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The abelian Higgs model is the textbook example for the superconductingtransition and the Anderson-Higgs mechanism, and has become pivotal in thedescription of deconfined quantum criticality. We study the abelian Higgs modelwith $n$ complex scalar fields at unprecedented four-loop order in the$4-\epsilon$ expansion and find that the annihilation of the critical andbicritical points occurs at a critical number of $n_c \approx 182.95\left(1 -1.752\epsilon + 0.798 \epsilon^2 + 0.362\epsilon^3\right) +\mathcal{O}\left(\epsilon^4\right)\nonumber$. Consequently, below $n_c$, thetransition turns from second to first order. Resummation of the series toextract the result in three-dimensions provides strong evidence for a critical$n_c(d=3)$ which is significantly below the leading-order value, but theestimates for $n_c$ are widely spread. Conjecturing the topology of therenormalization group flow between two and four dimensions, we obtain a smoothinterpolation function for $n_c(d)$ and find $n_c(3)\approx 12.2\pm 3.9$ as ourbest estimate in three dimensions. Finally, we discuss Miransky scalingoccurring below $n_c$ and comment on implications for weakly first-orderbehavior of deconfined quantum transitions. We predict an emergent hierarchy oflength scales between deconfined quantum transitions corresponding to different$n$.

Abelian Higgs model at four loops, fixed-point collision and deconfined criticality