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 $\bar{q}{\gamma }^{[{\mu }_1}\dots {\gamma }^{{\mu }_n]}q$ 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 $f_{{\mathrm{B}}^{\ast }}/f_{\mathrm{B}}=1-2{\alpha }_{\mathrm{s}}(m_b)/3\pi -K_b{\alpha }_{\mathrm{s}}^2/{\pi }^2+\mathrm{O}({\alpha }_{\mathrm{s}}^3,1/m_b)$ 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, ${\Delta }_c=0.18\pm 0.01$, that takes account of the non--zero ratio $m_c/m_b=0.28\pm 0.03$. Fastest apparent convergence would entail ${\alpha }_{\mathrm{s}}(\mu )$ 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 ${\beta }_0=11-\frac{2}{3}{N}_l$. Consistent cancellation between infrared-- and ultraviolet--renormalon ambiguities is demonstrated.

Comments: 26 pages, preprint OUT-4102-52, LaTeX with epsf style option, uuencoded PostScript figure (compressed by gzip) appended at the end