[Paper Review] Chiral condensates and QCD vacuum in two dimensions

H. R. Christiansen|arXiv (Cornell University)|Apr 2, 1997

Black Holes and Theoretical Physics3 citations

TL;DR

This paper investigates chiral condensates and the QCD vacuum structure in two-dimensional quantum chromodynamics using the path-integral formalism. It demonstrates the existence of chiral condensates in both massless and massive cases across arbitrary finite and infinite numbers of colors, raising critical topological questions when matter transforms in the fundamental representation of the gauge group.

ABSTRACT

We analyze the chiral symmetries of flavored quantum chromodynamics in two dimensions and show the existence of chiral condensates within the path-integral approach. The massless and massive cases are discussed as well, for arbitrary finite and infinite number of colors. Our results put forward the question of topological issues when matter is in the fundamental representation of the gauge group.

Motivation & Objective

• To analyze chiral symmetries in two-dimensional QCD with multiple flavors and colors.
• To determine the existence and nature of chiral condensates in massless and massive QCD in 2D.
• To explore topological constraints arising when matter fields transform in the fundamental representation of the gauge group.
• To establish a field-theoretic framework for understanding the QCD vacuum structure in low-energy, reduced spacetime dimensions.

Proposed method

• Employing the path-integral formulation of quantum field theory to study the partition function and vacuum structure of 2D QCD.
• Analyzing the behavior of the chiral condensate under chiral symmetry transformations in the presence of quark masses.
• Using large-Nc and finite-Nc expansions to examine the dependence of condensates on the number of colors.
• Applying functional integral techniques to derive effective actions and analyze symmetry breaking patterns.
• Investigating the role of topological sectors in the vacuum structure through the Atiyah-Singer index theorem and related anomalies.
• Considering the interplay between gauge invariance, chiral symmetry, and the emergence of condensates in two dimensions.

Experimental results

Research questions

• RQ1 Does a chiral condensate form in two-dimensional QCD with massless quarks, and how is it stabilized?
• RQ2 How do chiral condensates behave in the massive quark case across different numbers of colors?
• RQ3 What is the role of topological configurations in the formation of the QCD vacuum in 2D QCD with fundamental matter?
• RQ4 How does the path-integral approach reveal the structure of the QCD vacuum in 2D, especially in the presence of chiral symmetry breaking?
• RQ5 What are the implications of the fundamental representation for the topological structure of the vacuum in 2D QCD?

Key findings

• Chiral condensates exist in two-dimensional QCD for both massless and massive quarks, regardless of the number of colors.
• The path-integral approach confirms the spontaneous breaking of chiral symmetry through the non-vanishing vacuum expectation value of the quark bilinear.
• The analysis reveals that the vacuum structure in 2D QCD is sensitive to topological configurations, especially when matter fields are in the fundamental representation.
• The results suggest that topological effects—such as instantons or zero modes—play a crucial role in stabilizing the chiral condensate in 2D.
• The study establishes a consistent framework for computing condensates in 2D QCD using functional methods, valid for arbitrary finite and infinite Nc.
• The paper raises the unresolved question of how topological invariants influence the vacuum structure when quarks transform in the fundamental representation, pointing to deeper anomalies or constraints.