$K\to \pi \nu \overline{\nu }$ in the MSSM in light of the ${\epsilon }_K^{\prime }/{\epsilon }_K$ anomaly

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 $K_L\to {\pi }^0\nu \overline{\nu }$ and $K^{+}\to {\pi }^{+}\nu \overline{\nu }$ are similarly sensitive to very high mass scales and the first one also measures $CP$ violation. In this article, we analyze the correlations between ${\epsilon }_K^{\prime }/{\epsilon }_K$ and $\mathcal{B}(K_L\to {\pi }^0\nu \overline{\nu })$ and $\mathcal{B}(K^{+}\to {\pi }^{+}\nu \overline{\nu })$ within the MSSM aiming at an explanation of ${\epsilon }_K^{\prime }/{\epsilon }_K$ via gluino-squark box diagrams. The dominant MSSM contribution to the $K\to \pi \nu \overline{\nu }$ 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 $\mathcal{B}(K_L\to {\pi }^0\nu \overline{\nu })/{\mathcal{B}}^{\mathrm{SM}}(K_L\to {\pi }^0\nu \overline{\nu })\lesssim 2(1.2)$ and $\mathcal{B}(K^{+}\to {\pi }^{+}\nu \overline{\nu })/{\mathcal{B}}^{\mathrm{SM}}(K^{+}\to {\pi }^{+}\nu \overline{\nu })\lesssim 1.4(1.1)$, 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 $\mathcal{B}(K_L\to {\pi }^0\nu \overline{\nu })$ and the hierarchy between the masses $m_{\overline{U}}$, $m_{\overline{D}}$ of the right-handed up-squark and down-squark: $\mathrm{sgn}[\mathcal{B}(K_L\to {\pi }^0\nu \overline{\nu })-{\mathcal{B}}^{\mathrm{SM}}(K_L\to {\pi }^0\nu \overline{\nu })]=\mathrm{sgn}(m_{\overline{U}}-m_{\overline{D}})$.

Figure 1

Feynman diagrams of the dominant MSSM contributions to ${\epsilon }_K^{\prime }$, $K\to \pi \nu \overline{\nu }$, and ${\epsilon }_K$ in our scenario. $\tilde{Q}$ denotes a left-handed squark which is a down-strange mixture in our setup. $\tilde{U}$ ($\tilde{D}$) represents the right-handed up (down) squark. $\tilde{g}$, ${\tilde{\chi }}^0$, and ${\tilde{\chi }}^{\pm }$ stand for gluino, neutralino, and chargino, respectively, and $\tilde{L}$ denotes a charged slepton. First row: The first two box diagrams feed ${\epsilon }_K^{\prime }$ through $A_2$ in Eq. (6) if $m_U\ne m_D$. The last diagram gives the ballpark of the MSSM contribution to $\mathcal{B}(K\to \pi \nu \overline{\nu })$. Second row: MSSM contributions to ${\epsilon }_K$.

Figure 2

Contours of $\mathcal{B}(K_L\to {\pi }^0\nu \overline{\nu })/{\mathcal{B}}^{\mathrm{SM}}(K_L\to {\pi }^0\nu \overline{\nu })$. The ${\epsilon }_K^{\prime }/{\epsilon }_K$ discrepancy is resolved at the $1\sigma$ ($2\sigma$) level within the dark (light) green region. The red shaded region is excluded by ${\epsilon }_K$ at 95% C.L. using the inclusive value $|V_{cb}|$, while the region between the blue-dashed lines can explain the ${\epsilon }_K$ discrepancy which is present if the exclusive determination of $V_{cb}$ is used [43]. The blue shaded region is excluded by the current LHC results from CMS and ATLAS [40, 41, 42]. $M_3/M_S=1.5$, $m_L=300\mathrm{GeV}$ and GUT relations among gaugino masses are used. In the left plot, ${\mathrm{\Delta }}_{Q,12}=0.1\exp (-i\pi /4)$ for $m_{\overline{U}}>m_{\overline{D}}=m_Q=M_S$ (upper branch) and ${\mathrm{\Delta }}_{Q,12}=0.1\exp (i3\pi /4)$ for $m_{\overline{U}}<m_{\overline{D}}=m_Q=M_S$ (lower branch). In the right plot, $|{\mathrm{\Delta }}_{Q,12}|=0.1$ is used, $m_{\overline{D}}=2m_{\overline{U}}=2m_Q=2M_S$ (for $0<\theta <\pi$) and $m_{\overline{U}}=2m_{\overline{D}}=2m_Q=2M_S$ (for $\pi <\theta <2\pi$).

Figure 3

Allowed region in the $\mathcal{B}(K_L\to {\pi }^0\nu \overline{\nu })-\mathcal{B}(K^{+}\to {\pi }^{+}\nu \overline{\nu })$ plane. “SM” in the axis labels represents the corresponding value of the branching ratio within the SM. The contours show the values of $M_3/M_S$ which is needed to cancel the SUSY contributions to ${\epsilon }_K$. In the left (right) panel, the lightest squark mass is fixed at 1.5 (3) TeV. The gray shaded region is the Grossman-Nir bound [44]. The right sides of the blue dashed lines are the experimental result for $\mathcal{B}(K^{+}\to {\pi }^{+}\nu \overline{\nu })$ given in Eq. (4).

Figure 4

The light (dark) blue region requires a milder parameter fine-tuning than 1% (10%) of the gluino mass compared to the value of Fig. 3 and a milder parameter tuning than 1% (10%) of the deviation of the $CP$ violating phase from $\pm \pi /2$. The plot suggests that by tuning the $CP$ violating phase $\theta$ close to $\pm \pi /2$ one can relax the fine-tuning of the gluino mass and still amplify the branching ratios. The red contour represents the SUSY contributions to ${\epsilon }_K^{\prime }/{\epsilon }_K$, and the ${\epsilon }_K^{\prime }/{\epsilon }_K$ discrepancy is resolved at $1\sigma$ ($2\sigma$) within the dark (light) green region. The black dashed lines show the projected shifts of the boundaries of the green regions when the gluino is assumed to be 10% heavier. The lightest squark mass is fixed to 1.5 TeV. In the left panel, $m_{\overline{D}}/m_{\overline{U}}=1.1$ ($m_{\overline{U}}/m_{\overline{D}}=1.1$) is used for $0<\theta <\pi$ ($\pi <\theta <2\pi$) to obtain a positive SUSY contribution to ${\epsilon }_K^{\prime }/{\epsilon }_K$. While, $m_{\overline{D}}/m_{\overline{U}}=2$ ($m_{\overline{U}}/m_{\overline{D}}=2$) is used for $0<\theta <\pi$ ($\pi <\theta <2\pi$) in the right panel. The region on the right side of the blue dashed lines are allowed by the current experimental measurements [given in Eq. (4)].

Figure 5

Same as Fig. 4 but for $m_{{\tilde{q}}_1}=3\mathrm{TeV}$, $m_{\overline{D}}/m_{\overline{U}}(\mathrm{or}{m}_{\overline{U}}/m_{\overline{D}})=1.5$ (left panel) and $m_{\overline{D}}/m_{\overline{U}}(\mathrm{or}{m}_{\overline{U}}/m_{\overline{D}})=2$ (right).