Abelian scalar theory at large global charge

We elaborate on Abelian complex scalar models, which are dictated by natural actions (all couplings are of order one), at fixed and large global $U(1)$ charge in an arbitrary number of dimensions. The ground state $| \upsilon\rangle$ is coherently constructed by the zero modes and the appearance of a centrifugal potential is quantum mechanically verified. Using the path integral formulation we systematically analyze the quantum fluctuations around $| \upsilon\rangle$ in order to derive an effective action for the Goldstone mode, which becomes perturbatively meaningful when the charge is large. In this regime we explicitly show that the whole construction is stable against quantum corrections, in the sense that any higher derivative couplings to Goldstone's tree-level action are suppressed by appropriate powers of the large charge.