[Paper Review] The Spontaneous Breaking of Chiral Symmetry without Goldstone Bosons

She-Sheng Xue|arXiv (Cornell University)|Jan 31, 1994

Atomic and Subatomic Physics Research1 references3 citations

TL;DR

This paper demonstrates that chiral symmetry can be spontaneously broken in a lattice-regularized theory of mirror fermions without producing Goldstone bosons, due to the explicit violation of Lorentz invariance. The absence of Goldstone modes is explained by the breakdown of a key condition of the Goldstone theorem—Lorentz invariance—thereby providing a mechanism for dynamical mass generation without the usual Nambu-Goldstone mode spectrum.

ABSTRACT

Considering a self-interaction only of mirror fermions in the context of a lattice-regularized fermion field theory, we show that the system undergoes spontaneous breaking of chiral symmetry and mirror-fermion masses are generated. However, it is explicitly shown that there are no Goldstone bosons appearing together with this spontaneous symmetry breaking phenomenon, since Lorentz invariance, one of very general prerequisites of the Goldstone theorem, is violated. The result and its possible application are briefly discussed.

Motivation & Objective

• To investigate whether chiral symmetry breaking can occur without the concomitant appearance of Goldstone bosons in a lattice-regularized fermion system.
• To examine the role of Lorentz invariance in the context of the Goldstone theorem and its implications for spontaneous symmetry breaking.
• To analyze a self-interacting mirror fermion model on a lattice to determine the conditions under which dynamical mass generation occurs without Goldstone modes.
• To explore the theoretical implications of this mechanism for models of dynamical fermion mass generation in non-Lorentz-invariant field theories.

Proposed method

• Formulate a lattice-regularized field theory with self-interactions among mirror fermions, preserving chiral symmetry at the Lagrangian level.
• Introduce a non-Lorentz-invariant fermion action to break Lorentz invariance explicitly, thereby violating a necessary condition of the Goldstone theorem.
• Analyze the vacuum structure of the theory to identify spontaneous chiral symmetry breaking via the formation of a non-zero fermion bilinear expectation value.
• Use the lattice regularization to avoid anomalies and ensure a well-defined path integral measure, enabling non-perturbative analysis.
• Demonstrate that the absence of Goldstone bosons arises because the Goldstone theorem's assumptions—specifically Lorentz invariance—are not satisfied.
• Confirm that mirror fermion masses are dynamically generated despite the lack of Goldstone modes, through analysis of the effective potential and symmetry breaking pattern.

Experimental results

Research questions

• RQ1 Can chiral symmetry be spontaneously broken in a lattice-regularized fermion theory without the appearance of Goldstone bosons?
• RQ2 What is the role of Lorentz invariance in the emergence of Goldstone modes following spontaneous symmetry breaking?
• RQ3 How does the presence of self-interactions among mirror fermions lead to dynamical mass generation in a non-Lorentz-invariant framework?
• RQ4 To what extent does the breakdown of Lorentz invariance invalidate the standard Goldstone theorem in field theories with spontaneous symmetry breaking?
• RQ5 Can a consistent mechanism for fermion mass generation be constructed without the standard Nambu-Goldstone mode spectrum?

Key findings

• Spontaneous chiral symmetry breaking occurs in the lattice-regularized mirror fermion model, as indicated by a non-zero vacuum expectation value of the fermion bilinear.
• Despite the spontaneous breaking of chiral symmetry, no Goldstone bosons appear in the spectrum due to the explicit violation of Lorentz invariance.
• The absence of Goldstone modes is a direct consequence of the breakdown of Lorentz invariance, which invalidates a fundamental assumption of the Goldstone theorem.
• Dynamical masses are generated for the mirror fermions through self-interactions, even in the absence of explicit mass terms or Goldstone modes.
• The mechanism demonstrates that the Goldstone theorem does not apply in non-Lorentz-invariant field theories, allowing for symmetry breaking without massless modes.
• The result suggests a new pathway for dynamical fermion mass generation in theories where Lorentz symmetry is not preserved, such as in certain lattice or condensed matter analogs.