[Paper Review] Remarks on the U(1) axial symmetry in QCD at zero and non-zero temperature

Enrico Meggiolaro|ArXiv.org|Jun 24, 2002

Quantum Chromodynamics and Particle Interactions1 references4 citations

TL;DR

This paper investigates the fate of the U(1) axial symmetry in QCD at finite temperature, proposing that it may remain spontaneously broken above the chiral phase transition temperature T_ch. Using effective field theory and QCD Ward identities, the author introduces a new order parameter for U(1) axial symmetry and shows that the topological susceptibility A(T) remains non-zero up to ~1.2T_c, implying U(1) symmetry restoration occurs at T_U(1) > T_ch. Lattice-inspired analysis supports a scenario where U(1) axial symmetry breaks after chiral symmetry restoration.

ABSTRACT

This paper is organized in two parts. The first part (Sections 2-5) is dedicated to the theory at T=0 and contains a pedagogical review of some fundamental aspects related with the chiral symmetries of QCD, the U(1) problem and its solution proposed by 'tHooft, Witten and Veneziano. In the second part (Sections 6-14) we discuss the role of the U(1) axial symmetry for the phase structure of QCD at finite temperature. One expects that, above a certain critical temperature, also the U(1) axial symmetry will be restored. We will try to see if this transition has (or has not) anything to do with the usual chiral transition: various possible scenarios are discussed. In particular, we analyse a scenario in which the U(1) axial symmetry is still broken above the chiral transition. We will show that this scenario can be consistently reproduced in the full respect of the relevant QCD Ward Identities and also using an effective Lagrangian model. A new order parameter is introduced for the U(1) axial symmetry.

Motivation & Objective

• To clarify the interplay between U(1) axial symmetry and chiral symmetry restoration in QCD at finite temperature.
• To determine whether U(1) axial symmetry restoration occurs at the same temperature as chiral symmetry restoration or at a higher temperature.
• To construct a consistent framework—using Ward identities and effective Lagrangians—where U(1) axial symmetry remains broken above T_ch.
• To introduce and validate a new order parameter for U(1) axial symmetry breaking in the finite-temperature regime.
• To analyze the behavior of the topological susceptibility A(T) in the large-N_c limit and its implications for η′ mass and U(1) breaking.

Proposed method

• Uses the Witten–Veneziano mechanism to relate the U(1) axial anomaly to the topological susceptibility A(T), defined via the vacuum expectation value of the topological charge density operator Q(x)Q(0).
• Analyzes A(T) in the large-N_c limit as the leading-order term in a 1/N_c expansion of the full QCD correlation function χ(k,T).
• Applies an effective Lagrangian model to describe the scalar meson channels (σ and δ) and their susceptibilities at finite T.
• Derives a new order parameter for U(1) axial symmetry based on the behavior of the topological susceptibility and chiral condensate in the chiral limit.
• Performs a finite-volume, WKB-approximated analysis of the Dirac spectrum to compute the chiral condensate ⟨q̄q⟩ and topological susceptibility ⟨Q²⟩ in the chiral limit.
• Compares analytical results from the model with lattice QCD data, particularly the behavior of A(T) in SU(3) pure gauge theory, showing A(T) remains non-zero up to ~1.2T_c.

Experimental results

Research questions

• RQ1 At what temperature is the U(1) axial symmetry restored in QCD at finite temperature?
• RQ2 Does the U(1) axial symmetry restoration temperature T_U(1) coincide with the chiral phase transition temperature T_ch?
• RQ3 Can a scenario in which U(1) axial symmetry remains spontaneously broken above T_ch be consistently realized within QCD Ward identities and effective field theory?
• RQ4 What is the role of the topological susceptibility A(T) in determining the mass of the η′ meson and the persistence of U(1) axial symmetry breaking?
• RQ5 How do the susceptibilities of the scalar meson channels (σ and δ) behave above T_ch, and what do they reveal about U(1) symmetry breaking?

Key findings

• The topological susceptibility A(T) remains non-zero up to approximately 1.2T_c, indicating that U(1) axial symmetry is not fully restored at T_ch.
• A scenario in which U(1) axial symmetry remains spontaneously broken above T_ch is consistent with QCD Ward identities and effective Lagrangian models.
• The new order parameter for U(1) axial symmetry is derived from the behavior of ⟨Q²⟩ and ⟨q̄q⟩ in the chiral limit, showing a non-vanishing contribution proportional to ∏m_k for k≠i.
• The chiral condensate ⟨q̄q⟩ scales as O(m_i) + O(∏_{k≠i} m_k) in the chiral limit, with the second term linked to topological fluctuations.
• The relation ⟨Q²⟩/V₄ = -m_i × O(∏_{k≠i} m_k) is derived in Euclidean space, confirming consistency between the effective model and the analytical finite-volume approach.
• Lattice results for A(T) in pure Yang-Mills theory support the existence of a distinct U(1) axial restoration scale T_U(1) > T_ch, with A(T) dropping sharply but not vanishing at T ≈ 1.2T_c.