KL → π 0 νν̄ decay correlating with K in high-scale SUSY

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . We have studied the contribution of high-scale SUSY to the KL → π0νν̄ and K + → π+νν̄ processes by correlating with the CP-violating parameter K . Taking account of the recent LHC results for Higgs discovery and SUSY searches, we consider high-scale SUSY at the 10–50 TeV scale in the framework of non-minimal squark (slepton) flavor mixing. The Z penguin mediated chargino dominates the SUSY contribution for these decays. At the 10 TeV SUSY scale, the chargino contribution can enhance the branching ratio of KL → π0νν̄ by eight times compared with SM predictions, whereas the predicted branching ratio BR ( K + → π+νν̄) increases by up to three times that of the SM. The gluino box diagram dominates the SUSY contribution of K up to 30%. If down-squark mixing is neglected compared with up-squark mixing, the Z penguin mediated chargino dominates both SUSY contributions of BR ( KL → π0νν̄ ) and K . Then, a correlation between them is found, but the chargino contribution to K is at most 3%. Even if the SUSY scale is 50 TeV, the chargino process still enhances the branching ratio of KL → π0νν̄ from the SM prediction by a factor of two, and K is deviated from the SM prediction byO(10%). We also discuss the chargino contribution to the KL → π0e+e− process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

PTEP 2015, 053B07 M. Tanimoto and K. Yamamoto In the estimation of the branching ratio of K → πνν, the hadronic matrix elements can be extracted with the isospin symmetry relation [20,21]. These processes are theoretically clean because the longdistance contributions are small [12], and then the theoretical uncertainty is estimated to be below several percent. On the other hand, K has a different flavor-mixing structure from these processes since it is induced by the box diagram of K 0 -K 0 mixing. Therefore, the NP is expected to appear in both K → πνν and K with different magnitudes.
On the experimental side, the upper bound of the branching ratio of K L → π 0 νν is given by the KEK E391a experiment [22]. The branching ratio of K + → π + νν measured by the BNL E787 and E949 experiments is consistent with the SM prediction [23]: At present, the J-PARC KOTO experiment is an in-flight measurement of K L → π 0 νν approaching the SM-predicted precision [24,25], while the CERN NA62 experiment [26] studies the K + → π + νν process.
On the theoretical side, supersymmetry (SUSY) is one of the most attractive candidates for the NP. However, SUSY signals have not been observed yet, and the recent searches for new particles at the LHC have given us important constraints for SUSY. Since the lower bounds of the masses of the SUSY particles increase gradually, the squark and the gluino masses are supposed to be at a higher scale than 1 TeV [27][28][29]. Moreover, the SUSY models have been seriously constrained by the Higgs discovery, in which the Higgs mass is 126 GeV [30,31].
These facts suggest a class of SUSY models with heavy sfermions. If the squark and slepton masses are expected to be O(10-100) TeV, the lightest Higgs mass can be pushed up to 126 GeV, whereas all the SUSY particles will be out of the reach of the LHC experiment. Therefore, the indirect search for the SUSY particles becomes important in low-energy flavor physics [32][33][34].
So far, the effects of SUSY on the K + → π + νν and K L → π 0 νν processes have been intensively studied in the framework of the Minimal Supersymmetric Standard Model (MSSM) with the minimal flavor violation (MFV) scenario [8,10]. Since the SUSY mass scale is pushed up higher than the 1 TeV region at present, the effect of the MSSM with MFV is expected to be very small. These processes are also discussed in the framework of the general SUSY model [9,[35][36][37][38][39][40] at the O(500) GeV scale.
We have studied the SUSY contribution to the CP violation of the B meson and K induced by K 0 -K 0 mixing under the relevant SUSY particle spectrum constrained by the observed Higgs mass [34]. It is found that the SUSY contribution could be up to 40% in the observed K ; on the other hand, it is minor in the CP violation of the B meson at the high scale of 10-50 TeV. Therefore, in this paper, we investigate the high-scale SUSY contribution to K + → π + νν and K L → π 0 νν by correlating with K in the framework of the mass eigenstate of the SUSY particles, which is consistent with the updated experimental situations like the direct SUSY searches and the Higgs discovery, with non-minimal squark (slepton) flavor mixing.
Our paper is organized as follows. Sect. 2 gives the basic framework of K + → π + νν, K L → π 0 νν, and K in the SM and the MSSM. In Sect. 3, we present the setup of the high-scale SUSY. In Sect. 4, we discuss our numerical results. Sect. 5 is devoted to the summary. The SUSY mass spectra and the Z penguin amplitude mediated chargino are given in Appendices A and B, respectively.

Basic framework
In this section, we present the basic formulae for the K → πνν decay and the CP violating parameter K , which correspond to | S| = 1 and | S| = 2 processes, respectively. The K + → π + νν and K L → π 0 νν processes are clean ones theoretically since the hadronic matrix elements can be extracted, including isospin-breaking corrections, by taking the ratio to the leading semileptonic decay of K + → π 0 e + ν. Moreover, the long-distance contributions to these rare decays are negligibly small. Therefore, 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.
On the other hand, the CP-violating parameter K is measured with enough accuracy. The major theoretical ambiguity comes from the hadronic matrix element factorB K . Recent lattice calculations give us a reliable value forB K [41,42]. The more accurate estimate of the SM contribution enables us to search the NP for such a SUSY because we know the accurate observed value of K . Actually, the non-negligible SUSY contribution has been expected in K at the scale of O(100) TeV [32][33][34]. Consequently, it is necessary to examine the high-scale SUSY contribution in K → πνν by correlating with K .
2.1. Basic framework : K + → π + νν and K L → π 0 νν 2.1.1. K + → π + νν and K L → π 0 νν in the SM. Let us start by discussing the framework of the K + → π + νν and K L → π 0 νν processes in the SM [1]. The effective Hamiltonian for K → πνν in the SM is given by: which is induced by the box and the Z penguin mediated W boson. The dominant box contrition is derived by the top-quark exchange; on the other hand, the charm-quark exchange contributes to the Z penguin process as well as the top-quark one. The up-quark contribution is negligible due to its small mass. So, 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. (3). Let us define the function F as follows: The branching ratio of K + → π + νν is given in terms of F. Taking its ratio to the branching ratio of K + → π 0 e +ν , which is the tree-level process, we obtain a simple form: The K + → π 0 e +ν decay is precisely measured as BR K + → π 0 e +ν exp = (5.07 ± 0.04) × 10 −2 [43], and its hadronic matrix element is related to that of K + → π + νν with isospin symmetry: where the coefficients are determined by the Clebsch-Gordan coefficient. By using this relation, the hadronic matrix element has been removed in Eq. (5). Now the branching ratio for K + → π + νν is expressed as follows: where r K + is the isospin-breaking correction between K + → π 0 e +ν and K + → π 0 e +ν [20,21], 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, K 0 -K 0 mixing should be taken account, and one obtains In going from the first line to the second in (10), we use and then, after using the CP transition relation in the second line, we obtain the equation in the third line. In the final line, we neglect the CP violation in K 0 -K 0 mixing,¯ , due to its smallness |¯ | ∼ 10 −3 .
Taking the ratio between the branching ratio of K + → π 0 e +ν and K L → π 0 νν, we have the simple form: Therefore, the branching ratio of K L → π 0 νν is given as follows: where r K L and r K + denote the isospin-breaking effect [20,21]. Note that the branching ratio of K L → π 0 νν depends on the imaginary part of F. Since the charm-quark contribution is negligible due to the small imaginary part of V * cs V cd , it is enough to consider only the top-quark exchange in this decay. In the SM, K + → π + νν and K L → π 0 νν are related to the UT fit. We write down the branching ratio in terms of the Wolfenstein parameters. Since Re F and Im F are given as we can express the branching ratio of these decays as where and Note that BR K + → π + νν in Eq. (17) is approximately a circle centered atρ = ρ 0 1.2,η = 0 on theρ-η plane. On the other hand, BR K L → π 0 νν in Eq. (19) just depends on η and it can determine the height of the UT directly. In this way, the precise measurements of K + → π + νν and K L → π 0 νν become crucial tests for the SM. Before going on to discuss the SUSY formulation, we present the general bound between K + → π + νν and K L → π 0 νν, the so-called Grossman-Nir bound [15]. As seen from the above formulations, since the two processes are determined by the imaginary part and the absolute value of the same coupling, the model-independent bound is obtained as: where we use the isospin symmetry A K + → π + νν = √ 2A K 0 → π 0 νν . This bound must be satisfied for any NP [15,16].

2.1.2.
K + → π + νν and K L → π 0 νν in the MSSM The effective Hamiltonian in Eq. (3) 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 index of the flavor of the neutrino final state. Here, C i j VLL,VRL is the sum of the box contribution and the Z penguin one: 21i j VL(R)L and P 21 ZL(R) denote the box contribution and the Z penguin contribution, respectively. V , L, and R denote the vector coupling, the left-handed one, and the right-handed one, respectively. In addition to the W boson contribution, there are the gluinog, the chargino χ ± , and the neutralino χ 0 mediated ones. 1 We write each contribution as follows: where (i, j) denote the neutrinos of the final state. Explicit expressions are given in Ref. [44]. It is well known that the most dominant contribution comes from the Z penguin mediated chargino for the K + → π + νν and K L → π 0 νν decays [12]. The branching ratio of K + → π + νν and K L → π 0 νν are obtained by replacing the internal effect F in Eqs. (8) and (15)

K in the MSSM
It is well known that the CP-violating parameter K induced by the K 0 −K 0 oscillation gives us one of the most serious constraints on the NP. The general expression for K is given as where A 0 is the 0-isospin amplitude in the K → ππ decay, and M K 12 is the dispersive part of the K 0 -K 0 oscillations, and M K is the mass difference of the neutral K meson. The effects of ξ = 0 and φ < π/4 were estimated by Buras and Guadagnoli [45]. In the SM, the off-diagonal mixing amplitude M K 12 is obtained as The wino-higgsino mixing is tiny in our mass spectrum. where S(x) denotes the SM one-loop functions [46], and η cc,tt,ct are the QCD corrections [45]. Recent lattice calculations give us a precise determination of theB K parameter [41,42]. Taking account of the NP effect, the expression for M K 12 is modified. In the case of the SUSY, new contributions to the box diagrams are given by the gluinog, the charged Higgs H ± , the chargino χ ± , and the neutralino χ 0 exchanges: The explicit formula has been presented in Ref. [44].

Setup of the squark flavor mixing
We present the setup of our calculation in the framework of high-scale SUSY. Recent LHC results for the SUSY search may suggest high-scale SUSY, O(10-1000) TeV [32][33][34]47] since the lower bounds of the gluino mass and squark masses exceed 1 TeV. Taking account of these recent results, we consider the possibility of high-scale SUSY at 10, 50 TeV, in which the K → πνν decays and K are discussed.
Another important experimental result that should be mentioned is the Higgs discovery. The Higgs mass m H 126 GeV gives effect to the SUSY mass spectrum. In general, there are two possibilities for getting Higgs mass value: one is the heavy stop around 10 TeV, and the other is the large X t = A 0 − μ cot β given by the A-term. In the case that the SUSY scale is 10 to 50 TeV, we have already obtained the SUSY mass spectra which realize the Higgs mass at the electroweak scale with renormalization group equation (RGE) running in a previous work [34]. We use this numerical result for the SUSY particle mass spectrum. In this study, the first and second squarks are almost degenerate due to the assumption of universal soft masses. On the other hand, the third squark mass obtains a large contribution from RGE running due to the large Yukawa coupling of the top-quark. Therefore, mixing between the first and second is negligible, and it is taken account in the subsequent discussion for squark flavor mixing. The SUSY spectra at 10 and 50 TeV are given in Appendix A.
Once the SUSY mass spectrum is fixed, we can calculate the left-right mixing angle θ q , which is defined as In the case of the SUSY scale being 10 to 50 TeV, the left-right mixing angles of squarks and sleptons are very small, at θ d ∼ 0.0062, θ u ∼ 0.0024, θ e ∼ 0.014 and θ d ∼ 0.0009, θ u ∼ 0.0007, θ e ∼ 0.005 , respectively. The SUSY brings new flavor mixing through the quark-squark-gaugino and lepton-sleptongaugino couplings. The 6 × 6 squark mass matrix M 2 q in the super-CKM basis turns into the mass eigenstate basis by diagonalizing with rotation matrix (q) as where we use the abbreviations c , c θ q = cos θ q , and s θ q = sin θ q . Note that we take s q L,q R 12 = 0 due to the degenerate squark masses of the first and second families, as noted in Appendix A. The angle θ q is the left-right mixing angle betweenq L andq R , and they are calculable as mentioned above. Then, there are free mixing parameters θ As is well known, the charged Higgs and the chargino contributions dominate the K → πνν processes [12]. Since the SUSY scale is high in our scheme, the charged Higgs are heavy, O(10 TeV), so the charged Higgs contribution is suppressed in our framework. On the other hand, the dominant SUSY contribution to K comes from the gluino box diagram if the flavor mixing angles of the down-squark and the up-squark are comparable. In addition, the chargino box diagram is also non-negligible. Consequently, we will discuss both cases in which the down-squark mixing angles s d L(R) i j are negligibly small and are comparable to the up-squark mixing angles s uL(R) i j . We scan the phases of Eq. (30) for up-squarks, down-squarks, charged sleptons, and sneutrinos in the region of 0 ∼ 2π independently.
In our framework, the K → πνν processes are dominated by the Z penguin mediated chargino exchange, P sd ZL χ ± in Eq. (23), which occur through thet L s L (d L )χ ± andt R s L (d L )χ ± interactions, respectively. In our basis, the relevant mixing is given by where q = s, d, I = 1-6 for up-squarks, and α = 1, 2 for charginos. V CKM is the CKM matrix, and U + is the 2 ×

Numerical analysis
Let us discuss the high-scale SUSY contribution to the K → πνν processes by correlating with K [13]. At present, we cannot confirm whether the SM prediction SM K is in agreement with the experimental value exp K because there remains the theoretical uncertainty with an order of a few tens percent. However, the theoretical uncertainties of K are expected to be reduced significantly in the near future. Actually, the lattice calculations ofB K will be improved significantly [41,42], whereas |V cb | and the CKM phase γ will be measured more precisely in Belle-II. Therefore, we will be able to test the correlation between K → πνν and K .
In our previous work, we examined the sensitivity of the high-scale SUSY with 10 and 50 TeV to K . It was found that the SUSY contribution to K is allowed up to 40%. We begin to discuss the SUSY contribution at the 10 TeV scale. The present uncertainties in the SM prediction for K are due to the CKM elements V cb ,ρ, andη, and theB K parameter. We take the CKM parameters V cb , ρ, andη at the 90% C.L. of the experimental data: For theB K parameter, the recent result of the lattice calculations is given as [41,42]: which is used with the error bar of 90% C.L. in our calculation.
To start, we show the numerical results at the SUSY scale of 10 TeV. Figure 1 shows the predictions on the BR K L → π 0 νν vs BR K + → π + νν plane, where phase parameters are constrained by the observed | K | with the experimental error bar of 90% C.L. Here, we fix the mixing parameters in Eq. (30) by taking the common values s uL i3 = s u R i3 = s u = 0.1 (i = 1, 2) and s d L i3 = s d R i3 = s d = 0.1 (i = 1, 2) for the up-quark and the down-quark sectors, respectively. The Z penguin mediated chargino dominates the SUSY contribution to these branching ratios.
The SUSY contributions can enhance the branching ratio of K L → π 0 νν by eight times compared with the SM predictions in Eq. (1), 1.8 × 10 −10 , although it is much smaller than the Grossman-Nir bound. On the other hand, the predicted BR K + → π + νν increases up to three times, 2.1 × 10 −10 . It is also noticed that the predicted region of BR K L → π 0 νν is reduced to much smaller than 10 −11 due to the cancellation between the SM and SUSY contributions. The BR K + → π + νν could be reduced to 1.3 × 10 −11 .
We discuss the correlation between K and BR K L → π 0 νν in Fig. 2, in which (a) s u = s d = 0.1 and (b) s u = 0.1, s d = 0. The transverse axis denotes the SUSY contribution in | K |. If the down-squark mixing s d is comparable to the up-squark mixing s u , there is no correlation between them, as seen in Fig. 2(a), where the Z penguin mediated chargino dominates the SUSY contribution of K L → π 0 νν, and the gluino box diagram dominates the SUSY contribution of K . A gluino contribution of 30% is possible in K .
On the other hand, if the down-squark mixing s d is tiny compared with the up-squark mixing s u , the Z penguin mediated chargino dominates both SUSY contributions of K L → π 0 νν and K . Then, a correlation is found between them, as seen in Fig. 2(b), where the chargino contribution to K is at most 3%. This correlation is due to the difference of the phase structure between the penguin diagram and the box diagram of the chargino.
In conclusion, K could be deviated from the SM prediction by O(10%) due to the gluino box diagram, whereas the Z penguin mediated chargino could enhance the branching ratio of K L → π 0 νν from the SM prediction. Next, in order to see the mixing angle s u dependence of the branching ratios, we plot the predicted regions on the BR K L → π 0 νν vs s u and BR K + → π + νν vs s u planes, taking s u = 0 ∼ 0.3, in Fig. 3(a) and (b). We scan s d in the region of 0 ∼ 0.3 independent of s u , although the gluino contribution is much suppressed compared with the chargino one. In this plot, the SUSY contribution to K is free (0-40%), but the experimental constraint of | K | with the error bar of 90% C.L. is taken into account. We show the upper bound given by the Grossman-Nir bound together with the experimental upper bound of BR K + → π + νν with 3σ by the black line, at which the predicted BR K L → π 0 νν should be cut. Namely, the observed upper bound of BR K + → π + νν gives the constraint for the predicted BR K L → π 0 νν at s u larger than 0.2. The precise experimental measurement of BR K + → π + νν will lower the predicted upper bound of BR K L → π 0 νν . Let us discuss the case of a SUSY scale of 50 TeV. Figure 4 shows the predictions on the BR K L → π 0 νν and BR K + → π + νν plane at the SUSY scale of 50 TeV, where the mixing angle is fixed at s u = s d = 0.3. Although the predicted region is reduced considerably compared to the case of the 10 TeV scale in Fig. 1, the predicted branching ratio of K L → π 0 νν is enhanced by two times from the SM prediction, and the branching ratio of K + → π + νν could be enhanced from the SM prediction by three times.
To see the correlation between K and the predicted K L → π 0 νν branching ratio, we show the branching ratio of K L → π 0 νν versus the SUSY contribution of K in Fig. 5  where the gluino contribution to K is still possible up to 10%. However, a correlation is found between them as seen in Fig. 5(b), where the Z penguin mediated chargino dominates both SUSY contributions of K L → π 0 νν and K since the down-squark mixing s d vanishes by keeping s u = 0.3. The chargino contribution to K is at most 2%. This correlation is understandable from the difference of the phase structure between the penguin diagram and the box diagram of the chargino.  Thus, even if the SUSY scale is 50 TeV, K could be deviated from the SM prediction by O(10%) due to the gluino box diagram, whereas the chargino process deviates the branching ratio of K L → π 0 νν from the SM prediction by a factor of two. Figure 6 shows the s u dependence of BR K L → π 0 νν and BR K + → π + νν taking s u = 0 ∼ 0.5. We also scan s d in the region of 0 ∼ 0.3 independent of s u . In this plot, the SUSY contribution to K is free (0-40%), but the experimental constraint of K with the error-bar of 90% C.L. is taken into account. The predicted BR K L → π 0 νν could be large, up to 8 × 10 −11 , and BR K + → π + νν is up to 1.5 × 10 −10 . Thus, the enhancement from the SM prediction could be detectable even if the SUSY scale is 50 TeV.
Before closing our numerical study, we would like to discuss correlations to other quantities which are sensitive to the NP. They are the K L → π 0 e + e − process and the neutron electric dipole moment d n . The K L → π 0 e + e − process is induced in a similar way to K L → π 0 νν. The distinguishing feature of the K L → π 0 e + e − mode is the contribution of the photon penguin. Moreover, one cannot neglect the long-distance effect from the photon exchange process [48]. Thus, the decay amplitude of K L → π 0 e + e − has both a short-distance effect and a long-distance effect, and the SM prediction of the branching ratio is around 3 × 10 −11 , which is comparable to the SM prediction of K L → π 0 νν. Since our interest here is to check whether the SUSY effect does not exceed the experimental bound of K L → π 0 e + e − , we only consider the short-distance contribution in our analysis. The experimental bound of the branching ratio K L → π 0 e + e − is BR(K L → π 0 e + e − ) exp < 2.8 × 10 −10 [43]. In Fig. 7, the predicted BR K L → π 0 e + e − vs BR(K L → π 0 νν) plane is plotted with s u = 0 ∼ 0.3 and s d = 0 ∼ 0.3 at the 10 TeV SUSY scale. There are two predicted lines in this figure. Because the decay amplitude A K L → π 0 e + e − is described by the sum of the SM and the SUSY contributions, there are two ways of taking the relative phase of ± such as A K L → π 0 e + e − = A K L → π 0 e + e − : SM ± A K L → π 0 e + e − : SUSY , which has two solutions giving the same absolute value of the decay amplitude. Then, we have two predicted values of BR K L → π 0 e + e − for particular BR K L → π 0 νν . Both decay processes are dominated by the Z penguin mediated charginos, so the branching ratios are determined by the final state couplings of Z νν and Ze + e − , that is, the weak charges Q Z L . Moreover, three flavors of neutrinos are summed for K L → π 0 νν. Therefore, BR(K L → π 0 νν) is significantly larger than BR K L → π 0 e + e − . On the other hand, in the SM, there are some contributions to K L → π 0 e + e − such as the photon exchange processes. So, BR(K L → π 0 e + e − ) is comparable to BR K L → π 0 νν in the SM. In conclusion, the experimental upper bound of BR(K L → π 0 e + e − ) excludes the region larger than BR(K L → π 0 νν) = 1.7 × 10 −9 . However, if the long-distance effect is properly included [48], this 12 constraint becomes somewhat tight or loose depending on the relative sign between the SUSY contribution and the long-distance one.
The neutron electric dipole moment (EDM) d n is well known as a sensitive probe for the NP, and so we have studied the correlation between the neutron EDM and the K → π 0 νν branching ratio. It is found that our predicted K → π 0 νν does not correlate with d n . Suppose the SUSY contribution to the chromo-EDM of quarks through the gluon penguin mediated gluino [49][50][51][52][53], where the leftright mixing term of the down-squark is dominant. In our SUSY mass spectra, the left-right mixing is suppressed, as discussed in Sect. 3. Moreover, the CP-violating phase dependence of d n comes from the down-squark mixing matrix, whereas the phase of K → π 0 νν comes from the up-squark mixing matrix. In other words, those phase dependences are completely different each other. Therefore, we do not take account of the constraint from the experimental upper bound of the neutron EDM in our analyses.

Summary
We have studied the contribution of the high-scale SUSY to the K L → π 0 νν and K + → π + νν processes by correlating with the CP-violating parameter K . These rare decays have an important role in the decision concerning the CP phase in the CKM matrix; furthermore, they are also sensitive to the flavor structure of the NP.
Taking account of the recent LHC results for the Higgs discovery and SUSY searches, we consider the high-scale SUSY at the 10-50 TeV scale. We have then discussed the SUSY effects on K + → π + νν, K L → π 0 νν, and K in the framework of the mass eigenstate basis of the SUSY particles, assuming non-minimal squark (slepton) flavor mixing.
We have calculated the SUSY contribution to the branching ratios of K L → π 0 νν and K + → π + νν, where phase parameters are constrained by the observed K . The Z penguin mediated chargino dominates the SUSY contribution for these decays. At the 10 TeV SUSY scale, its contribution can enhance the branching ratio of K L → π 0 νν by eight times compared with the SM predictions, whereas the predicted branching ratio BR K + → π + νν increases up to three times the SM prediction in the case of up-squark mixing s u = 0.1.
We have investigated the correlation between K and the K L → π 0 νν branching ratio. Since the gluino box diagram dominates the SUSY contribution of K up to 30%, there is no correlation between them. However, if the down-squark mixing is neglected compared with the up-squark mixing, the chargino process dominates both SUSY contributions of K L → π 0 νν and K . Then a correlation is found between them, but the chargino contribution to K is at most 3%. It is concluded 13/18 PTEP 2015, 053B07 M. Tanimoto and K. Yamamoto that K could be deviated significantly from the SM prediction by O(10%) due to the gluino box process, whereas the chargino process could enhance the branching ratio of K L → π 0 νν by several times from the SM prediction.
Our predicted branching ratios depend on the mixing angle s u significantly. The observed upper bound of BR K + → π + νν gives the constraint for the predicted BR K L → π 0 νν at s u larger than 0.2.
Even if the SUSY scale is 50 TeV, the chargino process still enhances the branching ratio of K L → π 0 νν from the SM prediction by a factor of two, and the K is deviated from the SM prediction by O(10%) unless the down-squark mixing s d is suppressed.
We also discuss correlations to the K L → π 0 e + e − process and the neutron electric dipole moment d n , which are sensitive to the NP.
We expect the measurement of these processes will be improved by the J-PARC KOTO experiment and CERN NA62 experiment in the near future.
Suppose that the MSSM matches with the SM at the SUSY mass scale Q 0 ≡ m 0 . Then the smaller one m 2 − is identified to be the mass squared of the SM Higgs H with a tachyonic mass. The larger one m 2 + is the mass squared of the orthogonal combination H, which is decoupled from the SM at Q 0 , that is, m H Q 0 . Therefore, we have with which leads to the mixing angle between H 1 andH 2 , β, as follows:  Table A1. Input parameters at and the obtained SUSY spectra at Q 0 = 10 and 50 TeV. Thus, the Higgs mass parameter m 2 is expressed in terms of m 2 1 , m 2 2 , and tan β: Below the Q 0 scale, in which the SM emerges, the scalar potential is the SM one as follows: Here, the Higgs coupling λ is given in terms of the SUSY parameters at the leading order as λ Q 0 = 1 4 g 2 + g 2 cos 2 2β + 3h 2 t 8π 2 X 2 t 1 − and h t is the top Yukawa coupling of the SM. The parameters m 2 and λ run with the SM renormalization group equation down to the electroweak scale Q E W = m H , and then give It is easily seen that the VEV of Higgs, H , is v, and H = 0, taking account of H 1 = v cos β and H 2 = v sin β, where v = 246 GeV. Let us fix m H = 126 GeV, which gives λ Q 0 and m 2 Q 0 . This experimental input constrains the SUSY mass spectrum of the MSSM. We consider some universal soft breaking parameters at the SUSY-breaking scale as follows: 10% level, which leads to flavor mixing of order 0.1. We take these flavor-mixing angles as free parameters at low energies. Now, we have the five SUSY parameters , tan β, m 0 , m 1/2 , A 0 , where Q 0 = m 0 . In addition to these parameters, we take μ = Q 0 . By fixing , Q 0 , and tan β, we tune m 1/2 and A 0 in order to obtain m 2 (Q 0 ) and λ H (Q 0 ), which realizes the correct electroweak vacuum with m H = 126 GeV. Then, we obtain the SUSY particle spectrum. We consider the two cases of Q 0 = 10 TeV and 50 TeV. The input parameter set and the obtained SUSY mass spectra at Q 0 are summarized in Table A1, where we use m t (m t ) = 163.5 ± 2 GeV [43,56].