A Primer on Instantons in QCD

TL;DR

This primer surveys instantons in quantum mechanics and QCD, illustrating how semiclassical path-integral methods capture tunneling between degenerate vacua and the nonperturbative vacuum structure of Yang-Mills theory. In QM, it derives instanton solutions in a double-well potential, computes fluctuation determinants, zero modes, and dilute instanton gas contributions, and shows how tunneling induces energy splittings and band formation in periodic potentials. In QCD, it identifies YM instantons as finite-action, self-dual gauge-field configurations carrying topological charge Q, connects them to the axial anomaly and quark zero modes via the index theorem and 't Hooft vertices, and explains the θ-angle as a topological vacuum parameter arising from Gauß' law and cluster decomposition. The discussion extends to instanton ensembles beyond the dilute gas (the instanton-liquid picture), their impact on hadron phenomenology through direct instanton contributions and IOPE-based sum rules, and the broader significance for nonperturbative QCD and hadron structure. Overall, the text links topology, semiclassical methods, and hadronic physics to show how instantons shape the QCD vacuum and influence observable hadron properties.

Abstract

These are the (twice) extended notes of a set of lectures given at the ``12th Workshop on Hadronic Interactions'' at the IF/UERJ, Rio de Janeiro (31. 5. - 2. 6. 2000). The lectures aim at introducing essential concepts of instanton physics, with emphasis on the role of instantons in generating tunneling amplitudes, vacuum structure, and the induced quark interactions associated with the axial anomaly. A few examples for the impact of instantons on the physics of hadrons are also mentioned.

Figures (9)

• Figure 1: The two paths from $\left( x,t\right) =\left( 0,0\right)$ to $\left( 1,1\right)$ whose action is compared.
• Figure 2: A typical tunneling potential with nondegenerate minima. The total energy (horizontal line) is smaller than the hump so that a classically forbidden region exists.
• Figure 3: The double-well potential with $\frac{\alpha^{2}m}{2x_{0}^{2}}=10$ and $x_{0}=1$.
• Figure 4: The instanton solution in the double well potential of Fig. \ref{['doubwell']}.
• Figure 5: A multi-instanton solution in the periodical potential. The abscissa denotes the Euclidean time $\tau$ while the ordinate gives the position variable $x$. The integer values of $x$ correspond to the minima $x_{0,n}$ of the periodical potential.