### Renormalization group approach to scalar quantum electrodynamics on de Sitter *[Cross-Listing]*

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We consider the quantum loop effects in scalar electrodynamics on de Sitter space by making use of the functional renormalization group approach. We first integrate out the photon field, which can be done exactly to leading (zeroth) order in the gradients of the scalar field, thereby making this method suitable for investigating the dynamics of the infrared sector of the theory. Assuming that the scalar remains light we then apply the functional renormalization group methods to the resulting effective scalar theory and focus on investigating the effective potential, which is the leading order contribution in the gradient expansion of the effective action. We find symmetry restoration at a critical renormalization scale $\kappa=\kappa_{\rm cr}$ much below the Hubble scale $H$. When compared with the results of Serreau and Guilleux [arXiv:1306.3846 [hep-th], arXiv:1506.06183 [hep-th]] we find that the photon facilitates symmetry restoration such that it occurs at an RG scale $\kappa_{\rm cr}$ that is higher than in the case of a pure scalar theory. The true effective potential is recovered when $\kappa\rightarrow 0$ and in that limit one obtains the results that agree with those of stochastic inflation, provided one interprets it in the sense as advocated by Lazzari and Prokopec [arXiv:1304.0404 [hep-th]].