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Goldstone Boson Effects on Vacuum Decay

We study the effects of Goldstone modes on the stability of the vacuum in a $U(1)$ theory for a complex scalar field. The dynamics of the field resemble those of Keplerian motion in the presence of time-dependent friction, whose equations of motion imply a conserved quantity, $L$, reminiscent of conserved angular momentum. They also imply a persistent infinite barrier at $ρ=0$ and a divergent field value at the origin of coordinates in flat spacetime, rendering any solution physically unattainable. However, in a spacetime punctured at the origin of coordinates, we find finite-action solutions to the equations of motion, which correspond to the size of the hole $a_0$, which in turn determines the tunneling point $ρ_0$ and $L$. We find that the rates of vacuum decay get drastically enhanced by many orders of magnitude for all possible orderings in which the false and true vacua are placed in the potential. Finally, we show how Goldstone modes provide the necessary energy to overcome drag forces yielding finite-action solutions for any potential, including those that no such solutions exist for real scalar fields.