The Isgur-Wise function in a relativistic model for $q\bar Q$ system

Abstract

We use the Dirac equation with a ``(asymptotically free) Coulomb + (Lorentz scalar) linear '' potential to estimate the light quark wavefunction for $ q\bar Q$ mesons in the limit $m_Q\to \infty$. We use these wavefunctions to calculate the Isgur-Wise function $ξ(v.v^\prime )$ for orbital and radial ground states in the phenomenologically interesting range $1\leq v.v^ \prime \leq 4$. We find a simple expression for the zero-recoil slope, $ξ^ \prime (1) =-1/2- ε^2 <{r_q}^2>/3$, where $ε$ is the energy eigenvalue of the light quark, which can be identified with the $\barΛ$ parameter of the Heavy Quark Effective Theory. This result implies an upper bound of $-1/2$ for the slope $ξ^\prime (1)$. Also, because for a very light quark $q (q=u, d)$ the size $\sqrt {<{r_q}^2>}$ of the meson is determined mainly by the ``confining'' term in the potential $(γ_\circ σr)$, the shape of $ξ_{u,d}(v.v^\prime )$ is seen to be mostly sensitive to the dimensionless ratio $\bar Λ_{u,d}^2/σ$. We present results for the ranges of parameters $150 MeV <\bar Λ_{u,d} <600 MeV$ $(\barΛ_s \approx \barΛ_{u,d}+100 MeV)$, $0.14 {GeV}^2 \leq σ\leq 0.25 {GeV}^2$ and light quark masses $m_u, m_d \approx 0, m_s=175 MeV$ and compare to existing experimental data and other theoretical estimates. Fits to the data give: ${\barΛ_{u,d}}^2/σ=4.8\pm 1.7 $, $-ξ^\prime_{u,d}(1)=2.4\pm 0.7$ and $\vert V_{cb} \vert \sqrt {\frac {τ_B}{1.48 ps}}=0.050\pm 0.008$ [ARGUS '93]; ${\barΛ_{u,d}}^2/σ= 3.4\pm 1.8$, $-ξ^\prime_{u,d}(1)=1.8\pm 0.7$ and $\vert V_{cb} \vert \sqrt { \frac {τ_B}{1.48 ps}}=0.043\pm 0.008$ [CLEO '93]; ${\barΛ_{u,d}}^2/