Non-perturbative tests of Heavy Quark Effective Theory

Abstract

We consider QCD with one massless quark and one heavy quark in a finite volume of linear extent L 0 ≈ 0.2 fm. In this situation, HQET represents an expansion in terms of 1/z = 1/(mL 0 ), which we test by a non-perturbative computation of quenched current matrix elements and energies, taking the continuum limit of lattice results. These are seen to approach the corresponding renormalization group invariant matrix elements of the static effective theory as the quark mass becomes large. We are able to obtain estimates of the size of the 1/m-corrections to the static theory, which are also of practical relevance in our recent strategy to implement HQET non-perturbatively by matching to QCD in a finite volume.

Non-perturbative tests of continuum HQET through small-volume two-flavour QCD

“…Extrapolations to the static limit are performed, which not only gives numerical evidence of the correctness of static order HQET but also allows assessing the size of the 1/m h -effects. As such, the present study constitutes a non-trivial non-perturbative test of HQET being an effective theory of QCD and extends earlier work in quenched QCD [15] to the physically more realistic situation with dynamical light quarks. In particular, we investigate a considerable set of observables at varied kinematics, which in the spirit of the general non-perturbative matching strategy mentioned above are usually employed to define suitable matching relations.…”

“…For the present purpose of testing HQET we want to express all observables in terms of matrix elements computed in a finite volume. The interested reader can find the corresponding notation in terms of the traditional SF correlation functions in appendix A and reference [15]. In passing we just note that only states, which are eigenstates of spatial momentum with eigenvalue zero, enter them.…”

“…A generic conversion function C X (z) carries all the logarithmic dependence of a given quantity X to some order in perturbation theory such that only power corrections in 1/z remain in the effective theory. For more details, we refer to appendix B and appendix B of [15]. To avoid a remnant renormalization scheme dependence in the (static) effective theory, we favour to fully express the asymptotic behaviour in terms of renormalization group invariants (RGIs).…”

“…From earlier studies [15,31] it is known that an exclusion limit of aM > 0.7 has to be imposed on the data entering in the extrapolating fits, in order to avoid contaminations which are potentially dangerous for a reliable continuum limit. To assure stability in our CL results, different fit ansaetze have also been studied (e.g., allowing for a cubic term in a/L with mass-dependent coefficient or omitting some of the lightest masses, say z ≤ 4); these lead to consistent results.…”

“…appendix B, and refs. [15,32,35] for more details). 4 The crucial observation at this point is now that, in the transition to renormalization group invariants leading to (3.8), the perturbative running enters in C PS (M/Λ) = Y PS (m * )/X RGI through the only perturbative knowledge of the beta-function and the anomalous dimensions of the quark mass and A stat 0 .…”