## Abstract

We have studied the contribution of high-scale SUSY to the $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ and $$K^+ \to \pi ^+ \nu {\bar {\nu }}$$ processes by correlating with the CP-violating parameter $$\epsilon _K$$. 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 $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ by eight times compared with SM predictions, whereas the predicted branching ratio $${\textit {BR}}(K^+ \to \pi ^+ \nu {\bar {\nu }})$$ increases by up to three times that of the SM. The gluino box diagram dominates the SUSY contribution of $$\epsilon _K$$ up to $$30{\%}$$. If down-squark mixing is neglected compared with up-squark mixing, the Z penguin mediated chargino dominates both SUSY contributions of $${\textit {BR}}(K_L \to \pi ^0 \nu {\bar {\nu }})$$ and $$\epsilon _K$$. Then, a correlation between them is found, but the chargino contribution to $$\epsilon _K$$ is at most $$3{\%}$$. Even if the SUSY scale is $$50$$TeV, the chargino process still enhances the branching ratio of $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ from the SM prediction by a factor of two, and $$\epsilon _K$$ is deviated from the SM prediction by $$\mathcal {O}(10{\%})$$. We also discuss the chargino contribution to the $$K_L \to \pi ^0 e^+e^-$$ process.

## Introduction

$$K$$ meson physics has provided important information in the indirect search for new physics (NP). In particular, the rare decay processes $$K^+ \to \pi ^+ \nu {\bar {\nu }}$$ and $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ are known as clean theoretically [1, 2]. Therefore, both these processes have been considered to be powerful probes of NP [3–14], whereas these decay widths are bounded by the so-called Grossman–Nir bound for the NP [15, 16].

The $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ process is the CP-violating one and provides a direct measurement of the CP-violating phase in the Cabibbo–Kobayashi–Maskawa (CKM) matrix [17, 18]. In addition, the CP-conserving process $$K^+ \to \pi ^+ \nu {\bar {\nu }}$$ is also the physical quantity related to the unitarity triangle (UT). On the other hand, the CP-violating parameter $$\epsilon _K$$, which is induced by the $$K^0$$–$${\bar K^0}$$ mixing, also constrains the height of the UT. Hence these measured variables give us information on the UT fit as well as the CP-violating quantity $$\sin 2 \phi _1$$ induced by the $$B^0$$–$${\bar B^0}$$ mixing. Furthermore, the $$K \to \pi \nu {\bar {\nu }}$$ processes are expected to open the NP window in the CP-violating flavor structure.

The $$K^+ \to \pi ^+ \nu {\bar {\nu }}$$ and $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ decay processes are governed by the Z penguin diagram in the Standard Model (SM) [19], which predicts

On the experimental side, the upper bound of the branching ratio of $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ is given by the KEK E391a experiment [22]. The branching ratio of $$K^+ \to \pi ^+ \nu {\bar {\nu }}$$ measured by the BNL E787 and E949 experiments is consistent with the SM prediction [23]:

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 [27–29]. 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 $$\mathcal {O}(10$$–$$100)$$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 [32–34].

So far, the effects of SUSY on the $$K^+ \to \pi ^+ \nu {\bar {\nu }}$$ and $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ 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, 35–40] at the $$\mathcal {O}(500)$$GeV scale.

We have studied the SUSY contribution to the CP violation of the $$B$$ meson and $$\epsilon _K$$ induced by $$K^0$$–$${\bar K^0}$$ 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 $$\epsilon _K$$; 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 $$K^+ \to \pi ^+ \nu {\bar {\nu }}$$ and $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ by correlating with $$\epsilon _K$$ 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 $$K^+ \to \pi ^+ \nu {\bar {\nu }}$$, $$K_L \to \pi ^0 \nu {\bar {\nu }}$$, and $$\epsilon _K$$ 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 $$K \to \pi \nu {\bar {\nu }}$$ decay and the CP violating parameter $$\epsilon _K$$, which correspond to $$|\Delta S|=1$$ and $$|\Delta S|=2$$ processes, respectively. The $$K^+ \to \pi ^+ \nu {\bar {\nu }}$$ and $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ 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 $$K^+ \to \pi ^0 e^+ \nu$$. 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 $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ 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 $$\epsilon _K$$ is measured with enough accuracy. The major theoretical ambiguity comes from the hadronic matrix element factor $$\hat {B}_K$$. Recent lattice calculations give us a reliable value for $$\hat {B}_K$$ [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 $$\epsilon _K$$. Actually, the non-negligible SUSY contribution has been expected in $$\epsilon _K$$ at the scale of $${\mathcal O}(100)$$TeV [32–34]. Consequently, it is necessary to examine the high-scale SUSY contribution in $$K \to \pi \nu {\bar {\nu }}$$ by correlating with $$\epsilon _K$$.

### Basic framework: $$K^+ \to \pi ^+ \nu {\bar {\nu }}$$ and $$K_L \to \pi ^0 \nu {\bar {\nu }}$$

#### $$K^+ \to \pi ^+ \nu {\bar {\nu }}$$ and $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ in the SM.

Let us start by discussing the framework of the $$K^+ \to \pi ^+ \nu {\bar {\nu }}$$ and $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ processes in the SM [1]. The effective Hamiltonian for $$K \to \pi \nu {\bar {\nu }}$$ in the SM is given by:

Let us define the function $$F$$ as follows:

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 $$K^+ \to \pi ^+ \nu {\bar {\nu }}$$ is expressed as follows:

For the $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ decay, $$K^0$$–$${\bar K^0}$$ mixing should be taken account, and one obtains

Taking the ratio between the branching ratio of $$K^+ \to \pi ^0 e^+ {\bar {\nu }}$$ and $$K_L \to \pi ^0 \nu {\bar {\nu }}$$, we have the simple form:

In the SM, $$K^+ \to \pi ^+ \nu {\bar {\nu }}$$ and $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ are related to the UT fit. We write down the branching ratio in terms of the Wolfenstein parameters. Since $${\rm Re}\,F$$ and $${\rm Im}\,F$$ are given as

Before going on to discuss the SUSY formulation, we present the general bound between $$K^+ \to \pi ^+ \nu {\bar {\nu }}$$ and $$K_L \to \pi ^0 \nu {\bar {\nu }}$$, 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:

#### $$K^+ \to \pi ^+ \nu {\bar {\nu }}$$ and $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ 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 the weak neutral-current coupling $$Q_{ZL}^{(\nu )}=1/2$$, and $$B_{\rm VL(R)L}^{21ij}$$ and $$P_{\rm ZL(R)}^{21}$$ 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 gluino $$\tilde {g}$$, the chargino $$\chi ^{\pm }$$, and the neutralino $$\chi ^{0}$$ mediated ones.^{1}

We write each contribution as follows:

The branching ratio of $$K^+ \to \pi ^+ \nu {\bar {\nu }}$$ and $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ are obtained by replacing the internal effect $$F$$ in Eqs. (8) and (15) with $$C_{\rm VLL}^{ij}+C_{\rm VRL}^{ij}$$, as follows:

### $$\epsilon _K$$ in the MSSM

It is well known that the CP-violating parameter $$\epsilon _K$$ induced by the $$K^0-{\bar K^0}$$ oscillation gives us one of the most serious constraints on the NP. The general expression for $$\epsilon _K$$ is given as

where $$S(x)$$ denotes the SM one-loop functions [46], and $$\eta _{cc,tt,ct}$$ are the QCD corrections [45]. Recent lattice calculations give us a precise determination of the $$\hat {B}_K$$ parameter [41, 42].

Taking account of the NP effect, the expression for $$M_{12}^K$$ is modified. In the case of the SUSY, new contributions to the box diagrams are given by the gluino $$\tilde {g}$$, the charged Higgs $$H^{\pm }$$, the chargino $$\chi ^{\pm }$$, and the neutralino $$\chi ^{0}$$ exchanges:

## 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, $${\mathcal O}(10$$–$$1000)$$TeV [32–34, 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 $$10,~50\,{\rm TeV}$$, in which the $$K \to \pi \nu {\bar {\nu }}$$ decays and $$\epsilon _K$$ are discussed.

Another important experimental result that should be mentioned is the Higgs discovery. The Higgs mass $$m_H \simeq 126$$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 $$X_t = A_0 - \mu \cot \beta$$ 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 $$\theta ^q$$, which is defined as

The SUSY brings new flavor mixing through the quark–squark–gaugino and lepton–slepton–gaugino couplings. The $$6\times 6$$ squark mass matrix $$M_{\tilde q}^2$$ in the super-CKM basis turns into the mass eigenstate basis by diagonalizing with rotation matrix $$\Gamma ^{(q)}$$ as

For the lepton sector, the mixing matrices $$\Gamma _{L(R)}^{(\ell )}$$ have the same structure as the quark one in the charged-lepton flavors; however, there is only the left-handed $$\Gamma _{L}^{(\nu )}$$ for neutrinos.

As is well known, the charged Higgs and the chargino contributions dominate the $$K \to \pi \nu {\bar {\nu }}$$ processes [12]. Since the SUSY scale is high in our scheme, the charged Higgs are heavy, $$\mathcal {O}(10\,{\rm TeV})$$, so the charged Higgs contribution is suppressed in our framework. On the other hand, the dominant SUSY contribution to $$\epsilon _K$$ 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 $$s_{ij}^{dL(R)}$$ are negligibly small and are comparable to the up-squark mixing angles $$s_{ij}^{uL(R)}$$. We scan the phases of Eq. (30) for up-squarks, down-squarks, charged sleptons, and sneutrinos in the region of $$0 \sim 2\pi$$ independently.

In our framework, the $$K \to \pi \nu {\bar {\nu }}$$ processes are dominated by the Z penguin mediated chargino exchange, $$P_{\rm ZL}^{sd}(\chi ^{\pm })$$ in Eq. (23), which occur through the $$\tilde t_L s_L(d_L)\chi ^\pm$$ and $$\tilde t_R s_L(d_L)\chi ^\pm$$ interactions, respectively. In our basis, the relevant mixing is given by

## Numerical analysis

Let us discuss the high-scale SUSY contribution to the $$K \to \pi \nu {\bar {\nu }}$$ processes by correlating with $$\epsilon _K$$ [13]. At present, we cannot confirm whether the SM prediction $$\epsilon _K^{\rm SM}$$ is in agreement with the experimental value $$\epsilon _K^{\rm exp}$$ because there remains the theoretical uncertainty with an order of a few tens percent. However, the theoretical uncertainties of $$\epsilon _K$$ are expected to be reduced significantly in the near future. Actually, the lattice calculations of $$\hat B_K$$ will be improved significantly [41, 42], whereas $$|V_{cb}|$$ and the CKM phase $$\gamma$$ will be measured more precisely in Belle-II. Therefore, we will be able to test the correlation between $$K \to \pi \nu {\bar {\nu }}$$ and $$\epsilon _K$$.

In our previous work, we examined the sensitivity of the high-scale SUSY with $$10$$ and $$50$$TeV to $$\epsilon _K$$. It was found that the SUSY contribution to $$\epsilon _K$$ 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 $$\epsilon _K$$ are due to the CKM elements $$V_{cb}$$, $$\bar \rho$$, and $$\bar \eta$$, and the $$\hat B_K$$ parameter. We take the CKM parameters $$V_{cb}$$, $$\bar \rho$$, and $$\bar \eta$$ at the 90% C.L. of the experimental data:

To start, we show the numerical results at the SUSY scale of $$10$$TeV. Figure 1 shows the predictions on the $${\textit {BR}}(K_L \to \pi ^0 \nu {\bar {\nu }})$$ vs $${\textit {BR}}(K^+ \to \pi ^+ \nu {\bar {\nu }})$$ plane, where phase parameters are constrained by the observed $$|\epsilon _K|$$ with the experimental error bar of $$90{\%}$$ C.L. Here, we fix the mixing parameters in Eq. (30) by taking the common values $$s_{i3}^{uL}=s_{i3}^{uR}=s^u=0.1 \ (i=1,2)$$ and $$s_{i3}^{dL}=s_{i3}^{dR}= s^d=0.1 \ (i=1,2)$$ 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 $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ by eight times compared with the SM predictions in Eq. (1), $$1.8\times 10^{-10}$$, although it is much smaller than the Grossman–Nir bound. On the other hand, the predicted $${\textit {BR}}(K^+ \to \pi ^+ \nu {\bar {\nu }})$$ increases up to three times, $$2.1\times 10^{-10}$$. It is also noticed that the predicted region of $${\textit {BR}}(K_L\to \pi ^0 \nu {\bar {\nu }})$$ is reduced to much smaller than $$10^{-11}$$ due to the cancellation between the SM and SUSY contributions. The $${\textit {BR}}(K^+ \to \pi ^+ \nu {\bar {\nu }})$$ could be reduced to $$1.3\times 10^{-11}$$.

We discuss the correlation between $$\epsilon _K$$ and $${\textit {BR}}(K_L \to \pi ^0 \nu {\bar {\nu }})$$ 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 $$|\epsilon _K|$$. 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 $$K_L \to \pi ^0 \nu {\bar {\nu }}$$, and the gluino box diagram dominates the SUSY contribution of $$\epsilon _K$$. A gluino contribution of $$30{\%}$$ is possible in $$\epsilon _K$$.

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 $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ and $$\epsilon _K$$. Then, a correlation is found between them, as seen in Fig. 2(b), where the chargino contribution to $$\epsilon _K$$ 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, $$\epsilon _K$$ could be deviated from the SM prediction by $$\mathcal {O}(10{\%})$$ due to the gluino box diagram, whereas the Z penguin mediated chargino could enhance the branching ratio of $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ 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 $${\textit {BR}}(K_L \to \pi ^0 \nu {\bar {\nu }})$$ vs $$s^u$$ and $${\textit {BR}}(K^+ \to \pi ^+ \nu {\bar {\nu }})$$ vs $$s^u$$ planes, taking $$s^u=0 \sim 0.3$$, in Fig. 3(a) and (b). We scan $$s^d$$ in the region of $$0\sim 0.3$$ independent of $$s^u$$, although the gluino contribution is much suppressed compared with the chargino one. In this plot, the SUSY contribution to $$\epsilon _K$$ is free ($$0$$–$$40{\%}$$), but the experimental constraint of $$|\epsilon _K|$$ 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 $${\textit {BR}}(K^+ \to \pi ^+ \nu {\bar {\nu }})$$ with $$3\sigma$$ by the black line, at which the predicted $${\textit {BR}}(K_L \to \pi ^0 \nu {\bar {\nu }})$$ should be cut. Namely, the observed upper bound of $${\textit {BR}}(K^+ \to \pi ^+ \nu {\bar {\nu }})$$ gives the constraint for the predicted $${\textit {BR}}(K_L \to \pi ^0 \nu {\bar {\nu }})$$ at $$s^u$$ larger than $$0.2$$. The precise experimental measurement of $${\textit {BR}}(K^+ \to \pi ^+ \nu {\bar {\nu }})$$ will lower the predicted upper bound of $${\textit {BR}}(K_L \to \pi ^0 \nu {\bar {\nu }})$$.

Let us discuss the case of a SUSY scale of $$50$$TeV. Figure 4 shows the predictions on the $${\textit {BR}}(K_L \to \pi ^0 \nu {\bar {\nu }})$$ and $${\textit {BR}}(K^+ \to \pi ^+ \nu {\bar {\nu }})$$ 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 $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ is enhanced by two times from the SM prediction, and the branching ratio of $$K^+ \to \pi ^+ \nu {\bar {\nu }}$$ could be enhanced from the SM prediction by three times.

To see the correlation between $$\epsilon _K$$ and the predicted $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ branching ratio, we show the branching ratio of $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ versus the SUSY contribution of $$\epsilon _K$$ in Fig. 5, in which (a) $$s^u=s^d=0.3$$ and (b) $$s^u=0.3, \ s^d=0$$. We do not find any correlation between them in Fig. 5(a), where the gluino contribution to $$\epsilon _K$$ 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 $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ and $$\epsilon _K$$ since the down-squark mixing $$s_d$$ vanishes by keeping $$s_u=0.3$$. The chargino contribution to $$\epsilon _K$$ 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, $$\epsilon _K$$ could be deviated from the SM prediction by $$\mathcal {O}(10{\%})$$ due to the gluino box diagram, whereas the chargino process deviates the branching ratio of $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ from the SM prediction by a factor of two.

Figure 6 shows the $$s^u$$ dependence of $${\textit {BR}}(K_L \to \pi ^0 \nu {\bar {\nu }})$$ and $${\textit {BR}}(K^+ \to \pi ^+ \nu {\bar {\nu }})$$ taking $$s^u=0 \sim 0.5$$. We also scan $$s^d$$ in the region of $$0\sim 0.3$$ independent of $$s^u$$. In this plot, the SUSY contribution to $$\epsilon _K$$ is free ($$0$$–$$40{\%}$$), but the experimental constraint of $$\epsilon _K$$ with the error-bar of 90% C.L. is taken into account. The predicted $${\textit {BR}}(K_L \to \pi ^0 \nu {\bar {\nu }})$$ could be large, up to $$8\times 10^{-11}$$, and $${\textit {BR}}(K^+ \to \pi ^+ \nu {\bar {\nu }})$$ is up to $$1.5\times 10^{-10}$$. 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 $$K_L \to \pi ^0 e^+e^-$$ process and the neutron electric dipole moment $$d_n$$. The $$K_L \to \pi ^0 e^+e^-$$ process is induced in a similar way to $$K_L \to \pi ^0 \nu {\bar {\nu }}$$. The distinguishing feature of the $$K_L \to \pi ^0 e^+e^-$$ 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 $$K_L \to \pi ^0 e^+e^-$$ has both a short-distance effect and a long-distance effect, and the SM prediction of the branching ratio is around $$3\times 10^{-11}$$, which is comparable to the SM prediction of $$K_L \to \pi ^0 \nu {\bar {\nu }}$$. Since our interest here is to check whether the SUSY effect does not exceed the experimental bound of $$K_L \to \pi ^0 e^+e^-$$, we only consider the short-distance contribution in our analysis. The experimental bound of the branching ratio $$K_L \to \pi ^0 e^+e^-$$ is $${\rm BR}(K_L \to \pi ^0 e^+e^-)_{\rm exp}\lt 2.8 \times 10^{-10}$$ [43]. In Fig. 7, the predicted $${\textit {BR}}(K_L \to \pi ^0 e^+e^-)$$ vs $${\textit {BR}}(K_L \to \pi ^0 \nu {\bar {\nu }})$$ plane is plotted with $$s^u=0\sim 0.3$$ and $$s^d=0\sim 0.3$$ at the $$10$$TeV SUSY scale. There are two predicted lines in this figure. Because the decay amplitude $$A(K_L \to \pi ^0 e^+e^-)$$ is described by the sum of the SM and the SUSY contributions, there are two ways of taking the relative phase of $$\pm$$ such as $$A(K_L \to \pi ^0 e^+e^-) = A(K_L \to \pi ^0 e^+e^-: {\rm SM}) \pm A(K_L \to \pi ^0 e^+e^-: {\rm SUSY})$$, which has two solutions giving the same absolute value of the decay amplitude. Then, we have two predicted values of $${\textit {BR}}(K_L \to \pi ^0 e^+e^-)$$ for particular $${\textit {BR}}(K_L \to \pi ^0 \nu {\bar {\nu }})$$. Both decay processes are dominated by the Z penguin mediated charginos, so the branching ratios are determined by the final state couplings of $$Z\nu \bar \nu$$ and $$Ze^+e^-$$, that is, the weak charges $$Q_{ZL}^{(\nu )}$$ and $$Q_{ZL}^{(e)}$$. Moreover, three flavors of neutrinos are summed for $$K_L \to \pi ^0 \nu {\bar {\nu }}$$. Therefore, $${\textit {BR}}(K_L \to \pi ^0 \nu {\bar {\nu }})$$ is significantly larger than $${\textit {BR}}(K_L \to \pi ^0 e^+e^-)$$. On the other hand, in the SM, there are some contributions to $$K_L \to \pi ^0 e^+e^-$$ such as the photon exchange processes. So, $${\textit {BR}}(K_L \to \pi ^0 e^+e^-)$$ is comparable to $${\textit {BR}}(K_L \to \pi ^0 \nu {\bar {\nu }})$$ in the SM. In conclusion, the experimental upper bound of $${\textit {BR}}(K_L \to \pi ^0 e^+e^-)$$ excludes the region larger than $${\textit {BR}}(K_L \to \pi ^0 \nu {\bar {\nu }})=1.7\times 10^{-9}$$. However, if the long-distance effect is properly included [48], this 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 $$K \to \pi ^0 \nu {\bar {\nu }}$$ branching ratio. It is found that our predicted $$K \to \pi ^0 \nu {\bar {\nu }}$$ does not correlate with $$d_n$$. Suppose the SUSY contribution to the chromo-EDM of quarks through the gluon penguin mediated gluino [49–53], where the left–right 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 $$K \to \pi ^0 \nu {\bar {\nu }}$$ 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 $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ and $$K^+ \to \pi ^+ \nu {\bar {\nu }}$$ processes by correlating with the CP-violating parameter $$\epsilon _K$$. 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 $$K^+ \to \pi ^+ \nu {\bar {\nu }}$$, $$K_L \to \pi ^0 \nu {\bar {\nu }}$$, and $$\epsilon _K$$ 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 $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ and $$K^+ \to \pi ^+ \nu {\bar {\nu }}$$, where phase parameters are constrained by the observed $$\epsilon _K$$. 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 $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ by eight times compared with the SM predictions, whereas the predicted branching ratio $${\textit {BR}}(K^+ \to \pi ^+ \nu {\bar {\nu }})$$ 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 $$\epsilon _K$$ and the $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ branching ratio. Since the gluino box diagram dominates the SUSY contribution of $$\epsilon _K$$ 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 $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ and $$\epsilon _K$$. Then a correlation is found between them, but the chargino contribution to $$\epsilon _K$$ is at most $$3{\%}$$. It is concluded that $$\epsilon _K$$ could be deviated significantly from the SM prediction by $$\mathcal {O}(10{\%})$$ due to the gluino box process, whereas the chargino process could enhance the branching ratio of $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ by several times from the SM prediction.

Our predicted branching ratios depend on the mixing angle $$s^u$$ significantly. The observed upper bound of $${\textit {BR}}(K^+ \to \pi ^+ \nu {\bar {\nu }})$$ gives the constraint for the predicted $${\textit {BR}}(K_L \to \pi ^0 \nu {\bar {\nu }})$$ at $$s^u$$ larger than $$0.2$$.

Even if the SUSY scale is $$50$$TeV, the chargino process still enhances the branching ratio of $$K_L \to \pi ^0 \nu {\bar {\nu }}$$ from the SM prediction by a factor of two, and the $$\epsilon _K$$ is deviated from the SM prediction by $$\mathcal {O}(10{\%})$$ unless the down-squark mixing $$s^d$$ is suppressed.

We also discuss correlations to the $$K_L \to \pi ^0 e^+e^-$$ 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.

## Funding

Open Access funding: SCOAP^{3}.

## Acknowledgments

We would like to thank T. Kurimoto for comments at the early stage of this work. This work is supported by JSPS Grants-in-Aid for Scientific Research, 24654062 and 25-5222.

#### SUSY spectrum

In the framework of the MSSM, one obtains the SUSY particle spectrum which is consistent with the observed Higgs mass. The numerical analyses have been given in Refs. [54, 55]. At the SUSY-breaking scale $$\Lambda$$, the quadratic terms in the MSSM potential are given as

Thus, the Higgs mass parameter $$m^2$$ is expressed in terms of $$m_1^2$$, $$m_2^2$$, and $$\tan \beta$$:

Let us fix $$m_H=126$$GeV, which gives $$\lambda (Q_0)$$ 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 $$\Lambda$$ as follows:

Now, we have the five SUSY parameters $$\Lambda$$, $$\tan \beta$$, $$m_0$$, $$m_{1/2}$$, $$A_0$$, where $$Q_0=m_0$$. In addition to these parameters, we take $$\mu =Q_0$$. By fixing $$\Lambda$$, $$Q_0$$, and $$\tan \beta$$, we tune $$m_{1/2}$$ and $$A_0$$ in order to obtain $$m^2(Q_0)$$ and $$\lambda _H(Q_0)$$, 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 $$\overline m_t (m_t)=163.5\pm 2$$GeV [43, 56].

Input at $$\Lambda$$ and $$Q_0$$ | Output at $$Q_0$$ |
---|---|

at $$\Lambda =10^{17}$$ GeV, | $$m_{\tilde g}=12.8$$ TeV, $$m_{\tilde W}=5.2$$ TeV, $$m_{\tilde B}=2.9$$ TeV |

$$m_0=10$$ TeV, | $$m_{\tilde b_L}=m_{\tilde t_L}=12.2$$ TeV |

$$m_{1/2}=6.2$$TeV, | $$m_{\tilde b_R}=14.1$$ TeV, $$m_{\tilde t_R}=8.4$$ TeV |

$$A_0=25.803$$ TeV; | $$m_{\tilde s_L, \tilde d_L}=m_{\tilde c_L, \tilde u_L}=15.1$$ TeV |

at $$Q_0=10$$ TeV, | $$m_{\tilde s_R,\tilde d_R}\simeq m_{\tilde c_R,\tilde u_R}=14.6$$ TeV, $$m_{\mathcal {H}}=13.7$$ TeV |

$$\mu =10$$ TeV, | $$m_{\tilde \tau _{L}}=m_{\tilde {\nu _\tau }_L}=10.4$$ TeV, $$m_{\tilde \tau _R}=9.3$$ TeV |

$$\tan \beta =10$$ | $$m_{\tilde \mu _L, \tilde e_{L}}=m_{\tilde {\nu _\mu }_L, \tilde {\nu _e}_L}=10.8$$ TeV, $$m_{\tilde \mu _R, \tilde e_R}=10.3$$ TeV |

$$X_t=-0.22$$, $$\lambda _H=0.126$$ | |

$$m_1^2=1.84857\times 10^8\,{\rm GeV}^2$$, $$m_2^2=1.83996\times 10^6\,{\rm GeV}^2$$, $$m^2=8691\,{\rm GeV}^2$$ | |

at $$\Lambda =10^{16}$$ GeV, | $$m_{\tilde g}=115.6$$ TeV, $$m_{\tilde W}=55.4$$ TeV, $$m_{\tilde B}=33.45$$ TeV |

$$m_0=50$$ TeV, | $$m_{\tilde b_L}=m_{\tilde t_L}=100.9$$ TeV |

$$m_{1/2}=63.5$$ TeV, | $$m_{\tilde b_R}=104.0$$ TeV, $$m_{\tilde t_R}=83.2$$ TeV |

$$A_0=109.993$$ TeV; | $$m_{\tilde s_L, \tilde d_L}=m_{\tilde c_L, \tilde u_L}=110.7$$ TeV, $$m_{\tilde s_R,\tilde d_R}=110.7$$ TeV |

at $$Q_0=50$$ TeV, | $$m_{\tilde c_R,\tilde u_R}=105.0$$ TeV, $$m_{\mathcal {H}}=83.1$$ TeV |

$$\mu =50$$ TeV, | $$m_{\tilde \tau _{L}}=m_{\tilde {\nu _\tau }_L}=63.6$$ TeV, $$m_{\tilde \tau _R}=54.6$$ TeV |

$$\tan \beta =4$$ | $$m_{\tilde \mu _L, \tilde e_{L}}=m_{\tilde {\nu _\mu }_L, \tilde {\nu _e}_L}=63.8$$ TeV, $$m_{\tilde \mu _R, \tilde e_R}=55.0$$ TeV |

$$X_t=-0.65$$, $$\lambda _H=0.1007$$ | |

$$m_1^2=6.49990\times 10^9\,{\rm GeV}^2$$, $$m_2^2=4.06235\times 10^8\,{\rm GeV}^2$$, $$m^2=8840\,{\rm GeV}^2$$ |

Input at $$\Lambda$$ and $$Q_0$$ | Output at $$Q_0$$ |
---|---|

at $$\Lambda =10^{17}$$ GeV, | $$m_{\tilde g}=12.8$$ TeV, $$m_{\tilde W}=5.2$$ TeV, $$m_{\tilde B}=2.9$$ TeV |

$$m_0=10$$ TeV, | $$m_{\tilde b_L}=m_{\tilde t_L}=12.2$$ TeV |

$$m_{1/2}=6.2$$TeV, | $$m_{\tilde b_R}=14.1$$ TeV, $$m_{\tilde t_R}=8.4$$ TeV |

$$A_0=25.803$$ TeV; | $$m_{\tilde s_L, \tilde d_L}=m_{\tilde c_L, \tilde u_L}=15.1$$ TeV |

at $$Q_0=10$$ TeV, | $$m_{\tilde s_R,\tilde d_R}\simeq m_{\tilde c_R,\tilde u_R}=14.6$$ TeV, $$m_{\mathcal {H}}=13.7$$ TeV |

$$\mu =10$$ TeV, | $$m_{\tilde \tau _{L}}=m_{\tilde {\nu _\tau }_L}=10.4$$ TeV, $$m_{\tilde \tau _R}=9.3$$ TeV |

$$\tan \beta =10$$ | $$m_{\tilde \mu _L, \tilde e_{L}}=m_{\tilde {\nu _\mu }_L, \tilde {\nu _e}_L}=10.8$$ TeV, $$m_{\tilde \mu _R, \tilde e_R}=10.3$$ TeV |

$$X_t=-0.22$$, $$\lambda _H=0.126$$ | |

$$m_1^2=1.84857\times 10^8\,{\rm GeV}^2$$, $$m_2^2=1.83996\times 10^6\,{\rm GeV}^2$$, $$m^2=8691\,{\rm GeV}^2$$ | |

at $$\Lambda =10^{16}$$ GeV, | $$m_{\tilde g}=115.6$$ TeV, $$m_{\tilde W}=55.4$$ TeV, $$m_{\tilde B}=33.45$$ TeV |

$$m_0=50$$ TeV, | $$m_{\tilde b_L}=m_{\tilde t_L}=100.9$$ TeV |

$$m_{1/2}=63.5$$ TeV, | $$m_{\tilde b_R}=104.0$$ TeV, $$m_{\tilde t_R}=83.2$$ TeV |

$$A_0=109.993$$ TeV; | $$m_{\tilde s_L, \tilde d_L}=m_{\tilde c_L, \tilde u_L}=110.7$$ TeV, $$m_{\tilde s_R,\tilde d_R}=110.7$$ TeV |

at $$Q_0=50$$ TeV, | $$m_{\tilde c_R,\tilde u_R}=105.0$$ TeV, $$m_{\mathcal {H}}=83.1$$ TeV |

$$\mu =50$$ TeV, | $$m_{\tilde \tau _{L}}=m_{\tilde {\nu _\tau }_L}=63.6$$ TeV, $$m_{\tilde \tau _R}=54.6$$ TeV |

$$\tan \beta =4$$ | $$m_{\tilde \mu _L, \tilde e_{L}}=m_{\tilde {\nu _\mu }_L, \tilde {\nu _e}_L}=63.8$$ TeV, $$m_{\tilde \mu _R, \tilde e_R}=55.0$$ TeV |

$$X_t=-0.65$$, $$\lambda _H=0.1007$$ | |

$$m_1^2=6.49990\times 10^9\,{\rm GeV}^2$$, $$m_2^2=4.06235\times 10^8\,{\rm GeV}^2$$, $$m^2=8840\,{\rm GeV}^2$$ |

#### Z penguin amplitude mediated charginos

We present the expression for the Z penguin amplitude mediated chargino, $$P_{\rm ZL}^{sd}(\chi ^{\pm })$$, in our basis [44] as follows: