Conformal QEDd, F-theorem and the ε expansion

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 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 four-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 - 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 > 3. For N-f at or below some critical value N-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-crit <= 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.