Symmetry principles have turned out to be very important in the theory of fundamental interactions. In some cases the symmetries are exact within the limits of present knowledge. However, equally important and interesting are the situations involving symmetries that are broken or hidden in some manner. In fact, symmetry techniques are often more useful in these cases. There are many possible fates for a symmetry. It may be exact, explicitly broken, or dynamically or spontaneously broken.
Symmetries arise when a theory has an invariance under some transformation of the basic fields of the theory. This invariance may be either discrete or continuous. Continuous symmetries involve transformations where the magnitude of the transformation can take on a continuous range of values, in particular it can be close to zero (i.e., no change). An example of this is a transformation that shifts the position of an object, that is, translation invariance. In contrast, discrete symmetries involve finite non-continuous transformations. An example of a discrete symmetry is parity invariance, which involves changing all the spatial coordinates into the negative of their value, x → -x .
It is possible that a symmetry is clearly realized in all states seen in nature. This is referred to as an exact symmetry. An example is the invariance under electromagnetic gauge transformations. This symmetry allows a redefinition of the phase of the electron's wavefunction, different at each point in space, as long as one simultaneously makes a change in the scalar and vector potentials of electromagnetism, leaving all physics invariant. The existence of this symmetry predicts that electric charge is conserved. As far as is known, all particles and interactions respect this symmetry, and electric charge is conserved.
Another possibility is that a symmetry may be explicitly broken. This occurs if the theory contains an interaction that does not obey the proposed symmetry. The symmetry would be valid if this interaction vanished. Even though the symmetry is not fully present, the use of symmetry techniques could still be useful if the interaction that breaks the symmetry is in some sense small. In that case, in a first approximation one can analyze the theory in the limit where the symmetry is valid and then treat the breaking interaction as a perturbation. An example of this is isospin symmetry, which in the Standard Model would reflect the invariance of transformations among linear combinations of up and down quarks—a continuous symmetry. A consequence of this symmetry would be the equality of the masses of the neutron and the proton, which are made of the up and down quarks. However, the electromagnetic interaction spoils this symmetry, as it is different for the up and down quarks because of their different charge. The different masses of the up and down quarks also explicitly break the isospin symmetry. However, both electromagnetism and the quark mass differences have only small effects on the masses of the nucleons. This can be seen from the mass difference of the neutron and proton, which is only 1 percent of their average mass. Isospin symmetry also predicts other regularities in the interactions of particles, and these are generally valid predictions at the level of a few percent.
The most subtle case concerns dynamically or spontaneously broken symmetry. This situation occurs when the symmetry reflects a continuous invariance of the underlying theory, yet the observed spectrum of particles does not display such a symmetry. The most important state to consider is the ground state. In these theories, the ground state is not unique, and there is a continuous family of possible ground states. A common visual analogy is the lowest energy state of a classical particle in a vertical wine bottle. The particle could be at rest anywhere on the circle that defines the bottom of the bottle. The different ground states would be the different positions around the bottom of the bottle, and the symmetry would reflect invariance of the physics under rotations around the circle. A consequence of the symmetry is that each of the possible ground states possesses the same energy. Nevertheless, despite this symmetry, only one ground state can exist at any time, and any one ground state breaks the symmetry by choosing a preferred direction. A similar situation occurs in quantum field theories. In this case, the ground state is defined by some configurations of the fields, and there could be a continuous family of configurations related to each other by the symmetry—all with the same energy. However, any one of these ground states would break the symmetry by itself.
Whenever this phenomenon occurs, a massless particle generally exists as a consequence. This can be seen from the initial premise that there are many different states with the same energy. Once one of these states is selected as the ground state, the other configurations would be other states with the same energy. In field theory all excitations are described as particles, and only massless particles can be excited with no cost in energy (assuming they also carry zero momentum). This requirement of massless particles is called Goldstone's theorem, and the particles themselves are often referred to as Goldstone particles or Goldstone bosons.
The archetypal case involves a spinless (scalar) field that is allowed to take on both real and imaginary values, that is, it is a complex field. The symmetry involves the transformation of the field by any complex phase. This is an invariance of the theory if only the absolute value of the field enters the theory. Such a theory could result in a symmetry that is either exact or broken. If the ground state of the theory was a state where the value of the field was zero, such a state would be invariant under a change of phase, and the symmetry would be preserved. However, if the energetics of the theory favored a nonzero value of the field in the state of lowest energy, then the symmetry would be spontaneously broken. The initial invariance tells us that the ground state could occur for any value of the phase (much like the particle at the bottom of the wine bottle), but once a specific value for the phase occurs the symmetry is broken. A complex field has two components, that is, real and imaginary parts. Only one combination is fixed by the ground state condition, and it is the orthogonal combination that becomes the Gold-stone boson. Which of these two outcomes occurs depends on the nature of the potential energy for the theory in question, but commonly either scenario could result for different values of the parameters of such a theory.
The only exception to the Goldstone's theorem occurs through what is called the Higgs mechanism. When a gauge symmetry is broken in this fashion, instead of obtaining a massless Goldstone particle, the gauge bosons of the theory acquire a mass. Instead of the two spin states of a massless gauge boson (like the photon), the massive one has three spin states. The degree of freedom that would have been the Goldstone boson has transformed into this extra component of the gauge boson.
The terms spontaneous symmetry breaking and dynamical symmetry breaking both refer to the phenomenon previously discussed, in which the theory has a continuous invariance but the ground state does not. The phrases are not quite identical, with spontaneous symmetry breaking being used most often to refer to the situation where a scalar field is responsible for the symmetry breaking and dynamical most often used when there are no fundamental scalar fields involved.
The Standard Model reveals all forms of broken symmetries. The theory involves an SU(2) × U(1) gauge symmetry that describes the interactions of quarks and leptons with the gauge bosons (the W and Z bosons, and the photon). The SU(2) portion gauge symmetry is spontaneously broken by the vacuum state of the Higgs scalar field, leading to massive W and Z fields through the Higgs mechanism. There is a residual exact symmetry, that of the electromagnetic gauge symmetry. If the up and down quarks had the same mass and charge, the isospin symmetry mentioned above would exist: it is an example of a useful explicitly broken symmetry. There is also an example of a dynamically broken symmetry: chiral symmetry. This symmetry is an extension of isospin symmetry—if the up and down quark masses were both equal to zero, an independent isospin invariance of each of the two spin states of these quarks would exist. The Goldstone bosons would be the pions. Because the quark masses are not exactly zero, this symmetry is also explicitly broken. The pions are then not strictly massless, but they are, in fact, the lightest of the observed strongly interacting particles. The rich and varied symmetry of the Standard Model is one of the reasons that symmetry techniques have been so fruitful in exploring fundamental interactions in the physical universe.
See also:Symmetry Principles
Donoghue, J. F.; Golowich, E.; and Holstein, B. R. Dynamics of the Standard Model (Cambridge University Press, Cambridge, UK, 1992).
Kane, G. L. Modern Elementary Particle Physics (Addison Wesley, Redwood City, CA, 1987).
Zee, A. Fearful Symmetry: The Search for Beauty in Modern Physics (Macmillan, New York, 1986).
John F. Donoghue