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.
Spontaneous symmetry breaking
On the previous page we mentioned that spontaneous
symmetry breaking was the phenomenon that allowed gauge bosons to acquire
mass without spoiling the gauge invariance that protects quantum consistency
of the theory. But this trick is not special to electroweak theory;
spontaneous symmetry breaking is a powerful phenomenon that is tremendously
important in understanding unified particle theories in general. So
we will explain this phenomenon in more detail here.
The simplest example begins with a complex scalar
field f(x) with the Lagrangian
The potential V(f) has a strange
looking shape: the minimum is not at the center, but in a circle around
the center, as shown below.
The scalar field f(x) can be
written in terms of real and imaginary components, as below top, or
expressed in terms of radial and angular degrees of freedom, shown on
the bottom.
The minimum values of the potential lie along the
circle where
The problem with describing f(x)
in terms of f1 and f2
is that f1 and f2
don't describe the normal modes of oscillation around the minimum of
the potential. The normal modes for this potential are illustrated in
the animation above by the two distinct motions of the yellow ball.
One normal mode goes around and around the circle at the bottom of the
potential. The other normal mode bobs up and down in the radial direction
at a fixed value of angle, oscillating about the minimal value. Written
in terms of the normal modes, the field becomes
The physical states in the theory are the massive
field r(x) with mass r_{0}, and the massless field b(x).
The radial oscillations are resisted by the curved sides of the scalar
potential in the radial direction. That's why the radial field is massive.
But the minimum of the potential is flat in the angular direction. That's
why the angular mode is massless. This is called a flat
direction. Flat directions in the surface that forms the minimum
of the scalar potential lead to massless scalars. This issue comes up
again in string theory in not a good way.
The most crucial chapter in this story is what happens
when this scalar field is coupled to a massless gauge boson A with a
local U(1) gauge invariance. The Lagrangian
is
The story for the scalar field is as before. The
physical scalar fields that oscillate as normal modes about the potential
minimum are the massless angular mode and the massive radial mode. But
the plot thickens with the addition of the massless gauge boson. At
the minimum of the scalar potential, the Lagrangian above remains invariant
under the transformation
This transformation relates the normal modes of both
the scalar and vector fields so that they can be written as
The most important thing to notice about the redefined
fields above is that the angular oscillations b(x)
of the scalar field end up as part of the the physical gauge boson Ã(x).
This is the secret behind the power of spontaneous symmetry breaking.
The massless normal mode of the scalar field winds up mixed into the
definition of the physical gauge boson, because of gauge symmetry.
The oscillations of the scalar field around the flat
angular direction of the scalar potential turn into longitudinal oscillations
of the physical gauge field. A massless particle travels at the speed
of light and cannot oscillate in the direction of motion. Therefore,
the addition of a longitudinal mode of oscillation
means the gauge field has become massive.
The gauge field has a mass, but the gauge invariance
has not been spoiled in the process. The value of the scalar field at
the potential minimum determines the mass of the gauge boson, and hence
the range of the force carried by the gauge boson.
This whole coupled system is called the Higgs
mechanism, and the massive scalar field that remains in the end
is called a Higgs field.
