Sum rules in the heavy quark limit of QCD and Isgur-Wise functions

Abstract

Using the OPE, we formulate new sum rules in the heavy quark limit of QCD. These sum rules imply that the elastic Isgur-Wise function $\xi (w)$ is an alternate series in powers of $(w-1)$. Moreover, one gets that the $n$-th derivative of $\xi (w)$ at $w=1$ can be bounded by the $(n-1)$-th one, and an absolute lower bound for the $n$-th derivative $(-1{)}^n{\xi }^{(n)}(1)\ge \frac{(2n+1)!!}{2^{2n}}$. Moreover, for the curvature we find ${\xi }^{\prime \prime }(1)\ge \frac{1}{5}[4{\rho }^2+3({\rho }^2{)}^2]$ where ${\rho }^2=-{\xi }^{\prime }(1)$. We show that the quadratic term $\frac{3}{5}({\rho }^2{)}^2$ has a transparent physical interpretation, as it is leading in a non-relativistic expansion in the mass of the light quark. These bounds should be taken into account in the parametrizations of $\xi (w)$ used to extract $|V_{cb}|$. These results are consistent with the dispersive bounds, and they strongly reduce the allowed region of the latter for $\xi (w)$. The method is extended to the subleading quantities in $1/m_Q$, namely ${\xi }_3(w)$ and $\bar{\Lambda }\xi (w)$.}]

Comments: Talk given at the ICHEP04 Conference (Beijing, August 2004)