Minimally subtracted six loop renormalization of $O(n)$-symmetric $φ^4$ theory and critical exponents

TL;DR

The authors compute the six-loop renormalization-group functions for the $O(n)$-symmetric $\phi^4$ theory in $4-2\varepsilon$ dimensions within the minimal subtraction scheme, for general $n$. They implement advanced diagrammatic techniques, IRR, and a one-scale scheme to obtain exact $\varepsilon$-expansions and verify them against known results, including a detailed asymptotic analysis of the beta function and its primitive contributions up to 11 loops. Using a comprehensive Borel resummation with conformal mapping, they extract high-precision estimates of the critical exponents $\eta$, $\nu$, and $\omega$ in $D=3$ and $D=2$, across several $O(n)$ universality classes, with careful error assessment and parameter stability analysis. The results are broadly consistent with Monte Carlo and conformal bootstrap benchmarks, while highlighting areas (notably $\omega$ in some cases) where further loop data could tighten agreement. This work provides a robust, cross-validated six-loop benchmark for RG predictions and underscores the potential of continued high-order perturbative calculations for critical phenomena.

Abstract

We present the perturbative renormalization group functions of $O(n)$-symmetric $φ^4$ theory in $4-2\varepsilon$ dimensions to the sixth loop order in the minimal subtraction scheme. In addition, we estimate diagrams without subdivergences up to 11 loops and compare these results with the asymptotic behaviour of the beta function. Furthermore, we perform a resummation to obtain estimates for critical exponents in three and two dimensions.

Figures (9)

• Figure 1: These plots demonstrate that the perturbative coefficients of the $\beta$ function ($n=1$) are rather far from their asymptotic values: crosses show the known coefficients $\beta^{\mathrm{MS}}_k$ ($k \leq 7$) and circles indicate the estimates for the primitive contributions $\beta^{\mathrm{prim}}_k$ up to loop order $11$ (see table \ref{['tab:primitive-estimates']}). They are normalized by the predictions from the asymptotic formulas \ref{['eq:beta-asymptotics']} (left plot) and \ref{['eq:beta-asymptotics-gamma']} (right plot). The abscissa is $1/k$, such that in the limit $k\rightarrow \infty$ the points should approach $1.0$ on the vertical axis. Note that there are no primitives with two loops, resulting in the point $(1/3,0)$.
• Figure 2: Dependence of the coefficients of the beta function on $n$: The dashed curves show the primitive contributions $\beta^{\mathrm{prim}}_k/(k!\cdot k^{3+n/2})$ at $6$ and $11$ loops. For comparison with the full beta function, our result for $\beta^{\mathrm{MS}}_k/(k!\cdot k^{3+n/2})$ at $6$ loops is included as the dotted line. The solid line shows the limiting curve for $k\rightarrow \infty$, namely $C_{\beta}$ from \ref{['eq:beta-asymptotics-constant']}.
• Figure 3: Dependence of the resummations $\tilde{\eta}^{b,\lambda,q}_{\ell}$ of $\eta$ on $b$ for different values of $\lambda$, at $n=1$ in $D=3$ dimensions ($\varepsilon=0.5$) with $q=0.2$. The loop order $\ell$ is indicated by the label $\varepsilon^{\leq \ell}$.
• Figure 4: Dependence of the resummation of $\eta$ on $\lambda$ for different values of $b$, for $n=1$ in $D=3$ dimensions with $q=0.2$. The loop order $\ell$ is indicated by the label $\varepsilon^{\leq \ell}$.
• Figure 5: For different values of $q \in \left\{ 0,0.2,0.4,0.6 \right\}$, we plot the dependence of the resummation on $b$. In each case, we adjusted $\lambda$ to find the best apparent stability with respect to $b$. The loop order $\ell$ is indicated by the label $\varepsilon^{\leq \ell}$.

Theorems & Definitions (1)

• Definition 1