On $C_{J}$ and $C_{T}$ in Conformal QED

TL;DR

The authors analyze conformal QED$_d$ in the large-$N$ limit to compute the leading $1/N$ corrections to $C_T$ and $C_J$ with explicit $d$-dependent formulas for $C_{T1}(d)$ and $C_{J1}(d)$, including detailed checks against $d=2$ and the $4-\epsilon$ expansion. They derive a concise expression for the conformal Maxwell theory's $C_T$ in higher even dimensions and determine $C^{\mathrm{top}}_J$ for the topological current in $d=3$, illustrating how gauge-field dynamics enter the conformal data. The paper also analyzes 4-$\epsilon$ expansions of $C_J$ and $C_T$, presents a novel $C_T$-based inequality to bound symmetry breaking in QED$_3$, and extends the framework to large-$N_f$ QCD$_d$, linking central charges to RG flow properties and lattice results. Together, these results illuminate how central charges encode universal information across dimensions and contribute to understanding conformal phases, symmetry breaking, and RG flows in gauge theories.

Abstract

QED with a large number $N$ of massless fermionic degrees of freedom has a conformal phase in a range of space-time dimensions. We use a large $N$ diagrammatic approach to calculate the leading corrections to $C_T$, the coefficient of the two-point function of the stress-energy tensor, and $C_J$, the coefficient of the two-point function of the global symmetry current. We present explicit formulae as a function of $d$ and check them versus the expectations in 2 and $4-ε$ dimensions. Using our results in higher even dimensions we find a concise formula for $C_T$ of the conformal Maxwell theory with higher derivative action $F_{μν} (-\nabla^2)^{\frac{d}{2}-2} F^{μν}$. In $d=3$, QED has a topological symmetry current, and we calculate the correction to its two-point function coefficient, $C^{\textrm{top}}_{J}$. We also show that some RG flows involving QED in $d=3$ obey $C_T^{\rm UV} > C_T^{\rm IR}$ and discuss possible implications of this inequality for the symmetry breaking at small values of $N$.

Figures (9)

• Figure 1: Feynman rules for the Large $N$ QED .
• Figure 2: Diagramatic representation for $T=T_{\psi}+T_{A}$ and $J^{a}$.
• Figure 3: Diagrams contributing to $C_{J}$ up to order $1/N$.
• Figure 4: Plot of $C_{J1}$.
• Figure 5: Diagrams contributing to $C_{T}$ up to $N^{0}$ order.