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 μ {\displaystyle A^{\mu }} is described by
( F μ ν ) 2 4 + μ 2 2 ( A μ ) 2 {\displaystyle {(F^{\mu \nu })^{2} \over 4}+{\mu ^{2} \over 2}(A^{\mu })^{2}}
for F μ ν = ∂ μ A ν − ∂ ν A μ {\displaystyle F^{\mu \nu }=\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }} and the boson mass μ {\displaystyle \mu } must be strictly positive; the free fermion field ψ {\displaystyle \psi }
is described by
ψ ¯ ( i ∂ / − m ) ψ {\displaystyle {\overline {\psi }}(i\partial \!\!\!/-m)\psi }
where the fermion mass m {\displaystyle m} can be positive or zero. And the interaction term is
q A μ ( ψ ¯ γ μ ψ ) {\displaystyle qA^{\mu }({\bar {\psi }}\gamma ^{\mu }\psi )}
Although not required to define the massive vector field, there can be also a gauge-fixing term
α 2 ( ∂ μ A μ ) 2 {\displaystyle {\alpha \over 2}(\partial ^{\mu }A^{\mu })^{2}}
for α ≥ 0 {\displaystyle \alpha \geq 0}
There is a remarkable difference between the case α > 0 {\displaystyle \alpha >0} and the case α = 0 {\displaystyle \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 {\displaystyle m=0} ), the model is exactly solvable. One solution was found, for α = 1 {\displaystyle \alpha =1} , by Thirring and Wess using a method introduced by Johnson for the Thirring model; and, for α = 0 {\displaystyle \alpha =0} , two different solutions were given by Brown and Sommerfield. Subsequently Hagen showed (for α = 0 {\displaystyle \alpha =0} , but it turns out to be true for α ≥ 0 {\displaystyle \alpha \geq 0} ) that there is a one parameter family of solutions.
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