On the six-loop scaling dimensions of the $(\ddot{I}{•}^2{)}^n$ operators in $d=3$

Abstract

We consider a class of singlet operators $(\ddot{I}{•}^2{)}^n$ in the three-dimensional $O(N)$ model with $\hat{I}{»}^2\ddot{I}{•}^6$ interaction. Recently, the corresponding anomalous dimensions ${\hat{I}}_{2n}^3$ were computed by semiclassical methods and the all-loop result for the leading-$n$ corrections in the small $\hat{I}»$ 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 ${\hat{I}}_{2n}^3$ in the $O(N)$ case.

34 pages, 1 figure, 8 tables, version accepted in NPB, results in updated ancillary file