Feynman Path Integral Formulation

3.4 Self-coupled Fermion Model 833.4 Self-coupled Fermion ModelThe non-linear sigma model is not an isolated example of a field theory that isnot perturbatively renormalizable above two dimensions, although it is certainly byfar the most thoroghly explored one. Here it seems worthwhile to mention a secondexample of a theory which naively is not perturbatively renormalizable in d > 2, andyet 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 setof 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 theappearance of a fermion mass term. Interest in the model resides in the fact that itexhibits a mechanism for dynamical mass generation and chiral symmetry breaking.In two dimensions the fermion self-coupling constant is dimensionless, and aftersetting d = 2+ε one is again ready to develop the full machinery of the perturbativeexpansion in u and ε, as was done for the non-linear σ-model, since the model isagain believed to be multiplicatively renormalizable in the framework of the 2 + εexpansion. For the β-function one finds to three loopsβ(u)=εu − ¯N − 22π u2 + ¯N − 24π 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 vanishesidentically.As for the case of the non-linear σ-model, the solution of the renormalizationgroup equations involves an invariant scale, which can be obtained (up to a constantwhich cannot be determined from perturbation theory alone) by integratingEq. (3.70)[ ∫ uξ −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)uand thus non-analytic in the bare coupling u. Above two dimensions a non-trivialultraviolet fixed point appears atu c =2π¯N − 2 ε + 2π( ¯N − 2) 2 ε2 + ( ¯N + 1)π2( ¯N − 2) 3 ε3 + ... (3.73)