Conformal QED_<i>d</i>,<i>F</i>-theorem and the<i>ϵ</i>expansion

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 $\frac{d-4}{2}\mathrm{log}R$ 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 four-component fermions, it acquires an interacting conformal phase, which in $d\lt 4$ 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-\epsilon $ 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 ${N}_{f}\gt 3$. For N ( )f( ) at or below some critical value ${N}_{{\rm{crit}}}$, the ${SU}(2{N}_{f})$ 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 ${N}_{{\rm{crit}}}\leqslant 4$. As another application of our results, we calculate the one loop beta function in conformal QED(6), where the gauge field has a four-derivative kinetic term. We show that this theory coupled to N ( )f( ) massless fermions is asymptotically free.