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 \ge 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 $$ \alpha \ge 0 $$) that there is a one parameter family of solutions.