# 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 [1] using a method introduced by Johnson for the Thirring model; and, for $\alpha = 0$, two different solutions were given by Brown[2] and Sommerfield.[3] Subsequently Hagen [4] 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

1. ^ Thirring, W; Wess J (1964). "Solution of a field theoretical model in one space one time dimensions". Annals Phys. 27: 331–337.
2. ^ Brown, L (1963). "Gauge invariance and Mass in a Two-Dimensional Model". N.Cimento. 29.
3. ^ Sommerfield, C (1964). Annals Phys. 26.
4. ^ Hagen, C (1967). "Current definition and mass renormalization in a Model Field Theory". N. Cimento A 51.