[Paper Review] Goldstone bosons in the massless Thirring model. Witten's criterion

M. Faber, A. N. Ivanov|ArXiv.org|May 20, 2003

Quantum Chromodynamics and Particle Interactions7 references3 citations

TL;DR

This paper demonstrates that the quanta of the free massless (pseudo)scalar field in the massless Thirring model—bosonizing the fermionic theory in the chirally broken phase—satisfy Witten's criterion for Goldstone bosons. By showing that the low-energy theorems and Ward identities for the axial-vector current are non-trivially fulfilled only through contributions from this field, the authors confirm that these quanta are Goldstone bosons, despite the absence of infrared divergences in the quantum field theory framework used.

ABSTRACT

We discuss the Ward identity and the low-energy theorem for the divergence of the axial-vector current in the massless Thirring model with fermion fields quantized in the chirally broken phase (Eur. Phys. J. C20, 723 (2001)). The Ward identity and the low-energy theorem are analysed in connection with Witten's criterion for Goldstone bosons (Nucl. Phys. B145, 110 (1978)). We show that the free massless (pseudo)scalar field bosonizing the massless Thirring model in the chirally broken phase satisfies Witten's criterion to interpret quanta of this field as Goldstone bosons. As has been shown in hep-th/0210104 and hep-th/0212226, Goldstone's criterion, the non-invariance of the wave function of the ground state, is also fulfilled.

Motivation & Objective

• To resolve the apparent contradiction between the existence of massless (pseudo)scalar bosons in the massless Thirring model and Coleman’s theorem on spontaneous symmetry breaking in 1+1 dimensions.
• To verify whether the quanta of the free massless (pseudo)scalar field satisfy Witten’s criterion for Goldstone bosons, which requires their contributions to be essential in satisfying Ward identities and low-energy theorems.
• To establish consistency between the low-energy theorems for the axial-vector current divergence and the gap equation that generates a dynamical fermion mass in the chirally broken phase.
• To show that the quantum field theory of the massless (pseudo)scalar field, defined on the Schwartz class with zero Fourier mode, avoids infrared divergences and remains mathematically consistent, thus evading Coleman’s theorem.

Proposed method

• Derive the Ward identity and low-energy theorem for the divergence of the axial-vector current in the massless Thirring model, distinguishing between the chiral symmetric and chirally broken phases.
• Use the bosonization map to express the fermionic theory in the chirally broken phase as a free massless (pseudo)scalar field theory, with the field $\vartheta(x)$ describing the Goldstone mode.
• Apply Witten’s criterion by analyzing whether the quanta of $\vartheta(x)$ contribute non-trivially to the Ward identities and low-energy theorems, especially when the axial current is not conserved.
• Construct a quantum field theory of $\vartheta(x)$ using Wightman’s observables defined on the subspace $\mathcal{S}_0(\mathbb{R}^2)$, where $\tilde{h}(0) = 0$, to avoid infrared divergences and ensure positivity of the inner product.
• Verify consistency between the low-energy theorem and the gap equation by showing that the Källén–Lehmann spectral function $\rho_2(m^2)$ must be $M\delta(m^2 - M^2)$ for consistency.
• Confirm the non-invariance of the ground state wave function under chiral symmetry, satisfying Goldstone’s criterion, and show that the vacuum expectation value $\langle \cos\beta\vartheta(0) \rangle = 1$ implies a non-vanishing fermion condensate $\langle \bar{\psi}\psi \rangle = -M/g$.

Experimental results

Research questions

• RQ1 Do the quanta of the free massless (pseudo)scalar field in the massless Thirring model satisfy Witten’s criterion for Goldstone bosons?
• RQ2 Can the low-energy theorems and Ward identities for the axial-vector current be non-trivially satisfied in the chirally broken phase without violating the constraints of 1+1-dimensional quantum field theory?
• RQ3 How can a consistent quantum field theory of a free massless (pseudo)scalar field be constructed in 1+1 dimensions without infrared divergences?
• RQ4 Is the existence of Goldstone bosons in the massless Thirring model compatible with Coleman’s theorem on the non-existence of spontaneous symmetry breaking in 1+1 dimensions?
• RQ5 What is the role of the vacuum expectation value $\langle \cos\beta\vartheta(0) \rangle$ in generating a dynamical fermion mass and ensuring consistency with the gap equation?

Key findings

• The quanta of the free massless (pseudo)scalar field $\vartheta(x)$ satisfy Witten’s criterion for Goldstone bosons because they provide the only non-trivial contribution to the Ward identity and low-energy theorems in the chirally broken phase.
• In the chiral symmetric phase, the Ward identity reduces to $0=0$, but in the chirally broken phase, the non-vanishing divergence of the axial-vector current is satisfied only through contributions from the $\vartheta(x)$ field.
• The vacuum expectation value $\langle \cos\beta\vartheta(0) \rangle = 1$ is non-zero, implying a non-vanishing fermion condensate $\langle \bar{\psi}\psi \rangle = -M/g$, which confirms spontaneous chiral symmetry breaking.
• The Källén–Lehmann spectral function $\rho_2(m^2)$ must be $M\delta(m^2 - M^2)$ for consistency between the low-energy theorems and the gap equation, confirming the dynamical mass generation.
• The quantum field theory of $\vartheta(x)$ is consistent with Wightman’s axioms when observables are defined on $\mathcal{S}_0(\mathbb{R}^2)$, avoiding the infrared divergences that invalidate the standard $\mathcal{S}(\mathbb{R}^2)$ formulation.
• The ground state wave function of $\vartheta(x)$ is not invariant under chiral symmetry, satisfying Goldstone’s criterion, and thus the quanta of $\vartheta(x)$ are confirmed as Goldstone bosons by both Goldstone’s and Witten’s criteria.