LQCD

Lattice quantum chromodynamics (LQCD) is a non-perturbative approach to quantum chromodynamics (QCD) that formulates the theory on a finite, four-dimensional spacetime lattice. By discretizing spacetime into a grid with lattice spacing a, LQCD makes the QCD path integral amenable to numerical evaluation using Monte Carlo methods. The formulation preserves gauge invariance and provides a first-principles framework for studying the strong interaction in regimes where perturbation theory is unreliable.

In LQCD, gauge fields are defined on the links between lattice sites as SU(3) matrices, while quark fields reside on the lattice sites. The action is written as a sum of a gauge part and a fermion part, with the gauge action commonly based on the Wilson plaquette term and various fermion discretizations to address the fermion doubling problem. Different fermion formulations exist, such as Wilson, staggered (Kogut-Susskind), domain-wall, and overlap fermions, each with trade-offs in chiral symmetry, computational cost, and systematic errors. Observables are computed as ensemble averages over gauge configurations weighted by e^{-S}.

Simulations rely on algorithms like Hybrid Monte Carlo and require substantial high-performance computing resources. Results are obtained on ensembles with multiple lattice spacings and volumes to control discretization and finite-size effects, followed by extrapolations to the continuum limit (a → 0) and infinite volume, as well as to physical quark masses. Observables produced by LQCD include the hadron spectrum, decay constants, quark masses, and matrix elements relevant for weak decays and hadronic structure. Scale setting uses quantities such as the Sommer parameter or Wilson flow. LQCD provides a first-principles, non-perturbative window into strong-interinteraction physics and its connection to experimental data.