title: Thirring–Wess model
text: # 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 $\alpha \geq 0$
  
  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 $\alpha \geq 0$) that there is a one parameter family of solutions.
  
  