Determination of the CP violating phase γ by a sum over common decay modes toB _s and$\bar B_s$

Abstract

To help the difficult determination of the angle γ of the unitarity triangle, Aleksan, Dunietz and Kayser have proposed the modes of the typeK ⁻ D ⁺ _s , common toB _s and$\bar B_s$. We point out that it is possible to gain in statistics by a sum over all modes with ground state mesons in the final state, i.e.K ⁻ D ⁺ _s ,K ^*− D ⁺ _s ,K ⁻ D ^*+ _s ,K ^* D ^* _s . The delicate point is the relative phase of these different contributions to the dilution factorD of the time dependent asymmetry. Each contribution toD is proportional to a product$F^{cb} F^{ub} f_{D_s } f_K$ whereF denotes form factors andf decay constants. Within a definite phase convention (i.e. for example the one defined by the heavy quark symmetry in the limit of heavy quarks), lattice calculations do not show any change in sign when extrapolating to light quarks the form factors and decay constants. Then, we can show that all modes contribute constructively to the dilution factor, except theP-waveK ^* D ^*+ _s , which is small. Quark model arguments based on wave function overlaps also confirm this stability in sign. By summing over all these models we find a gain of a factor 6 in statistics relatively toK ⁻ D ⁺ _s . The dilution factor for the sumD _tot is remarkably stable for theoretical schemes that are not in very strong conflict with data onB→ψK(K ^*) or extrapolated from semileptonic charm form factors, givingD _tot≥0.6, always close toD(K ⁻ D ⁺ _s ).