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${}_c$ ≈ 182.95(1-1.752ϵ+0.798ϵ${}^2$+0.362ϵ${}^3$)+O(ϵ${}^4$). 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 nc are widely spread. Conjecturing the topology of the renormalization group flow between two and four dimensions, we obtain a smooth interpolation function for nc(d) and find n${}_c$(3)≈12.2±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.