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

A. V. Bednyakov, M. V. Kompaniets, A. V. Trenogin

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 letter, 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 the 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.

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

Abstract

We consider a class of singlet operators in the three-dimensional model with interaction. Recently, the corresponding anomalous dimensions were computed by semiclassical methods and the all-loop result for the leading- corrections in the small limit was found. In this letter, we obtain the six-loop expressions not only for the leading- 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 the all-loop expressions. As an important by-product of our calculation, we provide a full dependence on of the four-loop in the case.

Paper Structure

This paper contains 9 sections, 53 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Renormalization of $(\phi^2)^n$: leading-$n$ and subleading-$n$ orders
  3. Computation technique
  4. Results
  5. Conclusions
  6. Feynman rules of $O^{(n)}_{n-2,i}$ operators and the conformal primary
  7. Counterterms of graphs
  8. $\mathcal{KR}^\prime$ for quadratically-divergent graphs
  9. Tables of $\mathcal{KR}^\prime$

Figures (1)

  • Figure 1: Examples of Feynman diagrams giving rise to divergent contribution to $\Gamma_{2n}[O^{(n)}_{n}]$ at two loops $(a)$, and to $\Gamma_{2n-4}[O^{(n)}_{n}]$ at four $(b)$ and six $(c,d)$ loops. The "spectator" legs of the operator are marked by the red color. The corresponding auxiliary graphs $\Gamma_{3}[\tilde{O}^{(n)}_{3}]$, $\Gamma_{1}[\tilde{O}^{(n)}_{5}]$, $\Gamma_3[\tilde{O}^{(n)}_7]$, and $\Gamma_2[\tilde{O}^{(n)}_{6}]$ for $(a)$, $(b)$$(c)$, and $(d)$, respectively, are obtained by stripping off the "spectator" lines.