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 [math]\displaystyle{ A^\mu }[/math] is described by

[math]\displaystyle{ {(F^{\mu\nu})^2 \over 4} +{\mu^2\over 2} (A^\mu)^2 }[/math]

for [math]\displaystyle{ F^{\mu\nu}= \partial^\mu A^\nu - \partial^\nu A^\mu }[/math] and the boson mass [math]\displaystyle{ \mu }[/math] must be strictly positive; the free fermion field [math]\displaystyle{ \psi }[/math] is described by

[math]\displaystyle{ \overline{\psi}(i\partial\!\!\!/-m)\psi }[/math]

where the fermion mass [math]\displaystyle{ m }[/math] can be positive or zero. And the interaction term is

[math]\displaystyle{ qA^\mu(\bar\psi\gamma^\mu\psi) }[/math]

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

[math]\displaystyle{ {\alpha\over 2} (\partial^\mu A^\mu)^2 }[/math]

for [math]\displaystyle{ \alpha \ge 0 }[/math]

There is a remarkable difference between the case [math]\displaystyle{ \alpha \gt 0 }[/math] and the case [math]\displaystyle{ \alpha = 0 }[/math]: 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 ([math]\displaystyle{ m = 0 }[/math]), the model is exactly solvable. One solution was found, for [math]\displaystyle{ \alpha = 1 }[/math], by Thirring and Wess using a method introduced by Johnson for the Thirring model; and, for [math]\displaystyle{ \alpha = 0 }[/math], two different solutions were given by Brown and Sommerfield. Subsequently Hagen showed (for [math]\displaystyle{ \alpha = 0 }[/math], but it turns out to be true for [math]\displaystyle{ \alpha \ge 0 }[/math]) that there is a one parameter family of solutions.