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 $\mathcal{O}(\epsilon^2)$.
Highlights
One of the tools to study conformal field theories (CFTs) is to realize them as the endpoint of a renormalization group (RG) flow
In this paper we investigate the problem of obtaining the IR scaling dimension of physical operators beyond leading order (LO) in ε
In our companion paper [31], we focus on 3d QED and use the next-to-leading order (NLO) eigenvalues obtained here to estimate the scaling dimensions of four-fermion operators in d 1⁄4 3
Summary
One of the tools to study conformal field theories (CFTs) is to realize them as the endpoint of a renormalization group (RG) flow. In this paper we investigate the problem of obtaining the IR scaling dimension of physical operators beyond LO in ε These correspond to eigenvalues of the ADM evaluated at the fixed point of the theory and they are observables of the 4 − 2ε dimensional theory. The rest of the paper is organized as follows: in Sec. II we review the general setup of the ε-expansion, fix our notation, and relate the CFT scaling dimensions at NLO to renormalization constants; in Sec. III we discuss the transformation rules of the beta function and the ADM under a change of renormalization scheme, first to all orders in perturbation theory and more explicitly at the two-loop order, illustrating the scheme-independence of the scaling dimensions; in Sec. IV we explain the block structure of the mixing between evanescent and physical. The IR scaling dimension of the ith operator with UV dimension ΔUV equals ðΔIRÞi 1⁄4 ΔUV þ εðΔ1Þi þ ε2ðΔ2Þi þ Oðε3Þ; ð22Þ with the definitions ðΔ1Þi ≡ ðγÃ1Þi and ðΔ2Þi ≡ ðUγÃ2U−1Þii: ð23Þ
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