production cross section given in [64, 65]. For comparison, we also show the constraints assuming the SUSY production cross section for the degenerate squarks and gluino (the lighter blue shaded regions). In the actual spectrum, the gluino is heavier than the lightest squark in most of the favored parameter space, and hence, the production cross section of the colored SUSY particles [63] is smaller than the degenerated case. As the figure shows, the constraints via the colored SUSY particle production are at most comparable or weaker than those from the electroweakino productions for the heavy stable charged particle searches.^{14}
3.3.2 Nonuniversal gaugino masses at the GUT scale
In the case of the nonuniversal gaugino masses, the constraints from the collider searches can be weakened by several reasons. For the bino LSP cases, for example, the production cross section of the colored SUSY particles which are relevant for multijets plus missing transverse energy search is reduced if the gluino and squarks are heavy. The constraints from the heavy stable charged particle searches can be also evaded since the sneutrino can be lighter than the charged sleptons. In the following, we again take \(M_3=2.6\,\mathrm{TeV}\) or \(2.5\,\mathrm{TeV}\) at the GUT scale, which suppresses the colored SUSY particle production cross section. The choice of the sign of \(M_3\) will be relevant in the discussion in Sect. 5. We also take \(M_1=1.725\times M_2\) or \(M_1=1.74\times M_2\) with which the sneurinos are lighter than the charged sleptons in the favored parameter space.
On the right side of the black dashed line in Fig. 4, the neutralino is the LSP. As we have mentioned, there is no stringent collider constraint because the squarks and the gluinos become heavy by a rather large \(M_3\). It should be also noted that the constraints on the missing transverse momentum from the electroweakino production are far less relevant due to the rather degenerate electroweakino spectrum, \((m_{\chi _1^\pm }m_{\chi _0^1})/m_{\chi _1^\pm }\lesssim 30\% \) [66, 67].
Eventually, the searches for the missing transverse energy with the charged leptons put the most stringent constraints on the neutralino LSP region. In the figure, the blue shaded regions in Fig. 4 are excluded by the constraint in [68] (95% CL).^{15} In Table 2, we show a sample spectrum which evades the constraints while explains the muon \(g2\) within \(1\sigma \) in the neutralino LSP scenario.
3.4 Cosmology
As we have seen above, most parameter region favored by the muon \(g2\) has been excluded for the universal gaugino mass by the LHC searches. In this subsection, we discuss cosmology focusing on the nonuniversal gaugino masses.
A sample parameter point in the case of the nonuniversal gaugino masses. The neutralino is the LSP. \(\Omega h^2\) and \(\sigma ^{\mathrm{SI}}\) denote the current thermal relic abundance of the bino and the spinindependent binonucleon crosssection
\(m_0,~m_3\)  \(650\,\mathrm{GeV},~11\,\mathrm{TeV}\) 
\(M_{1},~M_{2},~M_3\)  \(1.725\times 480\,\mathrm{GeV},~ 480\,\mathrm{GeV},~2.6\,\mathrm{TeV}\) 
\(\mathrm{tan}\beta \)  40 
\(m_{\mathrm{Higgs}}\)  \(124.6\,\mathrm{GeV}\) 
\(a_{\mu }\)  \(2.69\times 10^{9}\) 
LSP  bino 
\(\Omega h^2\)  0.119 
\(\sigma ^{\mathrm{SI}}\)  \(2.4\times 10^{14}\,\mathrm{pb}\) 
\({\varDelta } M_K\)  \(2.0\times 10^{21}\,\mathrm{GeV}\) 
\(\epsilon _K\)  \(1.2\times 10^{7}\) 
\({\varDelta } M_D\)  \(1.5\times 10^{17}\,\mathrm{GeV}\) 
\(\mu \)  \(9108\,\mathrm{GeV}\) 
\(m_{\mathrm{gluino}}\)  \(5529\,\mathrm{GeV}\) 
\(m_{{\tilde{Q}}}\)  \(4442\,\mathrm{GeV}\) 
\(m_{{\tilde{e}}_L}(m_{\tilde{\mu }_L})\)  \(481\,\mathrm{GeV}\) 
\(m_{{\tilde{e}}_R}(m_{\tilde{\mu }_R})\)  \(635\,\mathrm{GeV}\) 
\(m_{\chi _0^1}\)  \(404\,\mathrm{GeV}\) 
\(m_{\chi _0^2}\)  \(478\,\mathrm{GeV}\) 
\(m_{\chi _1^\pm }\)  \(478\,\mathrm{GeV}\) 
For the sneutrino LSP, on the other hand, the relic abundance is much smaller than the observed dark matter density (\(\Omega h^2 =10^{2}\)) due to its large annihilation cross section in the parameter region favored by the muon \(g2\). Even with such a small relic abundance, however, the sneutrino LSP contribution to dark matter has been excluded by the direct detection experiments since it has a large scattering cross section with the nucleons, \(\sigma _{\mathrm{SI}} \simeq {\mathcal {O}}(10^{5})\,\mathrm{pb}\).
As a result, we find that tiny Rparity violation is required in the case of the sneutrino LSP as in the case of the charged slepton LSP (see the previous discussion in Sect. 3.3.1). In those cases, we need dark matter candidate other than the LSP.
4 FCNC constraints in SplitFamily SUSY model
In the SplitFamily model, there is a nontrivial enhancement of the FCNC by the nonuniversality of the sfermion masses. In this section, we investigate the FCNC constraints on the model for the minimal and small mixing scenarios defined in Sect. 2. In the minimal mixing scenario, the CKM matrix is the only source of the flavor mixing. We show that the FCNC constraints on the minimal scenario are not stringent. For the small mixing scenario, we demonstrate how large flavor mixing in the supersymmetry breaking parameters is allowed.
4.1 Experimental FCNC limits
Let us first summarize the FCNC constraints relevant to the mixing parameters.
4.1.1 Meson mixing
4.1.2 Lepton flavor violation
4.2 FCNC Constraints in the Minimal Mixing Scenario
Here, we investigate the FCNC constraints in the minimal mixing scenario. We search for the parameter space which is the same as the previous section in Figs. 1 and 2. In this scenario, the CKM matrix appearing in the Yukawa couplings in Eq. (6) leads to the flavor mixing in the squark mass matrices at the weak scale due to the nonuniversality of the split family structure at the GUT scale.
Such flavor mixing is constrained by \(\epsilon _K\), where the CP violation comes solely from the CP phase of the CKM matrix. In Fig. 3, the red hatched regions are excluded. There, the region with the small \(m_0\) and \(M_{1/2}\) is excluded due to the light gluino and squarks. Notice that the SUSY FCNC contribution is larger for a larger \(m_3^2/m_0^2\), and thus the constraint is severe for the left figure.^{19} In Fig. 4, on the other hand, no constraint appears from \(\epsilon _K\). This is because the SUSY contributions are suppressed due to the heavy gluino and squarks for \(M_3\simeq 2.52.6\) TeV.
As a result, the SplitFamily SUSY model is not stringently constrained by the flavor violation in the minimal mixing scenario. We have also confirmed that \({\varDelta } M_{K_{\mathrm{SUSY}}}\) and \({\varDelta } M_{D_{\mathrm{SUSY}}}\) become much smaller than the observed one (See e.g. Tables 1 and 2).^{20}
Several comments are in order. As we have mentioned earlier, we may consider another simple family basis in Eq. (7). Since the flavor mixing is dominated by the renormalization group effects, we obtained similar constraint even in this case for the large \(\mathrm{tan}\beta \).
One may also wonder how large SUSY FCNC contributions are expected if the first and the second generation soft SUSY breaking masses are not degenerate. We discuss this possibility in appendix B. There, the model is severely constrained by the FCNC even for the minimal mixing scenario.
4.3 FCNC constraints in small mixing scenario
4.3.1 FCNC constraints on slepton flavor mixing
In Fig. 5, we show how large slepton mixing is tolerable from the current and future constraints on \({\mathcal {B}}(\mu ^+\rightarrow e^+\gamma )\). There, we consider the input parameters in Tables 1 and 2 for examples. The blue curve line denotes the model predictions for \({\mathcal {B}}(\mu ^+\rightarrow e^+\gamma )\). The horizontal (dashed) lines are the current (future expected [85, 86]) experimental bounds. From the figures, the constraint from \({\mathcal {B}}(\mu ^+\rightarrow e^+\gamma )\) requires \(\epsilon \lesssim 0.06\).^{21}
Before closing this section, let us comment on the effect of the PMNS matrix. As in the case of the squark mixing, the PMNS matrix could lead to large flavor mixing in the slepton mass matrix at the weak scale even if we assume a diagonal soft masses (see Eq. (2)). However, the FCNC constraints due to the PMNS matrix are not stringent if the neutrino Yukawa matrix is small enough as long as the righthanded neutrino mass \(M_R\lesssim 10^{10}\,\hbox {GeV}\) (See Appendix C for more details).
4.3.2 FCNC constraints on squark flavor mixing
In the small mixing scenario, we can also introduce \(V_{\mathrm{mix}}\) to the squarks with the mixing angles given by Eq. (23). In this case, however, the FCNC constraints on the mixing angle \(\epsilon \) are much weaker than the slepton case, and hence, we do not discuss them any further.
5 Bottomtau unification
In the SplitFamily SUSY model, the muon \(g2\) can be explained by the small \(M_1,~M_2,~m_0^2\), and the large \(\mathrm{tan}\beta \). In addition, the stringent limits from the LHC experiments require the rather large \(M_3\). Interestingly, these parameter sets are found to be appropriate to realize the bottomtau unification [87].
6 Conclusion
In this paper, we have revisited the SplitFamily SUSY model. In the model, the sfermion masses of the first two generations are in the hundreds GeV range, while that of the third generation is in the tens TeV range. With this spectrum, the deviation of the muon \(g2\) and the observed Higgs boson mass are explained simultaneously.
In Sect. 3, we have first shown the parameter space to explain the muon \(g2\) and the Higgs boson mass. In our analysis, we have searched for two cases of the universal gaugino mass and the nonuniversal gaugino masses. For the universal gaugino mass, almost the entire region to explain the muon \(g2\) within \(2\sigma \) is excluded by the collider searches as shown in Fig. 3. This is due to the lightness of the squarks, the gluino, and the wino masses. For the nonuniversal gaugino masses, the gluino can be heavier with which the collider constraints can be easily evaded (See Fig. 4).^{25} We have also found the parameter space where the bino LSP can explain the observed dark matter density thanks to the coannihilation with the wino.
In Sect. 4, we have studied the FCNC problems in the SplitFamily SUSY model. We have searched for two scenarios, i.e. the minimal mixing scenario and the small mixing scenario (see Sect. 4 for details). In the minimal scenario, we have assumed that the CKM matrix is the only source of the flavor mixing. There, we have shown that the SUSY FCNC contributions are small enough to evade the problem.
For the small mixing scenario, we have assumed the CKM like mixing matrix to the soft mass parameters (See Eq. (22) and around it). Then, we have demonstrated how large flavor mixing is allowed in the slepton sector. There, the most stringent constraint comes from the lepton flavor violation decay of \(\mu ^+ \rightarrow e^++\gamma \). We have shown that the mixing angles have to be relatively small, \(\epsilon \lesssim 0.06\).
In Sect. 5, we have discussed one bonus feature of the model, the bottomtau unification. For the successful bottomtau unification, the large threshold correction is required. Interestingly, such parameter space is compatible with the one favored by the muon \(g2\). In Fig. 6, we have shown that the bottomtau unification is significantly improved for the large \(\mathrm{tan}\beta \).
Several comments are in order. First, it is possible to achieve a small \(\mu \) term in the SplitFamily SUSY model. In fact, for \(m_{H_{u,d}}^2=m_3^2\), the focus point mechanism [95, 96] results in a small \(\mu \) term. In such a case, the neutralino LSP can have a sizable Higgsino contribution, so that the dark matternucleon cross section becomes large.
Throughout this paper, we have assumed that the SUSY breaking parameters do not have any CPviolating phases. In fact, these are strong assumptions and it is highly nontrivial to achieve such soft SUSY breaking parameters from high energy theory. The CP violating phases in the SUSY breaking parameters are constrained by the measurements of the EDMs, which will be discussed elsewhere.
Footnotes
 1.
 2.
 3.
 4.
We also assume that the hierarchy of the soft masses and the Yukawa couplings are aligned.
 5.
Even if we include the PMNS effect, the lepton flavor mixing is not so large for the model with the degenerate righthanded masses \(M_R\lesssim 10^{10}\,\hbox {GeV}\) (See the Appendix C for more details). Thus, we ignore the finite neutrino masses in the superpotential.
 6.That is, we may take the basis,while the soft masses are given by Eq. (2).$$\begin{aligned} W=(V_{\mathrm{CKM}}^T{\hat{f}}_U)^{ij} Q_i {{\bar{U}}}_j H_u+({\hat{f}}_D)^{ii} Q_i{{\bar{D}}}_i H_d, \end{aligned}$$(7)
 7.
These assumptions are motivated by the SU(5) GUT.
 8.
When the required value of \(\mu \) is as small as \(m_0\), the Higgsino also contributes to the muon \(g2\) [23].
 9.
We slightly modify SPheno4.0.3 to calculate the SUSY spectrum on the basis in Eq. (6).
 10.
We added the two loop corrections for the large \(\mathrm{tan}\beta \) given in [57] for the SUSY contributions to the muon \(g2\). Although we use FeynHiggs to calculate the muon \(g2\), the following results are not changed even if we use SPheno.
 11.
 12.
There are two kinds of blue regions (the blue and the lighter blue regions). The blue plus light blue shaded regions are excluded here. We will explain this difference soon.
 13.
The muon \(g2\) can be explained for a larger \(M_{1/2}\) for a larger \(\mathrm{tan}\beta \). In this case, however, the CPodd neutral Higgs scalar becomes tachyonic.
 14.
For the searches of multijets plus missing transverse energy which is relevant for the bino search, the colored SUSY production plays the dominant roles.
 15.
The constraint from the latest result in [69] is not stringent, where it is assumed that the slepton decays into a lepton and a neutralino with a 100% branching ratio. In our scenario, a slepton also decays into a chargino and a neutrino. Furthermore, a charged lepton from a chargino decay becomes soft. Thus, the constraint is weakened.
 16.
In some of the neutralino LSP region, the wino is the LSP. There, we have confirmed that the constraints from the direct detection are negligible due to the small thermal relic abundance of the wino.
 17.
 18.
We do not use the theoretical prediction on \(\epsilon _K\) from the lattice QCD [81].
 19.
In our analysis, we fully diagonalize the squark masses without using the mass insertion technique (See Appendix A for more details).
 20.
In the minimal mixing scenario, we have also confirmed that the neutron electric dipole moment (EDM) from the CKM phase does not lead to the stringent constraint. See also Ref. [84] for the EDM constraints.
 21.
We checked that the constraints are not significantly changed even if we put additional CP phases in \(V_{\mathrm{mix}}\) of the sleptons. Detail analysis will be done elsewhere.
 22.
In our notation, \(y_b\) is real positive (see Eq. (6)).
 23.
The threshold correction from the gluino is sensitive to the matching scale, which requires careful treatments of the decoupling and the renormalization group effects [92].
 24.
On the left side of the dotted line, the sneutrino is the LSP as discussed in Sect. 3.
 25.
 26.
Notes
Acknowledgements
This work is supported by JSPS KAKENHI Grant Numbers JP26104001 (T.T.Y), JP26104009 (T.T.Y), JP16H02176 (T.T.Y), JP17H02878 (M. I. and T. T. Y.), No. 15H05889 and No. 16H03991 (M. I.), JP15H05889 (N.Y.), JP15K21733 (N.Y.), JP17H05396 (N.Y.), JP17H02875 (N.Y.), and by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan (T.T.Y.). The work of M. S. is supported in part by a Research Fellowship for Young Scientists from the Japan Society for the Promotion of Science (JSPS).
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