The two-flavor Schwinger model at 50: Solving Coleman's puzzles

TL;DR

This paper analytically and numerically resolves three longstanding puzzles in the two-flavor Schwinger model, clarifying its phase structure, mass gap behavior, and isospin-breaking effects.

Contribution

It provides new solutions to Coleman's three puzzles using advanced analytical methods and lattice simulations, deepening understanding of the two-flavor Schwinger model.

Findings

Spontaneous C-symmetry breaking and deconfinement at  for equal masses.

Mass gap behaves as m e^{-0.111 g^2/m^2} in strong coupling.

Confirmed level crossing between isosinglet particles at  through integrability and numerics.

Abstract

In his 1976 paper "More about the massive Schwinger model", Coleman introduced $1+1$-dimensional Quantum Electrodynamics coupled to two charged massive fermions. By applying Abelian bosonization, he elucidated much of the physics of this two-flavor Schwinger model, but he listed three puzzles at the end of his paper. We present new analytical and numerical calculations to solve Coleman's three puzzles and thereby deepen our understanding of this model. These puzzles pertain to the theory with equal fermion masses at $\theta = 0$ and at $\theta = \pi$, as well as the size of isospin-breaking effects when the fermion masses are unequal. For the puzzle at $\theta = \pi$, the solution is related to the structure of the zero-temperature phase diagram arXiv:2305.04437: for equal fermion masses $m$, the model exhibits spontaneous breaking of charge conjugation symmetry and absence of confinement for any value of the gauge coupling $g$, so that there is a smooth interpolation from weak to strong coupling. Using two-loop Renormalization Group and integrability methods, we show that the mass gap behaves as $\sim m e^{-0.111 g^2/m^2}$ in the strong coupling regime $m\ll g$. Our numerical results using the lattice Hamiltonian are in good agreement with this behavior. For the puzzle at $\theta = 0$, the solution is related to a level crossing between two isosinglet particles with different discrete quantum numbers; we demonstrate the necessity of such a crossing by comparing integrability and weak coupling calculations, and we also exhibit the crossing numerically. Finally, we provide a new estimate for the size of isospin-breaking effects caused by different fermion masses at strong coupling.