Conformal QED$_d$, $F$-Theorem and the $ε$ Expansion

TL;DR

The paper analyzes sphere free energies in U(1) gauge theories to probe conformal phases and RG flows across dimensions. It combines exact results for Maxwell theory on S^d with large-N_f and ε-expansion techniques to study conformal QED_d, and uses the F-theorem to bound the conformal window, finding N_crit ≤ 4 for QED_3 and refining estimates via Padé resummation. It also explores higher-dimensional behavior, showing asymptotic freedom in a 6d higher-derivative Abelian gauge theory and discussing non-unitary CFTs for d>4. Together, these results provide a consistent, cross-validated picture of conformal phases, fixed points, and transitions in QED-like theories using sphere free energies as a diagnostic.

Abstract

We calculate the free energies $F$ for $U(1)$ gauge theories on the $d$ dimensional sphere of radius $R$. For the theory with free Maxwell action we find the exact result as a function of $d$; it contains the term $\frac{d-4}{2} \log R$ consistent with the lack of conformal invariance in dimensions other than 4. When the $U(1)$ gauge theory is coupled to a sufficient number $N_f$ of massless 4 component fermions, it acquires an interacting conformal phase, which in $d<4$ describes the long distance behavior of the model. The conformal phase can be studied using large $N_f$ methods. Generalizing the $d=3$ calculation in arXiv:1112.5342, we compute its sphere free energy as a function of $d$, ignoring the terms of order $1/N_f$ and higher. For finite $N_f$, following arXiv:1409.1937 and arXiv:1507.01960, we develop the $4-ε$ expansion for the sphere free energy of conformal QED$_d$. Its extrapolation to $d=3$ shows very good agreement with the large $N_f$ approximation for $N_f>3$. For $N_f$ at or below some critical value $N_{\rm crit}$, the $SU(2N_f)$ symmetric conformal phase of QED$_3$ is expected to disappear or become unstable. By using the $F$-theorem and comparing the sphere free energies in the conformal and broken symmetry phases, we show that $N_{\rm crit}\leq 4$. As another application of our results, we calculate the one loop beta function in conformal QED$_6$, where the gauge field has a 4-derivative kinetic term. We show that this theory coupled to $N_f$ massless fermions is asymptotically free.

Figures (6)

• Figure 1: Plot of the smooth function $A_0(d)$ from eq. (\ref{['tF-largeN']}). It has values $\tilde{F}=-55\pi/168$ ($a=-55/84$) in $d=6$, $\tilde{F}=31\pi/90$ ($a=31/45$) in $d=4$, and $\tilde{F} = -\pi/6$ ($c=-1$) in $d=2$.
• Figure 2: Padé$_{[1,3]}$ on $\delta \tilde{F}_{d}(N_{f})$ for various $N_{f}$
• Figure 3: Comparison of the Padé resummation of the $\epsilon$ expansion, and the large $N$ result (\ref{['confQED']}) for the sphere free energy of conformal QED$_3$.
• Figure 4: Plot of $\Delta (N_{f})=F_{{\rm conf}}(N_f)-F_{\rm SB}(N_f)$, using Padé$_{[1,3]}$.
• Figure 5: Schematic picture of RG flows for $N_f\gtrsim N_{\rm crit}$ (a) and $N_f\lesssim N_{\rm crit}$ (b). The QED$_3$ and QED$_3^*$ fixed points merge at $N_{\rm crit}$ and acquire small imaginary parts for $N_f\lesssim N_{\rm crit}$. In the latter case, the interacting conformal behavior is no longer possible, but the RG flow from the UV can "hover" near the complex fixed points before running away to large quartic coupling and presumably leading to the broken symmetry phase.