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

Abstract

Recent LHCb data for time-dependent CP violation in $B_d^0\to {\pi }^{+}{\pi }^{-}$ 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 ${\mathcal{A}}_{K^{+}{K}^{-}}^{\Delta \Gamma }$ violate quantum mechanic CPT invariant sum rule (CPT sum rule) prediction of $|{\mathcal{C}}_{K^{+}{K}^{-}}{|}^2+|{\mathcal{S}}_{K^{+}{K}^{-}}{|}^2+|{\mathcal{A}}_{K^{+}{K}^{-}}^{\Delta \Gamma }{|}^2=1$ (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 (SU(3)/U) relation ${\mathcal{C}}_{K^{+}{K}^{-}}/{\mathcal{C}}_{{\pi }^{-}{\pi }^{+}}=-\mathcal{B}(B_d^0\to {\pi }^{-}{\pi }^{+}){\tau }_{B_s^0}/\mathcal{B}(B_s^0\to K^{+}{K}^{-}){\tau }_{B_d^0}$ (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 $B_s\to K^0{\bar{K}}^0,{\pi }^{+}{\pi }^{-},{\pi }^0{\pi }^0$ to further test the CPT sum rule. Assuming that the sum rule holds, we propose that violation of the SU(3)/U relation may indicate a large FSI phase difference in the ${\pi }^{+}{\pi }^{-}$ and $K^{+}{K}^{-}$ re-scattering. We suggest several other U-spin pairs of $B\to PP$ decays to further test SU(3)/U relations.

arXiv comment12 pages, 1 figure; ACP of $B_d^0\to {\pi }^{+}{\pi }^{-}$ updated to the latest HFAG average, figures slightly changed, a few comments and refs added

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