Thirring–Wess model

Not to be confused with Thirring 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 ${\displaystyle A^{\mu }}$ is described by

${\displaystyle {(F^{\mu \nu })^{2} \over 4}+{\mu ^{2} \over 2}(A^{\mu })^{2}}$

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

${\displaystyle {\overline {\psi }}(i\partial \!\!\!/-m)\psi }$

where the fermion mass ${\displaystyle m}$ can be positive or zero. And the interaction term is

${\displaystyle qA^{\mu }({\bar {\psi }}\gamma ^{\mu }\psi )}$

Although not required to define the massive vector field, there can be also a gauge-fixing term

${\displaystyle {\alpha \over 2}(\partial ^{\mu }A^{\mu })^{2}}$

for ${\displaystyle \alpha \geq 0}$

There is a remarkable difference between the case ${\displaystyle \alpha >0}$ and the case ${\displaystyle \alpha =0}$: 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 (${\displaystyle m=0}$), the model is exactly solvable. One solution was found, for ${\displaystyle \alpha =1}$, by Thirring and Wess ^[1] using a method introduced by Johnson for the Thirring model; and, for ${\displaystyle \alpha =0}$, two different solutions were given by Brown^[2] and Sommerfield.^[3] Subsequently Hagen^[4] showed (for ${\displaystyle \alpha =0}$, but it turns out to be true for ${\displaystyle \alpha \geq 0}$) that there is a one parameter family of solutions.