Physics:Thirring–Wess model

Short description: Solvable 1+1 dimensional quantum field theory

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

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

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

$\bar{\psi }(i\partial /-m)\psi$

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

$q{A}^{\mu }(\overline{\psi }{\gamma }^{\mu }\psi )$

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

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

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.