Thirring–Wess model

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

ψ ¯ ( i ∂ / − m ) ψ {\displaystyle {\overline {\psi }}(i\partial \!\!\!/-m)\psi } {\displaystyle {\overline {\psi }}(i\partial \!\!\!/-m)\psi }

where the fermion mass m {\displaystyle 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 )} {\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}} {\displaystyle {\alpha \over 2}(\partial ^{\mu }A^{\mu })^{2}}

for α ≥ 0 {\displaystyle \alpha \geq 0} {\displaystyle \alpha \geq 0}

There is a remarkable difference between the case α > 0 {\displaystyle \alpha >0} {\displaystyle \alpha >0} and the case α = 0 {\displaystyle \alpha =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} {\displaystyle m=0}), the model is exactly solvable. One solution was found, for α = 1 {\displaystyle \alpha =1} {\displaystyle \alpha =1}, by Thirring and Wess [1] using a method introduced by Johnson for the Thirring model; and, for α = 0 {\displaystyle \alpha =0} {\displaystyle \alpha =0}, two different solutions were given by Brown[2] and Sommerfield.[3] Subsequently Hagen[4] showed (for α = 0 {\displaystyle \alpha =0} {\displaystyle \alpha =0}, but it turns out to be true for α ≥ 0 {\displaystyle \alpha \geq 0} {\displaystyle \alpha \geq 0}) that there is a one parameter family of solutions.

References

  1. 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.
  2. 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. S2CID 122285105.
  3. 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.
  4. 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. S2CID 58940957.