Tests for CPT sum rule and U-spin violation in Time-dependent CP violation of $B_s^0\to K^{+}{K}^{-}$ and $B_d^0\to \ddot{I}{€}^{+}\ddot{I}{€}^{-}$

Abstract

Recent LHCb data for time-dependent CP violation in $B_d^0\to \ddot{I}{€}^{+}\ddot{I}{€}^{-}$ 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 ${\mathcal{A}}_{K^{+}{K}^{-}}^{\hat{I}”\hat{I}“}$ violate quantum mechanic CPT invariant sum rule (CPT sum rule) prediction of $|{\mathcal{C}}_{K^{+}{K}^{-}}{|}^2+|{\mathcal{S}}_{K^{+}{K}^{-}}{|}^2+|{\mathcal{A}}_{K^{+}{K}^{-}}^{\hat{I}”\hat{I}“}{|}^2=1$ (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 ${\mathcal{C}}_{\ddot{I}{€}^{+}\ddot{I}{€}^{-}}=-0.24\pm 0.07\pm 0.01$ also show large violation of SU(3) or its U-spin sub-group symmetry (SU(3)/U) relation ${\mathcal{C}}_{K^{+}{K}^{-}}/{\mathcal{C}}_{\ddot{I}{€}^{-}\ddot{I}{€}^{+}}=-\mathcal{B}(B_d^0\to \ddot{I}{€}^{-}\ddot{I}{€}^{+})\ddot{I}{„}_{B_s^0}/\mathcal{B}(B_s^0\to K^{+}{K}^{-})\ddot{I}{„}_{B_d^0}$ (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 $B_s\to K^0{\bar{K}}^0,\ddot{I}{€}^{+}\ddot{I}{€}^{-},\ddot{I}{€}^0\ddot{I}{€}^0$ to further test the CPT sum rule. Assuming that the sum rule holds, we propose that violation of the SU(3)/U relation may indicate a large FSI phase difference in the $\ddot{I}{€}^{+}\ddot{I}{€}^{-}$ and $K^{+}{K}^{-}$ re-scattering. We suggest several other U-spin pairs of $B\to PP$ decays to further test SU(3)/U relations.

12 pages, 1 figure; ACP of $B_d^0\to \ddot{I}{€}^{+}\ddot{I}{€}^{-}$ updated to the latest HFAG average, figures slightly changed, a few comments and refs added