[Paper Review] Chiral symmetry breaking in nearly conformal gauge theories

Zoltán Fodor, Kieran Holland+3 more|arXiv (Cornell University)|Nov 12, 2009

Quantum Chromodynamics and Particle Interactions22 references9 citations

TL;DR

This paper investigates chiral symmetry breaking (χSB) in nearly conformal SU(3) gauge theories with fundamental fermions using staggered lattice QCD simulations. By applying chiral perturbation theory and random matrix theory in finite volumes, it identifies χSB for Nf = 4 to 12 and observes enhanced chiral condensate with increasing Nf when the electroweak scale F is fixed. At Nf = 16, inside the conformal window, it reveals non-trivial vacuum structure via Polyakov loop dynamics and zero-mode gauge field effects.

ABSTRACT

We present new results on chiral symmetry breaking in nearly conformal gauge theories with fermions in the fundamental representation of the SU(3) color gauge group. The number of fermion flavors is varied in an extended range below the conformal window with chiral symmetry breaking ($χ{ m SB}$) for all flavors between $N_f=4$ and $N_f=12$. To identify $χ{ m SB}$ we apply several methods which include, within the framework of chiral perturbation theory, the analysis of the Goldstone spectrum in the p-regime and the spectrum of the fermion Dirac operator with eigenvalue distributions of random matrix theory in the $\eps$-regime. Chiral condensate enhancement is observed with increasing $N_f$ when the electroweak symmetry breaking scale $F$ is held fixed in technicolor language. Important finite-volume consistency checks from the theoretical understanding of the $SU(N_f)$ rotator spectrum of the $δ$-regime are discussed. We also consider these gauge theories at $N_f=16$ inside the conformal window. The importance of understanding finite volume, zero momentum gauge field dynamics inside the conformal window is pointed out. Staggered lattice fermions with supressed taste breaking are used throughout the simulations.

Motivation & Objective

• To determine the phase structure of nearly conformal gauge theories below the conformal window by identifying chiral symmetry breaking (χSB) across varying fermion flavors.
• To test the consistency of finite-volume methods—specifically chiral perturbation theory (χPT) and random matrix theory (RMT)—in identifying χSB in lattice simulations.
• To investigate the behavior of the theory inside the conformal window at Nf=16, focusing on zero-momentum gauge field dynamics and vacuum structure.
• To examine the role of finite-volume effects, including the SU(Nf) rotator spectrum in the δ-regime and Polyakov loop dynamics, in understanding the vacuum structure of strongly interacting gauge theories.

Proposed method

• Simulations use staggered fermions with exponential (stout) smearing to suppress taste-breaking artifacts in the Goldstone spectrum.
• χSB is identified via analysis of the Goldstone pion spectrum in the p-regime using chiral perturbation theory (χPT).
• The spectrum of the Dirac operator is analyzed using eigenvalue distributions from random matrix theory (RMT) in the ε-regime.
• The δ-regime is used for consistency checks via the SU(Nf) rotator model, testing theoretical predictions for the spectrum of the effective Hamiltonian.
• For Nf=16, the effective potential for spatially constant abelian gauge fields is computed using one-loop fermion fluctuations, with boundary conditions tuned to probe vacuum degeneracy.
• Polyakov loop distributions are monitored over time to identify vacuum structure and spontaneous CP violation in small volumes under periodic boundary conditions.

Experimental results

Research questions

• RQ1 Does chiral symmetry breaking persist across Nf = 4 to 12 in SU(3) gauge theories with fundamental fermions, and how does the chiral condensate scale with Nf when F is held fixed?
• RQ2 How do finite-volume effects, particularly in the ε- and δ-regimes, affect the identification of χSB and the validity of χPT and RMT predictions?
• RQ3 What is the vacuum structure of the Nf=16 theory inside the conformal window, and how do zero-momentum gauge field modes influence the effective potential and Polyakov loop dynamics?
• RQ4 How do periodic versus antiperiodic boundary conditions affect the degeneracy and stability of vacua in small-volume SU(3) gauge theories with Nf=16?

Key findings

• Chiral symmetry breaking is confirmed for all Nf = 4, 8, 9, and 12 in the SU(3) gauge theory with fundamental fermions, using multiple finite-volume consistency checks.
• The chiral condensate increases with Nf when the electroweak symmetry breaking scale F is held constant, indicating enhanced dynamical symmetry breaking in the nearly conformal regime.
• At Nf=16, the theory lies inside the conformal window, and simulations confirm the existence of eight degenerate vacua under periodic boundary conditions, with non-trivial Polyakov loop values Pj = exp(±2πi/3).
• Under antiperiodic boundary conditions, the vacuum is unique and trivial (Pj=1), and the system evolves from a randomized configuration to the trivial vacuum via metastable complex minima.
• The measured fermion-antifermion spectra and Dirac operator eigenvalue distributions are consistent with the predicted vacuum structure and spontaneous breaking of P and C symmetries in small volumes.
• The effective potential for constant gauge fields shows that fermion loops dramatically alter the vacuum structure, with minima corresponding to non-trivial center elements only under periodic boundary conditions.