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

Abstract

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\ifmmode\bar\else\textasciimacron\fi{}${\ensuremath{\gamma}}_{1}^{[\mathrm{\ensuremath{\mu}}}$...${\ensuremath{\gamma}}_{\mathit{n}]}^{\mathrm{\ensuremath{\mu}}}$q is obtained, yielding a new result for the tensor current. The dependence of matching coefficients on ${\ensuremath{\gamma}}_{5}$ 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${\mathrm{\ensuremath{\alpha}}}_{\mathit{s}}$(${\mathit{m}}_{\mathit{b}}$)/3\ensuremath{\pi}-${\mathit{K}}_{\mathit{b}}$${\mathrm{\ensuremath{\alpha}}}_{\mathit{s}}^{2}$/${\mathrm{\ensuremath{\pi}}}^{2}$ +O(${\mathrm{\ensuremath{\alpha}}}_{\mathit{s}}^{3}$,1/${\mathit{m}}_{\mathit{b}}$) is ${\mathit{K}}_{\mathit{b}}$=83/12+4/81${\mathrm{\ensuremath{\pi}}}^{2}$+2/27${\mathrm{\ensuremath{\pi}}}^{2}$ln2-1/9\ensuremath{\zeta}(3)-19/54${\mathit{N}}_{\mathit{l}}$ +${\mathrm{\ensuremath{\Delta}}}_{\mathit{c}}$=6.37+${\mathrm{\ensuremath{\Delta}}}_{\mathit{c}}$, with ${\mathit{N}}_{\mathit{l}}$=4 light flavors, and a correction ${\mathrm{\ensuremath{\Delta}}}_{\mathit{c}}$=0.18\ifmmode\pm\else\textpm\fi{}0.01 that takes account of the nonzero ratio ${\mathit{m}}_{\mathit{c}}$/${\mathit{m}}_{\mathit{b}}$=0.28\ifmmode\pm\else\textpm\fi{}0.03. Convergence of the perturbative series is poor: the fastest apparent convergence would entail ${\mathrm{\ensuremath{\alpha}}}_{\mathit{s}}$(\ensuremath{\mu}) at \ensuremath{\mu}=370 MeV. ``Naive non-Abelianization'' of large-${\mathit{N}}_{\mathit{l}}$ results, via ${\mathit{N}}_{\mathit{l}}$\ensuremath{\rightarrow}${\mathit{N}}_{\mathit{l}}$-33/2, gives reasonable approximations to exact two-loop results. All-order results for anomalous dimensions and matching coefficients are obtained at large ${\mathrm{\ensuremath{\beta}}}_{0}$=11=2/3${\mathit{N}}_{\mathit{l}}$. Consistent cancellation between infrared- and ultraviolet-renormalon ambiguities is demonstrated.