Scaling dimensions in QED$_3$ from the $\epsilon$-expansion

TL;DR

This paper uses epsilon-expansion to compute scaling dimensions of operators in QED$_3$, providing insights into the conformal phase and the effects of four-fermion interactions at the IR fixed point.

Contribution

It derives the scaling dimensions of four-fermion and bilinear operators beyond leading order, including a two-loop mixing calculation, and extrapolates results to three dimensions.

Findings

Next-to-leading order corrections significantly alter four-fermion operator dimensions.

Four-fermion operators do not become marginal for any N_f, indicating stability of the conformal phase.

The epsilon-expansion offers reliable estimates for scalar operator dimensions across N_f range.

Abstract

We study the fixed point that controls the IR dynamics of QED in $d = 4 - 2\epsilon$. We derive the scaling dimensions of four-fermion and bilinear operators beyond leading order in $\epsilon$-expansion. For the four-fermion operators, this requires the computation of a two-loop mixing that was not known before. We then extrapolate these scaling dimensions to $d = 3$ to estimate their value at the IR fixed point of QED$_3$ as function of the number of fermions $N_f$. The next-to-leading order result for the four-fermion operators corrects significantly the leading one. Our best estimate at this order indicates that they do not cross marginality for any value of $N_f$, which would imply that they cannot trigger a departure from the conformal phase. For the scaling dimensions of bilinear operators, we observe better convergence as we increase the order. In particular, $\epsilon$-expansion provides a convincing estimate for the dimension of the flavor-singlet scalar in the full range of $N_f$.