On the Determination of $a_1$ and $a_2$ from Hadronic Two Body $B$ Decays

Abstract

{}From Class I decays : $B^o \ra π^+ ( ρ^+ ) + D^- ( {D^*}^- )$, we determine $a_1$, and from Class III decays : $B^+ \ra π^+ ( ρ^+ ) + \ol{D}^o ( {\ol{D}^*}^o )$, we determine an allowed domain in the $( a_1, a_2 )$ plane. We find that within one standard deviation errors, the allowed band of $a_1$ from Class I decays is at least three standard deviations removed from the allowed domain $( a_1, a_2 )$ from Class III decays.If we expand the experimental errors to two standard deviations we do find a small intersection between the $a_1$ band and the allowed $( a_1, a_2 )$ domain. The results usually quoted in the literature lie in this intersection. We suggest : (1) an independent measurement of the branching ratio for the Class III decay, $B^+ \ra ρ^+ \ol{D}^o $, (2) a high-statistics measurement of the branching ratio of the Class I decay, $B \ra \ol{D} ( \ol{D}^* ) + D_s ( D_s^* )$ in both charged states, and (3) a measurement of the longitudinal polarization fraction in the Class III decay $B^+ \ra ρ^+ {\ol{D}^*}^o $ to shed more light on the questions involved .