The Subleading Isgur-Wise Form Factor $\chi_3(v\cdot v')$ to Order $\alpha_s$ in QCD Sum Rules

We calculate the contributions arising at order $\alpha_s$ in the QCD sum rule for the spin-symmetry violating universal function $\chi_3(v\cdot v')$, which appears at order $1/m_Q$ in the heavy quark expansion of meson form factors. In particular, we derive the two-loop perturbative contribution to the sum rule. Over the kinematic range accessible in $B\to D^{(*)}\ell\,\nu$ decays, we find that $\chi_3(v\cdot v')$ does not exceed the level of $\sim 1\%$, indicating that power corrections induced by the chromo-magnetic operator in the heavy quark expansion are small.