Thirring–Wess model

Not to be confused with Thirring 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

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

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Quantum field theories

Standard	Chern–Simons model Chiral model Kondo model Conformal field theory Noncommutative quantum field theory Non-linear sigma model Quartic interaction Quantum chromodynamics Quantum electrodynamics Quantum Yang–Mills theory Schwinger model sine-Gordon Standard Model String theory Toda field theory Topological quantum field theory Wess–Zumino model Wess–Zumino–Witten model Yang–Mills theory Yang–Mills–Higgs model Yukawa model Thirring–Wess model

Four-fermion interactions	Thirring model Gross–Neveu model Nambu–Jona-Lasinio model