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

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 $\Gamma(\bar{B}_d \to D ({1 \over 2}) \ell \nu) \ll \Gamma(\bar{B}_d \to D ({3 \over 2}) \ell \nu)$ 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, $f_{3\mathrm{/}2}^{(n)}=0$. 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 $\Gamma(\bar{B}_d \to \bar{D}_s ({3 \over 2}) D) \ll \Gamma(\bar{B}_d \to \bar{D}_s ({1 \over 2}) D)$,since$j = {1 \over 2}$ $P$-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 $\sum _n(\frac{f^{(n)}}{f^{(0)}}{)}^2$, $\sum _n(\frac{f_{1\mathrm{/}2}^{(n)}}{f^{(0)}}{)}^2$ (where $f^{(n)}$ and $f_{1\mathrm{/}2}^{(n)}$ 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 `a la Bakamjian and Thomas, that satisfy HQET and predict decays constants $f^{(n)}$ and $f_{1\mathrm{/}2}^{(n)}$ that do not decrease with the radial quantum number $n$.