Matching QCD and heavy-quark effective theory heavy-light currents at two loops and beyond

Heavy-light QCD currents are matched with heavy-quark effective theory (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 q¯${\gamma }_1^{[\mathrm{\mu }}$...${\gamma }_{\mathit{n}]}^{\mathrm{\mu }}$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 ${\mathit{f}}_{\mathit{B}}^{\mathrm{*}}$/${\mathit{f}}_{\mathit{B}}$ =1-2${\mathrm{\alpha }}_{\mathit{s}}$(${\mathit{m}}_{\mathit{b}}$)/3π-${\mathit{K}}_{\mathit{b}}$ ${\mathrm{\alpha }}_{\mathit{s}}^2$/${\mathrm{\pi }}^2$ +O(${\mathrm{\alpha }}_{\mathit{s}}^3$,1/${\mathit{m}}_{\mathit{b}}$) is ${\mathit{K}}_{\mathit{b}}$=83/12+4/81${\mathrm{\pi }}^2$+2/27${\mathrm{\pi }}^2$ln2-1/9ζ(3)-19/54${\mathit{N}}_{\mathit{l}}$ +${\mathrm{\Delta }}_{\mathit{c}}$=6.37+${\mathrm{\Delta }}_{\mathit{c}}$, with ${\mathit{N}}_{\mathit{l}}$=4 light flavors, and a correction ${\mathrm{\Delta }}_{\mathit{c}}$=0.18±0.01 that takes account of the nonzero ratio ${\mathit{m}}_{\mathit{c}}$/${\mathit{m}}_{\mathit{b}}$=0.28±0.03. Convergence of the perturbative series is poor: the fastest apparent convergence would entail ${\mathrm{\alpha }}_{\mathit{s}}$(μ) at μ=370 MeV. ‘‘Naive non-Abelianization’’ of large-${\mathit{N}}_{\mathit{l}}$ results, via ${\mathit{N}}_{\mathit{l}}$→${\mathit{N}}_{\mathit{l}}$-33/2, gives reasonable approximations to exact two-loop results. All-order results for anomalous dimensions and matching coefficients are obtained at large ${\mathrm{\beta }}_0$=11=2/3${\mathit{N}}_{\mathit{l}}$. Consistent cancellation between infrared- and ultraviolet-renormalon ambiguities is demonstrated.