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Introduction to Lattice Field Theory

Raúl A. Briceño

TL;DR

The chapter addresses the non-perturbative regime of QCD and the challenge of accessing hadron spectra from first principles. It frames lattice QCD as a UV regulator with lattice spacing $a$ and IR regulator via finite volume $V$, employing a Wick-rotated Euclidean path integral with Monte Carlo sampling to evaluate observables. Key ideas include gauge links, discretized fermionic and gluonic actions, and correlation functions (two- and three-point) for spectroscopy and matrix elements, together with scale setting and mass tuning. The chapter surveys modern developments across precision SM tests, hadron structure, finite-temperature and finite-density QCD, and algorithmic advances, illustrating the broad impact and future potential of lattice methods for SM and beyond.

Abstract

This chapter provides a pedagogical introduction to lattice quantum field theory, with strong emphasis on lattice quantum chromodynamics. The chapter reviews key foundational concepts of lattice quantum chromodynamics, as well as a broad summary of ongoing research in the field.

Introduction to Lattice Field Theory

TL;DR

The chapter addresses the non-perturbative regime of QCD and the challenge of accessing hadron spectra from first principles. It frames lattice QCD as a UV regulator with lattice spacing and IR regulator via finite volume , employing a Wick-rotated Euclidean path integral with Monte Carlo sampling to evaluate observables. Key ideas include gauge links, discretized fermionic and gluonic actions, and correlation functions (two- and three-point) for spectroscopy and matrix elements, together with scale setting and mass tuning. The chapter surveys modern developments across precision SM tests, hadron structure, finite-temperature and finite-density QCD, and algorithmic advances, illustrating the broad impact and future potential of lattice methods for SM and beyond.

Abstract

This chapter provides a pedagogical introduction to lattice quantum field theory, with strong emphasis on lattice quantum chromodynamics. The chapter reviews key foundational concepts of lattice quantum chromodynamics, as well as a broad summary of ongoing research in the field.
Paper Structure (17 sections, 24 equations, 2 figures)

This paper contains 17 sections, 24 equations, 2 figures.

Table of Contents

  1. Introduction to Lattice Field Theory
  2. Introduction
  3. Lattice: The most natural regulator
  4. Wick rotating to Euclidean spacetime
  5. Symmetry breaking
  6. Discretization of the Action
  7. Gauge links
  8. Fermionic Action
  9. Gluonic Action
  10. Chiral Symmetry and Discretization
  11. Correlation functions and propagators
  12. Two-point correlators and spectroscopy
  13. Tuning quark masses and scale setting
  14. Wick contractions and propagators
  15. Propagators
  16. ...and 2 more sections

Figures (2)

  • Figure 1: Shown is a cartoon depiction of a possible contribution to the $\pi^+$ two-point correlator discussed in the main text.
  • Figure 2: Shown are some key highlights from the last decade. ($a$) Shown is the mass splitting between different isospin partners, including the neutron-proton, as calculated by Ref. Borsanyi:2014jba. ($b$) Shown is the first percent-level determination of the nucleon axial charge Chang:2018uxx. For a recent update of this calculation, see Ref. Hall:2025ytt. ($c$) Shown is a continuum-limit determination of the unpolarized isovector unpolarized PDF of the nucleon at a fixed value of the quark masses, corresponding to $m_\pi\approx 370$ MeV Bhat:2022zrw, compared to the phenomenological determination by the NNPDF Collaboration NNPDF:2017mvq. ($d$) Shown is the quark-mass evolution of the $\sigma$ resonance Rodas:2023nec, constrained from a series of calculations Briceno:2016mjcBriceno:2017qmbRodas:2023gmaRodas:2023nec.