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Strong CP problem, theta term and QCD topological properties

Claudio Bonanno, Claudio Bonati, Massimo D'Elia

TL;DR

The chapter analyzes theta-dependence and topology in QCD and its connection to the strong CP problem, framing non-perturbative effects through topological sectors labeled by $Q$ and a theta-term $\theta Q$. It surveys analytical approaches, including the Dilute Instanton Gas Approximation, large-$N$ scaling, and chiral perturbation theory, and compares these with lattice QCD results across low and high temperature regimes. Leading topological quantities, notably the topological susceptibility $\chi$, are tied to phenomenology via the Witten–Veneziano mechanism and axion physics through $m_a^2=\chi/f_a^2$, while the temperature dependence $\chi(T)$ informs cosmological constraints. The work also highlights open questions—precise $\chi(T)$ at temperatures up to several GeV and the treatment of topology on the lattice—emphasizing the need for algorithmic advances to achieve robust, continuum-control results relevant for axion cosmology.

Abstract

In this chapter we introduce the $θ$-dependence and the topological properties of QCD, features of the strongly interacting sector which give rise to the strong CP problem in the more general context of the Standard Model of particle physics. We discuss the analytical approaches that can be used to obtain qualitative, or in some cases quantitative, information on the $θ$-dependence of QCD and QCD-like models, discussing their range of validity and comparing their predictions with the numerical results obtained by means of lattice simulations.

Strong CP problem, theta term and QCD topological properties

TL;DR

The chapter analyzes theta-dependence and topology in QCD and its connection to the strong CP problem, framing non-perturbative effects through topological sectors labeled by and a theta-term . It surveys analytical approaches, including the Dilute Instanton Gas Approximation, large- scaling, and chiral perturbation theory, and compares these with lattice QCD results across low and high temperature regimes. Leading topological quantities, notably the topological susceptibility , are tied to phenomenology via the Witten–Veneziano mechanism and axion physics through , while the temperature dependence informs cosmological constraints. The work also highlights open questions—precise at temperatures up to several GeV and the treatment of topology on the lattice—emphasizing the need for algorithmic advances to achieve robust, continuum-control results relevant for axion cosmology.

Abstract

In this chapter we introduce the -dependence and the topological properties of QCD, features of the strongly interacting sector which give rise to the strong CP problem in the more general context of the Standard Model of particle physics. We discuss the analytical approaches that can be used to obtain qualitative, or in some cases quantitative, information on the -dependence of QCD and QCD-like models, discussing their range of validity and comparing their predictions with the numerical results obtained by means of lattice simulations.

Paper Structure

This paper contains 11 sections, 59 equations, 2 figures.

Table of Contents

  1. Strong CP problem, theta term and QCD topological properties
  2. Introduction
  3. theta-dependence and the strong CP problem
  4. Why theta-dependence
  5. The strong CP problem
  6. General aspects of theta-dependence
  7. A semiclassical model: the Dilute Instanton Gas Approximation
  8. Low temperature: large-N and chiral scaling
  9. The Lattice QCD approach: basics and results
  10. High temperature: theta-dependence across the phase transition
  11. Conclusions

Figures (2)

  • Figure 1: Left: extrapolation of $\chi/\sigma^2$, where $\sigma$ is the string tension, towards the large-$N$ limit, using the fit function $\chi/\sigma^2=\bar{\chi}/\sigma^2+k/N^2$; the best fit yields $\bar{\chi}/\sigma^2=0.0199(10)$ and $k=0.082(17)$. (source: Ref. Bonanno:2020hht). Right: Extrapolation of $b_2$ towards the large-$N$ limit (source: Ref. Bonanno:2020hht); a best fit according to $b_2=\bar{b}_2/N^c$ with $c = 2$ yields $\bar{b}_2 = - 0.193(10)$ (dashed line), and is stable within errors by making $c$ a free parameter ($c=2.17(26)$, solid line) or by adding $1/N^4$ corrections (dotted line).
  • Figure 2: Left: continuum extrapolation of $\chi^{1/4}$ at $T = 0$, for $N_{{{\mathrm{f}}}} = 2+1$ QCD at the physical point (discretized via staggered fermions) and various definitions of topological observables; SP stands for Spectral Projectors, with $M/m_s$ being the cutoff on the Dirac spectrum used for the projection; figure adapted from Ref. Athenodorou:2022aay. Right: for the same theory and discretization, continuum extrapolated values of $\chi$ for different values of the light-to-strange quark mass ratio, and extrapolation to the chiral limit; figure taken from Ref. Bonanno:2023xkg.