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

The abelian Higgs model is the textbook example for the superconducting transition and the Anderson-Higgs mechanism, and has become pivotal in the description of deconfined quantum criticality. We study the abelian Higgs model with $n$ complex scalar fields at unprecedented four-loop order in the $4-ϵ$ expansion and find that the annihilation of the critical and bicritical 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$\mathrm{.}Consequently,below$n_c$, the transition turns from second to first order. Resummation of the series to extract the result in three-dimensions provides strong evidence for a critical $n_c(d=3)$ which is significantly below the leading-order value, but the estimates for $n_c$ are widely spread. Conjecturing the topology of the renormalization group flow between two and four dimensions, we obtain a smooth interpolation function for $n_c(d)$ and find $n_c(3)\approx 12.2\pm 3.9$ as our best estimate in three dimensions. Finally, we discuss Miransky scaling occurring below $n_c$ and comment on implications for weakly first-order behavior of deconfined quantum transitions. We predict an emergent hierarchy of length scales between deconfined quantum transitions corresponding to different $n$.