What this means in less technical terms is that the
potential energy density, as a
function of
, looks like the bottom of a wine bottle: a hump in the middle and
a circular valley around it. (One visualizes the complex field value
as a 2-dimensional plane, the Argand diagram, and the potential as the
height above the plane.)
The point
is symmetric with respect to the U(1)
symmetry that changes the complex
phase of
as
for some randomly chosen value of
. This induces an asymmetry of the vacuum, in the sense that the
ground state is not invariant under the U(1) symmetry, which
transforms one value of
to a different one.
The problem in using a spontaneous symmetry-breaking model in particle
physics is that, according to a theorem of Jeffrey Goldstone, it
predicts a massless scalar particle, which is the quantum excitation
along the direction of
, a so-called Nambu-Goldstone boson. There is no potential energy
cost to move around the bottom of the circular valley, so the energy
of such a particle is pure
kinetic energy, which in
quantum field theory implies that its
mass is zero. But no massless scalar particles were detected.
A similar problem in Yang-Mills theory, a.k.a. nonabelian
gauge theory, was the existence of
massless gauge bosons, which (apart from the
photon) were also not detected. It
was Higgs' insight that when you combined a gauge theory with a
spontaneous symmetry-breaking model, the two problems solved
themselves rather elegantly. Higgs had found a loophole in the
Goldstone theorem: when you couple the scalar to the gauge theory, the
massless
mode of the Higgs combines with the vector boson to form a massive
vector boson.
Higgs' original article presenting the model was rejected by
Physical Review Letters when first
submitted, apparently because it didn't predict any new detectable
effects. So he added a sentence at the end, mentioning that it implies
the existence of one or more new, massive scalar bosons, which don't
form complete representations of the symmetry. These are the
Higgs bosons.
Before the symmetry-breaking, all elementary particles (except the
Higgs boson itself) are massless and the symmetry is unbroken, much
like the rotational symmetry of a pencil that stands on its tip.
However, the scalar field spontaneously slides from the point of
maximum energy in a randomly chosen direction into a minimum - much
like the pencil that eventually falls. Important is that the symmetry
doesn't disappear, it is just hidden. One says that the original
symmetry is broken and elementary particles - such as the
leptons,
quarks, W boson, and Z boson acquire
masses. The origin of the masses can be interpreted as a result of the
interactions of the other particles with the "Higgs ocean".
The Higgs mechanism was incorporated into modern particle physics by
Steven Weinberg and is an essential
part of the
Standard Model.
A slightly more technical presentation of the Higgs mechanism, which
presumes at least an elementary knowledge of quantum field theory, is
reviewed in the article on the
Yukawa interaction.
