Anomalous dimensions from conformal field theory: Generalized $\phi^{2n+1}$ theories

TL;DR

This paper studies anomalous dimensions in generalized $\, ext{phi}^{2n+1}$ theories using conformal bootstrap and extends the analysis to models with higher derivatives and symmetry groups, revealing new constraints and consistency checks.

Contribution

It introduces a conformal multiplet recombination approach to compute anomalous dimensions in generalized $\, ext{phi}^{2n+1}$ theories and extends the analysis to models with higher derivatives and symmetries.

Findings

Leading anomalous dimensions are computed for $n=1$ cases.

For $n>1$, some anomalous dimensions contain an unconstrained constant.

Results are consistent with crossing symmetry and bootstrap methods.

Abstract

We investigate $\phi^{2n+1}$ deformations of the generalized free theory in the $\epsilon$ expansion, where the canonical kinetic term is generalized to a higher-derivative version. For $n=1$, we use the conformal multiplet recombination method to determine the leading anomalous dimensions of the fundamental scalar operator $\phi$ and the bilinear composite operators $\mathcal J$. Then we extend the $n=1$ analysis to the Potts model with $S_{N+1}$ symmetry and its higher-derivative generalization, in which $\phi$ is promoted to an $N$-component field. We further examine the Chew-Frautschi plots and their $N$ dependence. However, for each integer $n>1$, the leading anomalous dimensions of $\phi$ and $ \mathcal{J}$ are not fully determined and contain one unconstrained constant, which in the canonical cases can be fixed by the results from the traditional diagrammatic method. In all cases, we verify that the multiplet-recombination results are consistent with crossing symmetry using the analytic bootstrap methods.