Introduction to Lattice QCD

TL;DR

This work surveys the lattice QCD framework, detailing discretization strategies, gauge and fermion actions, and the improvements needed to control discretization errors. It outlines how lattice simulations, via Monte Carlo methods on Euclidean spacetime, yield nonperturbative calculations of hadron spectra, decay constants, and weak matrix elements, while addressing systematic errors such as finite volume, lattice spacing, and quenching effects. The notes connect foundational QCD concepts like confinement and asymptotic freedom to practical lattice techniques (improvement programs, RG trajectories, and operator renormalization), and illustrate their application to extracting αs and light/heavy quark masses. Overall, the document emphasizes the maturity of LQCD as a first-principles tool with growing impact on Standard Model phenomenology and CKM-parameter determinations. The discussions on improved actions, renormalization, and finite-temperature behavior highlight the path toward increasingly precise, continuum-extrapolated results.

Abstract

These notes aim to provide a pedagogical introduction to Lattice QCD. The topics covered include the scope of LQCD calculations, lattice discretization of gauge and fermion (naive, Wilson, and staggered) actions, doubling problem, improved gauge and Dirac actions, confinement and strong coupling expansions, phase transitions in the lattice theory, lattice operators, a general discussion of statistical and systematic errors in simulations of LQCD, the analyses of the hadron spectrum, glueball masses, the strong coupling constant, and the quark masses.

Figures (35)

• Figure 1: The CKM matrix in the Wolfenstein parameterization. I show examples of physical processes that need to be measured experimentally and compared with theoretical predictions via the "master" equation to estimate various elements of the CKM matrix.
• Figure 2: An illustration of the difference in the binding energy between the electron and the proton in a hydrogen atom (interacting via electromagnetic forces) and the quarks in a proton (strong force).
• Figure 3: The Feynman diagram for the semi-leptonic decay $D^- \to K^0 e^- \overline \nu_e$. The QCD corrections are illustrated by the various gluons being exchanged between the initial and final hadrons. The leptonic vertex can be calculated reliably using perturbation theory, whereas the hadronic vertex requires non-perturbative methods.
• Figure 4: A schematic of the pion 2-point correlation function for (A) local and (B) non-local interpolating operators.
• Figure 5: The two gauge invariant quantities. a) An ordered string of $U's$ capped by a fermion and an anti-fermion and b) closed Wilson loops.