[Paper Review] TASI Lectures on The Strong CP Problem

Michael Dine|ArXiv.org|Nov 29, 2000

Particle physics theoretical and experimental studies8 citations

TL;DR

This paper provides a comprehensive pedagogical overview of the strong CP problem in quantum chromodynamics (QCD), explaining how the $ heta$ parameter—introduced via a topological term in the QCD Lagrangian—leads to a neutron electric dipole moment that is experimentally constrained to be extremely small ($\theta \ll 10^{-9}$). It analyzes the role of anomalies in forbidding field redefinitions that would eliminate the $ heta$ term, and evaluates key solutions such as the axion, a massless up quark, and spontaneous CP violation, with the axion emerging as a leading candidate due to its naturalness and cosmological viability.

ABSTRACT

These lectures discuss the $θ$ parameter of QCD. After an introduction to anomalies in four and two dimensions, the parameter is introduced. That such topological parameters can have physical effects is illustrated with two dimensional models, and then explained in QCD using instantons and current algebra. Possible solutions including axions, a massless up quark, and spontaneous CP violation are discussed.

Motivation & Objective

• To explain the origin and physical consequences of the $θ$ parameter in QCD, which arises from a topological term in the Lagrangian and violates CP symmetry.
• To clarify why the $θ$ parameter cannot be removed by field redefinitions due to the chiral anomaly, using 4D and 2D model calculations.
• To analyze the experimental constraints on $θ$ from the neutron electric dipole moment, which require $θ \ll 10^{-9}$, forming the core of the strong CP problem.
• To evaluate proposed solutions to the strong CP problem, including the axion mechanism, a massless up quark, and spontaneous CP violation.
• To assess the cosmological viability of the axion, particularly its production and dynamics in the early universe, and the resulting constraints on the axion decay constant $f_a$.

Proposed method

• Uses 2D and 4D quantum field theory models to illustrate how topological terms like $\theta F\tilde{F}$ lead to physical CP-violating effects despite being total divergences.
• Applies path integral methods and anomaly calculations to show that field redefinitions that would remove the $\theta$ term are obstructed by the chiral anomaly.
• Performs one-loop triangle diagram calculations (e.g., with massive fermions) to derive the effective $\theta F\tilde{F}$ term in the low-energy effective action.
• Introduces the Peccei-Quinn symmetry and the axion as a dynamical solution to the strong CP problem, where the axion field rolls to minimize its potential.
• Analyzes the axion's cosmological production via the misalignment mechanism in a radiation-dominated early universe, solving the axion equation of motion with damping.
• Uses the axion potential $V(\phi) \propto \Lambda^4 \left(1 - \cos(\phi/f_a)\right)$ and computes the energy density from coherent oscillations to derive constraints on $f_a$.

Experimental results

Research questions

• RQ1 Why does the $\theta$ parameter in QCD lead to CP violation despite being a total divergence in the Lagrangian?
• RQ2 How does the chiral anomaly prevent the removal of the $\theta$ term via field redefinitions, and what is the role of the triangle diagram in this mechanism?
• RQ3 What are the cosmological constraints on the axion decay constant $f_a$, and how do they arise from axion production and oscillations in the early universe?
• RQ4 Why is the axion considered a natural solution to the strong CP problem, and how does its dynamics naturally suppress the $\theta$ parameter?
• RQ5 What are the implications of the axion's weak coupling for its detection, and how does string theory provide a natural framework for axion physics?

Key findings

• The $\theta$ parameter in QCD leads to a neutron electric dipole moment that is experimentally bounded to $\theta \ll 10^{-9}$, making the smallness of $\theta$ a fundamental problem.
• The chiral anomaly prevents the removal of the $\theta$ term via field redefinitions, as shown by the non-invariance of the path integral measure under such transformations.
• The one-loop triangle diagram with massive fermions generates an effective $\theta F\tilde{F}$ term in the low-energy action, confirming the physical relevance of the $\theta$ parameter.
• The axion solution dynamically drives $\theta$ to zero via the Peccei-Quinn symmetry and the rolling of the axion field to its minimum, solving the strong CP problem naturally.
• Cosmological constraints from axion production and coherent oscillations limit the axion decay constant to $f_a < 10^{11}$ GeV, assuming the axion is the dominant dark matter component.
• String theory provides a natural setting for axion physics, with exact symmetries in perturbation theory broken only by exponentially small non-perturbative effects, supporting high-scale $f_a \sim M_{\text{GUT}}$ or $M_p$.