Matching QCD and HQET heavy--light currents at two loops and beyond

Abstract

Heavy--light QCD currents are matched with HQET currents at two loops and leading order in $1/m$. A single formula applies to all current matchings. As a by--product, a master formula for the two--loop anomalous dimension of the QCD current $\bar{q}\gamma^{[\mu_1}\ldots\gamma^{\mu_n]}q$ is obtained, yielding a new result for the tensor current. The dependence of matching coefficients on $\gamma_5$ prescriptions is elucidated. Ratios of QCD matrix elements are obtained, independently of the three--loop anomalous dimension of HQET currents. The two--loop coefficient in $f_{{\rm B}^*}/f_{\rm B} =1-2\alpha_{\rm s}(m_b)/3\pi-K_b\alpha_{\rm s}^2/\pi^2 +{\rm O}(\alpha_{\rm s}^3,1/m_b)$ is \[K_b=\frac{83}{12}+\frac{4}{81}\pi^2+\frac{2}{27}\pi^2\log2-\frac19\zeta(3) -\frac{19}{54}N_l+\Delta_c=6.37+\Delta_c\] with $N_l=4$ light flavours, and a correction, $\Delta_c=0.18\pm0.01$, that takes account of the non--zero ratio $m_c/m_b=0.28\pm0.03$. Fastest apparent convergence would entail $\alpha_{\rm s}(\mu)$ at $\mu=370$~MeV. ``Naive non--abelianization'' of large--$N_l$ results, via $N_l\to N_l-\frac{33}{2}$, gives reasonable approximations to exact two--loop results. All--order results for anomalous dimensions and matching coefficients are obtained at large $\beta_0=11-\frac23N_l$. Consistent cancellation between infrared-- and ultraviolet--renormalon ambiguities is demonstrated.