Tests for $CPT$ sum rule and $U$-spin violation in time-dependent $CP$ violation of $B_s^0\to K^{+}{K}^{-}$ and $B^0\to {\pi }^{+}{\pi }^{-}$

Recent LHCb data for time-dependent $CP$ violation in $B^0\to {\pi }^{+}{\pi }^{-}$ and $B_s^0\to K^{+}{K}^{-}$ show deviations from theoretical predictions. Besides their central values for ${\mathcal{C}}_{K^{+}{K}^{-}}$, ${\mathcal{S}}_{K^{+}{K}^{-}}$ and ${\mathcal{A}}_{K^{+}{K}^{-}}^{\mathrm{\Delta }\mathrm{\Gamma }}$ violate quantum mechanic $CPT$ invariant sum rule ($CPT$ sum rule) prediction of $|{\mathcal{C}}_{K^{+}{K}^{-}}{|}^2+|{\mathcal{S}}_{K^{+}{K}^{-}}{|}^2+|{\mathcal{A}}_{K^{+}{K}^{-}}^{\mathrm{\Delta }\mathrm{\Gamma }}{|}^2=1$ (LHCb data imply the sum to be $0.67\pm 0.20$.), their values for ${\mathcal{C}}_{K^{+}{K}^{-}}=0.24\pm 0.06\pm 0.02$ and ${\mathcal{C}}_{{\pi }^{+}{\pi }^{-}}=-0.24\pm 0.07\pm 0.01$ also show large violation of SU(3) or its U-spin sub-group symmetry ($\mathrm{SU}(3)/\mathrm{U}$) relation ${\mathcal{C}}_{K^{+}{K}^{-}}/{\mathcal{C}}_{{\pi }^{-}{\pi }^{+}}=-\overline{\mathcal{B}}(B^0\to {\pi }^{-}{\pi }^{+}){\tau }_{B_s^0}/\overline{\mathcal{B}}(B_s^0\to K^{+}{K}^{-}){\tau }_{B^0}$ (LHCb data imply the ratio of left-side to right-side to be $4.67\pm 1.88$.). The LHCb results need to be further confirmed to be taken seriously. We suggest to use time-dependent $CP$ violation in $B_s\to K^0{\overline{K}}^0,{\pi }^{+}{\pi }^{-},{\pi }^0{\pi }^0$ to further test the $CPT$ sum rule. Assuming that the sum rule holds, we propose that violation of the $\mathrm{SU}(3)/\mathrm{U}$ relation may indicate a large FSI phase difference in the ${\pi }^{+}{\pi }^{-}$ and $K^{+}{K}^{-}$ rescattering. We suggest several other U-spin pairs of $B\to PP$ decays to further test $\mathrm{SU}(3)/\mathrm{U}$ relations.

Figure 1

Allowed region of $|T|$, $|P|$ and $({\varphi }_P-{\varphi }_T)$ for $B^0\to {\pi }^{+}{\pi }^{-}$ (red) and $B_s^0\to K^{+}{K}^{-}$ (green) decays under the constraints of branching ratio $\mathcal{B}$ and $CP$ violation parameter ${\mathcal{C}}_f$. Here, the difference between $|T_d|$ and $|T_s|$ and $|P_d|$ and $|P_s|$ should be less than 20% is required. The dark and light regions correspond to 68% CL and 95% CL, respectively. The best-fit points for $B_d^0\to {\pi }^{+}{\pi }^{-}$ and $B_s^0\to K^{+}{K}^{-}$ correspond to ${\chi }_{\min }^2=0.00$ and 0.13, respectively.