Supersymmetric Explanation of $CP$ Violation in $K\to \pi \pi$ Decays

Recent progress in the determination of hadronic matrix elements has revealed a tension between the measured value of ${\epsilon }_K^{\prime }/{\epsilon }_K$, which quantifies direct $CP$ violation in $K\to \pi \pi$ decays, and the standard-model prediction. The well-understood indirect $CP$ violation encoded in the quantity ${\epsilon }_K$ typically precludes large new-physics contributions to ${\epsilon }_K^{\prime }/{\epsilon }_K$ and challenges such an explanation of the discrepancy. We show that it is possible to cure the ${\epsilon }_K^{\prime }/{\epsilon }_K$ anomaly in the minimal supersymmetric standard model with squark masses above 3 TeV without overshooting ${\epsilon }_K$. This solution exploits two features of supersymmetry: the possibility of large isospin-breaking contributions (enhancing ${\epsilon }_K^{\prime }$) and the Majorana nature of gluinos (permitting a suppression of ${\epsilon }_K$). Our solution involves no fine-tuning of $CP$ phases or other parameters.

Figure 1

Supersymmetric gluino box contribution to ${\epsilon }_K^{\prime }/{\epsilon }_K$ (called a Trojan penguin in Ref. [25]). It contributes to $\mathrm{Im}{A}_2$ for $m_{\overline{U}}\ne m_{\overline{D}}$ and is the largest contribution in our scenario. There are also crossed box diagrams.

Figure 2

The left plot shows ${\epsilon }_K^{\mathrm{SUSY}}/{\epsilon }_K^{\exp }$ as a function of the gluino-squark mass ratio $m_{\tilde{g}}/M_S$, where we take $M_S=m_Q=m_{\overline{U}}=m_{\overline{D}}=10\mathrm{TeV}$. The red line shows the gluino-gluino box contribution (with the zero crossing near $m_{\tilde{g}}/M_S=1.5$ [26]), while the blue line denotes the sum of the box contributions with one or two winos. The total contribution is shown in black. The red (blue) regions are excluded by the measurement of ${\epsilon }_K$ at the 95 % confidence level (C.L.), if the SM prediction uses the inclusive (exclusive) measurement of $|V_{cb}|$ [28]. On the right, the black lines show $|{\epsilon }_K^{\mathrm{SUSY}}|$ for several gluino-squark mass ratios as a function of the squark mass.

Figure 3

Individual supersymmetric contributions to $|{\epsilon }_K^{\prime }/{\epsilon }_K|$ as a function of $M_S=m_Q=m_{\overline{D}}$. $\tilde{g}\tilde{g}$, $\tilde{g}{\tilde{\chi }}^0$, $\tilde{g}g$, $\tilde{g}\gamma$, $\tilde{g}Z$, ${\tilde{\chi }}^{-}Z$, ${\tilde{\chi }}^{0}Z$, and $\tilde{g}{g}^{*}$ represent the gluino-gluino and gluino-neutralino boxes, gluino gluon, photon, and $Z$ penguins, chargino and neutralino $Z$ penguins, and chromomagnetic contributions, respectively. The thick lines show the case of universal squark masses $m_{\overline{U}}=M_S$. The broken black lines are the gluino-gluino box contributions for $m_{\overline{U}}/M_S=0.5$, 2.0, 0.8, 1.2 from top to bottom. The ${\epsilon }_K^{\prime }/{\epsilon }_K$ discrepancy is resolved at $1\sigma$ ($2\sigma$) in the dark (light) green band.

Figure 4

Contours of the supersymmetric contributions to ${\epsilon }_K^{\prime }/{\epsilon }_K$ in units of ${10}^{-4}$. The ${\epsilon }_K^{\prime }/{\epsilon }_K$ discrepancy is resolved at $1\sigma$ ($2\sigma$) in the dark (light) green region. The red shaded region is excluded by ${\epsilon }_K$ with inclusive $|V_{cb}|$ at 95% C.L., while the region between the blue-dashed lines can explain the ${\epsilon }_K$ discrepancy which is there for the exclusive $|V_{cb}|$. The green regions labeled with negative ${\epsilon }_K^{\prime }/{\epsilon }_K$ correspond to the change ${\mathrm{\Delta }}_{Q,12,13,23}=0.1\exp (-i\pi /4)\to {\mathrm{\Delta }}_{Q,12,13,23}=0.1\exp (i3\pi /4)$, which flips the sign of ${\epsilon }_K^{\prime \mathrm{SUSY}}$ (making it positive) while leaving ${\epsilon }_K$ essentially unchanged.