This section uses units where (Planck's constant)/2p
and the speed of light = 1. This choice of units is called natural
units. With this choice, mass has units of inverse length,
and vice versa. The conversion factor is 2x10-7 eV = 1/meter.
The Higgs mechanism forms the basis of the experimentally
well-tested theory of the weak and electromagnetic interactions that
is referred to as electroweak theory.
The initial gauge invariance in the theory is SU(2)xU(1),
with three massless gauge bosons from SU(2)
and one from U(1). In the end there
has to be only one massless gauge boson -- the photon that carries the
electromagnetic force -- and three massive gauge bosons mediating the
short range weak nuclear force.
Therefore, three massless scalar normal modes (also
known as Goldstone bosons) are needed
to serve as longitudinal modes to turn the four
massless gauge bosons into one massless
gauge boson and three massive gauge
Remember that for a single complex scalar field, the
massless mode, or Goldstone boson, comes
from the angular normal mode that oscillates
around the flat circle at the potential minimum.
A circle is just a one-dimensional sphere, or a "one sphere".
In general, an N-dimensional sphere
has N angular directions, and for oscillations
about the sphere, there is one radial direction.
We need a set of scalar fields that transform under the group SU(2)
with a potential whose minimum has the geometry of a three sphere. This
can be accomplished by using two complex scalar fields, transforming
as a two-component object under transformations by the group SU(2),
so that f(x) is given by
The potential minimum is at
which is the equation of a three sphere in f-space.
The normal modes for this potential will consist of
one radial mode and three angular modes, just enough to create one massive
Higgs boson, and give mass to the three of the four massless gauge bosons
in the SU(2)xU(1) theory. This leaves
leaving one massless gauge boson for the remaining unbroken U(1)
A complicating factor in electroweak theory is the
presence of electroweak mixing. The four massless gauge bosons in the
unbroken SU(2)xU(1) theory are the three
SU(2) bosons, let's called them W+,
W- and W0,
and the massless U(1) gauge boson, let's
call it B. The spontaneous symmetry
breaking winds up mixing the W0
and the B, into two different gauge
bosons -- the massless photon that carries
the electromagnetic force, and the massive Z0
boson that carries the weak nuclear force. The mixing is described by
the weak mixing angle qw
as shown below
The final physical states of this theory are the massless photon, and
the massive neutral weak boson, the Z0.
The distance scale of the electroweak mixing is roughly
100 GeV, or about 10-17 m. At scales smaller than that distance
scale, or equivalently, at energy scales much above 100 GeV, the weak
gauge bosons look massless and the full SU(2)xU(1)
symmetry is restored. But at larger distance scales, or lower energy,
only the U(1) symmetry of electromagnetism
is apparent in the conservation laws and amplitudes.
The mathematical beauty and experimental success
of this idea have led physicists to extend it to higher energies and
possible higher symmetries, as will be described below.
Running coupling constants
In quantum field theory, when computing a particle
scattering amplitude, one has to sum over all possible intermediate
interactions, including those that happen at zero distance, or, expressed
in terms of momentum space according to the de Broglie rule, at infinite
momentum. These calculations lead to integrals of the form
which diverge at infinite momentum for n=0,1,2. The limit has to be
approached through the use of a momentum cutoff of some kind. But the
physical quantities must be independent of the cutoff, so that they
remain finite as the cutoff is removed.
This procedure is called renormalization,
and it cannot be done for any quantum field theory, just those theories
whose divergences obey certain patterns that allow them to be added
consistently to the definition of a finite number of physical quantities,
namely the masses and coupling constants, or charges, in the theory.
The end result is that the masses and charges of elementary
particles are dependent on the momentum scale at which they are measured.
For example, the coupling strength of a renormalizable gauge theory
has the mass dependence
where M and m are two mass scales at which the coupling strength is
being measured and compared. The function f(n) depends on the number
of degrees of freedom in the theory. For electromagnetism, f(n) = 1,
but for QCD with six flavors of quarks, f(n) =-5.25.
Notice that this means electromagnetism gets stronger
at higher energies, while the strong nuclear force gets weaker as the
energy of the particle scattering increases. This is very important
for understanding what physics might look like at higher energies than
we can currently measure, see below.
Quantum field theories whose divergences can be hidden
in a finite number of physical quantities are called renormalizable
quantum field theories. Quantum field theories that are not renormalizable
are regarded as being physically realizable theories. Note that the
list of unrenormalizable quantum field theories includes Einstein's
theory of gravity, which is one reason why string theory became