Remarks on sum rules in the heavy quark limit of QCD

Abstract

We underline a problem existing in the heavy quark limit of QCD concerning the rates of semileptonic B decays into P-wave $D_J(j)$ mesons, where $j=\frac{1}{2}$ (wide states) or $j=\frac{3}{2}$ (narrow states). The leading order sum rules of Bjorken and Uraltsev suggest $\Gamma [\bar{B}\to D_{0,1}(\frac{1}{2})\ell \nu ]\ll \Gamma [\bar{B}\to D_{1,2}(\frac{3}{2})\ell \nu ]$, in contradiction with experiment. The same trend follows also from a sum rule for the subleading $1/m_Q$ curent matrix element correction ${\xi }_3(1)$. The problem is made explicit in relativistic quarks models à la Bakamjian and Thomas, that give a transparent physical interpretation of the latter as due, not to a $L\cdot S$ force, but to the Wigner rotation of the light quark spin. We point out moreover that the selection rule for decay constants of $j=\frac{3}{2}$ states, $f_{3/2}=0$, predicts, assuming the model of factorization, the opposite hierarchy $\Gamma [\bar{B}\to {\bar{D}}_{s_{1,2}}(\frac{3}{2})D^{(\ast )}]\ll \Gamma [\bar{B}\to {\bar{D}}_{s_{0,1}}(\frac{1}{2})D^{(\ast )}]$.

Comments: Contribution to the International Europhysics Conference on HEP, Budapest, July 2001 (presented by L. Oliver); 5 pages