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78) m2R = m20 + λ0 (n + 2) C0 a−2 − C2 m20 + 1 m2 ln(a2 m20 ) . 83) A renormalized coupling constant λR may be defined in terms of Γ(4) at zero momentum, by writing ΓR α1 α2 α3 α4 (0, 0, 0, 0) = −2λR sα1 α2 α3 α4 . 85) we find n+8 [ln(a2 m20 ) + c], 16π 2 1 16π 2 C2 − c=− . 86), m20 = m2R − λR (n + 2) C0 a−2 − C2 m2R + 1 m2 ln(a2 m2R ) 16π 2 R + O(λ2R ), n+8 [ln(a2 m2R ) + c] + O(λ3R ). 91) which are indeed independent of the lattice spacing a. Notice that the constants C0 , C1 and C2 are absent: all reference to the lattice has disappeared from the renormalized vertex functions.
The particles π ± and π 0 are described by the fields πk (x). The σ field (after which the model is named the σ model) corresponds to the very broad σ resonance around 900 MeV. The model loses its validity at such energies, for example the ρ mesons with mass 770 MeV are completely neglected. 2 O(n) models as spin models We continue in the quantum theory. The lattice regularized action will be taken as α α 1 2 ∂µ ϕx ∂µ ϕx S=− α α α 2 1 + 12 m20 ϕα x ϕx + 4 λ0 (ϕx ϕx ) . 13) x We have changed the notation for the parameters: µ2 → m20 , λ → λ0 .
67) Note that cϕ is dimensionless. The dimension of ϕ follows from the requirement that the action S is dimensionless. In d space–time dimensions, [ϕ] = a−(d−2)/2 , c = a(d−2)/2 . 80). The lattice action was chosen such that for smooth functions f (x), S(f ) → Scont (f ) in the classical continuum limit a → 0. However, it is useful to keep in mind that typical field configurations ϕx contributing to the path integral are not smooth at all on the lattice scale. The previous sentence is meant in the following sense.