A COMPARED ANALYSIS OF THE SUSCEPTIBILITY IN THE O(N) THEORY
Abstract
The longitudinal susceptibility χL of the O(N) theory in the broken phase is analyzed by means of three different approaches, namely the leading contribution of the 1/N expansion, the Functional Renormalization Group flow in the Local Potential approximation and the improved effective potential via the Callan–Symanzik equations, properly extended to d = 4 dimensions through the expansion in powers of ϵ = 4-d. The findings of the three approaches are compared and their agreement in the large N limit is shown. The numerical analysis of the Functional Renormalization Group flow equations at small N supports the vanishing of 
in d = 3 and d = 3.5 but is not conclusive in d = 4, where we have to resort to the Callan–Smanzik approach. At finite N as well as in the limit N→∞, we find that 
vanishes with J as Jϵ/2 for ϵ> 0 and as (ln(J))-1 in d = 4.
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