Conformal QEDd, F-theorem and the 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 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 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 and higher. For finite N f, following arXiv:1409.1937 and arXiv:1507.01960, we develop the expansion for the sphere free energy of conformal QEDd. Its extrapolation to d = 3 shows very good agreement with the large N f approximation for Nf > 3. For N f at or below some critical value , the symmetric conformal phase of QED3 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 . As another application of our results, we calculate the one loop beta function in conformal QED6, where the gauge field has a four-derivative kinetic term. We show that this theory coupled to N f massless fermions is asymptotically free.