Operator mixing in $\boldsymbol{\epsilon}$-expansion: scheme and evanescent (in)dependence

TL;DR

This paper investigates how scheme choices affect the anomalous dimensions of operators in fermionic theories near four dimensions, revealing scheme dependence beyond leading order and proposing a truncation method for eigenvalue extraction.

Contribution

It clarifies the scheme dependence of anomalous dimensions in theories with evanescent operators and introduces a truncation approach to compute observable scaling dimensions.

Findings

Eigenvalues at the fixed point depend on the renormalization scheme beyond leading order.

A truncation method allows for the extraction of observable eigenvalues.

The scheme dependence cancels out for physical scaling dimensions.

Abstract

We consider theories with fermionic degrees of freedom that have a fixed point of Wilson-Fisher type in non-integer dimension $d = 4-2\epsilon$. Due to the presence of evanescent operators, i.e., operators that vanish in integer dimensions, these theories contain families of infinitely many operators that can mix with each other under renormalization. We clarify the dependence of the corresponding anomalous-dimension matrix on the choice of renormalization scheme beyond leading order in $\epsilon$-expansion. In standard choices of scheme, we find that eigenvalues at the fixed point cannot be extracted from a finite-dimensional block. We illustrate in examples a truncation approach to compute the eigenvalues. These are observable scaling dimensions, and, indeed, we find that the dependence on the choice of scheme cancels. As an application, we obtain the IR scaling dimension of four-fermion operators in QED in $d=4-2\epsilon$ at order $\mathcal{O}(\epsilon^2)$.