On the six-loop scaling dimensions of the $(\phi^2)^n$ operators in $d=3$

We consider a class of singlet operators $(\phi^2)^n$ in the three-dimensional $O(N)$ model with $\lambda^2 \phi^6$ interaction. Recently, the corresponding anomalous dimensions $\gamma_{2n}$ were computed by semiclassical methods and the all-loop result for the leading-$n$ corrections in the small $\lambda$ limit was found. In this paper, we obtain the six-loop expressions not only for the leading-$n$ contribution but also for the subleading one. While the leading correction confirms the predictions of recent semiclassical calculation, the subleading one is a new result and will serve as a future welcome check for all-loop expressions. As an important by-product of our calculation, we provide a full dependence on $n$ of the four-loop $\gamma_{2n}$ in the $O(N)$ case.