Thirring–Wess model
# 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.

