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 r₀, 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.