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

A. V. Bednyakov; M. V. Kompaniets; A. V. Trenogin

doi:10.1016/j.nuclphysb.2026.117331

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arXiv · 2512.05059

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

A. V. Bednyakov·M. V. Kompaniets·A. V. Trenogin

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Abstract

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

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A. V. Bednyakov, M. V. Kompaniets, A. V. Trenogin. 2026-01-29. On the six-loop scaling dimensions of the $(ϕ^2)^n$ operators in $d=3$. https://doi.org/10.1016/j.nuclphysb.2026.117331

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