### 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 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 , the 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 . 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.

Original language | English (US) |
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Article number | 135403 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 49 |

Issue number | 13 |

DOIs | |

State | Published - Feb 19 2016 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Statistics and Probability
- Modeling and Simulation
- Mathematical Physics
- Physics and Astronomy(all)

### Keywords

- F-theorem
- conformal field theory
- renormalization group

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## Cite this

_{d}, F-theorem and the expansion.

*Journal of Physics A: Mathematical and Theoretical*,

*49*(13), [135403]. https://doi.org/10.1088/1751-8113/49/13/135403