Tests for CPT sum rule and U-spin violation in Time-dependent CP violation of $B^0_s \to K^+ K^-$ and $B^0_d \to π^+ π^-$

Abstract

Recent LHCb data for time-dependent CP violation in $B_d^0 \to π^+π^-$ and $B^0_s\to K^+K^-$ show deviations from theoretical predictions. Besides their central values for $\mathcal C_{K^+K^-}$, $\mathcal S_{K^+K^-}$ and $\mathcal A^{ΔΓ}_{K^+K^-}$ violate quantum mechanic CPT invariant sum rule (CPT sum rule) prediction of $|\mathcal C_{K^+K^-}|^2 + |\mathcal S_{K^+K^-}| ^2 + |\mathcal A^{ΔΓ}_{K^+K^-}|^2 = 1$ (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 $\mathcal C_{π^+ π^-} = - 0.24\pm 0.07\pm 0.01$ also show large violation of SU(3) or its U-spin sub-group symmetry (SU(3)/U) relation $\mathcal C_{K^+ K^-} /\mathcal C_{π^- π^+} = - \mathcal B(B_d^0 \to π^- π^+)τ_{B^0_s}/\mathcal B(B^0_s \to K^+ K^-)τ_{B_d^0}$ (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 $B_s\to K^0\bar K^0, π^+π^-, π^0π^0$ 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 $π^+π^-$ 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 $B^0_d \to π^+ π^-$ updated to the latest HFAG average, figures slightly changed, a few comments and refs added