Counting of States in Higgs Theories
  We enumerate the micro-states in Higgs theories, addressing (i) the number of
vacuum states and (ii) the appropriate measure in the quantum path integral. To
address (i) we explicitly construct the set of ground state wave-functionals in
the field basis focussing on scalar modes $\theta(x)$. Firstly, we show that in
the limit in which the gauge coupling is zero, we obtain an infinite set of
degenerate 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 unique
ground 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 conserved
charges or symmetries in the Higgs phase; so the Higgs mechanism is the removal
of symmetry from the theory. We show how physical features of defects, such as
cosmic 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 integral
is 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 unitary
gauge. We prove that the need to avoid quartic divergences demands a tower of
counter terms that resum to exactly give the volume measure. So the size of the
Hilbert space in the zero gauge coupling case and finite gauge coupling case
are in one-to-one correspondence, despite the degeneracy of the ground state
being lifted in the latter. As a cosmological application, we point out that
the volume measure can make it exponentially more unlikely in $N(=4)$ for the
Standard Model Higgs to relax to the electroweak vacuum in the early universe.
