Phase transitions in two dimensions and multiloop renormalization group expansions

We discuss using the field theory renormalization group (RG) to study the critical behavior of two-dimensional (2D) models. We write the RG functions of the 2D $\lambda\phi^4$ Euclidean $n$-vector theory up to five-loop terms, give numerical estimates obtained from these series by Padé–Borel–Leroy resummation, and compare them with their exact counterparts known for $n=1,0,-1$. From the RG series, we then derive pseudo-$\epsilon$-expansions for the Wilson fixed point location $g^*$, critical exponents, and the universal ratio $R_6=g_6/g^2$, where $g_6$ is the effective sextic coupling constant. We show that the obtained expansions are “friendler” than the original RG series: the higher-order coefficients of the pseudo-$\epsilon$-expansions for $g^*$, $R_6$, and $\gamma^{-1}$ turn out to be considerably smaller than their RG analogues. This allows resumming the pseudo-$\epsilon$-expansions using simple Padé approximants without the Borel–Leroy transformation. Moreover, we find that the numerical estimates obtained using the pseudo-$\epsilon$-expansions for $g^*$ and $\gamma^{-1}$ are closer to the known exact values than those obtained from the five-loop RG series using the Padé–Borel–Leroy resummation.