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

:$+ (A^\mu)^2$

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

:$\overline(i\partial\!\!\!/-m)\psi$

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
:$(\partial^\mu A^\mu)^2$
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.

References

External links

{{DEFAULTSORT:Thirring-Wess Model
Quantum field theory
Exactly solvable models