Decay constants in the heavy quark limit in models à la Bakamjian and Thomas

Abstract

In quark models à la Bakamjian and Thomas, that yield covariance and Isgur-Wise scaling of form factors in the heavy quark limit, we compute the decay constants $f^{(n)}$ and $f_{1/2}^{(n)}$ of S-wave and P-wave mesons composed of heavy and light quarks. Heavy quark limit scaling $\sqrt{M}f=Cst$ is obtained, and it is shown that this class of models satisfies the sum rules involving decay constants and Isgur-Wise functions recently formulated by us in the heavy quark limit of QCD. Moreover, the model also satisfies the selection rules of the type $f_{3/2}^{(n)}=0$ that must hold in this limit. We discuss different Ansätze for the dynamics of the mass operator at rest. For non-relativistic kinetic energies $\frac{p^2}{2m}$ the decay constants are finite even if the potential $V(r)$ has a Coulomb part. For the relativistic form $\sqrt{p^2+m^2}$, the S-wave decay constants diverge if there is a Coulomb singularity. Using phenomenological models of the spectrum with relativistic kinetic energy and regularized short distance part (Godfrey-Isgur model or Richardson potential of Colangelo et al.), that yield ${\rho }^2\simeq 1$ for the elastic Isgur-Wise function, we compute the decay constants in the heavy quark limit, and obtain $f_B\simeq$ 300 MeV, of the same order although slightly smaller than in the static limit of lattice QCD. We find the decay constants of $D^{\ast \ast }$ with $j=1/2$ of the same order of magnitude. The convergence of the heavy quark limit sum rules is also studied.

Comments: 21 pages, 2 figures, LaTeX2e, amsbsy (from the AMS-LaTeX package), epsf