DETERMINATION OF THE CP VIOLATING PHASE GAMMA BY A SUM OVER COMMON DECAY MODES TO B-S AND (B)OVER-BAR(S)

To help the difficult determination of the angle gamma of the unitarity triangle, Aleksan, Dunietz and Kayser have proposed the modes of the type K-Ds+ ,common to B-s and ($) over 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-Ds+ ,K*(-) D-s(+), K-Ds*(+), K*D--(s)*(+). The delicate point is the relative phase of these different contributions to the dilution factor D of the time dependent asymmetry. Each contribution to D is proportional to a product F(cb)F(ub)f(Ds)f(K) where F denotes form factors and f 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 the P-wave K*(-) 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 to K-Ds+. The dilution factor for the sum D-tot is remarkably stable for theoretical schemes that are not in very strong conflict with data on B-->psi K(K*) or extrapolated from semileptonic charm form factors, giving D(tot)greater than or equal to 0.6, always close to D(K-Ds+).