On P-wave meson decay constants in the heavy quark limit of QCD

Abstract

In previous work it has been shown that, either from a sum rule for the subleading Isgur-Wise function ${\xi }_3(1)$ or from a combination of Uraltsev and Bjorken SR, one infers for $P$-wave states $|{\tau }_{1/2}(1)|\ll |{\tau }_{3/2}(1)|$. This implies, in the heavy quark limit of QCD, a hierarchy for the {\it production} rates of $P$-states $\Gamma ({\bar{B}}_d\to D(\frac{1}{2})\ell \nu )\ll \Gamma ({\bar{B}}_d\to D(\frac{3}{2})\ell \nu )$ that seems at present to be contradicted by experiment. It was also shown that the decay constants of $j=\frac{3}{2}$ $P$-states vanish in the heavy quark limit of QCD, $f_{3/2}^{(n)}=0$. Assuming the {\it model} of factorization in the decays ${\bar{B}}_d\to {\bar{D}}_s^{\ast \ast }D$, one expects the opposite hierarchy for the {\it emission} rates $\Gamma ({\bar{B}}_d\to {\bar{D}}_s(\frac{3}{2})D)\ll \Gamma ({\bar{B}}_d\to {\bar{D}}_s(\frac{1}{2})D)$, since $j=\frac{1}{2}$ $P$-states are coupled to vacuum. Moreover, using Bjorken SR and previously discovered SR involving heavy-light meson decay constants and IW functions, one can prove that the sums $\sum _n(\frac{f^{(n)}}{f^{(0)}}{)}^2$, $\sum _n(\frac{f_{1/2}^{(n)}}{f^{(0)}}{)}^2$ (where $f^{(n)}$ and $f_{1/2}^{(n)}$ are the decay constants of $S$-states and $j=\frac{1}{2}$ $P$-states) are divergent. This situation seems to be realized in the relativistic quark models à la Bakamjian and Thomas, that satisfy HQET and predict decays constants $f^{(n)}$ and $f_{1/2}^{(n)}$ that do not decrease with the radial quantum number $n$.

Comments: 7 pages, Latex