Critical exponents from the two-particle irreducible1/Nexpansion

Abstract

We calculate the critical exponent $\ensuremath{\nu}$ of the $O(N)$ symmetric ${\ensuremath{\varphi}}^{4}$ model within the $1/N$ expansion of the two-particle-irreducible effective action, which provides us with a self-consistent approximation scheme for the correlation function. The exponent $\ensuremath{\nu}$ controls the behavior of a two-point function $⟨\ensuremath{\varphi}\ensuremath{\varphi}⟩$ near the critical point $T\ensuremath{\ne}{T}_{c}$ through the correlation length $\ensuremath{\xi}\ensuremath{\sim}|T\ensuremath{-}{T}_{c}{|}^{\ensuremath{-}\ensuremath{\nu}}$, but we notice that it appears also in the scaling form of the three-point vertex function ${\ensuremath{\Gamma}}^{(2,1)}\ensuremath{\sim}⟨\ensuremath{\varphi}\ensuremath{\varphi}{\ensuremath{\varphi}}^{2}⟩$ at the critical point $T={T}_{c}$; in the momentum space, ${\ensuremath{\Gamma}}^{(2,1)}\ensuremath{\sim}{k}^{2\ensuremath{-}\ensuremath{\eta}\ensuremath{-}1/\ensuremath{\nu}}$. We derive a self-consistent equation for ${\ensuremath{\Gamma}}^{(2,1)}$ from the two-particle-irreducible effective action including the skeleton diagrams up to the next-leading-order in the $1/N$ expansion, and solve it to the leading-log accuracy (i.e., keeping the leading $\mathrm{ln}k$ terms) to obtain $\ensuremath{\nu}$. Our results turn out to improve those obtained in the standard one-particle-irreducible calculation at the next-leading-order.