Matching QCD and HQET heavy--light currents at two loops and beyond
D. J. Broadhurst, A. G. Grozin
TL;DR
This work develops a universal framework to match heavy–light QCD currents with HQET currents to two loops at leading order in 1/m, providing a general CΓ(m) for arbitrary Dirac structures and a master formula for the two–loop anomalous dimension γJ. It clarifies γ5 prescription effects, analyzes ratios of QCD matrix elements and their independence from three–loop HQET anomalous dimensions, and delivers all–order large–β0 results with explicit renormalon cancellations. The authors apply the formalism to phenomenologically relevant ratios such as f_B^*/f_B and tensor current couplings, revealing sizable two–loop corrections and near–decoupling of heavy flavors, with insights useful for lattice and sum–rule determinations of heavy–meson properties. Overall, the paper provides a comprehensive, universally applicable toolkit for precise heavy–quark current matching and its perturbative and nonperturbative implications.
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 $\bar{q}γ^{[μ_1}\ldotsγ^{μ_n]}q$ is obtained, yielding a new result for the tensor current. The dependence of matching coefficients on $γ_5$ 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 $f_{{\rm B}^*}/f_{\rm B} =1-2α_{\rm s}(m_b)/3π-K_bα_{\rm s}^2/π^2 +{\rm O}(α_{\rm s}^3,1/m_b)$ is \[K_b=\frac{83}{12}+\frac{4}{81}π^2+\frac{2}{27}π^2\log2-\frac19ζ(3) -\frac{19}{54}N_l+Δ_c=6.37+Δ_c\] with $N_l=4$ light flavours, and a correction, $Δ_c=0.18\pm0.01$, that takes account of the non--zero ratio $m_c/m_b=0.28\pm0.03$. Fastest apparent convergence would entail $α_{\rm s}(μ)$ at $μ=370$~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 $β_0=11-\frac23N_l$. Consistent cancellation between infrared-- and ultraviolet--renormalon ambiguities is demonstrated.