Instantons in QCD

TL;DR

The review analyzes instantons in QCD as topologically nontrivial tunneling events that generate strong nonperturbative interactions among light quarks, encapsulated by the ’t Hooft vertex and zero modes. It develops and validates the instanton liquid paradigm, detailing mean-field and interacting ensembles, Dirac spectra, and hadronic correlators that reproduce light hadron phenomenology while connecting to lattice results. At finite temperature, the work shows instantons reorganize into correlated molecules, impacting chiral symmetry restoration and the temperature dependence of hadronic correlators, with lattice data supporting a nontrivial, non-perturbative mechanism near $T_c$. The review also situates instantons in broader contexts (two-dimensional models, electroweak theory, SUSY QCD) and outlines open questions regarding confinement, topological screening, and the precise nature of the QCD vacuum. The central framework—where instanton size $\rho$, density $n$, and action $S_{\text{inst}}$ set scales and drive chiral dynamics—provides a consistent, testable picture linking vacuum structure to hadron physics via $S_{\text{inst}} = 8\pi^2/g^2$ and the Dirac spectrum near zero. $

Abstract

We review the theory and phenomenology of instantons in QCD. After a general overview, we provide a pedagogical introduction to semi-classical methods in quantum mechanics and field theory. The main part of the review summarizes our understanding of the instanton liquid in QCD and the role of instantons in generating the spectrum of light hadrons. We also discuss properties of instantons at finite temperature and the chiral phase transition. We give an overview over the importance of instantons in some other models, in particular two dimensional sigma models, electroweak theory and supersymmetric QCD.

Figures (42)

• Figure 1: Schematic picture of the instanton liquid at zero temperature (a) and above the chiral phase transition (b). Instantons and anti-instantons are shown as open and shaded circles. The lines correspond to fermion exchanges. Figures (c) and (d) show the schematic form of the Dirac spectrum in the configurations (a) and (b).
• Figure 2: Instanton contribution to hadronic correlation functions Fig. a) shows the pion, b) the nucleon and c) the rho meson correlator. The solid lines correspond to zero mode contributions to the quark propagator.
• Figure 3: Feynman diagrams for the energy (a) and the Green's function (b) of the anharmonic oscillator.
• Figure 4: Feynman diagrams for the two-loop correction to the tunneling amplitude in the quantum mechanical double well potential. The first three correspond to the diagrams in Fig. \ref{['fig_qm_graphs']}a, but with different propagators and vertices, while the forth diagram contains a new vertex, generated by the collective coordinate Jacobian.
• Figure 5: Streamline configurations in the double well potential for $\eta=1.4$ and $\lambda=1$ ($\omega\simeq 4$, $S_0\simeq 5$), adapted from Shu_88c. The horizontal axis shows the time coordinate and the vertical axis the amplitude $x(\tau)$. The different paths correspond to different values of the streamline parameter $\lambda$ as the configuration evolves from a well separated pair to an almost perturbative path. The initial path has an action $S=1.99S_0$, the other paths correspond to a fixed reductions of the action by $0.2S_0$.