Three Loop Analysis of the Critical $O(N)$ Models in $6-ε$ Dimensions

TL;DR

This work extends the cubic $O(N)$ model with a $ frac{1}{2}g_1\sigma(\phi^i)^2 + \tfrac{1}{6}g_2\sigma^3$ interaction to three-loop order in $d=6-\epsilon$, deriving the full renormalization group functions and extracting IR fixed point data and operator dimensions up to $O(\epsilon^3)$. By introducing rescaled couplings, the authors determine the IR fixed point $(x_*,y_*)$ and the critical value $N_{\rm crit}(\epsilon)$, showing a strong reduction of $N_{\rm crit}$ as $\epsilon$ increases (e.g., $N_{\rm crit}(\epsilon=1) \approx 64$). They compute dimensions of $\phi$, $\sigma$, and mixed quadratic/cubic operators, confirming agreement with known large-$N results and providing detailed spectra at the fixed points. The paper also maps the unitary and non-unitary conformal windows, identifies a special $N=1$ $Z_2$-symmetric fixed point with connections to the 3-state Potts model and non-unitary minimal models in lower dimensions, and discusses potential bootstrap approaches to sharpen the non-perturbative picture in $d=5$ and below.

Abstract

We continue the study, initiated in arXiv:1404.1094, of the $O(N)$ symmetric theory of $N+1$ massless scalar fields in $6-ε$ dimensions. This theory has cubic interaction terms $\frac{1}{2}g_1 σ(φ^i)^2 + \frac{1}{6}g_2 σ^3$. We calculate the 3-loop beta functions for the two couplings and use them to determine certain operator scaling dimensions at the IR stable fixed point up to order $ε^3$. We also use the beta functions to determine the corrections to the critical value of $N$ below which there is no fixed point at real couplings. The result suggests a very significant reduction in the critical value as the dimension is decreased to $5$. We also study the theory with $N=1$, which has a $Z_2$ symmetry under $φ\rightarrow -φ$. We show that it possesses an IR stable fixed point at imaginary couplings which can be reached by flow from a nearby fixed point describing a pair of $N=0$ theories. We calculate certain operator scaling dimensions at the IR fixed point of the $N=1$ theory and suggest that, upon continuation to two dimensions, it describes a non-unitary conformal minimal model.

Figures (9)

• Figure 1: The $O(\epsilon^3)$ in $\Delta_{\sigma}$ as a function of $N$ for $N \geq 1039$. The $1/N$ expansion approaches the exact result as we include more terms.
• Figure 2: The zeroes of the one loop $\beta$ functions and the RG flow directions for $N=2000$. The red dots correspond to the stable IR fixed points, while the black dots are unstable fixed points. As $N\rightarrow N_{\rm crit}$, the red dot merges with the nearby black dot, and the two fixed points move into the complex plane.
• Figure 3: Feynman rules.
• Figure 4: The Special $KITE$ and $ChT$ diagrams, the numbers labeling each propagator denote its index.
• Figure 5: The $LADDER$ diagram can be reduced to five master integrals