NON-PERTURBATIVE QCD: RENORMALIZATION, O(A)-IMPROVEMENT AND MATCHING TO HEAVY QUARK EFFECTIVE THEORY

Abstract

We give an introduction to three topics in lattice gauge theory:I. The Schrödinger functional and O(a) improvement.O(a) improvement has been reviewed several times. Here we focus on explaining the basic ideas in detail and then proceed directly to an overview of the literature and our personal assessment of what has been achieved and what is missing.II. The computation of the running coupling, running quark masses and the extraction of the renormalization group invariants.We focus on the basic strategy and on the large effort that has been invested in understanding the continuum limit. We point out what remains to be done.III. Non-perturbative Heavy Quark Effective Theory.Since the literature on this subject is still rather sparse, we go beyond the basic ideas and discuss in some detail how the theory works in principle and in practice.

The determination of α s by the ALPHA collaboration

“…Our scheme [9] for the high energy region, reviewed in [16,17,12], is based on the so-called Schrödinger functional (SF) [8,18]. Here, we just summarize what is needed to judge our findings below.…”

Non-perturbative renormalization of the static vector current and its O(a)-improvement in quenched QCD

“…As for the theoretical derivation of the axial WI, we follow the approach of [18,19]. For the moment no attention is paid to the specific regularization of the theory.…”

Non-perturbative improvement of the axial current with three dynamical flavors and the Iwasaki gauge action

“…One is then sure that the O(a) effects are entirely removed. Details of this programme as well as the present status have recently been reviewed [26]. Here we just mention that the coefficient c sw of the Sheikholeslami-Wohlert term [21], the only dimension five correction to the action, 1 has been determined non-perturbatively for different gauge actions and numbers of flavors [25, 27 -29].…”

“…However, as explained in some detail in refs. [26,28,33,34], in such a situation it is important to impose improvement conditions on a line of constant physics. This means that as the lattice spacing a is varied, all other physical scales are kept fixed.…”