Probing SUSY with 10 TeV stop mass in rare decays and CP violation of kaon

...................................................................................................................WeprobeSUSYatthe10TeVscaleintheraredecaysandCPviolationofkaon.Wefocusontheprocessesof K L → π 0 ν ¯ ν and K + → π + ν ¯ ν , combined with the CP-violating parameters (cid:4) K and (cid:4) (cid:4) K /(cid:4) K . The Z-penguin mediated by the chargino loop cannot enhance K L → π 0 ν ¯ ν and K + → π + ν ¯ ν because the left–right mixing of the stop is constrained by the 125 GeV Higgs mass. On the other hand, the Z-penguin mediated by the gluino loop can enhance the branching ratios of both K L → π 0 ν ¯ ν and K + → π + ν ¯ ν . The former increases up to more than 1.0 × 10 − 10 , which is much larger than the SM prediction even if the constraint of (cid:4) K is imposed. It is remarkable that the Z-penguin mediated by the gluino loop can simultaneously enhance (cid:4) (cid:4) K /(cid:4) K and the branching ratio of K L → π 0 ν ¯ ν , which increases up to 1.0 × 10 − 10 . We also study the decay rates of K L → μ + μ − , B 0 → μ + μ − , and B s → μ + μ − , which correlate with the K L → π 0 ν ¯ ν decay through the Z-penguin. It is important to examine the B 0 → μ + μ − process since we expect enough sensitivity of this decay mode to SUSY at LHCb.

In particular, the K L → π 0 νν process is the CP-violating one and provides direct measurement of the CP-violating phase in the CKM matrix. On the other hand, the indirect CP-violating parameter PTEP 2016, 123B02 M. Tanimoto and K. Yamamoto those numerical results should be revised with those of this paper since the relevant constraints are not imposed enough there. In this paper, we also reexamine them comprehensively by taking account of the gluino contribution, as well as the chargino one, with a large left-right mixing angle of squarks.
Our paper is organized as follows. In Sect. 2, we discuss the formulation of the rare decays, K L → π 0 νν, K + → π + νν, K L → μ + μ − , B 0 → μ + μ − , and B s → μ + μ − , and CP violations of K and K / K . Section 3 gives our setup of SUSY with 10 TeV squark masses. In Sect. 4, we present our numerical results. Section 5 is devoted to discussions and a summary. The relevant formulae are presented in Appendices A, B, and C.
2. Observables 2.1. K L → π 0 νν and K + → π + νν Let us begin by discussing the kaon rare decays K L → π 0 νν and K + → π + νν, which are dominated by the Z-penguin process in the SM. In the estimation of the branching ratios of K → πνν, the hadronic matrix elements can be extracted with the isospin symmetry relation [47,48]. These processes are theoretically clean because the long-distance contributions are small [14], and then the theoretical uncertainty is estimated below several percent.Accurate measurements of these decay processes provide crucial tests of the SM. In particular, the K L → π 0 νν process is a purely CP-violating one, which can reveal the source of the CP-violating phase. The basic formulae are presented in Appendix C.1. The SM predictions have been discussed Refs. [4,49,50]. They are given as 1 BR(K L → π 0 νν) SM = (3. 36  ( On the experimental side, the upper bound of the branching ratio of K L → π 0 νν is given by the KEK E391a experiment [18], and the branching ratio of K + → π + νν was measured by the BNL E787 and E949 experiments as [19]: BR(K L → π 0 νν) exp < 2.6 × 10 −8 (90% C.L.), BR(K + → π + νν) exp = (1.73 +1. 15 −1.05 ) × 10 −10 .
At present, the J-PARC KOTO experiment is an in-flight measurement of K L → π 0 νν approaching the SM-predicted precision [20,21], while the CERN NA62 experiment [22] is expected for the precise measurement of the K + → π + νν decay. The SUSY contribution has been studied in many works [11,[51][52][53][54][55]. The sizable enhancement of these kaon decays was expected through large left-right mixing of the chargino interaction in s Lti χ − and d Lti χ − at the SUSY scale of O(1) TeV [51,54]. We find that even at the O(10) TeV scale, these decays are enhanced through the Z-penguin mediated by the gluino with large left-right mixing.

K
Let us discuss another CP-violating parameter K , which has been measured precisely. Its hadronic matrix elementB K is reliably determined by the lattice calculations as [56,57] B K = 0.766 ± 0.010. (4) Another theoretical uncertainty in K is also reduced by removing the QCD correction factor of the two charm box diagram [58]. Thus, the accurate estimate of the SM contribution enables us to search for NP such as SUSY. A nonnegligible SUSY contribution has been expected in K even at a scale of O(100) TeV [36][37][38]. Consequently, K gives us one of the most important constraints to predict the SUSY contribution in the K → πνν decays. In our calculation of K , we investigate the SUSY contributions for the box diagram, which is correlated with the K L → π 0 νν process directly.

K / K
The direct CP violation K / K is also important to constrain the NP. The basic formula for K / K is given as [12,23,59] where with Functions B, C, D, and E denote the loop functions including SM and SUSY effects, which come from boxes with external dd(B (d) ), uū(B (u) ), Z-penguin (C), photon-penguin (D), and gluon-penguins (E). The coefficients P i are given by with the nonperturbative parameters B (1/2) 6 and B (3/2) 8 defined as The numerical values of r (0,8,6) i are presented in Ref. [23].
The most important parameters to predict K / K are the non-perturbative parameters B (1/2) 6 and B (3/2) 8 . Recently, the RBC-UKQCD lattice collaboration [60,61] gave which predict ( K / K ) SM = (1.9 ± 4.5) × 10 −4 in the SM [23]. This SM prediction is much smaller than the experimental result [62]  with 3σ in our calculation. The dominant contribution to the Z-penguin C comes from a chargino-mediated one and a gluinomediated one if the large left-right mixing of squarks is allowed. On the other hand, the effects of neutralinos are suppressed [51][52][53]. The chargino-mediated Z-penguin C(χ ± ) and the gluinomediated Z-penguin C(g) are given as where c 2 w = cos 2 θ W and s 2 w = sin 2 θ W , with the Weinberg angle θ W , and the Z-penguin amplitudes P sd ZL(R) (χ ± ) and P sd ZL(R) (g) are given in Eqs. (B1) and (B4) in Appendix B. The box diagram effect is suppressed compared with the penguin diagram if the SUSY-breaking scale M S satisfies M S m W [11]. Thus, the dominant SUSY contribution to K / K is given by the Z-penguin mediated by the chargino and gluino. Therefore, we should consider the correlation between K / K and the branching ratio of K L → π 0 νν.
Let us write K / K as where the second and third terms denote the Z-penguin induced by the left-handed and right-handed interactions of SUSY, respectively. The contributions are written as follows [24] In order to see the correlation between K / K and the K L → π 0 νν decay, it is helpful to write down the K L → π 0 νν amplitude induced by the chargino-and gluino-mediated Z-penguin in terms of sd L(R) (Z) as as seen in Appendix C.1. The Z-penguin amplitude mediated by the chargino dominates the left-handed coupling of the Z boson. Therefore, the chargino contribution to K / K is opposite to K L → π 0 νν. If the Z-penguin mediated by the chargino enhances K / K , the K L → π 0 νν decay is suppressed considerably. On the other hand, the Z-penguin amplitude mediated by the gluino gives equal left-handed and righthanded Z couplings. Then, the right-handed Z coupling of the Z-penguin amplitude is a factor of 5/21  3.3 larger than the left-handed one. Therefore, we can obtain the SUSY contribution, which can enhance, simultaneously, K / K and the branching ratio for K L → π 0 νν. Actually, by choosing Im sd L (Z) > 0 and Im sd R (Z) < 0, the region can enhance both K / K and the branching ratio for K L → π 0 νν. We discuss this case in our numerical results.

K L
The Z-penguin also contributes K L → μ + μ − , B 0 → μ + μ − , and B s → μ + μ − decays. These decay amplitudes are governed by the axial semileptonic operator O 10 , which is created by the Z-penguin top loop and the W box diagram in the SM. Those general formulae are presented in Appendix C.2. The CMS and LHCb Collaborations have observed the branching ratio for B s → μ + μ − , and B 0 → μ + μ − has also been measured [26]: The SM predictions have been given as [63] BR On the other hand, the long-distance effect is expected to be large in the K L → μ + μ − process [64]. Therefore, it may be difficult to extract the effect of the Z-penguin process. The SM prediction of the short-distance contribution was given as [24] BR( The experimental data for from which the constraint on the short-distance contribution has been estimated as [64] BR Thus, the SUSY contribution through the Z-penguin is expected to be correlated among the rare decays of K L → π 0 νν,

SUSY flavor mixing
Recent LHC results for the SUSY search may suggest high-scale SUSY, O(10-1000) TeV [36][37][38] since the lower bounds of the gluino mass and squark masses are close to 2 TeV. Taking account of these recent results, we consider the possibility of high-scale SUSY at 10 TeV, in which the K → πνν decays and K / K with the constraint of K are discussed.
We also consider the split-family model, which has a specific spectrum of the SUSY particles [39,40]. This model is motivated by the Nambu-Goldstone hypothesis for quarks and leptons in the 6 [41]. Therefore, the third family of squarks/sleptons is heavy, e.g., O(10) TeV, while the first and second family of squarks/sleptons have relatively low masses of O(1) TeV. Close to the experimental lower bound, the masses of bino and wino are assumed to be small, less than 1 TeV. The model was first discussed in the B s -B s mixing [39]. It successfully explained both the 125 GeV Higgs mass and the muon g − 2 simultaneously [40]. The stop mass with O(10) TeV pushes up the Higgs mass to 125 GeV. The deviation of the muon g − 2 is explained by the sleptons of the first and second families with mass less than 1 TeV. Since the squark masses of the first and second families are also relatively low, as well as the sleptons, we expect the SUSY contribution in the kaon system to become large.
The new flavor mixing and CP violation effect are induced through the quark-squark-gaugino and the lepton-slepton-gaugino couplings. The 6 × 6 squark mass matrix M 2 q in the super-CKM basis is diagonalized to the mass eigenstate basis in terms of the rotation matrix (q) as where (q) is a 6 × 6 unitary matrix, and it is decomposed into 3 × 6 matrices as (q) = (  13 , where q = u, d for squark mixing. In addition, we also introduce left-right (LR) mixing angles θ t,b LR . In practice, we take s qL,qR 12 = 0, which is motivated by the almost-degenerate squark masses of the first and the second families to protect the large contribution to the K 0 −K 0 mass difference M K . It is also known that the single mixing effect of s qL, qR 12 to K → πνν is minor [51]. Actually, we have checked numerically that the contribution of s qL,qR 12 = 0 ∼ 0.3 is negligibly small. There also appear the phases φ qL ij and φ qR ij associated with the mixing angles, which bring new sources of CP violations. In our work, we treat those mixing parameters and phases as free parameters in the framework of the non-MFV scenario.
Since the Z-penguin processes give the dominant contribution for K → πνν and K / K , we calculate the Z-penguin mediated by the chargino and gluino. The interaction is presented in Appendix B. The relevant parameters are presented in the following section.

Numerical analysis 4.1. Setup of parameters
Let us discuss the decay rates of K L → π 0 νν and K + → π + νν processes by choosing a sample of the mass spectrum in the high-scale SUSY model at O(10) TeV. The enhancements of these kaon rare decays require large left-right mixing with large squark-flavor mixing. In order to show our results clearly, we take a simple setup for the high-scale SUSY model as follows: • We fix the gluino, wino, and bino masses M i (i = 3, 2, 1) with μ and tan β as for high-scale SUSY. • We take the masses of stopt 1 ,t 2 , and sbottomb 1 ,b 2 as a sample set On the other hand, we take the masses of the first and second family up-type and down-type squarks around 15 TeV within 5-15% as relevant. This mass spectrum of the first and second where θ t LR is estimated by input of the stop masses in Eq. (26) with the large A term, which is constrained by the 125 GeV Higgs mass due to the large radiative correction [35]. On the other hand, there is no strong constraint for the left-right mixing of the down-squarks from the B meson experiments in the region of O(10) TeV. 2 Therefore, we take rather large values to see the enhancement of the K L → π 0 νν decay.
where the upper bound 0.3 is given by the experimental constraint of the K 0 −K 0 mass difference M K .As discussed in the previous section, we ignore mixing between the first and second family of squarks, s qL 12 , and then can avoid the large contribution from s qL 12 to M K . This single mixing effect of s qL 12 to the Z-penguin mediated by the chargino is known to be minor compared with the double mixing effect [51,54]. Namely, the SUSY contributions of the K L → π 0 νν and K + → π + νν processes are dominated by the double mixing of the stop and sbottom. are also free parameters. We scan them in −π ∼ π randomly.
• We neglect the minor contribution from the sleptons and sneutrinos. We also neglect the charged Higgs contribution, which is tiny due to the CKM mixing. We use the CKM elements |V cb |, |V ub |, |V td | in Ref. [50] with 3σ error bars, which are obtained in the framework of the SM. If there is a large SUSY contribution to the kaon and the B meson systems, the values of the CKM elements may be changed. Actually, the SUSY contribution is comparable to the SM one for K in our following numerical analyses, although very small for CP violations and the mass differences of the B mesons at the O(10) TeV scale of squarks [38]. We use the CKM element in the study of the unitarity triangle, including the data of CP asymmetries and the mass differences of B mesons without inputting K (strategy S1 in Ref. [50]).

Results in SUSY at 10 TeV
Let us discuss the case of high-scale SUSY, where all squarks/sleptons are at the 10 TeV scale.
First, we discuss the contribution of the Z-penguin induced by the chargino to the K L → π 0 νν and K + → π + νν processes. In this case, the left-right mixing of the up-squark sector controls the magnitude of the Z-penguin amplitude. Since the A term is considerably constrained by the  125 GeV Higgs mass, the left-right mixing angle cannot be large in our mass spectrum, at most θ t LR = 0.07 as presented in the above setup. Therefore, we cannot obtain the enhancement of those processes. 3 Actually, the predicted branching ratios of K L → π 0 νν and K + → π + νν deviate from the prediction of the SM with order 10%. Thus, we conclude that the Z-penguin mediated by the chargino cannot bring a large enhancement for the K L → π 0 νν and K + → π + νν decays due to the constraint of the 125 GeV Higgs mass. This result is consistent with the recent work in Ref. [66], where the metastability of vacuum constrains the left-right mixing for the up-squark sector.
On the other hand, the Z-penguin induced by the gluino could be large due to the large downtype left-right mixing θ b LR = 0.1-0.3. In our setup of parameters, we show the predicted branching ratios, BR(K L → π 0 νν) versus BR(K + → π + νν) in Fig. 1, where the mixings s dL,dR 13 and s dL,dR 23 are scanned in 0-0.3 and the left-right mixing angle θ b LR is fixed at 0.3. Here the Grossman-Nir bound is shown by the slanted green line [67]. In order to see the θ b LR dependence, we also present the BR(K L → π 0 νν) versus BR(K + → π + νν) in Figs. 2 and 3, in which θ b LR is fixed at 0.2 and 0.1 respectively. As seen in Figs. 1-3, the branching ratio of BR(K L → π 0 νν) depends considerably on the left-right mixing angle θ b LR . The enhancement of BR(K L → π 0 νν) requires the left-right mixing angle to be larger than 0.1.
Though the constraint of the experimental value of K is important, it is not imposed in Figs. 1-3. Let us take account of K . The gluino contribution to K depends on the phase differences of φ dL(dR) 13 and φ dL(dR) 23 , which are associated with flavor mixing angles. In order to avoid the large contribution of the relatively light squarks to K , the phases φ dL,dR 13 − φ dL,dR 23 should be tuned near n × π/2 (n = −2, −1, 0, 1, 2). 4 For the phase cycle in the branching ratio, BR(K L → π 0 νν) is half of the one in K . Therefore, the enhancement of BR(K L → π 0 νν) is realized at φ dL 13 − φ d 23 π/2 and φ dR 13 −φ dR 23 −π/2, where K is sufficiently suppressed. At φ dL 13 −φ dL 23 −π/2 and φ dR 13 −φ dR 23 π/2, the SUSY contribution to the K L → π 0 νν process is opposite to the SM one, and then the branching ratio is suppressed compared with the SM prediction.  The predicted region for BR(K L → π 0 νν) versus BR(K + → π + νν), without imposing K , where θ b LR = 0.2. Notation is the same as in Fig. 1.   Fig. 3. The predicted region for BR(K L → π 0 νν) versus BR(K + → π + νν), without imposing K , where θ b LR = 0.1. Notation is the same as in Fig. 1.   Fig. 4. The predicted region for BR(K L → π 0 νν) versus BR(K + → π + νν), with imposing K , where θ b LR = 0.3. Notation is the same as in Fig. 1.
We show the predicted region for BR(K L → π 0 νν) versus BR(K + → π + νν), imposing K where θ b LR = 0.3 is fixed in Fig. 4. There are two directions in the predicted plane of BR(K L → π 0 νν) versus BR(K + → π + νν). The direction of the enhancement of BR(K L → π 0 νν) corresponds to φ dL 13 − φ dL 23 −π/2 and φ dR 13 − φ dR 23 π/2, and the enhancement of BR(K + → π + νν) to φ dL,dR 13 − φ dL,dR 23 0, π. As a result, it is found that BR(K L → π 0 νν) can be enhanced up to 4 × 10 −10 , which is much larger than the SM enhancement, with the K constraint satisfied. 10  We comment on the constraint from the K 0 −K 0 mass difference M K . Our SUSY contribution of M K (SUSY) is comparable with the SM contribution M K (SM). It is possible to fit the following condition, keeping the enhancement of BR(K L → π 0 νν): which is the criterion of the allowed NP contribution in Ref. [68]. We also estimate the SUSY contributions to M B 0 and M B s , which are at most 10% of the SM. Let us discuss the correlation between BR(K L → π 0 νν) and K / K . As discussed in Sect. 2.3, both processes come from the imaginary part of the same Z-penguin, and can be enhanced simultaneously once the condition of Eq. (18) is imposed. In Fig. 5, we show the correlation between BR(K L → π 0 νν) and K / K , where Zsd coupling satisfies the condition of Eq. (18). The constraint from K is also imposed. It is remarkable that the Z-penguin mediated by the gluino enhances K / K and the branching ratio for K L → π 0 νν simultaneously. While the estimated K / K fits the observed value, the branching ratio of K L → π 0 νν increases up to 1.0 × 10 −10 . In this region, the phase of Im sd L and Im sd R becomes opposite, so the enhanced region of BR(K L → π 0 νν) is somewhat reduced by the cancelation between the left-handed coupling of Z and the right-handed one partially, compared with the result in Fig. 4.
The real parts of sd L and sd R are sufficiently small since φ dL,dR 13 − φ dL,dR 23 ±π/2 is taken. Therefore, the SUSY contribution does not spoil the agreement between the real part of the K → ππ amplitude in the SM and the experimental data.
In Fig. 6, we show the correlation between BR(K L → π 0 νν) and BR(K + → π + νν). In the parameter region where BR(K L → π 0 νν) and K / K are enhanced, the branching ratio of K + → π + νν does not deviate from the SM. It is understandable because φ dL,dR 13 − φ dL,dR 23 ±π/2 is taken in order to enhance BR(K L → π 0 νν) with the K constraint. On the other hand, BR(K + → π + νν) is dominated by the considerably sizable real part of the SM. The addition of the imaginary part of the SUSY contribution does not change the SM prediction significantly.
The Z-penguin process also contributes to another kaon rare decay K L → μ + μ − , and the B meson rare decays, B 0 → μ + μ − and B s → μ + μ − . Therefore, we expect them to correlate with the K → πνν decays. In the K L → μ + μ − process, the long-distance effect is estimated to be large in Ref. [64]. Therefore, we discuss only the short-distance effect, which is dominated by the Zpenguin. We show BR(K L → π 0 νν) versus BR(K L → μ + μ − ) in Fig. 7, where the constraint from 11 Fig. 6. The predicted region for BR(K L → π 0 νν) versus BR(K + → π + νν), where the Zsd coupling satisfies the condition of Eq. (18). Notation is the same as in Fig. 1.   Fig. 7. The predicted BR(K L → π 0 νν) versus BR(K L → μ + μ − ). The pink area indicates the SM with 3σ . The solid red line denotes the bound for the short-distance contribution.
K is imposed. It is noticed that the predicted value almost satisfies the bound for the short-distance contribution in Eq. (23), presented as the red line.
The clear correlation between two branching ratios is understandable because BR(K L → μ + μ − ) is sensitive only to the real part of Z-couplings. When the enhancement of BR(K L → π 0 νν) is found in the future, BR(K L → μ + μ − ) will remain less than 10 −9 . On the other hand, when BR(K L → μ + μ − ) is larger than 10 −9 , there is no enhancement of BR(K L → π 0 νν). This relation is testable in future experiments.
We also show BR(K L → π 0 νν) versus BR(B 0 → μ + μ − ) in Fig. 8. We can expect the enhancement of BR(B 0 → μ + μ − ) in our setup even if BR(K L → π 0 νν) is comparable to the SM one. Since LHCb will observe the BR(B 0 → μ + μ − ) [69], this result is the attractive one in our model.
On the other hand, we do not see the correlation between BR(K L → π 0 νν) and BR(B s → μ + μ − ) since the SM component of BR(B s → μ + μ − ) is relatively large compared with B 0 → μ + μ − . The enhancement of the K L → π 0 νν decay rate is still consistent with the present experimental data of BR(B s → μ + μ − ).

Results in the split-family model with 10 TeV stop and sbottom
Let us discuss the case of the split-family SUSY model with 10 TeV stop and sbottom, where first and second family squark masses are around 2 TeV. The constraint of K is seriously tight for CP-violating phases associated with squark mixing in the split-family SUSY model. Moreover, the | F| = 2 processes receive overly large contributions from the the first and second squarks because In addition, the large left-right mixing generates large contributions to the b → sγ decay, therefore, the left-right mixing angle is severely constrained by the experimental data for b → sγ . Therefore, it is impossible to realize the enhancement of BR(K L → π 0 νν) in the split-family model satisfying constraints | F| = 1, 2 transitions in the kaon and the B meson systems.

EDMs of neutron and mercury
Finally, we add a comment on the electric dipole moments (EDMs) of the neutron and mercury (Hg), d n and d Hg , which arise through the chromo-EDM of the quarks, d C q due to gluino-squark mixing [70][71][72][73][74][75]. If both left-handed and right-handed mixing angles are taken to be large, such as s dL 13 = s dR 13 0.3 or s dL 23 = s dR 23 0.3, with large left-right mixing, d n and d Hg are predicted to be one and two orders larger than the experimental upper bound [62], respectively, |d n | < 0.29×10 −25 e · cm and |d Hg | < 3.1 × 10 −29 e · cm.
However, there still remains the freedom of phase parameters. For example, by tuning φ dL i3 and φ dR i3 (i = 1, 2) under the constraint from K , we can suppress the EDMs sufficiently. This tuning does not spoil our numerical results above.

Summary and discussions
In order to probe SUSY at the 10 TeV scale, we have studied the processes of K L → π 0 νν and K + → π + νν combined with the CP-violating parameters K and K / K . The Z-penguin mediated by the chargino loop cannot enhance K L → π 0 νν and K + → π + νν because the left-right mixing of the stop is constrained by the 125 GeV Higgs mass. On the other hand, the Z-penguin mediated by the gluino loop can enhance the branching ratios of both K L → π 0 νν and K + → π + νν, where the former increases more than 1.0 × 10 −10 , much larger than the SM prediction even if the constraint of K is imposed. Thus, the K L → π 0 νν and K + → π + νν decays provide us with very important information to probe the SUSY. It is remarkable that the Z-penguin mediated by the gluino loop can simultaneously enhance K / K and the branching ratio for K L → π 0 νν. While the estimated K / K fits the observed value, the branching ratio of K L → π 0 νν increases up to 1.0 × 10 −10 . We have also studied the decay rates of K L → μ + μ − , B 0 → μ + μ − , and B s → μ + μ − , which correlate with the K L → π 0 νν decay through the Z-penguin. In particular, it is important to examine the B 0 → μ + μ − decay carefully since we can expect enough sensitivity of the SUSY in this decay mode at LHCb .
We have also discussed them in the split-family model of SUSY, where the third family of squarks/sleptons is heavy, of O(10) TeV, while the first and second families of squarks/sleptons and the gauginos have relatively low masses of O(1) TeV. The constraint of K is much seriously tight for CP-violating phases associated with the squark mixing in the split-family SUSY model. Moreover, the | F| = 2 processes receive overly large contributions from the first and second family squarks because they are relatively light, at O(1) TeV. Therefore, it is impossible to realize the enhancement of BR(K L → π 0 νν) in the split-family model.
LR is given as The left-right mixing angles θ q LR are given approximately as

Appendix B. Chargino-and gluino-interactions-induced Z-penguin
The Z-penguin amplitude mediated by the chargino, P sd ZL (χ ± ) in our basis [78] is given as where and with q =s, d, I = 1-6 for up-squarks, and α = 1, 2 for charginos. Here, (U ± ) α i denotes the mixing parameters between the wino and the higgsino.
The right-handed Z-penguin P sd ZR (χ ± ) is also given simply by replacements between L and R [78]. The Z-penguin amplitude mediated by the gluino, P sd ZL (g) [78] is written as The right-handed Z-penguin P sd ZR (g) is also given simply by replacements between L and R.
Appendix C. Basic formulae C.1. K + → π + νν and K L → π 0 νν The effective Hamiltonian for K → πνν in the SM is given as [3] H SM eff = which is induced by the box and the Z-penguin mediated by the W boson. The loop function X c denotes the charm-quark contribution of the Z-penguin, and X t is the sum of the top-quark exchanges of the box diagram and the Z-penguin in Eq. (C1). Let us define the function F as The branching ratio of K + → π + νν is given in terms of F. Taking the ratio of it to the branching ratio of K + → π 0 e + ν, which is the tree level transition, we obtain a simple form: Here the hadronic matrix element has been removed by using the fact that the hadronic matrix element of K + → π 0 e + ν, which is well measured as BR(K + → π 0 e + ν) exp = (5.07 ± 0.04) × 10 −2 [62], is related to that of K + → π + νν with isospin symmetry: Finally, the branching ratio for K + → π + νν is expressed as where r K + is the isospin breaking correction between K + → π + νν and K + → π 0 e + ν [47,48], and the factor 3 comes from the sum of three neutrino flavors. It is noticed that the branching ratio for K + → π + νν depends on both the real and imaginary parts of F.
For the K L → π 0 νν decay, the K 0 -K 0 mixing should be taken into account, and one obtains with We neglect CP violation in K 0 -K 0 mixing,¯ , due to its smallness, |¯ | ∼ 10 −3 . Taking the ratio between the branching ratios of K + → π 0 e + ν and K L → π 0 νν, we have the simple form (C10) Therefore, the branching ratio of K L → π 0 νν is given as where r K L denotes the isospin breaking effect [47,48]. It is remarked that the branching ratio of K L → π 0 νν depends on the imaginary part of F. The effective Hamiltonian in Eq. (C1) is modified due to new box diagrams and penguin diagrams induced by SUSY particles. Then, the effective Lagrangian is given as where i and j are the indices of the flavor of the neutrino final state. Here, C ij VLL,VRL is the sum of the box and Z-penguin contributions: where the weak neutral-current coupling Q (ν) ZL = 1/2, and B sdij VL(R)L and P sd ZL(R) denote the box contribution and the Z-penguin contribution, respectively, and V, L, and R denote the vector, lefthanded, and right-handed couplings, respectively. In addition to the W boson contribution, there are the gluino-(g), the chargino-(χ ± ), and the neutralino-(χ 0 ) mediated contributions.
The branching ratios of K + → π + νν and K L → π 0 νν are obtained by replacing internal effect F in Eqs. (C6) and (C11) to C ij VLL + C ij VRL : BR(K L → π 0 νν) = κ · r K L r K + The Z-penguin process appears in B s → μ + μ − and B 0 → μ + μ − decays. We show the branching ratio for B s → μ + μ − , which includes the Z-penguin amplitude [78]: On the other hand, the SM component of the Z-penguin amplitude is where B 0 (x t ) and C 0 (x t ) are well-known loop functions depending on x t = m 2 t /m 2 W . We have neglected other amplitudes such as the Higgs-mediated scalar amplitude since we focus on NP in the Z-penguin process.
The branching ratio of B 0 → μ + μ − is given by a similar expression. For the K L → μ + μ − decay, its branching ratio is given as [79] BR(K L → μ + μ − ) SD = κ μ Re λ t λ 5 Y (x t ) + where λ is the Wolfenstein parameter, λ i = V * is V id , and the charm-quark contribution P c is calculated in NNLO as P c = 0.115 ± 0.018, and Y is the same as in Eq. (7). We use its SM value as Y (x t ) = 0.950 ± 0.049 (x t ≡ m 2 t /M 2 W ).