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We enumerate the micro-states in Higgs theories, addressing (i) the number ofvacuum states and (ii) the appropriate measure in the quantum path integral. Toaddress (i) we explicitly construct the set of ground state wave-functionals inthe field basis focussing on scalar modes $\theta(x)$. Firstly, we show that inthe limit in which the gauge coupling is zero, we obtain an infinite set ofdegenerate ground states at large volume distinguished by$\theta(x)\to\theta(x)+\theta_0$, spontaneously breaking the global symmetry,as is well known. We then show that at finite gauge coupling there is a uniqueground state at large volume since the wave-functional only depends on$\nabla\theta$ in the IR, and we explain this at the level of the Lagrangian.Since gauge fields fall off exponentially from sources there are no conservedcharges or symmetries in the Higgs phase; so the Higgs mechanism is the removalof symmetry from the theory. We show how physical features of defects, such ascosmic strings in the abelian Higgs model, are best understood in this context.Since there is a unique ground state, we address (ii) whether the path integralis a volume measure for the radial Higgs field $\mathcal{D}\rho\,\rho^{N-1}$from the $N$ components of the Higgs multiplet, or a line measure$\mathcal{D}\rho$ as the $N-1$ would-be Goldstones can be removed in unitarygauge. We prove that the need to avoid quartic divergences demands a tower ofcounter terms that resum to exactly give the volume measure. So the size of theHilbert space in the zero gauge coupling case and finite gauge coupling caseare in one-to-one correspondence, despite the degeneracy of the ground statebeing lifted in the latter. As a cosmological application, we point out thatthe volume measure can make it exponentially more unlikely in $N(=4)$ for theStandard Model Higgs to relax to the electroweak vacuum in the early universe.

Counting of States in Higgs Theories