Correlations among new CP violating effects in Delta F = 2 observables

TL;DR

The paper investigates whether the SM can accommodate current CP-violation measurements in $K$ and $B_d$ systems given new lattice inputs for $\hat B_K$ and a suppression factor $\kappa_{\epsilon}$. It derives the corrected $\epsilon_K$ formula including $\phi_\epsilon$ and $\xi$, and demonstrates that the SM prediction for $|\epsilon_K|$ can be systematically below the observed value, motivating new CP-violating contributions. The authors explore three NP scenarios—NP in the $K$ system, in the $B_d$ system, or in both with a common phase—showing that a modest negative NP phase in Bd and Bs could reconcile $|\epsilon_K|$ with $S_{\psi K_s}$ and potentially enhance $S_{\psi\phi}$, depending on CKM parameters. They emphasize correlations among $\epsilon_K$, $S_{\psi K_s}$, and $S_{\psi\phi}$ and stress the crucial role of precise $V_{cb}$, $\hat B_K$, and $\xi_s$ determinations to identify or constrain new CP-violating effects in $\Delta F=2$ transitions.

Abstract

We point out that the observed CP violation in Bd - bar{Bd} mixing, taking into account the measured ratio Delta Md / Delta Ms, the recently improved lattice value of the non-perturbative parameter BK and an additional effective suppression factor kappa_epsilon ~= 0.92 in epsilon_K neglected sofar in most analyses, may be insufficient to describe the measured value of epsilon_K within the Standard Model (SM), thus hinting at new CP violating contributions to the K - bar{K} and/or Bd - bar{Bd} systems. Furthermore, assuming that Delta Md / Delta Ms is SM-like, the signs and the magnitudes of new physics effects in epsilon_K and in the CP asymmetries S_{psi Ks} and S_{psi phi} may turn out to be correlated. For example, in a scenario with new CP-phases in Bd and Bs mixings being approximately equal and negative, a common new phase ~= - 5 degrees could remove the tension between epsilon_K and S_{psi Ks} present in the SM and simultaneously accommodate, at least partly, the recent claim of S_{psi phi} being much larger than the SM expectation. We emphasize the importance of precise determinations of Vcb, BK, FK and xi_s, to which the parameter epsilon_K and its correlation with the CP violation in the Bd - bar{Bd} system are very sensitive.

Figures (5)

• Figure 1: Left panel: $|\epsilon_K^{\rm SM}|$ vs. $\sin 2 \beta$ with only $\hat{B}_K$ errors included. $\hat{B}_K \in \{0.65, 0.70, 0.75, 0.80\} \pm 3$% are shown as blue areas (darker to lighter). Vertical green areas display $\sin 2 \beta \in \{ 0.681, 0.75, 0.88 \} \pm 3.7$% (see text). Right panel: $\gamma$ vs. $|V_{ub}|$ for $\sin 2 \beta \in \{0.681,0.75,0.88\} \pm 3.7$% (green areas). Displayed in blue is the area corresponding to $R_t = R_t^{\rm SM}$, while orange lines represent the contours of $\epsilon_K^{\rm exp}$.
• Figure 2: $|\epsilon_K^{\rm SM}|$ vs. $\sin 2 \beta$ with inclusion of all input uncertainties. Left panel assumes present $\hat{B}_K$ and $\xi_s$ errors, whereas right panel shows the situation with errors on both quantities shrunk to 2.5%.
• Figure 3: $\hat{B}_K$ ranges compatible with the experimental $\epsilon_K$ result as a function of $\sin 2 \beta$. $\hat{B}_K$ and/or $\xi_s$ are taken with present or 2.5% uncertainties (see legend). Comparing red with blue areas one can note the role of a decrease in the $\xi_s$ error.
• Figure 4: CP asymmetry $S_{\psi \phi}$ as a function of $S_{\psi K_s}$ for a common NP phase $\phi_B \in [-12, +2]^\circ$.
• Figure 5: $|V_{ub}|$ ranges implied by a given NP phase in the $B_d$ system, $\phi_d$. The green band displays the most recent $|V_{ub}|$ average quoted in the PDG PDG08.