Unconventional large-N limit of the Gaussian effective potential and the phase transition in λφ ⁴ theory

Abstract

In the conventional large-N limit the coupling constant λ is required to scale as 1/N. While the Gaussian effective potential (GEP) is known to contain the exact result in this limit, it shows a phase transition only when λ≫1/N (in units of the renormalized mass in the symmetric vacuum). Here we determine the asymptotic behaviour, asN→∞, of λ and other quantities at the phase transition of the GEP. We find λ_crit to be finite in 0+1 dimensions; of order 1/lnN in 1+1 dimensions; 1/N ^1/3 in 2+1 dimensions; and$1/\sqrt N$ in 3+1 dimensions. The GEP's first-order phase transition is shown to become asymptotically second-order in 1+1 dimensions and below. We also discuss non-integer dimensions and the approach to the non-trivial “autonomous” theory in 3+1 dimensions.