Six-loop anomalous dimension of the $\phi^Q$ operator in the $O(N)$ symmetric model

TL;DR

This paper calculates the six-loop anomalous dimension of the $\,\phi^Q$ operator in the $O(N)$ model, extending previous results to higher charge operators and enabling comparison with simulations and large-$N$ predictions.

Contribution

It introduces a method to compute six-loop anomalous dimensions for operators with arbitrary charge $Q$, including new diagrams with five or more legs, in scalar theories.

Findings

Computed the anomalous dimension of $\,\phi^{Q=5}$ at six loops.

Derived general-$Q$ anomalous dimensions combining multiple approaches.

Compared critical exponents with Monte-Carlo and large-$N$ results.

Abstract

A technique of large-charge expansion provides a novel opportunity for calculation of critical dimensions of operators $\phi^Q$ with fixed charge $Q$. In the small-coupling regime the polynomial structure of the anomalous dimensions can be fixed from a number of direct perturbative calculations for a fixed $Q$. At the six-loop level one needs to include new diagrams that correspond to operators with five or more legs. The latter never appeared before in scalar-theory calculations. Here we show how to compute the anomalous dimension of the operator $\phi^{Q=5}$ at the six-loop order. In combination with results for operators with $Q<5$, which are extracted from the six-loop beta-functions for general scalar theory, and with predictions from the large-charge expansion, our calculation allows us to derive the answer for general-$Q$ anomalous dimensions. At the critical point resummation in three dimensions enables us to compare the critical exponents with results of Monte-Carlo simulations and large-$N$ predictions.