Schwinger Model

The Schwinger model is QED with one massless Dirac fermion in $1+1$ dimensions. On the infinite line its gauge-invariant local dynamics reduces exactly to a free massive scalar, with mass $e/\sqrt\pi$ . It provides a sharp example of mass generation and charge screening without a weak-coupling expansion.

Massless two-dimensional electrodynamics

Take $e>0$ and

\mathcal L=\bar\psi i\gamma^\mu(\partial_\mu-ieA_\mu)\psi -\frac14F_{\mu\nu}F^{\mu\nu}.

There is no bare fermion mass. In two spacetime dimensions, $[\psi]=1/2$ , $[A_\mu]=0$ , and $[e]=1$ . A mass proportional to $e$ is therefore dimensionally possible without generating a scale from a dimensionless coupling.

A free Maxwell field in one spatial dimension has no transverse propagating photon. Coupling it to the fermion nevertheless produces a propagating gauge-invariant massive mode. This is not the same mechanism as a Higgs expectation value.

The exact massive mode

Bosonization represents the fermion current by a derivative of a scalar. With its canonical kinetic normalization, integrating the gauge field gives the local scalar Lagrangian

\mathcal L_{\mathrm{bos}} =\frac12(\partial\varphi)^2-\frac12m_\gamma^2\varphi^2, \qquad m_\gamma^2=\frac{e^2}{\pi}.

This statement concerns the massless one-flavor theory with the relevant vacuum and global sector fixed. Its anomaly and gauge dynamics produce the mass; neither gauge fixing alone nor classical Maxwell counting produces it. The explicit bosonized action and the mass follow in Tong, undated lecture notes, §7.5.6, PDF.

Screening a pair of static probes

For external opposite charges of magnitude $q$ separated by $R$ on the infinite line, the massive Green function gives the potential relative to coincident probes,

V(R)=\frac{q^2}{2m_\gamma} \left(1-e^{-m_\gamma R}\right).

At small separation, $V(R)=q^2R/2+O(R^2)$ , recovering the linear electrostatic energy. At large separation it saturates at $q^2/(2m_\gamma)$ . The force magnitude decays exponentially. The result is a statement about external probes in this specified massless model, not a claim that every two-dimensional gauge theory screens.

A bare fermion mass introduces a cosine interaction in the bosonized description. Multiple flavors, compact space, boundary conditions, and background electric-flux sectors also change the problem. Those modifications cannot inherit the single free massive-boson answer unchanged.

Develop the calculation

The Schwinger Model, Screening, and Bosonization gives the current conventions, anomaly, and probe-potential derivation. The course lesson Schwinger Model and Gauge-Invariant Correlators provides a study route. Compare Gross–Neveu for a mass generated from a dimensionless interaction.

References

Tong, David. Gauge Theory. University of Cambridge lecture notes, undated online revision, §7.5.6. Author’s site.