Six loop analytical calculation of the field anomalous dimension and the critical exponent $η$ in $O(n)$-symmetric $\varphi^4$ model

TL;DR

This work delivers a completely analytical six-loop calculation of the field anomalous dimension $\gamma_{\varphi}$ and the critical exponent $\eta$ for the $O(n)$-symmetric $\varphi^4$ theory, pushing the frontier of analytic critical-point computations. It leverages infrared rearrangement and the $R^*$-operation, exploiting the TV-Reducible diagram structure to reduce most six-loop contributions to tractable lower-loop integrals, with two TV-irreducible diagrams treated via advanced IBP and dual-graph techniques. The authors provide the explicit six-loop expression for $\gamma_{\varphi}(g)$, derive the corresponding $\varepsilon$-expansion for $\eta$, perform rigorous consistency checks against the $1/n$ expansion, and offer resummation-based predictions for higher-loop terms that agree closely with known benchmarks. They also compare the $n=1$ case to Borel-resummed conformal-mapping predictions and supply public data for the diagram-by-diagram counterterms, underscoring the approach’s reliability and utility as a benchmark for future high-loop analyses in scalar field theory.

Abstract

We report on a completely analytical calculation of the field anomalous dimension $γ_{\varphi}$ and the critical exponent $η$ for the $O(n)$-symmetric $\varphi^4$ model at the record six loop level. We successfully compare our result for $γ_{\varphi}$ with $n=1$ with the predictions based on the method of the Borel resummation combined with a conformal mapping. Predictions for seven loop contribution to the field anomalous dimensions are given.

Figures (6)

• Figure 1: No IR safe IRR (with one softened line) is possible for this graph.
• Figure 2: IRR performed using vertex cut, dashed line represent corresponding softening factor.
• Figure 3: The only TVI diagram contributing to the field self-energy at five loop.
• Figure 4: (a) and (b): TVI diagrams contributing to the field self-energy at six loop level.
• Figure 5: Graph that corresponds to Nickel index (NI) equal to $ee12|223|3|ee|$.