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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 \( {\displaystyle A^{\mu }} \) is described by

\( {\displaystyle {(F^{\mu \nu })^{2} \over 4}+{\mu ^{2} \over 2}(A^{\mu })^{2}} \)

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

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

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

\( {\displaystyle qA^{\mu }({\bar {\psi }}\gamma ^{\mu }\psi )} \)

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

\( {\displaystyle {\alpha \over 2}(\partial ^{\mu }A^{\mu })^{2}} \)

for \( {\displaystyle \alpha \geq 0}

There is a remarkable difference between the case \( {\displaystyle \alpha >0} \) and the case \( {\displaystyle \alpha =0} \): 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 ( \( {\displaystyle m=0}) \), 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 \( {\displaystyle \alpha =0} \) , two different solutions were given by Brown[2] and Sommerfield.[3] Subsequently Hagen[4] showed (for \( {\displaystyle \alpha =0} \), but it turns out to be true for \( {\displaystyle \alpha \geq 0}) \) that there is a one parameter family of solutions.

References

Thirring, WE; Wess, JE (1964). "Solution of a field theoretical model in one space one time dimensions". Annals of Physics. 27 (2): 331–337. Bibcode:1964AnPhy..27..331T. doi:10.1016/0003-4916(64)90234-9.
Brown, LS (1963). "Gauge invariance and Mass in a Two-Dimensional Model". Il Nuovo Cimento. 29 (3): 617–643. Bibcode:1963NCim...29..617B. doi:10.1007/BF02827786.
Sommerfield, CM (1964). "On the definition of currents and the action principle in field theories of one spatial dimension". Annals of Physics. 26 (1): 1–43. Bibcode:1964AnPhy..26....1S. doi:10.1016/0003-4916(64)90273-8.
Hagen, CR (1967). "Current definition and mass renormalization in a Model Field Theory". Il Nuovo Cimento A. 51 (4): 1033–1052. Bibcode:1967NCimA..51.1033H. doi:10.1007/BF02721770.

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