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.

Electroweak unification

   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 bosons.
   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) gauge invariance.
   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 q_w 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 popular.