Non-perturbative QCD: renormalization, O(a)-improvement and matching to Heavy Quark Effective Theory

TL;DR

This work surveys how lattice QCD achieves non-perturbative control over renormalization, discretization errors, and effective-field theory matching. It details the Schrödinger functional as a finite-volume renormalization framework, the non-perturbative determination of the running coupling and quark masses, and the non-perturbative HQET program, emphasizing the necessity of non-perturbative renormalization to obtain a well-defined continuum limit. Through Symanzik’s local effective theory and Ward identities, it outlines systematic ${\rm O}(a)$-improvement via improved actions and currents, including the non-perturbative determination of improvement coefficients such as $c_{\rm sw}$ and $c_{\rm A}$. The document also discusses practical aspects like boundary terms, lattice artifacts, and constant-physics prescriptions, illustrating how these techniques enable precision predictions for heavy-quark physics (e.g., $1/m_b$ corrections) and the connection between non-perturbative QCD and the perturbative regime at high energies. Altogether, it underscores the interplay between finite-volume renormalization, non-perturbative improvement, and HQET matching as essential for reliable, first-principles QCD calculations.

Abstract

We give an introduction to three topics in lattice gauge theory: I. The Schroedinger 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.

Figures (8)

• Figure 1: Illustration of the Schrödinger functional.
• Figure 2: $f_{\rm P}$ (left) and $f_1$ (right) in terms of quark propagators.
• Figure 3: Dependence of current quark mass $m$ on the boundary condition and the time coordinate impr:lett. The calculation is done in the quenched approximation on a $16 \times 8^3$ lattice at $\beta=6.4$, which corresponds to a lattice spacing of $a \approx 0.05\,{\rm fm}$. "Boundary values" refer to the gauge field boundary conditions in the SF. Their values are given in impr:lett.
• Figure 4: Lattice spacing dependence of the step scaling function of the LWW coupling in the 2-d O(3) sigma modelsigma:nonstandard for coupling $u_0=1.0595$. The data points with the smaller cutoff effects are for the standard nearest neighbor action.
• Figure 5: Coefficient of $1/N$ in the $1/N$-expansion of the cutoff effects of the step scaling function of the LWW coupling of 2-d O($N$) sigma models. Graph prepare by U. Wolff based on sigma:largeNaeff.