Feynman Path Integral Formulation

3.4 Self-coupled Fermion Model 83
3.4 Self-coupled Fermion Model
The non-linear sigma model is not an isolated example of a field theory that is
not perturbatively renormalizable above two dimensions, although it is certainly by
far the most thoroghly explored one. Here it seems worthwhile to mention a second
example of a theory which naively is not perturbatively renormalizable in d > 2, and
yet whose critical properties can again be worked out both in the 2 + ε expansion,
and in the large N limit. It is described by an U(N)-invariant action containing a set
of N massless self-coupled Dirac fermions (Wilson, 1973; Gross and Neveu, 1974)
∫
S(ψ, ¯ψ)=− d d x[ ¯ψ· ̸∂ψ+ 1 2 Λ d−2 u( ¯ψ · ψ) 2 ] . (3.69)
In even dimensions the discrete chiral symmetry ψ → γ 5 ψ, ¯ψ →−¯ψγ 5 prevents the
appearance of a fermion mass term. Interest in the model resides in the fact that it
exhibits a mechanism for dynamical mass generation and chiral symmetry breaking.
In two dimensions the fermion self-coupling constant is dimensionless, and after
setting d = 2+ε one is again ready to develop the full machinery of the perturbative
expansion in u and ε, as was done for the non-linear σ-model, since the model is
again believed to be multiplicatively renormalizable in the framework of the 2 + ε
expansion. For the β-function one finds to three loops
β(u)=εu − ¯N − 2
2π u2 + ¯N − 2
4π 2 u3 + ( ¯N − 2)( ¯N − 7)
32π 3 u 4 + ... (3.70)
with the parameter ¯N = N tr1, where the last quantity is the identity matrix in the γ-
matrix algebra. In two dimensions ¯N = 2N and the model is asymptotically free; for
¯N = 2 the interaction is proportional to the Thirring one and the β-function vanishes
identically.
As for the case of the non-linear σ-model, the solution of the renormalization
group equations involves an invariant scale, which can be obtained (up to a constant
which cannot be determined from perturbation theory alone) by integrating
Eq. (3.70)
[ ∫ u
ξ −1 du ′ ]
(u)=m(u)=const.Λ exp −
β(u ′ . (3.71)
)
In two dimensions this scale is, to lowest order in u, proportional to
[
m(u) ∼ Λ exp −
2π ]
, (3.72)
u→0 ( ¯N − 2)u
and thus non-analytic in the bare coupling u. Above two dimensions a non-trivial
ultraviolet fixed point appears at
u c =
2π
¯N − 2 ε + 2π
( ¯N − 2) 2 ε2 + ( ¯N + 1)π
2( ¯N − 2) 3 ε3 + ... (3.73)