# Thirring–Wess model

The Thirring–Wess model or Vector Meson model is an exactly solvable quantum field theory describing the interaction of a Dirac field with a vector field in dimension two.

## Definition

The Lagrangian density is made of three terms:

the free vector field $A^{\mu }$ is described by

${(F^{\mu \nu })^{2} \over 4}+{\mu ^{2} \over 2}(A^{\mu })^{2}$ for $F^{\mu \nu }=\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }$ and the boson mass $\mu$ must be strictly positive; the free fermion field $\psi$ is described by

${\overline {\psi }}(i\partial \!\!\!/-m)\psi$ where the fermion mass $m$ can be positive or zero. And the interaction term is

$qA^{\mu }({\bar {\psi }}\gamma ^{\mu }\psi )$ Although not required to define the massive vector field, there can be also a gauge-fixing term

${\alpha \over 2}(\partial ^{\mu }A^{\mu })^{2}$ for $\alpha \geq 0$ There is a remarkable difference between the case $\alpha >0$ and the case $\alpha =0$ : the latter requires a field renormalization to absorb divergences of the two point correlation.

## History

This model was introduced by Thirring and Wess as a version of the Schwinger model with a vector mass term in the Lagrangian .

When the fermion is massless ($m=0$ ), the model is exactly solvable. One solution was found, for $\alpha =1$ , by Thirring and Wess  using a method introduced by Johnson for the Thirring model; and, for $\alpha =0$ , two different solutions were given by Brown and Sommerfield. Subsequently Hagen showed (for $\alpha =0$ , but it turns out to be true for $\alpha \geq 0$ ) that there is a one parameter family of solutions.