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

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.