$K\to \pi \nu\overline{\nu}$ in the MSSM in Light of the $\epsilon^{\prime}_K/\epsilon_K$ Anomaly

TL;DR

This paper investigates how the MSSM can explain the $ ext{ε'}_K/ ext{ε}_K$ anomaly through gluino-squark box diagrams and explores the resulting correlations with rare kaon decay branching ratios, emphasizing implications for future experiments.

Contribution

It provides a detailed analysis of MSSM contributions to kaon decay processes and their correlation with the $ ext{ε'}_K/ ext{ε}_K$ anomaly, highlighting distinctive patterns from $Z$-penguin scenarios.

Findings

MSSM can enhance $K o uar{ u}$ decay rates up to 2 times the SM prediction.

Correlations depend on squark mass hierarchy and CP-violating phases.

Sign of $ ext{ε'}_K$ contribution correlates with the mass difference between right-handed up- and down-squarks.

Abstract

The Standard-Model (SM) prediction for the CP-violating quantity $\epsilon_K^{\prime}/\epsilon_K$ deviates from its measured value by 2.8 $\sigma$. It has been shown that this tension can be resolved within the Minimal Supersymmetric Standard Model (MSSM) through gluino-squark box diagrams, even if squarks and gluinos are much heavier than 1 TeV. The rare decays $K_L \to \pi^0\nu\bar{\nu}$ and $K^+ \to \pi^+\nu\bar{\nu}$ are similarly sensitive to very high mass scales and the first one also measures CP violation. In this article, we analyze the correlations between $\epsilon^{\prime}_K/\epsilon_K$ and $B(K_L \to \pi^0\nu\bar{\nu})$ and $B(K^+ \to \pi^+\nu\bar{\nu})$ within the MSSM aiming at an explanation of $\epsilon_K^{\prime}/\epsilon_K$ via gluino-squark box diagrams. The dominant MSSM contribution to the $K \to \pi\nu\bar{\nu}$ branching fractions stems from box diagrams with squarks, sleptons, charginos, and neutralinos, and the pattern of the correlations is different from the widely studied $Z$-penguin scenarios. This is interesting in light of future precision measurements by KOTO and NA62 at J-PARC and CERN, respectively. We find $B(K_L \to \pi^0\nu\bar{\nu})/B^{SM} (K_L \to \pi^0\nu\bar{\nu})\lesssim 2\,(1.2)$ and $B(K^+ \to \pi^+\nu\bar{\nu})/B^{SM}(K^+ \to \pi^+\nu\bar{\nu}) \lesssim 1.4\,(1.1)$, if all squark masses are above 1.5 TeV, gaugino masses obey GUT relations, and if one allows for a fine-tuning at the $1\%\,(10\%)$ level for the gluino mass. Larger values are possible for a tuned CP violating phase. Furthermore, the sign of the MSSM contribution to $\epsilon_K^{\prime}$ imposes a strict correlation between $B(K_L \to \pi^0\nu\bar{\nu})$ and the hierarchy between the masses $m_{\bar{U}}$, $m_{\bar{D}}$ of the right-handed up-squark and down-squark: sgn$[B(K_L \to \pi^0\nu\bar{\nu})-B^{SM} (K_L \to \pi^0\nu\bar{\nu})] = $sgn$(m_{\bar{U}}-m_{\bar{D}}) $.

Figures

Equations

$$\text{Re}{\frac{{ϵ}_K^{\prime }}{{ϵ}_K}|}_{\exp }=(16.6\pm 2.3)\times 10^{-4}(\text{PDG cite[cite]@@bibrefAuthors Phrase1YearPhrase2Olive:2016xmw@@citephrase(@@citephrase)}).$$

$${\frac{{ϵ}_K^{\prime }}{{ϵ}_K}|}_{\mathrm{SM}}=(1.06\pm 5.07)\times 10^{-4},$$

$$\mathcal{B}(K_L\to {\pi }^0\nu \overline{\nu }{)}_{\mathrm{SM}}$$

$$\mathcal{B}(K^{+}\to {\pi }^{+}\nu \overline{\nu }{)}_{\mathrm{SM}}$$

$$\mathcal{B}(K^{+}\to {\pi }^{+}\nu \overline{\nu }{)}_{\exp }=(17.3_{-10.5}^{+11.5})\times 10^{-11},$$

$$\mathcal{B}(K_L\to {\pi }^0\nu \overline{\nu }{)}_{\exp }<2.6\times 10^{-8}.$$

$$\frac{{ϵ}_K^{\prime }}{{ϵ}_K}=\frac{{\omega }_{+}}{\sqrt{2}|{ϵ}_K^{\text{exp}}|\text{Re}{A}_0^{\text{exp}}}\left\{\frac{\text{Im}{A}_2}{{\omega }_{+}}-\left(1-{\hat{\Omega }}_{\text{eff}}\right)\text{Im}{A}_0\right\},$$

$$m_Q,|{\Delta }_{Q,12}|,\theta ,M_3,M_2,M_1,m_{\bar{U}}/m_{\bar{D}},m_L.$$

$$|{\Delta }_{Q,12}|,\theta ,M_3,m_{\bar{U}}/m_{\bar{D}},$$

Full text

††preprint: PSI-PR–17–03††preprint: TTP–17–013

I Introduction

Flavor-changing neutral current (FCNC) decays of $K$ mesons are extremely sensitive to new physics (NP) and probe virtual effects of particles with masses far above the reach of future colliders, especially if the corresponding observable is CP violating. Prime examples of such observables are $\epsilon_{K}$ and $\epsilon_{K}^{\prime}$ measuring indirect and direct CP violation in $K\to\pi\pi$ decays and also $K_{L}\to\pi^{0}\nu\overline{\nu}$. While indirect CP violation was already found in 1964 Christenson:1964fg , it took 35 more years to establish a nonzero value of $\epsilon_{K}^{\prime}$ in 1999 by the NA48 and KTeV Collaborations epspexp :

[TABLE]

Until recently, large theoretical uncertainties precluded reliable predictions for $\textrm{Re}\,(\epsilon_{K}^{\prime}/\epsilon_{K})$. Calculating the hadronic matrix elements with the large-$N_{c}$ (dual QCD) method one finds a Standard-Model (SM) value well below the experimental range given in Eq. (1) dual . A major breakthrough has been the recent lattice-QCD calculation of Ref. Bai:2015nea , which gives support to the large-$N_{c}$ result. The current status is Kitahara:2016nld

[TABLE]

which is consistent with $(\epsilon_{K}^{\prime}/\epsilon_{K})_{\rm SM}=(1.9\pm 4.5)\times 10^{-4}$ Buras:2015yba . Both results are based on the lattice numbers in Refs. Bai:2015nea ; latticeA2 and further use CP-conserving $K\to\pi\pi$ data to constrain some of the hadronic matrix elements involved. The SM prediction in Eq. (2) lies below the experimental value in Eq. (1) by 2.8 $\sigma$.#1#1#1Calculations using chiral perturbation theory instead are consistent with both the measurement and Eq. (2), because they have larger errors chipt .

This tension can be explained by NP effects like $Z^{\prime}$ gauge bosons Buras:2015yca ; Buras:2015kwd ; Buras:2015jaq ; Buras:2016dxz ; Bobeth:2016llm , models with modified $Z$-couplings Buras:2015yca ; Buras:2015jaq ; Endo:2016tnu ; Bobeth:2017xry , by a right-handed coupling of quarks to the $W$ Cirigliano:2016yhc , within the littlest Higgs model Blanke:2015wba , but also within the Minimal Supersymmetric Standard Model (MSSM) Kitahara:2016otd ; Endo:2016aws .

When pursuing such NP interpretations of the tension in $\epsilon_{K}^{\prime}$ it is natural to look for signatures in other $s\to d$ transitions which are, in general, correlated in UV complete models. To this end the rare decays $K_{L}\to\pi^{0}\nu\overline{\nu}$ and $K^{+}\to\pi^{+}\nu\overline{\nu}$ play an important role. Within the SM, the branching ratios are predicted to be Buras:2006gb ; Brod:2010hi ; Buras:2015qea

[TABLE]

The first error summarizes the uncertainty from CKM parameters, the second one denotes the remaining theoretical uncertainties (in $\mathcal{B}(K_{L}\to\pi^{0}\nu\overline{\nu})_{\rm SM}$, $0.04$ is rounded to $0.0$). The numbers in Eq. (3) are based on the best-fit result for the CKM parameters in Ref. ckmfitter . Experimentally, we have Artamonov:2008qb

[TABLE]

and the $90\%$ C.L. upper bound Ahn:2009gb ,

[TABLE]

In the future, these measurements will be significantly improved. The NA62 experiment at CERN Rinella:2014wfa ; Romano:2014xda is aiming to reach a precision of 10 % compared to the SM already in 2018. In order to achieve $5\%$ accuracy more time is needed. Concerning $K_{L}\to\pi^{0}\nu\overline{\nu}$, the KOTO experiment at J-PARC aims in a first step at measuring $\mathcal{B}(K_{L}\to\pi^{0}\nu\overline{\nu})$ around the SM sensitivity Komatsubara:2012pn ; Shiomi:2014sfa . Furthermore, the KOTO-step2 experiment will aim at 100 events for the SM branching ratio, implying a precision of 10 % of this measurement KOTO .

In our MSSM scenario—in which the desired effect in $\epsilon_{K}^{\prime}$ is generated via gluino-squark boxes Kitahara:2016otd — correlations with $\mathcal{B}(K_{L}\to\pi^{0}\nu\overline{\nu})$ and $\mathcal{B}(K^{+}\to\pi^{+}\nu\overline{\nu})$ are not unexpected, since sizable box contributions also occur in these rare decays Buras:2004qb (see Fig. 1). Reference Kitahara:2016otd achieves sizable effects in $\epsilon_{K}^{\prime}$ gkn together with a simultaneous efficient suppression of the supersymmetric QCD contributions to $\epsilon_{K}$ Crivellin:2010ys . The suppression occurs because crossed and uncrossed gluino box diagrams cancel if the gluino mass is roughly 1.5 times the squark masses. With appropriately large left-left squark mixing angle and a CP phase one can reconcile the measurements of $\epsilon_{K}$ and $\Delta M_{K}$ with the large value in Eq. (1) and squark and gluino masses in the multi-TeV range, so that there is no conflict with collider searches.

However, there is no such cancellation in the (dominant) chargino box contribution to $K_{L}\to\pi^{0}\nu\overline{\nu}$ and $K^{+}\to\pi^{+}\nu\overline{\nu}$ which permits potentially large effects.

This article is organized as follows: In the next section, we will review $\epsilon^{\prime}_{K}/\epsilon_{K}$ and $K\to\pi\nu\overline{\nu}$ within the MSSM. In Sec. III, we then perform the phenomenological analysis highlighting the correlations before we conclude in Sec. IV.

II Preliminaries

$\epsilon_{K}^{\prime}/\epsilon_{K}$ is given by Buras:2015yba

[TABLE]

with $\omega_{+}=(4.53\pm 0.02)\times 10^{-2}$, $|\epsilon_{K}^{\rm exp}|=(2.228\pm 0.011)\cdot 10^{-3}$, $\hat{\Omega}_{\textrm{eff}}=(14.8\pm 8.0)\times 10^{-2}$, and the amplitudes $A_{I}=\langle(\pi\pi)_{I}|\mathcal{H}^{\left|\Delta S\right|=1}|K^{0}\rangle$ involving the effective $|\Delta S|=1$ Hamiltonian $\mathcal{H}^{\left|\Delta S\right|=1}$. Short-distance physics enters $\textrm{Im}\,A_{0}$ and $\textrm{Im}\,A_{2}$ through the Wilson coefficients in $\mathcal{H}^{\left|\Delta S\right|=1}$. The SM prediction of the renormalization-group (RG) improved Wilson coefficients is known to the next-to-leading order (NLO) of QCD and QED corrections epspnlo and the next-to-next-to-leading-order QCD calculation is underway Cerda-Sevilla:2016yzo . Equation (2) is based on a novel analytic formula for the NLO RG evolution.

The Wilson coefficients multiply the four-quark operators $Q_{j}$ whose hadronic matrix elements $\langle{(\pi\pi)_{I}}|Q_{j}|{K^{0}}\rangle$ must be calculated by nonperturbative methods. For some time these calculations for the matrix elements entering $\textrm{Im}\,A_{2}$ are in good shape, thanks to precise results from lattice QCD latticeA2 . However, $\textrm{Im}\,A_{0}$ has become tractable with lattice QCD only recently Bai:2015nea .

CP-conserving data determine $\textrm{Re}\,A_{0}$ and $\omega_{+}$ in Eq. (6). $\omega_{+}$ is essentially equal to the ratio $\textrm{Re}\,A_{2}/\textrm{Re}\,A_{0}$, except that it is calculated from charged rather than neutral kaon decays. The smallness of $\omega_{+}$ encodes the famous “$\Delta I=1/2$” rule $\textrm{Re}\,A_{0}\gg\textrm{Re}\,A_{2}$. It leverages the $\textrm{Im}\,A_{2}$ term in Eq. (6) and leads to the above-mentioned high sensitivity of $\epsilon_{K}^{\prime}$ to new physics in this amplitude.

Following the approach of Ref. Kitahara:2016otd we aim at explaining the discrepancy in $\epsilon_{K}^{\prime}/\epsilon_{K}$ with contributions to the Wilson coefficients $c_{1,2}^{\prime q}$. Therefore, we need the flavor (and CP) violation in the left-handed squark sector while the mass difference between the right-handed up- and down-squarks accounts for the necessary isospin violation.

The small errors in Eq. (3) show that the $K\to\pi\nu\overline{\nu}$ branching ratios are theoretically very clean. While $K_{L}\to\pi^{0}\nu\overline{\nu}$ is only sensitive to the CP violating part of the amplitude, $K^{+}\to\pi^{+}\nu\overline{\nu}$ is dominated by the CP conserving part. In principle, many diagrams contribute to $K\to\pi\nu\overline{\nu}$ in the MSSM with generic sources of flavor violation Buras:2004qb . However, since we are interested in a scenario with $s-d$ flavor violation in the left-handed squark sector, chargino-box contributions are numerically most important.

III Phenomenological analysis

Although the correlations between $\epsilon_{K}^{\prime}/\epsilon_{K}$ and $K\to\pi\nu\overline{\nu}$ in the MSSM have already been discussed in detail in Refs. Buras:1998ed ; Buras:2004qb ; Tanimoto:2016yfy ; Endo:2016aws , our study has several novelties. First of all, Refs. Buras:1998ed ; Buras:2004qb were written before the appearance of the $\epsilon_{K}^{\prime}$ anomaly, while we take into account the implication of the current deviation from the SM prediction. With the progress on the SM prediction, $\epsilon_{K}^{\prime}$ implies a much sharper constraint on the MSSM parameters, resulting in tighter bounds on the deviations of $\mathcal{B}(K\to\pi\nu\overline{\nu})$ from the SM prediction. In addition, in our analysis we employ $m_{\bar{U}}\neq m_{\bar{D}}$ to generate large gluino box (Trojan penguin) gkn contributions to $\epsilon_{K}^{\prime}$, while Refs. Tanimoto:2016yfy ; Endo:2016aws enhance $\epsilon_{K}^{\prime}$ through $Z$ penguins. Furthermore, we consider the latest LHC limits on the supersymmetric (SUSY) masses TheATLAScollaboration:2013hha ; Aad:2015iea ; ATLAS:2016kts ; CMS:2016xva .

Defining the bilinear terms for the squarks as $M^{2}_{X,ij}=m_{X}^{2}(\delta_{ij}+\Delta_{X,ij})$ for $X=Q,\bar{U},\bar{D}$, the numerically relevant parameters entering $\epsilon_{K}^{\prime}$, $\epsilon_{K}$ and $K\to\pi\nu\overline{\nu}$ in our analysis are

[TABLE]

Here $m_{Q}$ is the universal mass parameter for the bilinear terms of the left-handed squarks which we define in the down-quark basis (i.e. the up-squark mass matrix is obtained via a CKM rotation from $M^{2}_{Q}$). $\theta\equiv\textrm{arg}(\Delta_{Q,12})$, $M_{3}$ is the gluino mass, $M_{2}$ ($M_{1}$) the wino (bino) mass, and $m_{L}$ is the (universal) mass for the left-handed sleptons, respectively. The trilinear $A$-terms as well as the off-diagonal elements of the bilinear terms $\Delta_{X,ij}$ are set to [math] except for $\Delta_{Q,12}$ which generates the required flavor and CP violation in our setup. The values of the other (SUSY) parameters barely affect our results.#2#2#2We use the fixed values $\tan\beta=10$, $\mu=M_{A}=m_{Q}$, $A_{ij}=0$. We also fix $B_{G}=1$, which parameterizes the matrix element of the chromomagnetic penguin operator $Q_{8g}$.

The SUSY contribution to $\epsilon_{K}$ ($\epsilon_{K}^{\rm SUSY}$) and $\Delta M_{K}$, originates from one-loop boxes with all possible combinations of gluinos, winos, and binos. For $K^{+}\to\pi^{+}\nu\overline{\nu}$ and $K_{L}\to\pi^{0}\nu\overline{\nu}$ we take into account all MSSM one-loop contributions Buras:2004qb . However, numerically the chargino boxes turn out to be by far dominant in our setup. In $\epsilon^{\prime}_{K}/\epsilon_{K}$, we include all SUSY QCD (SQCD) contributions as well as $Z$-penguin contributions originating from chargino diagrams to the $I=0,\,2$ amplitudes with hadronic matrix elements evaluated at $1.3\,\,\mbox{GeV}$ Kitahara:2016nld ; Kitahara:2016otd . In the calculation of all contributions, we perform an exact diagonalization of the squark mass matrices.

In the SM contributions, we fix the relevant CKM elements to their best-fit values ckmfitter , in particular we set $V_{td}^{*}V_{ts}=(-3.22+1.41i)\cdot 10^{-4}$. In this way, we assume that the MSSM contributions to the standard unitarity-triangle analysis are small, so that the change in $V_{td}^{*}V_{ts}$ is unimportant compared to the explicit MSSM contributions to $\epsilon_{K}^{\prime}$ and $\mathcal{B}(K\to\pi\nu\overline{\nu})$. This is justified in typical MSSM scenarios with generic flavor violation.

First, we show a typical prediction for $\mathcal{B}(K_{L}\to\pi^{0}\nu\overline{\nu})/\mathcal{B}^{\rm SM}(K_{L}\to\pi^{0}\nu\overline{\nu})$ as a function of the squark masses in the left panel of Fig. 2. Here we assume universal diagonal elements for the left-handed and right-handed down squark mass matrices $M_{S}=m_{Q}=m_{\bar{D}}$ and use $m_{L}=300\,\,\mbox{GeV}$. We also choose $\Delta_{Q,12}=0.1\exp(-i\pi/4)$ ($0.1\exp(i3\pi/4)$) for $m_{\bar{U}}>M_{S}$ ($m_{\bar{U}}<M_{S}$) regions to obtain a positive contribution to $\epsilon^{\prime}_{K}$. We impose $M_{3}/M_{S}=1.5$ in order to obtain an efficient suppression of $\epsilon_{K}^{\rm SUSY}$ Crivellin:2010ys ; Kitahara:2016otd . In addition, the GUT relation for $M_{2}$ and $M_{1}$ are imposed. The $\epsilon_{K}^{\prime}/\epsilon_{K}$ discrepancy between Eq. (1) and the second prediction in Eq. (2) is resolved at $1\,\sigma$ ($2\,\sigma$) within the dark (light) green region. The red shaded region is excluded by $\epsilon_{K}$ at 95 % C.L. if the inclusive value of $|V_{cb}|$ is used, while the region between the blue-dashed lines can explain the $\epsilon_{K}$ discrepancy present if the exclusive determination of $V_{cb}$ is used Bailey:2015tba .#3#3#3The difference compared to Fig. 4 of Ref. Kitahara:2016otd comes from $\Delta_{Q,13,23}$. Note that $\theta=\pm\pi/4$ maximizes the effect in $\epsilon_{K}^{\rm SUSY}$, while the SUSY contributions to $\epsilon^{\prime}_{K}/\epsilon_{K}$ is maximized at $\theta=\pm\pi/2$ resulting instead in a vanishing effect in $\epsilon_{K}^{\rm SUSY}$. The blue shaded region is excluded by the current LHC results ATLAS:2016kts ; CMS:2016xva ; Aad:2015iea . Here, in order to be conservative, we use the most stringent one, i.e. we maximize the bound which is a function of the neutralino mass. In this setup, we find that $\mathcal{B}(K_{L}\to\pi^{0}\nu\overline{\nu})/\mathcal{B}^{\rm SM}(K_{L}\to\pi^{0}\nu\overline{\nu})\simeq 1.05$–$1.1$ is predicted in light of the $\epsilon_{K}^{\prime}/\epsilon_{K}$ discrepancy (and the potential $\epsilon_{K}$ discrepancy) if $m_{\bar{U}}>m_{\bar{D}}$.

In the right panel of Fig. 2, the dependence on the CP-violating phase ($\theta$) is shown. Here, we chose $|\Delta_{Q,12}|=0.1$, and $m_{\bar{D}}=2m_{\bar{U}}=2m_{Q}=2M_{S}$ ($m_{\bar{U}}=2m_{\bar{D}}=2m_{Q}=2M_{S}$ ) for $0<\theta<\pi$ ($\pi<\theta<2\pi)$. It can be seen that if $\theta$ is close $\pm\pi/2$, the constraint from $\epsilon_{K}$ is weakened while $\epsilon^{\prime}_{K}$ as well as $\mathcal{B}(K_{L}\to\pi^{0}\nu\overline{\nu})$ is enhanced.

Next, let us investigate upper and lower limits on $\mathcal{B}(K_{L}\to\pi^{0}\nu\overline{\nu})$ and $\mathcal{B}(K^{+}\to\pi^{+}\nu\overline{\nu})$. In the following analysis, we fix the slepton mass close to the experimental limit ($m_{L}=300\,\,\mbox{GeV}$) TheATLAScollaboration:2013hha and use GUT relations among all three gaugino masses. Therefore, when one fixes the lightest squark mass, the relevant free parameters are only

[TABLE]

with $0<|\Delta_{Q,12}|<1$ and $0<\theta<2\pi$. In Fig. 3, the blue solid line encloses the maximally allowed region in the $\mathcal{B}(K_{L}\to\pi^{0}\nu\overline{\nu})$–$\mathcal{B}(K^{+}\to\pi^{+}\nu\overline{\nu})$ plane (normalized by their SM values). The maximal values are obtained whenever the SUSY contributions to the $\Delta S=2$ amplitude exactly cancel. The contour lines in the figures show the required value of $M_{3}/M_{S}$ (imposing again GUT relations) for this cancellation. The maximal and minimal values for $\mathcal{B}(K\to\pi\nu\overline{\nu})$ are obtained by the decoupling one of the left-handed mixed down-strange squark while simultaneously maximizing their mixing. Since we assume equal diagonal entries of the bilinear terms this corresponds to the limit $m_{Q}\to\infty$ and $|\Delta_{Q,12}|\to 1$ which implies one light squark which is an equal admixture of the first and second generation of interaction eigenstates. Note that these results are independent of $m_{\bar{U}}/m_{\bar{D}}$, but $m_{\bar{U}}/m_{\bar{D}}$ is important when considering the correlation with $\epsilon^{\prime}_{K}/\epsilon_{K}$. In the left and right panels, the lightest squark mass is fixed to 1.5  TeV and 3  TeV, respectively. The latest searches for first-generation squarks at the LHC imply $m_{\tilde{q}_{1}}\gtrsim 1.4\,\,\mbox{TeV}$ if the gluino is heavy and the neutralino is light ATLAS:2016kts ; CMS:2016xva . We find that the upper allowed values for the branching ratios differ significantly from the SM predictions. However, in order to achieve these maximal values, severe fine-tuning of the gluino mass (with respect to the squark masses) or tuning of the CP violating phase is necessary: e.g. around $\theta=3\pi/2$, $\epsilon_{K}^{\rm SUSY}$ is much suppressed while $\mathcal{B}(K_{L}\to\pi^{0}\nu\overline{\nu})$ is enhanced.

Let us now investigate the degree of fine-tuning of the gluino mass needed to suppress $\epsilon^{\rm SUSY}_{K}$. In Fig. 5, the necessary amount of the fine-tuning in the gluino mass with respect to the value for the exact cancellation is shown, again in the $\mathcal{B}(K_{L}\to\pi^{0}\nu\overline{\nu})$–$\mathcal{B}(K^{+}\to\pi^{+}\nu\overline{\nu})$ plane like Fig. 3. In the light (dark) blue regions, the amount of fine-tuning is milder than 1 % (10 %), respectively while in the regions outside more sever fine-tuning is required in order to satisfy constraints from $\epsilon_{K}$ (using the inclusive $|V_{cb}|$ Bailey:2015tba ) and $\Delta M_{K}$ at the $2\,\sigma$ level. This means that the gluino mass can be shifted from its value necessary for an exact cancellation in $\epsilon_{K}$ and $\Delta M_{K}$ (given by the contours in Fig. 3) by 1 % ($\sim 20\,\,\mbox{GeV}$) and 10 % ($\sim 200\,\,\mbox{GeV}$) without violating the constraints. Here we have scanned over all values of the CP violating phase $\theta$. Alternatively, one can satisfy $\epsilon_{K}$ by tuning $\theta$ to values different from $\pm 90^{\circ}$ by at most $0.9^{\circ}$ ($9^{\circ}$). In our plot we have discarded such a tuned CP phase, which explains the white dents in the blue regions around $\theta=\pm 90^{\circ}$. However, in the case of a tuned $\theta$ the correlation between the two $K\to\pi\nu\overline{\nu}$ branching ratios is stronger. The red contour show the SUSY contributions to $\epsilon^{\prime}_{K}/\epsilon_{K}$ and the current $\epsilon_{K}^{\prime}/\epsilon_{K}$ discrepancy is resolved at $1\,\sigma$ ($2\,\sigma$) within the dark (light) green region. The black dashed lines indicate the shifts of the boundaries of the green regions when the gluino is taken to be $10\%$ heavier than in Fig. 3. The lightest squark mass is fixed to 1.5  TeV. In the left (right) panel, we used $m_{\bar{D}}/m_{\bar{U}}=1.1$ (2) with $m_{\bar{U}}=m_{Q}$ for $0<\theta<\pi$, and $m_{\bar{U}}/m_{\bar{D}}=1.1$ (2) with $m_{\bar{D}}=m_{Q}$ for $\pi<\theta<2\pi$. The same results are depicted in Fig. 5 but for a lightest squark mass of 3  TeV, and $m_{\bar{D}}/m_{\bar{U}}=1.5$ (2) with $m_{\bar{U}}=m_{Q}$ is used for $0<\theta<\pi$, or $m_{\bar{U}}/m_{\bar{D}}=1.5$ (2) with $m_{\bar{D}}=m_{Q}$, in the left (right) panel.

Comparing Fig. 5 to Fig. 5 we can see that if $m_{\bar{U}}/m_{\bar{D}}$ (or $m_{\bar{D}}/m_{\bar{U}}$) differs more strongly from 1, $|\mathcal{B}(K_{L}\to\pi^{0}\nu\overline{\nu})-\mathcal{B}^{\rm SM}(K_{L}\to\pi^{0}\nu\overline{\nu})|$ is predicted to be smaller in light of the $\epsilon^{\prime}_{K}/\epsilon_{K}$ discrepancy. Figs. 5 and 5 also illustrate an important finding: There is a strict correlation between $\mathcal{B}(K_{L}\to\pi^{0}\nu\overline{\nu})$ and $m_{\bar{U}}/m_{\bar{D}}$: $\mbox{sgn}\,(\mathcal{B}(K_{L}\to\pi^{0}\nu\overline{\nu})-\mathcal{B}^{\rm SM}(K_{L}\to\pi^{0}\nu\overline{\nu}))=\mbox{sgn}\,(m_{\bar{U}}-m_{\bar{D}})$. This finding is easily understood by recalling that $\mbox{sgn}\,(m_{\bar{U}}-m_{\bar{D}})$ determines whether we must choose the CP phase $\theta$ between 0 and $\pi$ or instead between $\pi$ and $2\pi$ to generate the desired positive contribution to $\epsilon_{K}^{\prime}$. Now the sign of the MSSM contribution to $\mathcal{B}(K_{L}\to\pi^{0}\nu\overline{\nu})$ depends on the CP phase in the same way, but there is no explicit dependence of $\mathcal{B}(K_{L}\to\pi^{0}\nu\overline{\nu})$ on $m_{\bar{U},\bar{D}}$. The shape of the blue regions in Figs. 5 and 5 is a generic feature of NP models with FCNC transitions only among left-handed quarks and stems from the constraint of $\epsilon_{K}$ on the new CP phases Blanke:2009pq .

Numerically, we observed $\mathcal{B}(K_{L}\to\pi^{0}\nu\overline{\nu})/\mathcal{B}^{\rm SM}(K_{L}\to\pi^{0}\nu\overline{\nu})\lesssim 2\,(1.2)$ and $\mathcal{B}(K^{+}\to\pi^{+}\nu\overline{\nu})/\mathcal{B}^{\rm SM}(K^{+}\to\pi^{+}\nu\overline{\nu})\lesssim 1.4\,(1.1)$ in light of $\epsilon^{\prime}_{K}/\epsilon_{K}$ discrepancy, if all squark are heavier than $1.5\,\,\mbox{TeV}$ and if a $1\%\,(10\%)$ fine-tuning is permitted. Here and hereafter, the quoted fine-tuning corresponds to the fine-tuning of the gluino mass or, alternatively, the tuning of the CP violating phase. Similarly, $\mathcal{B}(K_{L}\to\pi^{0}\nu\overline{\nu})/\mathcal{B}^{\rm SM}(K_{L}\to\pi^{0}\nu\overline{\nu})\lesssim 1.1$ and $\mathcal{B}(K^{+}\to\pi^{+}\nu\overline{\nu})/\mathcal{B}^{\rm SM}(K^{+}\to\pi^{+}\nu\overline{\nu})\lesssim 1.02$ are predicted, if all squark masses are above $3\,\,\mbox{TeV}$ with a $10\%$ fine-tuning.

Note that if $m_{\bar{U}}/m_{\bar{D}}$ is close to $1$, the Trojan penguin contribution from the SUSY QCD box diagrams are suppressed and the gluino contribution to the chromomagnetic operator entering $\epsilon^{\prime}_{K}/\epsilon_{K}$ becomes dominant: for $m_{\bar{U}}/m_{\bar{D}}$ = 1.05 (1.02), 25  % (50 %) of the SUSY contribution comes from the chromomagnetic operator for $m_{\tilde{q}_{1}}=1.5\,\,\mbox{TeV}$ and larger values of $|\mathcal{B}(K_{L}\to\pi^{0}\nu\overline{\nu})-\mathcal{B}^{\rm SM}(K_{L}\to\pi^{0}\nu\overline{\nu})|$ are predicted. However, it is shown that such a case always requires fine-tuning at the $1\%$ level.

IV Discussion and Conclusions

In this article, we have studied the correlations between $\epsilon_{K}$, $\epsilon_{K}^{\prime}$, $K_{L}\to\pi^{0}\nu\overline{\nu}$ and $K^{+}\to\pi^{+}\nu\overline{\nu}$ in detail within the MSSM. In order to accommodate the $\epsilon_{K}^{\prime}/\epsilon_{K}$ anomaly, we generate isospin violation by a mass splitting between right-handed up and down-squark and flavor as well as CP violating by off-diagonal elements in the left-handed bilinear squark mass terms.

We find strong correlations between these observables depending (to a very good approximation) only on $m_{Q},~{}|\Delta_{Q,12}|,~{}\theta,~{}M_{3},~{}M_{2},~{}m_{\bar{U}}/m_{\bar{D}},~{}m_{L}$. In particular, we find the following prediction: $\mbox{sgn}\,(\mathcal{B}(K_{L}\to\pi^{0}\nu\overline{\nu})-\mathcal{B}^{\rm SM}(K_{L}\to\pi^{0}\nu\overline{\nu}))=\mbox{sgn}\,(m_{\bar{U}}-m_{\bar{D}})$. This is in contrast to generic $Z^{\prime}$ models where couplings to leptons are in general free parameters, decoupling $\epsilon_{K}^{\prime}/\epsilon_{K}$ from $K_{L}\to\pi^{0}\nu\overline{\nu}$ and $K^{+}\to\pi^{+}\nu\overline{\nu}$.

We show that $\mathcal{B}(K_{L}\to\pi^{0}\nu\overline{\nu})$ is expected to be shifted with respect to the SM value by 5%–10% within the typical parameter region of our scenario. Even a larger shift is possible if one allows for fine tuning: $\mathcal{B}(K_{L}\to\pi^{0}\nu\overline{\nu})/\mathcal{B}^{\rm SM}(K_{L}\to\pi^{0}\nu\overline{\nu})\lesssim 2\,(1.2)$ and $\mathcal{B}(K^{+}\to\pi^{+}\nu\overline{\nu})/\mathcal{B}^{\rm SM}(K^{+}\to\pi^{+}\nu\overline{\nu})\lesssim 1.4\,(1.1)$ for a fine-tuning at the $1\%\,(10\%)$ level.

It is also clearly shown that our scenario can be distinguished from those with dominant $Z$-penguins. In the latter scenarios, the $Z$-penguin contributions to $\epsilon^{\prime}_{K}$ is proportional to $(\textrm{Im}{\Delta_{L}}+3.3\,\textrm{Im}{\Delta_{R}})$ and $\mathcal{B}(K_{L}\to\pi^{0}\nu\overline{\nu})-\mathcal{B}^{\rm SM}(K_{L}\to\pi^{0}\nu\overline{\nu})$ is proportional to $-(\textrm{Im}{\Delta_{L}}+\textrm{Im}{\Delta_{R}})$. Therefore, a suppression of the branching ratio of $K_{L}\to\pi^{0}\nu\overline{\nu}$ (numerically $\mathcal{B}(K_{L}\to\pi^{0}\nu\overline{\nu})/\mathcal{B}^{\rm SM}(K_{L}\to\pi^{0}\nu\overline{\nu})\lesssim 0.7$ Endo:2016tnu ) is in general predicted if there is no cancellation between $\textrm{Im}{\Delta_{L}}$ and $\textrm{Im}{\Delta_{R}}$ Buras:2015yca . Here, $\Delta_{L(R)}$ denotes the effective coupling of $\bar{s}\gamma_{\mu}P_{L(R)}dZ^{\mu}$ originating from NP interactions. This means that an accurate measurement of $K_{L}\to\pi^{0}\nu\overline{\nu}$ would be able to distinguish these scenarios.

For our analysis, we assume GUT relations among the gauginos. Relaxing this assumption allows for larger, but less correlated, effects in $K_{L}\to\pi^{0}\nu\overline{\nu}$ and $K^{+}\to\pi^{+}\nu\overline{\nu}$. Such an analysis together with a presentation of the complete analytic expressions for $\epsilon_{K}^{\prime}/\epsilon_{K}$, $K_{L}\to\pi^{0}\nu\overline{\nu}$ and $K^{+}\to\pi^{+}\nu\overline{\nu}$ will be presented in a forthcoming article.

Finally, we discuss the Higgs boson mass within the MSSM. In our phenomenological analysis ($\tan\beta=10$ and $A_{ij}=0$), the Higgs boson mass of 125  GeV can be achieved for stop masses around 5  TeV Hahn:2013ria . To accommodate for the measured Higgs mass with lighter stops we have checked that one can choose large diagonal trilinear $A_{ii}$ terms (defined with the Yukawa couplings factored out) without relevant effect on the studied observables. In particular, diagonal $A_{ii}$ terms neither generate sizable $Z$-penguins nor effects in nucleon EDM. Furthermore, promoting the MSSM to the NMSSM or adding additional $D$-term contributions to the Higgs boson mass would leave our analysis unchanged. (The patterns of flavor observables in the MSSM and NMSSM are essentially identical.) Therefore, one can account for the measured value of 125 GeV for the light Higgs boson mass within our setup.

Acknowledgments — A. C. is supported by an Ambizione Grant (No. PZ00P2_154834) of the Swiss National Science Foundation. G. D. was supported in part by MIUR under Project No. 2015P5SBHT and by the INFN research initiative ENP. The work of U. N. is supported by the German Bundesministerium für Bildung und Forschung under Grant No. 05H15VKKB1.